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Modern Portfolio Management: from Markowitz to Probabilistic Scenario Optimisation

Modern Portfolio Management: from Markowitz to Probabilistic Scenario Optimisation Goal-Based and Long-Term Portfolio Choice

Paolo Sironi

Published by Risk Books, a Division of Incisive Media Investments Ltd Incisive Media Haymarket House 28–29 Haymarket London SW1Y 4RX Tel: + 44 (0)20 7484 9700 E-mail: [email protected] Sites: www.riskbooks.com www.incisivemedia.com © 2015 Incisive Media Investments Ltd ISBN 978-1-78272-204-5 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

Publisher: Nick Carver Commissioning Editor: Sarah Hastings Managing Editor: Lewis O’Sullivan Designer: Lisa Ling Copy-edited and typeset by T&T Productions Ltd, London Printed and bound in the UK by PrintonDemand-Worldwide Conditions of sale All rights reserved. No part of this publication may be reproduced in any material form whether by photocopying or storing in any medium by electronic means whether or not transiently or incidentally to some other use for this publication without the prior written consent of the copyright owner except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Limited of Saffron House, 6–10 Kirby Street, London EC1N 8TS, UK. Warning: the doing of any unauthorised act in relation to this work may result in both civil and criminal liability. Every effort has been made to ensure the accuracy of the text at the time of publication, this includes efforts to contact each author to ensure the accuracy of their details at publication is correct. However, no responsibility for loss occasioned to any person acting or refraining from acting as a result of the material contained in this publication will be accepted by the copyright owner, the editor, the authors or Incisive Media. Many of the product names contained in this publication are registered trade marks, and Risk Books has made every effort to print them with the capitalisation and punctuation used by the trademark owner. For reasons of textual clarity, it is not our house style to use symbols such as TM, ®, etc. However, the absence of such symbols should not be taken to indicate absence of trademark protection; anyone wishing to use product names in the public domain should first clear such use with the product owner. While best efforts have been intended for the preparation of this book, neither the publisher, the editor nor any of the potentially implicitly affiliated organisations accept responsibility for any errors, mistakes and or omissions it may provide or for any losses howsoever arising from or in reliance upon its information, meanings and interpretations by any parties.

To Nabi Zwei, because you are free!

Contents About the Author

ix

Foreword

xi

Introduction 1 Beyond Modern Portfolio Theory

PART I

RISK MANAGEMENT FRAMEWORK

xiii 1

13

2 A Modern Risk Management Perspective

15

3 The Probability Measure

43

4 Real Securities and Reinvestment Strategies: Fixed-Income and Inflation-Linked Securities and Structured Products

55

5 Derivation and Modelling of Risk–Return Time Profiles

85

PART II

PORTFOLIO OPTIMISATION METHODS

103

6 À la Markowitz: A Tale of Simple Worlds

105

7 The Black–Litterman Approach: A Tale of Subjective Views

123

8 Probabilistic Scenario Optimisation

139

PART III

161

PORTFOLIO OPTIMISATION CASE STUDIES

9 Case Studies: Mean–Variance and Black–Litterman

163

10 Case Studies: Probabilistic Scenario Optimisation

175

Symbols and Notation

195

Index

197

vii

About the Author Paolo Sironi is practice leader of wealth management solutions and risk content services at IBM Risk Analytics, where he is responsible for quantitative methods and asset allocation advisory for financial institutions (retail banking, private banking, ultra-high-net-worth and institutional advisory clients). Combining risk analytics and technology, Paolo’s expertise spans wealth management, asset management, investment banking, market and credit risk management, regulatory reporting, cognitive computing, on-cloud and banking digitalisation. Before joining IBM, Paolo worked as managing director of Capitects, the company (a provider of risk management solutions) that he founded in 2008 as a joint venture between Sal. Oppenheim Private Bank and Algorithmics and that became part of IBM following the Algorithmics acquisition. Prior to Capitects, Paolo worked as head of market and counterparty risk modelling at Banca Commerciale Italiana and Banca Intesa Sanpaolo.

ix

Foreword At the heart of every investment decision lies the question “what will be the value of a given portfolio at some future time horizon?”. By definition, the future value is uncertain. There are a range of future possible outcomes depending on the market scenarios that are possible. The decision to invest in a given portfolio will depend on the trade-off between the possible downside and upside, or risk and reward. This is subjective for each investor and is a function of their preferences: tolerance for risk and desire for performance. This book features an excellent description of Modern Portfolio Theory, which still forms the basis for many investment decisions. It also does an excellent job of describing the Black–Litterman methodology, a more modern enhancement. Paolo Sironi’s key contribution, however, is in making scenario analysis and the very general Markto-Future approach accessible to goal-based investing. He describes in great detail how to simulate investment strategies over time while accounting for an investor’s risk–return profile. This is not only a theoretical treatise but one based on many years of experience of real-world investment decision-making. I believe it will be an excellent addition to any portfolio manager’s library. Ron Dembo January 2015

xi

Introduction Investment banking, asset management and wealth management are sophisticated industries that correspond to the investment needs of a large population of investors (institutional and private individuals) who require quantitative but intuitive solutions for investment decision-making. Products with mathematically complex payoffs (eg, structured notes) are nowadays broadly traded on financial markets and distributed to final investors. Yet, institutional portfolio management is often based upon rules of thumb and simplifications, such as the usage of benchmarks to proxy real investments. This can affect the coherence of optimal portfolio analysis and lead to inefficient capital allocations across risk factors and asset classes. This book addresses a renewed interpretation of portfolio choice based upon a modern risk management perspective and a clearer definition of the investors’ risk–return profile. The probability of achieving a desired target return (ie, a return target for an investment fund, a return ambition for a private investor) or minimising risk (ie, a value-at-risk (VaR) limit for a trading desk, a potential capital loss for a private investment) is chosen as the statistical measure that enforces optimal portfolio allocations by explicitly stating investment goals and downside boundaries. Portfolio managers, asset managers and wealth managers, who engage in long-term and goal-based portfolio construction, are concerned not only about today’s perception of risk and opportunity, but also about the way risk and return evolve over time. Such investors might ponder over statistical analysis and institutional research discussing optimal allocation and diversification among global asset classes, and might adopt benchmark-based frameworks to represent actual portfolios. However, they ultimately trade actual products whose payoffs can no longer be disregarded when discussing portfolio choice. Modern optimisation techniques therefore must fully embrace the risk–return characteristics of fixed-income securities and derivatives to overcome the limitations of classical approaches. Modern Portfolio Theory relies on the Markowitz (1952) formulation, which combines the basic objectives of investing: maximising xiii

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

expected return or minimising risk. This leads to an efficient frontier that indicates the set of portfolios with the best combination of risk–return characteristics. Portfolio managers have certainly benefited from this insightful formulation, but they did not grant it the expected practical success due to known limitations: professional investors may be believed to possess asymmetrical information; mean and variance are very restrictive indicators of risk– return characteristics of fixed-income payoffs and derivatives; the mean–variance efficient frontier often indicates extreme portfolio weights; the uncertainty of the input variables is not embedded in the approach. Hence, the framework does not include the results of calculating these dynamics, leading to insufficient risk–return management for life-cycle portfolio insurance and goal-based investments. Black and Litterman (1992) proposed an elegant approach to alleviate some of these limitations and indicated the positive weights stemming from the market equilibrium as the initial reference portfolio, thus combining return expectations with investors’ subjective views of the market. Although the Black–Litterman approach helps individuals to identify a more reasonable, less extreme and less sensitive portfolio weighting scheme, it still cannot address some of the relevant risk management challenges posed by modern finance: the approach still relies on the dynamics of benchmarks, which are an incomplete representation of the full universe of risk factors and opportunities; real investments and the way they can change over time are neglected; embedding investors’ views in consistent formats is not an easy exercise; investors’ characteristics, denoted by profiles of return ambition and risk appetite, do not enter the optimisation method explicitly. This book discusses ways to mitigate such limitations and covers portfolio choice from the perspective of goal-based investing and probabilistic scenario optimisation (PSO). In particular, we address the challenges of long-term investments in order that a myopic approach to portfolio choice need not dominate the asset allocation exercise. Investors might not know enough about future states of the world, which is why they may focus on short-term moneymanagement although they express goals for longer investment horizons (eg, a yearly budget or a multi-year portfolio insurance strategy). The investment horizon, as well as the frequency of the intermediate steps of portfolio rebalancing, is important in portfolio xiv

INTRODUCTION

choice and cannot be disregarded. The introduction of multi-period stochastic simulations, common practice in counterparty risk measurement and credit value adjustment (CVA), may make behaviour less short-sighted by including professional knowledge about future potential returns, so that investment decisions can be tested ex ante and verified throughout the life of the investment. ORGANISATION OF THE BOOK The book is organised into an introductory chapter and three parts: (I) risk management framework; (II) portfolio optimisation methods; (III) portfolio optimisation case studies. Chapter 1: Beyond Modern Portfolio Theory The three parts are preceded by an introductory chapter that examines modern investment environments and outlines the reasons for portfolio choice to evolve beyond the Markowitz and Black– Litterman approaches. Probabilistic scenario optimisation is briefly reviewed as a valuable alternative and its main traits are discussed; these traits are linked across all subsequent chapters, guiding the reader in their studies throughout the book. Part I: Risk Management Framework The first section is a precursor to our review of goal-based optimisation principles and covers aspects of financial risk management. Chapter 2 describes the main characteristics of a modern risk management perspective based on scenario simulation. While investment banks have implemented enterprise-wide risk management architectures in order to comply with best practices and banking regulation, asset managers and wealth managers often rely upon simplified approaches for the risk management of portfolio exposures and the optimisation of the risk–return profile. Therefore, we start our discussion by presenting the most common risk measurement methods of computing VaR and expected shortfall (parametric, historical, bootstrapping and Monte Carlo) and comparing them with numerical examples. This discussion outlines why Monte Carlo scenarios are chosen to simulate the assets used in the multi-period optimisation. xv

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Chapter 3 introduces the a posteriori probability measure, which can be estimated by overlapping the investor’s risk–return profile with the density function of the portfolio potential returns generated by a multi-period Monte Carlo simulation. This emphasises the appealing advantages of choosing the probability measure as the objective function of goal-based optimisations, as it allows us to compare ex post and ex ante performance in a synthetic and graphical representation. Chapter 4 discusses the advantages of building portfolio choice on a risk management framework that directly models real securities as opposed to benchmarks and market indexes. Modelling the reinvestment rules of fixed-income and derivative products allows portfolio managers to supplement long-term simulations of maturity-bearing securities. This is particularly relevant for portfolio managers wanting to optimise long-term portfolio allocations with fixed-income holdings, derivatives, structured products and inflation-linked exposures. In Chapter 5, we present aspects of modelling the investors’ risk– return profiles, so that we can map the vectors of the actors’ preferences onto the full space of potential total returns of portfolios, which is a building block of probabilistic scenario optimisation. Part II: Portfolio Optimisation Methods This section examines portfolio choice from the point of view of the main approaches available to market practitioners: Markowitz, Black–Litterman and probabilistic scenario optimisation. Chapter 6 presents Modern Portfolio Theory, a classical diversification framework. The key traits of Markowitz-type optimisations are outlined, taking mean–variance as a starting point, tracking error minimisation as an alternative for asset managers, using semi-variance to overcome the statistical limitations of the volatility measure and expected shortfall as a more advanced formulation of the objective function. Chapter 7 relaxes the classical assumptions of information symmetry embedded in the Markowitz approach and reviews the Black– Litterman alternative. We start by formalising the market equilibrium portfolio (CAPM) and then introduce the investors’ views in order to estimate the posterior distribution of the expected returns of assets and optimal portfolios. xvi

INTRODUCTION

Chapter 8 discusses probabilistic scenario optimisation; this is seen as a turning point in goal-based investing, because it combines the mathematical properties of investment products with the preferences of actual investors. PSO is an exhaustive enumeration technique that aims at maximising the probability of achieving or beating an investment target, thus complying with a risk profile. Our description of the methodological steps is enriched by examining low-discrepancy sequences and lexicographical representations, which allow computational performance to be properly addressed. Part III: Case Studies of Portfolio Optimisation This section presents a set of case studies using numerical examples that allow us to compare the three optimisation methods presented in the previous section with respect to the model inputs and the outputs of the optimisation routines. Chapter 9 examines both a mean–variance case and a Black–Litterman optimisation, and Chapter 10 examines probabilistic scenario optimisation and compares the findings of the multi-period exercises for a set of alternative risk–return profiles: risk averse, risk mitigating and risk tolerant. SUMMARY OF THE BOOK Portfolio choice and goal-based investing are attractive cutting-edge topics for a large and international audience. We discuss the related aspects of quantitative finance with the intention to make them as digestible as possible. This book does not aim to provide direct advice to portfolio managers, private investors or their intermediaries. Instead, it provides an empirical framework based on probability measurement for those practitioners willing to apply their intuition together with an understanding of the dynamics of the trade-off between the portfolio risks and returns, as part of a decision-making process designed for long-term investments. Some limitations should be acknowledged. First, the risk management methods are only outlined. Second, trading costs are not formally discussed because the focus of the book is an argumentation of life-cycle optimal investments, as opposed to myopic trading, so that the cost implications due to short-term trading become less relevant. Third, taxation is generally ignored, although it can significantly influence decisions of wealth allocation. Last, inflation can xvii

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

play a key role in long-term investing but this topic is only partly discussed, leaving space for further applications. ACKNOWLEDGMENTS I thank the numerous colleagues and friends that have inspired my professional activity. In particular, I am grateful to Gabor Topa, Dominik Flierl and Andres Hernandez for their thoroughness and dedication as they contributed to this work with open discussions, formalisation and constructive criticism. I am indebted to Ron Dembo, Michael Zerbs and colleagues at Algorithmics: their visionary work in risk management has inspired my career. A sincere thank you to Sarah Hastings, Commissioning Editor, and to Lewis O’Sullivan, Managing Editor, for believing in this project. Most importantly, I am grateful to my family, who helped me to dedicate time to this work. This book contains the formulations, evidence and opinions of the author alone; these do not necessarily represent the practice or the views of his current or previous employer, or the beliefs of his present and past colleagues. January 2015

REFERENCES

Black, F., and Litterman, R., 1992, “Global Portfolio Optimization”, Financial Analysts Journal, pp. 28–43. Markowitz, H. M., 1952, “Portfolio Selection”, Journal of Finance 7(1), pp. 77–91.

xviii

1

Beyond Modern Portfolio Theory An investment in knowledge pays the best interest. Benjamin Franklin (1706–90)

This chapter sketches the main arguments of this book, which are related to portfolio choice for long-term and goal-based investing, and provides a summary of Modern Portfolio Theory, the Black–Litterman approach, probabilistic scenario optimisation and knowledge-based principles of optimal investing. INTRODUCTION Financial markets underwent a profound transformation during the last decade of the 20th century. The integration of international markets, fostered by broader deregulation of cross-border capital flows, was accompanied by strong financial innovation: the landscape of investment opportunities became more accessible yet heterogeneous (ie, derivatives, structured products, securitisation) and also more interdependent, as revealed by the contagion risk that characterised the global financial crisis in 2007–12. This affected the dynamics of the correlations among global asset classes, as it appeared not only that risks become over-concentrated more often than expected, instead of being diversified away across a larger number of players, but also that asset classes co-move faster than forecasted, as capital flows in and out of international markets. A direct consequence is that there is a growing demand from investors to shift their priorities in the direction of more customised asset–liability management and to be more ambitious in modelling risk appetite; this ambition cannot be addressed by existing market equilibrium approaches, elegant in nature as they are. Regulators also demand more risk transparency in financial intermediation, stimulating the financial services industry to revise existing methodologies of portfolio choice towards risk-based approaches. These elements reinforce the call for optimal portfolio modelling to be based on actual products, actual investors’ preferences and actual investment goals over the life cycle. 1

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Yet, financial markets get more and more sophisticated and volatile, so investors are invited to make increasingly complex decisions about their wealth allocation and require greater knowledge of quantitative finance. Can the optimal portfolio be the same for longterm investors and short-term players? Is cash a risk-free heaven when looking at longer investment horizons, in which reinvestment occurs at today’s unknown real interest rates? Can money managers provide long-term capital protection but yield returns stemming from tactical opportunities, in such a way that investments are always optimal during all periods? Behavioural finance has documented patterns of individual behaviour that do not reconcile with rational models, so that actual portfolios tend to be a function of short-term market opportunities only, making it unfeasible to optimise portfolios over the life cycle. It is now acknowledged that conditions for the market to be efficient, in the sense that investors have accurate information and use it correctly to their advantage, and the statement that the market portfolio is an efficient portfolio should be discussed differently. In fact, market efficiency implies portfolio efficiency only under some specific assumptions which are proved to be inappropriate: transaction costs and liquidity constraints must not be ignored, most investors do not hold efficient portfolios or the same (correct) beliefs about the risk– return profiles of securities and cannot lend or borrow without limits at the risk-free rate. Investment practices at established investment banks and assetmanagement firms often rely upon overly simplified rules of thumb to assess the trade-off between investment risks and potential returns, which leads to the indication of strategic asset allocations that do not always reconcile with real investment opportunities. The divergence between an enlarged set of investment requirements and the need for consistent responses has widened the information gap between the so-called optimal market portfolio and an investor’s attainable portfolio. The strategic market portfolio, which arises from a theoretical asset allocation, emerges from optimisation exercises based on the statistical properties of the market variables (ie, expected return and standard deviation) and the asset classes that map to them. The investor’s attainable portfolio instead emerges from the operational asset allocation, which is the result of a selfdirected or an intermediated process that implicitly bounds the 2

BEYOND MODERN PORTFOLIO THEORY

investors’ choices to a defined set of real products (out of the larger set available) bearing non-linear risk–return profiles. The provision of more intuitive and consistent information about potential future states of the world and the simulation of investment returns (net of commissions, transaction costs and possibly tax) can contribute to improved market efficiency and reconcile operational and theoretical portfolio allocations. As a matter of fact, a new interpretation of Modern Portfolio Theory based on scenario optimisation seems to be emerging: probabilistic scenarios, which are part of established risk management practices, grant investment managers the chance to employ time-varying characteristics of investors’ preferences and achieve a more consistent risk–return description. Explicit modelling of the investor’s profile can change the traditional landscape of optimisation models, whose main inputs are market variables or their subjective reinterpretation (equity tilt) at a single point in time. The inclusion of the investor’s profile makes it easier to realign the actors’ preferences not only with the prevailing market outlook, but also with the most appropriate mix of long-term costs/ benefits that originate from the simulation of the potential returns of real investments. Empowering individual investors to take transparent care of their own assets, directly or indirectly via the professional work of financial advisors, is also a developing idea. Keynes (1931) had already imagined central bankers as orthodontists, intervening with humble fiscal and monetary policy to optimise the dynamics of the economy at large: “If economists could manage to get themselves thought of as a humble, competent people, on a level with dentists, that would be splendid”. As Campbell and Viceira (2002) brightly indicated, it is now common wisdom that dentists shall also pursue the goal of advising on oral hygiene, rather than simply intervening once the pain becomes unbearable. Similarly, investors should be given the tools and the means to reallocate investments with an ex ante view of the potential drawbacks and opportunities, which is the essence of proactive risk management. Probabilistic scenarios are the cornerstone of this new interpretation. By simulating total returns of actual investments and liabilities over time, we are granted direct access to the information hidden in the potential dynamics of the probability densities of actual products. Thus, we can verify whether a given set of an 3

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

individual’s constraints complies with the simulated total return space of portfolios, by measuring the probability of achieving or underperforming a defined investment goal so that the probability measure becomes the key variable of the min/max objective function. Furthermore, the evolution of the potential total returns of optimal portfolios can be stress tested with subjective views or alternative hypotheses of market behaviour to strengthen the risk management aspects. This chapter introduces the main traits of probabilistic scenario optimisation (PSO), a risk-based optimisation framework for longterm and goal-based investing. First, the main traits of portfolio theory are outlined by reviewing the essential elements of the Markowitz and Black–Litterman approaches to portfolio choice. Then, scenario optimisation is introduced as an exhaustive enumeration technique requiring Monte Carlo simulation of actual products, modelling of actual investors’ risk–return profiles, low-discrepancy sequences and lexicographical representations to achieve computational efficiency. Finally, five knowledge-based principles are outlined to address goal-based portfolio investing in the long term. THE MAIN TRAITS OF MODERN PORTFOLIO THEORY Modern Portfolio Theory relies on Markowitz’s (1952) formulation, which combines the basic objective of investing: maximising expected return while minimising risk. This leads to an efficient frontier that indicates the set of portfolios with the best combination of risk–return characteristics. The theory suggests that investors, who care only about the mean and the variance of portfolio returns over a single period, can choose an optimal portfolio that is the unique combination of risky assets combined with an appropriate amount of risk-free cash, so that personal propensity to risk can be dealt with. Portfolio managers have certainly benefited from this insight, but they did not grant it the expected practical success, due to known limitations. First, professional investors might be believed to possess superior information about financial markets, or require more customised decision-making to better reflect personal elements. Second, portfolio managers might not have a complete set of return expectations for the entire universe of asset classes that is required to generate optimal portfolio weights of global asset 4

BEYOND MODERN PORTFOLIO THEORY

allocations. Third, the mean–variance efficient frontier often indicates extreme portfolio weights, either long or short, which are excessively sensitive to changes in the estimate of expected returns. Moreover, the uncertainty in the input variables is not embedded in the approach (estimation error). Last, a realistic and practical asset allocation that encompasses global investment opportunities, especially fixed-income securities and derivative payoffs, cannot easily be identified. Although market practice has improved the original mean–variance proposition with the use of better risk measurements such as regret, expected shortfall, semi-variance and tracking error, these approaches tend to be restricted to an oversimplified representation of real securities by means of benchmarks and market indexes. Therefore, Markowitz-type optimisations are not fully suited to addressing risk–return management for life-cycle portfolio insurance and goal-based investments, since the implications of total return dynamics of actual securities stand outside the framework. THE MAIN TRAITS OF THE BLACK–LITTERMAN APPROACH Black and Litterman have further extended the original mean– variance formulation (1992) and have indicated the positive weights stemming from the market equilibrium as the initial reference portfolio, thus combining return expectations with investors’ subjective views of the market. Portfolio managers have been given the chance to indicate a confidence level for each view and re-optimise the equilibrium portfolio by shifting the asset weights towards the preferred strategies. Although the Black–Litterman approach helps individuals to identify a more reasonable, less extreme and less sensitive portfolio weighting scheme, it still cannot address some of the relevant risk management challenges posed by modern finance. First, the market equilibrium is a theoretical formulation of how financial markets function, and it relies on the dynamics of benchmarks, which are an incomplete representation of the full universe of risk factors and opportunities. Second, the distinctive risk–return properties of real investments and the way they can change over time are neglected. Third, embedding investors’ views in consistent formats is not an easy exercise, so institutionalised processes of portfolio choice cannot be enforced. Last, investors’ characteristics, denoted 5

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

by profiles of return ambition and risk appetite, do not enter the optimisation method explicitly. THE MAIN TRAITS OF PROBABILISTIC SCENARIO OPTIMISATION PSO is an exhaustive enumeration technique that allows us to mitigate some of the limitations of Markowitz-type and Black–Litterman approaches. Owing to advances in computing power it has become increasingly accessible, allowing institutional investors and wealth managers to find solutions to the problems of multi-period portfolio choice based on discrete-state approximations. This technique requires the simulation of the potential returns of real securities over time, which permits fixed-income products, derivatives and structured products to be conveniently represented in making optimal allocations. The introduction of dynamic reinvestment strategies allows us to make long-term simulations of optimal portfolios beyond the contractual expiry of maturity-bearing securities, so that portfolio choice can be made conveniently across asset classes and payoffs. Investors’ ambitions and fears can also be elicited, so that their risk–return profile over time can be drawn and overlapped with the potential total return space of strategic and tactical asset allocations. Stress tests and investment views can be modelled freely, and the potential dynamics of actual payoffs can be reviewed without loss of information. This allows portfolio managers to complement strategic portfolio optimisation with asymmetrical opinions and make decisions regarding “suboptimal” portfolios (with respect to the theory) through a clearer understanding of the confidence levels associated with stressed market changes. The key element of PSO is a reinterpretation of the objective function, which becomes the maximal probability of achieving (or beating) an investment target while complying with a given risk limit, so that goal-based investing is supported. The explicit statement of the probability measure helps to combine past and future performance and track the deviation from optimality, as time passes and investment goals become more likely or less likely to be attained. PSO is a step-by-step process of portfolio filtering and ordering according to probability measurement criteria, as constructed in Table 1.1. 6

BEYOND MODERN PORTFOLIO THEORY

Table 1.1 Probabilistic Scenario Optimisation process ΦU , ϕU (generate all potential portfolios) ↓ ΨU (identify only the admissible portfolios) ↓ ΘU (filter the risk-adequate portfolios) ↓ ΘU∗ (indicate the optimal goal-based portfolio)

The computing power challenges posed by such an exhaustive enumeration technique are still relevant for the treatment of large portfolio allocations. However, this is no longer a limitation in the context of wealth management and portfolio insurance, as the optimal portfolio is generated out of a reduced universe of investment opportunities. Quasi-random methods can be applied in such a way that the resulting space of the admissible portfolio compositions is made of equidistant outcomes that represent well all possible portfolio combinations, thus avoiding large gaps and clustering. Halton (1960), Sobol (1967), Faure (1982) and Niederreiter (1987) are all wellknown alternatives, among the variety of low-discrepancy methods proposed by this growing field of mathematical research. We argue for the non-binding adoption of Halton sequences in the making of the examples and case studies presented in the following chapters. Halton sequences are deterministic sequences of numbers based on increasingly fine prime-based division (eg, 2, 3, 7, 11, 13, . . . ) of subunit intervals, which produce well-spaced draws from the unit interval so that the quasi-random variables sampled from a larger population are ex post evenly spread (equidistant). Quasi-random methods still require that the optimisation routine generates the full explicit list of the ordered portfolios from which to sample. One way to further improve the calculation efficiency is offered by computational science, as we can model a lexicographical representation with a more parsimonious tree of the relationships among the ordered numbers, so that the explicit list of all possible allocations can be sampled by a smaller number of iterations, without having to generate the full space. More importantly, knowing that the objective 7

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 1.1 Example of PSO portfolio simulation Optimisation node CIO view H1 CIO view H2

Monte Carlo Ambition profile Risk profile 1.0 0.8

Risk/return

0.6 0.4 0.2 0 –0.2

Probability

–0.4 1.0 0.5 0

1Y

2Y

3Y

4Y

5Y

Time Positive return

Beating target

function is not convex, we can make use of genetic algorithms to surf the multi-dimensional space generated by the verification of the objective function with even greater speed and accuracy. In such a case, the step-by-step approach indicated in Table 1.1 would be different. Exhaustive enumeration techniques are very unrestrictive methods and can be applied to any type of investment problem. However, for the convenience of the applications, in the remainder of this book we often refer to simpler cases of model portfolio optimisation that optimise private wealth. Figure 1.1 shows an example of PSO portfolio analysis (as in Chapter 10). FIVE KNOWLEDGE-BASED PRINCIPLES Decision-making for goal-based investments is in itself a thorough exercise that certainly requires dedication, knowledge and time, as if planning a journey. In the course of this work, the reader will 8

BEYOND MODERN PORTFOLIO THEORY

be taken through this journey, composed of five knowledge-based steps. 1. Know the products: we identify the universe of potential investments and simulate them over time with stochastic scenarios in order to investigate their risk–return properties. 2. Know the investor: we identify the optimisation constraints according to the investor’s preferences, such as minimum allocation, investment step size and maximum exposure to a certain market, and generate the set of all potential portfolios that comply with the given constraints. Also, we indicate the investor’s risk–return profile and the time discretisation along which optimisation should be performed (investment horizon, liquidity term and reallocation steps). 3. Know the portfolio risks: we discard all potential portfolios that do not comply with the investor’s risk appetite and focus on the potential allocations that are risk-adequate. 4. Know the portfolio returns: we measure the probability of each potential risk-adequate portfolio to beat/achieve the investor’s ambitions and order portfolio results in terms of probability levels to indicate the optimality. We can also stress test the chosen optimal allocation and challenge its robustness and meaningfulness. 5. Know the performance: we track the performance of the newly invested portfolio by drawing the ex post (historical) and ex ante (prospective) dynamics of total returns. The probability of reaching a chosen target is a function of the ex post performance (net capital loss or gain plus/minus cashflows) and the density of potential future risks/returns. We can therefore identify the most appropriate time steps for revising the asset allocation and rolling forward the financial bets. An appealing feature of PSO is that we can operate multiple problems without having to recalibrate the full set of simulation inputs: we can redefine time horizons, time steps, allocation constraints, client ambitions or risk appetite levels and operate on the same stochastic distribution of the total returns of individual products. This should facilitate the institutionalisation of the methodology across advisory networks, well outside the specialised desks of 9

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

quantitative professionals, and hence allow sell-side institutions to deliver better and transparent support to buy-side players. Moreover, we can identify an intuitive metric that allows comparison of strategic and tactical asset allocations with a certain level of intuition. The probability, such as that of beating a financial goal, of yielding a minimum total return or avoiding a capital loss, is such a measure. CONCLUSIONS Modern financial environments require mitigation of the limitations of Modern Portfolio Theory to make portfolio choice easier in the context of long-term and goal-based investing. PSO is emerging as an alternative to the classical methods, such as the Markowitz-type and Black–Litterman approaches. The adoption of probabilistic scenarios requires a thorough understanding of modern risk management techniques, based upon full revaluation methods of actual securities by means of multi-period stochastic simulations. Therefore, the first section of this book will introduce the reader to the principles of portfolio simulation, the generation of scenarios and scenario paths, the calculation of product and portfolio total returns, the difference between the most common risk management approaches (parametric, historical, bootstrapping and Monte Carlo), the time properties of fixed-income securities, the asymmetry of the potential returns of derivatives and structured products as well as the added-value of dynamic strategies for the simulation of maturitybearing securities. The elicitation of the risk–return profile of actual investors (being professional traders or private individuals) is also presented in such a way, that it constitutes fundamental input to the Probabilistic Scenario Optimisation exercise.

REFERENCES

Barberis, N., 2000, “Investing in the Long Run when Returns Are Predictable”, Journal of Finance 55(1), pp. 225–64. Campbell, J. Y., and L. M. Viceira, 2001, Strategic Asset Allocation: Portfolio Choice for LongTerm Investors (Oxford University Press). Dembo, R., and A. Freeman, 1998, Seeing Tomorrow: Rewriting the Rules of Risk (Chichester: John Wiley & Sons). Faure, H., 1982, “Discrépance de Suites Associées à un Système de Numération (en Dimension s)”, Acta Arithmetica 41, pp. 337–51.

10

BEYOND MODERN PORTFOLIO THEORY

Halton, J. H., 1960, “On the Efficiency of Certain Quasi-Random Sequences of Points in Evaluating Multi-Dimensional Integrals”, Numerische Mathematik 2(1), pp. 84–90. Keynes, J. M., 1931, Essays in Persausion (New York: Norton). Kahneman, D., P. Slovic and A. Tversky, 1982, Judgment under Uncertainty: Heuristics and Biases (Cambridge University Press). Litterman, R., and G. He, 1999. “The Intuition behind Black–Litterman Model Portfolios”, Report, Goldman Sachs Investment Management Series. Markowitz, H. M., 1952, “Portfolio Selection”, Journal of Finance 7(1), pp. 77–91. Markowitz, H. M., 2000, “Foundations of Portfolio Theory”, Nobel Lecture (New York: Baruch College, The City University of New York). Niederreiter, H., 1987, Random Number Generation and Quasi-Monte Carlo Methods, CBMSNSF Regional Conference Series in Applied Mathematics (Philadelphia, PA: SIAM). Sobol, I., 1967, “On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals”, USSR Computational Mathematics and Mathematical Physics 7(4), pp. 86–112. Zimmermann, H., W. Drobetz and P. Oertmann, 2003, Global Asset Allocation: New Methods and Applications (Wiley Finance).

11

Part I

Risk Management Framework

2

A Modern Risk Management Perspective It does not do to leave a live dragon out of your calculations, if you live near him. J. R. R. Tolkien (1892–1973)

This chapter discusses risk management of real products to enrich optimal portfolio choice: defining scenarios and scenario paths. It covers the parametric and historical scenario, bootstrapping and Monte Carlo methods, geometric Brownian motion (GBM) and the Hull–White one-factor model and gives scholastic examples. INTRODUCTION Financial markets have been characterised by a sustained path of financial innovation that, coupled with the broader market imbalances that culminated in the global financial crisis of 2007, has sparked a debate about the appropriateness of the methods dedicated to constructing optimal asset allocations. This debate questions both sides of the investment equation: the risk measure and the expected returns. On the risk side, the commonly used measure of volatility is known to be a limiting risk management estimate, as it fails to represent the asymmetry and the long tail of potential returns. Quantitative research has investigated more refined measures of risk, such as value-at-risk (VaR) and expected shortfall (ES), in an attempt to remove some of the limitations of Modern Portfolio Theory. However, featuring more refined risk measurement can add value only if the distribution of the investment returns that underlies the optimisation exercise is based on actual securities instead of benchmarks. VaR and ES are also known to be incomplete measures of risk; hence, best practices would complement portfolio analysis by means of stress testing. Yet, for a stress test to be meaningful, the payoff of actual products should be simulated. 15

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

On the return side, expected returns are oversimplified representations of the potential returns of financial investments and fail to properly differentiate among investment alternatives. The Black– Litterman approach allows the formulation of subjective views of excess expected returns, thus introducing an exogenous bias in the modelling of risk perception. However, investors’ views should not modify risk–return measurement per se, but rather enhance financial analysis by drawing stress test scenarios to verify the model assumptions. Again, for scenario analysis to be truly informative, the payoff of actual products should be simulated instead of benchmarks. Therefore, the techniques dedicated to optimal portfolio allocation need to be upgraded with respect not just to the statistics available, but also to the inputs to the models. Investment managers primarily use indexes (ie, asset class representations) owing to their attractive simplicity and affordability. However, adopting a modern risk management approach to strengthen portfolio optimisation requires a move outside the comfort zone of indexes in favour of the simulation of real payoffs. This does not limit the possibility of representing the resulting allocations by means of asset classes, as is often seen in marketing and regulatory fact sheets, because asset classes can be treated as aggregation tags. By starting from the quantitative modelling of real products and then aggregating portfolios by descriptive tags, users can augment the simplified definitions of asset classes with the asymmetrical payoffs of real securities, avoiding the loss of relevant risk-based information. Risk management of real products is a key element of long-term and goal-based portfolio choice. This chapter examines the most common approaches to risk measurement of financial securities and portfolios: parametric, historical, bootstrapping and Monte Carlo. Monte Carlo will be extensively used throughout this book to generate potential returns of financial investments over time; thus, some common models of stochastic simulation (geometric Brownian motion and the Hull–White one-factor model) are discussed. Scholastic examples are included at the end of the chapter. SCENARIOS AND SCENARIO PATHS The value of an asset (or portfolio) varies over time. Fair-value pricing allows us to represent market values conditional on the level of the risk factors (eg, equity prices, values of indexes or interest 16

A MODERN RISK MANAGEMENT PERSPECTIVE

Figure 2.1 Simulation of market variables over time

Path H1

Scenario return

Scenario h (H1, 3Y)

Path H2

Path H3

1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

Time

Figure 2.2 Scenario paths: potential evolution of a market risk factor

Percentage returns

0.60 Equity market cycle Deflationary market

0.40 0.20 0 –0.20 –0.40 –0.60

1Y

2Y

3Y

4Y

5Y 6Y Time

7Y

8Y

9Y 10Y

rates) that contribute to the evaluations. By simulating the market variables over time, as in Figure 2.1, we can create a set of scenarios in which we can restate the potential value of any security under perturbed market conditions. A scenario h is a potential state of the world at a particular time, featuring a defined set of risk factors that take values that are potentially different from their respective evaluation at the beginning of the holding period. A scenario path H is a sequence over time of scenarios h ∈ H, and models the potential evolution of a set of risk factors at any point t along the investment horizon Γ . The set of all scenario paths H is denoted by S. The base scenario path is defined 17

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

as the scenario path that describes the current and future state of any risk factor assuming no exogenous influences. According to the simulation model chosen, the base scenario can imply constant evolution of risk factors or a forward growth modification, given today’s trading consensus or given estimates of risk premiums. In the following, the base scenario path is indicated by H = 0. Scenarios can also represent instantaneous shocks corresponding to historical realisations. The measurable uncertainty about the realisation of scenario returns is called risk. In Figure 2.2 we represent graphically a scenario set containing two distinct paths representing the potential evolution of a market risk factor (eg, market index) along the investment horizon Γ . For example, we can generate a path that implies a constant devaluation of an equity index year on year, or can model an economic cycle of market booms and busts. Any real security, whose market value depends upon the level of such a risk factor, can be reevaluated at any future time by shocking the value of the underlying risk factor by a quantity equal to the return indicated by the scenario that corresponds to the chosen time step and scenario path. MEASURING RISKS AND RETURNS OF REAL INVESTMENTS The price of any security is linked to the level of the underlying variables or risk factors (eg, equity prices, credit spreads, interest rates), which can be simulated by scenario analysis. Risk is the measurable amount of price uncertainty due to the fluctuations of these variables, which include the passage of time and its interaction with the fixed or callable maturity of an instrument, as in general risk increases with increasing maturity. The issuer’s creditworthiness, or that of the guarantor of the investment, also plays a relevant role, as governments and companies can be perceived to hold higher or lower default risk, which in turn affects the time value of the investment claims. In addition, the legal priority of these claims matters, as senior debt is less risky than subordinated debt. The market risk factors that combine the fair-value pricing of all claims embedded in a security are key variables, among which are interest rates, implied volatilities, foreign exchange rates, equity prices, inflation expectations, commodity prices. Finally, the level of, and changes in, market and idiosyncratic liquidity can affect the width of bid–ask spreads. 18

A MODERN RISK MANAGEMENT PERSPECTIVE

A common indicator of financial risk is the standard deviation (square root of variance), which measures the dispersion of investment returns around the mean value of their distribution. As standard deviation is known to be an insufficient risk indicator, quantile measurement and tail statistics are usually preferred: the most common measures are VaR and ES. VaR is the estimate of the maximum potential loss of an investment over a holding period at a specified level of confidence. ES instead estimates the mean of the potential losses of an investment beyond the VaR threshold over a holding period, so, given the same holding period and level of confidence, ES is a more conservative risk estimate than VaR. The elements of the sample distribution of the negative/positive potential returns of a portfolio U are indicated by RU,H,t . The VaR of a portfolio U, measured at time t over the sample distribution of returns simuα lated with a scenario set S, is denoted by ξU,S,t , where α indicates the complement to the confidence interval. Throughout the book, the left 5% quantile of one tail of the distribution identifies VaR95 α −α VaR1U,S,t = ξU,S,t

(2.1)

The probability of the portfolio returns breaching the confidence threshold at time t is α PU,S,t (R  ξU,S,t )=α

(2.2)

It follows that −α ES1U,S,t =

1 α



(PU,H,t RU,H,t )

(2.3)

α s∈{s∗ : RU,H,t ξU,S,t }

−α ES1U,S,t denotes the expected shortfall of portfolio U, at time t, over

the sample distribution of the returns (simulated with a scenario set α S) that are beyond the VaR threshold indicated by ξU,S,t . Investment managers can choose different methodologies to estimate and investigate the properties of return distributions of risk factors and products. However, as the following sections will indicate, our preference in this book is for Monte Carlo simulations. PARAMETRIC METHOD: VARIANCE–COVARIANCE The variance–covariance method assumes that returns are normally distributed. In other words, VaR can be measured by estimating only 19

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 2.3 Parametric density

95%

99% –2.32 –1.65 –1.00 0 Multiples of standard deviation

three sets of parameters: the expected (or average) returns, the historical standard deviations and the correlations among the historical returns. As mean and variance are sufficient to define a normal distribution, we can conveniently operate with parametric forms instead of dealing with sample distributions. In this section the time subscript is dropped, as no explicit simulation is performed into the future but potential returns are estimated directly on the time series of the risk factors. Therefore, S indicates the series of n historical returns, where each historical return can be indicated directly by h. The standard deviation of the S distribution, made of the n total returns Rj,h of a jth asset, is given by σj,S

in which

   =

1



¯ j,S ) (Rj,h − R

n − 1 h∈S

 ¯ j,S = 1 (Rj,h ) R n h∈S

(2.4)

(2.5)

The advantages of the normal distribution are that we can easily calculate where the α value of the total return of an asset lies on the bell curve, as a function of the chosen confidence interval (1 − α) and the standard deviation σj,S as in Figure 2.3. The time dimension is not explicitly stated because it is a function of the size of the returns sample (eg, daily, weekly, monthly): 20

A MODERN RISK MANAGEMENT PERSPECTIVE

distributions of daily returns allow us to estimate daily VaR; weekly returns allow weekly VaR, etc. However, it might be inconvenient to use sampling lengths longer than a few days, since time series might not be readily available for sufficiently long periods and because adopting long time series might weaken the model sensitivity to current changes in market equilibriums. Therefore, market practitioners tend to estimate short-term VaR and then scale up the results by using the square-root-of-time rule of thumb (which assumes returns are independent and identically distributed). The time scaling of risk is performed similarly to the time scaling of volatility in the Black– Scholes equation of option pricing. Hence, assuming a quarter has 62 working days, the following formula indicates how to move the VaR estimate of a jth asset from a one-day (1D) to a quarterly (3M) figure 95,3M

VaRj,S

√ 1D = (−1. 65σj,S ) 62

(2.6)

Portfolio VaR can be estimated by introducing into the equations the correlation matrix between the asset returns and the allocation weights of the individual securities. For the sake of simplicity, we can assume that all relationships between asset returns and risk factor returns are linear, so that each jth asset corresponds to an individual jth risk factor, the volatility of the jth risk factor corresponds to the volatility of the jth asset and the correlation between any pair of jth and ith risk factors corresponds to the correlation between any pair of jth and ith assets in the portfolio. Hence, portfolio VaR can be calculated as (1−α)

VaRU (1−α)

 = − VaR (1−α) Σ(VaR (1−α) )T

(2.7)

in which VaRU is the portfolio VaR with a (1 − α) confidence (1−α) interval, VaR is the vector of the |VaR| of all assets in the portfolio and Σ is the correlation matrix of any pair of assets in portfolio U, so that    1 ρ(1,2) · · · ρ(1,i)      ρ(2,1) 1 · · · ρ (2,i)    σ = . .. ..  ..  .. . .  .      ρ(j,1) ρ(j,2) · · · 1 

Although the variance–covariance approach is appealing for its simplicity and inexpensiveness, it is a very limited risk management

21

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

technique because it makes strong assumptions on the form of the return distribution, it does not properly represent non-linear payoffs (especially derivatives) and it does not facilitate modelling of the passage of time consistently. A numerical example is provided towards the end of the chapter. HISTORICAL SCENARIOS METHOD The historical method is a scenario-based approach, ie, it requires the investigation of investment risk–return properties directly using the simulated distribution of profits and losses stemming from historical realisations: investment profits and losses are simulated by drawing directly from the history of the actual returns of the underlying risk factors. Therefore, in the simple case of linear products, the resulting distributions will be identical to actual historical distributions. To allow for more sensitivity to recent market events compared with past events, historical returns can be time weighted with various schemas, among which the most common is the exponentially weighted moving average. Historical simulations are generated with a straightforward methodology, which can be summarised in the following three steps: selection of risk factors; generation of scenarios (calculation of the historical returns); full revaluation of the securities under each scenario. • The time series of the risk factors that underlie the fair-value

pricing are selected for a given historical window.

• The returns for the selected risk factors over the chosen time

window are calculated (eg, 250 daily returns for the one-day VaR calculation with one-year historical simulation).

• Fair-value pricing based upon these scenarios is performed for

all securities in the investment universe.

The resulting distribution of the investment profits and losses can be ordered from worst to best, and quantiles can be estimated given (1−α) confidence levels. The methodology allows for a single investment horizon, which is usually fairly short, as historical scenarios are better suited for short-term analysis and do not allow us to build consistent risk–return profiles of investments over time. All methods based upon historical time series are very sensitive to the choice of the return type that builds the scenario set: differences or ratios. Scenarios can be expressed as additive shocks or multiplicative shocks, 22

A MODERN RISK MANAGEMENT PERSPECTIVE

(a) –9M

–6M

–3M

0.06 0.03 0 –0.03 –0.06 –0.09 –1Y

–9M

–6M

–3M

0.06 0.03 0 –0.03 –0.06 –0.09 –1Y

–9M

–6M

–3M

Daily returns

Daily returns

0.06 0.03 0 –0.03 –0.06 –0.09 –1Y

Daily returns

Daily returns

Figure 2.4 Historical scenarios

0.06 0.03 0 –0.03 –0.06 –0.09

Today

(b) Today

(c) Today

(d) VaR95

(a) Historical scenarios, stock A. (b) Historical scenarios, stock B. (c) Historical scenarios, stock A plus stock B. (d) Scenarios in ascending order.

which leads to over- or underestimation of measurable risks if the time series indicate a clear increasing or decreasing trend, as can often occur in financial markets. Therefore, additive scenarios tend to underestimate risks in up-trending markets and overestimate risks in down-trending markets, because historical basis point volatility might not comply with the most recent level of the risk factors. This is especially evident when moving to or stepping out from zero rate environments (eg, market participants would trade interest rates with different basis point (bp) ticks at a 10bp level than at a 500bp level). Multiplicative scenarios tend to overestimate risks in up-trending markets and underestimate risks in down-trending markets, because historical percentage returns might represent a different basis point volatility according to the level of the risk factors that enter the return calculation (eg, a 20% return applied to a 10bp interest rate leads to a 2bp increase scenario, but when applied to a 500bp rate leads to a 100bp change scenario). 23

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Moving from the risk estimate of individual securities to the risk appraisal of investment portfolios can be performed by aggregating the weighted monetary sum of the returns of each individual asset belonging to a portfolio or a subset, conditional on the same scenario path and time definition. Figure 2.4 shows this graphically, and a numerical example is provided towards the end of the chapter. HISTORICAL BOOTSTRAPPING METHOD Scenario bootstrapping is an alternative technique of creating historical scenario sets that is usually preferred to direct historical methods when time horizons longer than one day or one week are required (eg, one month) but only a limited amount of historical data is available, so that sampling of longer returns is not feasible. The method is based on the assumption that day-to-day returns are independent, which is a common assumption for financial time series although it does not always hold. The following example demonstrates how scenario bootstrapping works. We might want to bootstrap 1,000 scenarios to estimate historical VaR over three months from a five-year time series (about 1,250 observations). The idea of scenario bootstrapping is to calculate 1,250 daily returns and place them into a “pool”. The returns for all risk factors are calculated over the same period; hence, the risk factors are properly correlated within each item in the pool. Based on the assumption that day-over-day returns are independent, the items in the pool are independent. Drawing 60 items from the pool at random, with replacement, and accumulating them would create a three-month scenario. All draws are made with replacement to preserve the properties of the historical distribution. This means that there is a small chance that the same return could occur twice in a single scenario. Repeating this procedure 1,000 times would create 1,000 independent scenarios, thereby achieving the stated goal. Portfolio simulations can be easily performed, as the portfolio distribution is a linear combination of the risk–return realisations of all the individual products belonging to the portfolio, conditional on the same scenario path and time, as described for the historical scenarios method. A numerical example is provided towards the end of the chapter. 24

A MODERN RISK MANAGEMENT PERSPECTIVE

MONTE CARLO METHOD: STOCHASTIC SCENARIOS The Monte Carlo method is certainly the most advanced of the methods described in this chapter, because it can find solutions to problems that require probability analysis and it is suited to solving highly complex financial statements (eg, the simulation of derivatives over time). The Monte Carlo method requires multiple hypothetical trials to be run through an explicit model of the future distribution of returns of available risk factors, while separating the behaviour of security prices from the joint properties of the risk factors themselves. In this section, Monte Carlo refers to a generic method that estimates a set of random trials without being restricted to any specific model (eg, geometric Brownian motion or Hull– White). Investment managers are asked to decide on the model by which the stochastic movement of risk factors over time is to be described, as well the way in which the risk factors are jointly related. The process for generating Monte Carlo multi-period scenarios can be described in six steps: selection of simulation times; selection of risk factors; choice of the simulation model for each risk factor; calibration of the model; draws from the model; full revaluation of the securities under each time step and scenario. This allows us to model both the stochasticity and the dynamics of market risk factors, as well as the passage of time towards a selected investment horizon. • The investment manager chooses the length of the holding

period and the time steps for the simulation into the future.

• The time series of the risk factors are selected for a given

length, according to the preference towards more reactive risk measurement or through-the-cycle analysis.

• Each risk factor is assigned a simulation model that specifies

the potential returns distribution.

• Scenarios are generated by drawing from the model on all risk

factors (multivariate) and time steps (multi-period).

• The fair value of any financial security can be revaluated by

applying the “perturbed” risk factors contained in the set of stochastic scenarios. This technique is replicated for every scenario path at every time step, allowing securities to pay cashflows and to be revalued with reducing maturity. Fixed-income 25

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 2.5 Portfolio distribution Suboptimal portfolios

Frequency

Optimal portfolio

tn

on

riz

Sc

Loss

en

o eh

ari

os

Tim (1,

...,

t1

m) Profit

t0

securities, in particular, benefit from this technique as pullto-parity elements can contribute significantly to shaping the potential evolution of risk–return estimates over time. The resulting distribution of the investment profits and losses can be ordered from worst to best at any point in time of the corresponding investment horizon, and quantiles can be estimated given (1 −α) confidence levels. The portfolio distribution is a linear combination of the potential return realisations of all individual products belonging to the portfolio, conditional on the same scenario path and time, as in Figure 2.5. The numerical analysis provided throughout this book complies with either geometric Brownian motion or the Hull–White onefactor model. These models have been chosen from the many alternatives available in the financial literature, primarily for their tractability, and the reader is not restricted to this choice. Some quantitative formalisation follows, while a numerical example is provided towards the end of this chapter. Geometric Brownian motion GBM is a stochastic process that corresponds to a Markov process, ie, a continuous-time random walk with independent observations at each time step t across the investment horizon Γ . The model requires 26

A MODERN RISK MANAGEMENT PERSPECTIVE

us to estimate two parameters: the drift and the variance of the variable, ie, the risk factor underlying the Monte Carlo simulation. GBM satisfies the following stochastic differential equation dSt = µ St dt + σ St dWt

(2.8)

in which Wt is a Wiener process (Brownian motion), µ is a percentage drift that controls the “trend” and σ is a percentage volatility that controls the “random noise”. By applying separation of the variables we get dSt = µ dt + σ dWt St

(2.9)

and by integrating on both sides we have 



dSt = (µ dt + σ dWt ) dt St

(2.10)

Lemma 2.1 (Ito’s Lemma). If a random variable x follows an Ito process with drift a(x,t) and variance b2(x,t) , ie, dx = a(x,t) dt + b2(x,t) dWt , then a function G that depends on x and t can be defined so that the rate by which it changes respects Ito’s Lemma dG =



∂G ∂G ∂G 1 ∂2G 2 a(x,t) + b(x,t) dWt + b dt + ∂t ∂x 2 ∂ x2 (x,t) ∂x

(2.11)

∂2G 1 = − 2, ∂ x2 x

(2.12)

in which ∂G 1 = , ∂x x

∂G =0 ∂t

Therefore, by rewriting the function on the variable S (eg, a stock price) and setting a(x,t) = µ S and b2(x,t) = σ 2 S2 , we can see that the rate of change of the function G = ln(S) is indicated by 

dG = 0 +



1 1 2 2 1 1 µS − σ S dt + σ S dWt S 2 S2 S

1 µ dt − 12 σ 2 dt + σ dWt S = (µ − 12 σ 2 ) dt + σ dWt =

(2.13)

By defining St as the initial condition, the discrete case of a variation in ∆t between time steps t and t + 1 is given by 

dSt dG = ln St



= (µ − 21 σ 2 ) dt + σ dWt

(2.14) 27

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 2.6 Geometric Brownian motion for µ = 0% and σ = 18.5% 2.00

Total returns

1.50 1.00 0.50 0 –0.50 –1.00 1M 2M 3M 4M 5M 6M 7M 8M 9M 10M 11M 1Y Time

Figure 2.7 Geometric Brownian motion for µ = 12% and σ = 18.5% 2.00

Total returns

1.50 1.00 0.50 0

–0.50 –1.00

1M 2M 3M 4M 5M 6M 7M 8M 9M 10M 11M 1Y Time

Hence

ln(St+1 ) = ln(St ) + (µ − 12 σ 2 )∆t + σ ∆tϑ

(2.15)

Taking the exponential on both sides yields the equation of the stochastic time simulation

St+1 = St exp[(µ − 12 σ 2 )∆t + σ ∆tϑ]

(2.16)

in which µ is drift that can be arbitrary or calibrated to the history of the variable, (− 12 σ 2 ) centres to zero the log-return distribution of √ the GBM and σ ∆tϑ is a Brownian motion in ∆t, so that

exp[(µ − 21 σ 2 )∆t + σ ∆tϑ] has a normal distribution. Thus, by estimating µ and σ , a GBM simulation can be produced over the intervals ∆t, as in Figures 2.6 and 2.7. 28

A MODERN RISK MANAGEMENT PERSPECTIVE

Hull–White one-factor model: modelling the domestic interest rates The Hull–White one-factor model of interest rates (Hull and White 1990, 1993) assumes that the instantaneous short-rate r(t) evolves under the risk-neutral measure according to the process dr(t) = [θ(t) − κ r(t)] dt + σ dW (t)

(2.17)

in which σ is a positive constant that represents the instantaneous volatility, κ is a positive constant that represents the speed of mean reversion and θ is a time-dependent parameter that allows us to exactly fit the initial term structure of the interest rates, ie, the forwards observed in the market, and is determined by σ and κ . If we denote by F(0, T ) the instantaneous forward rate for maturity T with respect to time 0, the discount factor of any maturity T can then be defined as P(0, T ) = e−F(0,T )

(2.18)

so that the initial term structure of interest rates can be fitted by F(0, T ) F(0, T ) = −

∂ ln[P(0, T )] ∂t

(2.19)

The function θ(t) is given by θ(t) =

σ2 ∂ F(0, t) (1 − e−2κ t ) + κ F(0, t) + ∂T 2κ

(2.20)

in which ∂ F(0, t)/∂ T represents the partial derivative of F(0, T ) with respect to T. By integrating the function dr(t) for each s < t, the process is given by r(t) = r(s)e−κ(t−u) +

t s

e−κ(t−u) θ(u) du + σ

t

= r(s)e−κ(t−s) + α(t) − α(s)e−κ(t−s) + σ

s

e−κ(t−u) dW (u)

t s

e−κ(t−u) dW (u) (2.21)

in which α(t) = F(0, t) +

σ2 (1 − e−2κ t )2 2κ 2

(2.22) 29

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

The Hull–White model assumes that the short rate r(t) follows a normal distribution conditional on F(0, s) r (t) ∼ N (E[(t) | r(s)], var[r(t) | r (s)])

(2.23)

in which E[r(t) | F(0, s)] = r(s)e−κ(t−s) + α(t) − α(s)e−κ(t−s) var[r(t) | F(0, s)] =

σ2 (1 − e−2κ(t−s) ) 2κ

(2.24)

The behaviour of the short-rate is primarily determined by the volatility parameter, σ , while the behaviour of the long-term interest rate is affected both by σ and by κ , which is usually very low. This model is often used in risk management practices due to its tractability for modelling long-term simulations of financial derivatives (eg, counterparty credit risk and credit value adjustments), although it is characterised by some known drawbacks. First, the short rate is assumed to be normally distributed, which implies that the spot and the forward rates can become negative. Historically, such non-zero probability has often been disregarded by market practitioners, as it has been fairly low due to the non-zero initial level of the interest rates. However, financial markets exhibited near-zero rates during the Global Financial Crisis starting in 2007, making the results of the model more difficult to interpret economically. Second, the model does not handle steepening or flattening of the term structures since all forward rates are determined by the evolution of the short rate. This is fairly limiting in option pricing, particularly for complex payoffs that have high convexity due to non-parallel moves in the structure of forward rates. Figures 2.8 and 2.9 give an example of a stochastic simulation that complies with a Hull–White one-factor model, in which the euro-based term structure of interest rates has been perturbed with 1,000 stochastic scenarios for a multiplicity of time steps: the figures represent some scenario realisations over one-year and five-year simulation time steps. The calibration of parameters σ and κ is not always an easy exercise. Traditionally, the parameters have been calibrated to bond and option prices. However, liquid estimates of implied volatility might not always be available except for major currencies (eg, euro, US 30

A MODERN RISK MANAGEMENT PERSPECTIVE

Figure 2.8 Hull–White one-factor model with a simulation time of one year 0.06

Base curve Scenario #56 Scenario #677 Scenario #652

Zero rates

0.05 0.04 0.03 0.02 0.01 0

0 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y10Y 12Y Terms

15Y

20Y

Figure 2.9 Hull–White one-factor model with a simulation time of five years Base curve Scenario #230

Scenario #3 Scenario #41

0.06

Zero rates

0.05 0.04 0.03 0.02 0.01 0

0 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y10Y 12Y Terms

15Y

20Y

dollars), raising concerns over the generation of risk-neutral scenarios for simulating global market portfolios. Moreover, the calibration to implied volatility surfaces makes long-term stochastic scenarios particularly sensitive to daily market events, leading to excessive output volatility in the long run. Market practitioners are resorting to regression-based calibration methods as in Meng et al (2013) since historical data of interest rates is more widely available 31

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 2.10 Model sensitivity to the estimation of σ (κ is assumed constant at 0.01)

Total returns

2.0 1.5 1.0 0.5

Total returns

0.0855

0.224

0

1.5

0.0452

–0.0846

–0.5 –1.0 2.0

1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

9Y

10Y

9Y

10Y

(b)

1.0 0.122

0.5

0.26

0

–0.5 –1.0 2.0

Total returns

(a)

1.5

0.00742

–0.12 1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

(c)

1.0 0.5

0.181

0.272

0

–0.5 –1.0

–0.0539 –0.176 1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

US Government bond (2024, 7.50% US dollars): (a) σ = 0.07; (b) σ = 0.1; (c) σ = 0.15.

than implied volatility estimates, and leads to more stable long-term estimates. These approaches seem to become mainstream in risk management practices at major financial institutions, which have to generate stochastic scenarios for a broad set of term structures referring to currency denominations for which caps/floors, bond options or swaptions markets have not been developed. Two-factor models are also available in the literature and can help to overcome some of the original limitations of single-factor models, such as non-parallel modelling of the term structures. 32

A MODERN RISK MANAGEMENT PERSPECTIVE

Figure 2.11 Model sensitivity to the estimation of κ (σ is assumed constant at 0.10)

Total returns

2.0 1.5

(a)

1.0 0.5

–1.0

2.0 Total returns

0.269

–0.126

–0.000613

0 –0.5

1.5

1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

9Y

10Y

9Y

10Y

(b)

1.0 0.26

0.5

0.122

0

–0.12

–0.5 –1.0

2.0 Total returns

0.134

1.5

1Y

2Y

0.00742 3Y

4Y

5Y

6Y

7Y

8Y

(c)

1.0 0.5 0 –0.5 –1.0

0.203

0.074 –0.0714 1Y

2Y

0.0643 3Y

4Y

5Y

6Y

7Y

8Y

US Government bond (2024, 7.50% US dollars): (a) κ = 0.0001; (b) κ = 0.01; (c) κ = 0.1.

Figure 2.10 provides an example of model sensitivity to the estimation of σ (κ is assumed constant at 0.01) for a Monte Carlo multi-step simulation of the total returns of a fixed-rate government bond. Figure 2.11 provides an example of model sensitivity to the estimation of κ (σ is assumed constant at 0.10). The application of the Hull–White one-factor model, here described for interest rates, can easily be extended to model the evolution of inflation curves, foreign exchange rates and equity prices. 33

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

The Hull–White one-factor model: modelling foreign interest rates and the corresponding foreign exchange rates The implementation of the model for multi-currency frameworks requires the estimation of more than one interest rate stochastic process and more than one foreign exchange process. We assume for simplicity that only two term structures of the interest rate are available, the domestic rate r$ (t) and foreign rate r€ (t), and the corresponding foreign exchange between the two economies f$/€ (t). The model for the domestic interest rate given in the previous paragraph is dr$ (t) = [θ$ (t) − κ r$ (t)] dt + σ$ dW$ (t)

(2.25)

The model for the foreign currency rate is given by dr€ (t) = [θ€ (t) − κ r€ (t) − σ$/€ σ€ ] dt + σ€ dW€ (t)

(2.26)

in which σ$/€ σ€ is the quantised adjustment. Therefore, the process of calculating the foreign exchange risk factor with respect to the domestic base currency ($) is given by the following extension of the interest rate parity relationship df$/€ (t) = [r$ (t) − r€ (t)] dt + σ$ dW$ (t) f$/€ (t)

(2.27)

EXAMPLE OF VALUE-AT-RISK AND EXPECTED SHORTFALL MEASUREMENT The VaR and ES of an investment in the IBM stock are estimated by applying the four alternative methodologies. Parametric method Daily VaR95 reported in Table 2.1 corresponds to the standard deviation of daily returns of the IBM stock prices, here calculated by taking a one-year time series, and it is equal to −1.17%. Daily VaR95 can be scaled up to a higher confidence interval (ie, 99%) and to a longer investment horizon (ie, 3M) as indicated previously. Historical scenarios method The histograms in Figure 2.12 illustrate the one-year time series of the historical daily total returns of IBM stock, from the perspective of a US dollar investor (no currency risk). 34

A MODERN RISK MANAGEMENT PERSPECTIVE

Table 2.1 Parametric method Confidence interval 95% 99%

One day

Three months

−1.17% × 1.65 = 1.93% −1.17% × 1.65 × 7.75 = −14.95% −1.17% × 2.32 = 2.71% −1.17% × 2.32 × 7.75 = −21.02%

The VaR95 is −1.69% (see Table 2.2), which indicates that the potential negative return in one day will not exceed −1.69% with 95% confidence. In monetary terms, if US$100 is invested, there is a 95% probability that the potential loss in one day will not exceed US$1.69. At the 99% confidence level the VaR is −2.35%, which has a similar interpretation. The ES95 is equal to −2.85%. In other words, the mean value of the potential negative tail returns beyond the VaR estimated at 95% confidence level is −2.85%. Similarly to the mean–variance approach, the historical simulation does not facilitate explicit modelling of the passage of time, which is a relevant element of the total-return dynamics for fixed-income securities due to cashflow payments and pull-to-parity features. Market participants would therefore approximate longer time horizons by applying the squareroot-of-time rule of thumb although, as outlined earlier, this method neglects all cashflows occurring in the intermediate period and disregards pull-to-parity and convexity embedded in investment payoffs. In Table 2.2, we can see the impact of the change in currency denomination between the investment and the investor’s portfolio, by taking the perspective of a US dollar or a euro investor. Historical bootstrapping method VaR and ES are calculated for the same one-year time series of the IBM stock, by generating 1,000 bootstrapping 3M scenarios, as indicated previously, and displayed in Figure 2.13 and Table 2.3. Monte Carlo method The distribution in Figure 2.14 represents the frequency of the potential 1,000 profits and losses of an investment in the IBM stock, estimated with a Monte Carlo process whose parameters are taken from the statistics of a one-year time series, with a US dollar investor perspective and a three-month time horizon. The shaded bars highlight the “left tails” of the histogram, indicating the 95% and 99% confidence levels. These are the lowest 5% and 35

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Daily returns

Figure 2.12 Historical scenarios method: one-year time series of the daily total returns of IBM stock 0.06 0.03 0 –0.03 –0.06 –0.09

(a)

Daily returns

–1Y

0.06 0.03 0 –0.03 –0.06 –0.09

–9M

–6M

–3M

Today

(b)

VaR95 (a) Historical scenarios. (b) Scenarios in ascending order.

Table 2.2 Historical scenarios method: impact of the changes in currency denomination between the investment and the investor’s portfolio US$ investor (1D) US$ investor (3M) € investor (3M)    Confidence interval VaR (%) ES (%) VaR (%) ES (%) VaR (%) ES (%) 95% 99%

−1.69 −2.35

−2.85 −5.86

−13.31 −18.50

−22.44 −46.14

−15.50 −24.80 −27.48 −47.42

1% of quarterly returns, as the returns are ordered from worst to best. VaR values of −13.99% and −19.28% are indicated at the 95% and 99% confidence levels, respectively (see Table 2.4), for a three-month horizon. VaR is a statistical measure, not an absolute certainty, and allows for potential negative returns that are worse than −13.99% with 95% confidence. To increase confidence, we need to “move to the left” on the graph, where, for example, −19.28% represents the lowest 1% of quarterly returns. Alternatively, ES can be estimated beyond the quantile set for the VaR confidence interval. This allows for a more conservative risk measure that summarises the length of the tail by using its mean value. Hence, ES99 is −21.66%. Risk and return can also be investigated from an offshore perspective, like the view of a euro-based investor. Foreign exchange risk is a change in 36

A MODERN RISK MANAGEMENT PERSPECTIVE

Figure 2.13 Historical bootstrapping density

95% 99% –0.50 –0.40 –0.30 –0.20 –0.10 0 0.10 0.20 0.30 0.40 0.50 Quarterly returns

Table 2.3 Historical bootstrapping method: Impact of the change in currency denomination between the investment and the investor’s portfolio

Confidence interval 95% 99%

US$ investor (3M) 

€ investor (3M) 

VaR (%)

ES (%)

VaR (%)

ES (%)

−12.93 −20.58

−17.04 −22.91

−15.06 −28.57

−22.09 −32.75

numéraire for each scenario result, where USD/EUR rates are jointly simulated with the other risk factors. Thus, any product or portfolio risk–return profile can be expressed in any different numéraire, by diversifying foreign exchange risk with the other risk components. ABOUT THE DYNAMICS OF THE RISK MEASURES Risk measurement is a professional exercise that requires the choice of a quantitative methodology from the available alternatives, knowing that no model will hold under all the stressed market conditions. Moreover, the length and the consistency of the inputs of the various models might vary across market practices: two investment managers might assess the level of financial risk of the same financial product in a different way. We shall observe that each method may indicate a different level of risk perception and could evolve differently depending on its statistical properties. However, in the long 37

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 2.14 Monte Carlo density

95%

99%

–0.50 –0.40 –0.30 –0.20 –0.10 0 0.10 0.20 0.30 0.40 0.50 Quarterly returns

Table 2.4 Monte Carlo method: Impact of the change in currency denomination between the investment and the investor’s portfolio

Confidence interval 95% 99%

US$ investor (3M) 

€ investor (3M) 

VaR (%)

ES (%)

VaR (%)

ES (%)

−13.99 −19.28

−17.17 −21.66

−15.13 −29.32

−20.80 −35.41

run the dynamics of the different methods should tend to track one another, although with different speed and intensity. Traders often adopt a highly sensitive risk measure in order to manage hedging strategies, and hence favour historical or Monte Carlo methods with short time series of the risk factors and exponential decays. Portfolio and insurance managers, on the other hand, who are willing to optimise invested portfolios in the longer run, tend to favour longer time series as inputs to Monte Carlo methods, for their stability and suitability for probability measurement through time. For example, we can compare historical VaR and Monte Carlo VaR as calculated with the following risk factors: US Treasury zero rate 10Y, US dollar swap rate 10Y, EUR/USD foreign exchange rate, Standard and Poor’s (S&P 500) market index. VaR95 has been chosen as the risk measure and has been estimated at every observation date within a two-year period, using a oneyear rolling time series over a two-year time frame (2012–13) for 38

A MODERN RISK MANAGEMENT PERSPECTIVE

Figure 2.15 S&P 500 index –0.0070

Daily VaR95

–0.0092

Historical Monte Carlo

–0.0114 –0.0136 –0.0158 –0.0180 –1Y

–9M

–6M Time

–3M

Today

–3M

Today

Figure 2.16 EUR/USD foreign exchange rate –0.0070

Daily VaR95

–0.0092 –0.0114 –0.0136 –0.0158 –0.0180 –1Y

Historical Monte Carlo –9M

–6M Time

Figure 2.17 US Treasury rate 10Y –0.0070

Daily VaR95

–0.0092 –0.0114 –0.0136 Historical Monte Carlo

–0.0158 –0.0180 –1Y

–9M

–6M Time

–3M

Today

39

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 2.18 US dollar swap rate 10Y –0.0070

Daily VaR95

–0.0092 –0.0114 –0.0136 Historical Monte Carlo

–0.0158 –0.0180 –1Y

–9M

–6M Time

–3M

Today

both methods. In this example, the passage of time has not been explicitly modelled (one-day horizon assumption) to facilitate the direct comparison between the two methods. We can observe that the Monte Carlo simulation leads to more continuous measurement than the Historical simulation, as the latter displays more abrupt jumps in the risk appraisal due to changes in the scenario set, which are instead “smoothed” out with Monte Carlo measurement, as can be seen in Figures 2.15–2.18. CONCLUSIONS Monte Carlo methods have been selected to simulate over time the potential returns of real investment opportunities since fixed-income products and derivatives need to be fully revalued to reveal the risk–return properties of asymmetrical payoffs and pull-to-parity features, as indicated in subsequent chapters. This allows portfolio managers to overcome the simplifications of classical portfolio theory as it traditionally refers to simplified benchmarks, thus enhancing the coherence and efficiency of portfolio optimisations for long-term and goal-based investments. REFERENCES

Brigo, D., and F. Mercurio, 2006, Interest Rate Models – Theory and Practice: With Smile, Inflation and Credit Risk, Second Edition (Springer). Hull, J., and A. White, 1990, “Pricing Interest Rate Derivatives Securities”, Review of Financial Studies 3(4), pp. 573–92.

40

A MODERN RISK MANAGEMENT PERSPECTIVE

Hull, J., and A. White, 1993, “One Factor Interest Rate Models and the Valuation of Interest Rate Derivative Securities”, Journal of Financial and Quantitative Analysis 28(2), pp. 235–54. Markowitz, H. M., 1952, “Portfolio Selection”, Journal of Finance 7(1), pp. 77–91. Meng, Q., A. Kaplin and A. Levy, 2013, “Estimating Parameters in the Single-Factor Hull–White Model Using Historical Data”, Working Paper, Moody’s Analytics.

41

3

The Probability Measure A poker player may believe that a deck of cards is well shuffled. Yet he may not know all the implications of this belief. He is not likely to know, offhand, the probability of beating 3 aces and 2 jacks; or of beating 4 eights and a king if deuces are wild. It is usually not polite or convenient to employ a computing machine to calculate probabilities during the course of a poker game. In portfolio selection, however, the stakes are higher and decisions should be made on the basis of thorough analysis. Markowitz (1959)

In this chapter we investigate the probability measure: a priori and a posteriori probability, the probability distribution function, goalbased probability and the analysis of ex post and ex ante performance. INTRODUCTION Risk-based portfolio management requires estimation of the probability density function of a multitude of risk factors to model the potential dynamics of asset prices and inform investment decisionmaking within a consistent risk management framework. Classical approaches to portfolio choice have relied upon the measurement of specific moments of these distributions or tail loss estimates in order to formulate the objective function of optimisation routines. Yet, the a posteriori probability measure emerges as a convenient alternative because of its tractability in shaping optimal goal-based investing. This chapter starts with a basic review of probability theory: a priori and a posteriori estimates. Then, the probability density function is introduced so that we can draw the risk–return profile as an α quantile representation of the investment goals or loss limits over time. The last section investigates the concept of goal-based probability, which is a convenient measure for reconciling ex post and ex ante performance along the continuum of the investment life. THE PROBABILITY A PRIORI AND A POSTERIORI Games based upon bets and draws appeared in Bruges (“lotene”) and Milan (“fortuna”, “lotto”) around the 16th century, and led to 43

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

the first elementary formulations of probability problems. In 1539 the French king François I approved the first public lottery in the French kingdom, and in 1640 Jules Mazarin formally established the “jeu de la blanque royale”. In the game of chance, whatever its format (tossing a coin or rolling a dice) the outcome of the trial is unknown but we can calculate the probability of such an outcome. The term a priori probability refers to the probability of an event based upon deductive reasoning referring to existing knowledge, rather than calculation. For example, the probability that a balanced coin will turn up heads is 50% because, by deductive reasoning, we can argue that if the coin is not unbalanced there are only two ways that the coin can fall, either heads or tails. As this is a book about financial investments, we can say that the probability of a stock price increasing or strictly decreasing in any given minute of open trade is 50% because, by deductive reasoning, and without the need to take into account the intentions of the market participants, there are only two outcomes that can occur: stability or an increase; a decrease. The a priori probability here does not require any calculation or any coin to be tossed, as it can be based solely upon deductive reasoning. Hence, in a random experiment of N mutually exclusive and equally likely events, if a certain outcome occurs n < N times, then the a priori probability P of this outcome is equal to n/N. The a priori probability definition suffers from a series of drawbacks that make it tractable only for simple cases, in which the elements “mutually exclusive”, “equally likely” and “random” can hold. Now we can consider the case of tossing an unbalanced coin so that the two outcomes, heads and tails, are no longer equally likely. In such a case, deductive reasoning cannot help in finding the probability that the coin will turn up heads, so a frequentist approach is required. We can therefore assume that the coin can be tossed for a series of random trials (observations) occurring under uniform conditions so that the observations are individually unpredictable. As we can observe that in many cases the coin falls heads up (for example), the probability of this outcome is the ratio of the number of outcomes in which a head occurs to the total number of trials, also known as a posteriori probability, so that the relative frequency approaches the true 44

THE PROBABILITY MEASURE

probability as the number of trials increases. The important thing is that we can conceive a series of observations under rather uniform conditions (which is, by the way, quite a restrictive assumption in financial markets compared with traditional fields of science). A way to represent financial events is to construct mathematical models that adequately describe the real world. Hence, a probability model, ie, a probability distribution function, also known as a density function, can be constructed so that the properties of a set of outcomes can be described, although nothing much can be said about each individual outcome of the random financial variable. THE PROBABILITY DISTRIBUTION FUNCTION AND THE α-QUANTILE PROFILE Monte Carlo simulations have been indicated as a convenient approach to estimate the density function of the potential returns of any financial variable, and thus the potential total returns of the securities that map into these risk factors. Monte Carlo methods have two appealing features: 1. the separation between the simulation of risk factors and the full revaluation of financial products; 2. the ability to simulating financial variables over time, through a series of explicit time steps, which depict the potential patterns of financial returns along the investment horizon. By separating the simulation of the securities from the simulation of the underlying risk factors, we can implement a simulation model of total returns over time that does not lose accuracy: pricing formulas of real securities are a function of market variables (eg, interest rate term structures, inflation expectation curves or equity spot prices) which can be perturbed with stochastic scenarios at different time steps (scenario paths) to derive the space of the potential profits and losses, hence returns, of actual payoffs. By applying multi-period Monte Carlo simulations, securities can be revalued in a consistent multivariate fashion against potential future states of the world, pull-to-parity elements can be revealed as time goes by and total return analysis is enabled as cashflows can be tracked and reinvested. Investment professionals can choose among alternative models to calibrate stochastic simulations. In this book we express our non-restrictive preference for a mixture of risk-neutral 45

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

and real-world calibrations for the making of the case studies and examples. Once the simulated total return space of real securities has been drawn, quantiles, statistics and probabilities can be measured to shape the objective function of the probabilistic scenario optimisation presented in Part II. We denote by Vj,H,t and CFj,H,t , respectively, the value and the cashflows of a security j ∈ U conditional on a scenario path H ∈ S and a time step t ∈ Γ , with VU,H,t the value of portfolio U and Rj,H,t and RU,H,t the corresponding total returns generated by a multi-period stochastic process. The probability distribution function of portfolio returns (or individual securities), also indicated by a cumulative distribution function, defines a function of a set of values of a realvalued random variable R (indicating the total returns) that can be denoted by FR (R) = PU,S,t (R  b)

(3.1)

where the right-hand side represents the probability that the random variable R for a given time step t ∈ Γ across all scenario paths H ∈ S takes a value greater than or equal to b. Similarly, the probability that R lies in the closed interval [a, b] is FR (b) − FR (a) = PU,S,t (b  R  a)

(3.2)

In the continuous case, a probability density function (pdf) or density of a continuous random variable is a function that describes the relative likelihood of this random variable taking a given value. The probability of the random variable falling within a closed interval of values [a, b] is given by the integral of this variable’s density over that range. That is, it is given by the area under the density function between the lowest and the highest values of the range. The probability density function is non-negative everywhere, and its integral over the entire space is equal to 1 by definition. In the case of a discrete distribution, which is the outcome of stochastic simulations, the probability would be calculated by counting the number of scenarios that are above (or below) a given target b normalised by the total number of scenarios. Figure 3.1 gives an example of multi-period Monte Carlo simulation of an equity price and the computation of the probability of beating an assigned total return target. Figure 3.2 represents the densities of the simulation plotted in Figure 3.1, conditional on some individual time steps t into the future. 46

THE PROBABILITY MEASURE

Figure 3.1 Monte Carlo simulation

Total returns

Probability of beating target 3Y

1Y

2Y

3Y

4Y

5Y Time

6Y

7Y

8Y

9Y

10Y

Figure 3.2 Densities of the Monte Carlo simulation plotted in Figure 3.1 Distribution 45

45

(a)

25

(b)

25

0 –1 45

Target

0

1

2

45

(c)

25

0 –1

0 –1

0

1

2

0

1

2

(d)

25

0

1

2

0 –1

(a) 1Y. (b) 3Y. (c) 6Y. (d) 10Y.

α Quantiles ξU,S,t are points taken at regular intervals from the same cumulative distribution function of an ordered real-valued random variable R with confidence level α ∈ [0, 1] and t ∈ Γ . An α quantile α ξU,S,t takes a value such that the probability that the random variα able is less than ξU,S,t is equal to α (eg, 5%) and the probability that α the random variable is more than ξU,S,t is (1 − α). Therefore, for a

47

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

α population of discrete values, ξU,S,t is the data point where the cumuα is an α quantile lative distribution function crosses α. That is, ξU,S,t for the variable R at a given time t ∈ Γ over the scenario paths H ∈ S, defined by α )=α PU,S,t (R  ξU,S,t

(3.3)

The quantile definition can easily be extended to the concept of α a quantile profile over time. The α quantile profile ξU,S,T for a timedependent random variable RU,S,t with α ∈ [0, 1] is indicated by the set α α ξU,S, Γ = {ξU,S,t | t ∈ Γ }

(3.4)

This allows us to investigate the potential dynamics of portfolio total returns, induced by portfolio compositions ΦU,H,t (as defined in Part II) and income/consumption streams cU,H,t , based on the α implied α quantiles’ profile ξU,S, Γ of the portfolios. While quantile analysis (eg, VaR) requires us to indicate the confidence interval so that the corresponding quantile return can be read off from the distribution in order to check the adequacy of the risk appetite profile, the probability measure requires us to indicate the return level and expresses the probability of crossing such a threshold so that the target profile can be identified. A key characteristic of optimisation routines that work directly with probabilities is that all the information contained in the tail of the distribution (above or below a threshold) contributes equally to the solution of the optimisation exercise. Thus, the aim of PSO is to maximize the probability of achieving or beating a certain return target while complying with an α quantile risk limit. Clearly, for a given definition of the investor’s risk– return profile (as defined in Chapter 5), the optimisation methodology is sensitive to the choice of the simulation model applied to the underlying risk factors of the simulated securities. THE LINK BETWEEN EX POST AND EX ANTE PERFORMANCE A relevant characteristic of the probability measure is related to performance attribution. It might be informative for a portfolio manager to understand via a simple representation how well the portfolio has performed ex post, starting from the inception of the investment (or 48

THE PROBABILITY MEASURE

Table 3.1 Evolution of ex post and ex ante performance for an investment horizon with target return 50%



Start

Q1

Time Q2

Q3



Q4

Ex post performance — 12.58 35.46 66.16 104.94 Ex ante at 1Y (LT5 ) −61.10 −27.34 3.80 40.59 104.94 100.92 68.62 80.98 95.19 104.94 Ex ante at 1Y (RT5 ) Prob of beating target at 1Y 14.59 11.89 23.98 83.82 100.00 Prob of incurring loss at 1Y 67.64 36.56 3.00 — — All values are given in percent. Data refers to Figures 3.3 and 3.4.

the declaration of the goal), and compare it with the ex ante performance, which indicates the remaining potential of a portfolio to yield a certain level of cumulated total return. While the evolution of the ex post performance can be plotted with certainty, as cashflows have been collected and all securities in the portfolio can be marked to market or marked to model, ex ante performance is uncertain and can only be described by plotting the probability density function that stems from a stochastic simulation. We can imagine plotting both ex ante and ex post elements of the portfolio total performance against time. As time passes, we can recalculate ex ante performance at every time step where stochastic scenarios are recalibrated according to the prevailing market conditions, while the ex post performance can be easily tracked at the same time. By eliciting a target return (eg, a constant value or a time profile) and imposing it on the simulated performance space, a solution to the problem of quantitative goal-based investing can be found. Thus, the probability of such a portfolio achieving or beating the declared investment goal can be estimated at the beginning of the investment period (with respect to every time between today and final maturity or by looking at the end of the investment horizon only). Over time, we can estimate how close we are to achieving our final goal, conditional on historical market dynamics featuring positive portfolio returns, or how much the performance of a portfolio deviates from the original goal, conditional on historical market dynamics featuring negative portfolio performance. Clearly, the probability of achieving our target return at the end of the investment 49

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 3.3 Evolution of ex post and ex ante performance for an investment over the investment horizon (Q1 and Q2) with target total return 50% Ex ante (scenarios)

Total returns (%)

Total returns (%)

Total returns (%)

Ex post 200 150 (a) 100 50 0 –50 –100 Today 1M

2M

3M

Ex ante (quantiles)

4M

5M

6M

7M

8M

9M 10M 11M 1Y

200 150 (b) 100 50 0 –50 –100 –3M –2M –1M Today1M

2M

3M

4M

5M

6M

7M

8M

9M

200 150 (c) 100 50 0 –50 –100 –6M –5M –4M –3M –2M –1M Today1M

2M

3M

4M

5M

6M

(a) Trade date. (b) Q1. (c) Q2.

period Γ becomes a function of how well the portfolio has performed so far in terms of returns, denoted as Rex post , and how much still remains to be achieved. At inception of the investment, such a probability measure can be estimated as follows FR (R) = PU,S,t (RΓ  (bΓ ))

(3.5)

At any subsequent investment period ex post

FR (R) = PU,S,t (RΓ  (bΓ − Rt∗ ∈{Γ |t∗ t} ))

(3.6)

in which RΓ is the random return of the portfolio along the investment horizon. Rex post is the total return cumulated up to the time of calculation. bΓ is the investment target at final investment horizon. 50

THE PROBABILITY MEASURE

Figure 3.4 Evolution of ex post and ex ante performance for an investment over the investment horizon (Q3 and Q4) with target total return 50% Total returns (%)

200 (a) 150 100 50 0 –50 –100 –9M –8M –7M –6M –5M –4M –3M –2M –1M Today1M

2M

3M

Probability at investment horizon

Total returns (%)

200 (b) 150 100 50 0 –50 –100 –1Y –11M–10M –9M –8M –7M –6M –5M –4M –3M –2M –1M Today 1

0

(c)

Beating target Capital loss

0

Q1

Q2

Q3

Q4

Time (a) Q3. (b) Q4. (c) Probability.

The example summarised in Table 3.1 can clarify this mechanism for reporting the performance of optimal investments. We can assume that a US investor wants to engage in an open long position in a US dollar denominated common stock traded on the New York Stock Exchange. The investor risk–return profile is indicated by a single time step equal to one year, the risk appetite corresponds to a capital guarantee (no-loss case) at the end of the investment period and the investment goal is equal to a 50% gain at the final date. We assume that the potential returns of the equity holdings can be simulated via a multi-period set of stochastic scenario paths. We can simulate the potential future outcomes of the equity investment at the start of every quarter, until the investment horizon is reached, 51

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

starting from the market performance at the end of the previous subperiod. Thus, every quarter we can measure the α quantile profiles with 95% confidence, indicated as left (LT5 ) and right (RT5 ) quantiles, the probability of achieving/beating the cumulated target return at the one-year investment time and the probability of incurring a loss at every intermediate time step. We can clearly see in Figures 3.3 and 3.4 that the total probability of beating a target (or incurring into a capital loss) is a function of the actual market realisations (ex post performance) and the potential realisations (ex ante stochastic simulation). This is the essence of PSO, which allows us to graphically represent seemingly complex mathematical relationships (stochastic processes, value-at-risk, confidence intervals, etc) and depict the journey that an investors’ wealth has made so far (since declaration of the investment goal) and might still be making into the future (modern risk–return management). CONCLUSIONS Accessing the full density of potential total return distributions can add value to the optimisation exercise, especially when dealing with real payoffs, since the risk–return profile of derivatives and structured products cannot be reduced to the main moments of the distributions. In the previous chapter we established the risk management framework and in this chapter we set the probability measure of achieving or beating a target as the key variable of the optimisation objective function. The next chapter elaborates on the importance of modelling real payoffs so that fixed-income and derivatives products can adequately be included in the making of optimal portfolios and discusses reinvestment strategies, which allow us to simulate portfolios made of different types of asset beyond the natural or legal maturity of the products themselves, to comply with the requirements of long-term investment analysis.

REFERENCES

Belmas, É., 2006, Jouer Autrefois: Essai sur Le Jeu dans La France Moderne (XVIe–XVIIIe Siècle), Champ Vallon Époques (Ceyzérieu: Éditions Champ Vallon). Culp, C. L., and R. Mensink, 1999, “Measuring Risk for Asset Allocation, Performance, Evaluation and Risk Control: Different Problems, Different Solutions”, Journal of Performance Measurement 4(1), pp. 55–73.

52

THE PROBABILITY MEASURE

Markowitz, H. M., 1959, Portfolio Selection: Efficient Diversification of Investments (Chichester: John Wiley & Sons). Mood, A. M., F. A. Graybill and D. C. Boes, 1974, Introduction to the Theory of Statistics, Third Edition (McGraw-Hill).

53

4

Real Securities and Reinvestment Strategies: Fixed-Income and Inflation-Linked Securities and Structured Products The Black Swan is what we leave out of simplification. Nassim Taleb (1960–)

In this chapter we discuss the importance of simulating the risk– return profile of real investments, as opposed to benchmarks, the definition of total asset value and return, the fair-value concept and how to model fixed-income securities, reinvestment strategies of bonds and structure funds, structured products and inflation-linked securities. INTRODUCTION Modern Portfolio Theory refers to the contribution of Harry Markowitz, whose intuitive mean–variance paradigm has permeated portfolio management practices ever since its appearance in the 1950s. Although equity portfolio managers have made extensive use of this method and its derivations, fixed-income traders and derivative desks have made much less use of the theory because it cannot conveniently treat the non-linearity of real securities, or the pullto-parity characteristics of bonds returns. Thus, portfolio managers seeking optimal mean–variance allocations have often resorted to benchmarks to represent fixed-income and derivatives exposures, potentially missing out on relevant risk management information. These limitations are particularly relevant for long-term investment. Compared with the early 1950s, modern advances in mathematical finance have significantly increased the way investment opportunities are packaged to private and institutional investors (eg, structured finance), augmenting the opportunities to model non-linear relationships in portfolio returns. Additionally, financial markets 55

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.1 S&P 500 2,000

Price (US$)

1,500

1,000

500

0

1954 1960 1965 1971 1976 1982 1987 1993 1998 2004 2009

have recently shown more frequent and severe swings than in the 1950s (as described in Figure 4.1), making the task of forecasting the expected returns of products and portfolios increasingly complex. Probabilistic scenarios are meant to address the risk management needs of long-term investors dealing with non-linear investments. Simulating real investments over time is a mathematically complex exercise for both equity-like investments (eg, stocks, equity funds) and non-equity-like products (eg, fixed-income, structured products, derivatives). However, the latest products face further challenges due to the treatment of legal maturities, as further quantitative methods need to be featured to roll portfolios over time and simulate the reinvestment of non-linear holdings into meaningful future exposures conditional on stressed market conditions. In this chapter we investigate the relevance of enriching portfolio optimisation by the simulation of real securities. First, we identify the universe of investment securities that we shall primarily refer to throughout the book. Then, we define total asset values and returns as the main outputs of Monte Carlo multi-period simulations. We discuss fair-value modelling with reference to fixedincome discounted cashflow models, structured funds featuring capital guarantees and inflation-linked securities. We also discuss reinvestment strategies; these are a key feature of long-term simulations because they allow us to roll over investments beyond the contractual maturity whenever it occurs before the final investment horizon. 56

REAL SECURITIES AND REINVESTMENT STRATEGIES

INVESTMENT SECURITIES UNIVERSE A security is a contract representing the right to receive future benefits under a stated set of conditions. Bonds, stocks, investment funds, structured notes, certificates and Treasury notes are all securities. The set of financial securities primarily considered in this book is limited to those contracts that are traded in organised markets and that qualify as assets in the investment management process. However, the probabilistic scenarios framework is generic, so overthe-counter derivatives and liability exposures can be introduced without modifications to the theory. A common way of classifying financial securities is the hierarchy in Figure 4.2. An investor can build direct exposure into any one of a number of different securities, many of which represent a different type of claim on a financial, corporate or government entity. Alternatively, an investor can hold exposures through an intermediary (fund) that offers quotes in the portfolio of financial instruments it holds. In practice, financial innovation has blurred the line between direct and indirect investments, by means of outright or embedded derivatives that can add via direct investment indirect exposure to any set of investment claims. Direct investments often bear an explicit time horizon (maturity). Rate-sensitive investments in low-risk debt with a short life are generally called money-market or liquidity instruments. Rate- and credit-sensitive investments with longer maturities are classified as fixed-income products. Equity and commodity claims can also be classified separately, while fixed-income investments are further divided according to whether they are issued by a government or private entity or they embed structured elements (such as covered bonds or mortgage-backed securities). An alternative classification of financial securities refers to the risk– return characteristics of the investment products themselves, as in Figure 4.3. Benchmark products are typically indirect investments (eg, equity funds) whose mandate is to track the evolution of a specific market segment by passive or active asset allocation. In the first case, we want the financial assets to replicate the risk–return characteristics of a given benchmark, while in the second case asset managers choose a different asset mix from the benchmark and adhere to specific views about the reference market to seek alpha (ie, extra performance). Risk-target products are typically indirect investments (eg, balanced 57

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.2 Financial securities classification by type Financial assets

Direct investments

Indirect investments

Derivatives

Investment funds

Commodity

Hedge funds

Equity

Fixed income

MM and liquidity

Government bonds

Credit

Structured notes



Figure 4.3 Financial securities classification by style Financial assets

Benchmarking

Risk taking

Return targeting

funds) with which we want to obtain the maximum return under a specific target in terms of risk exposure. Return-target products are typically a synthetic combination of risk-free assets and risky assets (eg, financial bonds or structured products) and can take a variety of forms according to the financial engineering embedded in their modelling. Global allocation strategies permit us to bundle together investments in any one of the above classifications and to hold the investments until the desired investment horizon is reached. Certainly, if the period is longer than the investment maturity date, the simulation of the potential distribution of investment returns requires clear assumptions on the reinvestment strategies of these claims. Similarly, portfolio holdings can also change over time, given opportunistic or strategic investment decisions that can be modelled with dynamic trading strategies. As discussed in the remainder of 58

REAL SECURITIES AND REINVESTMENT STRATEGIES

this chapter, for simplicity we here assume that the investment horizon is equal to or shorter than the time to maturity, while cashflows received during the holding period are not reinvested. With reference to credit markets, fixed-income trading has continuously grown in importance from the early 1970s, with significant expansion in the amount of outstanding notional in the US and other Group of Seven markets. However, as opposed to equity markets, where the adoption of stock indexes can be traced back to the beginning of the 20th century, total return bond indexes became the mainstream portfolio reference benchmark only in the 1980s, with the development of international government bond markets accompanied by the creation of international indexes, followed by the introduction of market references for financial and corporate issues (investment grade and high yield). The creation of a bond index is more difficult than the computation of an equity index. To create an equity index we need to select a basket of eligible stocks, define a weighting schema and adopt a computational method. As equities are usually listed on stock exchanges, their evaluation is quite simple and, apart from the management of mergers between companies, stock splits or relevant underperformance, they might remain in the index for a long time. To create fixed-income indexes instead, we face a series of difficulties. First, the universe of bonds is more variegated than the universe of stocks across credit levels, industries and currencies. Second, individual institutions can issue very different claims in fixed-income markets regarding coupon, maturity, sinking or call features, embedded optionality and special indexations. Third, individual bonds are generally less liquid than individual stocks, which requires the bond sample to be marked-tomodel. Lastly, the passage of time and changes in the level of market rates and credit spreads continuously affect the time to maturity and duration of bonds, requiring more frequent rebalancing. Consequently, it may be less restrictive to represent equity exposures by means of indexes and multi-factor sensitivities than to approximate individual bonds by the risk–return characteristics of fixed-income indexes. MODELLING TOTAL ASSET VALUES AND RETURNS The time simulation of the fair value of a security must be coupled with the treatment of the proceeds that might occur within 59

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 4.1 Time evolution of the market prices and cashflow payments Issue

0.5Y

1.0Y

1.5Y

2.0Y

2.5Y

3.0Y

Prices 100.00 96.46 100.83 101.20 101.61 101.11 100.00 Changes (%) — −3.540 0.008 0.012 −0.016 −0.011 — Coupons — 3.00 3.00 3.00 3.00 3.00 3.00 Total P&L (%)



−0.540 6.830 10.202 13.611 16.110 18.00

Figure 4.4 Time evolution of the market prices and cashflow payments 115

Market price

110 105 100 95 90

Coupons

85 80 5 0

0.5

1.0

1.5 Time (years)

2.0

2.5

3.0

the investment horizon, so that a stochastic total return analysis can be derived. As opposed to counterparty credit risk, which is concerned with the potential point-in-time exposure of a financial claim in order to estimate the credit loss should the counterparty default, portfolio management cares about the total return dynamics of the investments, which include inter-temporal cashflows, conditional on scenario realisations, and any potential change in the simulated fair values. We can consider a fixed-rate bond issued at par, paying US$3.00 semi-annual coupons with three years’ maturity. In the absence of default, the return that an investor can receive by holding the position up to the investment horizon is a function of the capital gains/ losses plus any intra-period cashflow paid by the bond issuer, as in Table 4.1 and Figure 4.4. Thus, the total asset value of a security equals, at any point in time, the sum of the simulated value plus any compounded proceeds from the inception of the investment until a 60

REAL SECURITIES AND REINVESTMENT STRATEGIES

given future time, and is indicated by Vj,H,t , which is the mapping that assigns a value V to each asset j in the universe U ( j ∈ U) subject to a scenario path H at a time t along the investment horizon Γ (t ∈ Γ ). It is assumed that the initial asset values of each security are not zero, when t = 0 and H = 0 (indicated as base scenario path) Vj,H,t |t=0,H =0 ≡ Vj,0,0 = 0

for all j ∈ U

(4.1)

Hence, the total asset return indicates the total return of an investment over a particular span of time, also called holding period, starting at t = 0 and referring to any subsequent t ∈ Γ . As time goes by, investments can change in value due to shorter time to maturity or changes in market variables affecting the quoted prices, payments of coupons or dividends. Therefore, total asset return is calculated as the sum of the change in the market price and any cashflow received during the holding period, divided by the market price of the security at the beginning of the period. The total asset return Rj,H,t equals the percentage change between the value Vj,H,t of any asset j ∈ U, inclusive of cashflows CFj,H,t received/paid during the period, conditional on a scenario path H at a specific time step t ∈ Γ , and the initial non-zero value Vj,0,0 , such that Rj,H,t =

Vj,H,t +



t∗ ∈{Γ |t∗ t}

CFj,H,t∗

Vj,0,0

−1

(4.2)

As U is the universe of the investable assets, any combination of the constant allocations xj,0,0 of any asset belonging to U can make an attainable portfolio. An exchange rate transformation of an instrument currency $j into a given portfolio currency $U might be required, in order to allow for portfolio construction and valuation, where exchange rates are allowed to vary over time and across scenario paths, eg VU,H,t =



j ∈U

Vj,H,t +



t∗ ∈{Γ |t∗ t}

CFj,H,t∗ xj,0,0 $j/U,H,t

(4.3)

and RU,H,t =

VU,H,t −1 VU,0,0

(4.4)

in which xj,0,0 is the constant position of the asset in the portfolio U and $j/U is the exchange rate that transforms the denomination of the jth asset into the denomination of U. We can see that the 61

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

adoption of dynamic trading strategies can play a pivotal role in relaxing the assumption of constant portfolio allocation. Individual positions could be allowed to vary over time and across scenario paths, so that VU,H,t =



j ∈U

Vj,H,t +





CFj,H,t∗ xj,H,t $j/U,H,t

t∗ ∈{Γ |t∗ t}

(4.5)

MODELLING FAIR VALUES OF REAL SECURITIES Financial securities, whose modelling and contribution to optimal asset allocations are the main focus of this book, are usually traded on organised exchanges, so we know the corresponding market value of the holdings at any point in time during the life of the products or the length of the investment horizon, conditional on sufficient market liquidity. While the history of market prices is generally considered adequate information for simulating the value of equity-like products to a future date, non-equity-like products require the adoption of pricing formulas to indicate the fair-value price and to simulate the exposures to potential market values in the future. Thus, it is the simulation of the underlying financial variables that enables us to draw the density function of the security returns over time. As indicated by Minenna (2011), the fair value of a security is the expected value (eg, a bond), under the risk-neutral measure, of the future cashflows until maturity discounted at the risk-free rate. Therefore, at maturity, the final value of the product is derived from the riskneutral density of a random variable Vt , where t ∈ Γ . In the simple case of a fixed-rate bond, the density function of Vt is indicated by two determinants: the paths in which a default event occurs and the value converges towards the recovery estimate; the paths that are not affected by a credit event, so that the returns depend on the cashflow structure of the bond and on the level of the credit spread paid to the investors to compensate for credit risk. This is given by V0 = E

Q



VT BT



(4.6)

in which V0 is the fair value of the contingent claim, VT is the random variable corresponding to the stochastic process of the value of the contingent claim St for t ∈ Γ at maturity date T and BT is the random 62

REAL SECURITIES AND REINVESTMENT STRATEGIES

variable corresponding to the stochastic process of the risk-free rate for t ∈ Γ at maturity date T. Thus BT = exp

T 0

rs ds



(4.7)

The fair value of the product is the discounted expected value of its future cashflows until maturity, where the expectation is taken from the risk-neutral measure, that is without making any assumption on investors’ preferences. The density of VT can be obtained through a Monte Carlo simulation of the underlying variables that make up the fair-value pricing formula (eg, underlying asset prices, market indexations, implied volatility, credit spreads). The risk-neutral density of BT is determined accordingly in a correlated process. The fair value is, by definition, a synthetic indicator of the riskneutral density of the possible final profits and losses; therefore, it ignores information provided by moments of order higher than 1 and it does not allow for the appreciation of the degrees of randomness characterising the performance of a given product. In fact, the same discounted expected value may be obtained by density functions with very different shapes. This justifies our argument for working on the full density function of the potential returns of real products, without reducing their representation to the first two moments of such distributions, the mean value and the variance. Minenna (2011) also argues for the use of probabilistic scenarios to unbundle the risk and return determinants of any contingent claim in such a way that a direct comparison between securities, although generated with different financial engineering features, can be performed by juxtaposing the probability of achieving reference thresholds, linked to the dynamics of the risk-free asset, and the conditional expected value of the density partitions between the securities. We argue similarly that tactical asset allocations and optimal portfolios can be directly compared by juxtaposing their probability density functions against the set of predefined thresholds that constitute the risk–return profile of the investor and are expressed in terms of return targets and risk limits over time, as opposed to expected return and variance. Our main simplification pertains to the treatment of default risk, which is a relevant component in determining the fair value of contingent claims such as risky bonds. The computational challenges 63

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

required to integrate the differential equations of the market variables with the stochastic simulation of the credit events make the task particularly onerous for today’s risk management architectures. The number of scenarios required to fully integrate market and credit risks, by means of Monte Carlo simulations with the full valuation over time of real securities, may not be easily treatable. Therefore, the examples featured in this book embed the credit spread risk of credit-sensitive products but disregard the stochasticity of default events (particularly relevant for high-yield securities). We are aware that such an assumption is restrictive. MODELLING FIXED-INCOME SECURITIES Bond issues are claims on a series of future payments: the buyer of a bond lends the issuer an amount of money equal to the prevailing market price and expects to receive the repayment of the notional amount at maturity (if not amortising) plus periodic interest payments (eg, fixed coupons, floating coupons, spreads, capped/ floored rates, constant maturity indexed coupons). We can consider the case of two simple non-amortising bonds (a fixed-rate bond and a floating-rate bond), and decompose their potential total return into two sets of elements: coupon payments, denoted by CFj,H,t , and final notional reimbursement, denoted by Nj,0,0 . In the case of a non-amortising fixed-rate bond (eg, constant coupon), coupons are known at the beginning of the product’s life and do not change over time or across scenario paths Vj,s,t∗ =



t∗ ∈{Γ |t∗ t}

CFj,0,t DFH,(t−t∗ ) + Nj,0,0 DFs,(T −t∗ )

(4.8)

Here DFH,(t−t∗ ) indicates the discount factor between reference time t∗ and future time step t, conditional on scenario path H. For simplicity, bond claims are assumed to occur with the same frequency as the discretisation of t ∈ Γ (eg, yearly). In case of a plain vanilla floating-rate bond (eg, London Interbank Offered Rate (Libor) indexation), coupons are reset to the prevailing financial market conditions, so they might be different across scenario paths and time Vj,s,t∗ = 64



t∗ ∈{Γ |t∗ t}

CFj,H,t DFH,(t−t∗ ) + Nj,0,0 DFs,(T −t∗ )

(4.9)

REAL SECURITIES AND REINVESTMENT STRATEGIES

Figure 4.5 Price–yield relationship for a fixed coupon at 6% with a 10Y maturity 160

Price (US$)

140 120 100 80 60 0

2

4

6 Risky yield (%)

8

10

12

Therefore, the time dynamics of bond returns depend on the evolution of the forward rates and the discount factors. The main risk factors that affect bond prices are the level of the interest rates (or the indexation) and the level of the credit spreads representing the credit risk of the issuer. Clearly, market liquidity and trading volumes, subordination levels and taxation also contribute to determine the price of a bond, among the many variables. With respect to a fixed-rate bond, at any given pricing date the coupons are known to both the issuer and the buyers; the latter also know the remaining time to maturity, so a consistent discounting function (embedding credit risk) can be estimated. The return that an investor wants to receive from holding such a claim is known as the yield to maturity (YTM) and it is typically indicated by an annual rate. Here it is assumed that the YTM rewards the investors for the risks due to potential changes in the general level of interest rates and idiosyncratic credit spreads. Consequently, as the risky yield changes, the price of the bond changes to compensate the investor for the modification of the underlying conditions. This is the socalled price–yield relationship (as in Figure 4.5): when the risky yield increases, the price of the bond decreases and vice versa (at least in the general case of a fixed-rate bond). In frictionless markets, bonds would trade at prices that make the risky YTM equal to the relevant market rate and credit spread of any given issuer. We can interpret the risky YTM as a proxy of the full discounting of the bond’s claims, comprising interest rates 65

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.6 Time path of bond prices 120 115 Price (US$)

110

Above par

105 100

At par

95 90

Below par

85 80 –10Y

–9Y –8Y –7Y –6Y –5Y –4Y –3Y –2Y –1Y Maturity Remaining time to maturity

plus credit spreads. Should the risky YTM be larger than the coupon rate, the bond would be priced at the discount compared with par value (ie, the notional amount), since the investor would require additional compensation for buying the bond, which would come from capital appreciation (and vice versa for premium bonds). This is called the pull-to-parity relationship (described in Figure 4.6), ie, the relationship between the bond price and the remaining time to maturity. The risky YTM represents the internal rate of return (IRR) of a bond, ie, the interest rate that results in a zero present value when used to discount all future claims of the bond to be bought today. The relationships described by the definition of the risky YTM are a very simplistic indication of the risk–return profile of a bond, as they assume that interest rates and credit spreads do not change over the life of the investment. In reality, credit spreads and market rates differ according to different maturities, indicating different risky YTMs for fixed-rate bonds issued by the same entity and traded with different residual lives. Moreover, interest rates and credit spreads vary over time, conditional on changes in market conditions, economic cycles and idiosyncratic events that make the path of bond prices towards maturity uncertain, like any other financial variable. Still, in the absence of default, pull-to-parity conditions would hold. Probabilistic scenarios allow us to represent coherently the total risk– return space of fixed-income securities, as shown in Figure 4.7, which 66

REAL SECURITIES AND REINVESTMENT STRATEGIES

Figure 4.7 Time path of prices for fixed-income securities 110

Price (US$)

Above par 105

100

95

Below par

–10Y –9Y

–8Y

–7Y

–6Y

–5Y

–4Y

–3Y

–2Y

–1Y Maturity

Remaining time to maturity

illustrates the potential evolution of the market value of two bonds that initially trade above par and below par. As fixed-income securities differ from equity-like asset classes, Monte Carlo simulations based on full revaluation techniques of real securities can add significant value compared with models based on benchmarks, although the latter are more elegant in their formulation. This is especially evident when projecting investment returns in the long run. Accessing the full density of the potential risks and returns of a real security stemming from stochastic scenario generation of the underlying risk factors allows us to preserve the asymmetry of actual investments in portfolio allocations and augments the informative power of the classical asset allocation to a level where we can discuss time-dependent risk exposures. We can identify a fixed-income benchmark and decide to use such an approximation to describe the risk–return profile of a bond according to three dimensions: the risk measure, the potential return and the passage of time. Cartesian axes can be used to visualise the interaction between these elements, where the amount of attainable return with a certain confidence interval is plotted on the x-axis, and the amount of risk on the y-axis and the time horizon parameterises the Cartesian pairs (return, risk). More time usually relates to more risk, ie, more potential return. This is evident for seemingly linear products, such as indexes. However, maturity-bearing securities like “non-defaulting” fixed-rate bonds might show that residual risk is 67

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.8 Pull-to-parity properties over time Fixed-rate bond Fixed-income asset class 0.35 8Y→

0.25 0.20

4Y→

0.15 0.10

4Y ←

Standard deviations

0.30

1Y → ←1Y

0.05 0 0

← 8Y 0.05

0.10 0.15 Expected returns

0.20

0.25

“mitigated” as time goes by and total return converges to the IRR. We are aware that the “non-defaulting” assumption can be restrictive. Figure 4.8 plots the relationship between the standard deviation of the potential total return of a fixed-rate bond and its expected total return, estimated across the whole investment horizon (multiperiod Monte Carlo simulation), and compares it with the long-term estimation of a theoretical benchmark featuring identical volatility and expected return, by adopting the square-root rule-of-thumb to project the risk–return relationship over longer terms. Hence, the potential total returns of non-equity securities, in the absence of the issuer’s default, reveal pull-to-parity properties, which are relevant in investment decision-making. Stochastic simulation over time allows us to plot these properties graphically. Having discussed the main determinants of fair-value pricing, we now consider a numerical example of the fair value changes of a high-rated Treasury bond (eg, a ten-year German Bund), estimated with a set of multi-period stochastic scenario paths, as in Figure 4.9 and in Table 4.2. The risk–return simulation, represented by the full density of potential total returns or simplified by the left (LT5 ) and right (RT5 ) quantiles of the distribution, shows that the risk–return relationship is not constant over time. This is in accordance with the interaction of the simulated future states of the world and the remaining time to maturity of the fixed-income investment. 68

REAL SECURITIES AND REINVESTMENT STRATEGIES

Figure 4.9 Monte Carlo simulation of Treasury bond 1.0 0.8 Total returns

0.6 0.4 0.2 0

–0.2 –0.4 –0.6

1Y

2Y

3Y

4Y

5Y 6Y Time

7Y

8Y

9Y

10Y

Table 4.2 Monte Carlo simulation of Treasury bond Time step

RT5

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

0.169 0.224 0.224 0.234 0.247 0.247 0.247 0.247 0.247 0.247

LT5 −0.159 −0.187 −0.185 −0.177 −0.147 −0.118 −0.058 0.013 0.112 0.245

P > target (%) 38 22 17 12 9 — — — — —

Certainly, potential risks naturally evolve over time, while a reduction in maturity decreases the interest rate sensitivity; moreover, the cumulation of cashflows (not reinvested in this example) shifts the investment total return towards the internal rate of return target. This is a pull-to-parity benefit that reduces, over time, both the cumulated and the point-in-time risk–return variability. In the absence of default, the German Bund chosen would display more risk in the short term, and less or no risk towards maturity, here indicated from the euro investor’s perspective. The selection of a cumulated return target that crawls up to 40% over the ten-year period allows us to assess the probability of achieving or beating such an investment goal, as indicated in Table 4.2. 69

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.10 Monte Carlo simulation of Treasury bond with FX risk 1.0 0.8 Total returns

0.6 0.4 0.2 0

–0.2 –0.4 –0.6

1Y

2Y

3Y

4Y

5Y 6Y Time

7Y

8Y

9Y

10Y

Table 4.3 Monte Carlo simulation of Treasury bond with FX risk Time step

RT5

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

0.156 0.195 0.230 0.292 0.366 0.439 0.533 0.673 0.835 1.040

LT5 −0.155 −0.180 −0.187 −0.197 −0.201 −0.196 −0.199 0.199 0.207 0.214

P > target (%) 32 22 17 15 14 15 19 23 28 36

Since the risk of a security depends not solely on its own financial engineering (which is equal for all investors) but also on the currency denomination of the final investor’s portfolio, we can observe the modification of the risk–return perception of the same fixed-rate bond by introducing currency risk, as in Figure 4.10 and Table 4.3, in which the point of view of a US dollar investor is considered for the same level of return ambition. MODELLING REINVESTMENT STRATEGIES OF FIXED-INCOME SECURITIES Traditionally, portfolio managers have used benchmarks to proxy real securities in portfolio optimisation exercises, as they do not 70

REAL SECURITIES AND REINVESTMENT STRATEGIES

hold maturity-like features and their linear risk–return profile can be more easily projected into the future. However, this can lead to the loss of important risk–return information embedded in the asymmetry of real payoffs, which can no longer be disregarded in optimal portfolio choice. Modern risk management techniques can support the formalisation of reinvestment strategies of individual securities to facilitate the simulation of portfolios into the future. In this book we discuss such reinvestment strategies, which are different from the techniques dedicated to dynamic trading strategies of portfolios. Reinvestment strategies have the sole purpose of rolling products into identical or fairly similar bets, without explicit dependency on the state of the overall portfolio: if the simulation horizon lies beyond the legal maturity of an investment product, we can assume that an investor will reallocate the individual claims into a similar or identical bet, regardless of how the remaining portfolio has evolved during the same period. Investment decision-making is assumed to conform with a “constant preference” assumption, as a model for multi-period decisions, so that reinvestment strategies are an enrichment of the classical “buy-and-hold” assumption. We consider the case of a simple non-amortising fixed-rate bond and decompose its total return potential into two sets of elements: coupon payments, denoted by CFj,H,t , and final notional reimbursement, denoted by Nj,0,0 . Coupons are known at the beginning of the product life and do not change across scenario paths Vj,s,t∗ =



t∗ ∈{Γ |t∗ t}

CFj,0,t DFH,(t−t∗ ) + Nj,0,0 DFs,(T −t∗ )

(4.10)

where DFH,(t−t∗ ) indicates the discount factor between reference time t∗ and future time step t, conditional on scenario path H. For simplicity, bond claims are assumed to occur with the same frequency as the discretisation of t ∈ Γ (eg, yearly). The time dynamics of fixed-income returns depend on the evolution of the discount factors. We can then assume that, at maturity, any fixed-income security is reinvested into an equivalent bet, which can be represented by a virtual new issue at par of the same maturity, credit risk and payment frequency of the original note. Reinvestment might take place into an equivalent bet with the same remaining time to maturity as the original security with respect to current time, as opposed to the 71

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.11 Reinvestment strategy: fixed income 1.0

Total returns

0.8 0.6 0.4 0.2 0

–0.2 –0.4 –0.6

1Y

2Y

3Y

4Y

5Y 6Y Time

7Y

8Y

9Y 10Y

Figure 4.12 Reinvestment strategy: fixed income 1.0

Total returns

0.8 0.6 0.4 0.2 0

–0.2 –0.4 –0.6

Real product Reinvestment (low yield) Reinvestment (high yield) 1Y

2Y

3Y

4Y

5Y 6Y Time

7Y

8Y

9Y 10Y

original time to maturity. This enables us to avoid rolling the exposure of maturing short-term duration products into longer durations than originally envisaged, as we can assume that the initial portfolio was created so that these products had to provide money-marketlike contributions over time. Moreover, investors can also collect the multi-period claims (cashflows) as time passes, and might have to decide what to do with them. Investors’ behaviour can be modelled by designing one of the following strategies, among the many alternatives available. • Cash in the coupons and reinvest the notional: the investor

wants to make investment decisions only when notional amounts are paid and let the inter-temporal coupons reside in unrewarding cash accounts (as can be the case for private investors in the absence of inter-temporal consumption).

72

REAL SECURITIES AND REINVESTMENT STRATEGIES

Figure 4.13 Five-year reinvestment strategy low yields

Total returns

0.40 0.20 0 (a)

–0.20

1Y

2Y

3Y

4Y

5Y Time

6Y

7Y

8Y

9Y

10Y

Zero rates

0.06 Base curve Scenario #358

0.04 0.02

(b) 0

0

2Y

4Y

6Y

8Y

10Y 12Y Terms

15Y

20Y

(a) Reinvestment strategy. (b) Interest rate curve simulation.

• Cash in the coupons into virtual money-market funds and

reinvest the notional rebates: the investor wants to reward the coupon payments by investing in a reference money-market fund or deposit as the cashflows come due, while the final notional is reinvested into the same original bet.

• Total return reinvestment of all proceeds: the investor wants

to reinvest any proceeds of the original bet into a similar bet and act as a total return fund manager.

The graphs in Figure 4.11 represent the stochastic simulation of the potential returns of a fixed-rate bond featuring reinvestment, with original maturity around five years and internal rate of return of 5%. As time goes by, the cashflows are collected and reinvested with final notional into a similar bond, which is issued at par under the prevailing interest rate conditions. In the context of a multi-period Monte Carlo simulation such a theoretical bond is scenario dependent, eg, in every future stochastic scenario a new theoretical bond is virtually issued to yield higher or lower IRR according to the simulated level and shape of the term structures of interest rates and 73

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.14 Five-year reinvestment strategy: increasing yields

Total returns

0.40

(a)

0.20 0

–0.20

1Y

2Y

3Y

4Y

5Y Time

6Y

7Y

8Y

9Y

10Y

0.06 Zero rates

(b) 0.04 0.02 Base curve Scenario #249 0

0

2Y

4Y

6Y

8Y

10Y 12Y Terms

15Y

20Y

(a) Reinvestment strategy. (b) Interest rate curve simulation.

credit spreads. This is reflected in the shape of the cloud that represents the simulation of the potential total returns after maturity, as in Figure 4.12. Figure 4.13 represents the perturbed interest rate term structure conditional on a specific Monte Carlo scenario out of the 1,000 generated at the five-year time step, in which interest rates remain low at the time of simulation. Figure 4.14 instead represents an example of potentially increasing interest rates. We can certainly observe the informative value of this approach compared with the treatment of fixed-income securities by means of fixed-income indexes, as in classical portfolio theory. MODELLING REINVESTMENT STRATEGIES OF STRUCTURED FUNDS Modelling stochastic simulations of structured funds, such as equity wrappers with capital guarantees, can also benefit from the adoption of reinvestment strategy techniques, so investors can use models of current portfolios to aid them in buying into investment strategies as opposed to stand-alone products. 74

REAL SECURITIES AND REINVESTMENT STRATEGIES

Figure 4.15 Reinvestment strategy: structured fund

Total returns

4.0 3.6 3.2 2.8 2.4 2.0 1.6 1.2 0.8 0.4 0 –0.4 –0.8

1Y

2Y

3Y

4Y

5Y 6Y Time

7Y

8Y

9Y

10Y

We consider the case of a mutual fund featuring a capital protection plan. The investor is exposed to any swing in the net asset value of the fund during the holding period, but is protected at a legal amortising date M, since the seller of the product has committed to amortising the fund and returning the total performance if it is positive (net of cumulated fees at time t, denoted by fj,H,t ) or just the original net asset value Vj,0,0 if the performance is negative. The payoff of this product is very asymmetrical from the point of view of the private investor: they cannot realise the value of the embedded option (capital protection) in the secondary markets, since the option is vested only if the investment is amortised at the legal protection date. A portfolio manager can simulate a reinvestment strategy over time: the proceeds from the investment are reinvested into a similar fund with identical characteristics. Assuming that the amortisation date corresponds to a time step M = 3Y of the multi-period stochastic simulation (Figure 4.15), every time the fund performance ends up negative, the original amount is collected and reinvested in a similar product; every time the fund performance ends up positive, the full amount net of fees is collected and reinvested in a similar product

Vj,H,t

⎧ ⎪ ⎪ V − fj,H,t ⎪ ⎨ j,H,t = Vj,H,M − fj,H,t ⎪ ⎪ ⎪ ⎩V j,0,0

for all t = M

if Vj,H,M  V0,0,0

(4.11)

if Vj,H,M < V0,0,0

Hence, long-term investment goals can be investigated by modelling the rollover of shorter-term products or strategies, since longer-term 75

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

securities might not always be adequate or available to build optimal portfolios. MODELLING DERIVATIVES AND STRUCTURED PRODUCTS A derivative is a claim on future payments, agreed by the buyer and the seller of a financial contract, whose computation depends on an underlying asset and whose amount is not certain at the time the investor enters into the contract. Options are derivatives whose realisation is further uncertain, as one of the parties may own the right to exercise the final transaction or let it expire. The final payment of the derivative contract defines its intrinsic value and it is computed using a given formula for the payoff. Therefore, the effective payment depends upon the formula and the dynamics of an underlying value or set of values (eg, basket derivatives). Investment banks have been particularly innovative and have provided financial intermediaries with a broad range of investment payoffs, some of which can be evaluated by means of closed-form solutions (eg, Black–Scholes), while others require more complex mathematical models such as trees and stochastic simulations. A simple example of a derivative contract is a European call option on a given stock. Its value at maturity T depends on the difference between the value ST of the underlying stock at fixing date and the strike, ie, the price X at which the option buyer is given the option to buy the underlying security. Hence, the buyer is willing to exercise the right to buy the underlying at the given strike price only in the event that the underlying value is higher than the strike price, so that payoff = max(ST − X, 0) We observe that the return that can be attained by the option buyer is very asymmetrical because the investor can either lose a fixed amount, equal to the option premium, or gain a variable amount of money, equal to the difference between stock value and strike price. This implies that, although we may assume that the underlying stock returns are normally distributed, clearly the option payouts are not. Hence, the distribution of the option returns cannot be conveniently approximated by the first two moments of the underlying risk factor distribution: variance and expected return. The characteristics of option prices make them particularly difficult to treat as part of optimisation methods based upon Modern 76

REAL SECURITIES AND REINVESTMENT STRATEGIES

Table 4.4 Monte Carlo simulation: left and right quantiles Time-step

RT5

LT5

1Y 2Y 3Y 4Y

0.031 0.060 0.112 0.114

−0.319 −0.565 −0.679 −0.710

Portfolio Theory, although the non-linear payoffs of financial derivatives can be quite appealing for optimal asset allocation. Portfolio managers can benefit from the use of derivatives in portfolio allocations because they allow us to model a more desirable asymmetry of potential investment returns, and allow us to diversify among asset classes by accessing markets and financial relationships that might otherwise not be readily available (eg, hedging credit exposures by holding credit default swaps). Derivatives can be traded as direct investments or can be packaged within structured products. The latter have been designed by investment banks to offer private and institutional investors both an appealing way of achieving extra yield (eg, stock market returns) and a secure capital guarantee (eg, issuer risk-protected fixed-income notional), without entering fully leveraged bets. Derivatives are broadly used by portfolio managers to enhance investment returns. Insurers and pension funds use derivatives to replicate efficiently the life-cycle properties of assets and liabilities. Therefore, portfolio optimisation techniques that allow full valuation of financial derivatives by means of probabilistic scenarios become essential in advanced portfolio theory. Furthermore, including the full density of the asymmetrical distribution of investment returns in the optimisation exercise enables portfolio managers to preserve the relevant risk management information. Certainly, financial derivatives can add significant value when seeking long-term rewards and embedding capital insurance into portfolio modelling. Oversimplifying the representation of asymmetrical portfolios and products by adopting benchmarks would significantly limit our understanding of the risks and returns of real investments, hindering our capability to manage a bank’s capital allocation, a fund performance or a private investor’s financial wealth under stressed market conditions. 77

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 4.16 Payoff at maturity of auto-callable discrete barrier certificate 150.0

Notional (%)

117.5 100.0

50.0

0

–100

–50

0 50 100 Performance of the underlying

150

Figure 4.17 Monte Carlo simulation: auto-callable discrete barrier certificate 0.2

Total returns

0 –0.2 –0.4 –0.6 –0.8 –1.0

1Y

2Y Time

3Y

4Y

The graphical representation of the results of scenario simulations allows investment managers to visualise the embedded risk–return characteristics of structured products without having to understand all aspects of the underlying financial engineering. We can simulate an auto-callable discrete barrier certificate on a market index (eg, Eurostoxx Banks) with a four-year maturity, which pays three yearly coupons equal to 3.50%. In the third year, corresponding to the third coupon payment date, if the market index falls below the initial value, the 3.50% coupon is paid out and the certificate is not called; otherwise the certificate is called and reimburses the notional augmented by a premium equal to 8.65%. If the certificate is not called in 78

REAL SECURITIES AND REINVESTMENT STRATEGIES

the third year and if the value of the market index has fallen to less than 50% of the initial value, then the full notional is reimbursed at maturity and augmented by a premium equal to 17.30%; otherwise the investor takes the full loss stemming from the underlying negative performance, as the certificate reimburses only a percentage of the notional multiplied by the performance of the underlying market index. We can plot the payoff of this structured product (Figure 4.16) and perform a Monte Carlo simulation over time (Figure 4.17) whose main findings can be summarised in Table 4.4. Clearly, the simulation of derivatives and structured products should not be reduced to a mean–variance investigation. MODELLING INFLATION-LINKED SECURITIES Investors seeking long-term performance (eg, portfolio managers of insurance companies and pension funds) face more relevant challenges than short-term investors, due not only to the higher uncertainty in the nominal dynamics of the financial variables over a longer period of time, but also to the influence of inflation rates on the level of real prices. Inflation is a phenomenon that plays in the background and can significantly modify the long-term attractiveness of financial investments. Cash holdings, for example, although widely considered to be risk-free in the short term, can become risky in the long run because they will be reinvested at unknown interest rates. Furthermore, inflation-linked bonds possess uncertain capital value in nominal terms, as notional and coupons are linked to the changes in consumer price index (CPI), but they provide a fixedincome stream of cashflows in real terms over the investment horizon. “Non-defaulting” fixed-rate bonds, on the other hand, provide a stream of fixed cashflows in nominal terms, but also give variable returns in real environments. We can perform a stochastic simulation of US interest rates and the corresponding CPI inflation index. The application of the Hull– White one-factor model (described in Chapter 2) allows us to perform a full revaluation over time of two investments bearing the same credit risk and comparable nominal interest rate sensitivity, but having different sensitivity to inflation realisations: a US Treasury bond (10-year maturity, 2% annual coupon) and a US Treasury inflation-linked protected security (TIPS) (10-year maturity, linked 79

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Risk/return

Figure 4.18 US Treasury fixed-income 10Y 1.2 1.0 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8

Nominal Real

1Y

2Y

3Y

4Y

5Y Time

6Y

7Y

8Y

9Y

10Y

4Y

5Y Time

6Y

7Y

8Y

9Y

10Y

Risk/return

Figure 4.19 US TIPS 10Y 1.2 1.0 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8

Nominal Real

1Y

2Y

3Y

to CPI). In the absence of default, an investor might want to investigate the risk–return profile of the investments in both nominal and real terms, by expressing the invested portfolio in US dollars (original nominal currency) or CPI (virtual real currency). This reveals the properties of inflation adjustments on long-term investment returns and indicates that life-cycle portfolio choice must be carefully considered in the presence of inflation risk (Figures 4.18 and 4.19). We see that a US Treasury bond has pull-to-parity characteristics only if nominal environments are represented, while a US TIPS complies with pull-to-parity relationships only if real environments are considered. Modern Portfolio Theory cannot embed inflation expectations consistently in the making of portfolio choice. Probabilistic 80

REAL SECURITIES AND REINVESTMENT STRATEGIES

scenario optimisation, instead, can be configured to comply with either nominal or real interest rate environments, which leads to the identification of different optimal portfolios, as in Sironi (2012). The relationship between nominal and real environments is described by the Fisher equation, where the difference between nominal and real rates equals the expected inflation. Mathematically, this is equivalent to trading inflation as a virtual currency between two virtual economies, where the CPI is the exchange rate. • The nominal economy: fixed-income nominal amounts are

expressed in the original currency denomination (eg, US dollars).

• The real economy: real amounts are expressed in the inflation-

adjusted foreign currency (eg, CPI).

The future exchange rate between the two economies, perturbed by stochastic simulation, is defined by the interest rate parity, where the nominal curve represents the domestic forward rates and the real curve represents the foreign forward rates. By including such a virtual currency in the set of the financial variables, investment managers can generate both real and nominal stochastic scenarios, so the simulated value of any financial instrument can be expressed either in nominal terms or in real terms and reveal higher or lower risks due to the realisation of inflation. In fact, empirical evidence suggests that long-term portfolio performance is sensitive to the following. • Inflation trend: investment performance relates to the direc-

tion of inflation rate changes, not to absolute rates. Environments of spiking inflation seem to favour gold-, silver-, energy- and inflation-linked bonds, while disinflation favours Treasuries and fixed-income securities.

• Deflation: Modern Portfolio Theory might struggle to lock in

profitable allocations of financial wealth when capital preservation is the key goal and economies are deflationary, so guarantee products and highly rated Treasuries would become unattractive investments.

• Negative real yields: inflation-linked securities themselves

can become unappealing long-term investments if their real yield can become negative, which occurs when inflation rates are higher than nominal interest rates (the Fisher equation is negative). 81

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

The adoption of a modern risk management framework that supports long-term investments and makes use of stochastic simulations of real products in both real and nominal environments, as opposed to benchmarks, allows us to adjust optimal asset allocations to embed the effects of inflationary environments and add informative power to our investment decision-making in the context of goal-based investing. CONCLUSIONS We have described the limitations of using benchmarks rather than simulation of real investments and given some numerical and graphical examples. In particular, the assumption of symmetry of the investment returns seems too restrictive for long-term, goal-based or buy-and-hold portfolios that are made of fixed-income investments or derivatives. Also, consistent modelling of inflation-linked products can be quite relevant in assessing ex ante the portfolio performance in the long run. Dealing with real securities requires proficiency in scenario modelling and poses further difficulties to portfolio managers who operate across asset classes, eg, full integration of market and credit risk. Moreover, the chosen investment horizon might be longer than the maturity of some of the individual investments. Therefore, portfolio managers need to design reinvestment strategies to reinvest cashflows and notional amounts until the final optimisation horizon has been reached. Reinvestment strategies appear to be relevant for modelling fixed-income products and derivatives, since they enable us to take advantage of longterm stochastic simulations of the financial variables and products in creating optimal portfolios. So far in this book we have indicated the main mechanics of modern risk management frameworks based on Monte Carlo simulations, the relevance of the probability measure for goal-based investing, the relevance of simulating real securities and the appealing properties of reinvestment strategies, which allow us to simulate current portfolios beyond the legal maturities of the individual claims, until the final investment horizon has been reached. The next chapter complements our introduction to the theory by providing examples of modelling investment goals, ie, risk–return profiles of institutional or private investors. We then discuss optimisation methods. 82

REAL SECURITIES AND REINVESTMENT STRATEGIES

REFERENCES

Fabozzi, F. J., 2006, The Handbook of Fixed Income Securities, Seventh Edition (McGraw-Hill). Minenna, M., 2011, A Quantitative Framework to Assess the Risk–Reward Profile of Non-Equity Products (London: Risk Books). Ravi, B., M. Dahlquist and C. R. Harvey, 2004, “Dynamic Trading Strategies and Portfolio Choice”, Working Paper 10820, National Bureau of Economic Research. Sironi, P., 2012, “Advanced Simulation Techniques to Address the Inflation Challenge and Gold-Based Investments”, Risk magazine, IBM Sponsored Statement, November 30. Theiler, U., 2011, “Risk Minimising Investment Strategies: Embedding Portfolio Optimisation into a Dynamic Insurance Framework”, Journal of Risk Management in Financial Institutions 4(4), pp. 334–69.

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5

Derivation and Modelling of Risk–Return Time Profiles Time is money. Benjamin Franklin (1706–90)

In this chapter we present the risk–return profile as a function of time, discuss investors’ styles and personalities, derive the risk and return profiles of risk aversion, and summarise risk mitigation, risk tolerance and other elements of goal-based investing. INTRODUCTION The representation of real investors’ preferences and constraints is a key element in making investment decision-making more transparent and intuitive. Modern Portfolio Theory does not allow us to model real investors’ risk–return profiles consistently. The Markowitz approach combines the investors’ utility function and the efficient frontier with reference to a defined single point in time after the optimal market mix of investment opportunities has been indicated, whereas the Black–Litterman approach allows for information asymmetry by introducing investors’ views on excess return (not on risk appetite or on consumption) as a posterior modification of the vector of expected excess returns implied by market equilibrium portfolios. Yet, investors are economic individuals who, rationally or irrationally, express ambitions and fears over time. With this in mind, the optimal portfolio is not only the portfolio that “minimises risk or maximises expected return”, but also the one that maximises over time the probability of hitting the investor’s target and complies with the investor’s risk profile. The investor’s so-called risk–return profile is the multi-period representation of an individual’s goals and risk appetite (risk being what can endanger investment goals). A consistent risk management representation of the risk–return profile is relevant to portfolio choice, along with the simulation of total returns of real securities, 85

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

and it allows us to measure the probability of achieving or beating an investment target, as well as the probability of failing to hit the target, or falling short of a minimum return requirement. Yet, the process of eliciting risk–return preferences might not be trivial. This is particularly relevant for institutions managing portfolio hierarchies of different and complex strategies (eg, fixed-income portfolios, equity desks, volatility trading), as well as for wealth managers required to allocate money between a variety of seemingly isolated but potentially conflicting goals (eg, fulfilling a retirement plan yet providing above average returns in the short term). We start this chapter by examining investment styles and personalities and review the ways to derive risk–return profiles, with direct reference to the case of private investors (although an institutional risk–return derivation can be clearly accommodated). We discuss the relevance of time: the risk–return profile becomes a function of time along the investment horizon. Without being restrictive, we give some typical examples of investor type: risk averse, risk mitigating and risk tolerant. Finally, we apply the risk–return profile to the outcomes of Monte Carlo multi-period simulations in order to develop goal-based investing in the long run. INVESTMENT STYLE AND PERSONALITY The volatility of financial markets and the broad range of innovative products that a portfolio manager can trade, or that a financial advisor can discuss with a private investor, have made investment decision-making increasingly complex. In particular, with increasing life expectancy, investors are consciously seeking long-term investment opportunities: they are required to manage asset allocations not only under varying market conditions, but also with personal preferences that change over time. Since institutions and individuals are meant to take decisions over the course of a lifetime, they must address short-term ambitions but also optimise mediumto long-term needs in both nominal and real terms. With regards to private investments, the process of portfolio allocation is a concatenation of steps in a person’s life and requires periodic revision to meet changes in market variables, in personal preferences, in personal status, in economic environments and mitigate the influence of exogenous events. Investors are asked to carefully think of the evolving mix of assets and liabilities, as well as potential impacts on 86

DERIVATION AND MODELLING OF RISK–RETURN TIME PROFILES

their net worth, and thoroughly forecast future personal income and expenditure. Investors’ decision-making is based on knowledge of today’s worth, plus tomorrow’s desired worth, but should also take into account unpredictable changes in worth. A person’s net worth comes from the composition of their investment assets, which are meant to generate positive investment claims into the future, and their liabilities, which generate negative claims into the future to finance today’s consumption needs. Assets can be physical (eg, real estate) or financial (eg, stocks and bonds). Liabilities can be loans or mortgages. Income and expenses (eg, entertainment, childcare or education) shape the marginal increments/decrements in a person’s wealth, inflows or outflows. Exogenous conditions such as recession, inflation or inheritance can modify a person’s net worth or change their capability of contributing to marginal wealth. Investors are not a uniform set of individuals acting with investment rationality. Yet, the investor community can be aggregated into subsets of general styles and personalities: • trading (proactive management) or long-term holding (buy

and hold);

• risk aversion (conservative), mitigation (balanced) or tolerance

(aggressive);

• global diversification or local investing; • multiple asset classes or specialised investing (eg, equity

markets only);

• vanilla or leveraged investing (eg, derivatives); • direct or indirect investing (eg, investment funds, discre-

tionary portfolios).

A consistent expression of target risk–return profiles makes it easier to indicate how comfortable an investor is with investment risks, and hence the variability of returns. However, risk–return appetite is not constant and investment behaviour can change according to the individual’s life cycle or the investment goal itself. As a rule of thumb, young professionals might be more risk prone than retirees, whom tend to be more risk averse. Personal elements can also contribute to the investment decisions made over the course of an individual’s lifetime. 87

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

• Objective elements: social traits such as age, income, tax,

family situation.

• Subjective elements: emotions (such as reactiveness to finan-

cial information) and cognition (such as education, knowledge and beliefs).

The combination of such objective and subjective elements can influence the perception of exogenous factors, such as market events (eg, the level or the change in inflation). RISK–RETURN EXPRESSION The way risk and return are expressed is relevant for the transparent and effective risk management of institutional portfolios at large asset-management funds, as well as for private wealth management. In this section we refer to private investments for convenience. Clearly, we can directly extend the argument to asset management and investment banking: the risk–return profile corresponds to the capital allocation and risk-optimal budgeting, as opposed to a person’s ambitions and fears. The most common approach for determining the risk–return profile is to interview the investment stakeholders. With respect to private wealth, financial institutions have adopted relationship management tools to capture the characteristics of actual investors and create model profiles by submitting questionnaires to the clientele. Technological developments in cognitive computing, big data analytics and social media are also creating new techniques to improve profiling of investors. International market regulators have been sensitive to the need to address transparency and have incentivised financial advisors to know more about their respective customers and provide more accurate, albeit intuitive, information about investment risks. Whatever the language, format or length of the questionnaires, the following list corresponds to the most common questions. • Investor’s age.

• Current worth by aggregating savings and capital. • Expected income for the coming years.

• Existing or expected family needs, such as child education or

house rental.

88

DERIVATION AND MODELLING OF RISK–RETURN TIME PROFILES

• Track record of health expenses and its projection. • Investment experience (eg, knowledge of equity markets or

derivatives).

• Liquidity horizon, indicated as the time when money needs to

be withdrawn.

• Emotional reaction to information about financial market

fluctuations.

• Return ambition and risk appetite, to assess the range between

the target investment return and the minimum acceptable return (or maximum acceptable loss).

Clearly, sufficient time needs to be dedicated to understanding the implications of risk and return on personal wealth and to customise the asset allocation to the investment attitude of actual clients. However, deriving a quantification of the risk–return profile from interviews is not always unambiguous and has its own pitfalls as indicated by Taylor (2000): • investors are usually not neutral to the initiation of the invest-

ment process, as they might be asked to make investment decisions by adjusting from an existing portfolio (anchoring);

• investors are risk averse when facing gains but become risk

seeking when facing losses (prospect theory);

• investors’ perception of risk and return is influenced by

the frequency with which goals are monitored (myopic loss aversion);

• investors have the tendency to ignore underlying probability

distributions;

• investors, advisors and portfolio managers tend to be overcon-

fident.

Anchoring is a relevant aspect in goal-based investing and it requires us to jointly consider past performance and ex ante market potential. These have been discussed in previous chapters. Prospect theory also plays a relevant role in private wealth management, with particular relevance to the processes of risk–return elicitation. Slovic et al (1988) made an experiment in which a group of individuals was asked to choose between two alternatives. 89

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

1. A 100% chance of losing US$50. 2. A 25% chance of losing US$200 and a 75% chance of a zero loss. About 80% of individuals chose the second, which is consistent with the prospect theory: individuals tend to be risk-seeking when they are presented with choices between losses. Thus, the experiment was repeated with a rewording of the first option: 1. an insurance premium of US$50 to avoid a 25% chance of losing US$200; 2. a 25% chance of losing US$200 and a 75% chance of a zero loss. Interestingly enough, about 65% of the respondents chose the first option instead, meaning that individuals were quite sensitive to the framing of the questionnaire. This implies that questionnaires need to be designed thoughtfully and the findings carefully interpreted. PSO intends to also address the characteristics of myopic loss aversion, by featuring a multi-period representation of investment goals and fears, so that the frequency of the discrete risk–return appraisal itself contributes to the making of optimal portfolios. With respect to probability distributions, although probability is a simple concept that the majority of the investors seems to understand, behavioural theory shows that individuals can be easily mislead by the presentation of a financial problem. Kahneman et al (1982) conducted an experiment in which a group of respondents was confronted with the description of an unknown individual who was said to be drawn from a universe made of 70% engineers and 30% lawyers. The individual’s description was phrased in a very neutral way, so that it would not convey any information about the individual’s profession. The experiment proved that the respondents estimated the probability of the individual to be an “engineer” at 50%, although they knew that the a priori probability was 70%. This implies that a priori probabilities tend to be disregarded when descriptive information is provided as part of the judgemental request. This suggests graphical representation of such probabilities is better at conveying the correct risk–return appraisal of investment risks and potential returns. 90

DERIVATION AND MODELLING OF RISK–RETURN TIME PROFILES

TIME, RISK AND RETURN Financial investments can be made in a rush or under different personal conditions. This book assumes that individual investors (or their advisors) have the time and knowledge for informed decisionmaking and are supported by professional financial planning software. A typical financial planning workflow would start with a questionnaire whose final scoring is compared with ranges of investors’ subfamilies indicated with descriptive labels (eg, conservative, balanced, aggressive). Investors would be required to indicate personal ambition (eg, the financial goal or target) and propensity to risk (eg, acceptance of a loss threshold). The risk–return characteristics of any investments would be presented (by simulating real products or by using generic indexes, as often seen in practice). The comparison of alternative investment proposals against an existing asset allocation would indicate how investment potentials could be improved. The preferred investment plan would be executed, observing that investors’ preferences and constraints can force actual asset allocations to deviate from theoretical optimums. The performance of the investment plan would be monitored over time, by rolling over investments and reallocating wealth under modified personal and market conditions. Whether at the start of a new investment process or during a portfolio reallocation exercise, individuals will invest in any new opportunity only if the risk–return characteristics (individual or diversified in a broader portfolio) comply with the following criteria: • return consistency between the individual’s ambitions and the

potential returns of indicated markets and products;

• risk adequacy of the products and portfolios, given a personal

or a target risk profile;

• time compliance with investment and liquidity horizons,

which are indicated, respectively, by the longest acceptable time for achieving the individual’s ambitions and the minimum holding period before money can be withdrawn or portfolios get reshuffled.

If there are inconsistencies, investors will revise these latest declarations in order to recalibrate their ambitions and goals to the prevailing economic equilibriums. 91

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

We can assume the case of a conservative German investor, engaged in the local Treasury market at the beginning of 2013, declaring the ambition to achieve 10% return in one year’s time. Since Bund yields reached or even breached the zero level in 2012, the investor’s declared mix of preferences would appear to be inconsistent with economic reality. The investor will ponder the following alternative cases to match their preferences to reality. • Revise return ambitions: the investor will learn that 10% is

not a feasible return given current market conditions and personal constraints (risk appetite, preferred type of investments and financial market of reference) and will reduce their return ambitions.

• Revise risk appetite: the investor will learn that conservative

investments, although seemingly adequate to a risk profile, are not consistent with the declared ambition under current market conditions and will revise the risk appetite definition, by accepting a raise in the stakes.

• Stretch time: the investor will learn that conservative invest-

ments might be available in the reference market but will accept that it takes much longer than originally planned to achieve the target return.

Risk, return and time are the main determinants that represent the risk–return profile of investors and portfolios simulated with multiperiod stochastic scenarios. Probabilistic portfolio optimisation can be performed by mapping investors’ preferences to the stochastic dynamics of portfolio returns. SHAPING THE RISK–RETURN PROFILE OVER THE TIME HORIZON An investor is an individual who allocates capital by undertaking a quantity of risk, with the financial goal of achieving a target return over a given investment horizon. The term “ambition” is preferred to “expectation”, since the full density of potential profits and losses can be drawn by means of multi-period stochastic simulations, as opposed to considering only the expectation, as in classical portfolio theory. The concept of “time horizon” is more relevant than in Markowitz and Black–Litterman. 92

DERIVATION AND MODELLING OF RISK–RETURN TIME PROFILES

Figure 5.1 Investment horizon, Γ Current 0

Liquidity step

Reallocation step

t1

t2

Final step (...)

T

Figure 5.2 Connecting past and future potential performance 0.6

Total returns

0.4 0.2 0 –0.2 –0.4 Ex post –0.6

Ex ante

Ex post Ex ante (scenarios) Ex ante (quantiles)

–0.8 –3Y –2Y –1Y Today1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y Time

The investment horizon is the full reference period for which an individual holds an investment or engages into a trading strategy to reach a financial goal. It is indicated by Γ (Figure 5.1), ie, a set composed of discrete elements indicated as time steps t ∈ {0, 1, . . . , T }, where T is the upper boundary of the investment horizon itself. Time steps can be expressed in calendar terms (eg, days, years), in cardinal terms (eg, 1, 2, 3) or in ordinal terms (eg, first, second, third). The first time step can qualify as a liquidity step, which is the first time t when an investor may need to revise the asset allocation or assess performance. Therefore, the liquidity step qualifies as the minimum time to verify a constraint of risk adequacy. Any other step in between the current time and T is indicated as an intermediate reallocation step. Along the time axis, investors modify their preferences and financial markets enter into different equilibriums, which can turn initially adequate portfolios into inadequate investments. Therefore, portfolio allocation is never a once-for-always exercise but is rather an inter-temporal set of investment decisions that affect risks and returns in a cyclical way. In the time dimension events must be ordered from the past, through the present, into the future. 93

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

While the span of time between past and present is the dimension in which ex post performance is formulated, ex ante portfolio simulation refers to the span of time between the present and the future: the history is a vector, while future dynamics are a matrix or a space of possible realisations. Past performance and future performance can be plotted on the time axis, as in the example of a non-default fixedincome security depicted in Figure 5.2. Clearly, the ex post measure will dominate the final total return estimate as time goes by and, in this example, it would reduce the width between risk and return as simulated stochastically. Measuring how much the likelihood of hitting an investment goal varies, or the time required to fulfil personal investment ambitions, requires us to connect ex post and ex ante performance, which is facilitated by the probability measure. Behavioural finance studies the effects of cognitive and emotional factors on individual investment decisions and can help to investigate multi-period choices, as it formulates that investment behaviour tends to be inconsistent with basic models of rational expectations. Among behavioural models, hyperbolic discounting describes the tendency to evaluate return opportunities in the near future, such as at the next reallocation step, more than those that are more distant, such as at end of the investment horizon, hence the tendency for private investors to focus on short-term asset allocations more than long-term equilibriums: immediate risk–return tends to dominate investors’ asset allocations, notwithstanding the fact that investors themselves seem to indicate (through questionnaires) a preference for long-term risk mitigation (capital guarantees). Time matters. Long-term investors judge risk in different ways from short-term investors. However, over the longer investment horizon they make a series of investment decisions at intermediate reallocation steps that can be either myopic short-term trades (reconciling – or not – with long-term strategies) or rebalancing of long-term asset allocations (adapting to market changes, to modifications in goals, preferences and income streams that create personal wealth, and to consumption draws that reduce the amount of wealth available for financial investments). Planning ahead for these events becomes a winning basis for efficient asset allocations, especially when structured products are traded to make life-cycle management easier. Inflation itself might become a relevant driver for long-term investment decision-making; inflation-indexed securities 94

DERIVATION AND MODELLING OF RISK–RETURN TIME PROFILES

Figure 5.3 Risk–return profiles

Risk/return (%)

80 60

f3(t )

40

f2(t )

20 f1(t )

0

f4(t )

–20 –40

1Y

2Y

3Y

4Y

5Y 6Y Time

7Y

8Y

9Y

10Y

a = 0.03. rf = 0.03. ralpha = 0.02. σ = 0.10.

might seem unappealing investments in the short run, but they allow us to build consistent returns when real rates (as opposed to nominal rates) are projected far into the future. Investors may not know enough about future states of the world, so they would tend to focus more on ex post performance and short-term opportunities at any reallocation step. The introduction of multi-period scenario simulation techniques of financial products makes it easier to realign the potentially myopic behaviour of investors with professionals’ intuition about future risk–return potential, so investment decision-making can be tested in advance and verified in the making, whether by modelling a rebalancing strategy or by accepting simpler buy-and-hold assumptions. An investor’s risk–return profile indicates how comfortable an individual is with investment risk, ie, the variability of returns. Since this can be simulated along the time axis, the representation of how much ambition an investor has, and how much risk they are willing to face, can also be graphically represented along the time axis: • the ambition line (λA t ) is the vector of desired cumulated total

returns;

• the risk line (λLt ) is the vector of declared maximum bear-

able losses measured by standard deviation, value-at-risk or expected shortfall. Risk appetite can be designed as a maximum probabilistic loss boundary or as a probabilistic loss

95

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

range, in which the risk of the investment lies at any point in time. The ambition line and the risk line will not be constant over the investment horizon (eg, f1 (t)), as time is a key element for both financial goals and financial returns. Financial goals are usually modelled as monotonic functions with positive drift, while risk appetites are often modelled as convex functions (as in Figure 5.3). We might want to tailor the risk profile by freely drawing a step function or by using a mathematical definition. This later would require us to state the reference risk-free rate and the alpha premium (eg, f2 (t), f3 (t)), or the average and the standard deviation of the desired function (eg, f4 (t)). In this book we present the risk–return profile as a step function, without wanting to be prescriptive: the risk–return target is kept constant in between two subsequent reallocation steps f1 (t) = a

f2 (t) = (1 + rf + ralpha )t/365 − 1   t −1 f3 (t) = exp (rf + ralpha ) 365 f4 (t) = exp



 t µ + σ F −1 ( p )

365

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ t ⎪ ⎭ − 1⎪

(5.1)

365

in which µ is the drift, σ is the standard deviation, F−1 (p) is the inverse of the standard normal function that solves for a probability p. The actual risk–return profile of an individual can take any given form. This book refers to some typical profiles that are not exhaustive: conservative, balanced and aggressive. In reality financial institutions are known to use more granular representations that would require different labels (eg, conservative, moderate–conservative, moderate, moderate–aggressive, aggressive). Moreover, investors may possess more than one portfolio and goal, so their fears and ambitions might be combined differently according to the mandate of each portfolio. Risk-averse (conservative) profile A conservative investor is defined as an individual who professes low return ambitions over time and strong aversion to financial loss (as in Figure 5.4). Typically, their ambition takes the form of a linear growth in total return targets, as the investor might be content to 96

DERIVATION AND MODELLING OF RISK–RETURN TIME PROFILES

Risk/return

Figure 5.4 Risk-averse profile 0.6 0.5 0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4

Ambition profile Risk appetite profile

1Y

2Y

3Y

4Y

5Y Time

6Y

7Y

5Y Time

6Y

7Y

8Y

9Y

10Y

Risk/return

Figure 5.5 Risk-mitigating profile 0.6 0.5 0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4

Ambition profile Risk appetite profile

1Y

2Y

3Y

4Y

8Y

9Y

10Y

receive close to the risk-free rate over time. The risk appetite will be modelled as a zero-loss threshold or even a minimum performance requirement. Money-market mutual funds or reinvestment in short-term highly rated government paper might comply with a risk-averse profile. Risk-mitigating (balanced) profile A balanced investor is defined as an individual who expresses an intermediate level of total return ambition, since they are willing to achieve moderate extra return above the risk-free rate (as in Figure 5.5). Therefore, their ambition profile can be modelled as a linear function of risk-free plus a risk premium target. The risk appetite is 97

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Risk/return

Figure 5.6 Risk-tolerant profile 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8

Ambition profile Risk appetite profile

1Y

2Y

3Y

4Y

5Y Time

6Y

7Y

8Y

5Y 6Y Time

7Y

8Y

9Y

10Y

Figure 5.7 Target VaR profile 0.6 Ambition profile Target VaR profile

Risk/return

0.4 0.2 0

–0.2 –0.4 –0.6

1Y

2Y

3Y

4Y

9Y

10Y

instead a convex function of time, as the investor might be willing to accept some risks in the medium-to-long term as they do not want to miss out on market opportunities, but want to receive capital insurance in the medium-to-long term. Balanced investment funds might be modelled in accordance with a risk-mitigating profile, where the fixed-income properties and the equity-like characteristics allow us to model a positive drift to yield a capital loss mitigation. Risk-tolerant (aggressive) profile An aggressive investor is defined as an individual who has above average ambitions in terms of total return targets and can commit to an open long-term risk profile to leverage on all market 98

DERIVATION AND MODELLING OF RISK–RETURN TIME PROFILES

Figure 5.8 Connecting past and future performance 1.2 1.0

Total returns

0.8 0.6

Ex post Ex ante (scenarios) Ambition profile Risk appetite profile

0.4 0.2 0 –0.2 –0.4

Ex post

–0.6 –0.8 –3Y

–2Y

–1Y

Ex ante

Today

1Y Time

2Y

3Y

4Y

5Y

opportunities that can lead to the desired result (as in Figure 5.6). Portfolio allocation of speculative investments might comply with a risk-tolerant profile. Target risk profile In some cases, risk appetite can also take the form of a target range instead of a single line (as in Figure 5.7). This is frequently demanded by institutional fund managers who create investment opportunities with an explicit target VaR. APPLYING AN INVESTOR’S PROFILE TO PORTFOLIO SIMULATIONS Once the desired risk–return profile has been drawn, portfolio managers can apply its representation to the simulated total returns over time of the invested portfolio (as in Figure 5.8) and assess: • the adequacy of the asset allocation for the investor’s risk

appetite, with a given level of confidence;

• the feasibility of the investor’s ambitions or budgets, with

respect to the potential evolution of portfolio returns;

• the presence of portfolio added value in the long run (convex-

ity and capital guarantee);

• the minimum time required to achieve an investment goal,

with a given level of confidence.

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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

This is a fundamental step in the process of goal-based investing: the choice of an optimal portfolio becomes a function of the dynamics of the market variables (which generate the density function of the potential returns of actual securities), the investor’s risk–return profile (which sets the constraints, targets and boundaries of acceptable risk–return characteristics of the resulting portfolio) and the investor’s preferred time line for reallocation (consisting of the liquidity step and the following reallocation steps). Thus, investors facing the same market conditions and the same risk–return profile would choose a different portfolio according to individual propensity for more- or less-frequent portfolio rebalancing, especially as the risk constraint can take more or less relevance for shorter terms. CONCLUSIONS The modelling of capital adequacy and risk-optimal budgeting, as well as an investor’s ambitions and fear, effectively take centre stage in portfolio management. By plotting a representation of the desired risk–return profile over time, portfolio managers can optimise portfolios by combining the time simulated dynamics of financial markets and products with the life-cycle profile of investment goals. Some of the methods of generating multi-period stochastic scenarios and simulating real securities have been discussed in Chapters 2 and 4, so we can draw the ex ante cloud of potential total risks and returns of investment opportunities and combine it with ex post performance, and thus compare this with the final investment goal. The next section of this book is dedicated to the discussion of alternative methods of portfolio optimisation. First the key features of the Markowitz and the Black–Litterman approaches are recalled. Then we present Probabilistic Scenario Optimisation as an alternative solution to goal-based and long-term investing. Case studies are featured in Chapters 9 and 10, which compare the results of the three approaches.

REFERENCES

Dembo R., and A. Freeman, 1998, Seeing Tomorrow: Rewriting the Rules of Risk (Chichester: John Wiley & Sons). Kahneman, D., P. Slovic and A. Tversky, 1982, Judgment under Uncertainty: Heuristics and Biases (Cambridge University Press).

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Ruhm, D. L., R. Goldfarb, R. E. Kreps, K. Rogers, P. Schoolman and J. VanOpdorp, 2005, “Elicitation and Elucidation of Risk Preferences”, Working Paper, Casualty Actuarial Society Working Party on the Elicitation and Elucidation Risk Preferences. Slovic, P., B. Fischhoff and S. Lichtenstein, 1988, “Response Mode, Framing, and Information-Processing Effects in Risk Assessment ”, in D. E. Bell, H. Raiffa and A. Tversky, Decision Making: Descriptive, Normative and Prescriptive Interactions (Cambridge University Press). Taylor, N., 2000, “Making Actuaries Less Human: Lessons from Behavioural Finance”, Paper Presented to the Staple Inn Actuarial Society, January 18. URL: www.sias.org/.

101

Part II

Portfolio Optimisation Methods

6

À la Markowitz: A Tale of Simple Worlds A good portfolio is more than a long list of stocks and bonds. It is a balanced whole, providing the investor with protections and opportunities with respect to a wide range of contingencies. Markowitz (1959)

We discuss Modern Portfolio Theory as a diversification framework, define the efficient frontier and the mean–variance objective function and introduce the tracking error and semi-variance approach, before going on to discuss expected shortfall. INTRODUCTION Investors act under uncertainty. This is a fundamental assumption of portfolio theory. If investors could gain advantage by not facing uncertainty, they would be indifferent about allocating available capital to any combination of securities giving the same maximum return. Instead, given the existence of uncertainty in future returns, rational investors must seek portfolio diversification and optimal asset allocations. Modern Portfolio Theory assumes that investors (private or professional) know the probability distributions of future returns, whether these distributions are objective or subjective. Starting from the simplified belief that a probability distribution can be described by the first two statistical moments, expected return and variance, portfolio optimisation becomes a mathematical exercise in finding the global minimum/maximum in convex or non-convex portfolio functions. Investors can then look at the complete set of portfolios that solve a given investment problem, corresponding to a global minimum/maximum and satisfying declared targets and constraints: this set is known as the efficient frontier. Any element can be indicated on the efficient frontier in order that a Pareto optimal expected return is achieved. 105

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Expected returns

Figure 6.1 Set of mean–variance portfolios

Standard deviations

Based on simple mean–variance assumptions, Markowitz’s brilliant intuition has inspired Modern Portfolio Theory since the early 1950s. The belief that optimal portfolios need to solve a quantification problem, related to the maximisation of a measure for expected return or the minimisation of a measure of risk (variance), is now commonplace. Markowitz’s original approach starts from the analysis of the time series of the risky securities that belong to the investable universe and estimates the expected return, the variance and the covariances among any given pair. The generation of all possible combinations of the securities in a portfolio, whose expected return and variance is derived from the estimates of the individual securities, allows us to plot the set of all attainable portfolios on the Cartesian axes with the x-axis being the standard deviation (for convenience), and the y-axis the expected return. Figure 6.1 depicts the full set of possible portfolios with 10 assets, with a maximum 30% investment in each asset, and a 5% investment tick. The investor cannot create any other portfolio outside the Cartesian space bounded by the perimeter of this set. The efficient frontier consists of a segment of this perimeter such that, for any given level of risk, there is no other portfolio with a higher expected return, or for any given expected return there is no higher level of risk. Knowing the efficient frontier, the investor will choose a portfolio that corresponds to their risk–return target, ie, that corresponds to the point on the efficient frontier where the portfolio is tangential to the investor’s utility function. 106

À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

Markowitz himself outlined limitations and weaknesses of the mean–variance approach (such as variance being a convenient but imperfect risk measure), which have restricted its effectiveness in real investment practices. Moreover, the model shows excessive sensitivity to the historical estimates of the variances and covariances. Small changes in the estimates of the parameters, due to a refresh of the inputs or a modification to comply with subjective beliefs, can profoundly transform the composition of the efficient portfolio, leading to impractical asset allocations. However, technical and computational convenience have contributed to making the mean–variance approach an appealing theoretical reference at numerous institutions. In time, the original formulation has been further enriched, by finding mathematical solutions (eg, integer programming) to extend the theory and introduce more refined risk measures (eg, semi-variance, tracking error, expected shortfall) or by adding/relaxing the set of investment constraints (eg, short selling, limited lending, turnover), without hindering the level of generality of the original work. Hence, the “à la Markowitz” (or “Markowitz type”). Notwithstanding the limitations of the approach, what certainly remains of the Markowitz formulation in portfolio theory is the explanation of the importance of portfolio diversification: optimality is a combination of risky and non-risky assets, so that a suitable return is sought, while risk is diversified away as much as possible. Markowitz’s approach is indeed a model of asset correlations, whose relevance constitutes the real foundation of Modern Portfolio Theory. In the real world, correlation drifts between risk factors are possibly more relevant than volatility changes in making up for abrupt shifts in portfolio returns. However, pairwise asset correlations are complex variables to estimate, and their dynamics cannot easily be investigated by means of classical asset allocation models. Only stress tests can reveal the importance of correlation changes to diversified portfolios; thus, modelling real products instead of benchmarks becomes crucial. This chapter is dedicated to the traditional Markowitz approach. We start with a brief presentation of the principles of asset diversification, and then review some common Markowitz-type approaches. First, we examine the mean–variance formulation, as it is the most frequent approach. Second, tracking error minimisation allows us to optimise the relative performance of our portfolios against the 107

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

benchmarks. Third, the semi-variance measure is introduced to overcome the limitations of variance as a symmetrical measure. Finally, we discuss the objective functions based on the minimisation of the expected shortfall. ASSET DIVERSIFICATION AND THE EFFICIENT FRONTIER We assume that the universe U of possible investments is made of only two assets (namely, 1 and 2) denominated in the same currency ($), so that the discussion is initially kept at an intuitive level. It fol¯ U,S,T of portfolio lows that the value VU,0,0 and the expected return R U, estimated on the time series of the two asset returns indicated by the set S, are indicated by VU,0,0 = V1,0,0 x1,0,0 + V2,0,0 x2,0,0

(6.1)

¯ U,S,T = R ¯ 1,S,T w1,0,0 + R ¯ 2,S,T w2,0,0 R

(6.2)

and

We note that the investment horizon is denoted by T, as only a single time step is allowed. The expected return must be estimated explicitly for a given horizon; therefore, moving from a short-term representation to a longer-term representation requires the inconvenient re-estimation of all parameters. Wealth mangers dealing with model calibration for different investment horizons might estimate the parameters by using different lengths of time series, under the assumption that, the longer the time series, the more “appropriate” the estimation of long-term expected returns. Hence, having indicated T as the single investment horizon, the value of a security at time T can be given by ⎫

where

¯ 1,S,T ⎬ V1,0,T = V1,0,0 R ¯ 2,S,T ⎭ V2,0,T = V2,0,0 R w1,0,0 w2,0,0

(6.3)



V1,0,0 x1,0,0 ⎪ ⎪ ⎪ = ⎪ VU,0,0 ⎪ ⎪ ⎪ ⎬ V2,0,0 x2,0,0 = ⎪ ⎪ VU,0,0 ⎪ ⎪ ⎪ ⎪

⎪ w1,0,0 + w2,0,0 = 1 ⎭

(6.4)

Since U is made of any allocation of only two assets, it follows that extreme portfolios can be constructed by investing 100% in one of the 108

À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

two given assets and nothing in the other. While the return of U is a linear combination of the returns of the underlying assets, weighted by the relative w1,0,0 and w2,0,0 contributions to total portfolio value VU,0,0 , the standard deviation σU,S,T is not a linear measure but rather a quadratic function of the assets’ volatility. Therefore, given the volatility and the portfolio weight of each asset, portfolio risk is indicated by  2 2 2 2 σU,S,T = σ1,S,T w1,0,0 + σ2,S,T w2,0,0 + 2w1,0,0 w2,0,0 cov(1,2),S,T

(6.5)

where

ρ(1,2),S,T =

cov(1,2),S,T σ1,S,T σ2,S,T

(6.6)

We can therefore express the above equation as  2 2 2 2 σU,S,T = σ1,S,T w1,0,0 + σ2,S,T w2,0,0 + 2w1,0,0 w2,0,0 σ1,S,T σ2,S,T ρ(1,2),S,T

(6.7)

The correlation coefficient ρ has a maximum value of +1 and a minimum of −1. A value of +1 means that the two assets are perfectly correlated, ie, they move in perfect unison. Conversely, a value of −1 means that they are perfectly negatively correlated, ie, their movements are the opposites of each other. To understand the full spectrum of portfolio risk due to the level of the pairwise correlation, we can investigate the extreme cases when correlation is equal to −1, is null or equal to +1. It is here assumed that we can make assumptions on the correlation between two assets independently from the estimate of the individual volatilities, which are constant throughout the following exercise. If pairwise correlation is equal to +1, the covariance between the two assets will equal the product of the two volatilities: the volatility of the portfolio becomes a linear combination of the volatility of the underlying assets. Hence, plotting the relationship between portfolio returns and portfolio risk on the Cartesian axes, for any given combination of asset weights, will lead to a straight line  2 2 2 2 σU,S,T = σ1,S,T w1,0,0 + σ2,S,T w2,0,0 + 2w1,0,0 w2,0,0 σ1,0,T σ2,0,T  = (σ1,S,T w1,0,0 + σ2,S,T w2,0,0 )2 = σ1,S,T w1,0,0 + σ2,S,T w2,0,0

(6.8) 109

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 6.1 Portfolio with two assets

w1

w2

— 10 20 30 40 50 60 70 80 90 100

100 90 80 70 60 50 40 30 20 10 —

ρ = −1  R σ

0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140

0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080

R

ρ=0  σ

0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140

0.030 0.028 0.029 0.032 0.037 0.043 0.049 0.057 0.064 0.072 0.080



ρ = +1  R σ

0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140

0.030 0.019 0.008 0.003 0.014 0.025 0.036 0.047 0.058 0.069 0.080

All values are given in percent.

Similarly, when the correlation equals −1, we can plot on the Cartesian axes a segment that, although monotonic in the expected returns, can solve portfolio returns for two different attainable but equally likely asset allocations, which depend on the portfolio relevance of the exposure of each asset relative to the other  2 2 2 2 w1,0,0 + σ2,S,T w2,0,0 − 2w1,0,0 w2,0,0 σ1,S,T σ2,S,T σU,S,T = σ1,S,T  = (σ1,S,T w1,0,0 − σ2,S,T w2,0,0 )2 = |σ1,S,T w1,0,0 − σ2,S,T w2,0,0 |

(6.9)

When instead the correlation is 0, the resulting function is nonlinear  2 2 2 2 σU,S,T = σ1,S,T w1,0,0 + σ2,S,T w2,0,0

(6.10)

Table 6.1 shows an example of the estimate of the expected return and standard deviation of a portfolio made of two assets, conditional on different estimates of the correlations between the asset returns, in which the two assets as characterised as follows: the expected returns are respectively R1 = 14% and R2 = 4%; the standard deviations are respectively σ1 = 8% and σ2 = 3%. We can observe in Figure 6.2 that, for any value of ρ(1,2) , the line representing the risk– return characteristics of a potential portfolio in which the two assets are invested will always be contained in the space identified by the 110

À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

Figure 6.2 Portfolio risk–return combinations for two assets

Expected returns

0.14

Asset 2

0.04

0

Correlation = 1 Correlation = 0 Correlation = –1

Asset 1

0

0.03

Standard deviations

0.08

Figure 6.3 Portfolio risk–return combinations for three assets Asset 1

Expected returns

0.14

Asset 3

0.04

0

Asset 2 0

0.03

Standard deviations

0.08

extreme cases, where the assets are perfectly correlated or perfectly uncorrelated. This means that, for any given estimate of ρ(1,2) and any given portfolio composition, we can identify the minimum return portfolio, the maximum return portfolio and the minimum variance portfolio. As ρ(1,2) is usually an input in the optimisation exercise, the objective function would solve for a suitable combination of portfolio weights. When there are more than two assets in the universe (as in the three-assets case represented in Figure 6.3), pairwise correlations are indicated by a variance–covariance matrix. Moreover, since allocation constraints are usually specified, the exercise turns into 111

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 6.4 Objective function (a)

(b) Local max Global max Global min

Local min Global min

(a) Convex. (b) Non-convex.

a multi-dimensional problem and we need to adopt mathematically advanced routines to efficiently identify the global minimum/ maximum of the optimisation objective function. The best mix of risky assets for every maximum level of expected return is the efficient frontier curve. An optimisation problem can be described by indicating two sets of characteristics: the objective function (see, for example, Figure 6.4), which is the minimisation of volatility for a given level of return (quadratic function), the maximisation of return for a given level of variance (linear function) or the deviation of other variables (tracking error); the set of constraints imposed on the continuum of portfolio weights (eg, min–max allocations on asset classes) or that refer to binary statements of given variables (eg, minimum trade size, minimum number of trades). More than one alternative can be adopted with regards to the definition of the risk measure and the mathematics required to find the global minimum/maximum of the objective function. This book simply outlines the main advantages and limitations of the most common alternatives, without aiming to be a detailed compendium of Modern Portfolio Theory. Some of these common alternatives are • mean–variance optimisation, as in the original formulation, • semi-variance optimisation, which allows us to distinguish

between under- and overperformance of the mean (which is usually desirable),

112

À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

• minimisation of tracking error or regret, to optimise a port-

folio against a benchmark whose risk–return characteristics are accepted as the optimal reference,

• expected shortfall optimisation, which better accounts for long

tails of asymmetrical distributions.

MEAN–VARIANCE FORMULATION Finding the optimal portfolio for a given investor and a given set of constraints requires us to move along the efficient frontier to attain the desired expected return with the minimum portfolio variance. In the case of minimum portfolio variance, the efficient frontier is the “collection” of all portfolios that optimise the objective function, under the same set of constraints but for different target expected returns. The minimisation of this objective function can be solved with linear programming (if there are no constraints imposed on short sales) or with quadratic programming (if short sales are allowed). The latter is a more general formulation of the problem, as it can accommodate more objective functions alongside those solvable with linear programming. We can assume that the universe U is made up of all available ¯ j,S,T securities j. Each security is associated with an expected return R corresponding to the mean distribution return indicated by the set of the corresponding historical scenarios S, while wj,0,0 denotes each security’s fair-value exposure weight in the portfolio. As wj,0,0 can equal zero, we can identify the portfolio denoted U as the set of all securities in which we can invest (or not invest). We can also assume that σj,S,T is the volatility of the jth security, while cov(i,j),S,T refers to the covariance between the returns of any pair of securities in the universe U. The minimisation of the variance takes the form of a quadratic programming problem 2 min σU,S,T = min w

w



j ∈U

2 2 wj,0,0 σj,S,T +





j∈U i∈U, j=i

wj,0,0 wi,0,0 cov(i,j),S,T



(6.11)

subject to the following conditions: 113

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

¯∗ • the resulting portfolio yields at least a target R U,S,T  ¯∗ ¯ j,S,T wj,0,0  R R U,S,T

(6.12)

j ∈U

• all portfolio weights sum to 1  j ∈U

wj,0,0 = 1

(6.13)

• short selling is not allowed

wj,0,0  0

(6.14)

∗ ¯ U,S,T between the return of the minimum variance portfolio Varying R and the return of the maximum return portfolio indicates the efficient frontier, as in Figure 6.5. Short sales can also be allowed by relaxing the last constraint. We can also impose constraints on the fraction of the portfolio U invested in a particular category A ⊂ U (eg, asset class, sector, currency) so that the exposure in A is no more than a percentage b of VU,0,0



wj,0,0  b

(6.15)

j ∈A

The reader will also find it useful to define the problem in terms of position units, instead of wj,0,0 exposure weights. We can denote by Vj,0,0 the unit value of the jth security and by xj,0,0 the number of units, which becomes the decision variable indicating the optimal allocation of a portfolio U whose market value equals (VU,0,0 ). The weighted market value of each jth security, subject to a non-zero portfolio value, becomes wj,0,0 =

Vj,0,0 xj,0,0 VU,0,0

(6.16)

Therefore, the minimum of the objective function takes a new form 2 = min min σU,S,T w

w

1 2 VU,0,0



2 2 2 Vj,0,0 xj,0,0 σj,S,T

j ∈U

+





j∈U i∈U, j=i

subject to the following conditions: 114

Vj,0,0 xj,0,0 Vi,0,0 xi,0,0 cov(i,j),S,T



(6.17)

À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

Expected returns

Figure 6.5 Mean–variance efficient frontier

Optimal portfolio Asset i

Asset j Suboptimal portfolio

Standard deviations

¯∗ • the resulting portfolio yields at least a target R U,S,T ¯ j,S,T Vj,0,0  R

j ∈U

VU,0,0

¯∗ xj,0,0  R U,S,T

• all products’ values sum to VU,0,0  j ∈U

Vj,0,0 xj,0,0 = VU,0,0

(6.18)

(6.19)

• short selling is not allowed (in this example)

xj,0,0  0

(6.20)

The optimisation problem described in this section corresponds to the most simple and usual case. Although it often restricts the representation of real securities to asset classes and indexes, it deals with one investment horizon at a time, it accepts expected returns as a measure of future profitability and it adopts variance as the measure of risk. TRACKING ERROR MINIMISATION The expansion of the classical framework introduces the concept of benchmarking, which is a common practice in asset management and wealth management and refers to the comparison of the return space of a portfolio against a reference index or market portfolio. The goal is to construct a portfolio that best mimics one or more 115

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

characteristics of a target financial variable within a given set of scenarios, conditional on an assigned set of restrictions. We can aim to rebalance an existing portfolio U so that its value tracks that of a specified benchmark B. The possible values for U and B, VU,H,T and VB,H,T , are available in a set of scenario paths H ∈ S. We can also assume that Vj,H,T is the value of the jth security and VB,H,T is the value of the benchmark B under scenario path H at the single investment horizon T. Certainly, tracking the exact value of a benchmark might not be possible under all scenarios, as the constituents of U and B are not necessarily identical, due to discrepancies in the universe of securities or to the constraints set applied to U. Hence, we can adopt a measure for such a deviation (tracking) and estimate + − the variables eH,T and eH,T , which are the differences in the values of U and B conditional on scenario paths H ± eH,T =



j ∈U

wj,0,0 Vj,H,T − VB,H,T

±

(6.21)

± The original holding of the jth security is denoted by wj,0,0 ; cj,0,0 denotes the instantaneous negative or positive changes in the initial positions required in order to achieve the optimal portfolio state U ∗ ; PH,T denotes the probability of an individual scenario path H at a given horizon T. The objective function requires us to minimise the expected mismatch between the portfolio value and the benchmark value measured at a given investment time T

min w



H ∈S

− PH,T eH,T +



+ PH,T eH,T

H ∈S



(6.22)

subject to the following conditions (for all H ∈ S): • errors are given in absolute values − + eH,T , eH,T 0

• the sum of changes and errors is VB,H,T  − + − + {Vj,H,T (xj,0,0 + cj,0,0 + cj,0,0 ) − eH,T + eH,T } = VB,H,T

(6.23)

(6.24)

j∈U

We note that the Markowitz framework has been extended to use discrete distributions instead of parametric representations. 116

À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

SEMI-VARIANCE MODIFICATION As mentioned earlier, Markowitz indicated that one of the main pitfalls of the mean–variance approach is its failure to distinguish between under- and overperformance of the mean. One way to account for downside risk only is to formulate an optimisation problem that minimises the semi-variance of portfolio returns, subject to ¯∗ achieving at least an expected return R U,S,T . Computing the semivariance of returns can be performed in scenario analysis, by tracking the deviations from the mean in each scenario path H ∈ S and applying a probability-weighted sum of squares to the downside deviations. We assume that the portfolio value equals a non-zero constant VU,0,0 , so the minimisation problem becomes − eH,T

=



wj,0,0 Vj,H,T ¯ U,S,T −R VU,H,T

j ∈U

min w





(6.25)

− PH,T (eH,T )2

(6.26)

H ∈S

subject to the following conditions (for all s ∈ S): • all values sum to VU,0,0 

j ∈U

Vj,0,0 xj,0,0 = VU,0,0

(6.27)

∗ ¯ U,S,T • the expected return of U is at least a target R

¯ j,S,T Vj,0,0  R

j ∈U

VU,0,0



¯∗ xj,0,0  R U,S,T

(6.28)

• the downside deviation of portfolio returns from the mean

under scenario H is denoted by ¯ j,S,T Vj,0,0  R

j ∈U

VU,0,0

− xj,0,0 + eH,T 

¯ j,S,T Vj,0,0  R

j ∈U

VU,0,0

xj,0,0

(6.29)

Although semi-variance has appealing properties with respect to the representation of risk compared with variance, the industry has developed more advanced risk measures, such as VaR and ES estimates, as discussed in the following section. 117

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

EXPECTED SHORTFALL FORMULATION The introduction of the expected shortfall measure, also known as tail loss, is also a recent advance in portfolio optimisation. VaR and ES have been defined in previous sections, which indicate that ES is a more conservative measure of risk; portfolios with relatively low ES must also have relatively low VaR. However, as the length of the tail loss can vary with the asset mix, the portfolio with minimum ES among the set of all potential portfolios might not correspond to the portfolio with minimum VaR, as in Gilli et al (2006). By definition, with respect to a specified (1 − α) confidence level, the VaR of a portfolio is the lowest potential return RU,H,T across all scenario paths H ∈ S at a single investment step T such that the α potential loss return will not exceed ξU,S,T with probability 1 − α. Given VaR, ES is the conditional expectation of all potential loss α returns below the amount ξU,S,T . Although VaR is a more common measure of risk, it has undesirable mathematical properties within optimisation techniques, such as the lack of sub-additivity when dealing with non-normal return distributions (for a normal distribution VaR is proportional to standard deviation). That is, when ensuring the model assumptions conform to reality in order to account for fat-tails and asymmetry of asset returns, portfolio VaR estimated on a finite sample distribution of returns, stemming from a linear combination of the potential returns of assets j ∈ U (where wj,0,0 are the corresponding weights inside the portfolio), can be greater than the sum of the VaR estimates of each individual asset. This implies that choosing VaR as a risk measure can lead to a complex optimisation exercise that has to deal with multiple local minimums and maximums. On the contrary, ES is sub-additive, ie, diversification should not increase portfolio risks. It can be shown that ES is a coherent risk measure. Moreover, when risk is measured on a finite set of scenario paths, denoted by H ∈ S, we can reduce the ES maximisation problem to one of linear programming, and hence build efficient solvers, as indicated by Rockafellar and Uryasev (2000). ES is often defined in the financial literature as a positive measure, or the absolute value α of ξU,S,T , so ES optimisation is commonly defined as a minimisation problem. In the course of this work, we prefer an unrestricted definition of ES, so that it refers to the expectation of the left tail of the potential returns distribution of the jth asset or a portfolio U. 118

À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

The elements of the sample distribution of the negative/positive potential returns of a portfolio U are indicated by RU,H,T . The VaR of the portfolio, measured at the single time step T over the sample distribution of returns simulated with scenario set S, is denoted by α ξU,S,T , where 1 − α indicates the confidence interval. In this book, the left 5% quantile of one tail of the distribution identifies VaR95 −α α VaR1U,S,T = ξU,S,T

(6.30)

It follows that the maximum expected shortfall is given by the following objective function −α max ES1U,S,T

w



α

= max ξU,S,T + w

1 α

S 

α s∈{s∗ : RU,H,T ξU,S,T }

PH,T (RU,H,T −



α ξU,S,T )

(6.31)

−α where ES1U,S,T denotes the ES of portfolio U over the sample distribution of the returns (simulated with a scenario set S) that are beyond α the VaR threshold indicated by ξU,S,T (which is a negative return).

CONCLUSIONS Although the Markowitz approach proved to be intuitively appealing and fairly flexible in accommodating more accurate measures of risk, it became more of a reference framework for investment managers and portfolio insurers than an effective decision-making tool. This is because the model remains extremely input sensitive, as the estimation of expected returns is a key driver of the optimisation exercise and even a small shift in the estimates can lead to very different and seemingly extreme asset allocations. Moreover, the model is well suited to dealing with product simplifications such as asset classes or indexes but cannot be easily calibrated in the presence of real investments exhibiting non-linear payoffs, notwithstanding the adoption of asymmetrical risk measures such as semi-variance or ES. In fact, expected returns are quite a biased and limiting measure of portfolio potentials in the long term. In summary: • a complete set of return expectations for different but single

time horizons would be required to generate optimal portfolio weights à la Markowitz;

119

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

• the generated efficient frontier often indicates extreme port-

folio weights, either long or short, which are excessively sensitive to changes in the estimation of expected returns (tilt sensitivity);

• as the approach does not encompass the uncertainty of all

input variables, this might lead to excessive errors from parameter estimation;

• a realistic and global asset allocation that spans multiple types

of investment opportunity in the long term (eg, fixed-income, derivatives, commodities, equities) cannot be identified, and therefore the analysis tends to be restricted to an oversimplified representation of asset classes;

• the passage of time stands outside the traditional framework,

leading to insufficient risk–return management, since the longterm risk–return time dynamics cannot be modelled.

Once an optimal portfolio has been indicated and the investor has realigned the composition of their tactical asset allocation with the theoretical optimum, the Markowitz-type approaches do not facilitate any mechanism to clearly track over time the deviation of portfolio performance from optimality, which can instead be conveniently addressed by PSO. In fact, this would require the key elements of goal-based investing to be introduced: target returns, risk appetite and a continuous comparison between actual portfolio performance (realised/potential) and the performance indicated by the optimal strategy by means of probability measurement.

REFERENCES

Bertsimas, D., G. J. Lauprete and A. Samarov, 2004, “Shortfall as a Risk Measure: Properties Optimization and Applications”, Journal of Economic Dynamics Control 28(7), pp. 1353–81. Dembo, R., and A. King, 1992, “Tracking Models and the Optimal Regret Distribution in Asset Allocation”, Applied Stochastic Models and Data Aanlysis 8(3), pp. 151–7. Elton, E. J., and M. J. Gruber, 1998, Modern Portfolio Theory and Investment Analysis, Fifth Edition (Chichester: John Wiley & Sons). Estrada, J., 2008, “Mean–Semivariance Optimization: A Heuristic Approach”, Journal of Applied Finance 18(1), pp. 57–72. Gilli, M., E. Këlezi and H. Hysi, 2006, “A Data-Driven Optimization Heuristic for Downside Risk Minimization”, Swiss Finance Institute Research Paper 06-2.

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À LA MARKOWITZ: A TALE OF SIMPLE WORLDS

Huelin, L., and M. Kheyam, 2011, Portfolio Optimization in a Downside Risk Framework: A Study of the Performance of Downside Risk Measures in Investment Management (Saarbrücken: Lap Lambert Academic Publishing). Kahneman, D., P. Slovic and A. Tversky, 1982, Judgment under Uncertainty: Heuristics and Biases (Cambridge University Press). Leland, H. E., 1999, “Beyond Mean–Variance: Performance Measurement in a NonSymmetrical World”, Financial Analysts Journal 55(1), pp. 27–36. Markowitz, H. M., 1952, “Portfolio Selection”, Journal of Finance 7(1), pp. 77–91. Rockafellar, T. R., and S. Uryasev, 2000, “Optimization of Conditional Value at Risk”, The Journal of Risk 2(3), pp. 21–42.

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7

The Black–Litterman Approach: A Tale of Subjective Views A man who wants the truth becomes a scientist; a man who wants to give free play to his subjectivity may become a writer; but what should a man do who wants something in between? Robert Musil (1880–1942)

In this chapter we present an optimisation model (the Black– Litterman approach) that embeds asymmetric information and the market equilibrium portfolio (the Capital Asset Pricing Model). We give investors’ views and discuss the posterior distribution of portfolio returns. INTRODUCTION In 1991 Fisher Black and Robert Litterman published their work on asset allocation which they had built for in-house use at Goldman Sachs. Their Bayesian portfolio construction model was widely considered a step forward from the original mean–variance approach and made use of three novel ideas: 1. information about financial returns is asymmetrical and can be divided into long-term market equilibrium (eg, CAPM) and short-term investors’ views; 2. both sets of information are uncertain and can be described by means of probability distributions; 3. a complete set of expected excess returns can be estimated by combining the investor’s views with the market equilibrium, which becomes the new input of the mean–variance model. There is an abundant literature describing the advantages of the Black–Litterman model over the original mean–variance approach. In particular, Litterman and He (1999) observed that the approach overcomes the tendency to generate non-tradable portfolios that are extreme and oversensitive to the updates of the parameters. 123

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

In this chapter we outline the attractive features of such an elegant model, and address some of the criticism by the investment community. The model is quite sophisticated and challenging to implement in investment practices and as part of long-term private wealth advice, especially if fixed-income investments and derivatives are a relevant component of the investment universe. First, we review the technicalities of the Black–Litterman approach and define the market equilibrium portfolio (CAPM) by using the capital market line (CML) and the security market line (SML). Next, we allow the investors’ views to modify the prior vectors of expected returns, to introduce asymmetry of the information. Finally, we use the resulting posterior distributions of asset returns entering the objective function to shape portfolio choice. THE TECHNICALITIES OF THE MODEL The starting point of the Black–Litterman approach to portfolio choice is the identification of the equilibrium market portfolio and the estimate of the expected excess returns for all assets or indexes that have been chosen to represent the investment universe. The vector of equilibrium expected excess returns does not necessarily need to be observed directly from the returns’ time series for the individual assets, as in the original mean–variance formulation, but could be the result of econometric analysis (eg, CAPM) that feeds the so-called reverse optimisation to indicate the initial equilibrium market weights, as indicated by Idzorek (2007). Investors can formulate personal views and related confidence levels about financial returns, modify the initial equilibrium and then re-optimise the objective function to solve for the portfolio weights that reflect the equilibrium excess returns, equilibrium weights, personal views and allocation constraints. The steps of the approach can be summarised as follows. 1. Preparation of the inputs: identify the investment universe, estimate excess returns and historical variance–covariances and the risk aversion coefficient. 2. Reverse optimisation: estimate the equilibrium expected excess returns (eg, CAPM) and indicate the equilibrium market weights by reverse optimisation. 124

THE BLACK–LITTERMAN APPROACH: A TALE OF SUBJECTIVE VIEWS

3. Declaration of investor’s beliefs: declare the investor’s views about excess returns of the assets, declare the confidence level of the views and estimate the distribution of these views. 4. Portfolio optimisation: estimate the posterior distribution of expected excess returns and optimise to give the optimal tilted weights. Our presentation of this approach begins with its foundations: the Capital Asset Pricing Model. THE MARKET EQUILIBRIUM PORTFOLIO The CAPM was formulated by Sharpe (1964) and Lintner (1965) as a direct continuation of the Markowitz theory and is concerned with describing the economic equilibrium conditional on the hypothesis of symmetrical information, ie, all market participants are rational investors holding the same complete set of knowledge about financial investments (ie, expected excess return, variance and covariance) and following the same paradigm of risk aversion as postulated by the mean–variance approach. Hence, given individual levels of risk aversion, each investor holds a risky portfolio on the same efficient frontier. The equilibrium market portfolio, denoted by M, is the result of the investment decision-making of all market participants: the investment community is assumed to hold symmetrical information about asset returns, variances and covariances, and continuous trading permits that the market adjusts towards a long-term equilibrium M are governed by frictionless value where the market weights wj,0,0 price discovery. We might estimate the equilibrium market weights by assessing the capital value of each asset and dividing it by the total capital value of the whole market, so that the equilibrium market portfolio can be described as a function of asset prices only M wj,0,0 =

Vj,0,0 VM,0,0

(7.1)

The investment recommendation would be to hold a combination of the risk-free portfolio and the market portfolio (eg, buy each asset in the universe according to the respective capital weights in the market): for any given level of risk no other portfolio can provide 125

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

higher expected excess return, because trading against the market equilibrium would not be valuable. Yet, since the number of assets is enormous in the real world, trying to construct a real market portfolio would be unrealistic. That is why the investors might indicate a representative market index B to approximate the initial equilibrium weights of M. Alternatively, we can initiate the Black–Litterman approach by estimating the CAPM betas in order to model the expected excess returns of individual asset classes. Thus, reverse optimisation formulas can derive the equivalent CAPM weights representing the initial market equilibrium. In this book we mainly refer to this approach, although the Black–Litterman approach can be modelled with alternative starting points. Recalling the efficient frontier introduced by Markowitz, if every investor has the same expectation about asset returns, then the portfolio that optimises any investor’s utility function would yield the risk-free rate plus the excess return provided by the optimal risky portfolio, which is indicated by the tangent to the Markowitz efficient frontier of the line originating from the risk-free rate investment. This is the so-called capital market line ¯ U,S,T = rf,S,T + R



RM,S,T − rf,S,T σM,S,T

σU,S,T

(7.2)

The notation S and T, indicating the historical scenario returns and the single step investment horizon, will be dropped in this chapter for simplicity, but will be reintroduced in Chapter 8 to indicate the individual nodes of the stochastic simulation over time. Hence ¯ RM − rf ¯RU = rf + σU (7.3) σM

This means that any portfolio on the CML (Figure 7.1), which can be achieved by a suitable mix between risk-free investments and optimal weights on the risky assets, has a superior risk–return combination to any other portfolio lying on the efficient frontier, except the market portfolio at the tangent point. The market portfolio is invested solely in risky assets and its composition reflects the equilibrium weights that originate from the assumption of frictionless price discovery. The slope of the CML identifies the price of risk, λ, also known as the Sharpe ratio, which represents the change in the expected 126

THE BLACK–LITTERMAN APPROACH: A TALE OF SUBJECTIVE VIEWS

Figure 7.1 Capital market line

Expected returns

Leveraged portfolio Mix of risk-free and market portfolios Wealth in market portfolio only Wealth in risk-free portfolio only Standard deviations

¯ U of the investor’s portfolio per unit change in portfolio return R volatility σU λ=

¯ M − rf ¯U R ∂R = ∂σU σM

(7.4)

¯ M − rf is the expected excess return of the market efficient portfolio. R The CAPM implies that investors are willing to trade individual securities only if they can add extra reward (eg, stock picking to generate alpha): buying assets with initially higher Sharpe ratio (ie, higher expected excess return) and selling assets with initially lower Sharpe ratio. But, according to the Efficient Market Hypothesis this would affect asset prices until the expected excess returns realigned so that any extra risk-adjusted reward is eliminated. The Efficient Market Hypothesis posit that for a portfolio to lie on the CML the ratio between marginal return and marginal variance must be equal for all invested securities: in equilibrium, all investors are optimally invested and the Sharpe ratio of the portfolio is constant. While the marginal increase in portfolio excess return due to a new investment is easily estimated, the marginal increase in portfolio variance is proportional to the covariance of the given assets with the portfolio. We can suppose that an investor, holding the market portfolio, decides to add an amount ∆q of the asset q that can ¯ q (hence higher variance σq2 ) at the expense provide extra reward R of the amount already invested in the market portfolio M, so that the expected return of the new (M + ∆q ) portfolio becomes ¯ q + (1 − ∆q )R ¯ M = ∆q (R ¯q −R ¯ M) + R ¯M ¯ M+∆q = ∆q R R

(7.5) 127

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

and the standard deviation σM+∆q  2 = ∆2q σq2 + (1 − ∆q )2 σM + 2∆q (1 − ∆q ) cov(q,M)  2 = ∆2q σq2 + σM + ∆2q σM − 2∆q σM + 2∆q cov(q,M) −2∆2q cov(q,M)

(7.6)

We can indicate the change in expected excess return of the port¯ M of the above ¯q −R ¯ M+∆q /∂∆q = R folio by the partial derivative ∂ R equation ¯ M+∆q ∂R

¯q −R ¯M =R

∂∆q

(7.7)

and the standard deviation ∂σM+∆q ∂∆q

=

2 ∆q σq2 + (1 − 2∆q ) cov(q,M) +(∆q − 1)σM

σM

(7.8)

For convenience later, we note that if the investment in q is null (∆q = 0), then ∂σM+∆q ∂∆q

2 cov(q,M) −σM

=

σM

(7.9)

remains unchanged as there is no ∆q element. Hence, we might want to estimate how much change in expected return is due to the change in the risk profile of the (M +∆q ) portfolio ¯ M+∆q /∂∆q ∂R

∂σM+∆q /∂∆q

=

¯ M+∆q ∂R

∂σM+∆q

=

¯M ¯q −R R

2 (cov(q,M) −σM )/σM

(7.10)

Clearly, the above equation has the same economic meaning as the slope of the CML, so the two definitions could be equated ¯M ¯q −R R

2 (cov(q,M) −σM )/σM

=

¯ M − rf R σM

(7.11)

Simplifying, by the market volatility ¯ M − rf ¯M ¯q −R R R 2 = 2 cov(q,M) −σM σM 128

(7.12)

THE BLACK–LITTERMAN APPROACH: A TALE OF SUBJECTIVE VIEWS

Figure 7.2 Security market line

Expected return

Asset C Asset B

Market portfolio

Asset A

β( j,M )

and rearranging the equation 2 ⎫ ¯ M − rf )(cov(q,M) −σM )⎪ (R ⎪ ⎪ ⎪ 2 ⎪ ⎪ σM ⎪ ⎪ ⎪ ⎪ 2 ¯ M − rf ) cov(q,M) +rf σM ⎬ (R ¯q = R ⎪ 2 ⎪ σM ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ cov(q,M) ⎪ ⎪ ¯ q = (R ¯ M − rf ) ⎭ + r R f 2 σM

¯q −R ¯M = R

we find that

¯ q − rf ) = β(q,M) (R ¯ M − rf ) (R

(7.13)

(7.14)

This is the relationship between the expected excess return of any individual asset with respect to the market portfolio (Figure 7.2), known as the security market line and indicating that the expected excess return of any jth asset equals the expected excess return of the market multiplied by a factor that represents the riskiness of the asset with respect to the correlation with the market as a whole, so that cov(j,M) (7.15) β(j,M) = 2 σM

The single-security case can be generalised by assuming that the investor holds a portfolio U, which is different from the market portfolio M, the tangential portfolio to which we cannot add return unless it is at the expense of more risk, so that ¯ M − rf ) ¯ U − rf = β(U,M) (R R

(7.16) 129

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

where β(U,M) =

cov(U,M) 2 σM

=



wj β(j,M)

(7.17)

j∈U

It follows that when the investor’s portfolio equals the market portfolio β(U,M) = β(M,M) = 1

(7.18)

so that the weights of the investor’s portfolio equal the equilibrium market weights, which are the starting point of the Black–Litterman approach. By reverse engineering the estimation of the betas of each asset class against the market portfolio, often approximated by a market index, we can estimate the CAPM equilibrium weights of each asset class. This book refers to the time series of the excess returns as the initial input for the estimation of the CAPM equilibrium, so that ¯ j = E[Rj − rf ] R ¯ M = E[RM − rf ] R cov(j,M)

β(j,M) =

2 σM

¯M ¯ capm = β(i,M) R R j

(7.19) (7.20) (7.21) (7.22)

The variance–covariance matrix of the excess returns of all assets in the universe is denoted by Σ and allows us to derive the vector of capm by so-called reverse optimisation the initial portfolio weights wj T ¯ capm − 1 λwcapmT Σ wcapm ) max(wcapm R 2

(7.23)

¯ capm = (λΣ)wcapm R

(7.24)

wcapm

ie

so that ¯ wcapm = (λΣ)−1 R

capm

(7.25)

λ indicates the CAPM risk aversion coefficient: the implied reverse optimised excess returns indicate the specified market risk premiums; hence, a larger λ implies more excess return per unit of risk adding a positive influence on the level of the estimated excess returns. Under the CAPM theory it is assumed that the idiosyncratic risks of the assets are uncorrelated so that risk can be reduced

130

THE BLACK–LITTERMAN APPROACH: A TALE OF SUBJECTIVE VIEWS

by diversification. Therefore, the coefficient of the linear regression analysis is assumed to be null since the investor holding the market portfolio will be rewarded only for the systemic risk estimated by β(j,M) . THE INVESTORS’ VIEWS The CAPM assumes that information about expected return, variance and covariance of individual securities is known to any investor and this knowledge allows us to identify the long-term equilibrium excess return for any security conditional on frictionless price discovery forces. Black and Litterman deviate from the assumption of symmetric information (ie, that all market participants do invest or are willing to invest in the same efficient frontier). A realistic formulation must recognise that investors allocate capital across multiple assets without much knowledge of other investors’ preferences. Some investors might believe they possess superior information compared with the market as a whole and, although they accept asset prices as a starting point to indicate initial optimal weights (prior), they might want their beliefs about the short-term dynamics of asset prices to be reflected in the construction of the final optimal portfolio (posterior). The individual investor’s expectations indicate the belief that in the short term a given asset or set of assets will not converge to the equilibrium but will yield a different return. This belief is uncertain, as it refers to a future state of the world, and it can be described by an expectation and a probability distribution, ie, a view. The CAPM equilibrium is the starting point; thus, investors’ views on expected excess returns are added, to build a new combined distribution that indicates a more informative set of market potentials. The posterior vector of combined expected excess returns and their related uncertainty is supplied to the final optimisation process so that the posterior vector of tilted asset weights is derived: this indicates the optimal portfolio. According to the original Markowitz approach, investors willing to include their own views in the optimisation process are asked to overwrite the market statistics, thus facing excessive sensitivity to the inputs, which in turn leads to extreme portfolios. According to Black and Litterman the views do not directly replace the original historical inputs, but add new information to the exercise as they are 131

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

combined with the prior expectation of market returns by means of a Bayesian model; this leads to more stable posterior asset allocations. The information contained in the views can be an absolute or a relative expectation, as investors can express their own beliefs about the performance of an individual asset or the expected performance relative to other investments. Bayes’s Theorem The Bayes Theorem expresses how a new piece of evidence affects the probability that a belief is true. Let us suppose a passenger reports to a flight attendant that a person sitting in economy class is smoking. The probability of this person being a man is 50%, conditional on the fact that the flight has an equal number of men (M) and women (W) travelling at that time in economy class P(M) = 0. 50,

P(W ) = 0. 50

(7.26)

Let us now suppose that it is also reported that the annoying passenger is smoking a cigar (event C). Let us assume that men are more likely to smoke cigars than women; thus, the Bayes Theorem can be applied to calculate the probability that the annoying passenger was indeed a man. Let us suppose it is known that 25% of men smoke cigars, compared with only 5% of women P(C | M) = 0. 25,

P(C | W ) = 0. 05

(7.27)

In order to calculate the probability that the smoking passenger was a man, given the fact that the passenger had been smoking a cigar, we can use the Bayes Theorem P(C | M)P(M) P(C) P(C | M)P(M) = P(C | M)P(M) + P(C | W )P(W )

P(M | C) =

(7.28)

Hence, P(M | C) = 132

0. 25 × 0. 50 ≅ 0. 83 (0. 25 × 0. 50) + (0. 05 × 0. 50)

(7.29)

THE BLACK–LITTERMAN APPROACH: A TALE OF SUBJECTIVE VIEWS

The views Let Q be a column vector with r rows, where each row element represents the subjective expected excess return attached to a view ⎡ ⎤

⎢ ⎢ Q=⎢ ⎣

q1 ⎥ .. ⎥ .⎥ ⎦ qr

(7.30)

This information comes with a level of confidence, representing the level of certainty that the investor has in the stated information. Let ε be a column vector with r rows, indicating the confidence intervals of each view ⎡ ⎤ ε ⎢ 1⎥ ⎢ .. ⎥ ε=⎢.⎥ ⎣ ⎦ εr

(7.31)

Let D be a matrix that contains the views on the individual assets, so each column element corresponds to a view and each row element corresponds to the asset or asset class affected by such a view ⎡

d1,1 ⎢ ⎢d2,1 ⎢ D=⎢ . ⎢ .. ⎣ dr,1

d1,2 d2,2 .. . dr,2

··· ···

..

.

···



d1,c ⎥ d2,c ⎥ ⎥ .. ⎥ . ⎥ ⎦ dr,c

(7.32)

In the original proposition, Black and Litterman seem to allow the investors to express directly their confidence in the views, so for each element of Q the investor will express a fluctuation band (a; b) and a probability P(a < qr < b) of falling inside the given band. However, no clear indication has been given on how this can be combined to estimate the final square-diagonal matrix of the confidences in the views, indicated by Ω ⎡

⎢ ⎢ ⎢ Ω=⎢ ⎢ ⎣

ς1,1

0

0 .. . 0

ς2,2

.. . 0

···

···

..

.

···

0 0 .. . ςr,r

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(7.33)

Clearly, it might not be trivial for an individual investor to express meaningful confidences for the views. Alternative methods can be 133

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

found in, for example, Litterman and He (1999) and Meucci (2010), who assume the Ω is proportional to the known variances estimated for the CAPM excess returns. In such a case, the diagonal elements of Ω would be estimated as follows ςr,c = D(r) Σ DT(r)

(7.34)

We note that a term τ is introduced in the definition of the posterior distribution in the section on page 135, indicating a coefficient of proportionality expressing the confidence interval of the estimation of the mean of the original prior distribution. There seems to be little consensus in the literature about how this term should be estimated. Therefore, those readers not interested in indicating a level of τ might want to include this parameter in the estimation of the elements of Ω , so that it cancels out in the subsequent application of the Bayesian model ςr,c = D(r) τΣ DT(r)

(7.35)

Example We assume that there are five assets, such that: in the absolute view asset 3 is expected to yield a 12% excess return, with full confidence in the view; in the relative view, assets 1 and 3 are expected to outperform assets 2 and 4 by yielding 5% more excess return, with low confidence in the view. It follows that "

#

view A view B

(7.36)

"

#

view A view B

(7.37)

#

(7.38)

0. 12 Q= 0. 05 0 ε= 0. 70 "

0 0 D= 0. 5 −0. 5

1 0 0. 5 −0. 5

0 0

view A: absolute view B: relative

We note that, when the view is absolute, the elements in each row need to add up to 1.00; when the view is relative, the result of the summation is 0.00. In the original Black–Litterman formulation, the views were presented as fairly generic statements, so the elements in each view were set as equally relevant. In subsequent formulations, academics have also refined the approach by allowing the views to reflect market capitalisation weights, eg, so that asset 1 would represent 90% of the market capitalisation of the combined outperforming 134

THE BLACK–LITTERMAN APPROACH: A TALE OF SUBJECTIVE VIEWS

relative view (composed of assets 1 and 3), and asset 2 would represent 70% of the combined underperforming relative view (composed of assets 2 and 4). Then the elements of the matrix would be "

0 D= 0. 9

0 −0. 7

#

1 0 0 0. 1 −0. 3 0

(7.39)

THE POSTERIOR DISTRIBUTION The aim of the Black–Litterman approach is to combine the probability density function (pdf) of the CAPM excess returns with the probability density function of the views, so that the posterior probability density function can be used as the input to the mean–variance optimisation. The original proposition assumes the distributions are normal, but this assumption could also be relaxed ¯ Fcapm (R) ∼ N (R

capm

; Σ)

(7.40)

Fviews (R) ∼ N (Q; Ω)

(7.41)

The Bayes Theorem allows us to construct the combined posterior distribution, so that ¯ Fpost (R) ∼ N ([(τΣ)−1 + DT Ω −1 D]−1 [(τΣ)−1 R

capm

+ DT Ω −1 Q];

[(τΣ)−1 + DT Ω −1 D]−1 )

(7.42)

We can analyse the mean of the pdf and rearrange the equation so that ¯ post = R ¯ capm ] ¯ capm + τΣ[(DT τΣ D) + Ω]−1 [Q − DR R

(7.43)

If Ω = 0, then all views are certain, so for all assets specified in the views the return is given by the views themselves. If Ω = ∞, then the views are totally uncertain, thus the equation simplifies to ¯ capm . ¯ post = R R We can analyse the covariance, which is the summation of the variance of the pdf (ie, the mean estimate around the real mean) and the initial CAPM variance, so that Σ post = Σ + [(τΣ)−1 + DT Ω −1 D]−1

= τΣ − τΣ DT (DτΣ DT + Ω)−1 DτΣ

(7.44)

If Ω = 0, then the views are certain, so ρ ∗ = 0. If Ω = ∞, then the views are totally uncertain and we stay where we were: ρ ∗ = λρ . 135

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Having derived the pdf of the posterior distribution of the excess returns, we can optimise the mean–variance objective function and estimate the posterior “tilted” weights of the assets that indicate the optimal portfolio. This is achieved by solving the following unconstrained maximisation problem post

¯U max{wT R w

− 12 λwT Σ post w}

(7.45)

CONCLUSIONS Black and Litterman have certainly contributed to the construction of advance portfolio optimisation techniques, to serve the ever growing needs of investment managers. In particular, they have addressed the need to reconcile the general theory of market efficiency with the extra information added by the subjective views of professional investors, so that optimisation routines allow for information asymmetry to be embedded in the composition of asset allocations. However, the original material and its further descriptions do not permit us to fully gauge the true nature of the first implementation; nor have the subsequent academic contributions managed to simplify the challenging steps required to calibrate the model. Thus, the approach has failed to become mainstream in investment management processes, particularly in the field of private wealth, as it is quite technical and complex to parameterise, and cannot lead to the construction of institutionalised frameworks. Moreover, this technique cannot accommodate direct modelling of the risk–return distributions of real products, which is highly relevant for long-term portfolio choice.

REFERENCES

Biondi, L., 2013, Il Modello Black & Litterman: Descrizione Teorica del Modello (Edizioni Accademiche Italiane). Black, F., and Litterman, R., 1992, “Global Portfolio Optimization”, Financial Analysts Journal, pp. 28–43. Gofman, M., and A. Manela, 2012, “An Empirical Evaluation of the Black–Litterman Approach to Portfolio Choice”, SSRN Working Paper, URL: http://dx.doi.org/10.2139/ ssrn.1782033. Idzorek, T. M., 2007, “A Step-by-Step Guide to the Black–Litterman Model: Incorporating User-Specified Confidence Intervals”, in S. Satchell, Forecasting Expected Returns in the Finnacial Markets, First Edition (Elsevier).

136

THE BLACK–LITTERMAN APPROACH: A TALE OF SUBJECTIVE VIEWS

Idzorek, T. M., and J. X. Xiong, 2010, “Mean–Variance versus Mean-Conditional Valueat-Risk Optimization: The Impact of Incorporating Fat Tails and Skewness into the Asset Allocation Decision”, Working Paper, Ibbotson. Lintner, J., 1965, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets”, Review of Economics and Statistics 47(1), pp. 13–37. Litterman, R., and G. He, 1999, “The Intuition behind Black–Litterman Model Portfolios”, Working Paper, Goldman Sachs Investment Management Division. Martellini, L., and V. Ziemann, 2007, “Extending Black–Litterman Analysis beyond the Mean–Variance Framework”, Working Paper, EDHEC Risk and Asset Management Research Centre. Michaud, R. O., D. N. Esch and R. O. Michaud, 2013, “Deconstructing Black–Litterman: How to Get the Portfolio You Already Knew You Wanted”, Journal of Investment Management 11(1), pp. 6–20. Meucci, A., 2010, “The Black–Litterma Approach: Original Model and Extensions”, Social Science Research Network. Sharpe, W. F., 1964, “Capital Asset Prices: A Theory of Market equilibrium under Conditions of Risk”, Journal of Finance 19(3), pp. 425–42. Walters, J., 2009, “The Black–Litterman Model in Detail”, SSRN Working Paper.

137

8

Probabilistic Scenario Optimisation Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting points and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up. Albert Einstein (1879–1955)

PSO is an exhaustive enumeration technique for designing goalbased and long-term optimal investing. In this chapter we present a step-by-step process from potential to admissible portfolios, and discuss low-discrepancy sequences and lexicographical representations, risk-adequate portfolios and the goal-based objective function. INTRODUCTION Markowitz has shaped the landscape of portfolio optimisation since the introduction of the mean–variance approach in the 1950s, and subsequent academic advances have further refined his original ideas, particularly regarding the risk measure definition (eg, semivariance and expected shortfall). The change in perspective introduced by Black and Litterman in the 1990s improved upon the mean–variance exercise by including the expected return measure, allowing for probabilistic investment views that are statements of individual beliefs. However, at the time of writing, the global nature of financial markets, growth in life expectancy and the use of derivatives across asset classes have raised the stakes, especially for longterm investors, and have sustained the debate about extending the techniques of portfolio choice beyond the classical approaches. Probabilistic scenario optimisation (PSO) is a risk-based approach designed to facilitate goal-based investing in institutions’ portfolio management processes. In particular, the decision-making for longterm investments can benefit from this approach in structuring 139

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

added-value for final investors, because PSO is an exhaustive enumeration technique that does not restrict portfolio analysis to a mean–variance representation, but rather includes the full valuation of actual securities conditional on stochastic scenarios, which is best practice for market and counterparty risk management at numerous financial institutions. Starting with a full revaluation of actual payoffs can enhance the creation of optimal asset allocations: real market variables can be simulated, and investments repriced, conditional on perturbed market conditions, so that stress tests can also be modelled to introduce individual views about the markets and criticise or validate the results of the theoretical optimums. The set of investors’ preferences takes centre stage and can be superimposed on the probability density function of portfolio total returns: return targets and risk boundaries can be tested on the space of the simulated risk–return measure, so that reinvestment strategies and wealth consumption can also be modelled over time. Thus, portfolio managers can represent past and future performance in a single graphic that helps investors visualise the risk–return characteristics of their investment intuitively. The probabilistic measure of achieving/beating investment targets becomes the key variable of the objective function. Only by taking into account the full information contained in the simulated and multi-period probability density of portfolio returns an informative optimisation space can be generated that combines market dynamics, product characteristics and target risk–return profiles so that goal-based investing resolves to be quite an intuitive step-bystep approach. A portfolio manager would depict the target risk– return profile of the investment and overlap it (we can imagine a set of thresholds) to the potential total return space of a set of equally likely asset allocations (we can imagine clouds of potential realisations) and assess the probability of achieving or beating total return targets embedded in the risk–return profile at given time steps (we can measure the density of the clouds above/below a threshold), while adhering to a limit of risk adequacy (we can imagine a capital guarantee or a minimum return requirement) and complying with a given set of allocation constraints (we can imagine a concentration limit on individual products or asset classes), so that the portfolio dominating the alternatives can be indicated as optimum. 140

PROBABILISTIC SCENARIO OPTIMISATION

Table 8.1 The main steps of the PSO process ΦU , ϕU (generate all potential portfolios) ↓ ΨU (identify only the admissible portfolios) ↓ ΘU (filter the risk-adequate portfolios) ↓ ΘU∗ (indicate the optimal goal-based portfolio)

Reinvestment strategies and multi-period investment decisions can be modelled as a sequence of time-dependent realisations, so that the set of multi-period preferences and constraints that define the optimisation exercise can be phrased intuitively: maximising the probability of achieving a desired return level year on year; minimising a left quantile of the risk–return distribution at given time steps; maximising the probability of achieving long-term reward while minimising short-term risk at any intermediate liquidity or reallocation step. PSO is a step-by-step process of portfolio filtering and ordering according to a series of probability measurement criteria, as illustrated in Table 8.1. First, we review the main elements of PSO. Second, we introduce allocation constraints to identify admissible portfolios from the broader universe of potential allocations. Third, we discuss lowdiscrepancy sequences and lexicographical representations to solve the computational demands. Fourth, we elicit the risk adequacy profile in order to filter out the inadequate portfolios from the set of asset allocations that contribute to the maximisation of the objective function. Finally, the probability of achieving or beating an investment target is examined as the optimisation variable. THE PROBABILISTIC SCENARIO OPTIMISATION PROCESS PSO aims to identify the asset allocation that shows the highest probability of beating an investment goal, while complying with given allocation constraints and risk limits, by means of exhaustive enumeration. Advanced risk management methods are required to deal 141

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

with the estimation of uncertainty about risk–return realisations of actual financial products. Such estimates involve the generation of tens of thousands of stochastic scenarios about the evolution of the market risk factors that drive fair-value pricing of financial securities. The accuracy and meaningfulness of this process are influenced by the quality of the input data and the methodology adopted to simulate the future states of the world of each class of risk factors. In this book we make our own set of assumptions in generating stochastic scenarios and state a non-binding preference for a mix of real-world and risk-neutral calibrations. The process can be represented by the following steps. • Define the optimisation problem: select the investment uni-

verse, indicate the allocation constraints and declare the investment risk–return profile in order to depict the investment goal and the risk limit.

• Generate the space of future total returns by simulating real

securities over time, conditional on probabilistic scenarios.

• Exhaustively generate the quasi-random space of the admis-

sible asset allocations and reduce this to the risk-adequate set: filter all admissible allocations that satisfy the asset allocation constraints and reduce these to the set of risk-adequate portfolios with respect to the investor’s risk profile.

• Measure the probability and portfolio ranking: optimise the

objective function and represent graphically the resulting asset allocation and its characteristics.

• Measure the performance and compare portfolios: investment

performance can be tracked over time, and the distance to optimality can be measured by computing the residual probability to achieve/beat a target return across time.

The traits of a modern risk management framework, the importance of working with real payoffs and the elicitation of investors’ preferences have been discussed in previous chapters. We can now review the definition of the allocation constraints and the mathematical properties of low-discrepancy sequences, to reduce the sample set and make exhaustive enumeration a viable optimisation alternative. We thus discuss the formalisation of the maximisation/ minimisation of the PSO objective functions. 142

PROBABILISTIC SCENARIO OPTIMISATION

Exhaustive enumeration techniques are very unrestricted methods and can be applied to any type of investment problem. However, for convenience, in this chapter we refer in particular to the simpler cases of model portfolios that optimise private wealth, because the set of investment opportunities is usually preselected by means of research and recommendations, so that it corresponds to a manageable universe. POTENTIAL AND ADMISSIBLE PORTFOLIOS: ALLOCATION CONSTRAINTS An investor has an initial amount of capital VU,0,0 to be invested across a universe of opportunities, U. ΦU,0,0 denotes the space of elements identifying all the unrestricted potential portfolio allocations (ie, the set of vectors of the potential allocation weights on each of the j assets in the universe). The elements of ΦU,0,0 correspond to the individual percentage weights wj,0,0 , which are constant over scenario paths and time, since only the fair value of each security j changes, conditional on scenario paths and time steps ΦU,0,0 : VU,H,t =



Vj,H,t wj,0,0

(8.1)

j ∈U

Advanced techniques of dynamic trading strategies allow us to relax the constant allocation assumption and model income/ consumption events which require the portfolio weights to be reallocated according to a given set of rules. Hence, percentage weights can change, conditional on events or investor’s decisions occurring at predefined time steps, so ΦU,H,t : VU,H,t =



Vj,H,t wj,H,t

(8.2)

j ∈U

Investments can be added and money can be withdrawn from existing allocations by modelling potential capital inflows and outflows (eg, income streams, dividends, real estate investments). We can identify two main approaches to model income and consumption. 1. Homogeneous income/consumption: money can be invested or withdrawn proportionally to the prevailing asset allocation, conditional on time and scenario. 143

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

2. Selective income/consumption: discretionary rules can drive inflows and outflows according to a waterfall schema (eg, invest new money into money markets, disinvest from the worst performing assets). Clearly, discretionary decisions can be modelled as well, conditional on the marginal contribution of the chosen assets to the resulting probability of achieving/ beating the investment goal. Reflecting the exogenous income/consumption strategy in our portfolio simulation model, we denote by c a multi-period stream of cashflows and by ϕU,H,t the adjusted portfolio ϕU,H,t (ΦU,H,t , cU,H,t ) after income/composition. For example, a multi-period income/consumption constraint that complies with proportional scaling in a model with no short selling allowed could be imposed on a portfolio allocation conditional on scenario simulation, so that all elements of ϕU,H,t > 0

for all H ∈ S, t ∈ Γ

(8.3)

The total portfolio value VU,H,t is calculated by assuming cU,H,t = 0, conditional on scenario paths and time steps for each element of ΦU,H,t

VU,H,t = VU (ΦU,H,t )

(8.4)

The income/consumption stream cU,H,t is added/subtracted, so that the adjusted portfolio composition becomes ϕU,H,t

⎧ c ⎪ ⎨ 1 + U,H,t ΦU,H,t VU,H,t := ⎪ ⎩

0

if VU,H,t  −cU,H,t

(8.5)

if VU,H,t < −cU,H,t

Portfolio managers can find it useful to restrict the size of the resulting optimal portfolios to a set of minimum/maximum percentage amounts and avoid portfolio concentration and portfolio fragmentation in creating investment proposals. This is particularly useful for the optimisation of strategic private wealth portfolios (ie, model portfolios). Hence, we can impose a minimum allocation step, which corresponds to a percentage of the initial wealth, so that each initial element in ΦU,0,0 is a multiple of a constant z, so that (Vj,0,0 wj,0,0 )/z is an integer. We can also assume that the amount invested in a particular category A of portfolio U (eg, asset class, sector or currency) is not 144

PROBABILISTIC SCENARIO OPTIMISATION

lower than a percentage a (fragmentation limit) and not higher than a percentage b (concentration limit) a

 Vj,0,0 wj,0,0

j ∈A

VU,0,0

b

(8.6)

If short selling is allowed, the a boundary can become negative and a constraint must be imposed on the ratio of long to short exposures in order to limit the space of the admissible asset allocations ΨU,0,0 , ie, the subset of the unrestricted potential portfolio compositions of ΦU,0,0 complying with the allocation constraints (assuming that income/consumption rules are not modelled). PSO is an exhaustive enumeration technique that requires the generation and contemporaneous evaluation of an exceptionally large number of potential portfolio allocations, whose space grows exponentially with the number of assets in U and the distance between the ticks, z. The number of the admissible portfolio allocations ΨU,0,0 can be estimated ex ante by making use of an iterative process that takes into account the number of actual products in the available universe, the min/max constraints and the minimum discretisation of the investment ticks z Ψ (U, N

max

,N

tot

)=

min( U,N tot )

i=|N tot /N max |

$ %

J i

=

$ %

U Ψ (i, N max − 1, N tot − i) (8.7) i

U! i!(U − i)!

(8.8)

in which U is the number of assets belonging to the universe (eg, 100), z is the increment in tick size of each investment as a percentage of portfolio value VU,0,0 , so wj,0,0 /z is an integer (eg, 0.20/0.05), bz is the maximum percentage exposure in which b is an integer, so bz  1 (eg, 0.30), N tot is the total number of ticks (eg, 1/0.05 = 20), N max is the maximum number of ticks allowed for each investment (eg, 0.30/0.05 = 6), N tot /N max is the lower bound, ie, the sum of the ceilings, and Ψ (U, N max , N tot ) is the number of possible allocations determined by the recursive formula, where Ψ (U, 0, N tot ) = 1. We assume, for example, that no more than 100 assets are part of an advisory catalogue, that short selling is not allowed and that the maximum investment in each asset is 30%. Two cases are presented in Table 8.2: z = 0.05, so that N tot = 20 and N max = 6; z = 0.10, so that N tot = 10 and N max = 3. 145

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 8.2 Dimension of admissible asset allocations Ψ (U, N max , N tot ) for bz = 30% U

z = 5%

z = 10%

10 50 100

5.27 × 106 1.15 × 1017 2.45 × 1022

4.48 × 104 6.14 × 1010 4.25 × 1013

Figure 8.1 Cumulative standard normal distribution 1.0

Probability

0.8 0.6 0.4

Entry of the draw from the unity interval

0.2 0 –5

–4

–3

–2

–1

0 1 Variable

2

3

4

5

Since admissible asset allocations amount to a significantly large set, ranging from a few millions to more than 1022 , low-discrepancy sequence techniques become extremely advantageous in reducing the computational burden. LOW-DISCREPANCY METHODS: HALTON SEQUENCES Quasi-random sequences are methods of generating representative samples from probability distributions and are a growing field of mathematical research. Halton (1960), Sobol (1967), Faure (1982) and Niederreiter (1987) are probably the best known quasi-random methods used in finance to improve the computational performance of Monte Carlo simulations, thus preserving high accuracy. The main concept is to build a series of numbers by sampling from the unit interval [0, 1] in such a way that every draw is further away from the preceding draw, ie, clustering (groups of numbers close to each other) is avoided, and that the draws are as far apart from each other as possible, ie, large gaps are avoided because the method fills in the gaps sequentially between the previous number and next number in 146

PROBABILISTIC SCENARIO OPTIMISATION

the sequence. The resulting uniform distribution in the unit interval is used to derive the samples from target distributions (eg, the normal distribution). A direct way to transform the uniform distribution into the target distribution is to operate on the inverse of the target cumulative distribution function (as in Figure 8.1): we can associate elements of the uniform distribution to the corresponding equally likely probability estimate which is read off the target cumulative distribution function, and thus estimate the inverse values. Low-discrepancy sequences can also be applied to the problem of reducing the space of the admissible portfolio compositions that PSO has to evaluate, so that the exhaustive enumeration technique is performed on a smaller set of equidistant asset allocations that are representative of all possible portfolio combinations, thus avoiding large gaps and clustering. We use Halton’s sequences (Halton 1960) based on low prime numbers for our optimisation examples and the case studies featured in this book. However, our preference is non-binding and the reader may want to use different techniques to improve calculation performance. Halton’s sequences are deterministic sequences of numbers based on increasingly finer prime-based divisions of subunit intervals (ie, 2, 3, 7, 11, 13, . . . ), which produce well-spaced draws from the unit interval so that the quasi-random variables sampled from a larger population are ex post evenly spread (equidistant). R indicates a prime number, so any other integer n can be written in radix-R notation n ≡ nM nM−1 · · · n2 n1 n0

= nM RM + nM−1 RM−1 + · · · + n2 R2 + n1 R1 + n0

(8.9)

in which M is an integral value M = ⌊ logR (n)⌋ =

&

ln(n) ln(R)

'

(8.10)

A fraction lying between 0 and 1 can be uniquely constructed by reversing the order of the digits in n ϕ = ϕR (n) = nm nm−1 · · · n2 n1 0 = nM R−M−1 + · · · + n1 R−2 + n0 R−1

(8.11)

A numerical example is provided for the one-dimensional and radix-2 notation (R = 2) to easily show the Halton mechanism. 147

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

1. We start the sequence by setting the integer n = 1, so M = ⌊ log2 (1)⌋ = 0 1 ≡ 1 · 20 = 1

ϕ2 (1) = 1 · 2(−0−1) = 0. 5 =

1 2

2. n = 2, so M = ⌊ log2 (2)⌋ = 1 2 ≡ 1 · 21 + 0 · 20 = 1 0

ϕ2 (2) = 1 · 2(−1−1) + 0 · 2(−1+1−1) = 2−2 = 0. 25 =

1 4

3. n = 3, so M = ⌊ log2 (3)⌋ = 1 3 ≡ 1 · 21 + 1 · 20 = 1 1

ϕ2 (3) = 1 · 2(−1−1) + 1 · 2(−1+1−1) = 2−2 + 2−1 = 0. 75 =

3 4

4. n = 4, so M = ⌊ log2 (4)⌋ = 2 4 ≡ 1 · 22 + 0 · 21 + 0 · 20 = 1 0 0

ϕ2 (4) = 1 · 2(−1−1) + 0 · 2(−2+1−1) + 0 · 2(−2+2−1) = 2−3 = 0. 125 =

1 8

and so on. The same mechanism for the one-dimensional and radix-2 notation (R = 2) can also be described more graphically. 1. We consider the unit interval 

1 0 −−−−−−−−−−−−−−− 0 1



2. The unit interval can be split into two equal parts 

0 1 −−−−−−−a−−−−−−− 0 1

so the sequence can start at 



1 2

0 1 1 −−−−−−− −−−−−−− 0 2 1



3. Each sub-interval can be further split into two equal parts 

148

0 1 1 −−−b−−− −−−c−−− 0 2 1



PROBABILISTIC SCENARIO OPTIMISATION

so that the sequence becomes 12 , 14 , 34 

0 −−− 0  1 −−− 2

1 −−− 4 3 −−− 4



1 2  1 1

4. Each sub-interval can be further split into two equal parts 



1 3 1 −d− −f − 2 4 1   1 3 1 −e− −g− 2 4 1 so, starting from the lowest sub-interval, the sequence becomes 1 1 3 1 5 3 7 2, 4, 4, 8, 8, 8, 8   0 1 1 − − 0 8 4   1 5 3 − − 2 8 4   1 3 1 − − 8 8 2   3 7 1 − − 4 8 1 This pattern can be repeated until a sequence of the desired length is generated. The resulting low-discrepancy sequence is then used to sample from the ordered series of the portfolio allocations (lexicographical representation). Figure 8.2 represents a high-discrepancy case (Monte Carlo) performed by drawing 1,000 and 5,000 random coordinates, and compares it with the low-discrepancy outcomes of Halton (Figure 8.3) and Sobol (Figure 8.4). The Halton sequence generates a sequence of draws from a unit interval. Thus, its application to the universe of the admissible portfolios requires that the set ΦU,0,0 also complies with an explicit ordered series. In this book we argue for the non-binding adoption of lexicographical ordering to generate an ordered series of portfolios in an unambiguous canonical order that is similar, but not restricted, to an alphabetical representation (from which the name is taken). Similarly, we can generate the explicit list of the ordered asset allocations that make ΦU,0,0 by ordering the compositions in such a way that the order associated to each individual asset allocation in the portfolio is preserved throughout the sequence. 149

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 8.2 Monte Carlo case (b) 1.0

0.8

0.8 Random Y

Random Y

(a) 1.0

0.6 0.4

0.4 0.2

0.2 0

0.6

0

0.2

0.4 0.6 Random X

0.8

1.0

0 0

0.2

0.4 0.6 Random X

0.8

1.0

0.2 0.4 0.6 0.8 Prime number 2

1.0

(a) 1,000 draws. (b) 5,000 draws.

Figure 8.3 Halton case (b) 1.0

0.8

0.8 Prime number 3

Prime number 3

(a) 1.0

0.6 0.4 0.2 0

0.6 0.4 0.2

0

0.2 0.4 0.6 0.8 Prime number 2

1.0

0

0

(a) 1,000 draws. (b) 5,000 draws.

LEXICOGRAPHICAL REPRESENTATIONS Lexicographical ordering can be adopted to generate an ordered series of portfolios in an unambiguous canonical order that is similar, but not restricted, to an alphabetical representation. 150

PROBABILISTIC SCENARIO OPTIMISATION

Figure 8.4 Sobol case (b) 1.0

0.8

0.8 Prime number 3

Prime number 3

(a) 1.0

0.6 0.4 0.2

0.4 0.2

0 0

0.6

0.2 0.4 0.6 0.8 Prime number 2

1.0

0

0

0.2 0.4 0.6 0.8 Prime number 2

1.0

(a) 1,000 draws. (b) 5,000 draws.

For example, in the English dictionary the word “briefcase” appears before the word “briefing” because the letter “c” appears after the prefix “brief” but before the letter “i” in the alphabet. Since the first five letters (“brief”) are identical in the two words, the first letter that makes a difference is the sixth, and its evaluation drives the final ordering. Similarly, we can generate the explicit list of the ordered asset allocations in such a way that the order associated to each individual asset allocation in the portfolio is preserved throughout the sequence. Once the explicit list has been generated, the ordering can be normalised to the unit interval and a reduced set of portfolio allocations that are eligible for the PSO can be sampled by means of a low-discrepancy sequence. We assume that the universe U of the eligible investment assets is made of four securities, denoted A, B, C and D. The set of the admissible portfolio compositions is made of a number of possible portfolios that comply with a tick size equal to 25%, so that each asset in the portfolio can take a finite number of possible percentage allocations from the following set (0. 00 — 0. 25 — 0. 50 — 0. 75 — 1. 00)

151

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

We can map this set into a numerical ordering (0 — 1 — 2 — 3 — 4)

and assume that (min — max) allocation conditions are imposed to each asset, so that A → [0. 25, 0. 50] → [1, 2]

B → [0. 25, 0. 75] → [1, 3]

C → [0. 00, 0. 75] → [0, 2]

D → [0. 00, 0. 75] → [0, 2] We can now build the explicit list of the ordered asset allocations by starting to allocate the minimum amount allowed for each asset in the alphabetical order and increase the last elements of the alphabet until the portfolio allocation adds up to 100%. Thus, the first allocation will be (0. 25 — 0. 25 — 0. 00 — 0. 50)

and the second (0. 25 — 0. 25 — 0. 25 — 0. 25)

so on, as in Table 8.3. The explicit list can be conveniently represented in lexicographical terms by associating to each allocation its corresponding numerical order, as in Table 8.4. The order of the possible portfolio allocations is represented well by the numerical sequence where portfolio “4” comes after portfolio “3” because the associate lexicographical index “1201” is larger than “1120”. By normalising the ranks of this list to the unit interval, portfolios can be sampled from the list by mapping the normalised ranking to a low-discrepancy sequence, thus generating an equidistant approximation of the original universe of the potential asset allocations. ADEQUATE PORTFOLIOS: RISK ADEQUACY It is computationally advantageous to perform risk-adequacy checks only on the space ΨU,H,t of the potential sample allocations that are compliant with the allocation constraints, instead of the full space 152

PROBABILISTIC SCENARIO OPTIMISATION

Table 8.3 Generation of ordered asset allocations Allocation

A

B

C

D

Total

#1 #2 #3 #4 #5 #6 #7 #8 #9

0.25 0.25 0.25 0.25 0.25 0.25 0.50 0.50 0.50

0.25 0.25 0.25 0.50 0.50 0.75 0.25 0.25 0.50

0.00 0.25 0.50 0.00 0.25 0.00 0.00 0.25 0.00

0.50 0.25 0.00 0.25 0.00 0.00 0.25 0.00 0.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Table 8.4 Lexicographical mapping of ordered asset allocations Allocation

A

B

C

D

Total

#1 #2 #3 #4 #5 #6 #7 #8 #9

1 1 1 1 1 1 2 2 2

1 1 1 2 2 3 1 1 2

0 1 2 0 1 0 0 1 0

2 1 0 1 0 0 1 0 0

4 4 4 4 4 4 4 4 4

ΦU,H,t since the discarded portfolios are not acceptable anyway. ΘU,H,t is the resulting subset of ΨU,H,t containing the allocations that also

comply with the risk limit definition at any chosen reallocation time step. We can impose a risk limit as a hard line or as a boundary condition, so that the VaR of the simulated portfolio at preselected time steps is lower than a risk limit or it falls in a target bandwidth. For a given confidence level 1 − α, λ is a risk–return limit function where λL : {λt | t ∈ Γ } indicates the risk limit and λA : {λt | t ∈ Γ } indicates the ambition target return. In the case of a hard limit, the probabilistic risk limit function α states a constraint on ξU,S,t , which is the α quantile profile of the investor applicable to portfolio returns RU,H,t α ξU,S,t  λLt

for all t ∈ Γ

(8.12) 153

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

α This translates into the following statement: the left quantile ξU,S,t of the optimal portfolio at any selected time point t, with confidence interval 1−α, will be contained within the investor’s risk appetite λLt . The probabilistic target return limit function instead states a conα , which is the α quantile profile of the investor straint on ξU,S,t applicable to portfolio returns RU,H,t α ξU,S,t  λA t

for all t ∈ Γ

(8.13)

This translates into the following statement: the right quantile of the optimal portfolio at any selected time point t, with confidence interval 1 − α, will be greater than or equal to the investor’s return ambition λA t . Thus, by choosing a low α-level we can impose a probabilistic risk cap, whereas by choosing a medium to high α-level we can impose a probabilistic return floor. We can note that the definition of the risk-limit function also incorporates the form of a single point-in-time constraint by setting it arbitrarily negative for all other time steps belonging to the investment horizon except the last one. Similarly, we could decide to impose the risk-limit function on a number of reallocation steps smaller than that contained in the simulation framework. α ξU,S,t

OBJECTIVE FUNCTION: PROBABILITY MAXIMISATION The optimisation journey started with an initial space ΦU,0,0 of quasirandom potential allocations, reduced the data set to a space of ΨU,0,0 admissible compositions and further reduced the set to the riskadequate initial allocations ΘU,0,0 . Thus, the objective function can now be imposed on the risk–return properties of ΘU,H,t . The optimisation problem can be performed in multiple periods, where t ∈ Γ . Hence, we need to have a notion of preference that determines the level at which one target function at a particular point in time dominates another. This can be achieved by introducing a multi-period weighting scheme K, which is a vector of k ∈ K that allocates a positive weight at each t ∈ Γ . The weighting scheme is defined rather generally and need not necessarily integrate to 1, since it might incorporate a normalisation of the target function over time as well. Alternatively, we could estimate a more refined indicator of the multi-period probability by computing at final investment horizon the conditional probability of reaching such a final step given the probability measurement at all previous allocation steps. 154

PROBABILISTIC SCENARIO OPTIMISATION

Figure 8.5 Probabilistic scenario optimisation

Risk/return

Monte Carlo Ambition profile 1.0 (a) 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6

1Y

2Y

Risk profile Optimisation node

3Y

4Y

5Y

3Y

4Y

5Y

Risk/return

Time 1.0 0.8 (b) 0.6 0.4 0.2 0 –0.2 –0.4 –0.6

1Y

2Y Time

(a) Inadequate. (b) Adequate and optimal.

Objective function Maximise the probability of complying with a minimum, timedependent ambition or target return λA , subject to a multi-period weighting scheme K over a time horizon Γ and across all elements ΘU,H,t so that ∗ ΘU,S, Γ : max Θ

 t∈Γ



kt PU,S,t (R > λA t )

(8.14)

Clearly, the multi-period optimisation can be turned into a discrete or single-point-in-time optimisation by assigning full weight to a discrete set of points or a single point only. However, if the weight is not allocated to a single point only, the construction of the weighting scheme should reflect the nature of the respective variable, and its term structure, to allow for meaningful results. 155

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Objective function

Figure 8.6 Quasi-random method: 1M sampling 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

0

1

2

3

4

5

Order

6

7

8

9

10

(x105)

One of the advantages of PSO is that its findings can be easily plotted graphically to support the decision-making of long-term goal-based investing (as in Figure 8.5). The objective function can be plotted against its lexicographical ordering, as in Figures 8.6 and 8.7, in which the x-axis represents the lexicographically ordered adequate portfolios, while the y-axis indicates the associated probability of achieving or beating a target at final investment horizon. A PSO process has been run twice against the same universe of assets, 1,000 Monte Carlo scenario paths on 21 time steps, no allocation constraints, no short-selling allowed. The two processes differ in the application of the quasi-random method adopted for the sampling: 1M (Figure 8.6) and 100M (Figure 8.7) sampling. The graphs indicate clearly that a local optimiser would be inadequate, as the plot is not a smooth function and the solver would be dominated by the initial points. There seems to be no significant extra information about the intensity of the objective function across the two runs, which differ by two orders of magnitude. This seems to indicate that Halton sampling can provide a sufficiently adequate representation of all potential portfolios. Genetic algorithms could provide a further advance over quasirandom methods. By investigating the broader space of potential portfolios with a genetic mechanism, the optimisation algorithm could focus on the neighbourhood of the most relevant areas shown in the lexicographical ordering representation, where peaks tend to occur. 156

PROBABILISTIC SCENARIO OPTIMISATION

Objective function

Figure 8.7 Quasi-random method: 100M sampling 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

0

1

2

3

4 5 6 Order (x107)

7

8

9

10

Figure 8.8 Example of probability measure (multi-period)

Prob at inv horizon

1.0 0.8

Positive return Beating target

0.6 0.4 0.2 0

1Y

2Y

3Y Time

4Y

5Y

Total returns

Figure 8.9 Connecting past and future performance 1.2 1.0 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –3Y

Ex post Ex ante (scenarios) Ambition profile Risk appetite profile

Ex post –2Y

–1Y Today

Ex ante 1Y

2Y

3Y

4Y

5Y

Time

157

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

The probability measures, such as the probability of beating the investment goal or falling short of the risk appetite limit, can be plotted for the optimal portfolio (as in Figure 8.8). Portfolio potential performance over time can also be investigated to highlight whether the investment goals are still attainable, have been achieved or are challenged by adverse market movements (as in Figure 8.9). Portfolio managers can indicate minimum probability targets per time step and verify ex ante and ex post the compliance of both the strategic and the tactical asset allocations with the ambitions and risk appetite of final investors. This allows us to anticipate the needs for a proactive revision of the asset allocation, in case the investment has performed better than expected, as the market has turned in favour of the chosen strategy (indicating the possibility of cashing in and entering into a new portfolio to enhance investment returns) or in case the investment has underperformed, as market has turned against the chosen strategy or might not provide enough drift or volatility to achieve the stated ambition within the investment horizon (indicating the need to revise the asset allocation and optimise the timing of such a decision). CONCLUSIONS Probabilistic scenario optimisation is an exhaustive enumeration technique designed to provide portfolio managers with a riskbased framework that supports goal-based and long-term portfolio choice. Low-discrepancy sequences and lexicographical representations have been introduced to ease the computational burden. Genetic algorithms can also be modelled to further improve the speed and accuracy of the process. Some case studies are presented in Chapters 9 and 10, which provide empirical evidence for the functioning of PSO and compare the findings with the Markowitz and the Black–Litterman approaches.

REFERENCES

Barberis, N., 2000, “Investing in the Long Run when Returns Are Predictable”, Journal of Finance 55(1), pp. 225–64. Campbell, J. Y., and L. M. Viceira, 2001, Strategic Asset Allocation: Portfolio Choice for LongTerm Investors (Oxford University Press).

158

PROBABILISTIC SCENARIO OPTIMISATION

Dembo, R., 1991, “Scenario Optimisation”, Annals of Operations Research 30(1), pp. 63–80. Dembo, R., and A. Freeman, 1998, Seeing Tomorrow: Rewriting the Rules of Risk (Chichester: John Wiley & Sons). Faure, H., 1982, “Discrépance de Suites Associées à un Système de Numération (en Dimension s)”, Acta Arithmetica 41, pp. 337–51. Halton, J. H., 1960, “On the Efficiency of Certain Quasi-Random Sequences of Points in Evaluating Multi-Dimensional Integrals”, Numerische Mathematik 2(1), pp. 84–90. Niederreiter, H., 1987, Random Number Generation and Quasi-Monte Carlo Methods, CBMSNSF Regional Conference Series in Applied Mathematics (Philadelphia, PA: SIAM). Sobol, I., 1967, “On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals”, USSR Computational Mathematics and Mathematical Physics 7(4), pp. 86–112.

159

Part III

Portfolio Optimisation Case Studies

9

Case Studies: Mean–Variance and Black–Litterman What can be asserted without evidence can also be dismissed without evidence. Christopher Hitchens (1949–2011)

We present a numerical application of mean–variance and Black– Litterman, conduct a review of the data set (market indexes, statistics) and compare the mean–variance optimisation with Black– Litterman optimisation. INTRODUCTION In this chapter we present two exercises corresponding to the mean– variance and Black–Litterman portfolio optimisations, which start from the same market data inputs: the statistical properties of a set of 31 market indexes that correspond to broader asset classes. These market indexes have been chosen to represent 42 distinct investment opportunities (funds, Treasury notes, financial and corporate bonds), which are direct inputs to the probabilistic scenario optimisation that is covered in the next chapter, so that we can make an intuitive comparison between different methods. The Black–Litterman approach requires us to indicate the market portfolio as the initial asset allocation. Thus, a synthetic non-traded index (indicated as the Global Market Index) has been engineered from the chosen market indexes. All case studies comply with the point of view of a US investor, holding financial assets in a US dollar denominated portfolio. January 2014 is the reference date of the calculations; thus, all time series correspond to a historical five-year period starting in January 2009. DATA SET: MARKET INDEXES The set of benchmarks corresponding to the investment universe is made up of 31 market indexes, which represent international equity 163

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 9.1 Five years of daily returns R, estimated from the point of view of a US investor Market index

SD R

Skewness Kurtosis 0.3554 −0.6231 1.5851 0.0731 −0.2940 −0.2923 −0.4842 0.1936 0.2626

Risk-free rate

0.0010 0.0025 0.0040 0.0008 0.0020 0.0033 0.0007 0.0033 0.0058 0.0075 0.0079 0.0081 0.0073 0.0074 0.0077 0.0076 0.0080 0.0089 0.0051 0.0053 0.0058 0.0199 0.0231 0.0201 0.0165 0.0225 0.0153 0.0278 0.0192 0.0202 0.0142 0.0128

Global Market Index

0.0061

USD fin 1–3Y∗ USD fin 5–7Y∗ USD fin 10–15Y∗ USD non-fin 1–3Y∗ USD non-fin 5–7Y∗ USD non-fin 10–15Y∗ USD Treas 1–3Y∗ USD Treas 5–7Y∗ USD Treas 10–15Y∗ EUR fin 1–3Y∗ EUR fin 5–7Y∗ EUR fin 10Y+∗ EUR non-fin 1–3Y∗ EUR non-fin 5–7Y∗ EUR non-fin 10Y+∗ EUR Treas 1–3Y∗ EUR Treas 5–7Y∗ EUR Treas 10Y+∗ USD GEMX 1–3Y∗ USD GEMX 5–7Y∗ USD GEMX 10–15Y∗ USD Germany∗∗ USD Brazil∗∗ USD France∗∗ USD UK∗∗ USD Italy∗∗ USD Japan∗∗ USD Russia∗∗ USD China∗∗ USD India∗∗ USD US∗∗

0.2072 0.1702 0.7770 0.3115 0.5243 0.4618 0.2090 0.3357 0.4198 −0.2225 −0.2513 −0.4412 0.1461 0.0974 0.1859 −0.1520 0.0405 −0.2089 −1.0007 −0.0707 0.3895 −0.3259 −2.2406 −0.0204

Min R

Max R

9.8424 −0.0048 6.8528 −0.0130 27.1688 −0.0229 11.2808 −0.0051 5.9766 −0.0097 6.5529 −0.0180 13.2767 −0.0065 6.2103 −0.0134 5.8072 −0.0203 6.6677 −0.0415 6.7184 −0.0388 12.9302 −0.0367 7.5377 −0.0411 9.6458 −0.0384 9.2851 −0.0358 6.7799 −0.0427 7.5647 −0.0405 8.9330 −0.0389 4.6585 −0.0207 4.8220 −0.0233 4.8783 −0.0328 7.1846 −0.0878 10.5640 −0.1544 6.5972 −0.0857 9.4228 −0.1008 5.7609 −0.0860 7.2356 −0.0881 19.1119 −0.2786 9.6531 −0.1310 13.8214 −0.1315 10.1745 −0.0940 42.0803 −0.1708

0.0066 0.0111 0.0506 0.0053 0.0110 0.0195 0.0043 0.0252 0.0392 0.0518 0.0545 0.0828 0.0552 0.0645 0.0675 0.0536 0.0612 0.0768 0.0221 0.0250 0.0240 0.1310 0.1625 0.1209 0.1109 0.1189 0.0764 0.1981 0.1406 0.1996 0.1006 0.1363

6.7242 −0.0335 0.0347

Daily standard deviation (SD R), skewness, kurtosis, minimum daily return (min R) and maximum daily return (max R). ∗ iBoxx, ∗∗ MSCI.

and fixed-income markets with a variety of target durations and credit risk exposures. 164

CASE STUDIES: MEAN–VARIANCE AND BLACK–LITTERMAN

Figure 9.1 Correlation map iBoxx USD financials 1–3Y iBoxx USD financials 5–7Y iBoxx USD financials 10–15Y iBoxx USD non-financials 1–3Y iBoxx USD non-financials 5–7Y iBoxx USD non-financials 10–15Y iBoxx USD Treasuries 1–3Y iBoxx USD Treasuries 5–7Y iBoxx USD Treasuries 10–15Y iBoxx EUR financials 1–3Y iBoxx EUR financials 5–7Y iBoxx EUR financials 10Y+ iBoxx EUR non-financials 1–3Y iBoxx EUR non-financials 5–7Y iBoxx EUR non-financials 10Y+ iBoxx EUR sovereigns 1–3Y iBoxx EUR sovereigns 5–7Y iBoxx EUR sovereigns 10Y+ iBoxx USD GEMX 1–3Y iBoxx USD GEMX 5–7Y iBoxx USD GEMX 10–15Y MSCI Germany MSCI Brazil MSCI France MSCI UK MSCI Italy MSCI Japan MSCI Russia MSCI US MSCI China MSCI India

–1.0

–0.5

0

0.5

1.0

• We chose Markit iBoxx indexes to represent fixed-income

investments in US, European and growth credit markets across the following asset classes. Financial foreign: Financial local: Non-financial foreign: Non-financial local: Treasury foreign: Treasury local: Credit growth markets:

EUR USD EUR USD EUR USD USD

1–3Y, 5–7Y, 10Y+ 1–3Y, 5–7Y, 10–15Y 1–3Y, 5–7Y, 10Y+ 1–3Y, 5–7Y, 10–15Y 1–3Y, 5–7Y, 10Y+ 1–3Y, 5–7Y, 10–15Y 1–3Y, 5–7Y, 10–15Y

• MSCI equity indexes represent potential investments in inter-

national equity funds across the following asset classes. Equity foreign:

Brazil, France, Germany, UK, Italy, Japan, Russia, China, India

Equity local:

US

• The USD Libor rate is assumed to model the dynamics of the

risk-free rate.

165

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 9.2 Historical dynamics: examples (rebased 2009) 300 200

iBoxx USD financials 5–7Y iBoxx EUR financials 5–7Y

iBoxx USD Non-financials 5–7Y iBoxx EUR Non-financials 5–7Y

100 0 2009 300 200

2010

2011

2012

2013

2014

2012

2013

2014

2013

2014

iBoxx USD treasuries 1–3Y iBoxx EUR treasuries 1–3Y iBoxx USD treasuries 10–15Y iBoxx EUR treasuries 10Y+

100 0 2009 300

2010

MSCI Germany MSCI Italy

2011

MSCI US MSCI China

200 100 0 2009

2010

2011

2012

Table 9.1 contains statistics (daily standard deviation (SD R), skewness, kurtosis, minimum daily return (min R) and maximum daily return (max R)) on five years of daily returns R, estimated from the point of view of a US investor (currency risk included). The correlation matrix of the indexes is represented graphically by the intensity map in Figure 9.1. We note that the diagonal is equal to 1 and observe that the five-year historical correlations are higher among the fixed-income indexes belonging to the euro area, and lower for US dollar fixed-income indexes: this reflects the contagion risk in the European Union (EU) financial markets during the 2007–12 global financial crisis. The correlations between equity and fixed-income markets seem to be slightly negative in the US, and slightly positive in EU markets. Equity markets (all indexes are US dollar denominated) seem to have signalled positive correlation 166

CASE STUDIES: MEAN–VARIANCE AND BLACK–LITTERMAN

Figure 9.3 Mean–variance efficient frontier (unconstrained)

Expected returns (annual)

0.16 12 2

0.12

3 11

25 14

5

0.08

0.04

28

30 24

23

27

Market

9 18 21 10 17 1 8 Optimal 13 4 20 16

0.06

22

15

6

0.10

0.02

29

31

0.14

19 26

7

0 Risk-free 0 0.1

0.2 0.3 0.4 0.5 Standard deviations (annual)

0.6

0.7

0.6

0.7

Numbers refer to indexes listed in Table 9.2.

Figure 9.4 Mean–variance efficient frontier (constrained)

Expected returns (annual)

0.16

31

0.14 2

0.12

3 6

0.10

1

21 8

Optimal

0.04

4

22

25

28

30 24

Market 9

0.06

29

14

5

0.08

0.02

12 15 11

23

27

18 10 17 13

20

16

19 26

7

0 Risk-free 0 0.1

0.2 0.3 0.4 0.5 Standard deviations (annual)

Numbers refer to indexes listed in Table 9.2.

overall: this reflects the trend in global equity markets as influenced by the quantitative easing, protracted low interest rates and hikes in commodity prices that have improved the performance in emerging markets. Figure 9.2 represents the time series of some of these market indexes, as an example, rebased to January 2009 to facilitate comparison. 167

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 9.2 Mean–variance optimisation results

Market index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Historical  R SD

USD fin 1–3Y∗ 5.99 USD fin 5–7Y∗ 12.35 USD fin 10–15Y∗ 12.25 4.42 USD non-fin 1–3Y∗ USD non-fin 5–7Y∗ 9.30 USD non-fin 10–15Y∗10.69 USD Treas 1–3Y∗ 1.46 5.05 USD Treas 5–7Y∗ USD Treas 10–15Y∗ 7.19 6.20 EUR fin 1–3Y∗ EUR fin 5–7Y∗ 11.34 13.60 EUR fin 10Y+∗ EUR non-fin 1–3Y∗ 5.50 EUR non-fin 5–7Y∗ 10.26 EUR non-fin 10Y+∗ 12.13 EUR sov 1–3Y∗ 3.48 6.21 EUR sov 5–7Y∗ EUR sov 10+∗ 7.15 3.54 USD GEMX 1–3Y∗ USD GEMX 5–7Y∗ 4.65 USD GEMX 10–15Y∗ 6.16 USD Germany∗∗ 12.93 9.66 USD Brazil∗∗ USD France∗∗ 10.13 USD UK∗∗ 11.80 1.87 USD Italy∗∗ USD Japan∗∗ 8.27 10.60 USD Russia∗∗ USD China∗∗ 15.86 11.15 USD India∗∗ USD US∗∗ 14.77 Portfolio return (annual) Portfolio SD (annual)

Unconstr optim  Wt R (port)

1.51 41.28 3.97 6.41 1.23 42.92 3.24 5.19 1.03 8.57 5.30 9.21 11.90 12.54 12.91 11.60 11.71 12.27 12.10 12.66 14.14 8.07 8.47 1.46 9.27 4.84 31.61 36.65 31.86 26.27 35.65 24.24 44.19 30.45 32.07 22.49 0.93

Constr optim  Wt R (port)

2.47

30.00 4.91

1.80 0.61

1.90

30.00 3.55

1.33 0.33

0.12

23.14

0.34

0.07 0.30

1.80 5.31

0.08 0.33

0.14

1.30

0.19

5.00

5.00

1.22

1.26

All values given in percent. Wt denotes weights. SD denotes standard deviation. ∗ iBoxx, ∗∗ MSCI.

168

CASE STUDIES: MEAN–VARIANCE AND BLACK–LITTERMAN

Table 9.3 Mean–variance optimisation results: aggregation by asset class Asset class

Unconstrained weights

Constrained weights

Financial foreign Financial local Non-financial foreign Non-financial local Sovereign foreign Sovereign local Equity foreign Equity local Credit growth markets

— 41.28 — 42.92 — 8.57 — 0.93 6.30

— 34.91 — 33.55 — 23.14 — 1.30 7.11

100.00

100.00

Total All values given in percent.

MEAN–VARIANCE PORTFOLIO The main inputs of the mean–variance optimisation are the expected returns, the standard deviations and the correlations of the market indexes, estimated on the five-year time series of daily returns: the standard deviations have been annualised by the root-of-time rule of thumb; the expected returns have been annualised as in Meucci (2001) and exogenous risk premiums have not been estimated. We ran two optimisations, both characterised by the objective function to minimise portfolio volatility and yield a target expected return equal to 5.00% annual. • Unconstrained optimisation: no constraints were imposed on

invested amounts, yet short selling was not allowed.

• Constrained optimisation: short selling was not allowed and

a 30% concentration constraint was imposed on invested amounts.

We summarise the optimisation results in Table 9.2. The unconstrained optimal portfolio shows a lower standard deviation than the constrained case, as expected. However, the overall standard deviation seems fairly low compared with the expected return measure. This is due to the positive drift that most of the indexes experienced 169

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 9.4 Black–Litterman optimisation results

Market index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Unconstr: no views Constr: views (post portfolio) (post portfolio)   Beta R (CAPM) Wt (%) R (post) Wt (%)

USD fin 1–3Y∗ 0.01 USD fin 5–7Y∗ 0.07 0.04 USD fin 10–15Y∗ USD non-fin 1–3Y∗ 0.01 0.02 USD non-fin 5–7Y∗ USD non-fin 10–15Y∗ 0.03 USD Treas 1-3Y∗ 0.00 −0.06 USD Treas 5–7Y∗ USD Treas 10–15Y∗ −0.13 1.02 EUR fin 1–3Y∗ EUR fin 5–7Y∗ 1.07 1.00 EUR fin 10Y+∗ EUR non-fin 1–3Y∗ 0.98 0.97 EUR non-fin 5–7Y∗ EUR non-fin 10Y+∗ 0.89 EUR sov 1–3Y∗ 1.03 1.05 EUR sov 5–7Y∗ EUR sov 10+∗ 1.03 0.02 USD GEMX 1–3Y∗ USD GEMX 5–7Y∗ 0.11 USD GEMX 10–15Y∗ 0.16 MSCI Germany USD∗∗ 2.87 2.73 MSCI Brazil USD∗∗ MSCI France USD∗∗ 2.92 MSCI UK USD∗∗ 2.27 3.14 MSCI Italy USD∗∗ MSCI Japan USD∗∗ 0.94 3.38 MSCI Russia USD∗∗ MSCI China USD∗∗ 1.43 1.74 MSCI India USD∗∗ MSCI US∗∗ 1.98 Market Index Portfolio return (annual) Portfolio SD (annual)

0.34 0.86 0.61 0.32 0.39 0.52 0.20 −0.27 −0.86 8.84 9.19 8.60 8.49 8.38 7.70 8.86 9.02 8.91 0.38 1.14 1.60 24.31 23.14 24.76 19.25 26.58 8.16 28.64 14.82 16.87 12.22 8.63

0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.02 0.09 1.12 13.11 0.07 0.23 16.63 0.08 0.19 7.36 0.01 0.01 0.01 8.04 7.24 13.10 0.06 10.23 4.25 6.88 4.72 6.44 0.02

0.36 0.79 0.94 0.34 0.51 0.67 0.27 0.25 0.13 4.69 4.89 4.56 4.51 4.48 4.16 4.73 4.89 4.90 0.33 0.71 0.98 13.16 11.35 13.63 10.43 14.81 4.77 13.82 6.63 7.78 6.44

0.04 1.81 23.68 0.03 0.10 3.99 0.03 0.09 12.18 0.11 0.28 0.07 0.10 2.58 10.01 0.12 2.18 4.30 0.02 0.02 0.03 5.72 2.20 9.58 3.86 7.50 6.26 0.16 0.04 1.73 1.16

16.93

5.52

19.47

11.30

All values given in percent. Wt denotes weights. SD denotes standard deviation. ∗ iBoxx, ∗∗ MSCI. Boldface denotes market indexes whose allocation is directly affected by the imposition of the investor’s views.

170

CASE STUDIES: MEAN–VARIANCE AND BLACK–LITTERMAN

Table 9.5 Black–Litterman optimisation results by aggregating per asset class Asset class Financial foreign Financial local Non-financial foreign Non-financial local Sovereign foreign Sovereign local Equity foreign Equity local Credit growth markets Total

Weight (no views)

Weight (views)

14.32 0.04 16.93 0.03 7.63 0.04 60.96 0.02 0.03

0.46 23.53 12.69 4.12 6.60 12.30 37.05 1.16 0.07

100.00

100.00

All values given in percent.

Table 9.6 Mean–variance versus Black–Litterman optimisation

Asset class Financial foreign Financial local Non-financial foreign Non-financial local Sovereign foreign Sovereign local Equity foreign Equity local Credit growth markets Total

Mean–variance  Unconstr Constr

Black–Litterman  No views∗ Views

— 41.28 — 42.92 — 8.57 — 0.93 6.30

— 34.91 — 33.55 — 23.14 — 1.30 7.11

14.32 0.04 16.93 0.03 7.63 0.04 60.96 0.02 0.03

0.46 23.53 12.69 4.12 6.60 12.30 37.05 1.16 0.07

100.00

100.00

100.00

100.00

All values given in percent. ∗ CAPM market portfolio.

in the five years preceding the analysis: equity markets reached historical heights, fixed-income markets were affected by the quantitative easing and the euro maintained sustained appreciation against the US dollar. The resulting optimal portfolio, indicating minimumvariance asset allocation to yield a 5.00% expected return target, was dominated by domestic fixed-income investments. The expected 171

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

return of the indicated fixed-income indexes was close to the target value, while the historical volatility tended to be lower than for equities and foreign investments. Table 9.3 shows the optimisation results on aggregating by asset class. Figures 9.3 and 9.4 show the impact of the allocation constraints on the shape of the efficient frontier. BLACK–LITTERMAN PORTFOLIO The Black–Litterman optimisation starts from the estimate of CAPM betas, excess returns and standard deviations on the five-year daily returns of the market indexes. The US investor was allowed to express market views, which have been applied jointly with the optimisation exercise. • Absolute view: the investor believed that the BRIC (Brazil,

Russia, India, China) economies would slow down in the upcoming period, leading to an expected return equal to 8.00% annual, conditional on a statistical confidence proportional to the variances.

• Relative view: the investor believed that European finan-

cials would outperform US financials in fixed-income markets by no more than a 2.00% annual return (accounting for foreign exchange risk), conditional on a statistical confidence proportional to the variances.

We performed two optimisations. • Unconstrained optimisation: no constraints have been im-

posed on invested amounts, yet short selling has not been allowed and no views have been indicated to alter CAPM prior estimates.

• Constrained optimisation: short selling has not been allowed,

a 30% constraint has been imposed on investment concentration, the indicated views have been modelled.

The optimal portfolio is indicated by the asset allocation that minimised portfolio volatility with the risk-aversion parameter λ = 4.00. The range of variation of the λ parameter was estimated on the time series of the risk-aversion coefficients. 172

CASE STUDIES: MEAN–VARIANCE AND BLACK–LITTERMAN

The optimisation results are summarised in Table 9.4. The Black– Litterman portfolio appeared less concentrated than the mean– variance portfolio indicated in the previous exercise, as theory suggests, while the allocation constraints did not play a significant role. The introduction of the views reduced the weights on the BRIC countries, while the euro financial exposures also lowered. Table 9.5 shows the optimisation results by aggregating per asset class. CONCLUSIONS We performed the mean–variance and Black–Litterman optimisations on a set of market indexes that represent international investment opportunities in equity and fixed-income markets. As theory suggests, the mean–variance optimisation led to a more concentrated portfolio than Black–Litterman (Table 9.6), in which the views played a relevant role in defining the optimal asset allocation.

REFERENCES

Meucci, A., 2001, “Common Pitfalls in Mean–Variance Asset Allocation”, Technical Article, URL: http://www.willmot.com.

173

10

Case Studies: Probabilistic Scenario Optimisation However beautiful the strategy, you should occasionally look at the results. Winston Churchill (1874–1965)

We demonstrate numerical application of PSO, review the data set (securities, risk factors, statistics) and compare multi-period optimisation with risk-averse and risk-mitigating profiles and multi-period and single-period optimisation with a risk-tolerant profile. INTRODUCTION In this chapter we present a set of optimal asset allocations identified by means of PSO. The set of investment opportunities is made up of 42 distinct securities (funds, Treasury notes, financial and corporate bonds) that cover the same risk factors represented by the market indexes introduced in the previous chapter, dedicated to the case studies of mean–variance and Black–Litterman optimisations. We generated Monte Carlo multi-variate stochastic scenarios over time to shock the market risk factors and derive the potential return distributions of these real products. The investment horizon We set the investment horizon to five years, as a subset of the total length of the 10Y multi-period Monte Carlo simulations performed with an uneven time-step discretisation to show higher frequency in the short-term than in the long-term, as can be seen in Figure 10.1: one day, every month up to one year and every six months up to ten years, for a total of 31 time steps. One thousand Monte Carlo scenarios were generated for each time step to estimate a total of 31,000 revaluations for each of the 42 securities. January 2014 is the reference date of the calculations; thus, all the market risk factors’ time series correspond to a historical five-year period that starts in January 2009. 175

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 10.1 Investment horizon

0 6M1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y Investment horizon (Γ ) over the simutation time frame

Figure 10.2 Monte Carlo simulations for a selection of securities 2.0 1.5

2.0

(a)

1.5

1.0

1.0

0.5

0.5

0

0

–0.5

–0.5

–1.0 2.0 1.5

2Y

4Y

6Y

8Y

10Y

–1.0 2.0

(c)

1.5

1.0

1.0

0.5

0.5

0

0

–0.5

–0.5

–1.0 2.0 1.5

2Y

4Y

6Y

8Y

10Y

4Y

6Y

8Y

10Y

2Y

4Y

6Y

8Y

10Y

2Y

4Y

6Y

8Y

10Y

(f)

4.0

1.0 0.5

2.0

0 –0.5 –1.0

2Y (d)

–1.0 6.0

(e)

(b)

2Y

4Y

6Y

8Y

10Y

–0 –1.0

(a) Bond Gov US 2020 8.75% USD. (b) Bond Gov US 2028 5.255% USD. (c) Bond Gov Spain 2020 43.00% EUR. (d) Bond Gov Italy 2027 6.50% EUR. (e) Bond Ford 2028 6.625% USD. (f) Fund Equity US USD.

The risk–return profiles All our exercises comply with the point of view of US investors, holding financial assets in US dollar denominated portfolios, and refer to three cases: a risk-averse profile, a risk-mitigating profile and a risk-tolerant profile. 176

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

• Risk-averse profile: the target return is set similar to the risk-

free rate of the US economy, modelled with an ambition line of around 1% year on year. The risk adequacy line is set very tight (ie, reset to zero or close to zero) to enforce 100% capital insurance starting from the third year.

• Risk-mitigating profile: the target return is set slightly higher

than US dollar Libor, approximated with an ambition line of around 2% year on year. As interest rates were particularly low at time of analysis, the risk adequacy line was modelled to accommodate some risk taking in the short term, but to enforce tight risk mitigation in the medium/long run (capital protection).

• Risk-tolerant profile: the target return is set to yield at least

6% year on year. The risk adequacy profile is set fairly loosely to allow for risk taking throughout the investment horizon.

We performed multi-period optimisation for all three cases, while we also included a single-period optimisation for the risk-tolerant profile, to examine model sensitivity to the time discretisation. • Single-time optimisation: we chose a single horizon equal

to five years, and did not model any intermediate riskverification time step.

• Multi-period optimisation: we set the liquidity step at the first

year, with every year until the fifth qualifying as an allocation step. Therefore, we verified the adequacy constraints at the end of every year. An equally weighted schema was applied to synthesise all probabilities into a single indicator, which is the maximisation variable of the PSO objective function.

The objective function The objective function was formulated to find out which portfolio is characterised by the highest weighed probability of yielding a total return higher than the ambition line, while complying with the allocation constraints and the risk limit. The risk measure used to verify the compliance with the risk limit was defined as the one-tail VaR95 . The optimisation exercise was imposed on a set of 10 million admissible asset allocations by applying a recursive Halton sequence technique, which allowed us to sample 10 million equidistant admissible portfolios out of the larger set potentially available. 177

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All optimisation exercises took less than ten minutes to perform, on a machine hosting a 2 GHz eight-core processor. The CIO views (stress tests) The resulting PSO portfolios, which correspond to the different risk– return profiles, were simulated over time by performing two stress tests that correspond to hypothetical Chief Investment Officer (CIO) views about the dynamics of financial market variables (ie, the general level of interest rates and the equity markets). Two alternative views (ie, H1 and H2 ) have been created in order to fully evaluate all portfolio holdings conditional on the corresponding scenario path. • View H 1 : a period of booming stock markets, modelled as a

sustained 60% growth of global equity evaluations over the first two years, is followed by a sharp decline, so all potential gains are wiped out by the end of the third year, leading to a period of market stagnation and then moderate growth until the fifth year. We assumed that interest rates rose over the economic cycle by more than 200bp on all maturities, and reverted to pre-boom levels during the last two years of the investment horizon.

• View H 2 : a period of two years of economic contraction, mod-

elled as a global decline in equity markets evaluations by around 50%, is followed by a period of protracted expansion until the fifth year, reaching a 20% increase in market indexes compared with the present. We assumed that the interest rates decreased during the economic contraction, reaching nearzero rates on short- and mid-term tenors, and rising again as the economic cycle improved after the third year of the stress test simulation.

DATA SET: UNIVERSE OF PRODUCTS The universe of financial securities is made up of 42 distinct products that correspond to a set of eight asset classes. 1. Financial foreign: • Bond Deutsche Bank 2026 0.00% EUR; • Bond Santander 2028 5.78% EUR; • Bond Intesa Sanpaolo 2015 3.10% EUR;

178

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

• Bond Deutsche Bank 2015 3.00% EUR; • Bond ING 2018 3.38% EUR; • Bond Nordea 2020 4.00% EUR.

2. Financial local: • Bond Goldman Sachs 2028 5.00% USD; • Bond Morgan Stanley 2015 3.45% USD; • Bond Goldman Sachs 2015 3.00% USD; • Bond Goldman Sachs 2019 4.25% USD.

3. Non-financial foreign: • Bond France Telecom 2015 3.625% EUR; • Bond Volkswagen 2015 3.50% EUR; • Bond RWE 2019 6.625% EUR; • Bond KPN 2020 3.75% EUR; • Bond ENEL 2027 5.625% EUR; • Bond EDF 2025 4.00% EUR.

4. Non-financial local: • Bond Ford 2028 5.00% USD; • Bond General Electric 2019 6.00% USD; • Bond General Electric 2016 5.375% USD; • Bond Ford 2014 8.70% USD.

5. Sovereign foreign: • Bond Gov Austria 2026 4.85% EUR; • Bond Gov Italy 2027 6.50% EUR; • Bond Gov Germany 2015 3.75% EUR; • Bond Gov France 2016 3.25% EUR; • Bond Gov Netherlands 2019 4.00% EUR; • Bond Gov Spain 2020 4.00% EUR.

6. Sovereign local: • Bond Gov US 2015 11.25% USD;

179

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• Bond Gov US 2016 9.25% USD; • Bond Gov US 2018 9.00% USD; • Bond Gov US 2020 8.75% USD; • Bond Gov US 2024 7.50% USD; • Bond Gov US 2028 5.25% USD.

7. Equity foreign: • Fund Equity Italy USD; • Fund Equity Russia USD; • Fund Equity Japan USD; • Fund Equity Germany USD; • Fund Equity Brazil USD; • Fund Equity France USD; • Fund Equity UK USD; • Fund Equity China USD; • Fund Equity India USD.

8. Equity local: • Fund equity US USD.

We simulated the distributions of the potential total returns, conditional on 1,000 Monte Carlo scenario paths and 31 uneven allocation time steps between the start date and five years into the future (one day, every month up to one year, every six months up to ten years). The stochastic simulation conformed with a mixture of Hull–White one-factor and geometric Brownian motion models (as described in Chapter 2). Securities with a maturity of less than ten years were reinvested into a virtual security with the same credit and interest rate risk, reissued at par conditional on the level of the risk factors at the relevant time step and scenario realisation. Simulated coupons were collected and kept in a non-interest-bearing account, and thus reinvested with the notional (as described in Chapter 4). Figure 10.2 shows the Monte Carlo simulation of some of the securities in their original currency (no foreign exchange risk). 180

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

Figure 10.3 Time series examples for 5Y term

Basis points

1,000 800

(a)

CS Deutsche Bank CS Santander

600 400 200

0 2009 1,000 800

Basis points

CS Goldman Sachs CS Banca Intesa Sanpaolo

2010 (b)

2011

CS General Electric CS Volkswagen

2012

2013

2014

CS France Telecom CS KPN

600 400 200

Basis points

0 2009 1,000 (c) 800 600

2010

2011

TRS USA TRS Austria TRS Germany

2012

2013

2014

2012

2013

2014

TRS France TRS Italy

400 200 0 2009

2010

2011

(a) Financial. (b) Corporate. (c) Government.

Figure 10.4 Correlation matrix for the fixed-income risk factors IBK USD 5Y IBK EUR 5Y TRS Austria 5Y TRS France 5Y TRS Germany 5Y TRS Netherlands 5Y TRS Italy 5Y TRS Spain 5Y CS Goldman Sachs 5Y CS Morgan Stanley 5Y CS Ford 5Y CS General Electric 5Y CS Deutsche Bank 5Y CS ING 5Y CS Nordea Bank 5Y CS Intesa Sanpaolo 5Y CS Santander 5Y CS EDF 5Y CS ENEL 5Y CS France Telecom 5Y CS KPN 5Y CS RWE 5Y CS Volkswagen 5Y –1.0

–0.5

0

0.5

1.0

181

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 10.1 Statistics for 5Y term: full revaluation Risk factor CS Goldman Sachs CS Morgan Stanley CS Banca Intesa SP CS Deutsche Bank CS Nordea Bank CS ING CS Santander CS General Electric CS Ford CS Volkswagen CS France Telecom CS RWE CS KPN CS EDF CS ENEL IBK USD IBK EUR TRS Austria TRS France TRS Netherlands TRS Italy TRS Germany TRS Spain Fund USD Germany Fund USD Brazil Fund USD France Fund USD UK Fund USD Italy Fund USD Japan Fund USD Russia Fund USD China Fund USD India Fund USD US

SD R

Skewness Kurtosis Min R

Max R

0.0424 0.0388 0.0413 0.0382 0.0234 0.0321 0.0409 0.0525 0.0341 0.0293 0.0252 0.0267 0.0267 0.0262 0.0402 0.0307 0.0242 0.0356 0.0294 0.0334 0.0235 0.0494 0.0241 0.0199 0.0231 0.0201 0.0165 0.0225 0.0153 0.0278 0.0192 0.0202 0.0142

0.6290 0.7770 −0.0995 −0.1641 0.2005 −0.1530 −0.7107 0.1684 −0.4049 0.0105 −0.2504 0.6344 −0.1516 0.9459 0.1228 0.2871 0.5702 0.2200 0.3416 0.3838 −0.2920 0.1546 −1.1574 0.1461 0.0974 0.1859 −0.1520 0.0405 −0.2089 −1.0007 −0.0707 0.3895 −0.3259

9.6155 −0.2005 10.4337 −0.1756 7.2396 −0.2979 7.6361 −0.2701 7.5977 −0.1393 6.0718 −0.1603 14.3183 −0.4179 13.7430 −0.3807 15.5652 −0.2915 9.3930 −0.2278 10.9154 −0.2108 14.8082 −0.1695 12.7937 −0.2252 11.8968 −0.1204 13.9004 −0.3508 6.5120 −0.1611 7.1859 −0.1296 6.1116 −0.1904 7.0493 −0.1337 6.8207 −0.1495 10.7424 −0.1780 6.9244 −0.2454 14.4786 −0.2371 7.1846 −0.0878 10.5640 −0.1544 6.5972 −0.0857 9.4228 −0.1008 5.7609 −0.0860 7.2356 −0.0881 19.1119 −0.2786 9.6531 −0.1310 13.8214 −0.1315 10.1745 −0.0940

0.3149 0.3212 0.2115 0.1944 0.1099 0.1714 0.1896 0.3540 0.2699 0.1470 0.1323 0.2326 0.1824 0.2031 0.3308 0.2235 0.1676 0.1964 0.1746 0.1920 0.1466 0.2800 0.1200 0.1310 0.1625 0.1209 0.1109 0.1189 0.0764 0.1981 0.1406 0.1996 0.1006

SD denotes standard deviation.

DATA SET: MARKET RISK FACTORS The full revaluation of the set of securities required us to generate Monte Carlo scenarios on a variety of risk factors: market indexes, the EUR/USD exchange rate, the EUR and USD term structures bootstrapped from the overnight indexed swaps, single-name credit 182

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

curves for government and credit issuers. Time series of all risk factors were collected and we computed the daily returns over a fiveyear period. Table 10.1 reviews some statistics calculated on daily returns of the market indexes representing the equity funds, tracking regional market indexes, and the 5Y nodes of interest rate and spread curves, as an example. Figure 10.3 shows an example of some of these time series. Figure 10.4 represents the correlation matrix for the 5Y nodes of the fixed-income risk factors. The historical correlation between Southern European Treasury notes, as well as between highly rated European Treasuries, seemed fairly positive, but the correlation between the two sets of Treasuries became negative: this is evidence of the fly-to-quality that characterised the eurozone during the 2007–12 global financial crisis. RISK-AVERSE PROFILE: MULTIPLE PERIOD The investment goal was set to yield at least the risk-free rate of the US economy, modelled with an ambition line of around 1% year on year. The risk adequacy line was set very conservatively to indicate the mitigation of capital risk starting from the third year. We modelled the following constraints. • Short selling: minimum 0% investment (no short selling). • Currency: maximum 30% in foreign currency risk.

• Asset allocation: maximum 30% in a single asset class.

• Product allocation: maximum 10% in an individual security,

maximum 30% in US stocks.

• Tick size increment: minimum 5% investment tick.

• Optimal times: each year.

The resulting optimal portfolio (Table 10.2) is dominated by fixedincome investments in short-term investment-grade bonds (in a nodefault risk world), equity exposures have not been allowed but some FX risk. This is in line with general portfolio theory. The ex ante risk–return profile of the optimal portfolio is plotted in Figure 10.5. Such a portfolio has shown a high probability of yielding a positive return year on year, and a large probability of yielding more than the given return target. 183

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 10.2 PSO optimisation results: risk-averse profile (multiple period) Investment product Bond Deutsche Bank EUR 2026∗ Bond Santander EUR 2028∗ Bond Intesa Sanpaolo 2015∗ Bond Deutsche Bank 2015∗ Bond ING 2018∗ Bond Nordea 2020∗ Bond Goldman Sachs USD 2028∗∗ Bond Morgan Stanley USD 2015∗∗ Bond Goldman Sachs USD 2015∗∗ Bond Goldman Sachs 2019∗∗ Bond France Telecom 2015∗ Bond Volkswagen 2015∗ Bond RWE 2019∗ Bond KPN 2020∗ Bond ENEL 2027∗ Bond EDF 2025∗ Bond Ford USD 2028∗∗ Bond General Electric 2019∗∗ Bond General Electric 2016∗∗ Bond Ford 2014∗∗ Bond Gov Austria 2026∗ Bond Gov Italy 2027∗ Bond Gov Germany 2015∗ Bond Gov France 2016∗ Bond Gov Netherlands 2019∗ Bond Gov Spain 2020∗ Bond Gov US 2015∗∗ Bond Gov US 2016∗∗ Bond Gov US 2018∗∗ Bond Gov US 2020∗∗ Bond Gov US 2024∗∗ Bond Gov US 2028∗∗ Fund Equity Italy∗∗ Fund Equity Russia∗∗ Fund Equity Japan∗∗ Fund Equity Germany∗∗ Fund Equity Brazil∗∗ Fund Equity France∗∗ Fund Equity UK∗∗ Fund Equity China∗∗ Fund Equity India∗∗ Fund Equity US∗∗ ∗ EUR. ∗∗ USD. Fin,

184

Weight (%)

Asset class

Weight (%)

— 5 — — 5 10 — 5 10 10 — — 5 — 5 — 10 5 10 5 — — — — — — 5 — — 5 5 — — — — — — — — — — —

Fin foreign

20

Fin local

25

Non-fin foreign

10

Non-fin local

30

Sov foreign



Sov local

15

Eq foreign



Eq local



financial. Eq, equity. Sov, sovereign.

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

Figure 10.5 Risk-averse profile (multiple period)

Risk/return

0.4

0.2

Monte Carlo Ambition profile Risk profile Optimisation node CIO view H1 CIO view H2

0

Probability

–0.2 1.0 0.5 0

1Y

2Y

Time

Positive return

3Y

4Y

5Y

Beating target

We can observe that the CIO views indicate portfolio sensitivity to interest rates and credit spread changes, which seem to be accounted for by the Monte Carlo simulation. RISK-MITIGATING PROFILE: MULTIPLE PERIOD The investment goal was set to yield slightly more than the risk-free rate of the US economy, approximated with an ambition of around 2% year on year. The interest rates were particularly low at the time of analysis (January 2014) and the risk adequacy line was modelled to accommodate some risk taking in the short term, but to enforce tight risk mitigation in the medium/long run (capital protection). We modelled the following constraints. • Short selling: minimum 0% investment (no short selling). • Currency: maximum 30% in foreign currency risk.

• Asset allocation: maximum 30% in a single asset class.

• Product allocation: maximum 10% in an individual security;

maximum 30% in US stocks.

• Tick size increment: minimum 5% investment tick.

• Optimal times: each year.

185

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 10.6 PSO portfolio: risk-mitigating profile (multiple period)

Risk/return

0.4

0.2

Monte Carlo Ambition profile Risk profile Optimisation node CIO view H1 CIO view H2

0

Probability

–0.2 1.0 0.5 0

1Y

2Y Positive return

Time

3Y

4Y

5Y

Beating target

The resulting portfolio (Table 10.3) featured lower exposure in US Treasury bonds than the risk-averse exercise, to accommodate for more credit and equity exposures. Still, we modelled capital preservation, which represents the total return dynamics of the optimal portfolio. Figure 10.6 shows the ex ante risk–return profile of the optimal portfolio. Such a portfolio has shown a high probability of yielding a positive return year on year, and a sufficiently large probability of yielding more than the given return target. We can observe that the CIO views indicate portfolio sensitivity to interest rates and credit spread changes, which seem to be accounted for by the Monte Carlo simulation. RISK-TOLERANT PROFILE: MULTIPLE PERIOD The investment goal was set to yield at least 6% year on year. The risk adequacy profile was set fairly loose, which allowed for risk taking throughout the full investment horizon. We modelled the following constraints. 186

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

Table 10.3 PSO optimisation results: risk-mitigating profile (multiple period) Investment product Bond Deutsche Bank EUR 2026∗ Bond Santander EUR 2028∗ Bond Intesa Sanpaolo 2015∗ Bond Deutsche Bank 2015∗ Bond ING 2018∗ Bond Nordea 2020∗ Bond Goldman Sachs USD 2028∗∗ Bond Morgan Stanley USD 2015∗∗ Bond Goldman Sachs USD 2015∗∗ Bond Goldman Sachs 2019∗∗ Bond France Telecom 2015∗ Bond Volkswagen 2015∗ Bond RWE 2019∗ Bond KPN 2020∗ Bond ENEL 2027∗ Bond EDF 2025∗ Bond Ford USD 2028∗∗ Bond General Electric 2019∗∗ Bond General Electric 2016∗∗ Bond Ford 2014∗∗ Bond Gov Austria 2026∗ Bond Gov Italy 2027∗ Bond Gov Germany 2015∗ Bond Gov France 2016∗ Bond Gov Netherlands 2019∗ Bond Gov Spain 2020∗ Bond Gov US 2015∗∗ Bond Gov US 2016∗∗ Bond Gov US 2018∗∗ Bond Gov US 2020∗∗ Bond Gov US 2024∗∗ Bond Gov US 2028∗∗ Fund Equity Italy∗∗ Fund Equity Russia∗∗ Fund Equity Japan∗∗ Fund Equity Germany∗∗ Fund Equity Brazil∗∗ Fund Equity France∗∗ Fund Equity UK∗∗ Fund Equity China∗∗ Fund Equity India∗∗ Fund Equity US∗∗ ∗ EUR. ∗∗ USD. Fin,

Weight (%)

Asset class

— 5 — — — — 10 — 5 10 — — — 5 5 — 10 10 5 5 — — — — 5 10 — — 10 — — — — — — — — — — 5 — —

Fin foreign

Weight (%) 5

Fin local

25

Non-fin foreign

10

Non-fin local

30

Sov foreign

15

Sov local

10

Eq foreign

5

Eq local



financial. Eq, equity. Sov, sovereign.

187

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 10.4 PSO optimisation results: risk-tolerant profile (multiple period) Investment product Bond Deutsche Bank EUR 2026∗ Bond Santander EUR 2028∗ Bond Intesa Sanpaolo 2015∗ Bond Deutsche Bank 2015∗ Bond ING 2018∗ Bond Nordea 2020∗ Bond Goldman Sachs USD 2028∗∗ Bond Morgan Stanley USD 2015∗∗ Bond Goldman Sachs USD 2015∗∗ Bond Goldman Sachs 2019∗∗ Bond France Telecom 2015∗ Bond Volkswagen 2015∗ Bond RWE 2019∗ Bond KPN 2020∗ Bond ENEL 2027∗ Bond EDF 2025∗ Bond Ford USD 2028∗∗ Bond General Electric 2019∗∗ Bond General Electric 2016∗∗ Bond Ford 2014∗∗ Bond Gov Austria 2026∗ Bond Gov Italy 2027∗ Bond Gov Germany 2015∗ Bond Gov France 2016∗ Bond Gov Netherlands 2019∗ Bond Gov Spain 2020∗ Bond Gov US 2015∗∗ Bond Gov US 2016∗∗ Bond Gov US 2018∗∗ Bond Gov US 2020∗∗ Bond Gov US 2024∗∗ Bond Gov US 2028∗∗ Fund Equity Italy∗∗ Fund Equity Russia∗∗ Fund Equity Japan∗∗ Fund Equity Germany∗∗ Fund Equity Brazil∗∗ Fund Equity France∗∗ Fund Equity UK∗∗ Fund Equity China∗∗ Fund Equity India∗∗ Fund Equity US∗∗ ∗ EUR. ∗∗ USD. Fin,

188

Weight (%)

Asset class

Weight (%)

— 10 — — — — 10 — — 5 — — — 5 5 — 10 — — — — 5 — — — — — — — — — — 5 10 — 5 5 — — 5 — 20

Fin foreign

10

Fin local

15

Non-fin foreign

10

Non-fin local

10

Sov foreign

5

Sov local



Eq foreign

30

Eq local

20

financial. Eq, equity. Sov, sovereign.

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

Figure 10.7 PSO portfolio: risk-tolerant profile (multiple period) Optimisation node CIO view H1 CIO view H2

Monte Carlo Ambition profile Risk profile 1.0 0.8

Risk/return

0.6 0.4 0.2 0 –0.2

Probability

–0.4 1.0 0.5 0

1Y

2Y

3Y

4Y

5Y

Time Positive return

Beating target

• Short selling: minimum 0% investment (no short selling). • Currency: maximum 30% in foreign currency risk. • Asset allocation: maximum 30% in a single asset class. • Product allocation: maximum 10% in an individual security,

max 30% in US stocks.

• Tick size increment: minimum 5% investment tick. • Optimal times: each year.

Table 10.4 indicates the asset allocation of the optimal portfolio, which consumed the full allowance for equity investments and featured more mid- to long-term fixed-income exposures than the previous exercises. The ex ante risk–return profile of the optimal portfolio is plotted in Figure 10.7. Such a portfolio shows a larger probability of yielding a positive return year on year as opposed to a loss, and an appreciable probability of yielding more than the given return target. 189

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Figure 10.8 PSO portfolio: risk-tolerant profile (single period) Monte Carlo Ambition profile Risk profile

Optimisation node CIO view H1 CIO view H2

1.0 0.8 0.6 0.4 0.2 0 –0.2

Probability

–0.4 1.0 0.5 0

Positive return

1Y

Beating target

2Y

Time

3Y

4Y

5Y

We can observe that the CIO views indicate portfolio sensitivity to the potential dynamics of equity markets, which seem to be accounted for by the Monte Carlo simulation. RISK-TOLERANT PROFILE: SINGLE PERIOD The investment goal was set to yield at least 6% year on year. The risk adequacy profile was set fairly loose to allow for risk taking throughout the full span of the investment horizon. We modelled the following constraints. • Short selling: minimum 0% investment (no short selling). • Currency: maximum 30% in foreign currency risk. • Asset allocation: maximum 30% in a single asset class. • Product allocation: maximum 10% in an individual security,

maximum 30% in US stocks.

• Tick size increment: minimum 5% investment tick. • Optimal time: fifth year only.

190

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

Table 10.5 PSO optimisation results: risk-tolerant profile (single period) Investment product Bond Deutsche Bank EUR 2026∗ Bond Santander EUR 2028∗ Bond Intesa Sanpaolo 2015∗ Bond Deutsche Bank 2015∗ Bond ING 2018∗ Bond Nordea 2020∗ Bond Goldman Sachs USD 2028∗∗ Bond Morgan Stanley USD 2015∗∗ Bond Goldman Sachs USD 2015∗∗ Bond Goldman Sachs 2019∗∗ Bond France Telecom 2015∗ Bond Volkswagen 2015∗ Bond RWE 2019∗ Bond KPN 2020∗ Bond ENEL 2027∗ Bond EDF 2025∗ Bond Ford USD 2028∗∗ Bond General Electric 2019∗∗ Bond General Electric 2016∗∗ Bond Ford 2014∗∗ Bond Gov Austria 2026∗ Bond Gov Italy 2027∗ Bond Gov Germany 2015∗ Bond Gov France 2016∗ Bond Gov Netherlands 2019∗ Bond Gov Spain 2020∗ Bond Gov US 2015∗∗ Bond Gov US 2016∗∗ Bond Gov US 2018∗∗ Bond Gov US 2020∗∗ Bond Gov US 2024∗∗ Bond Gov US 2028∗∗ Fund Equity Italy∗∗ Fund Equity Russia∗∗ Fund Equity Japan∗∗ Fund Equity Germany∗∗ Fund Equity Brazil∗∗ Fund Equity France∗∗ Fund Equity UK∗∗ Fund Equity China∗∗ Fund Equity India∗∗ Fund Equity US∗∗ ∗ EUR. ∗∗ USD. Fin,

Weight (%)

Asset class

Weight (%)

5 — — 5 — — 10 — 5 5 — — — — 10 — 5 — — — — — — — — — — — — — — — 5 5 — — 10 — — 5 5 25

Fin foreign

10

Fin local

20

Non-fin foreign

10

Non-fin local

5

Sov foreign



Sov local



Eq foreign

30

Eq local

25

financial. Eq, equity. Sov, sovereign.

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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Table 10.6 PSO optimisations: results comparison

Risk mitigating

Risk tolerant

Single period Risk tolerant

20 25 10 30 — 15 — —

5 25 10 30 15 10 5 —

10 15 10 10 5 — 30 20

10 20 10 5 — — 30 25

100

100

100

100



Asset class

Risk averse

Financial foreign Financial local Non-financial foreign Non-financial local Sovereign foreign Sovereign local Equity foreign Equity local Total

Multiple periods 

All values are given in percent.

Table 10.7 PSO optimisations: probability comparison Single period Risk Risk Risk Risk averse mitigating tolerant tolerant

Final return target (5Y) Probability of positive return (5Y) Probability of beating target (5Y)

5 100 95

Multiple periods  10 98 65

30 70 35

30 65 35

All values are given in percent.

Compared with the multi-period case discussed in the previous section the optimal portfolio picked up more risky exposures on equity markets as the risk constraints of the shorter terms were relaxed (as in Table 10.5). The ex ante risk–return profile of the optimal portfolio is plotted in Figure 10.8. Such a portfolio has shown a larger probability of yielding a positive return at the 5Y horizon as opposed to a loss, and an appreciable probability of yielding more than the given return target. We can observe that the CIO views indicate portfolio sensitivity to the potential dynamics of equity markets, which seem to be accounted for by the Monte Carlo simulation. 192

CASE STUDIES: PROBABILISTIC SCENARIO OPTIMISATION

Table 10.8 Optimisation results: comparison of mean–variance (MV), Black–Litterman (BL) and PSO BL PSO   MV  No Multiple Single Asset class Unconstr Constr views∗ Views period period Financial foreign Financial local Non-financial foreign Non-financial local Sovereign foreign Sovereign local Equity foreign Equity local Credit growth markets Total





14.32

0.46

10.00

10.00

41.28 —

34.91 —

0.04 16.93

23.53 12.69

15.00 10.00

20.00 10.00

42.92

33.55

0.03

4.12

10.00

5.00





7.63

6.60

5.00



23.14 — 1.30 7.11

0.04 60.96 0.02 0.03

12.30 37.05 1.16 0.07

— 30.00 20.00 —

— 30.00 25.00 —

8.57 — 0.93 6.30 100.00

100.00 100.00 100.00 100.00 100.00

All values are given in percent. ∗ CAPM market portfolio.

CONCLUSIONS AND COMPARISONS Probabilistic scenario optimisations allow us to perform goal-based long-term asset allocation by embedding the risk–return profile of the real securities as well as the time preferences of the investors, with respect to their investment ambitions and their risk concerns. The resulting asset allocations are in line with general theory of portfolio choice, when discussing the diverse mix of risks and returns stemming from different asset classes and durations, as can be seen in Table 10.6. The PSO approach permits a flexible framework for modelling time constraints as well as convex risk–return profiles, in such a way that myopic trading and long-term capital insurance can be modelled and considered jointly when defining optimal portfolio choice. Clearly, the higher the return ambition, the larger the risks. Therefore, we expect that the resulting probability of beating a higher target is lower for a risk-tolerant profile than for a risk-averse portfolio, 193

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

as can be seen in Table 10.7, which reviews the resulting probabilities of the four cases, measured at a final investment horizon of five years. Table 10.8 juxtaposes the optimisation results of the three methodologies described in this book: mean–variance and Black–Litterman (from the previous chapter), PSO (for the risk-tolerant case, as this is the most comparable due to a similar level of yearly target return). As noted in previous chapters, although the case studies refer to the same historical period and reference markets, the methods are not directly comparable due the different treatments of the market data, the different way of adding constraints and views, the differences in the objective functions and the diverse treatment of time.

194

Symbols and Notation We list below some notation used throughout this book. Portfolio or universe of the investment opportunities Individual asset/security in a portfolio Investment horizon Time step inside Γ Last time step inside Γ Scenario as a set of risk factor modifications Scenario path as a sequence over time of scenarios h ∈ H Scenario set as a collection of all scenario paths Value of an asset/security or a portfolio Total return of an asset/security or a portfolio Unit of position of an asset/security in a portfolio; random variable w Percentage exposure of an asset/security in a portfolio $ Foreign exchange rate σ Standard deviation of an asset/security or a portfolio cov(i, j) Covariance between the total returns of j and i 1 − α Confidence interval ξα α quantile indicating VaR(1 − α) or ES(1 − α) M Market equilibrium portfolio B Benchmark portfolio Negative (positive) tracking error es∓ P Probability measure CF Cashflow DF Discount factor c Change in value due to income/consumption ej∓ Negative (positive) change in asset allocations Σ Correlation or covariance matrix λA , λL Return ambition and risk appetite profile of an investor λ Coefficient of risk aversion Φ Initial set of portfolio compositions Ψ Set of potential allocations: constraint compliant Ω Set of admissible allocations: risk adequate Ω∗ Optimal portfolio Function of a real-valued random variable F (y ) a, b Thresholds µ Average of a distribution β Beta of CAPM equation rf Risk-free rate U i, j, q Γ t T h H S V R x

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Q, q D, d ε Ω N z

196

Set of excess expected returns that define a view Set of asset classes directly affected by a view Confidence level of an investment view Covariance matrix of error terms of the views Bond notional or total number of investment ticks of size z; number of mutually exclusive and equally likely events Tick size increment of security/asset j in a portfolio

Index (page numbers in italic type relate to tables or figures) A adequate portfolios: risk adequacy, 152–4 applying investor’s profile to portfolio simulations, 99–100 asset diversification and the efficient frontier, 108–13, 110, 111 B Bayes’s Theorem, 132, 135 Black, Fischer, see Black–Litterman approach Black–Litterman approach, 16, 123–36, 127, 129, 170, 171 elegance of, xiv and inputs, preparation of, 124 introduced and discussed, 123–4 and investor’s beliefs, declaration of, 125 and investors’ views, 131–5 and Bayes’s Theorem, 132 example, 134–5 limitations of, xiv main traits of, 5–6 and market equilibrium portfolio, 125–31 and mean–variance approach, 5, 123, 139 Black–Litterman portfolio, 172–3 case studies, 163–73, 164 data set: market indexes, 163–9 introduced and discussed, 163 mean–variance portfolio, 169–72

mitigated by PSO, 6 and portfolio optimisation, 125 and posterior distribution, 135 and reverse optimisation, 124 technicalities of, 124 Black–Litterman portfolio, 172–3 bonds: Deutsche Bank 2015, 179 Deutsche Bank 2026, 178 EDF 2025, 179 ENEL 2027, 179 Ford 2014, 179 Ford 2028, 179 France Telecom 2015, 179 General Electric 2016, 179 General Electric 2019, 179 Goldman Sachs 2015, 179 Goldman Sachs 2019, 179 Goldman Sachs 2028, 179 ING 2018, 179 Intesa Sanpaolo 2015, 178 KPN 2020, 179 Morgan Stanley 2015, 179 Nordea 2020, 179 RWE 2019, 179 Santander 2028, 178 Volkswagen 2015, 179 bonds (government): Austria 2026, 179 France 2016, 179 Germany 2015, 179 Italy 2027, 179 Netherlands 2019, 179 Spain 2020, 179 US 2015, 179 US 2016, 180 US 2018, 180 US 2020, 180 US 2024, 180 US 2028, 180 bootstrapping method, historical, 24, 35, 37

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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Brownian motion, geometric (GBM), 26–7, 28 C Capital Asset Pricing Model (CAPM), 123, 124, 125–6, 127, 130–1, 135, 172 and investors’ views, 131–5 and Bayes’s Theorem, 132 example, 134–5 see also market equilibrium portfolio case studies: mean–variance and Black–Litterman, 163–73 probabilistic scenario optimisation (PSO), 175–94 Churchill, Winston, 175 D deflation, 81 distribution function and alpha-quantile profile, 45–8 dynamics of risk measures, 37–40, 37 see also risk management E Einstein, Albert, 139 ex post and ex ante performance, link between, 48–52, 49, 50, 51 see also probability measure expected shortfall (ES), 15, 19 formulation, 118–19 and value-at-risk (VaR), 34–7, 35, 36, 37, 118 historical bootstrapping method, 35, 37 historical scenarios method, 34–5 Monte Carlo method, 35–7, 38 parametric method, 34, 35

198

F fair values of real securities, modelling 62–4 see also reinvestment strategies and real securities financial securities products, universe of: bonds: Deutsche Bank 2015, 179 Deutsche Bank 2026, 178 EDF 2025, 179 ENEL 2027, 179 Ford 2014, 179 Ford 2028, 179 France Telecom 2015, 179 General Electric 2016, 179 General Electric 2019,179 Goldman Sachs 2015, 179 Goldman Sachs 2019, 179 Goldman Sachs 2028,179 ING 2018, 179 Intesa Sanpaolo 2015, 178 KPN 2020, 179 Morgan Stanley 2015, 179 Nordea 2020, 179 RWE 2019, 179 Santander 2028 ,178 Volkswagen 2015, 179 bonds (government): Austria 2026, 179 France 2016, 179 Germany 2015, 179 Italy 2027, 179 Netherlands 2019, 179 Spain 2020, 179 US 2015, 179 US 2016,180 US 2018, 180 US 2020, 180 US 2024, 180 US 2028, 180 equities: foreign, 180 local, 180 equities (funds): Brazil, 180 China, 180

INDEX

France, 180 Germany, 180 India, 180 Italy, 180 Japan, 180 Russia, 180 UK, 180 US, 180 financials: foreign, 178–9 local, 179 non-financials: foreign, 179 local, 179 sovereigns: foreign, 179 local, 179 fixed-income securities, 65, 67 modelling, 64–70 modelling reinvestment strategies of, 70–4, 72 Franklin, Benjamin, 1, 85 Funds (equity): Brazil, 180 China, 180 France, 180 Germany, 180 India, 180 Italy, 180 Japan, 180 Russia, 180 UK, 180 US 180 G geometric Brownian motion (GBM), 26–7, 28 global financial crisis, 1, 15, 30, 166, 183 Goldman Sachs, 123 Group of Seven, 59 H Halton sequences, 146–9, 150 historical bootstrapping method, 24, 35, 37

historical scenarios method, 22–4, 23, 34–5 Hitchens, Christopher, 163 Hull–White one-factor model, 31, 79 modelling domestic interest rates, 29–33 modelling foreign interest rates and corresponding foreign-exchange rates, 34 I inflation-linked securities, modelling 79–82 inflation trend, 81 investment securities universe, 57–9, 58 investment style and personality, 86–8 K Keynes, John Maynard, 3 knowledge-based principles, 8–9 enumerated, 9 L lexicographical representations, 150–2, 153 Litterman, Robert, see Black–Litterman approach low-discrepancy methods: Halton sequences, 146–9 M market equilibrium portfolio, 123, 124, 125–31 see also Capital Asset Pricing Model Markowitz approach, 85, 105–20, 106, 110, 111, 112, 115 and asset diversification and the efficient frontier, 108–13, 110, 111 and expected shortfall (ES) formulation, 118–19

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MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

introduced and discussed, 105–8 and mean–variance formulation, 113–15 mitigated by PSO, 6 reliance on, xiii, 4, 55 and semi-variance modification, 117 and tracking error minimisation, 115–16 Markowitz, Harry, 55 on portfolios, 105 on probability measure, 43 mean–variance approach, 5, 106, 115, 125 and Black–Litterman, 5, 123, 139 Black–Litterman portfolio, 172–3 case studies, 163–73, 164, 165, 166, 167, 168, 169, 170, 171 data set: market indexes, 163–9 introduced and discussed, 163 mean–variance portfolio, 169–72 formulation, 113–15 pitfalls of, 117 weaknesses of, 107 see also Markowitz approach mean–variance portfolio, 169–72 modelling: fair values of real securities, 62–4 see also reinvestment strategies and real securities fixed-income securities, 64–70, 65 inflation-linked securities, 79–82 Modern Portfolio Theory: and Black–Litterman, xiv main traits of, 5–6 emergence of new interpretation of, 3

200

and five knowledge-based steps, 8–9 enumerated, 9 introduced and discussed, 1–10 and investor, knowing, 9 main traits of, 4–5 and Markowitz, reliance on, xiii, 4, 55 and performance, knowing, 9 and probabilistic scenario optimisation (PSO), main traits of, 6–8 and probability measure, 43–52 distribution function and alpha-quantile profile, 45–8 and ex post and ex ante performance, link between, 48–52, 49, 50 introduced and discussed, 43 Monte Carlo simulation, 47 a priori and a posteriori, 43–5 and products, knowing, 9 and returns, knowing, 9 and risks, knowing, 9 and rule of thumb, xiii, 2 Monte Carlo method, 35–7, 38, 47, 150 and auto-callable discrete barrier certificate, 78 densities of, 47 and left and right quantiles, 77 six steps of, 25 stochastic scenarios, 25–34 geometric Brownian motion (GBM), 26–7 Hull–White one-factor model: modelling domestic interest rates, 29–33 Hull–White one-factor model: modelling foreign interest rates and corresponding foreign-exchange rates, 34

INDEX

and Treasury bonds, 69 with foreign-exchange risk, 70 Musil, Robert, 123 N negative real yields, 81 nominal economy, 81 O objective function: probability maximisation, 154–8 P parametric method, 34, 35 variance–covariance, 19–22 portfolio simulations, applying investor’s profile to, 99–100 posterior distribution, 135 probabilistic scenario optimisation (PSO), 7, 8, 139–58, 141, 146, 150, 151, 153, 155, 156, 157 adequate portfolios: risk adequacy, 152–4 appealing features of, 9 case studies, 175–94, 176, 181, 182, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193 and CIO views (stress tests), 178 data set: market risk factors, 182–3 data set: universe of products, 178–82; see also financial securities products, universe of introduced and discussed, 175–8 and investment horizon, 175, 176 and objective function, 177–8 risk-averse profile: multiple period, 183–5, 184, 185

risk-mitigating profile: multiple period ,185–6, 186, 187 and risk–return profiles, 176–7 risk-tolerant profile: multiple period, 186–90, 188, 189 risk-tolerant profile: single period, 190–3, 190, 191 conclusions and comparisons, 193–4 introduced and discussed, 139–41 lexicographical representations, 150–2, 153 low-discrepancy methods: Halton sequences, 146–9, 150 main traits of, 6–8 objective function: probability maximisation, 154–8 potential and admissible portfolios: allocation constraints, 143–6 process of 141–3, 141 probability measure, 43–52 distribution function and alpha-quantile profile, 45–8 and ex post and ex ante performance, link between, 48–52, 49, 50, 51 introduced and discussed, 43 Monte Carlo simulation, 47 a priori and a posteriori, 43–5 products, universe of: bonds: Deutsche Bank 2015, 179 Deutsche Bank 2026, 178 EDF 2025, 179 ENEL 2027, 179 Ford 2014, 179 Ford 2028, 179 France Telecom 2015, 179 General Electric 2016, 179 General Electric 2019, 179 Goldman Sachs 2015, 179

201

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

Goldman Sachs 2019, 179 Goldman Sachs 2028, 179 ING 2018, 179 Intesa Sanpaolo 2015, 178 KPN 2020, 179 Morgan Stanley 2015, 179 Nordea 2020, 179 RWE 2019, 179 Santander 2028, 178 Volkswagen 2015, 179 bonds (government): Austria 2026, 179 France 2016, 179 Germany 2015 ,179 Italy 2027, 179 Netherlands 2019, 179 Spain 2020, 179 US 2015, 179 US 2016, 180 US 2018, 180 US 2020, 180 US 2024, 180 US 2028 180 equities: foreign, 180 local, 180 equities (funds): Brazil, 180 China, 180 France, 180 Germany, 180 India, 180 Italy,180 Japan, 180 Russia, 180 UK, 180 US, 180 financials: foreign, 178–9 local, 179 non-financials: foreign, 179 local, 179 sovereigns: foreign, 179 local, 179 PSO, see probabilistic scenario optimisation

202

R real economy, 81 real investments, measuring risks and returns of, 18–19 real securities, modelling fair values of, 62–4 see also reinvestment strategies and real securities reinvestment strategies and real securities, 55–82, 56, 58, 60, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 80 and derivatives and structured products, 76–9 and fixed-income securities, 70–4, 72 modelling, 64–70, 65 time path of prices for, 67 and inflation-linked securities, modelling, 79–82 introduced and discussed, 55–6 and investment securities universe, 57–9, 58 and modelling inflation-linked securities, 79–82 and real securities, modelling fair values of, 62–4 and structured funds, 74–7, 75 and total asset values and returns, modelling, 59–62 risk adequacy, 152–4 risk-averse profile, 96–7, 97, 177 multiple period, 183–5, 184, 185 risk management, 15–40, 17, 20, 23, 26, 28, 31, 32, 33, 35, 36, 37, 38, 39, 40 and Black–Litterman, 16 and expected shortfall (ES): historical bootstrapping method, 35, 37 historical scenarios method, 34–5 Monte Carlo method, 35–7, 38 parametric method, 34, 35

INDEX

and value-at-risk (VaR), 34–7 and historical bootstrapping method, 24 and historical scenarios method, 22–4, 23 introduced and discussed, 15–16 and Monte Carlo Method: stochastic scenarios, 25–34 geometric Brownian motion (GBM), 26–7 Hull–White one-factor model: modelling domestic interest rates, 29–33 Hull–White one-factor model: modelling foreign interest rates and corresponding foreign-exchange rates, 34 and parametric method, 35 variance–covariance, 19–22 and real investments, measuring risks and returns, of 18–19 and risk measures, dynamics of, 37–40, 37 and scenarios and scenario paths, 16–18, 17 and value-at-risk (VaR): and expected shortfall (ES), 34–7 historical bootstrapping method, 35, 37 historical scenarios method, 34–5 Monte Carlo method, 35–7, 38 parametric method, 34, 35 risk-mitigating profile, 97–8, 97, 177 multiple period, 185–6, 186, 187 risk–return expression, 88–90 risk–return time profiles, 85–100, 93, 95, 97, 98, 99 and applying investor’s profile to portfolio situations, 99–100

introduced and discussed, 85–6 and investment style and personality, 86–8 and risk–return expression, 88–90 shaping, over time horizon, 92–9 risk averse, 96–7, 97 risk mitigating, 97–8, 97 risk tolerant, 98–9, 98 target risk profile, 99 and time, risk and return, 91–2 risk-tolerant profile, 98–9, 98, 177, 187, 189 multiple period, 186–90, 188, 189 single period, 190–3, 190, 191 S scenarios and scenario paths, 16–18, 17 semi-variance modification, 117 shaping risk–return profile over time horizon, 92–9 risk averse, 96–7 risk mitigating, 97–8, 97 risk tolerant, 98–9, 98 target risk profile, 99 see also risk–return time profiles structured funds, modelling reinvestment strategies of, 74–7, 75 see also reinvestment strategies and real securities T Taleb, Nassim, 55 target risk profile, 99 see also risk–return time profiles time, risk and return, 91–2 Tolkien, J. R. R., 15 tracking error minimisation, 115–16

203

MODERN PORTFOLIO MANAGEMENT: FROM MARKOWITZ TO PSO

V value-at-risk (VaR), 15, 19, 21, 34, 35–6, 98, 118–19 and expected shortfall, 34–7, 36, 37, 118 historical bootstrapping method, 35, 37 historical scenarios method, 34–5 Monte Carlo method, 35–7, 38 parametric method, 34, 35 historical, 24 and Monte Carlo, 38

204