Nonlinear Modeling of Economic and Financial Time-Series [1 ed.] 9780857244901, 9780857244895

Citation preview

NONLINEAR MODELING OF ECONOMIC AND FINANCIAL TIME-SERIES

International Symposia in Economic Theory and Econometrics Series Editor:

William A. Barnett

Volume 14:

Economic Complexity Edited by W.A. Barnett, C. Deissenberg & G. Feichtinger

Volume 15:

Modelling Our Future: Population Ageing, Social Security and Taxation Edited by Ann Harding & Anil Gupta

Volume 16:

Modelling Our Future: Population Ageing, Health and Aged Care Edited by Anil Gupta & Ann Harding

Volume 17:

Topics in Analytical Political Economy Edited by Melvin Hinich & William A. Barnett

Volume 18:

Functional Structure Inference Edited by William A. Barnett & Apostolos Serletis

Volume 19:

Challenges of the Muslim World: Present, Future and Past Edited by William W. Cooper & Piyu Yue

International Symposia in Economic Theory and Econometrics Volume 20

NONLINEAR MODELING OF ECONOMIC AND FINANCIAL TIME-SERIES

EDITED BY

FREDJ JAWADI Universite´ d’E´vry Val d’Essonne & Amiens School of Management, E´vry Cedex, France

WILLIAM A. BARNETT University of Kansas, Lawrence, USA

United Kingdom – North America – Japan India – Malaysia – China

Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2010 Copyright r 2010 Emerald Group Publishing Limited Reprints and permission service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertisements. The opinions expressed in these chapters are not necessarily those of the Editor or the publisher. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-85724-489-5 ISSN: 1571-0386 (Series)

Emerald Group Publishing Limited, Howard House, Environmental Management System has been certified by ISOQAR to ISO 14001:2004 standards Awarded in recognition of Emerald’s production department’s adherence to quality systems and processes when preparing scholarly journals for print

Contents List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Editorial Advisory Board Members . . . . . . . . . . . . . . . . . . . . . . . . .

ix

About the Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

1 Collateralizable Wealth, Asset Returns, and Systemic Risk: International Evidence Ricardo M. Sousa

1

2 Nonlinear Stock Market Links between Mexico and the World Mohamed El Hedi Arouri and Fredj Jawadi

29

3 Dynamic Linkages between Global Macro Hedge Funds and Traditional Financial Assets Wafa Kammoun Masmoudi

41

4 Copula Theory Applied to Hedge Funds Dependence Structure Determination Rania Hentati and Jean-Luc Prigent

83

5 European Exchange Rate Credibility: An Empirical Analysis Iuliana Matei

111

6 Oil Prices and Exchange Rates: Some New Evidence Using Linear and Nonlinear Models Mohamed El Hedi Arouri and Fredj Jawadi

121

7 Sources of European Growth Externalities: A Two-Step Approach Se´bastien Pommier and Fabien Rondeau

143

8 Alternative Methods for Forecasting GDP Dominique Gue´gan and Patrick Rakotomarolahy

161

9 GARCH Models with CPPI Application Hachmi Ben Ameur

187

v

List of Contributors Arouri, M.E.H., LEO – Universite´ d’Orle´ans & EDHEC Business School, France (Chs 2, 6) Ben Ameur, Hachmi, Amiens School of Management, France (Ch. 9) Gue´gan, D., Universite´ Paris 1, France (Ch. 8) Hentati, R., Universite´ Paris 1, France (Ch. 4) Jawadi, F., University of E´vry and Amiens School of Management, France (Chs 2, 6) Kammoun Masmoudi, W., University of Rennes 1(France); University of Tunis El Manar, Tunisia (Ch. 3) Matei, I., EPEE – University of E´vry, France (Ch. 5) Pommier, S., University of Rennes 1, France (Ch. 7) Prigent, J.-L., The´ma – Universite´ de Cergy-Pontoise, France (Ch. 4) Rakotomarolahy, P., University of Paris 1, France (Ch. 8) Rondeau, F., CREM – University of Rennes 1, France (Ch. 7) Sousa, R.M., University of Minho, UK (Ch. 1) .

vii

Editorial Advisory Board Members Prof. W. Barnett, University of Kansas, USA Prof. Ma. Bellalah, University of Jules Verne Picardie, France Prof. P. Demediuk, University of Melbourne, Australia Prof. S. Gre´goir, EDHEC Business School, France Prof. C. Hommes, University of Amsterdam, The Netherlands Prof. F. Jawadi, University of E´vry & Amiens School of Management, France Prof. A. Kirman, University of Aix-Marseille III, France Prof. P. Le´oni, Euromed Management, France Prof. F. Lundtofte, Lund University, Sweden Prof. Th. Lux, University of Kiel, Germany Prof. V. Mignon, University of Paris Ouest Nanterre La De´fense, France Prof. G. Prat, EconomiX-CNRS, France Prof. E. Quintin, Federal Reserve Bank of Dallas, USA Prof. G. Talmain, University of Glasgow, UK

ix

About the Series The series International Symposia in Economic Theory and Econometrics publishes quality proceedings of conferences and symposia. Since all articles published in these volumes are refereed relative to the standards of the best journals, not all papers presented at the symposia are published in these proceedings volumes. Occasionally these volumes include articles that were not presented at a symposium or conference, but are of high quality and are relevant to the focus of the volume. The topics chosen for these volumes are those of particular research importance at the time of the selection of the topic. Each volume has different coeditors, chosen to have particular expertise relevant to the focus of that particular volume. William A. Barnett Series Editor

xi

Introduction During the global financial crisis of 2008–2009, most developed and emerging economies and financial markets have recorded important financial losses. Those economies have experienced momentous corrections, and their assets were significantly devaluated, implying many losses and bankruptcies for banks, investors, and firms. Overall, despite continuing efforts made by governments and central banks to support their financial systems, most financial markets (stock markets, derivative markets, monetary markets, and currency markets) have been strongly affected by this crisis. Furthermore, the rapid transmission of the US subprime crisis to several European and Asian developed and emerging countries and the transformation into a global financial and economic crisis have revealed a high level of financial integration and linkage with the US market. The financial shocks have also induced negative feedbacks to macroeconomic indicators, suggesting significant relationships between financial markets and macroeconomies. In the literature, several recent studies have focused on financial markets to investigate the causes and effects of this crisis and have identified the financial market dynamics within the global financial crisis. Mainly, those authors have, on the one hand, highlighted negative and significant feedbacks between the financial crisis and financial markets, while characterizing significant decreases of stock returns and important increases of risk and risk premia associated with investment in stock markets. On the other hand, their findings have shown that sectors, even those not directly involved in this crisis (e.g., oil markets and information technology markets), were implicitly affected by this global financial crisis. Interestingly, the authors have unanimously pointed out the existence of asymmetric and nonlinear feedbacks. Indeed, their results suggest the presence of several structural changes, breaks, and regime changes characterizing the dynamics of financial markets. Usual econometric modeling may fail to reproduce these nonlinear dynamics, because the most common models limit a cycle to be linear, which is not appropriate to represent asymmetric dynamics. This evolving feature has particularly led to a commensurate increase in sophistication of nonlinear modeling techniques used for understanding financial market dynamics. This volume aims at providing a comprehensive understanding of financial markets in various aspects using modern nonlinear financial econometric methods. It addresses the empirical techniques needed by economic agents to analyze the dynamics of these markets and illustrates xiii

xiv

Introduction

how they can be applied to the actual data. It also presents and discusses new research findings and their implications. It is possible to classify the chapters of this book into three groups regarding their applications and the implemented tools. This introduction first summarizes the chapters and then outlines some broad areas for future research. The first chapter in this book is by Ricardo Sousa. This chapter studies the role of collateralizable wealth and systemic risk in explaining and forecasting future asset returns. The author points out that nonlinear deviation of housing wealth from its cointegrating relationship with labor income helps to forecast expected future returns. Using data for a set of industrialized countries, the chapter finds that when the housing wealth-to-income ratio falls, investors demand a higher risk premium for stocks. It also shows that the occurrence of crisis episodes amplifies the transmission of housing market shocks to financial markets. In the second chapter, Mohamed El Hedi Arouri and Fredj Jawadi also focus on stock markets. This chapter investigates the stock market comovements between Mexico and the world capital market during the financial crisis. While applying recent nonlinear cointegration and nonlinear error correction models to investigate the comovements between stock prices over the recent period, this chapter specifies a nonlinear mechanism characterizing the comovement between the Mexican and world stock prices. It shows that a nonlinear relationship between stock prices is activated across economic regimes. These findings highlight strong evidence of significant comovements and integration that explain the global collapse of emerging stock markets in 2008–2009. In the third chapter, the study of Wafa Kammoun Masmoudi looks at the effect of the financial crisis on hedge funds through the investigation of the relationships between global macro hedge funds and traditional financial assets. To explore this relationship, the author applies several causality and cointegration tests. Her findings show significant long- and short-term relationships between global macro hedge funds and traditional financial assets for Canada, France, and Germany. Her findings imply, as a consequence of the financial crisis, a reduction of opportunities for international portfolio diversification. In the fourth chapter, Rania Hentati and Jean-Luc Prigent also focus on hedge funds. They have validated the non-normal distribution of univariate returns of hedge funds and show nonlinearity characterizing their distribution. Their results imply the inadequacy of using the linear correlation coefficients to describe the dependency between two variables in this context. The use of this coefficient is compromised outside the multivariate Gaussian case. After presenting methods for choosing dependency structures, three empirical studies have been conducted. The authors have shown that the

Introduction

xv

dependency structure between hedge funds, equities (represented by the MSCI Free Word index), and bonds (represented by the JPM Global Bond Index) can be modeled correctly by the Student copula. Their empirical validation for choosing the best structure of dependence also enables them to justify the use of copulas. Chapters 5 and 6 look at the exchange rate market. Chapter 5 by Iuliana Matei aims to study the determinants of realignment expectations for 14 European countries over the period 2001:01–2009:12. The author shows that standard macroeconomic phenomena and financial crises over the selected period exerted a significant and positive impact on European’s realignment expectations, indicating meaningful relationships between realignment expectations and financial crises (both systemic and nonsystemic crises) and macroeconomic variables. Chapter 6 by Mohamed El Hedi Arouri and Fredj Jawadi looks at the specification of the linkages between oil prices and exchange markets over the last period. Applying recent linear and nonlinear econometric techniques over the period 1973–2009, the authors point out significant linkages between oil price and the US currency exchange rate. They highlight strong evidence of nonlinear mean-reversion between the oil and currency markets. In particular, their findings show that exchange rates are not a fundamental of oil prices, but exchange rate changes help to forecast oil prices in the short run. Chapters 7 and 8 investigate modeling of economic time series with different econometric tools. In Chapter 7, Se´bastien Pommier and Fabien Rondeau investigate empirical evidence of the main sources of economic interdependencies in Europe. While applying cointegration tools and carrying out panel estimations, the authors show strong evidence in favor of a positive relationship between openness, country size, knowledge accumulation, and the long-run sensitivity to European income. The authors conclude that the European income spillovers are not explained by specialization of trade and production and that those countries which benefit the most from economic integration are the largest and invest the most in R&D. Chapter 8 by Dominique Gue´gan and Patrick Rakotomarolahy develops ‘‘Alternative methods for forecasting GDP.’’ In particular, this chapter aims to forecast GDP using the nonparametric technique known as the multivariate nearest neighbors method and to provide asymptotic properties for this method. Using the multivariate nearest neighbors method, the authors provide better forecasts of the Euro area monthly economic indicator and quarterly GDP than with a competitive linear VAR model. The authors also provide the asymptotic normality of this k-nearest neighbor regression estimator for dependent time series, and thereby produce confidence intervals for the point forecast in their time series. In the final chapter, Hachemi Ben Ameur examines the constant proportion portfolio insurance (CPPI) method, when the multiple is allowed

xvi

Introduction

to vary. In this framework, the author provides explicit values of the multiple as a function of the past asset returns and other state variables. He shows how the multiple can be chosen to satisfy the guarantee condition at a given level of probability and for particular market conditions. According to this chapter, a new multiple can be determined according to the distributions of the risky asset log return and volatility. Overall, the findings of these chapters yield several important implications regarding the dynamics of economic and financial series along with the financial crisis and its main consequences. Indeed, the empirical results point out strong evidence of nonlinearity and asymmetry in most time series under consideration. The presupposed hypotheses of independency and efficiency seem to be strongly rejected and further evidence of strong integration and comovements between financial markets (stock prices, exchange rates, hedge funds) is shown. Furthermore, emerging markets appear not to be spared from the global financial crisis, since they are economically and financially integrated with economies of developed countries. The financial crisis is explored in different types of financial assets, such as stocks, hedge funds, and bond yields. From a methodological point of view, it appears that nonlinear models fit the dynamics of economic and financial series better than linear models. This may result from the fact that nonlinearity better enables capture of structural breaks and discontinuities induced by market frictions and financial crises and the dynamics of economic and financial series along with the properties of business cycles through different regimes. The research topics discussed in these chapters can be seen as a rapid look into mainstream, cutting-edge areas of economic and financial time-series modeling. The findings should provide further areas of innovative research. For example, Chapter 1 opens new and challenging avenues for understanding the dynamics of the relationship between the housing sector, stock market and government bond developments, and the banking system. Chapter 2 analyzes the evolution of emerging and developed stock price comovements. Chapters 3 and 4 yield some evidence of hedge fund properties over the financial crisis. Chapters 5 and 6 offer an overview of exchange rate markets and their dynamics over the last decades. The last chapters develop new methods to describe and forecast the main economic cycle properties and the dynamics of macroeconomic time series.

Acknowledgement Fredj Jawadi thanks Amiens School of Management for financing the first International Symposium in Computational Economic and Finance

Introduction

xvii

(ISCEF2010), which was organized in Sousse (Tunisia), on February 25–27, 2010 (http://www.iscef.com). The chapters of this volume are selected from among the papers that were presented in the ISCEF2010. Fredj Jawadi William A. Barnett Editors

Chapter 1

Collateralizable Wealth, Asset Returns, and Systemic Risk: International Evidence Ricardo M. Sousaa,b a

Department of Economics and Economic Policies Research Unit (NIPE), University of Minho, Campus of Gualtar, 4710-057 – Braga, Portugal, e-mail: [email protected] b Financial Markets Group (FMG), London School of Economics, Houghton Street, London WC2 2AE, UK, e-mail: [email protected]

Abstract Purpose – The purpose of this chapter is to assess the role of collateralizable wealth and systemic risk in explaining future asset returns. Methodology/approach – To test this hypothesis, the chapter uses the residuals of the trend relationship among housing wealth and labor income to predict both stock returns and government bond yields. Specifically, it shows that nonlinear deviations of housing wealth from its cointegrating relationship with labor income, hwy, forecast expected future returns. Findings – Using data for a set of industrialized countries, the chapter finds that when the housing wealth-to-income ratio falls, investors demand a higher risk premium for stocks. As for government bond returns: (i) when they are seen as a component of asset wealth, investors react in the same manner and (ii) if, however, investors perceive the increase in government bond returns as signaling a future rise in taxes or a deterioration of public finances, then they interpret the fall in the housing wealth-to-income ratio as a fall in future bond premia. Finally, this work shows that the occurrence of crisis episodes amplifies the transmission of housing market shocks to financial markets. Originality/value of chapter – These findings are novel. They also open new and challenging avenues for understanding the dynamics of the relationship between the housing sector, stock market and government bond developments, and the banking system. International Symposia in Economic Theory and Econometrics, Vol. 20 F. Jawadi and W.A. Barnett (Editors) Copyright r 2010 by Emerald Group Publishing Limited. All rights reserved ISSN: 1571-0386/DOI: 10.1108/S1571-0386(2010)0000020006

2

Ricardo M. Sousa

Keywords: housing wealth, labor income, stock returns, government bond returns, crises JEL Classification: E21, E44, D12

1. Introduction The theoretical and empirical literature has shown that credit markets are not perfect and are characterized by the lack of arbitrage and rationing (Stiglitz and Weiss, 1981). Besanko and Thakor (1987) argue that these problems could be avoided if borrowers had enough collateralizable wealth. In fact, banks would be able to offer two different contracts to prospective customers: (i) one requiring a high collateral (and a low interest rate), therefore, attracting low-risk individuals; and (ii) another one requiring less collateral (and a high interest rate), thus favoring high-risk entrepreneurs. In addition, the efficiency of the housing finance system is of key interest to financial institutions, homeowners, and policy makers. Liquidity and collateralizable wealth play, therefore, a major role for asset pricing. First, liquidity shocks are positively correlated with shocks to returns (Jones, 2002). Second, assets have higher expected returns when they are positively correlated with aggregate market liquidity (Pastor and Stambaugh, 2003; Acharya and Pedersen, 2005). Third, assets with high transaction costs or illiquid assets normally trade at a discount (Brennan and Subrahmanyam, 1996). While differences in expected returns are typically explained by differences in risk, the covariance of portfolio returns and contemporaneous consumption growth does not fully explain the cross-sectional variation (Mankiw and Shapiro, 1986; Breeden et al., 1989). As a result, the identification of the economic sources of risk remains an important issue. Moreover, given the strong linkages between housing market developments and stock market dynamics, many authors started to consider features of those markets in asset pricing models (Lustig and van Nieuwerburgh, 2005; Yogo, 2006; FernandezCorugedo et al., 2007; Piazzesi et al., 2007; Sousa, 2007). This chapter addresses the role of collateralizable wealth in analyzing predictability of both stock and government bond returns for a set of industrialized countries. Specifically, I assess the forecasting power of the nonlinear deviations of housing wealth from its cointegrating relationship with labor income, hwy, for expected future returns. The rationale behind this linkage lies on the fact that a decrease in housing prices reduces the value of housing in providing collateral services and, therefore, increases household’s exposure to idiosyncratic risk.

Collateralizable Wealth, Asset Returns, and Systemic Risk

3

Consequently, a decrease in the ratio of asset wealth to human wealth predicts higher stock returns. As for government bond returns, one needs to understand the way government debt is perceived by the agents. If government bonds are seen as a component of asset wealth, then investors demand a higher bond risk premium when they face a fall in the ratio of housing wealth-to-income. If, however, the issuance of government debt is understood as leading to an increase in future taxation or as a symptom of public finance deterioration, then investors will interpret the fall in the housing wealth-to-income ratio as predicting a decrease in future government bond returns. I show that the housing wealth-to-income ratio, hwy, predicts both stock and government bond returns, which highlights the characteristic of housing as providing collateral to the banking system. It also emphasizes the important channel by which shocks originated in the housing sector are transmitted to risk premium in asset markets. The empirical findings suggest that the predictive power of hwy for real stock returns is substantial, ranging between 6% (the US), 8% (Finland and the UK), and 10% (Australia) over the next 4 quarters. With regard to government bond returns, the analysis shows that one can cluster the set of countries into two groups. In the first group (which includes Australia, Denmark, Finland, the Netherlands, and Spain), hwy has an associated coefficient with negative sign in the forecasting regressions. The predictive power is, particularly, large for the Netherlands (11%), Finland (13%), and Spain (49%). This, therefore, corroborates the idea that government debt is seen as part of the investor’s asset wealth, which implies that agents exhibit a non-Ricardian behavior. In the second group (which includes Belgium, Canada, France, Germany, Ireland, Italy, Japan, Sweden, the UK, and the US), the forecasting regressions show that hwy has a positive coefficient. Specifically, the predictive ability of hwy is large for Germany (11%), Ireland (12%), Belgium (28%), and the US (29%). Consequently, agents in these countries perceive the rise in government bond returns rather as a deterioration of public finances and as signaling an increase in future taxation, that is, they behave in a Ricardian way. Finally, I ask about the importance of episodes of crisis in amplifying the transmission of shocks in the housing market to the financial system. In particular, I assess whether the occurrence of systemic versus nonsystemic crises can help improving our understanding about the linkages between housing and financial markets. I show that the predictive power of future asset returns is indeed improved when one takes into account the presence of such phenomena, especially, the systemic ones. The robustness of the results is analyzed in several directions. In fact, I show that: (i) the inclusion of additional control variables does not change the predictive power of hwy and (ii) models that include hwy perform better

4

Ricardo M. Sousa

than the autoregressive and the constant expected returns benchmark models. The research presented in this chapter is indebted to the work of Lettau and Ludvigson (2001). However, the authors use the consumer’s intertemporal budget constraint to explore the predictive ability of the deviations of consumption from its long-run relationship with aggregate wealth and labor income, cay, for stock returns. In contrast, I use the structure of the preferences of the representative agent to assess the forecasting power of the deviations of housing wealth from its equilibrium relationship with labor income, hwy, for both stock returns and government bond yields. This work is organized as follows. Section 2 reviews the literature on the predictability of asset returns. Section 3 describes the theoretical approach. Section 4 discusses the empirical results from the forecasting regressions for stock returns and government bond yields. Section 5 provides the robustness analysis. Section 6 analyzes the role of systemic. Finally, in Section 7, I conclude and discuss the implications of the findings.

2. Literature Review In this section, I review the literature on the predictability of stock returns and government bond returns, in particular, by highlighting the works that focus on the transmission of housing market developments to the financial system. 2.1. Predictability of Stock Returns Risk premium is generally considered as reflecting the ability of an asset to ensure against consumption fluctuations. The empirical evidence has, however, shown that the covariance of returns across portfolios and contemporaneous consumption growth is not sufficient to justify the differences in expected returns. In fact, the literature on asset pricing has emphasized the role of market inefficiencies (Fama, 1998; Fama and French, 1996), the rational response of agents to time-varying investment opportunities that is driven by variation in risk aversion (Constantinides, 1990) and by changes in the joint distribution of consumption and asset returns (Duffee, 2005), and different models of economic behavior. These explanations also justify why expected excess returns on assets appear to vary with the business cycle. Therefore, different economically motivated variables have been developed to capture time variation in expected returns and document long-term predictability. Lettau and Ludvigson (2001) show that the transitory deviation from the common trend in consumption, aggregate wealth, and

Collateralizable Wealth, Asset Returns, and Systemic Risk

5

labor income is a strong predictor of stock returns, as long as the expected returns to human capital and consumption growth are not too volatile. Bansal and Yaron (2004) find that the long-run risk, that is, the exposure of assets’ cash flows to consumption is an important determinant of risk premium. Julliard (2004) emphasizes the role of labor income risk, while Parker and Julliard (2005) measure the risk of a portfolio by its ultimate risk to consumption, that is, the covariance of its return and consumption growth over the quarter of the return and many following quarters. Lustig and van Nieuwerburgh (2005) show that the housing collateral ratio can shift the conditional distribution of asset prices and consumption growth. Yogo (2006) and Piazzesi et al. (2007) emphasize the role of nonseparability of preferences in explaining the countercyclical variation in equity premium, while Fernandez-Corugedo et al. (2007) focus on the relative price of durable goods. Sousa (2007) shows that housing can be used as a hedge against wealth shocks. Chien and Lustig (2010) find that accounting for the importance of collateralizable wealth, namely, by allowing agents to file for bankruptcy, allows one to improve asset pricing predictions.

2.2. Predictability of Bond Returns In contrast with the literature on the predictability of stock returns, there are just a few studies that try to explain the factors undermining bond risk premia. Fama and Bliss (1987) show that the spread between the n-year forward rate and the one-year yield can forecast the n-year excess bond returns. Campbell and Shiller (1991) find that excess bond returns can be predicted by the Treasury yield spreads. More recently, Cochrane and Piazzesi (2005) highlight that a linear combination of forward rates explains up to 44% of the variation in next year’s excess returns on bonds with maturities ranging from one to five years. While these findings imply that bond risk premium is time varying, they are, in general, silent regarding its relationship with macroeconomic magnitudes. Campbell and Cochrane (1999) suggest that risk premia on equity reflects a slow-moving habit that is driven by shocks to aggregate consumption. Despite the linkages between equity risk premia and the macroeconomic fundamentals addressed in the above-mentioned works, their importance for bond risk premia has been typically neglected. Moreover, the existing empirical evidence tends to show that excess bond returns can be forecasted not by macroeconomic variables such as aggregate consumption or inflation, but rather by pure financial indicators, such as forward spreads and yield spreads. For instance, Ludvigson and Ng (2009) find marked countercyclical variation in bond risk premia.

6

Ricardo M. Sousa

3. Theoretical Framework and Empirical Approach 3.1. Theoretical Consideration: Housing Wealth and Risk Premium I assume that there is a continuum of agents who consume nondurable consumption, ct, and housing services (from which they derive utility or collateral services), hwt, and are endowed with stochastic labor income, gt(it, at), where it represents the idiosyncratic event and at denotes the aggregate event. The household maximizes utility, that is: 1 XX

Uðc; hwÞ ¼

bt pðst js0 Þu½ct ðst Þ; hwt ðst Þ,

(1)

st js0 t¼0

where b is the time discount factor, st represents the state of the economy, p(st|s0) denotes the probability of state st given the initial state s0, and preferences are specified by ð1Þ=

uðct ; hwt Þ ¼

½ct

ð1Þ=

þ chwt ð1gÞ=ð1Þ , 1g

(2)

where cW0 captures the importance of housing wealth in the utility function, e is the intratemporal elasticity of substitution between consumption and services from housing wealth, and g is the coefficient of risk aversion. The solvency constraints are restrictions on the value of the household’s consumption claim net of its labor income claim, that is: Lst ½ct ðst Þ þ rt ðat Þhwt ðst Þ  Lst ½yt ðst Þ,

(3)

where Lst ½d t ðst Þ represents the price of a claim to dt(st), and rt is the rental price of housing services. The strength of these constraints is determined by the ratio of asset wealth to human wealth (i.e., the housing wealth-to-income ratio), hwy, hwyt ðat Þ ¼

Lzt ½rhwa  , Lzt ½ca 

(4)

where hwa and ca correspond, respectively, to aggregate housing wealth and aggregate consumption. Equilibrium allocations and prices will depend on the consumption weight y as follows: (1) if the household does not switch to a state with a binding constraint, it is y0t ðy; st Þ; and (2) if it switches, then the new weight is the cutoff level yt ðyt ; at Þ. In order to obtain aggregate consumption, one integrates over the new R household weights, that is, zat ðat Þ ¼ y0t ðy; st ÞdFt ðy; at Þ, where Ft ð; at Þ represents the distribution over weights at the start of period t. The

Collateralizable Wealth, Asset Returns, and Systemic Risk

7

consumption share of an agent can then be represented as the ratio of his or her consumption weight to the aggregate consumption weight, ct ðy; st Þ ¼ y0t ðy; st Þ  cat ðat Þ=zat ðat Þ, and, similarly, for the housing wealth share of an agent, hwt ðy; st Þ ¼ y0t ðy; st Þ  hwat ðat Þ=zat ðat Þ; where zat ðat Þ defines a nondecreasing stochastic process. As the ratio of housing wealth-to-income, hwy, decreases, the cutoff levels for the consumption weights increase, yðyt ; at Þ=zat ðat Þ; and, if the consumer moves to a state where the constraint is binding, then the cutoff level for the consumption share equals the household’s labor income share. As a result, when the ratio of housing wealth-to-income, hwy, decreases, the household’s exposure to labor income shocks increases and a higher risk premium is demanded. Consequently, it should predict a rise in future stock returns. In contrast with stocks, an increase in government bond yields may not be seen as a rise in wealth, but merely perceived as signaling a future increase in taxes. Therefore, when agents see government debt as a wealth component, one should observe a behavior similar to the one found for stocks; otherwise, deviations in the long-term trend among housing wealth and income should be positively related with future government bond returns. 3.2. Empirical Counterpart: Housing Wealth-to-Income Ratio Real per capita housing wealth, hw, and labor income, y, are nonstationary. As a result, I estimate the following vector error-correction model (VECM): " # D logðhwt Þ ¼ a½logðhwt Þ þ $ logðyt Þ þ Wt þ w D logðyt Þ " # K D logðhwtk Þ X þ Dk þ t , D logðytk Þ k¼1 where t denotes the time trend and w is a constant. The K error-correction terms allow one to eliminate the effect of regressor’s endogeneity on the distribution of the least-squares estimators of [1, $, W, w]. The components log(hw) and log(y) are stochastically cointegrated with the cointegrating vector [1, $, w]. I also impose the restriction that the cointegrating vector eliminates the deterministic trends, so that logðhwt Þ þ $ logð yt Þ þ Wt þ w is stationary. Then, the ratio of housing wealth-to-income, hwy, is measured as the deviation from the cointegration relationship, that is: ^ þ w^ . ^ logðyt Þ þ Wt hwyt ¼ logðhwt Þ þ $

(6)

Given that the Ordinary Least Squares (OLS) estimators of the cointegration parameters are superconsistent, one can use the ratio of housing

8

Ricardo M. Sousa

wealth-to-income, hwy, as a regressor without needing an errors-in-variables standard error correction.

4. Results 4.1. Data The data are quarterly, post-1960, and include 16 countries (Australia, since 1970:1; Austria, since 1978:2; Belgium, since 1980:2; Canada, since 1965:1; Denmark, since 1977:1; Finland, since 1979:1; France, since 1970:2; Germany, since 1965:1; Ireland, since 1975:4; Italy, since 1971:4; Japan, since 1965:1; the Netherlands, since 1975:1; Spain, since 1978:1; Sweden, since 1977:1; the UK, since 1961:2; and the US, since 1965:1). It, therefore, cover the last 30–50 years of data. All series – with the obvious exceptions of stock returns and government bond yields – are deflated with consumption deflators, expressed in logs of per capita terms and seasonally adjusted. Labor income is approximated by the compensation series of the National Institute of Economic and Social Research (NIESR) Institute. In the case of the US, I follow Lettau and Ludvigson (2001). As for the UK, I follow Sousa (2010). Wealth includes financial and housing wealth, and data come from National Central Banks, the Eurostat, the Bank for International Settlements (BIS), and the United Nation’s Bulletin of Housing Statistics for Europe and North America. Stock returns are computed using the share price index provided by the International Financial Statistics (IFS) of the International Monetary Fund (IMF) and the dividend yield ratio provided by Datastream. The 10-year government bond yield data are also provided by the IFS of the IMF. The government finance data normally refer to the central government; that is, it excludes the local and/or the regional authorities. It is typically disseminated through the monthly publications of the General Accounting Offices, Ministries of Finance, National Central Banks, and National Statistical Institutes of the respective countries. The latest figures are also published in the Special Data Dissemination Standard (SDDS) section of the IMF website. Data for population are taken from Organisation for Economic CoOperation and Development’s (OECD) Main Economic Indicators and interpolated from annual series.

4.2. The Long-Run Relation I first use the Augmented Dickey and Fuller (1979), the Phillips and Perron (1988), and the Kwiatkowski et al. (1992) tests to determine the existence of

Collateralizable Wealth, Asset Returns, and Systemic Risk

9

unit roots in the series of housing wealth and labor income and conclude that they are first-order integrated, I(1). Next, I analyze the existence of cointegration among the two series using the methodology of Engle and Granger (1987), Johansen and Juselius (1990), Phillips and Ouliaris (1990), and MacKinnon (1996) and find evidence that supports that hypothesis. Finally, I estimate the VECM as expressed in Equation (5). Table 1 shows the estimates (ignoring the coefficient estimates on the constant and the time trend) for the shared relationship among housing wealth and income. It can be seen that, with the exceptions of Canada, France, and Spain, the long-run elasticity of housing wealth with respect to labor income is positive, implying that the two aggregates tend to share a positive long-run path. The table also presents the unit root tests to the residuals of the cointegration relationship and supports the idea that they are stationary. 4.3. Forecasting Stock Returns Section 3 shows that transitory deviations from the long-run relationship among housing wealth and income, hwyt, mainly reflect agents’ expectations of future changes in asset returns. Therefore, I look at real stock returns (denoted by SRt) for which quarterly data are available. They should provide a good proxy for the nonhuman component of asset wealth. Table 2 summarizes the forecasting power of hwyt for different horizons. It reports estimates from OLS regressions of the H-period real stock return, SRtþ1þ    þSRtþH, on the lag of hwyt. I estimate the following model: H X

SRtþh ¼ a þ bhwyt1 þ t .

(7)

h¼1

Note that long-horizon returns are calculated by summing the (continuously compounded) quarterly returns. This implies that the observations on longhorizon returns overlap, which possibly biases the different test statistics toward rejecting the null hypothesis of no predictability more often than is correct (Nelson and Kim, 1993; Stambaugh, 1999; Valkanov, 2003; Ang and Bekaert, 2006). Nevertheless, one should emphasize that these works focus on the predictive ability of the dividend yield and the price-to-earnings ratio, which are very persistent regressors. In contrast, I assess the forecasting power of the deviations from the equilibrium relationship between housing wealth and labor income, hwy, which exhibit much less persistence. Thus, the above-mentioned problems become less severe. Additionally, Lettau and Ludvigson (2001), Whelan (2008), and Sousa (2010) find that the bias does not impact on the predictive ability of a wide range of variables in the forecasting regressions for stock returns. Finally, the adopted methodology

10

Table 1:

Ricardo M. Sousa

^ þ w^ Cointegration Estimations: hwyt ¼ logðhwt Þ þ $ ^ logð yt Þ þ Wt $ ^

Australia Austria Belgium Canada Denmark Finland France Germany Ireland Italy Japan Netherlands Spain Sweden UK US

1.89*** (2.57) 27.75*** (4.62) 4.73*** (8.37) 10.20*** (2.93) 12.38*** (3.42) 1.80*** (3.81) 4.01*** (2.95) 0.54*** (2.87) 4.09*** (5.58) 1.25*** (3.00) 2.18*** (5.79) 4.17*** (8.31) 20.49* (1.40) 4.63*** (3.47) 2.59*** (3.73) 4.48*** (9.31)

Augmented Dickey and Fuller (1979) t-Statistic

MacKinnon (1996) Critical values

Kwiatkowski et al. (1992) LM-statistic

Kwiatkowski et al. (1992) Critical values

Lags: automatic based on SIC

5%

10%

Bandwidth: Newey–West using Bartlett kernel

5%

10%

1.98

2.88

2.58

0.15

0.46

0.35

3.90

2.88

2.58

0.64

0.46

0.35

5.22

2.88

2.58

0.47

0.46

0.35

3.11

2.88

2.58

0.49

0.46

0.35

2.13

2.89

2.58

0.74

0.46

0.35

3.22

2.88

2.58

0.59

0.46

0.35

3.55

2.89

2.58

0.59

0.46

0.35

6.27

2.88

2.58

0.34

0.46

0.35

2.79

2.88

2.58

0.73

0.46

0.35

3.31

2.89

2.58

0.34

0.46

0.35

2.80

2.88

2.58

0.68

0.46

0.35

4.88

2.88

2.58

0.22

0.46

0.35

1.79

2.91

2.59

0.70

0.46

0.35

3.05

2.88

2.58

0.52

0.46

0.35

3.09

2.88

2.58

0.10

0.46

0.35

2.97

2.89

2.58

0.27

0.46

0.35

Note: Newey-West (1987) corrected t-statistics appear in parentheses. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

0.20*** (2.61) [0.05] 0.00 (0.59) [0.00] 0.17*** (2.92) [0.08] 0.00 (0.34) [0.00] 0.03*** (2.33) [0.03] 0.11** (2.07) [0.02] 0.01 (0.43) [0.00] 0.27** (2.38) [0.04]

1

0.40*** (3.23) [0.13] 0.00 (0.43) [0.00] 0.34*** (3.80) [0.12] 0.00 (0.13) [0.00] 0.05*** (2.76) [0.04] 0.25*** (2.88) [0.05] 0.03 (0.61) [0.00] 0.56*** (3.27) [0.06]

2

0.53*** (3.58) [0.16] 0.00 (0.46) [0.00] 0.49*** (4.28) [0.15] 0.00 (0.13) [0.00] 0.08*** (2.85) [0.04] 0.39*** (3.12) [0.06] 0.03 (0.51) [0.00] 0.87*** (4.03) [0.08]

3

4 0.64*** (3.70) [0.10] 0.00 (0.20) [0.00] 0.62*** (4.41) [0.17] 0.01 (0.36) [0.00] 0.11*** (3.18) [0.05] 0.52*** (3.24) [0.08] 0.01 (0.22) [0.00] 1.18*** (4.75) [0.11]

Forecast horizon H

0.99*** (3.83) [0.12] 0.02* (1.92) [0.01] 0.93*** (4.71) [0.16] 0.04 (1.37) [0.02] 0.23*** (3.96) [0.11] 1.06*** (3.53) [0.13] 0.10 (1.06) [0.01] 2.10*** (7.39) [0.16]

8

US

UK

Sweden

Spain

Netherlands

Japan

Italy

Ireland

Forecasting Real Stock Returns: Estimated Effect of hwy

0.04 (0.87) [0.00] 0.25** (2.14) [0.06] 0.08 (1.20) [0.02] 0.02 (0.23) [0.00] 0.01 (1.38) [0.02] 0.16*** (2.80) [0.07] 0.20* (1.75) [0.06] 0.16* (1.81) [0.03]

1 0.10 (1.31) [0.01] 0.41** (2.07) [0.05] 0.13 (1.18) [0.02] 0.00 (0.04) [0.00] 0.01 (1.24) [0.02] 0.29*** (3.27) [0.09] 0.34* (1.71) [0.06] 0.33** (2.16) [0.04]

2 0.15 (1.53) [0.01] 0.47* (1.92) [0.04] 0.17 (1.22) [0.02] 0.03 (0.18) [0.00] 0.01 (1.08) [0.01] 0.38*** (3.44) [0.09] 0.46* (1.78) [0.08] 0.45** (2.32) [0.05]

3

4 0.18 (1.50) [0.01] 0.43 (1.63) [0.02] 0.19 (1.11) [0.02] 0.06 (0.30) [0.00] 0.02 (0.98) [0.01] 0.47*** (3.65) [0.10] 0.54* (1.82) [0.08] 0.62*** (2.71) [0.06]

Forecast horizon H

0.02 (0.10) [0.00] 0.23 (0.87) [0.00] 0.07 (0.31) [0.00] 0.28 (0.95) [0.01] 0.01 (0.39) [0.00] 0.86*** (5.25) [0.18] 0.59** (2.13) [0.05] 1.42*** (4.89) [0.16]

8

Note: Newey-West (1987) corrected t-statistics appear in parentheses. Adjusted R2 is reported in square brackets. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

Germany

France

Finland

Denmark

Canada

Belgium

Austria

Australia

Table 2:

Collateralizable Wealth, Asset Returns, and Systemic Risk 11

12

Ricardo M. Sousa

is standard in the empirical finance literature (Lettau and Ludvigson, 2001; Julliard, 2004; Lustig and van Nieuwerburgh, 2005; Santos and Veronesi, 2006; Yogo, 2006; Fernandez-Corugedo et al., 2007; Piazzesi et al., 2007; Sousa, 2010). Keeping these questions in mind, Table 2 shows that hwyt is statistically significant for a large number of countries and the point estimates of the coefficient are large in magnitude. Moreover, its sign is negative and statistically significant for Australia, Germany, Finland, Italy, the UK, and the US. These results are in line with the framework presented in Section 3, suggesting that investors expect a fall in future stock returns when they observe a rise in the housing wealth-to-income ratio. It can also be seen that the trend deviations explain an important fraction of the variation in future real returns (as described by the adjusted R2), in particular, at horizons spanning from 4 to 8 quarters. In fact, at the 4-quarter horizon, hwyt explains 5% (Japan), 6% (the US), 8% (Finland and the UK), and 17% (Belgium) of the real stock return. In contrast, its forecasting power is poor for countries such as Austria, Canada, France, Ireland, Japan, the Netherlands, and Spain. 4.4. Forecasting Government Bond Returns I now look at the power of hwyt in predicting government bond yields (denoted by BRt) for which quarterly data are available. Table 3 reports the forecasting ability of hwyt for different horizons. It provides estimates from OLS regressions of the H-period real government bond return, BRtþ1þyþBRtþH, on the lag of hwyt, as described by the model: H X

BRtþh ¼ a þ bhwyt1 þ t .

(8)

h¼1

One can see that hwyt is statistically significant for almost all countries (with the exception of Austria) and the associated coefficient are, in general, large in magnitude. At the 4-quarter horizon, hwyt explains 11% (Germany and the Netherlands), 12% (Ireland), 13% (Finland), 28% (Belgium), 29% (the US), and 49% (Spain) of the real bond returns. Interestingly, the results suggest that the sign of the coefficient associated to hwyt is negative for Australia, Finland, the Netherlands, and Spain, and positive for Belgium, Canada, France, Germany, Ireland, Italy, Japan, Sweden, the UK, and the US. This piece of evidence corroborates the idea that government debt is seen as part of investor’s wealth for the first set of countries: in the outcome of a fall in the ratio of housing wealth-to-income,

1

0.16* (1.94) [0.03] 0.00 (1.24) [0.01] 0.32*** (6.92) [0.28] 0.02*** (3.81) [0.07] 0.05 (1.50) [0.02] 0.32*** (5.73) [0.12] 0.04** (2.36) [0.03] 0.20*** (4.90) [0.08]

3

8

0.21** 0.38*** (1.96) (1.87) [0.03] [0.03] 0.00 0.00 (1.26) (0.07) [0.01] [0.00] 0.43*** 0.80*** (6.92) (6.65) [0.28] [0.27] 0.04*** 0.09*** (4.22) (6.02) [0.08] [0.14] 0.07* 0.18** (2.32) (1.72) [0.02] [0.03] 0.43*** 0.91*** (5.95) (6.68) [0.13] [0.15] 0.05** 0.10** (2.31) (2.26) [0.03] [0.03] 0.30*** 0.81*** (5.78) (9.42) [0.11] [0.26]

4

2

3

4

0.08*** 0.17*** 0.25*** 0.34*** (4.63) (4.59) (4.57) (4.61) [0.11] [0.11] [0.12] [0.12] Italy 0.05** 0.08* 0.10 0.11 (2.01) (1.79) (1.50) (1.18) [0.02] [0.02] [0.01] [0.01] Japan 0.05 0.10 0.14** 0.17*** (0.96) (1.38) (2.15) (3.41) [0.02] [0.03] [0.06] [0.09] Netherlands 0.06*** 0.13*** 0.20*** 0.25*** (3.04) (3.94) (4.07) (3.90) [0.08] [0.11] [0.11] [0.11] Spain 0.02*** 0.03*** 0.05*** 0.06*** (7.61) (7.98) (8.58) (7.69) [0.44] [0.46] [0.47] [0.49] Sweden 0.04* 0.05 0.08* 0.10** (1.82) (1.62) (1.84) (2.20) [0.03] [0.03] [0.03] [0.04] UK 0.01 0.03 0.06 0.10 (0.54) (0.77) (1.14) (1.46) [0.00] [0.00] [0.01] [0.01] US 0.21*** 0.42*** 0.63*** 0.81*** (7.44) (7.56) (7.60) (7.38) [0.30] [0.31] [0.30] [0.29]

Ireland

1

Forecast horizon H

0.79*** (5.30) [0.17] 0.04 (0.22) [0.00] 0.32*** (3.25) [0.09] 0.46*** (3.59) [0.10] 0.12*** (10.57) [0.49] 0.13 (1.35) [0.02] 0.31*** (2.59) [0.03] 1.41*** (6.28) [0.25]

8

Note: Newey-West (1987) corrected t-statistics appear in parentheses. Adjusted R2 is reported in square brackets. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

0.11** (1.95) [0.03] 0.00 (1.62) [0.01] 0.22*** (6.92) [0.28] 0.02*** (3.46) [0.06] 0.03 (1.29) [0.01] 0.21*** (5.25) [0.11] 0.03** (2.36) [0.03] 0.12*** (4.07) [0.06]

2

Forecast horizon H

Forecasting Real Government Bond Returns: Estimated Effect of hwy

0.05** (1.96) [0.03] Austria 0.00 (0.30) [0.00] Belgium 0.11*** (6.62) [0.25] Canada 0.01*** (3.12) [0.05] Denmark 0.01 (1.12) [0.01] Finland 0.10*** (4.37) [0.09] France 0.01** (2.35) [0.03] Germany 0.05*** (2.70) [0.03]

Australia

Table 3:

Collateralizable Wealth, Asset Returns, and Systemic Risk 13

14

Ricardo M. Sousa

agents allow consumption to rise as they expect future yields to increase. As for the second set of countries, agents perceive the rise in government bond returns as a deterioration of public finances and an increase in future taxation. As a result, they reduce consumption when they observe a fall in the ratio of housing wealth-to-income. In practice, these results largely reflect higher sustainability of public finances in the first set of countries. Additionally, they characterize the frequent swings in public deficits and government debt and the concerns about the long-term sustainability of public finances in the second group of countries.

5. Robustness Analysis 5.1. Additional Control Variables In this section, I assess the robustness of the previous results, namely, by considering additional control variables. Shiller (1984) and Campbell and Shiller (1988) show that the price-todividend ratio and the price-to-earnings ratio have predictive power for stock returns. Lamont (1998) suggests that the ratio of dividends to earnings is also a good predictor of stock returns at quarterly frequency. The relative T-bill rate, the term spread, and the default spread are also shown to have forecasting power (Fama and French, 1989; Hodrick, 1992). Table 4 reports the estimates from one-quarter-ahead forecasting regressions that include the dividend yield ratio (DivYldt) as an additional variable. It only displays information about countries for which data on the dividend yield ratio (DivYldt) is available. The results show that the coefficient estimates of hwy and their statistical significance do not change with respect to the findings of Table 2 where only hwy was included in the set of explanatory variables. Moreover, the dividend yield ratio (DivYldt) seems to provide some relevant information about future asset returns: it is statistically significant in a large number of regressions. By its turn, Table 5 summarizes the estimates from one-quarter-ahead forecasting regressions that include the inflation rate (Inflation) and the deficit-to-GDP ratio (Deficit) as potential determinants of future government bond yields. Brandt and Wang (2003) argue that the risk premium is driven by shocks to inflation and to aggregate consumption. Gale and Orszag (2003) highlight that budget deficits may raise nominal interest rates, because they reduce aggregate savings and increase the stock of government debt. Despite this, the literature has not provided a consensual answer yet (Engen and Hubbard, 2005).

Collateralizable Wealth, Asset Returns, and Systemic Risk

Table 4:

Australia Austria Belgium Canada Denmark Finland France Germany

15

Forecasting Real Stock Returns: Additional Control Variables hwyt1

DivYldt1

Adj. R2

0.19** (2.34) 0.00 (0.59) 0.12* (1.85) 0.00 (0.24) 0.03*** (2.33) 0.22*** (2.61) 0.01 (0.33) 0.68*** (2.98)

5.55** (2.22)

[0.08]

Ireland

[0.00]

Italy

[0.04]

Japan

[0.01]

Netherlands

[0.03]

Spain

[0.05]

Sweden

[0.01]

UK

[0.11]

US

0.43 (0.17) 3.13 (1.20)

1.65 (0.66) 1.28 (0.70) 11.33*** (2.75)

hwyt1 0.04 (0.87) 0.23** (2.02) 0.01 (0.12) 0.73** (2.44) 0.01 (1.38) 0.13** (2.46) 0.00 (0.01) 0.16* (1.68)

DivYldt1

Adj. R2 [0.00]

20.48*** (3.20) 9.94** (2.00) 6.24 (0.66)

[0.14] [0.04] [0.15] [0.02]

12.48*** (2.73) 3.60*** (0.01) 0.07 (0.04)

[0.12] [0.03] [0.03]

Note: Newey-West (1987) corrected t-statistics appear in parentheses. *, **, ***denote statistical significance at the 10%, 5%, and 1% level, respectively.

The results do not show substantial changes vis-a`-vis the findings reported in Table 3, where only hwy was considered in the set of regressors. In addition, both Inflation and the Deficit help forecasting bond returns. Therefore, this suggests that investors use government bonds to hedge against the risk of inflation. It also reveals that a deterioration of the fiscal stance is typically associated with a rise in future government bond yields.

5.2. Nested Forecast Comparisons Some recent studies (Bossaerts and Hillion, 1999; Goyal and Welch, 2003, 2004) expressed concerns about the apparent predictability of stock returns because, while a number of financial variables display significant in-sample forecasting power, they seem to have negligible out-of-sample predictive properties. In addition, the forecasting results presented so far could suffer from the ‘‘look-ahead’’ bias that arises from a long-term relationship estimated using the full sample (Brennan and Xia, 2005). In this context, some robust statistics such as the Clark and McCracken’s (2001) encompassing test (ENC-NEW), the McCracken’s (2006) equal forecast accuracy test (MSE-F), and the modified Diebold and Mariano (1995) encompassing test proposed by Harvey et al. (1998) could allow one to explore the out-of-sample performance of the forecasting model. Note,

0.05 (1.59) 0.00 (0.35) 0.04** (2.49) 0.00 (0.51) 0.01 (1.15) 0.05* (1.84) 0.02*** (4.01) 0.05** (2.47)

hwyt1

0.00 (1.10) 0.00 (0.85) 0.00 (0.28) 0.00** (2.08) 0.01*** (5.41) 0.00*** (3.87) 0.01*** (4.89) 0.00** (1.98)

Inflationt1

0.35*** (4.22) 0.02 (0.43) 0.19** (2.02)

0.11*** (4.93) 0.04 (0.05)

0.01 (0.26)

Deficitt1

[0.08]

[0.30]

[0.20]

[0.16]

[0.07]

[0.49]

[0.01]

[0.03]

Adj. R2

US

UK

Sweden

Spain

Netherlands

Japan

Italy

Ireland

0.05** (2.01) 0.06*** (3.88) 0.04 (0.47) 0.05*** (2.59) 0.01*** (6.42) 0.04 (1.51) 0.01 (0.51) 0.17*** (5.97)

hwyt1

Forecasting Real Government Bond Returns: Additional Control Variables

0.01*** (9.55) 0.01*** (4.67) 0.00 (1.33) 0.01** (2.19) 0.00 (0.62) 0.00*** (3.48) 0.02*** (7.49)

Inflationt1

0.33*** (10.02) 3.53** (2.06) 0.22*** (5.83) 0.29*** (3.11) 0.04 (0.30) 0.03 (0.46) 0.28 (1.35)

Deficitt1

[0.52]

[0.13]

[0.03]

[0.58]

[0.28]

[0.31]

[0.77]

[0.02]

Adj. R2

Note: Newey-West (1987) corrected t-statistics appear in parentheses. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

Germany

France

Finland

Denmark

Canada

Belgium

Austria

Australia

Table 5:

16 Ricardo M. Sousa

Collateralizable Wealth, Asset Returns, and Systemic Risk

17

however, that the in-sample and the out-of-sample tests are equally reliable under the null hypothesis of no predictability (Inoue and Kilian, 2004). Moreover, the results from out-of-sample forecasts where the cointegrating vector is reestimated every period using only the data available at the time of the forecast could strongly understate the predictive power of the regressor (Lettau and Ludvigson, 2001). Therefore, it would make it difficult for hwy to display forecasting power when the theory is true. Finally, Hjalmarsson (2006) shows that out-of-sample forecasting exercises are unlikely to generate evidence of predictability, even when the correct model is estimated and there is, in fact, predictability. With these caveats in mind and as a final robustness check, I make nested forecast comparisons, in which I look at the mean-squared forecasting error from a series of one-quarter-ahead out-of-sample forecasts obtained from a prediction equation that includes hwy as the sole forecasting variable and the mean-squared forecasting error from a variety of forecasting equations that do not include hwy. As a result, the unrestricted model nests the benchmark model. I consider two benchmark models: the autoregressive benchmark and the constant expected returns benchmark. In the autoregressive benchmark, I compare the mean-squared forecasting error from a regression that only includes the lagged asset return as the predictive variable with the meansquared error from regressions that include, in addition, hwy. In the constant expected returns benchmark, I compare the mean-squared forecasting error from a regression that includes a constant (as the only explanatory variable) with the mean-squared error from regressions that include, additionally, hwy. Table 6 summarizes the nested forecast comparisons for the equations of real stock returns and government bond yields. It shows that including hwy in the forecasting regressions, in general, improves over the benchmark models. This is particularly important when the benchmark model is the constant expected returns benchmark and, therefore, supports the existence of time variation in expected returns.

6. Does Systemic Risk Matter? Financial crises can be contagious and damaging and prompt quick policy responses, as they typically lead economies into recessions and sharp current account imbalances. Among the many causes of financial crises, one can refer: (i) credit booms, (ii) currency and maturity mismatches, (iii) large capital inflows, and (iv) unsustainable macroeconomic policies. Honohan and Laeven (2005) and Laeven and Valencia (2008) identify episodes of financial crises, and systemic crises include currency, debt, and banking crises. A systemic currency crisis corresponds to a nominal

18

Ricardo M. Sousa

Table 6: One-Quarter-Ahead Forecasts of Returns: hwy Model versus Constant/AR Models Real stock returns

Real government bond returns

MSEhwy/MSEconstant MSEhwy/MSEAR MSEhwy/MSEconstant MSEhwy/MSEAR Australia Austria Belgium Canada Denmark Finland France Germany Ireland Italy Japan Netherlands Spain Sweden UK US

0.978 1.003 0.964 1.003 0.990 0.992 1.003 0.986 1.002 0.976 0.996 1.005 0.996 0.969 0.975 0.991

0.980 1.004 0.990 1.004 0.995 0.994 1.003 0.988 1.002 0.992 0.996 1.004 0.997 1.002 0.969 0.997

0.988 1.003 0.870 0.979 1.000 0.957 0.988 0.990 0.945 0.994 0.995 0.962 0.753 0.992 1.003 0.842

1.004 1.004 1.005 1.000 1.005 0.997 1.004 0.994 1.004 1.003 1.000 1.005 1.005 1.005 1.003 0.997

Note: MSE – mean-squared forecasting error. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

depreciation of the currency of at least 30% and, simultaneously, at least a 10% increase in the rate of depreciation compared to the year before. A systemic debt crisis describes a situation where there are sovereign defaults to private lending and debt rescheduling programs. In a systemic banking crisis, there are a large number of defaults on corporate and financial sectors, nonperforming loans increase sharply, asset prices eventually depress, and real interest rates increase dramatically.

6.1. Systemic Crises In order to assess the importance of systemic crises, I estimate the following models: H X

SRtþh ¼ a þ bhwyt1 þ mhwyt1  SystemicCrisis þ t ,

(9)

BRtþh ¼ a þ bhwyt1 þ mhwyt1  SystemicCrisis þ t ,

(10)

h¼1 H X h¼1

Collateralizable Wealth, Asset Returns, and Systemic Risk

19

where SystemicCrisis is a dummy variable that takes the value of 1 in the presence of a systemic crisis and 0 otherwise, and H refers to the number of quarters-ahead of the forecasting exercise. Given that the effects of systemic crises may not be immediate, I consider H ¼ 4, therefore, allowing for a time lag from the date of occurrence of the crisis and the emergence of its effects. Tables 7 and 8 report the estimates from 4-quarters-ahead forecasting regressions as expressed by Eqs. (9) and (10), respectively. The results show that both the coefficient estimates of hwy do not change relative to the previous findings. Moreover, the coefficient associated to the interaction between hwy and the dummy variable for the systemic crisis is, in general, statistically significant. In addition, it has an opposite sign of the one associated with hwy, implying that investors demand a higher risk premium for both stocks and government bonds during systemic crises.

6.2. Nonsystemic Crises Finally, I analyze the impact of nonsystemic crises and regress the following equations: H X

SRtþh ¼ a þ bhwyt1 þ mhwyt1  NonSystemicCrisis þ t ,

(11)

BRtþh ¼ a þ bhwyt1 þ mhwyt1  NonSystemicCrisis þ t ,

(12)

h¼1

H X h¼1

where NonSystemicCrisis is a dummy variable that takes the value of 1 in the presence of a nonsystemic crisis and 0 otherwise, and H refers to the number of quarters-ahead of the forecasting exercise. Similarly, to the case of systemic crisis, I allow for a lag in the transmission of the effects of nonsystemic crises to financial markets and consider H ¼ 4. Tables 9 and 10 summarize the results from 4-quarters-ahead forecasting regressions. In general, the coefficient associated with the interaction between hwy and the dummy variable for the nonsystemic crisis is statistically significant and has, with the exception of Finland, the opposite sign of the one associated with hwy. Therefore, in the outcome of a nonsystemic crisis, investors demand a higher risk premium.

No episodes of systemic crisis 0.01 0.07 (0.43) (0.76) 0.17*** 0.13 (2.60) (1.56) No episodes of systemic crisis 0.01 4.45*** (0.19) (7.47) 1.22*** 0.38 (4.45) (0.92) [0.11]

[0.01]

[0.07]

[0.00]

(1.42)

[0.13]

Adj. R2

US

Sweden UK

Spain

Italy (0.20) Japan Netherlands

Ireland

hwyt1 * SystemicCrisis

0.10

No episodes of systemic crisis 1.10*** 1.13** (2.66) (2.18) 0.70*** 0.96** (2.81) (2.09)

No episodes of systemic crisis

No episodes of systemic crisis No episodes of systemic crisis

0.46

No episodes of systemic crisis

hwyt1

[0.07]

[0.16]

[0.02]

Adj. R2

Note: Newey-West (1987) corrected t-statistics appear in parentheses. *, **, *** denote statistically significance at the 10%, 5%, and 1% level, respectively.

Germany

Finland France

Denmark

Austria

Belgium Canada

hwyt1 * SystemicCrisis

0.84*** 0.972** (4.15) (2.05) No episodes of systemic crisis

hwyt1

Forecasting Real Stock Returns: Impact of Systemic Crises

Australia

Table 7:

20 Ricardo M. Sousa

No episodes of systemic crisis 0.04*** 0.38*** (5.55) (12.99) 0.24*** 0.34*** (2.97) (3.34) No episodes of systemic crisis 0.05** 0.88* (2.28) (1.88) 0.28*** 0.21* (4.99) (1.84) [0.12]

[0.03]

[0.09]

[0.26]

[0.11]

Adj. R2

US

Sweden UK

Spain

Japan Netherlands

Italy

Ireland

hwyt1 * SystemicCrisis

No episodes of systemic crisis 0.36*** 0.53*** (3.86) (3.93) 0.89*** 1.05*** (7.81) (4.83)

No episodes of systemic crisis

0.01 0.31 (0.05) (1.02) No episodes of systemic crisis No episodes of systemic crisis

No episodes of systemic crisis

hwyt1

[0.32]

[0.10]

[0.02]

Adj. R2

Note: Newey-West (1987) corrected t-statistics appear in parentheses. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

Germany

Finland France

Denmark

Austria

Belgium Canada

hwyt1 * SystemicCrisis

0.40*** 0.97*** (3.71) (4.40) No episodes of systemic crisis

hwyt1

Forecasting Real Government Bond Returns: Impact of Systemic Crises

Australia

Table 8:

Collateralizable Wealth, Asset Returns, and Systemic Risk 21

[0.12]

Adj. R2

UK US

Netherlands Spain Sweden

Ireland Italy Japan

No episodes No episodes 0.00 (0.02) No episodes No episodes 0.57*** (3.69) No episodes No episodes

hwyt1

of nonsystemic crisis of nonsystemic crisis 0.69** (2.13) of nonsystemic crisis of nonsystemic crisis 0.47 (1.40) of nonsystemic crisis of nonsystemic crisis

hwyt1 * SystemicCrisis

[0.12]

[0.04]

Adj. R2

Note: Newey-West (1987) corrected t-statistics appear in parentheses. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

France Germany

of nonsystemic crisis of nonsystemic crisis 1.74*** (3.04) of nonsystemic crisis of nonsystemic crisis

No episodes No episodes 0.42*** (2.68) No episodes No episodes

Canada Denmark Finland

hwyt1 * SystemicCrisis

No episodes of nonsystemic crisis No episodes of nonsystemic crisis No episodes of nonsystemic crisis

hwyt1

Forecasting Real Stock Returns: Impact of Nonsystemic Crises

Australia Austria Belgium

Table 9:

22 Ricardo M. Sousa

[0.24]

Adj. R2

UK US

Netherlands Spain Sweden

Ireland Italy Japan

No episodes No episodes 0.13*** (2.58) No episodes No episodes 0.12** (2.06) No episodes No episodes

hwyt1

of nonsystemic crisis of nonsystemic crisis 1.13*** (12.20) of nonsystemic crisis of nonsystemic crisis 0.09 (1.03) of nonsystemic crisis of nonsystemic crisis

hwyt1 * SystemicCrisis

[0.04]

[0.58]

Adj. R2

Note: Newey-West (1987) corrected t-statistics appear in parentheses. *, **, *** denote statistical significance at the 10%, 5%, and 1% level, respectively.

France Germany

of nonsystemic crisis of nonsystemic crisis 1.84*** (6.36) of nonsystemic crisis of nonsystemic crisis

No episodes No episodes 0.32*** (4.58) No episodes No episodes

Canada Denmark Finland

hwyt1 * SystemicCrisis

No episodes of nonsystemic crisis No episodes of nonsystemic crisis No episodes of nonsystemic crisis

hwyt1

Forecasting Real Government Bond Returns: Impact of Nonsystemic Crises

Australia Austria Belgium

Table 10:

Collateralizable Wealth, Asset Returns, and Systemic Risk 23

24

Ricardo M. Sousa

7. Conclusion This chapter explores the predictive power of the nonlinear deviations of housing wealth from its equilibrium relationship with labor income (summarized by the variable hwy) for expected future asset returns. The above-mentioned common trend summarizes agents’ expectations of both stock returns and government bond yields. In particular, when the housing wealth-to-income ratio falls (increases), forward-looking investors will demand a higher (lower) risk premium given that they will be exposed to larger (smaller) idiosyncratic shocks. As for bond yields, if government bonds are understood as another wealth component, then investors behave in the same way as for stocks. However, if the increase in government bond yields is perceived as a symptom of the deterioration of the fiscal stance, investors will interpret the fall in the wealth-to-income ratio as a fall in future bond risk premium. Using data for 16 industrialized countries, I show that the predictive power of hwy for real stock returns is particularly strong at horizons from 4 to 8 quarters. In what concerns bond returns, the analysis suggests that one can consider two sets of countries: (i) those where investors seem to behave in a non-Ricardian way (Australia, Denmark, Finland, the Netherlands, and Spain) and (ii) those where investors seem to be forward looking and to have a Ricardian behavior (Belgium, Canada, France, Germany, Ireland, Italy, Japan, Sweden, the UK, and the US). Finally, I show that systemic crises amplify the linkages between shocks in collateralizable wealth and financial markets. Therefore, the current work opens new and challenging avenues for understanding the dynamics of the relationship between the housing sector, stock market and government bond developments, and the banking system.

Acknowledgments I am extremely grateful to the Guest Editor, Fredj Jawadi, and to an anonymous referee for helpful comments and discussions. I also benefited from the comments of participants of the First International Symposium in Computational Economics and Finance, the First World Finance Conference, the Third Annual Brunel University PhD Conference in Economics and Finance, and the 6th Portuguese Finance Network Conference.

References Acharya, V.V. and Pedersen, L.H. (2005). Asset pricing with liquidity risk. Journal of Financial Economics 77, 375–410.

Collateralizable Wealth, Asset Returns, and Systemic Risk

25

Ang, A. and Bekaert, G. (2006). Stock return predictability: Is it there?. Review of Financial Studies 20 (3), 651–707. Bansal, R. and Yaron, A. (2004). Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59, 1481–1509. Besanko, D.A. and Thakor, A.V. (1987). Collateral and rationing: Sorting equilibria in monopolistic and competitive credit markets. International Economic Review 28, 671–689. Bossaerts, P. and Hillion, P. (1999). Implementing statistical criteria to select return forecasting models: What do we learn? Review of Financial Studies 12 (2), 405–428. Brandt, M.W. and Wang, K.Q. (2003). Time-varying risk aversion and unexpected inflation. Journal of Monetary Economics 50, 1457–1498. Breeden, D.T., Gibbons, M.R. and Litzenberger, R.H. (1989). Empirical tests of the consumption-oriented CAPM. Journal of Finance 44, 231–262. Brennan, M. and Xia, Y. (2005). tay’s as good as cay. Finance Research Letters 2, 1–14. Brennan, M.J. and Subrahmanyam, A. (1996). Market microstructure and asset pricing: On the compensation for illiquidity in stock returns. Journal of Financial Economics 41, 441–464. Campbell, J. and Cochrane, J. (1999). By force of habit: A consumption-based explanation of aggregate stock market behaviour. Journal of Political Economy 107, 205–251. Campbell, J.Y. and Shiller, R.J. (1988). The dividend-price ratio and expectations of future dividends and discount factors. Review of Financial Studies 1, 195–228. Campbell, J.Y. and Shiller, R.J. (1991). Yield spreads and interest rates: A bird’s eye view. Review of Economic Studies 58, 495–514. Chien, Y. and Lustig, H. (2010). The market price of aggregate risk and the wealth distribution. Review of Financial Studies 23, 1596–1650. Clark, T. and McCracken, M. (2001). Tests of equal forecast accuracy and encompassing for nested models. Journal of Econometrics 105, 85–110. Cochrane, J.H. and Piazzesi, M. (2005). Bond risk premia. American Economic Review 95 (1), 138–160. Constantinides, G. (1990). Habit-formation: A resolution of the equity premium puzzle. Journal of Political Economy 98, 519–543. Dickey, D.A. and Fuller, W.A. (1979). Distributions of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association 74, 427–431. Diebold, F. and Mariano, R. (1995). Comparing predictive accuracy. Journal of Economics and Business Statistics 13, 253–263. Duffee, G. (2005). Time variation in the covariance between stock returns and consumption growth. Journal of Finance 60 (4), 1673–1712. Engen, E.M. and Hubbard, R.G. (2005). Federal government debt and interest rates. NBER Macroeconomics Annual 19. Engle, R. and Granger, C. (1987). Co-integration and error-correction: Representation, estimation and testing. Econometrica 55 (2), 251–276. Fama, E. (1998). Market efficiency, long-term returns and behavioral finance. Journal of Financial Economics 49 (3), 283–306.

26

Ricardo M. Sousa

Fama, E. and French, K. (1996). Multifactor explanations of asset pricing anomalies. Journal of Financial Economics 51 (1), 55–84. Fama, E.F. and Bliss, R.H. (1987). The information in long-maturity forward rates. American Economic Review 77 (4), 680–692. Fama, E.F. and French, K.R. (1989). Business conditions and expected returns on stocks and bonds. Journal of Financial Economics 25, 23–49. Fernandez-Corugedo, E., Price, S. and Blake, A. (2007). The dynamics of consumers’ expenditure: The UK consumption ECM redux. Economic Modelling 24, 453–469. Gale, W.G. and Orszag, P.R. (2003). Economic effects of sustained budget deficits. National Tax Journal 56, 463–485. Goyal, A. and Welch, I. (2003). Predicting the equity premium with dividend ratios. Management Science 49, 639–654. Goyal, A. and Welch, I. (2004). A comprehensive look at the empirical performance of equity premium prediction. NBER Working Paper no. 10483. Harvey, D., Leybourne, S. and Newbold, P. (1998). Tests for forecast encompassing. Journal of Business and Economic Statistics 16, 254–259. Hjalmarsson, E. (2006). Should we expect significant out-of-sample results when predicting stock returns? Board of Governors of the Federal Reserve System, International Finance Discussion Paper No. 855. Hodrick, R.J. (1992). Dividend yields and expected stock returns: Alternative procedures for inference and measurement. Review of Financial Studies 5, 357–386. Honohan, P. and Laeven, L. (2005). Systemic Financial Distress: Containment and Resolution. Cambridge: Cambridge University Press. Inoue, A. and Kilian, L. (2004). In-sample or out-of-sample tests of predictability: Which one should we use? Econometric Reviews 23, 371–402. Johansen, S. and Juselius, K. (1990). Maximum likelihood estimation and inference on cointegration with applications to the demand for money. Oxford Bulletin of Economics and Statistics 52, 169–210. Jones, C.M. (2002). A century of stock market liquidity and trading costs. Columbia University, Columbia Business School, Working Paper. Julliard, C. (2004). Labor income risk and asset returns. Princeton University, Primary Job Market Paper. Kwiatkowski, D., Phillips, P.C.B., Schmidt, P. and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54 (1–3), 159–178. Laeven, L. and Valencia, F. (2008). Systemic banking crises: A new database. IMF Working Paper no. 224. Lamont, O. (1998). Earnings and expected returns. Journal of Finance 53, 1563–1587. Lettau, M. and Ludvigson, S. (2001). Consumption, aggregate wealth, and expected stock returns. Journal of Finance 41 (3), 815–849. Ludvigson, S. and Ng, S. (2009). Macro factors in bond risk premia. Review of Financial Studies 22 (12), 5027–5067. Lustig, H. and van Nieuwerburgh, S. (2005). Housing collateral, consumption insurance, and risk premia: An empirical perspective. Journal of Finance 60, 1167–1219.

Collateralizable Wealth, Asset Returns, and Systemic Risk

27

Mackinnon, J. (1996). Numerical distribution functions for unit-root and cointegration tests. Journal of Applied Econometrics 11, 601–618. Mankiw, N.G. and Shapiro, M.D. (1986). Risk and return: Consumption beta versus market beta. Review of Economics and Statistics 68, 452–459. McCracken, M. (2006). Asymptotics for out-of-sample tests of causality. Journal of Econometrics 140, 719–752. Nelson, C.R. and Kim, M.J. (1993). Predictable stock returns: The role of small sample bias. Journal of Finance 48 (2), 641–661. Newey, W. and West, K. (1987). A simple positive semi-definite, heteroskedasticity, and autocorrelation consistent covariance matrix. Econometrica 55 (3), 703–708. Pastor, L. and Stambaugh, R.F. (2003). Liquidity risk and expected stock returns. Journal of Political Economy 3, 642–685. Parker, J.A. and Julliard, C. (2005). Consumption risk and the cross section of expected returns. Journal of Political Economy 113, 185–222. Phillips, P. and Perron, P. (1988). Testing for a unit root in time series regression. Biometrika 75, 335–346. Phillips, P.C.B. and Ouliaris, S. (1990). Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165–193. Piazzesi, M., Schneider, M. and Tuzel, S. (2007). Housing, consumption and asset pricing. Journal of Financial Economics 83, 531–569. Santos, T. and Veronesi, P. (2006). Labor income and predictable stock returns. Review of Financial Studies 19, 1–44. Shiller, R.J. (1984). Stock prices and social dynamics. Brookings Papers on Economic Activity 84, 457–510. Sousa, R.M. (2010). Consumption, (dis)aggregate wealth, and asset returns. Journal of Empirical Finance 17 (4), 606–622. Sousa, R.M. (2007). Wealth shocks and risk aversion. University of Minho, NIPE Working Paper No. 28. Stambaugh, R.F. (1999). Predictive regressions. Journal of Financial Economics 54 (3), 375–421. Stiglitz, J. and Weiss, A. (1981). Credit rationing in markets with imperfect information. American Economic Review 71 (3), 393–410. Valkanov, R. (2003). Long-horizon regressions: Theoretical results and applications. Journal of Financial Economics 68, 201–232. Whelan, K. (2008). Consumption and expected asset returns without assumptions about unobservables. Journal of Monetary Economics 55 (7), 1209–1221. Yogo, M. (2006). A consumption-based explanation of expected stock returns. Journal of Finance 61 (2), 539–580.

Chapter 2

Nonlinear Stock Market Links between Mexico and the World Mohamed El Hedi Arouria and Fredj Jawadib a

LEO – Universite´ d’Orle´ans & EDHEC Business School, Rue de Blois BP 267-39 - 45067 Orle´ans Cedex 2, France, e-mail: [email protected] b Universite´ d’E´vry Val d’Essonne & Amiens School of Management, UFR Sciences Sociales et Gestion, Bat. La poste, Bureau 2282, rue du Facteur Cheval - 91025 E´vry Cedex, France, e-mail: [email protected]

Abstract Purpose – This chapter aims to investigate the stock market comovements between Mexico and the world capital market using nonlinear modeling tools. Methodology/approach – We apply recent nonlinear cointegration and nonlinear error correction models (NECMs) to investigate the comovements between stock prices over the recent period. Findings – While the previous literature only highlights some evidence of time-varying comovements, our chapter aims to specify the mechanism characterizing the comovement process through the comparison of two nonlinear error correction models (NECMs). It shows a nonlinear relationship between stock prices that are activated per regime. Originality – Studying the integration hypothesis between stock markets over the recent financial crisis, our findings highlight strong evidence of significant comovements that explain the global collapse of stock markets in 2008–2009. Keywords: stock market comovements, NECMs, Mexican stock market JEL Classification: C22, G15

International Symposia in Economic Theory and Econometrics, Vol. 20 F. Jawadi and W.A. Barnett (Editors) Copyright r 2010 by Emerald Group Publishing Limited. All rights reserved ISSN: 1571-0386/DOI: 10.1108/S1571-0386(2010)0000020007

30

Mohamed El Hedi Arouri and Fredj Jawadi

1. Introduction Since the 1980s, emerging stock markets have been widely seen as the most exciting and promising area for investment, especially because they are expected to generate high returns and to offer good portfolio diversification opportunities. Consequently, these markets have experienced considerable expansion with financial liberalization being largely implemented in several emerging countries through ongoing structural adjustment programs. As a prerequisite to the financial liberalization processes, stabilization policies have been designed to ensure macroeconomic stability, low inflation, and reduced budget deficits. As a result, emerging market capitalization has grown from 4% of world market capitalization in 1987 to 13% in 1996, and it reached around 20% in 2000. Overall, economic stability and good perspectives have been key factors in the development of emerging markets. These changes have stimulated the integration process of emerging markets into the world market and increased their comovements with developed stock markets. However, emerging markets are not homogeneous, and their financial dynamics differ significantly (Bekaert and Harvey, 1995). In other words, the nature of the financial dynamics depends on both internal and external factors: international; regional; and specific economic, financial, and political variables. In order to better understand the dynamics of emerging stock markets, this chapter focuses on a major emerging market whose financial market has increased in activity and relevance throughout the last two decades: Mexico. It is the biggest Latin American market almost fully accessible to foreign investors. In fact, in the last two decades, foreign investment barriers were reduced, country funds were introduced, and depository receipts (DR) were listed in order to improve the integration of Mexico into the world market. Integration should drive to a lower cost of capital, bigger investment opportunities, and higher economic growth (Bekaert and Harvey, 2003). Studying the Mexican stock market leads to a better view of the comovements between emerging and developed stock markets. Unlike previous studies on emerging stock markets’ comovements (De Santis and Imrohoroglu, 1997; Adler and Qi, 2003; Gerard et al., 2003; Phylaktis and Ravazzolo, 2005; Bekaert et al., 2005), our approach enables us to test whether the comovements are symmetric or asymmetric, continuous or discontinuous, constant or variable, and linear or nonlinear. It is clear that the answers to these questions are important because they help to understand the dynamics of interdependencies and correlations between emerging and developed stock markets, and thus provide crucial tools for international portfolio diversification. This chapter is developed according to the following outline. The second section briefly discusses the literature review. The econometric tools are

Nonlinear Stock Market Links between Mexico and the World

31

briefly presented before discussing the empirical results in the third section. Finally, the last section concludes.

2. What does the Literature Say? Several studies focus on financial integration and comovements between Mexico and developed stock markets using different tools and tests. Bekaert and Harvey (1995), Adler and Qi (2003), and Carrieri et al. (2006) make use of the capital asset pricing model (CAPM). Overall, even these studies show significant comovements between Mexico and the world market while suggesting a decrease in the links in 1994 and an increase over the last years; their results are rather mixed and rely on the validity of the CAPM. Another class of empirical studies has tested stock market comovements using econometric tools such as cointegration tests and Vectorial Autoregressive models (VAR) (Masih and Masih, 2001; Lim et al., 2003; Iwatsubo and Inagaki, 2007; Wang and Nguyen Thi, 2007). Their main results show that the US market is the most influential market and point out strong evidence of significant comovements between the stock prices in the long run. Nevertheless, the limitation of these studies consists of the fixation of the level of comovements. Indeed, as comovements between emerging and developed markets are governed by ongoing liberalization processes as well as regional and international factors, we can expect an unstable spillover effect between these markets and the world market yielding a time-varying interdependency. However, this may escape linear modeling and usual cointegration tools, which can only test for two polar cases of comovements and absence of comovements. In the rest of this chapter, we use a more original framework to investigate comovements between Mexico and the world stock market. More precisely, we make use of the nonlinear cointegration methodology to take into account time-varying aspects of comovements as well as potential asymmetries and nonlinearities.

3. Nonlinear Time-Varying Financial Integration This section aims to discuss the nonlinear tools to be used to test for nonlinear time-varying comovements between Mexico and the world market. In particular, we propose applying nonlinear cointegration techniques. The nonlinear cointegration methodology appears suitable for at least two reasons. First, this should enable to capture stock price comovements in the short and long terms. Second, this nonlinear framework allows reproducing the different regimes and states of stock market comovements. Thus, we suggest using two nonlinear error correction models (NECMs): the exponential smooth

32

Mohamed El Hedi Arouri and Fredj Jawadi

transition ECM (ESTECM) and the nonlinear ECM-rational polynomial (NECM-RP) in order to choose the most appropriate model to characterize the comovements. We briefly discuss these models, but we can refer to Escribano and Mira (2002) and Van Dijk et al. (2002) for more details about them. These models can be seen as a conjunction of two linear ECMs while including two adjustment terms and a nonlinear function. This function is bounded between 0 and 1 defining two different regimes, while the transition between these regimes is determined according to the statistical properties of this nonlinear function. Escribano (1997) suggests using the nonlinear function that satisfies the stability conditions such as the rational polynomial function, the cubic function, or the smoothing functions. In the literature, there are several types of NECMs depending on the definition of the nonlinear function. In this chapter, we propose using two specific kinds of NECMs: the ESTECM (Equation 1) and the NECM-RP (Equation 2). Dyt ¼ a0 þ r1 zt1 þ

q X

bi Dxti þ

i¼0

p X

dj Dytj þ r2 zt1

j¼1

(1)

2

 ½1  expfgðzt1  cÞ g þ et ,

Dyt ¼ a0 þ r1 zt1 þ

q X i¼0

bi Dxti þ

p X j¼1

dj Dytj þ r2 

ðzt1 þ aÞ3 þ b þ mt , ðzt1 þ cÞ2 þ d (2)

where yt is the endogenous variable and xt is a vector of K explanatory variables that are I(1). b0 i and d0 j are vectors of parameters, r1 and r2 are the adjustment terms, zt is the residual term of the linear cointegration relationship between the Mexican stock index and the world equity market. g and c are, respectively, the transition speed between regimes and the threshold parameter of the exponential function, whereas a, b, c, and d are real numbers defining the rational polynomial function. In order to check the market integration hypothesis between Mexico and the world market, we first need to test the mixing hypothesis using parametric and nonparametric tests. To do this, we use two mixing tests: the Kwiatkowski-Phillips-Schmidt-Shin tests (KPSS) (nonparametric test) and Lo’s(1991) R/S test (parametric test). Second, we estimate the potential integration process using these particular NECMs. The main advantage of the ESTECM is to capture smooth and gradual financial integration process while reproducing temporal paths governed by smooth-changing regimes and accounting for a slow adjustment mechanism. For the NECM-RP, it may reproduce several potential sources of nonlinearities (i.e., abrupt changes in adjustment speeds, effects of negative and positive shocks on stock price

Nonlinear Stock Market Links between Mexico and the World

33

adjustment, multiple long-run attractors, etc.). It can also reproduce asymmetric dynamics between the overvaluation and undervaluation regimes. In practice, gt in Equations (1) and (2) denotes the Mexican stock price and xt the Morgan Stanely Capital International (MSCI) world market index in logarithm. Our empirical investigation is conducted in several steps. First, we check the integration order of the stock indices with the usual unit root tests (Augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) tests). Second, we test the mixing and the nonlinear cointegration hypotheses while applying KPSS and R/S tests on the residual term (z^t ). Third, if we accept the mixing hypothesis, we retain the hypothesis of nonlinear mean-reversion and cointegration. We estimate in a final step the NECMs by the nonlinear least-squares (NLS) method.

4. Empirical Results 4.1. Data Preliminary Analysis Our data consist of monthly Mexican stock index and the world market index over the period December 1987–January 2008. Data were obtained from MSCI, and both indices are expressed in US dollars to take the currency risk into account. First, both MSCI indices are I(1) according to ADF and PP tests. Second, in addition, we computed the bilateral correlation between Mexican and world stock returns while taking the Tequila effect into account and reported the results in Table A.1. The correlation is significantly higher after the Mexican crisis, indicating that both markets became more correlated after 1994. Table A.2 points out the rejection of normality and symmetry hypotheses while suggesting significant leptokurtic distribution. The negativity of the skewness is a sign of nonlinearity in the dynamics of the Mexican stock market.

4.2. Linear and Nonlinear Cointegration Tests First, as suggested by Table A.3, we do not reject the linear cointegration hypothesis concluding in favor of financial dependence between the Mexican and the world markets. It also implies that the Mexican mean-reverting dynamics may be reproduced via a linear ECM. However, according to Taylor and Sarno (2001), linear modeling may not be appropriate to reproduce time series that are generated by nonlinear processes because it fails to reproduce asymmetry inherent to a time-varying financial integration process. Next, we check the cointegration hypothesis in a nonlinear framework using ‘‘mixing’’ tests, which are more powerful and

34

Mohamed El Hedi Arouri and Fredj Jawadi

robust than linear cointegration tests. If we accept the implications of the mixing and nonlinear cointegration hypotheses, we may conclude that the Mexican stock price is nonlinearly mean-reverting toward the world market index, which suggests that the Mexican market is a priori nonlinearly linked into the world market. In practice, we have applied two ‘‘mixing’’ tests: the KPSS and the R/S tests, which both test the null hypothesis of ‘‘mixing’’ against its ‘‘nonmixing’’ alternative. For the KPSS, we retain Schwert’s (1989) values given the following truncation parameter: "  "   #  # T 1=4 T 1=4 and l 12 ¼ int 12 l 4 ¼ int 4 100 100 where T is the number of observations.1 Concerning the choice of q for the R/S test, we retain the value of Andrews (1991): qt ¼ KT, where ^ K T ¼ ð3T=2Þ1=3 ðð2rÞ=ð1  r^ 2 ÞÞ2=3 ; [KT] ¼ int(KT), and r^ is the first-order autocorrelation coefficient. Both results of mixing tests are summarized in Table A.4. From Table A.4, the mixing hypothesis is accepted according to the KPSS and R/S tests, implying a nonlinear cointegration relationship between Mexican and world stock indexes. This indicates some evidence of timevarying and asymmetrical comovements for Mexico. Finally, we move to the implementation of the NECMs: the ESTECM and the NECM-RP. 4.3. Estimation of NECMs We estimate the NECMs by the NLS method using a nonlinear optimization program in accordance with the steps proposed by Escribano and Mira (2002) and Van Dijk et al. (2002). The number of lags (p) for the NECMs is then specified using the information criteria, the autocorrelation functions, and the Ljung–Box tests. The results indicated q ¼ 0, implying the absence of local time dependence for Mexico, and showed that its dynamics depends only on current world returns. Concerning the estimation, different initial values are tested with the NECM parameters, and the results retained (see Table A.5) are those for which the maximum is absolute. The empirical results suggest several interesting findings. First, we point out significant nonlinear relationship between the MSCI world index and Mexican market. The parameters g and c are statistically significant, validating the choice of the exponential function. Interestingly, we found 1

Int [.] designates the interior part.

Nonlinear Stock Market Links between Mexico and the World

35

Transition function

1.0 0.8 0.6 0.4 0.2 0.0 -1.40

-1.05

-0.70

-0.35

0.00

0.35

0.70

1.05

Transition variable

Graph 1:

Exponential Transition Function.

that r1 is positive and significant and r2 is negative and statistically significant at 5%. Furthermore, the sum (r1þr2) is negative, implying a significant and stable nonlinear error correction adjustment dynamic and indicating a nonlinear process regarding Mexico’s comovements with the world stock market. This indicates that in the first regime when Mexican deviations are small (before the Mexican crisis),2 Mexican stock price may deviate from the world market, its deviations remain uncorrected and approach a random walk, and the market is rather segmented. However, for a large disequilibrium (when deviations became large and exceed some threshold), a nonlinear mean-reversion mechanism is activated, ensuring the convergence of Mexican price toward the equilibrium and thus allowing for strong market links. Furthermore, as shown in Graph 1, the adjustment speed should increase with the stock price deviation size. Interestingly, Graph 1 shows two important features. First, the ESTECM captures the asymmetry characterizing the Mexican stock market dynamics. Second, this graph shows significant evidence of mean-reversion in the stock prices after 1994. Indeed, as shown in Graph 2, the transition function reaches the unity and persists in the upper regime (regime of integration), confirming the results of Adler and Qi (2003) for whom Mexican stock market integration has considerably increased in the recent years. This increase in financial integration may certainly be supported by the different arguments discussed in the chapter. However, it also implies that today’s Mexico market is progressively dependent on world equity, and this also explains the negative effect of the current international financial crisis on Mexico. The application of several misspecification tests has validated the

2

The threshold parameter corresponds approximately to the Mexican deviation price of May 1993, indicating that the transition occurred around this date.

36

Mohamed El Hedi Arouri and Fredj Jawadi

1.0

0.8

0.6

0.4

0.2

0.0 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Graph 2:

Evolution of the Exponential Transition Function. Rational Polynomial Function

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.45

Graph 3:

0.90

1.35

1.80

2.25

2.70

3.15

3.60

Histogram of the Rational Polynomial Function for Mexico.

estimation results of the ESTECM, which indicated, in particular, that the residuals of ESTECM are mixing. Finally, we also estimate the NECM-RP under the following restrictions: a ¼ c ¼ d ¼ 1 and b ¼ 0, in order to simplify the algorithm convergence. From Graph 3, the asymmetry, nonlinearity, and smoothness characterizing the Mexican stock price adjustment dynamic also seems to be captured by the NECM-RP (see Graph 3). The estimation results also show a significant correlation between the emerging Mexican market and the world market. Interestingly, the analysis of the histogram of the rational polynomial function (see Graph 3) that is plotted in accordance with the estimated misalignment values (z^t1 ) confirms the rejection of normality. The NECMRP has captured the asymmetry in comovements between the emerging Mexican market and the world equity market. Indeed, Graph 3 shows a bimodal density with two modes of unequal heights confirming the rejection of the symmetry and normality hypotheses. Furthermore, the presence of these unequal modes suggests important and extreme stock price deviations between the different regimes. This asymmetry in the distribution of the rational polynomial function also reproduces time-varying smoothness and persistence in the comovement’s process.

Nonlinear Stock Market Links between Mexico and the World

37

5. Conclusion This chapter aims at investigating stock market comovements between Mexico and the world stock market. Using nonlinear modeling, we propose an appropriate specification that not only reproduces the extreme cases of comovements (strong and weak comovements) but also specifies a continuum of intermediate states. Our results validate the hypothesis of nonlinear comovements and show that the latter have significantly increased after 1994. They also identify regimes of integration and those of segmentation. The approach we used in this chapter can naturally be extended to other emerging and developed stock markets to compare their financial convergence dynamics.

Appendix Table A.1:

Matrix of Bilateral Correlations

Series

RMSCI

RMEX

Before Mexican Crisis: January 1988–November 1994 RMEX 0.26 RMSCI 1.00

1.00 0.26

After Mexican Crisis: December 1994–January 2008 RMEX 0.60 RMSCI 1.00

1.00 0.60

The whole period: January 1988–January 2008 RMEX 0.47 RMSCI 1.00

1.00 0.47

Note: This table shows bilateral correlations between Mexican and world stock returns before and after the Mexican Crisis. RMSCI and RMEX are, respectively, the stock returns of the world and Mexican stock markets.

Table A.2:

Descriptive Statistics and Normality Test

Series

Mean

SD

Maximum

Minimum

Skewness

Kurtosis

Jarque-Bera (probability)

Mexico MSCI world index

0.0169 0.0053

0.0935 0.0398

0.2540 0.1055

0.4195 0.1444

0.9425 0.5733

6.1437 3.8673

134.92 (0.0) 20.75 (0.0)

Note: This table presents the descriptive statistics of stock returns from the world and Mexican stock markets.

38

Table A.3: Series

Mexico

Mohamed El Hedi Arouri and Fredj Jawadi

Linear Cointegration Tests Constant

LMSCI

4.71 (7.88)*

1.76 (19.85)*

2 R

0.62

ADF (p, model)

4.02 (1, a)

Note: This table reports the linear cointegration test. The values inside parentheses are the tratios. (*) and (a) designate, respectively, the significance at 5% and a model without a constant and linear trend.

Table A.4:

Mixing Tests

KPSS l4 0.45*

R/S l12

Andrews

0.12

1.1

Note: This table presents the results of mixing tests. (*) denotes the rejection of the null hypothesis at the 5% significance level.

Table A.5: Coefficients a0 r1 r2 b0 g g  sztd c ADFGLS R/S sNECM/sLECM

NECM Estimation Results for Mexico ESTECM (0,1)

NECM-RP

0.0045 (1.04) 0.0994* (2.36) 0.1315* (3.05) 1.105* (8.52) 26.06* (2.04) 7.64 0.6268* (16.979)

0.0094 (1.36) 0.032 (1.11) 0.0054 (0.36) 1.11* (8.38) – – –

9.83 1.4 0.73

9.79 1.5 0.88

Note: This table presents the estimation results of NECM for Mexico. The values inside parentheses are the t-statistic of nonlinear estimators. (*) denotes the significance at 5%.

Nonlinear Stock Market Links between Mexico and the World

39

References Adler, M. and Qi, R. (2003). Mexico’s integration into the North American capital market. Emerging Economic Review 4, 91–120. Andrews, D. (1991). Heteroscedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858. Bekaert, G. and Harvey, C. (1995). Time varying world market integration. Journal of Finance 50 (2), 403–444. Bekaert, G. and Harvey, C. (2003). Emerging markets finance. Journal of Empirical Finance 10, 3–55. Bekaert, G., Harvey, C. and NG, A. (2005). Market integration and contagion. Journal of Business 78, 39–69. Carrieri, F., Errunza, V. and Hogan, K. (2006). Characterizing world market integration through time. Journal of Financial and Quantitative Analysis 42, 915–941. De Santis, G. and Imrohoroglu, S. (1997). Stock returns and volatility in emerging financial markets. Journal of International Money and Finance 16 (4), 561–579. Escribano, A. (1997). Nonlinear error correction: The case of money demand in the U.K (1878–1970). , Universidad Carlos III, Madrid, Working Paper No. 96–55. Escribano, A. and Mira, S. (2002). Nonlinear error correction. Journal of Time Series Analysis 23, 509–522. Gerard, B., Thanyalakpark, K. and Batten, J. (2003). Are the East Asian markets integrated? Evidence from the ICAPM. Journal of Economics and Business 55. Iwatsubo, K. and Inagaki, K. (2007). Measuring financial market contagion using dually-traded stocks of Asian firms. Institute of Economic Research, Hitotsubashi University, CEI Working Paper Series No. 14. Lim, K., Lee H. and Liew, K. (2003). International diversification benefits in Asian stock markets: A revisit. Mimeo. Labuan School of International Business and Finance, Universiti Malaysia Sabah, Faculty of Economics and Management, Universiti Putra Malaysia. Lo, A.W. (1991). Long-term memory in stock market prices. Econometrica 59 (5), 1279–1313. Masih, A. and Masih, R. (2001). Long and short term dynamic causal transmission amongst international stock markets. Journal of International Money and Finance 20, 563–587. Phylaktis, K. and Ravazzolo, F. (2005). Stock market linkages in emerging markets: Implication for international portfolio diversification. Journal of International Markets and Institutions 15 (2), 91–106. Schwert, G.W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics 7, 147–159. Van Dijk, D., Tera¨svirta, T. and Franses, P.H. (2002). Smooth transition autoregressive models – A survey of recent developments. Econometric Reviews 21, 1–47. Wang, K.M. and Nguyen Thi, T.B. (2007). Testing for contagion under asymmetric dynamics: Evidence from the stock markets between US and Taiwan. Physica A 376, 422–432. Taylor, M.P. and Sarno, L. (2001). Real exchange rate dynamics in transition economies: A nonlinear analysis. Studies in Nonlinear Dynamics & Econometrics 5 (3).

Chapter 3

Dynamic Linkages between Global Macro Hedge Funds and Traditional Financial Assets Wafa Kammoun Masmoudia,b a

University of Rennes 1 – CREM, UMR CNRS 6211, 7 Place Hoche – CS 86514, 35065 Rennes Cedex (France), e-mail: [email protected]; [email protected] b URMOFIB, University of Tunis El Manar – University Campus – 2092 Tunisia

Abstract Purpose – The purpose of this chapter is to present an investigation on the dynamic linkages between global macro hedge funds and traditional financial assets of developed and emerging markets. Methodology/approach – To explore relationships among these price indices, we analyse Granger causality and vector autoregression (VAR) dynamics through impulse response functions. Besides, multivariate cointegration is used to know long-term relationships between assets and allows risk-averse investors to reduce uncertainty. Finally, a vector error correction model (VECM) provides active asset managers the opportunity to anticipate shortterm price movements. Findings – Our results show that in a Granger causality sense, we observe long- and short-term relationships between global macro hedge funds and traditional financial assets for Canada, France and Germany. This implies that opportunities for international portfolio diversification are significantly lower for countries having relationships between assets. For Canada, France and Germany, the risk-averse investors can reduce their long-term volatility by investing according to the cointegrating vector, whereas active managers can benefit from the knowledge of short-term asset price movements. The VEC Pairwise Granger causality in the short term confirms our analysis of causality according to VAR models.

International Symposia in Economic Theory and Econometrics, Vol. 20 F. Jawadi and W.A. Barnett (Editors) Copyright r 2010 by Emerald Group Publishing Limited. All rights reserved ISSN: 1571-0386/DOI: 10.1108/S1571-0386(2010)0000020008

42

Wafa Kammoun Masmoudi

Originality/value of paper – These results are original because they help the investor to understand the dynamics of the relationship between global macro hedge funds and traditional financial assets. Keywords: global macro hedge funds, portfolio diversification, vector autoregressions (VARs), Johansen cointegration test, vector error correction model (VECM), developed markets, emerging markets JEL Classification: C32, G11, G15

1. Introduction Alternative investments have experienced explosive growth in the past two decades and provide a fertile field to study active portfolio management. Interest in investing in hedge funds has fluctuated due to the perception among some investors both in developed and emerging countries that hedge funds exacerbated several financial crises including the attack on the British Pound led by George Soros and the exit of the pound sterling from the European Monetary System in 1992; the sharp decline of several Asian currencies during the Asian financial crisis of 1997–1998; the virtual collapse of Long-Term Capital Management in September 1998; and the financial crisis of 2008–2009.1 Simply put, there is a widespread belief (without significant empirical evidence) that large highly levered hedge funds could exert a significant impact on financial market prices and volatility. Hence they are able to hedge and diversify market risk while at the same time enhancing return performance (Fung and Hsieh, 1997, 2000; Liang, 1999, 2001). On the other hand, it is clear that the traditional financial assets have an impact on hedge funds returns as they try constantly to take bets on various markets such as equity, bond and currency markets. As noted by Fung and Hsieh (1997), hedge funds differ from traditional mutual funds by their more extensive use of dynamic trading strategies. Because of less regulatory oversight, hedge funds have flexibility to engage in short selling, leverage and various types of arbitrage activities. Hedge funds group many investment strategies such as global macro, market neutral, long–short and short selling (Fung and Hsieh, 1999; Capocci, 2006). Most of the large hedge funds in excess of US$1 billion in capital are global macro funds. This style of trading has long been associated with highly leveraged speculators aiding and abetting global market turmoil. Their 1

Source: Hedge Fund Research, Inc., Global Hedge Fund Industry Report Second Quarter 2009.

Global Macro Hedge Funds and Traditional Financial Assets

43

managers carry long and short positions in both developed and emerging markets and rely on macroeconomic analysis. The portfolios of these funds can include stocks, bonds, currencies and commodities in the form of cash or derivatives instruments. Global macro managers may elect to take outright directional positions, and depending on their own expertise and the risk–return profile of the markets in which they are investing, they may also elect to take relative value positions, where a long position or set of positions is dynamically paired off against a short position or set of positions. It is commonly known that returns from hedge funds exhibit non-linear behaviour, which may be attributed to highly leveraged and unconventional trading strategies adopted by hedge fund managers. In fact, Agarwal and Naik (2004), Favre and Galeano (2002) and Mitchell and Pulvino (2001) observe that hedge funds returns exhibit non-linear relationship with traditional asset classes. The non-linear feature of hedge fund returns can be captured by non-linear style factors which represent the returns of trading strategies on traditional asset classes that can explain the returns of a group of hedge funds. They link hedge fund returns to observed market prices. Therefore, in this chapter we attempt to detect dynamic linkages between global macro hedge funds and traditional financial assets of developed and emerging markets using vector autoregression (VAR) modelization, multivariate cointegration and vector error correction model (VECM) in order to know whether diversification benefits arise from adding this type of hedge funds to a bond/equity portfolio. The principal objective of this paper is focused on whether one obtains different results than previously reported in the financial economic literature if one examines an index of global macro hedge funds rather than ‘overall measures of hedge fund performance (across investing styles)’. In fact, global macro hedge funds trade globally in various asset classes and have multiple focus markets. Therefore, linkages between them and traditional financial assets are possible. In deed, hedge funds provide diversification benefits relative to stocks and bonds investments. Besides, the real risk and return benefits of hedge funds have less to do with the funds own standalone performance than with a classical portfolio performance. According to Chen (2005), several hedge fund categories show generally better timing ability – which can be one important source of the non-linear feature of hedge fund payoffs documented by Fung and Hsieh (1997) – than mutual funds. With a large sample of hedge funds during 1994–2002, Chen (2005) examines the ability of hedge funds in various investment categories to time their focus markets and finds that a few fund categories (e.g. global macro and managed futures) can time significantly in the focus markets, including bond, currency and equity markets; nevertheless, timing ability is sparse in the equity market.

44

Wafa Kammoun Masmoudi

Fung and Hsieh (2000) analyse hedge fund exposure during several major market events. They find that hedge funds are sometimes in a position to exert substantial market impact, as noted during the ERM Crisis in 1992,2 the European bond market rally in 1993 and subsequent decline in 1994. During the stock market crash in 1987, the Mexican peso crisis in 1994 and the Asian currency crisis in 1997, they had little impact. In the financial literature, several studies used a multivariate cointegration approach and focused on the relationship between hedge funds and traditional financial assets like stocks and bonds for developed and emerging markets. Fu¨ss and Kaiser (2007) analysed the short- and long-term relationships between hedge funds and traditional financial assets for the main emerging market regions of Asia, Latin America and Eastern Europe. These authors found that diversification benefits arise from adding emerging markets hedge funds to an emerging markets bond/equity portfolio. However, the profits are significantly less for Eastern Europe than for the other emerging market regions. Pan et al. (1999) examined common stochastic trends in Australia, Hong Kong, Japan, Malaysia or Singapore. They found no evidence of common trends.3 Gregoriou and Rouah (2001) investigated the long-term relationships between the 10 largest hedge funds in the Zurich database and 4 stock market indices over a 10-year period, using time series analysis and cointegration. The results show an evidence of cointegration with the stock market indices for only two hedge funds. Gregoriou and Rouah (2001) indicate that large hedge funds tend to allocate assets over a variety of investment instruments, such as currencies, futures, options, swaps and other derivatives. Consequently, their performance will not be as strongly correlated to standard benchmarks. Fu¨ss and Herrmann (2005) presented an investigation of the long- and short-term interdependence between different hedge fund strategy indices and the stock markets of France, Germany, Japan, North America and the United Kingdom from January 1994 through December 2003. The results indicate that there is no evidence of a long-term relationship among the prices of hedge funds strategies and stock markets in a bivariate cointegration framework. Consequently, opportunities exist to diversify an international portfolio by taking hedge funds into account in the short and long term.

2

According to Fung and Hsieh (2000), it is, however, difficult to determine whether the significant short position in Sterling ‘caused’ the devaluation, because it coincided with net capital outflows from the United Kingdom. 3 In Hung and Cheung (1995), Chaudhuri (1997) and Chen et al. (2002) papers, we find more studies on the interdependence of emerging markets, especially in Asia and Latin America.

Global Macro Hedge Funds and Traditional Financial Assets

45

Gregoriou and Rouah (2001) and Fu¨ss and Herrmann (2005) used the Engle and Granger’s (1987) two-step cointegration approach which is considered weaker than the multivariate Johansen approach, because it does not distinguish more than one cointegrating vector. Indeed, Johansen (1988) remedied this problem and presented a multivariate cointegration approach based on the likelihood method. Fu¨ss et al. (2006) incorporated hedge funds in both a traditional (which consists of S&P 500, value and large-cap stocks and bonds) and alternative portfolio (which includes NASDAQ, growth, small-cap, emerging markets, commodities and real estate) to detect the diversification effects of hedge funds. The authors confirm the existence of cointegration between hedge funds and traditional as well as alternative financial assets and suggest that, for a traditional portfolio, the hedge fund composite index not only enters the cointegrating vector, but the returns also react to the common trend. Hence, risk-averse investors with long-term investment horizons do not increase the risk by including hedge funds. Furthermore, for a portfolio consisting only of alternative assets, Fu¨ss et al. (2006) established that hedge funds share a common trend with NASDAQ-listed companies, small-cap stocks and real estate equities. Nevertheless, only hedge funds and stocks of emerging markets are influenced significantly by the common trend. Besides, because the adjustment coefficients in the alternative portfolio are much lower, active investors can anticipate the information on the short-term movements of asset prices. Overall, the authors conclude that investors can benefit from risk diversification both in traditional and alternative portfolios. Several studies (Gregoriou and Rouah, 2001; Fu¨ss and Herrmann, 2005; Fu¨ss et al., 2006; Fu¨ss and Kaiser, 2008) have analysed the dynamics of long- and short-term relationships between hedge funds returns and the returns on financial assets such as stocks and bonds for developed and emerging countries. This paper differs from these related studies in two important respects. First, we investigate the interdependence linkages between global macro hedge funds and traditional financial assets. In fact, these studies examined this issue, but they looked at overall measures of hedge fund performance (across investing styles) rather than simply at global macro hedge funds. Second, we conduct our analysis on developed and emerging markets, whereas many existing papers mainly focus either on developed markets or on emerging markets. In this chapter, we propose to answer the following questions: Is there any relationship between hedge fund returns and traditional financial assets returns? Are global macro hedge funds a good addition to a bond/equity portfolio? To answer the questions, we assume that the global macro strategy is known for its high return and low correlation with the market. One reason for the supposedly low correlation and potential diversification benefits is that hedge funds often employ skill-based investment strategies

46

Wafa Kammoun Masmoudi

that do not explicitly attempt to track a particular index. In fact, hedge funds try to maximize long-term returns independently of a proscribed traditional stock and bond index or currency markets. It is important to note that hedge funds often track a particular investment strategy or investment opportunity which is driven by common market factors such as changes in stock and bond returns or market volatility. In order to resolve this problem, we organize our chapter as follows: in the next section we describe our study data and present the descriptive statistics of the assets. Section 3 exposes the methodology and empirical results of VAR model, Johansen cointegration test and the VECM, and discusses our results. The final section recapitulates and offers conclusion remarks.

2. Data and Descriptive Statistics 2.1. Data In our database, we used the performance index of global macro hedge fund (HFRX). For developed countries, the evolution of the entire stock market of a country is represented by the index MSCI Morgan Stanley. The other financial asset class indices are represented by the S&P/International Finance Corporation Investable Indices (S&P/IFCI). We use the S&P/IFC Investable Indices because some of the funds are located outside the countries and hence are explicitly faced with capital restrictions. Concerning bond index, we use the EFFA-Datastream All Maturities for developed countries and the J.P. Morgan Emerging Markets Bond Indices (EMBI) index for regional and composite emerging bond markets.4 The database was extracted from Datastream. In our study, we used daily data in U.S. dollars. The data series range from 31 March 2003 through 11 March 2010. Thus, we obtained a total of 1814 observations. The returns are computed as the log difference between the two consecutive observations of index prices.

2.2. Descriptive Statistics Figure 1 shows the daily logarithmic levels denominated in U.S. dollars and indicates that the financial time series have exhibited similarly increasing trend behaviour since the end of March 2003 through December 2007. This 4

EMBI covers U.S. dollar denominated Brady Bonds, Eurobonds and local market debt instruments issued by sovereign and quasi-sovereign entities.

Global Macro Hedge Funds and Traditional Financial Assets

47

Figure 1: Evolution of Global Macro Hedge Funds, Stock and Bond Indices.

phenomenon is perfectly classic for the majority of the financial series which generally show growth trends. However, the MSCI Japan shows a little different evolving compared to the other stocks indices. In fact, the index increases at the beginning of the period in a constant way and reaches a peak at the end of 2005, followed by a progressive drop, which continues until the end of 2008. The graphs show that the stocks and bond indices series – except the Japanese and American bond indices – have exhibited similarly increasing trend behaviour since September 2003 and then a decline since the beginning of 2008 due to the financial crisis of the subprimes. The graphs in Figure 1 show that overall the variables belonging to the same unit seem to evolve in concert and answer in a similar way to the shocks, in particular. At the end of 2007, one notes a certain desynchronization in the

48

Wafa Kammoun Masmoudi

evolution of the indices compared with the global macro hedge funds index. Moreover, Figure 1 depicts the extreme volatility of the market indices at the end of 2007. It is plausible that the alternative strategies prove reliable here by amortizing the shocks which arrive from the market. Concerning the serial correlation, in accordance with the literature (Geltner, 1991, 1993; Asness et al., 2001; Getmansky et al., 2004), a large majority of hedge funds indices present smoothed data series which is highly autocorrelated. This autocorrelation bias leads to a gross underestimation of the expected volatility of these types of hedge funds. The autocorrelation concerns mainly the strategies implied in the assets whose liquidity is very weak, such as the strategies specialized in the emerging markets (Emerging Markets), or in arbitrage (Fixed-Income Arbitrage, Convertible Arbitration, Distressed Securities, Event Driven, etc.). The strategies operating on liquid assets such as Managed Future, Short Selling, Total Macro and Aggressive Growth have statistically nondependent returns. Thus, their autocorrelations are generally negative and small in absolute value, which does not lead us to carry out a desmoothing technique to the global macro hedge funds. Table 1 gives the descriptive statistics for the hedge fund time series and for the regional stock and bond indices. The majority of the classes of assets are characterized by a rejection of normality at the 1% level. This is due to significant excess kurtosis and a significant negative skewness. To some extent, this negative skewness can be related to extreme returns, since in general the minimum return is larger (in absolute terms) than the maximum return. These results show that the standard deviation is an incomplete risk measure and can alter portfolio performance. These results converge with those of Lo (2001), Geman and Kharroubi (2003) and Malkiel and Atanu (2005) who show that the distribution of hedge funds returns does not adhere to the normal distribution. In order to find the optimal portfolio weights in a mean–variance analysis, the return correlations between assets must be stable.5 Nevertheless, correlation analysis is valid only for stationary variables. Most financial data are made stationary by taking first differences, and valuable information can be lost in this process because detrending excludes any possible common price trends. Besides, hedge funds are typically non-linear functions of traditional markets (Lhabitant, 2002) and using correlation analysis is problematic since it measures only linear dependence. In addition, the results of correlation analysis are only significant when the multivariate distribution is elliptic (Embrechts et al., 1999, 2002; Kat, 2003). On the other hand, since most hedge fund and financial asset returns show negative 5 For example, Lhabitant (2002) notes that correlations between the majority of the hedge funds indices and between U.S. and European equity markets are much higher in down markets than in up markets.

Global Macro Hedge Funds and Traditional Financial Assets

Table 1:

49

Descriptive Statistics Mean

Standard Deviation

Skewness

Kurtosis

Jarque-Bera

HFRX MACRO

0.000120

0.004613

1.080267

9.799698

3845.358***

Stocks MSCI USA MSCI Canada MSCI UK MSCI France MSCI Germany MSCI Italy MSCI Japan S&P/IFCI Asia S&P/IFCI LA S&P/IFCI EE S&P/IFCI EM comp.

0.000176 0.000564 0.000211 0.000360 0.000487 0.000148 0.000255 0.000664 0.001044 0.000687 0.000745

0.013304 0.016507 0.015254 0.015886 0.016116 0.015759 0.015194 0.015051 0.019977 0.021903 0.013999

0.302923 0.911493 0.115526 0.006429 0.019026 0.006502 0.145756 0.426655 0.402639 0.530101 0.646901

15.43008 13.79139 14.63964 13.20379 11.19474 13.66070 7.793428 9.787042 13.00330 14.36061 11.39580

11699.44*** 9048.176*** 10238.51*** 7865.210*** 5073.013*** 8585.356*** 1742.137*** 3534.751*** 7608.136*** 9834.577*** 5451.344***

Bonds EFFA-D USA EFFA-D Canada EFFA-D UK EFFA-D France EFFA-D Germany EFFA-D Italy EFFA-D Japan JPM EMBI Global JPM EMBI Global JPM EMBI Global JPM EMBI Global

0.000164 0.000213 0.000173 0.000175 0.000174 0.000176 4.11E05 0.000343 0.000399 0.000370 0.000386

0.003097 0.002657 0.003509 0.002325 0.002287 0.002394 0.001434 0.005024 0.005370 0.005439 0.004683

0.014811 0.012404 0.133087 0.126420 0.126664 0.225659 0.288082 1.617909 2.034917 2.559425 2.108931

6.093842 4.180761 7.439692 4.616562 4.776790 4.797250 5.858524 80.45309 36.44968 74.34854 50.11168

723.1403*** 105.3664*** 1494.344*** 202.2401*** 243.3315*** 259.3946*** 642.3404*** 453964.0*** 85773.38*** 386532.8*** 169009.4***

Asia LA EE Comp.

*** Denotes significance at the 5% level (rejection of the normal distribution).

skewness and positive excess kurtosis, the joint distribution is not elliptic. Moreover, if the distribution is nonelliptic, the correlation does not exhaust the full interval [1, þ1], then this can lead to invalid results of very low dependence, even if the variables are perfectly correlated. Consequently, the cointegration approach is more adapted, because it applies directly on asset prices rather than returns, and it does not require the acceptance of stationarity of the asset value series. Table 2 shows the correlation matrix between the hedge funds Macro Global time series of stock and bond indices for the countries and regions indicated. Global macro hedge funds returns seem more correlated with the emerging markets stock indices than bond indices, while the hedge funds show weak comovements with the stocks and bonds markets of Canada, United Kingdom, France, Germany, Italy and Japan. In spite of the problems involved in the use of correlation, correlation and cointegration are related but distinct concepts. High correlation does not

50

Table 2:

Wafa Kammoun Masmoudi

Contemporaneous Correlations between Daily Returns HFRX MACRO

Stocks MSCI USA MSCI Canada MSCI UK MSCI France MSCI Germany MSCI Italy MSCI Japan S&P/IFCI Asia S&P/IFCI LA S&P/IFCI EE S&P/IFCI EM comp.

(0.004) 0.142 0.112 0.120 0.121 0.096 0.167 0.205 0.173 0.150 0.236

Bonds EFFA-D USA EFFA-D Canada EFFA-D UK EFFA-D France EFFA-D Germany EFFA-D Italy EFFA-D Japan JPM EMBI Global JPM EMBI Global JPM EMBI Global JPM EMBI Global

0.102 0.101 0.043 0.056 0.059 0.033 (0.092) 0.040 0.119 0.042 0.094

Asia LA EE Comp.

imply the presence of a cointegration. Yet, a high correlation is neither necessary nor sufficient for the existence of a cointegration relationship between assets. Indeed, cointegrated time series can have low correlations. Thus, a covariance significantly different from zero does not inevitably imply an economic or financial relationship. Before applying the cointegration analysis, it is necessary to test whether each series requires the same degree of differencing to achieve stationarity. In order to determine the order of integration, we apply the Augmented Dickey– Fuller t-tests (ADF). The null hypothesis of unit root is tested successively for each series by estimating regression equations for a random walk with a drift and trend, then a random walk with drift and finally a random walk. The ADF tests (Table 3) show that all the financial series are nonstationary in levels, and when we use first differences, the null hypothesis of a unit root is rejected at the 1% level for all asset classes. This means that all the indices are integrated of order 1, I(1) with constant term and drift parameter, a necessary condition to test the cointegration.

Global Macro Hedge Funds and Traditional Financial Assets

Table 3:

51

Augmented Dickey–Fuller Tests Standard model

Model with constant

Model with trend and constant

HFRX Macro

Index DIndex

0.950842

1.925440

1.737602 37.80852

USA

Stocks Bonds DStocks DBonds

0.591276 2.247696

1.966736 0.185034

1.935526 2.405663 35.57811 33.15422

Canada

Stocks Bonds DStocks DBonds

1.425183 3.395499

2.161290 1.155967

1.850848 3.280053 18.77973 32.95028

UK

Stocks Bonds DStocks DBonds

0.520171 2.088932

1.803525 0.511279

1.624609 2.964837 20.72594 40.76178

France

Stocks Bonds DStocks DBonds

0.847457 3.205411

2.230010 0.002908

1.724488 1.681630 19.85683 40.51368

Germany

Stocks Bonds DStocks DBonds

1.169541 3.016262

2.837198 0.079941

2.075860 1.854743 43.45628 39.74396

Italy

Stocks Bonds DStocks DBonds

0.252682 3.122175

1.642888 0.140461

1.621310 1.746912 19.44972 40.52331

Japan

Stocks Bonds DStocks DBonds

0.765308 1.219506

2.490788 0.001550

2.133117 2.613706 34.23094 42.77867

Asia

Stocks Bonds DStocks DBonds

1.509709 1.864538

2.240362 0.563406

1.987300 2.993395 37.68168 8.659935

Latin America

Stocks Bonds DStocks DBonds

1.901285 2.307530

1.864640 1.666167

1.917745 2.447096 39.44296 32.41010

52

Wafa Kammoun Masmoudi

Table 3. (Continued ) Standard model

Model with constant

Model with trend and constant

Eastern Europe

Stocks Bonds DStocks DBonds

0.978915 2.101755

2.186868 1.105101

1.707057 2.367560 37.35552 16.69596

Emerging Markets

Stocks Bonds DStocks DBonds

1.643423 2.122211

2.213464 1.162528

1.889178 2.516949 33.89265 11.51694

Critical value

1% 5% 10%

2.566240 1.940999 1.616582

3.433757 2.862932 2.567558

3.963121 3.412293 3.128080

3. Methodology and Empirical Results 3.1. Vector Autoregression Theory – Granger Causality Tests We can test Granger causality by running a VAR on the system of equations and testing for zero restrictions on the VAR coefficients. DHF t ¼ m1 þ g1 Z t1 þ

p X

y1i DStockti þ

i¼1

þ

p X

p X

d1i DBondsti

i¼1

l1i DHF ti þ 1t ;

i¼1

DStockt ¼ m2 þ g2 Z t1 þ

p X

y2i DStockti þ

i¼1

þ

p X

p X

d2i DBondsti

i¼1

(1)

l2i DHF ti þ 2t ;

i¼1

DBond t ¼ m3 þ g3 Z t1 þ

p X

y3i DStockti þ

i¼1

þ

p X

p X

d3i DBondsti

i¼1

l3i DHF ti þ 3t ;

i¼1

where m and et are N  1 vectors. The Granger (1969) approach to the question of whether x causes y is to see how much of the current y can be explained by the past values of y and to see whether adding lagged values of x can improve the explanation. The y

Global Macro Hedge Funds and Traditional Financial Assets

53

is said to be Granger-caused by x if x helps in the prediction of y, or equivalently if the coefficients on the lagged x’s are statistically significant. Note that the two-way causation is frequently the case; x Granger causes y and y Granger causes x (Table 4). The estimated VAR model parameters can be obtained by OLS estimation, equation by equation. The results of the estimation of VAR models for different countries and regions are not interpretable because of the structure of the VAR model which is considered atheoretical. Thus, it is difficult to interpret the effect of one variable on another. Nevertheless, some estimation results of VAR models for developed countries confirm the results highlighted in the literature and show that the funds return is explained both by its own past changes and by changes in stock market return.6 In a first step, to confirm the significance of variables lags, we apply the Granger causality test which provides significant power in terms of price distortion. The results of causality tests show that global macro hedge funds return is explained as well by its history as by stocks return for the United States, Canada, the United Kingdom, France, Germany, Italy, Latin America, Eastern Europe and emerging market regions. For Asia, the global macro hedge funds cause S&P IFCI Asia, since the null hypothesis (‘no Granger causality’) can be rejected at the 1% significance level. On another side, bidirectional causality exists between the bond indices and the hedge funds for Canada and Japan. Moreover, in Latin America, the EFFADatastream index causes the HFRX Macro. In addition, we observed another bidirectional causality between stocks and bond indices for France, Italy, Japan and all the emerging market regions. For Germany, MSCI index causes the EFFA-Datastream bond index, whereas this causality is reversed for the United Kingdom. As a second step, we study the dynamic behaviour of VAR models using the approach based on impulse response functions.7 An impulse response function traces the effect of a one standard deviation shock to one of the innovations on current and future values of the endogenous variables. A shock to the ith variable not only directly affects the ith variable but is also transmitted to all of the other endogenous variables through the dynamic (lag) structure of the VAR (Figure 2). To study the mechanisms of adjustment, it is interesting to analyse the response functions of the different indices associated with a positive shock (pulse unit) on the different indices returns. To determine the interaction of returns to a very short term, we use the Cholesky decomposition to identify ‘structural’ innovations.

6 7

The test results are available from the author upon request. Variance decomposition is also frequently used.

54

Table 4:

Wafa Kammoun Masmoudi

Pairwise Granger Causality Tests Pairwise Granger Causality Tests

Null Hypothesis:

FProbability Statistic

MSCI USA does not Granger Cause EFFA-D USA

0.25783

0.7728

EFFA-D USA does not Granger Cause MSCI USA

1.77161

0.1704

HFRX MACRO does not 1.42619 Granger Cause EFFA-D USA

0.2405

EFFA-D USA does not Granger Cause HFRX MACRO

2.41515

0.0896

HFRX MACRO does not Granger Cause MSCI USA

0.38804

0.6784

MSCI USA does not Granger Cause HFRX MACRO

6.38904

0.0017

MSCI UK does not Granger Cause EFFA-D UK

2.38853

0.0360

EFFA-D UK does not Granger Cause MSCI UK

0.52848

0.7549

HFRX MACRO does not Granger Cause EFFA-D UK

0.52768

0.7555

EFFA-D UK does not Granger Cause HFRX MACRO

0.99335

0.4203

HFRX MACRO does not Granger Cause MSCI UK

0.19692

0.9638

Null Hypothesis:

MSCI Canada does not Granger Cause EFFA-D Canada EFFA-D Canada does not Granger Cause MSCI Canada HFRX MACRO does not Granger Cause EFFA-D Canada EFFA-D Canada does not Granger Cause HFRX MACRO HFRX MACRO does not Granger Cause MSCI Canada MSCI Canada does not Granger Cause HFRX MACRO MSCI France does not Granger Cause EFFA-D France EFFA-D France does not Granger Cause MSCI France HFRX MACRO does not Granger Cause EFFA-D France EFFA-D France does not Granger Cause HFRX MACRO HFRX MACRO does not Granger Cause MSCI France

FProbability Statistic 1.76393

0.1171

1.87335

0.0959

2.52420

0.0276

2.13760

0.0584

0.44094

0.8201

4.08811

0.0011

3.34855

0.0052

2.74941

0.0176

1.34687

0.2417

1.19546

0.3089

0.36115

0.8752

Global Macro Hedge Funds and Traditional Financial Assets

55

Table 4. (Continued ) Pairwise Granger Causality Tests Null Hypothesis:

FProbability Statistic

MSCI UK does not Granger Cause HFRX MACRO

3.56923

0.0032

MSCI Germany does not 1.42636 Granger Cause EFFA-D Germany EFFA-D Germany does not 2.17966 Granger Cause MSCI Germany HFRX MACRO does not 1.22876 Granger Cause EFFA-D Germany

0.2116

EFFA-D Germany does not 1.20391 Granger Cause HFRX MACRO

0.3048

HFRX MACRO does not Granger Cause MSCI Germany MSCI Germany does not Granger Cause HFRX MACRO MSCI Japan does not Granger Cause EFFA-D Japan EFFA-D Japan does not Granger Cause MSCI Japan HFRXMACRO does not Granger Cause EFFA-D Japan EFFA-D Japan does not Granger Cause HFRX MACRO HFRX MACRO does not Granger Cause MSCI Japan MSCI Japan does not Granger Cause HFRX MACRO S&P IFCI Asia does not Granger Cause JPM EMBI Global Asia

0.97482

0.4318

4.30151

0.0007

4.99187

0.0019

4.75904

0.0026

2.61670

0.0495

2.91684

0.0331

2.20441

0.0857

1.06146

0.3643

1.98366

0.0648

0.0539

0.2930

Null Hypothesis:

MSCI France does not Granger Cause HFRX MACRO MSCI Italy does not Granger Cause EFFA-D Italy EFFA-D Italy does not Granger Cause MSCI Italy HFRX MACRO does not Granger Cause EFFA-D Italy EFFA-D Italy does not Granger Cause HFRX MACRO HFRX MACRO does not Granger Cause MSCI Italy MSCI Italy does not Granger Cause HFRX MACRO

FProbability Statistic 4.78426

0.0002

7.22463

1.E-06

4.32993

0.0006

1.46754

0.1973

1.21293

0.3005

0.84253

0.5194

4.09023

0.0011

S&P IFCI LA does 7.32266 not Granger Cause JPM EMBI Global LA

8.E-07

56

Wafa Kammoun Masmoudi

Table 4. (Continued ) Pairwise Granger Causality Tests Null Hypothesis:

FProbability Statistic

JPM EMBI Global Asia 9.77195 does not Granger Cause S&P IFCI Asia

1.E-10

HFRX MACRO does not Granger Cause JPM EMBI Global Asia

0.56100

0.7616

JPM EMBI Global Asia 0.63009 does not Granger Cause HFRX MACRO

0.7063

HFRX MACRO does not 4.01717 Granger Cause S&P IFCI Asia

0.0005

S&P IFCI Asia does not Granger Cause HFRX MACRO

0.81976

0.5545

S&P IFCI Eastern does not 6.84717 Granger Cause JPM EMBI Global EE

3.E-07

JPM EMBI Global Eastern 6.33554 does not Granger Cause S&P IFCI EE

1.E-06

HFRX MACRO does not Granger Cause JPM EMBI Global EE

0.47213

0.8294

JPM EMBI Global Eastern 1.77822 does not Granger Cause HFRX MACRO

0.0998

HFRX MACRO does not 0.59555 Granger Cause S&P IFCI EE

0.7341

S&P IFCI EE does not Granger Cause HFRX MACRO

0.0069

2.96602

Null Hypothesis:

JPM EMBI Global Latin Am does not Granger Cause S&P IFCI LA HFRX MACRO does not Granger Cause JPM EMBI Global LA JPM EMBI Global LA does not Granger Cause HFRX MACRO HFRX MACRO does not Granger Cause S&P IFCI LA S&P IFCI LA does not Granger Cause HFRX MACRO S&P IFCI EM does not Granger Cause JPM EMBI Global Comp. JPM EMBI Global Comp. does not Granger Cause S&P IFCI EM HFRX MACRO does not Granger Cause JPM EMBI Global Comp. JPM EMBI Global Comp. does not Granger Cause HFRX MACRO HFRX MACRO does not Granger Cause S&P IFCI EM S&P IFCI EM does not Granger Cause HFRX MACRO

FProbability Statistic 4.56694

0.0004

0.65340

0.6589

3.21577

0.0068

0.29173

0.9177

7.55677

5.E-07

2.24040

0.0287

6.93387

4.E-08

0.43044

0.8836

1.99943

0.0519

1.64073

0.1196

2.18502

0.0329

5

6

7

8

9

10

9

10

2

3

4

5

6

7

8

9

10

2

3

4

5

6

7

8

9

10

9

Figure 2:

10

1

3

4

5

6

7

8

9

10

Impulse Response Functions.

2

-.002

8

-.002

7

-.002

6

-.001

-.001

5

.000 -.001

.000

.000

4

.001

.001

.001

3

.002

.002

.002

2

.003

.003

.003

1

.004

-.004

.000

.004

.008

.012

.016

.004

Response of BOND_USA to STOCK_USA

1

2

3

4

5

6

7

8

9

10

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Response of BOND_USA to BOND_USA

1

Response of STOCK_USA to BOND_USA

-.001

.000

.001

.002

.003

.004

1

Response of GLOBAL_MACRO to BOND_USA .005

.004

Response of BOND_USA to GLOBAL_MACRO

8

-.004

7

-.004

6

.000

.000

5

.004

.004

4

.008

.008

3

.012

.012

2

.016

.016

1

1 Response of STOCK_USA to STOCK_USA

-.001

-.001

4

.000

.000

3

.001

.001

2

.002

.002

1

.003

.003

Response of STOCK_USA to GLOBAL_MACRO

.004

.004

Response of GLOBAL_MACRO to STOCK_USA .005

.005

Response of GLOBAL_MACRO to GLOBAL_MACRO

Response to Cholesky One S.D. Innovations ± 2 S.E.

Global Macro Hedge Funds and Traditional Financial Assets 57

7

8

9 10

1

2

3

4

5

6

7

8

9 10

8

9 10

1

2

3

4

5

6

7

8

9 10

6

7

8

9 10

-.001

5

-.001

4

.000

.000

3

.001

.001

2

.002

.002

1

.003

.003

2

3

4

Figure 2:

1

5

7

8

9 10

(Continued)

6

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

-.001

.000

.001

.002

.003

1

2

3

4

5

6

7

8

9 10

Response of BOND_CANADA to BOND_CANADA

Response of BOND_CANADA to STOCK_CANADA

7

Response of BOND_CANADA to GLOBAL_MACRO

6

.000 -.005

.000 -.005

.005

.010

.015

.020

.000

5

1

Response of STOCK_CANADA to BOND_CANADA

-.001

-.005

4

.005

.005

3

.010

.010

2

.015

.015

1

.020

.020

Response of STOCK_CANADA to GLOBAL_MACRO Response of STOCK_CANADA to STOCK_CANADA

6

-.001

-.001

5

.000

.000

4

.000

.001

.001

3

.001

.002

.002

2

.002

.003

.003

1

.004

.004

.004 .003

.005

.005

.005

Response of GLOBAL_MACRO to GLOBAL_MACRO Response of GLOBAL_MACRO to STOCK_CANADA Response of GLOBAL_MACRO to BOND_CANADA

Response to Cholesky One S.D. Innovations ± 2 S.E.

58 Wafa Kammoun Masmoudi

7

8

9

10

8

9

10

3

4

1

2

3

4

Figure 2:

-.002

-.002 10

2

5

6

7

8

9

10

6

7

8

9

(Continued)

5

10

Response of BOND_UK to STOCK_UK

1

.004

-.004

.000

.004

.008

.012

.016

-.002

-.001

.002

.003

.000

.000 -.001

.000

-.001 9

10

.001

8

9

.001

7

8

.001

6

7

.002

5

6

.002

4

5

.003

3

4

.004

2

3

-.001

.000

.001

.002

.003

.004

.003

1

2

Response of STOCK_UK to STOCK_UK

1 2

3

4

5

6

7

8

9

10

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Response of BOND_UK to BOND_UK

1

Response of STOCK_UK to BOND_UK

1

Response of GLOBAL_MACRO to BOND_UK .005

.004

Response of BOND_UK to GLOBAL_MACRO

7

-.004

6

-.004

5

.000

.000

4

.004

.004

3

.008

.008

2

.012

.012

1

.016

.016

Response of STOCK_UK to GLOBAL_MACRO

6

-.001

-.001

5

.000

.000

4

.001

.001

3

.002

.002

2

.003

.003

1

.004

.004

Response of GLOBAL_MACRO to STOCK_UK .005

.005

Response of GLOBAL_MACRO to GLOBAL_MACRO

Response to Cholesky One S.D. Innovations ± 2 S.E.

Global Macro Hedge Funds and Traditional Financial Assets 59

8

9 10

1

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

8

9 10

2

3

4

Figure 2:

1

5

7

8

9 10

(Continued)

6

-.001

7

-.001

6

-.001

5

.000

.000

.000

4

.001

.001

.001

3

.002

.002

.002

2

.003

1

1

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

Response of BOND_FRANCE to BOND_FRANCE

.003

Response of BOND_FRANCE to STOCK_FRANCE

.003

Response of BOND_FRANCE to GLOBAL_MACRO

.000 -.005

.000 -.005

.000

-.005

4

.005

.005

.005

3

.010

.010

.010

2

.015

.015

.015

1

.020

.020

.020

Response of STOCK_FRANCE to GLOBAL_MACRO Response of STOCK_FRANCE to STOCK_FRANCE Response of STOCK_FRANCE to BOND_FRANCE

7

-.001

-.001

6

.000 -.001

.000

.000

5

.001

.001

.001

4

.002

.002

.002

3

.003

.003

.003

2

.004

.004

1

.005

.005

.005

.004

Response of GLOBAL_MACRO to GLOBAL_MACRO Response of GLOBAL_MACRO to STOCK_FRANCE Response of GLOBAL_MACRO to BOND_FRANCE

Response to Cholesky One S.D. Innovations ± 2 S.E.

60 Wafa Kammoun Masmoudi

7

8

9 10

1

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

8

9 10

2

3

4

5

6

7

8

9 10

6

7

8

9 10

2

3

4

Figure 2:

1

5

7

8

9 10

(Continued)

6

-.001

5

-.001

4

-.001

3

.000

.000

.000

2

.001

.001

.001

.003 .002

1

1

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

Response of BOND_GERMANY to BOND_GERMANY

.002

Response of BOND_GERMANY to STOCK_GERMANY .003

1

.002

Response of BOND_GERMANY to GLOBAL_MACRO .003

7

-.005

6

-.005

5

-.005

4

.000

.000

.000

3

.005

.005

.005

2

.010

.010

.010

1

.015

.015

.015

Response of STOCK_GERMANY to GLOBAL_MACRO Response of STOCK_GERMANY to STOCK_GERMANY Response of STOCK_GERMANY to BOND_GERMANY .020 .020 .020

6

-.001

-.001

5

.000 -.001

.000

.000

4

.001

.001

.001

3

.002

.002

.002

2

.003

.003

.003

1

.004

.004

.004

Response of GLOBAL_MACRO to BOND_GERMANY .005

.005

.005

Response of GLOBAL_MACRO to STOCK_GERMANY

Response to Cholesky One S.D. Innovations ± 2 S.E.

Response of GLOBAL_MACRO to GLOBAL_MACRO

Global Macro Hedge Funds and Traditional Financial Assets 61

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

5

6

7

8

9 10

7

8

9 10

2

3

4

1

2

3

4

Figure 2:

-.001

6

-.001

5

.000

.000

4

.001

.001

3

.002

.002

2

.003

1

1

5

6

7

8

9 10

6

7

8

9 10

(Continued)

5

Response of BOND_ITALY to STOCK_ITALY

.003

Response of BOND_ITALY to GLOBAL_MACRO

4

.000 -.005

.000

-.005

3

.000

.005

.005

2

.005

.010

.010

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

-.001

.000

.001

.002

.003

1

2

3

4

5

6

7

8

9 10

Response of BOND_ITALY to BOND_ITALY

-.005

.010

.015

.015

.015

.020

.020

1

1

Response of STOCK_ITALY to BOND_ITALY

-.001

.000

.001

.020

Response of STOCK_ITALY to STOCK_ITALY

-.001

-.001

2

.000

.000

1

.001

.001

Response of STOCK_ITALY to GLOBAL_MACRO

.003

.002

.002

.002

.004

.003

.003

.005

.004

.004

Response of GLOBAL_MACRO to BOND_ITALY

.005

.005

Response of GLOBAL_MACRO to STOCK_ITALY

Response to Cholesky One S.D. Innovations ± 2 S.E.

Response of GLOBAL_MACRO to GLOBAL_MACRO

62 Wafa Kammoun Masmoudi

6

7

8

9 10

9 10

6

7

8

9 10

3

4

1

2

3

4

1

2

3

4

Figure 2:

-.0010

-.0010

5

-.0005

-.0005

4

.0000

.0000

3

.0005

.0005

2

.0010

.0010

1

2

5

6

7

8

9 10

5

6

7

8

9 10

6

7

8

9 10

(Continued)

5

Response of BOND_JAPAN to STOCK_JAPAN .0015

.0015

Response of BOND_JAPAN to GLOBAL_MACRO

8

-.004

7

-.004

6

.000

.000

5

.004

.004

4

.008

.008

3

.012

.012

2

.016

.016

1

1 Response of STOCK_JAPAN to STOCK_JAPAN

-.001

-.001

5

.000

.000

4

.001

.001

3

.002

.002

2

.003

.003

1

.004

.004

Response of STOCK_JAPAN to GLOBAL_MACRO

.005

.005

Response of GLOBAL_MACRO to STOCK_JAPAN

1

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

-.0010

-.0005

.0000

.0005

.0010

.0015

1

2

3

4

5

6

7

8

9 10

Response of BOND_JAPAN to BOND_JAPAN

-.004

.000

.004

.008

.012

.016

Response of STOCK_JAPAN to BOND_JAPAN

-.001

.000

.001

.002

.003

.004

.005

Response of GLOBAL_MACRO to BOND_JAPAN

Response to Cholesky One S.D. Innovations ± 2 S.E.

Response of GLOBAL_MACRO to GLOBAL_MACRO

Global Macro Hedge Funds and Traditional Financial Assets 63

3

4

5

6

7

8

9

10

9

10

7

8

9

10

4

1

2

3

4

1

2

3

4

Figure 2:

-.001

-.001

6

.000

.000

5

.001

.001

4

.002

.002

3

.003

.003

2

.004

.004

1

3

5

6

7

8

9

10

5

6

7

8

9

10

6

7

8

9

(Continued)

5

10

Response of BOND_ASIA to STOCK_ASIA .005

.005

Response of BOND_ASIA to GLOBAL_MACRO

8

-.004

7

-.004

6

.000

.000

5

.004

.004

4

.008

.008

3

.012

.012

2

.016

.016

1

2

Response of STOCK_ASIA to STOCK_ASIA

1

.000 -.001

.000

-.001

2

.001

.001

1

.000

.002

.002

Response of STOCK_ASIA to GLOBAL_MACRO

.001

.003

.003

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

-.001

.000

.001

.002

.003

.004

.005

1

2

3

4

5

6

7

8

9

10

Response of BOND_ASIA to BOND_ASIA

-.004

.000

.004

.008

.012

.016

Response of STOCK_ASIA to BOND_ASIA

-.001

.002

.003

.004

.004

.004

Response of GLOBAL_MACRO to BOND_ASIA .005

.005

.005

Response of GLOBAL_MACRO to STOCK_ASIA

Response to Cholesky One S.D. Innovations ± 2 S.E.

Response of GLOBAL_MACRO to GLOBAL_MACRO

64 Wafa Kammoun Masmoudi

8

9

10

8

9

10

7

8

9

10

3

4

1

2

3

4

1

2

3

4

Figure 2:

-.001

-.001

6

.000

.000

5

.001

.001

4

.002

.002

3

.003

.003

2

.004

.004

1

2

5

6

7

8

9

10

5

6

7

8

9

10

6

7

8

9

(Continued)

5

10

Response of BOND_LATIN to STOCK_LATIN .005

.005

Response of BOND_LATIN to GLOBAL_MACRO

7

-.005

-.005

6

.000

.000

5

.005

.005

4

.010

.010

3

.015

.015

2

.020

.020

1

1 Response of STOCK_LATIN to STOCK_LATIN .025

.025

Response of STOCK_LATIN to GLOBAL_MACRO

7

-.001

-.001

6

.000

.000

5

.001

.001

4

.002

.002

3

.003

.003

2

.004

.004

1

.005

Response of GLOBAL_MACRO to STOCK_LATIN

.005

Response of GLOBAL_MACRO to GLOBAL_MACRO

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

-.001

.000

.001

.002

.003

.004

.005

1

2

3

4

5

6

7

8

9

10

Response of BOND_LATIN to BOND_LATIN

-.005

.000

.005

.010

.015

.020

.025

Response of STOCK_LATIN to BOND_LATIN

-.001

.000

.001

.002

.003

.004

.005

Response of GLOBAL_MACRO to BOND_LATIN

Response to Cholesky One S.D. Innovations ± 2 S.E.

Global Macro Hedge Funds and Traditional Financial Assets 65

8

9 10 1

2

3

4

5

6

7

8

9 10

-.001

.000

.001

.002

.003

.004

.005

1

2

3

4

5

6

7

8

9 10

Response of GLOBAL_MACRO to BOND_EASTERN

6

7

8

9 10

6

7

8

9 10

-.001

-.001

5

.000

.000

4

.001

.001

3

.002

.002

2

.003

.003

1

.004

.004

.005

1

2

3

4

5

6

7

8

9 10

2

3

4

Figure 2:

1

5

7

8

9 10

(Continued)

6

Response of BOND_EASTERN to STOCK_EASTERN .005

-.005

-.005

5

.000

.000

4

.005

.005

3

.010

.010

2

.015

.015

1

.020

.020

Response of BOND_EASTERN to GLOBAL_MACRO

.025

.025

1

2

3

4

5

6

7

8

9 10

-.001

.000

.001

.002

.003

.004

.005

1

2

3

4

5

6

7

8

9 10

Response of BOND_EASTERN to BOND_EASTERN

-.005

.000

.005

.010

.015

.020

.025

Response of STOCK_EASTERN to GLOBAL_MACRO Response of STOCK_EASTERN to STOCK_EASTERN Response of STOCK_EASTERN to BOND_EASTERN

7

-.001

-.001

6

.000

.000

5

.001

.001

4

.002

.002

3

.003

.003

2

.004

.004

1

.005

Response of GLOBAL_MACRO to STOCK_EASTERN

.005

Response of GLOBAL_MACRO to GLOBAL_MACRO

Response to Cholesky One S.D. Innovations ± 2 S.E.

66 Wafa Kammoun Masmoudi

5

6

7

8

9 10

-.001

-.001

4

.000

.000

3

.001

.001

2

.002

.002

1

.003

.003

1

2

3

4

5

6

7

8

9 10

-.001

.000

.001

.002

.003

.004

1

2

3

4

5

6

7

8

9 10

Response of GLOBAL_MACRO to BOND_EMERGING .005

5

6

7

8

9 10

8

9 10

2

3

4

1

2

3

4

Figure 2:

-.001

7

-.001

6

.000

.000

5

.001

.001

4

.002

.002

3

.003

.003

2

.004

.004

1

1

5

6

7

8

9 10

-.004

.000

.004

.008

.012

.016

1

2

3

4

5

6

7

8

9 10

5

7

8

9 10

(Continued)

6

-.001

.000

.001

.002

.003

.004

1

2

3

4

5

6

7

8

9 10

Response of BOND_EMERGING to STOCK_EMERGING Response of BOND_EMERGING to BOND_EMERGING

-.004

-.004

4

.000

.000

3

.004

.004

2

.008

.008

1

.012

.012

Response of BOND_EMERGING to GLOBAL_MACRO

.016

.016

Response of STOCK_EMERGING to GLOBAL_MACRO Response of STOCK_EMERGING to STOCK_EMERGING Response of STOCK_EMERGING to BOND_EMERGING

.004

.004

Response of GLOBAL_MACRO to STOCK_EMERGING .005

Response to Cholesky One S.D. Innovations ± 2 S.E.

Response of GLOBAL_MACRO to GLOBAL_MACRO .005

Global Macro Hedge Funds and Traditional Financial Assets 67

68

Wafa Kammoun Masmoudi

For all the developed countries except Japan, the Latin America, Eastern Europe and the emerging markets, the results show that a shock on the stock index affects global macro but it is damped after seven periods maximum. The shock on HFRX Macro has an immediate impact on the stock index S&P IFCI Asia for the Asian region. In addition, a shock on the bond index is reflected on hedge funds index and conversely for Canada and Japan. Furthermore, in Latin America, a shock on the EFFA-Datastream bond index has an effect on HFRX Macro. The analysis of impulse response function for France, Italy, Japan and all emerging market regions shows that a shock on the stock index impacts the bond index. For developed countries indices, the reaction behaviour is quite similar, whereas the impact of the shock is larger and takes a little longer to damp for the emerging markets regions. For Germany, a shock on MSCI index touches the German bond index, whereas a shock on the EFFA-Datastream UK is reflected on the stock index of this country. It should be noted that these results are in conformity with those of Granger causality. Overall, the shock leads to a rise of the index then the price growth slowed and experienced a slight decline from the peak they had reached. In period 2, the market seems to correct this decline and pushes back the price increase, followed by a further decline. This could possibly be explained by the surajustement phenomenon. The shock that propagates market generates a process of adjustment by trial and error whose magnitude depends on the size of the initial shock. As a result, market corrections can be temporarily beyond what is necessary to make up the long-term average return. In addition, in an arbitrage optique, the liquidation of positions by arbitragist at the end of the day can lead to the decline. The dynamics of these shocks could reflect a liquid and efficient market supporting important price skids. At this level, we have shown causal relationships between the financial assets which could provide elements of reflection favourable with a better portfolio diversification. Nevertheless, knowing direction of causality is as important as the identification of relationships between assets.

3.2. Multivariate Cointegration Analysis Having determined that all the variables are integrated of the same order (I(1)), we can then test whether these variables are related together in the long term (i.e. have a common stochastic trend). The essence of cointegration is that the series cannot diverge arbitrarily far from each

Global Macro Hedge Funds and Traditional Financial Assets

69

other, implying that there exists a long-term relationship between these series and that they can be written in an error correction form. By definition, cointegrated markets thus exhibit common stochastic trends. This, in turn, limits the amount of independent variation between these markets. Hence, from the investors’ standpoint, markets that are cointegrated will present limited diversification opportunities. We will know then whether the global macro hedge funds offer diversification benefits for developed and emerging markets traditional portfolio. In the literature, two primary methods exist to examine the degree of cointegration among indices. The first is the Engle–Granger methodology (Engle and Granger, 1987) which is bivariate, testing for cointegration between pairs of indices. This approach is insufficient because it supposes the existence of only one relation of cointegration between the variables when the number of variables becomes higher than two. The second is the Johansen methodology, a maximum likelihood estimation of a fully specified error correction model (Johansen, 1988) which is a multivariate extension and allows us to test for the presence of more than one cointegration vector. This approach allows testing for the number as well as the existence of these common stochastic trends and involves determination of cointegrating vectors’ matrix rank. To implement the cointegration technique, we must estimate a VAR model used to capture the evolution and the interdependencies between multiple time series, generalizing the univariate AR models. All the variables in a VAR are treated symmetrically by including for each variable an equation explaining its evolution based on its own lags and the lags of all the other variables in the model. Formally, the Johansen methodology is based on the following n-dimensional cointegrated VAR(k) model in the vector equilibrium correction (VEC) form: DX t ¼ m þ G1 DX t1 þ    þ Gk1 DX tkþ1 þ PX t1 þ t ,

(2)

where m is an n  1 vector of intercept terms, DXt shows an n  1 vector of price changes in period t, Gi (i ¼ 1, y, k  1) represents the n  n coefficient matrix of short-term dynamics, P is the n  n long-term impact matrix and et is an n  1 iid Gaussian error vector (Maddala and Kim, 1998). When cointegration is present, the long-term response matrix can be decomposed as P ¼ abu, where the relevant elements of the a matrix are adjustment coefficients and the b matrix contains the cointegrating vectors. In other words, the expression buXt1 defines the stationary linear combinations (cointegration relationships) of the I(1) vector Xt, whereas the matrix a of the error correction terms (ECTs) describes how the system variables adjust to the equilibrium error from the previous period, buXt1.

70

Wafa Kammoun Masmoudi

Table 5:

VAR and VECM lag number USA Canada UK France Germany Italy Japan Asia L.A E.E E.M

VAR en niveau VECM

3 2

6 5

6 5

6 5

6 5

6 5

4 3

7 6

6 6

7 6

7 6

Thus, the cointegration relationships and long-term relationships act like recall forces on the short-term dynamics of the variables and are therefore qualified built-in error correction mechanisms. We continue the steps of the Johansen procedure by determining first of all the lags number (p) of Vector Autoregressive Models (VAR(p)) in levels and by taking the highest number of lags given by criteria AIC, BIC and HQ (Table 5). Johansen (1988, 1991) proposes two methods for testing the number of cointegration vectors: the trace statistic (ltrace) and the maximum eigenvalue test (lmax). The trace test equals a likelihood ratio test for maximum r cointegration vectors against the alternative of n vectors as follows: ltrace ¼ T

n X

lnð1  l~ i Þ,

(3)

i¼qþ1

where T is the sample size and l~ rþ1 ; ::::; l~ n equal the (n–r) smallest squared canonical correlations. If the statistic is bigger than the critical value, the null hypothesis of at most r cointegrating vectors is rejected. The maximum eigenvalue test possesses an identical null hypothesis, whereas the alternative hypothesis equals rþ1 cointegration vectors, i.e. if the statistic is bigger than the critical value, the null hypothesis of exactly r cointegrated vectors is rejected. The maximum eigenvalue statistic lmax is lmax ¼ T lnð1  l~ qþ1 Þ,

(4)

where l~ 1 ; ::::; l~ q are the largest squared canonical correlations. Given the time series used, we have ignored the trends of quadratic and linear data, but we have included a constant term in the long-term relationships, mainly because prices of different logarithmic asset classes are not driven by the same factor and thus do not show a similar pattern in the long term. Therefore, the model we choose for the trace and maximum eigenvalue tests has no deterministic trend, but it does have a constant in the cointegration vectors. The Johansen cointegration tests are based on the VAR lags number to show cointegration relationships for different countries and regions (Table 6). Hence, the VECM involves (k – 1) terms in differences. Note that

Global Macro Hedge Funds and Traditional Financial Assets

Table 6:

71

Johansen Cointegration Tests Eigen value

Trace statistic

5% critical value

H0: r

Max-eigen statistic

5% critical value

USA

0.007475 0.003465 0.002652

24.67027 11.08914 4.806782

35.19275 20.26184 9.164546

0 1 2

13.58112 6.282359 4.806782

22.29962 15.89210 9.164546

Canada

0.013706 0.004178 0.001855

35.85857** 10.92072 3.355456

35.19275 20.26184 9.164546

0 1 2

24.93785** 7.565268 3.355456

22.29962 15.89210 9.164546

UK

0.008036 0.003455 0.001537

23.61377 9.034552 2.780234

35.19275 20.26184 9.164546

0 1 2

14.57922 6.254318 2.780234

22.29962 15.89210 9.164546

France

0.013234 0.003259 0.001515

32.71038 8.637739 2.739666

35.19275 20.26184 9.164546

0 1 2

24.07264** 5.898073 2.739666

22.29962 15.89210 9.164546

Germany

0.013256 0.003257 0.001892

33.42970 9.316534 3.421707

35.19275 20.26184 9.164546

0 1 2

24.11317** 5.894827 3.421707

22.29962 15.89210 9.164546

Italy

0.011564 0.004004 0.001741

31.41539 10.39839 3.148753

35.19275 20.26184 9.164546

0 1 2

21.01700 7.249641 3.148753

22.29962 15.89210 9.164546

Japan

0.006251 0.002843 0.001634

19.45392 8.109888 2.959029

35.19275 20.26184 9.164546

0 1 2

11.34403 5.150859 2.959029

22.29962 15.89210 9.164546

Asia

0.007884 0.003783 0.001468

23.79264 9.497579 2.653337

35.19275 20.26184 9.164546

0 1 2

14.29506 6.844242 2.653337

22.29962 15.89210 9.164546

Latin America

0.010844 0.004380 0.002145

31.51293 11.81155 3.879953

35.19275 20.26184 9.164546

0 1 2

19.70138 7.931594 3.879953

22.29962 15.89210 9.164546

Eastern Europe 0.006279 0.003720 0.001122

20.13357 8.758416 2.028347

35.19275 20.26184 9.164546

0 1 2

11.37516 6.730069 2.028347

22.29962 15.89210 9.164546

Emerging Markets

20.50698 7.862921 2.802981

35.19275 20.26184 9.164546

0 1 2

12.64406 5.059941 2.802981

22.29962 15.89210 9.164546

0.006977 0.002798 0.001551

** Denotes that the null hypothesis of no cointegration can be rejected at the 5% significance levels. The ltrace and lmax statistics are computed under the assumption of no deterministic trend and without constant in the cointegrating vector; Eigenv is the eigenvalue associated with cointegrating vector.

72

Table 7:

Wafa Kammoun Masmoudi

Residual Analysis LM(1)

LM(6)

LM(10)

Canada

8.023717 0.5318

24.79807 0.0032

6.802155 0.6577

France

5.567225 0.7823

Germany

8.067386 0.5274

9.078853 0.4300 11.67497 0.2323

3.044634 0.9625 4.088909 0.9055

for Canada, France and Germany only the maximum eigenvalue tests reject the null hypothesis of the no-cointegration vector.8 For the other markets, there is no cointegration relationship between assets, which shows that diversification benefits increase from adding global macro to bond/equity portfolio. Normality test by the residual analysis (Table 7) shows that errors are not Gaussian. However, according to Gonzalo (1994), non-normality of the residuals does not bias the results for Johansen cointegration tests, and test results are valid. When the cointegrating vectors are normalized on the first component global macro hedge fund index, we obtain the following longterm relationships: Canada: HFRX Macro ¼ 6.47 þ 0.17 Stocks-0.33 Bonds France: HFRX Macro ¼ 4.73 þ 0.18 Stocks-0.65 Bonds Germany: HFRX Macro ¼ 4.70 þ 0.076 Stocks-0.53 Bonds These relations show that the global macro hedge funds tend to have a positive (negative) long-term relationship with stocks (bonds) market indices for Canada, France and Germany. We can explain the positive relationship by the fact that hedge funds are really net long stocks. It should be noted that the information drawn from these long-term relationships can be used in developing forecasts of hedge fund values and thus portfolio strategies for traditional assets by explicitly taking global macro hedge funds into account. It is important to note that the existence of one or more common stochastic trend(s) does not imply that all assets are a driving force in the common trend. In fact, it is possible that global macro hedge fund and

8

According to Banerjee et al. (1993), the Johansen test gives a different result and between the trace and maximum eigenvalue statistics, the latter is preferred.

Global Macro Hedge Funds and Traditional Financial Assets

73

traditional asset series do not enter the common stochastic trend. Besides, according to Kasa (1992) the relevance of diversification benefits depends on the speed of adjustment toward the common trend. Therefore, if returns do not react significantly to common trends, their existence will only slightly impact diversification benefits. To analyse the nature of the cointegrating vector and the adjustment coefficients, we perform a likelihood ratio (LR) test statistic. To carry out the LR test, we test the absence of deterministic trends by comparing the restricted model to the unrestricted model:

LR ¼ T

r X i¼1

1  l^ i log  1  l^

!w w2 ðrÞ,

(5)

i

 where l^ i and l^ i are the eigenvalues of constrained and unconstrained models. In the Table 8, we present the results of the restrictions tests on the composition of the cointegrating vector for France, Canada and Germany. This table gives also the results of the restrictions tests on the reaction of assets returns to the common trend. Results from Canada, France and Germany reveal that stocks and bonds do not share a common trend with the hedge funds. The results stem from the low significance level. Furthermore, for France and Canada, hedge funds do not adjust with the remaining asset prices to the cointegrating vector while all three time series adjust to the cointegrating vector for Germany. This implies that risk-averse investors with long-term investment horizons can decrease their level of risk even by adding hedge funds to a conservative equity/bond portfolio. Nevertheless the diversification benefits for Canadian and French markets investors are higher than those for Germany.

Table 8:

Entering in and Adjustment to the Relevant Cointegrating Vector

France

bHF ¼ 0 bstocks ¼ 0 bBonds ¼ 0

w2(1) ¼ 2.875350* w2(1) ¼ 1.506353 w2(1) ¼ 1.417923

aHF ¼ 0 astocks ¼ 0 aBonds ¼ 0

w2(1) ¼ 2.508745 w2(1) ¼ 8.194000*** w2(1) ¼ 5.658958**

Canada

bHF ¼ 0 bstocks ¼ 0 bBonds ¼ 0

w2(1) ¼ 0.813073 w2(1) ¼ 0.633083 w2(1) ¼ 0.167764

aHF ¼ 0 astocks ¼ 0 aBonds ¼ 0

w2(1) ¼ 1.543646 w2(1) ¼ 7.070475*** w2(1) ¼ 7.147318***

Germany

bHF ¼ 0 bstocks ¼ 0 bBonds ¼ 0

w2(1) ¼ 2.673633* w2(1) ¼ 0.291127 w2(1) ¼ 1.102569

aHF ¼ 0 astocks ¼ 0 aBonds ¼ 0

w2(1) ¼ 3.065726* w2(1) ¼ 7.466223*** w2(1) ¼ 5.817424***

***, ** and * indicate that the null hypothesis (no entering into and no adjustment to the cointegrating vector(s)) can be rejected at the 1%, 5%, and 10%.

74

Wafa Kammoun Masmoudi

3.3. VEC Granger Causality Granger causality from one variable to another means that the conditional forecast for the latter can be significantly improved by adding lagged variables of the former to the information set. If the series are I(1), the null hypothesis of no Granger causality can be tested with the standard Wald tests (e.g. Lu¨tkepohl, 1991). Engle and Granger (1987) indicate that if I(1) variables are cointegrated, a corresponding error correction model exists in which the short-term dynamics of the variables in this system are affected by the deviation from long-term equilibrium. Note that short-term causal effects are perceived by changes in other lagged explanatory variables, whereas the long-term relationship is implied by the lagged ECT. Hence, in the cointegration model, the proposition ‘Xk not Granger-causing Xl’ in the long term is equivalent to akl ¼ 0. In this case, Xl is weakly exogenous for parameter b; in other words, Xl does not react to equilibrium errors. Furthermore, the proposition ‘Xk do not Granger-cause Xl’ in the short term is equivalent to Gkl ¼ 0, where (L) is the lag operator. Thus, the VECM allows to detect short- and long-term Granger causality (Granger, 1969) and uncover the price relationships among the three assets within different countries and regions. The VECM for the different countries corresponding to Equation (1) is given as follows: DHF t ¼ m1 þ g1 Z t1 þ

p X

y1i DStockti þ

i¼1

þ

p X

p X

d1i DBondsti

i¼1

l1i DHF ti þ t ;

i¼1

DStockt ¼ m2 þ g2 Z t1 þ

p X

y2i DStockti þ

i¼1

þ

p X

p X

d2i DBondsti

i¼1

(6)

l2i DHF ti þ t ;

i¼1

DBond t ¼ m3 þ g3 Z t1 þ

p X

y3i DStockti þ

i¼1

þ

p X

p X

d3i DBondsti

i¼1

l3i DHF ti þ t ;

i¼1

where Zt1 is the error correction term derived from the cointegrating vector y, d and l are the parameters to be estimated, p is the lag length and et are supposed to be stationary random processes with mean zero and constant variance.

Global Macro Hedge Funds and Traditional Financial Assets

75

For each equation of the system (6), a pairwise short-term Granger causality is used in order to test whether endogenous variables can be considered as exogenous by the joint significance of the coefficients of each of the other lagged endogenous variables in that equation. Accordingly, we test H 0 : y11 ¼ y12 ¼ y13 ¼ 0 and the s and l1’s by using w2 (Wald) statistics. For block exogeneity, we use the w2 statistic for joint significance of all other lagged endogenous variables in the equation. Thus, we test whether the real variables (hedge funds-stock index or hedge fund-bond index or stock indexbond index) can be treated as an exogenous block. On the other hand, the long-term causality is appreciated by the significance of the t-test(s) of the lagged ECT(s), i.e. by testing g ¼ 0. Nevertheless the VECM shows econometric exogeneity of the variables if both the t-test(s) and the w2-tests are insignificant. Table 9 presents the Granger causality and block exogeneity test results.

Table 9:

VEC Pairwise Granger Causality/Block Exogeneity Wald Testsa

Dependent variable

Short-run causality w2 Statistics (for excluded variables) DHedge Funds

Canada DHedge Funds DStock DBond

– 3.445215 14.87509**

France DHedge Funds DStock DBond Germany DHedge Funds DStock DBond

DStocks

DBonds

Block exogeneity

Long-run causality t- statistics ECTt1

17.33466*** 10.94810* – 11.28149* 10.07594 –

33.7613*** 14.13924 25.93621***

1.39787 3.03253 3.09179***

– 2.247669 6.609483

20.63173*** 3.860402*** – 15.30151** 16.97540*** –

27.94143*** 17.87810 24.32440**

1.75764 3.10985 2.56469*

– 5.536050 6.424228

18.87147*** 4.212768 – 12.18017* 7.267206 –

26.05802*** 18.14627 14.23014

1.99116*** 2.94317*** 2.64086***

Note: The figures in the final column are the t-statistics testing the null hypothesis that the lagged ECT is statistically insignificant for each equation. ***, **, and * indicate that the null hypothesis (‘‘no Granger causality’’) can be rejected at the 1%, 5% and 10% significance levels, respectively. a All variables except for the lagged error correction terms ECTt1 are in first difference denoted by D. ECT is received from the first cointegrating vector (i.e. the highest eigenvalue) of the Johansen cointegration test, which is normalized on the hedge fund index. The reported statistic is the block exogeneity Wald-type causality test from the estimated VECM. Block exogeneity refers to the exclusion of all the endogenous variables from the VECM other than the lags of the dependent variable.

76

Wafa Kammoun Masmoudi

For Canada, the movements in the stock market influence the hedge funds in the short term. The fluctuations in the bonds market prices have effects on stocks, which imply unidirectional causality from bonds to stocks in the short term. Moreover, on the same horizon, we note bidirectional causality between hedge funds and bond index. Block exogeneity Wald test shows that groups of variables hedge funds-stocks and stocks-bonds are strongly endogenous. Besides, over the long term (see the last column of Table 9) hedge funds and stocks do Granger-cause bond index (ECT is highly significant in the equation for bond). For France, price series are strongly endogenous. In the short term, the variations in the prices of stocks and bond indices influence hedge funds. We also note bidirectional causality in the short term between stocks and bond indices. Over the long term, France presents the same relation as Canada at the 1% significance level. For France and Canada, stock markets are highly developed in terms of liquidity and market capitalization. This may make it difficult to detect any long-term causality with hedge funds. Through the results found for Germany, the stock index movements have an impact on global macro over the short term. There also exists a unidirectional causality from bonds to stocks at 10% significance level. Besides, when the dependent variable is hedge funds, the VECM indicates econometric endogenity of the variables because the t-test(s) and the w2-tests are highly significant. For Germany, the results seem quite different. Over the long term, all three index series act to clear price disequilibrium. Thus, we note the presence of long-term bidirectional Granger causality among all asset categories. It is evident that for risk-averse investors, the diversification benefits from adding global macro hedge funds to bond/equity portfolio are considerably less for Germany than for Canada and France. All results on the short term confirm our analysis of Granger causality tests according to VAR models. In conclusion, Figure 3 qualitatively summarizes the results of the Granger causality tests.

4. Concluding Remarks This chapter aims to analyse the dynamic linkages between hedge funds and traditional financial assets for the major developed and emerging countries and regions. We question whether diversification benefits arise from adding hedge funds to emerging and developed portfolios consisting of stocks and bonds by examining short- and long-term relationships among the asset categories and using daily data for the March 2003 to March 2010 time period.

Global Macro Hedge Funds and Traditional Financial Assets Canada

France

HF Global Macro

HF Global Macro

Stocks

Bonds

Stocks

77

Bonds

Germany HF Global Macro

Stocks

Bonds Short-run causality

Figure 3:

Long-run causality

Short- and Long-Term Dynamics among Asset Categories.

The results of causality tests show that global macro hedge funds return is explained as well by its history as by stocks return for the United States, Canada, the United Kingdom, France, Germany, Italy, Latin America, Eastern Europe and emerging market regions. For Asia, the global macro hedge funds cause the stocks, whereas a bidirectional causality exists between bond indices and hedge funds for Canada and Japan. Besides, in Latin America, bonds cause the global macro. Moreover, there is bidirectional causality between stocks and bond indices for France, Italy, Japan and all the emerging market regions. For Germany, MSCI index causes bond index, whereas this causality is reversed for the United Kingdom. The dynamic behaviour results of VAR models using the approach based on impulse response functions are in conformity with those of Granger causality. In general, the shock has the immediate effect to lead to a rise of the index then the price growth slowed and experienced a slight decline from the peak they had reached. In the second period, the market seems to correct this decline and pushes back the price increase, followed by a further decline. This could possibly be explained by the surajustement phenomenon. For the United States, United Kingdom, Italy, Japan, Asia, Eastern Europe, Latin America and the emerging markets, the analysis of multivariate cointegration shows no cointegration vector according to the Johansen test; thus, global macro hedge funds appear as a separate asset class. However, for Canada, France and Germany, a cointegration relationship is found. Our results suggest that the hedge funds tend to have a positive (negative) long-term relationship with the stocks (bond) index of these countries.

78

Wafa Kammoun Masmoudi

The existence of one or more common stochastic trend(s) does not imply that all assets are a driving force in the common trend. By testing whether an asset enters in and adjusts to the relevant cointegrating vector, we show that for France and Canada, the largest risk reduction is achieved when hedge funds are added to a portfolio consisting of stocks and bonds. In contrast, in Germany, the assets influence each other strongly in the long term. For better understanding the price discovery process, i.e. the short- and long-term relationships among the asset categories, a VECM is used. The VECM shows short-term causal effects and long-term relationships implied by the level of disequilibrium in the cointegration relationship expressed by the lagged ECT. The VECM also shows exogeneity between asset categories in the short term. For France, in the short term the fluctuations in the bonds market prices influence the global macro, while we note bidirectional causality between stocks and bond indices. Concerning Canada, the fluctuations on the bonds market prices have effects on the stocks and there is bidirectional causality between bonds and global macro. On the same horizon, for France and Canada, we note unidirectional causality from stocks to hedge funds. We also note that price series are strongly endogenous. In the long term, for Canada and France hedge funds and stocks do Granger-cause bond index. For Germany, there is unidirectional causality from bonds to stocks and from stocks to global macro over the short term, as well as long-term bidirectional Granger causality among all three assets. This result suggests that global macro prices can be forecasted by using the price information of other assets. Coming back to our initial question as to whether the benefits of diversification result or not from the addition of global macro hedge funds to the portfolio of the developed and emerging markets containing equity and bond indices, we can conclude that global macro can provide diversification benefits in the short and long term. However, the empirical results have to be more differentiated. For countries having no cointegrating relationship among the three assets, portfolio managers of traditional funds can profit by allocating in this category of hedge funds. Specifically, diversification benefits from adding global macro hedge funds to bond/equity portfolio are considerably less for Germany, Canada and France. Nevertheless, the riskaverse investors can reduce their long-term volatility by investing according to the cointegrating vector while active managers can benefit from the knowledge of short-term asset price movements.

Acknowledgments I am extremely grateful to the Guest Editor, Fredj Jawadi, the professor Franck Martin (University of Rennes 1) and an anonymous referee for their

Global Macro Hedge Funds and Traditional Financial Assets

79

constructive criticism and their helpful suggestions on the manuscript. I also benefited from the comments of participants of the First International Symposium in Computational Economics and Finance, and the 27th Symposium on Money, Banking and Finance in Bordeaux. Any remaining errors are, of course, my own.

References Agarwal, V. and Naik, N.Y. (2004). Risks and portfolio decisions involving hedge funds. The Review of Financial Studies 17 (1), 63–98. Asness, C.S., Krail, R. and Liew, J.M. (2001). Do hedge funds hedge? Journal of Portfolio Management 28, 6–19. Banerjee, A., Dolado, J.J., Galbraith, J.W. and Hendry, D.F. (1993). Co-integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. Oxford: Oxford University Press. Capocci, D.P. (2006). The neutrality of market neutral funds. Global Finance Journal 17 (2), 309–333. Chaudhuri, K. (1997). Cointegration, error correction and Granger causality: An application with Latin American stock markets. Applied Economics Letters 4 (8), 469–471. Chen, G.M., Firth, M. and Rui, O.M. (2002). Stock market linkages: Evidence from Latin America. Journal of Banking and Finance 26 (4), 385–390. Chen, Y. (2005). Timing ability in the focus market of hedge funds. Boston College Global Finance Journal 17 (2), 309–333.. Embrechts, P., McNeil, A. and Straumann, A. (1999). Correlation: Pitfalls and alternatives. Risk Magazine 12 (5), 69–71. Embrechts, P., McNeil, A. and Straumann, A. (2002). Correlation and dependence in risk management: Properties and pitfalls. In Dempster, M. (Ed.), Risk Management: Value at Risk and Beyond. Cambridge: Cambridge University Press, pp. 176–223. Engle, R.F. and Granger, C.W. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica 55 (2), 251–276. Favre, L. and Galeano, J. (2002). An analysis of hedge fund performance using Loess fit regression. Journal of Alternative Investments 14, 8–24. Fu¨ss, R. and Herrmann, F. (2005). Long-term interdependence between hedge fund strategy and stock market indices. Managerial Finance 31 (12), 29–45. Fu¨ss, R. and Kaiser, D.G. (2007). The tactical and strategic value of hedge fund strategies: A cointegration approach. Financial Markets and Portfolio Management 21 (4), 425–444. Fu¨ss, R. and Kaiser, D.G. (2008). Dynamic linkages between hedge funds and traditional financial assets: Evidence from emerging markets. European Business School (EBS), Oestrich-Winkel. Working Paper. Fu¨ss, R., Kaiser, D.G., Rehkugler, H. and Butina, I. (2006). Long-term comovements between hedge funds and financial asset markets. In Gregoriou, G.N. and Kaiser, D.G. (Eds), Hedge Funds and Managed Futures – A Handbook for Institutional Investors. London: Risk Books, pp. 397–428.

80

Wafa Kammoun Masmoudi

Fung, W. and Hsieh, D.A. (1997). Empirical characteristics of dynamic trading strategies: The case of hedge funds. Review of Financial Studies 10, 275–302. Fung, W. and Hsieh, D.A. (1999). A primer on hedge funds. Journal of Empirical Finance 6 (3), 309–331. Fung, W. and Hsieh, D.A. (2000). Measuring the market impact of hedge funds. Journal of Empirical Finance 7 (1), 1–36. Geltner, D. (1991). Smoothing in appraisal-based returns. Journal of Real Estate Finance and Economics 4 (3), 327–345. Geltner, D. (1993). Estimating market values from appraised values without assuming an efficient market. Journal of Real Estate Research 8 (3), 325–346. Geman, H. and Kharroubi, C. (2003). Hedge funds revisited: Distributional characteristics, dependence structure and diversification. Journal of Risk 5 (4), 55–73. Getmansky, M., Lo, A.W. and Makarov, I. (2004). An econometric model of serial correlation and illiquidity in hedge fund returns. Journal of Financial Economics 74 (3), 529–609. Gonzalo, J. (1994). Five alternative methods of estimating long-run equilibrium relationships. Journal of Econometrics 60, 203–233. Granger, C.W. (1969). Investigating causal relations by econometric methods and cross-spectral methods. Econometrica 34, 424–438. Gregoriou, G.N. and Rouah, F. (2001). Do stock market indices move the ten largest hedge funds? A cointegration approach. Journal of Alternative Investments 3 (3), 61–66. Hung, W.S. and Cheung, Y.L. (1995). Interdependence of Asian emerging equity markets. Journal of Banking and Finance 22 (2), 281–288. Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12 (2–3), 231–254. Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59 (6), 1551–1580. Kasa, K. (1992). Common stochastic trends in international stock markets. Journal of Monetary Economics 29 (1), 95–124. Kat, H.M. (2003). The dangers of mechanical investment decision-making: The case of hedge funds. City University London. Working Paper Lhabitant, F.S. (2002). Hedge Funds: Myths and Limits. London: Wiley. Liang, B. (1999). On the performance of hedge funds. Financial Analysts Journal 55, 72–85. Liang, B. (2001). Hedge fund performance: 1990–1999. Financial Analysts Journal 57 (1), 11–18. Lo, A.W. (2001). Risk management for hedge funds: Introduction and overview. Financial Analysts Journal 57, 16–33. Lu¨tkepohl, H. (1991). Introduction in Multiple Time Series Analysis. Berlin: Springer. Maddala, G.S. and Kim, I.M. (1998). Unit Roots, Cointegration and Structural Change. Cambridge: Cambridge University Press.

Global Macro Hedge Funds and Traditional Financial Assets

81

Malkiel, B.G. and Atanu, S. (2005). Hedge funds: Risk and return. Financial Analysts Journal 61, 80–88. Mitchell, M. and Pulvino, T. (2001). Characteristics of risk in risk arbitrage. Journal of Finance 56, 2135–2175. Pan, M.S., Liu, A.Y. and Roth, H.J. (1999). Common stochastic trends and volatility in Asian Pacific equity markets. Global Finance Journal 10 (2), 162–172.

Chapter 4

Copula Theory Applied to Hedge Funds Dependence Structure Determination Rania Hentatia and Jean-Luc Prigentb a

University of Paris 1 Panthe´on-Sorbonne, CES, Paris, France University of Cergy-Pontoise, ThEMA, Cergy-Pontoise, France

b

Abstract Purpose – In this chapter, copula theory is used to model dependence structure between hedge fund returns series. Methodology/approach – Goodness-of-fit tests, based on the Kendall’s functions, are applied as selection criteria of the ‘‘best’’ copula. After estimating the parametric copula that best fits the used data, we apply previous results to construct the cumulative distribution functions of the equally weighted portfolios. Findings – The empirical validation shows that copula clearly allows better estimation of portfolio returns including hedge funds. The three studied portfolios reject the assumption of multivariate normality of returns. The chosen structure is often of Student type when only indices are considered. In the case of portfolios composed by only hedge funds, the dependence structure is of Franck type. Originality/value of the chapter – Introducing goodness-of-fit bootstrap method to validate the choice of the best structure of dependence is relevant for hedge fund portfolios. Copulas would be introduced to provide better estimations of performance measures. Keywords: hedge funds, performance measure, copula, goodness-of-fit JEL Classification: C6, G11, G24, L10

International Symposia in Economic Theory and Econometrics, Vol. 20 F. Jawadi and W.A. Barnett (Editors) Copyright r 2010 by Emerald Group Publishing Limited. All rights reserved ISSN: 1571-0386/DOI: 10.1108/S1571-0386(2010)0000020009

84

Rania Hentati and Jean-Luc Prigent

1. Introduction Since the early 1990s, hedge fund industry has emerged as the most innovative alternative to traditional asset management. Its development has been quite impressive with average assets growth rate up to 25% p.a. (until the recent financial crisis). This keen interest for hedge fund industry has been illustrated in early 2000. Indeed, after years of steady growth, all equity world markets have fallen almost continuously between April 2000 and December 2002. During this period, hedge funds have provided investors with strong returns, decorrelated to traditional investment. Indeed, the growth of the hedge fund industry is mainly based on the search for alpha performance. Investors seek to enhance their portfolio profitability to benefit from the advantages provided by these alternative assets in portfolio diversification. This explains the changing profile of clients to whom this industry is originally addressed. Recently, new investors such as pension funds and insurance companies have emerged. The interest of pension funds for this type of management has been demonstrated for the first time when the US pension fund CALPERS in 1999 chose to allocate 6% of its assets to alternative investments. From the investor perspective, hedge funds provide a source of diversification within a portfolio of traditional assets. The access to hedge funds has not been reserved to high net worth individuals. Big asset managers offer a broad access to hedge fund through onshore vehicles with low minimum investment required. This pushes the regulator to impose more rules for the investment in hedge funds since then risk and performance evaluation has become crucial. The year 2008 was a very tough year for almost all hedge fund strategies (except Managed Futures). Hedge fund industry lived free fall returns period accelerated by massive outflows:  Average loss varied between 25% and 20%, except for strategies CTA/Futures, which have provided protection against liquidity crises;  High percentage of funds blowout;  High number of frauds (biggest ponzie scheme ever seen by Madoff ). During this market shakeout, hedge fund assets under management have registered for the first time a net withdrawal of US$150 billion. Therefore, the forecast growth of hedge funds has been revised sharply downward: assets under management, estimated at US$7 trillions for 2013 (before the crisis in 2008), lowered to US$3 trillions. Thus, the financial crisis has reinforced the need to better explore both marginal distributions and dependence structure of hedge funds (dependence between main hedge fund indices and also with other financial indices such as equity indices).

Hedge Funds Dependence Structure Determination

85

For marginal distributions, most of empirical studies show that the assumption of normality in return distribution is not justified, in particular when dealing with hedge funds that have significant positive or negative skewness and high kurtosis due to the extensive use of derivatives and dynamic strategies (see Fung and Hsieh, 1997; Ackerman et al., 1999; Brown et al., 1999; Caglayan and Edwards, 2001; Bacmann and Scholz, 2003; Agarwal and Naik, 2004). Modeling the multivariate distribution of hedge fund portfolio is a quite complex problem. The multidimensional Gaussian distribution has been intensively used to represent the probability law of financial asset returns. But it implies on one hand that all components are Gaussian distributed and, on the other hand, it means that their dependence is entirely determined by their linear correlations. For hedge fund returns, such assumptions are clearly not valid. It is well known that the linear correlation coefficient can characterize the dependency (or not) between two random variables but only for multivariate normal distribution. Embrechts et al. (2002) have shown that the concept of correlation entails several pitfalls, in particular when the assumption of normality in return distribution is not verified. In fact, correlation is not sufficient to describe precisely the dependence structure, and could be a source of confusion. For example, independence between two random variables implies that they are uncorrelated (when the linear correlation is equal to zero), but the inverse is not always true. In addition to that, it is not invariant under nonlinear strictly increasing transformations. In this framework, the copula theory was introduced to take account of marginal distributions and of their dependence structure. The concept of copula was introduced by Sklar (1959) and studied later by many authors such as Deheuvels (1979) and Genest and MacKay (1986). Nelsen (1999) and Joe (1997) provide details and complete references about copulas. We can define copula as a statistical tool that allows the aggregation of marginal distributions: for any given marginal cumulative distribution functions (cdf), the copula associates the corresponding cdf of the random vector. It can be seen as a function that links univariate marginals to their full multivariate distribution. Thus, an n-dimensional copula is basically a multivariate cdf with uniform distributed margins in [0,1]. This proves the interest of introducing copula theory to model statistical dependency in finance, as illustrated by Roncalli (2004) and Embrechts et al. (2002). Additionally, copula theory is a very powerful tool to generate multivariate distributions. Recently, many applications of the copula approach have been used in finance and risk management. For example, it was introduced in the pricing of structured credit products, given the ability of these functions to incorporate dependencies of the underlying credits. In this sense, copula functions allow better modelization of dependency and correlation structures when pricing credit products like credit default swaps, credits

86

Rania Hentati and Jean-Luc Prigent

swap indices, and collateralized debt obligation. Other applications of copulas concern the dependence between financial markets linked to the notions of portfolio diversification and financial integration. For instance, Malevergne and Sornette (2003) test dependence structure for pairs of currencies and pairs of major stocks. They find that the Gaussian copula hypothesis can be accepted, while this hypothesis can be rejected for the dependence between pairs of commodities. Lauprete et al. (2002) prove that introducing, for example, the Student-t distribution to model multivariate tail-dependence allows showing the impact of the heavy tails of marginal distributions on risk-minimizing portfolios. However, one of the main problems posed by this approach is the choice of the most adequate copula to represent the multivariate distribution, as illustrated by Genest and Rivest (1993). In fact, there are different families of copula functions that can be classified. The most common are the Gaussian and t-copulas that belong to the elliptical family and the Archimedean ones. Several methods are proposed to choose the best copula. But only few researches are focusing on the issue of copula specification testing. Genest and Rivest (1993) propose a graphical method. This method is based on Kendall function. More recently, adequacy tests for copula have been proposed. There exist several copula goodness-of-fit tests. Some of them involve nonparametric specification of the marginal distributions. This leads to arbitrary parameters, for example, the choice of a kernel and a bandwidth (see Scaillet, 2007). In Genest et al. (2007), goodness-of-fit tests are based on empirical copula and on Kendall’s probability integral transformation of the data. This method is based on two statistics: the first one is defined from Cramer–von Mises distance; the second one involves the Kolmogorov–Smirnov distance. This approach consists in estimating these two statistics and comparing them to the corresponding ones associated to given p-values. Genest et al. (2007) have provided a method to determine the asymptotic threshold of these two tests, based on asymmetric bootstrap of goodness-of-fit statistics. Robustness of such method has been proved in Genest and Re´millard (2008). In this chapter, we use copula theory to get more accurate estimations of hedge fund returns, then of funds of hedge funds. As a byproduct, we provide a survey of recent results about copula theory, in particular about goodness-of-fit tests. To test empirically the efficiency of such methodology, we examine the dependence structure of three portfolios: the first consists of three hedge fund indices, the second treats CTAs (Commodities Trading Advisors) and the last one is devoted to a portfolio composed of indices representing the following asset classes: hedge funds, equities, and bonds. The time period of the analysis lies between December 1993 and October 2008. For each of these portfolios, first we provide the estimation of its main statistical properties. Then, we estimate parametric copula that best fits the

Hedge Funds Dependence Structure Determination

87

data. Finally, we apply previous results to construct the cdf of the equally weighted portfolios. This chapter is organized as follows. Section 2 provides an overview of main definitions and properties of copula. Results about dependence structure determination are detailed in Section 3. Section 4 contains the main empirical results about the choice of the best copula. The first step consists in correcting smoothed data as in Geltner (1993) and Gallais Hammono and Nguyen-Thithanh (2008), based on autocorrelation analysis. Then, we fit marginal distributions of each of the index return. Second step concerns the determination of the dependence structure via copula. We implement goodness-of-fit tests to ensure the adequacy of selected copula.

2. Notions about Copula Theory 2.1. Definition and General Properties Definition. An n-dimensional copula is a function C: [0,1]d-[0,1] that has the following properties:  C(u) is increasing in each component uk with kA{1,2y, d}.  For every vector uA[0,1]d, C(u) ¼ 0 if at least one coordinate of the vector u is 0, and C(u) ¼ uk if all the coordinates of u are equal to 1 except the kth one.  For every a, bA[0,1]d, with arb given a hypercube B ¼ ½0; 1d ¼ ½a1 ; b1   ½a2 ; b2       ½ad ; bd  which lies in the domain of C, its volume VC(B)Z0. Sklar’s theorem (1959) proves the one-to-one correspondence between the copula function and the multivariate distribution function. Theorem. (Sklar, 1959) Let F be a multidimensional cdf with marginal cdf F1,y, Fd, then F can be represented by F(x1,y,xd) ¼ C[F1(x1),y, Fd(xd)], where C is a function, called the copula. This function is unique if the marginal distributions are continuous. Conversely, if C is a copula and if F1(x1),y, Fd(xd) are continuous, then F(x1,y,xd) ¼ C[F1(x1),y, Fd(xd)] is a cdf with marginal cdf F1(x1),y, Fd(xd). Copula verifies property of invariance. Thus, if x1 and x2 are two continuous random variables and F1 and F2 are the margins of their copula C and if h1 and h2 are two strictly increasing functions, then we have CðF 1 ðh1 ðx1 ÞÞ; F 2 ðh2 ðx2 ÞÞÞ ¼ C½F 1 ðx1 Þ; . . . ; F d ðxd Þ

(1)

88

Rania Hentati and Jean-Luc Prigent

2.2. Elliptical and Archimedean Copulas 2.2.1. Elliptical Copulas Elliptical copulas are defined simply as the copulas of elliptical distributions. The multivariate Gaussian and Student distributions are the most used families of copula functions. They are very practical in finance modeling, but do not have closed form expressions and are restricted to have radial symmetry. The Normal and Student t copulas are discussed below. Multivariate Gaussian copula: The multivariate Gaussian copula is an elliptical copula given by Cðu1 ; u2 ; . . . ; ud ; RÞ ¼ UR ðF1 ðu1 Þ; F1 ðu2 Þ; . . . ; F1 ðud ÞÞ

(2)

where R denotes a symmetric, positive definite matrix with diag(R) ¼ 1; UR is the standardized multivariate normal distribution with correlation matrix R; and F1(u) the inverse of the normal cdf. Multivariate Student copula: The multivariate Student copula is given by 1 1 Cðu1 ; u2 ; . . . ; ud ; R; vÞ ¼ TR;v ðt1 v ðu1 Þ; tv ðu2 Þ; . . . ; tv ðud ÞÞ

(3)

where R denotes a symmetric, positive definite matrix with diag(R) ¼ 1; TR,v(u) is the standardized multivariate Student distribution correlation with matrix R and v denotes the degree of freedom; t1 v ðuÞ is the inverse of the student cdf. 2.2.2. Archimedean Copulas Definition. Let f denote the strict generator and f1 (the pseudo inverse of f) is d-completely monotonic function (0,1]-(0,N]. Thus, for all l 2 f1; . . . ; dg; ð1Þl ð@l ðf1 ÞÞ=ð@tl ÞðtÞ40: The Archimedean copula C is defined by Cðu1 ; u2 ; . . . ; ud Þ ¼ f1 ðfðu1 Þ þ    þ fðu1 ÞÞ. Theorem. Let C denote an Archimedean copula with generator function f. Then, we have the following properties:  C is symmetric: C(u, v) ¼ C(v, u) for all u, vAI ¼ [0,1].  C is associative: C((u, v), w) ¼ C(u, C(v, w)), for all u, vAI.  If CW0 is constant, Cf is also a generator function. Archimedean copulas can better modelize financial and credit market data than usual Gaussian copula because they are tractable and provide a large variety of dependence structures. The main popular Archimedean copulas are: the Frank, Gumbel, and Clayton copulas. They can exhibit upper and/or lower tail dependence. The Frank copula models positive dependences as well as negative ones. The Gumbel copula can only model

Hedge Funds Dependence Structure Determination

89

positive dependences. It allows the risk modeling of upper tails of the distribution. As well as Frank structure, Gumble copula is also asymmetric, with more weight in the right tail (see Embrechts et al., 2002). The Clayton copula cannot take account of positive dependence, but it is able to model dependence of random events of weak intensity.1

2.3. Empirical Copula In this section, we present the notion of the empirical copula as introduced by Deheuvels (1979). Definition. Empirical copula (Nelsen, 1999) Let (xk, yk)k ¼ 1,y, d be a sample of a bivariate distribution. The empirical copula Ce is given by  Ce

 i j number of pairs ðx; yÞ in the sample with xoxðiÞ and yoyðiÞ ; ¼ n n n (4)

Let Xk ¼ (Xk,1,y, Xk,d) be a sequence of i.i.d. random vectors with common cdf F and marginal cdf F1,y, Fd. In order to guarantee the uniqueness of the copula C(F), the cdf F is assumed to be continuous. Let B denote the Dirac measure at B 2 1 is estimated recursively starting from the one-step-ahead formula. Using five years of vintage data, from the first quarter of 2003 to the third quarter of 2007, we provide RMSEs for the euro area flash estimates of GDP growth Y^ t in genuine real-time conditions. We have computed the RMSEs for the quarterly GDP flash estimates, obtained with the forecasting methods used to complete adequately in real time the monthly indicators, that is, VAR modeling and k-NN methods (d ¼ 1 and d>1). More precisely, we provide the RMSEs of the combined forecasts based on the arithmetic mean of the eight Diron equations. Thus, for a given forecast horizon h, j we compute Y^ t ðhÞ, which is the predictor stemming from the eight equations j ¼ 1; . . . ; 8, in which we have plugged the forecasts of the monthly economic indicators, and we compute the final estimate GDP at horizon h: P j Y^ t ðhÞ ¼ ð1=8Þ 8j¼1 Y^ t ðhÞ. The RMSE criterion used for the final GDP is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u T u1 X (9) ðY^ t ðhÞ  Y t Þ2 , RMSEðhÞ ¼ t T t¼1 where T is the number of quarters between the first quarter of 2003 and the second quarter of 2007 (in our exercise, T ¼ 18) and Yt the euro area flash estimate for quarter t. The RMSE errors for the final GDP are provided in Table 2 and comments follow. For both methods, VAR modeling and k-NN method, the RMSE becomes lower when the forecast horizon reduces from h ¼ 6 to h ¼ 1, illustrating the accuracy of the nowcasting and forecasting, which increases as soon as the information set becomes more and more efficient, thanks to the released monthly data. This is the strength of GDP forecasting based on monthly economic indicators, instead of considering only GDP itself, since each month new true values of economic indicators are available.

172

Dominique Gue´gan and Patrick Rakotomarolahy

Table 2: RMSE on GDP Growth from the Three Methods VAR, k-NN with d ¼ 1, and k-NN with dW1 Horizon

VAR

k-NN(d ¼ 1)

k-NN(dW1)

6 5 4 3 2 1

0.225 0.224 0.214 0.192 0.181 0.173

0.198 0.203 0.202 0.186 0.176 0.174

0.214 0.192 0.196 0.177 0.177 0.171

Note: RMSE for the estimated mean quarterly GDP growth Yt computed from Equation (9), using VAR(p) modeling (column 2) and k-NN predictions (d ¼ 1 (column 3), and dW1 (column 4)) for the monthly economic indicators X it ; i ¼ 1; . . . ; 13, h is the monthly forecast horizon. Values in boldface correspond to the smallest error for a given forecast horizon.

We remark that few days before the publication of the flash estimate (around 13 days with h ¼ 1), the lowest RMSE is obtained with the multivariate k-NN method (RMSE ¼ 0.171). Looking at forecast errors by comparing column 2 with columns 3 and 4 of Table 2, we find that forecast errors are always lower with the method of NN rather than with VAR modeling (except at horizon h ¼ 1 where VAR modeling gives better forecast error than univariate k-NN). One source of such gain comes from the use of nearest neighbors method that is adapted even with small samples, but this is not the case when working with VAR modeling that requires large samples to be robust. Finally, if we focus on nearest neighbors method, we obtain smaller errors when working with multivariate setting d>1 than with univariate d ¼ 1. This result shows the gain of the method developed in a space of higher dimension. We expect that in terms of predictions, any method developed in higher dimension improves the forecast accuracy. This is confirmed when we compare, for the same method, the forecast errors obtained only in R with the error calculated from a treatment in Rd : in this latter case, the errors are always smaller (e.g., comparing columns 3 and 4 of Table 2). This idea has already been developed in other empirical works that consider multivariate methods with factor models (Kapetanios and Marcellino, 2006), and methods with multivariate nonparametric techniques (Gue´gan and Rakotomarolahy, 2010). To see the evolution of the trajectory of forecasting both parametric and nonparametric methods, we provide, in Figures 1 and 2, the graphs of the observed and estimated GDP growth from k-NN methods and VAR modeling for forecast horizons varying from one to six quarters. These graphs show that the trajectories of GDP forecasts from both methods are very close for horizons hr3 and are able to follow the ‘‘true’’

Alternative Methods for Forecasting GDP

173

Figure 1: Quarterly Observed (in Black) and Forecasted GDP Growth Rate Computed from k-NN with d ¼ 1 (in Green), k-NN with dW1 (in Blue), and VAR (in Red) Models between the first quarter of 2003 and the second quarter of 2007 for Different Forecast Horizons. Panel (a): Graph of GDP Growth Rate Forecast at Horizon h ¼ 1 Using VAR and k-NN Methods. Panel (b): Graph of GDP Growth Rate Forecast at Horizon h ¼ 2 Using VAR and k-NN methods. Panel (c): Graph of GDP Growth Rate Forecast at Horizon h ¼ 3 Using VAR and k-NN Methods.

174

Dominique Gue´gan and Patrick Rakotomarolahy

Figure 2: Quarterly Observed (in Black) and Forecasted GDP Growth Rate Computed from k-NN with d ¼ 1 (in Green), k-NN with dW1 (in Blue), and VAR (in Red) Models between the first quarter of 2003 and the second quarter of 2007 for Different Forecast Horizons. Panel (a): Graph of GDP Growth Rate Forecast at Horizon h ¼ 4 Using VAR and k-NN methods. Panel (b): Graph of GDP Growth Rate Forecast at Horizon h ¼ 5 Using VAR and k-NN Methods. Panel (c): Graph of GDP Growth Rate Forecast at Horizon h ¼ 6 Using VAR and k-NN methods.

Alternative Methods for Forecasting GDP

175

trajectory. In addition, they permit to detect also some declines of euro area GDP, for example, in the second quarter of 2003. On the other hand, for forecast horizons h>3, the VAR modeling provides forecasts for GDP that converge to the sample mean when the forecast horizon increases. Conversely, the k-NN modeling provides forecasts that follow the observed GDP trajectory.

4. Conclusion Knowing the importance of the nowcast and the forecast of macroeconomic variables (such as GDP or inflation) when analyzing the current state of the economics and setting policy for the future economic conditions, we suggest in this chapter alternative methods based on nonparametric multivariate k-nearest neighbors method to improve the accuracy of GDP forecasts. We focus on detecting the best predictor for economic indicators using an RMSE criterion, working in an embedded space of dimension d, and focusing on the relevant set of data that helps solve this specific criterion (Han et al., 1997; Hoover and Perez, 1999). Our application used a new theoretical result that extends, for the multivariate k-nearest neighbors estimation method, the L2-consistent result obtained with uniform weighting (Yakowitz, 1987). Some factors are questionable: the use of the aggregated monthly economic indicators to match quarterly GDP; a specific test to decide a good strategy between parametric and nonparametric modeling; the trade-off between stationarity and nonlinearity when we work with nonparametric techniques.

Acknowledgments The authors are extremely grateful to the Guest Editor Fredj Jawadi for helpful comments and discussions. We have also benefited from the comments of participants of the 3rd International Conference on Computation and Financial Econometrics (CFE) in October 2009 in Limassol, Cyprus; of the 1st International Symposium on Computational Economics and Finance (ISCEF) in February 2010 in Sousse, Tunisia; of the 9th German Open Conference on Probability and Statistics (GOCPS) in March 2010 in Leipzig, Germany; and of the 16th Annual Conference on Computing in Economics and Finance (CEF) in July 2010 in London, UK.

Appendix. Euro Area Monthly Indicators We provide in Table 1 the list of the monthly economic indicators used in this study for the computation of GDP using bridge equations.

176

Dominique Gue´gan and Patrick Rakotomarolahy

The Bridge Equations We specify the bridge equations we use; details can be found in Diron (2008). Let us define Yt as Y t ¼ ðlog GDPt  log GDPt1 Þ  100, where GDPt is GDP at time t. The final GDP Yt used in the chapter is the mean of the eight values computed below: (1) EQ1. Y 1t ¼ a10 þ a11 ðlog X 1t  log X 1t1 Þ þ a12 ðlog X 2t  log X 2t1 Þ þ 1 3 a3 X t1 þ t . (2) EQ2. Y 2t ¼ a20 þ a21 ðlog X 1t  log X 1t1 Þ þ a22 ðlog X 2t  log X 2t1 Þ þ a23 4 ðlog X t  log X 4t1 Þ þ a24 ðlog X 5t  log X 5t1 Þ þ t . (3) EQ3. Y 3t ¼ a30 þ a31 X 7t þ a32 X 7t1 þ t . (4) EQ4. Y 4t ¼ a40 þ a41 ðX 6t  X 6t1 Þ þ a42 X 3t þ t . (5) EQ5. Y 5t ¼ a50 þ a51 ðX 6t  X 6t1 Þ þ a52 X 9t þ a53 X 8t þ t . 10 11 11 6 (6) EQ6. Y 6t ¼ a60 þ a61 ðlog X 10 t2  log X t3 Þ þ a2 ðlog X t1  log X t2 Þ þ t . 7 12 12 12 7 7 7 (7) EQ7. Y t ¼ a0 þ a1 ðlog X t  log X t1 Þ þ a2 ðlog X t2  log X 12 t3 Þ þ 7 7 a3 Y t1 þ t . (8) EQ8. Y 8t ¼ a80 þ a81 X 13 t þ t . Proofs of Theorem 1 and Corollary 1 We start giving the proof of Theorem 1. We first establish a preliminary lemma. Lemma 1. Under the hypotheses of Theorem 1, either the estimate mn ðxÞ is asymptotically unbiased or E½mn ðxÞ ¼ mðxÞ þ Oðnb Þ,

(A.1)

with b ¼ ðð1  QÞpÞ=d. Proof of Lemma 1 We denote Bðx; r0 Þ ¼ fz 2 Rd ; jj x zjj r0 g, with the ball centered at x with radius r0>0. We characterize the radius r ensuring that k(n) observations fall in the ball Bðx; rÞ; since the function h(  ) is p-continuously differentiable, for a given i the probability qi of an observation xi to fall in Bðx; rÞ is qi ¼ Pðxi 2 Bðx; rÞÞ Z Z hðxi Þdxi ¼ hðxÞ: ¼ Bðx;rÞ

¼ hðxÞcrd þ oðrd Þ,

Z dxi þ Bðx;rÞ

ðhðxi Þ  hðxÞÞdxi Bðx;rÞ

ðA:2Þ

where c is the volume of the unit ball and d x ¼ dx1 dx2    dxd . Thus, qi  qj ¼ oðrd Þ for all i6¼j. We consider now the k-NN vectors xðkÞ and we denote q the probability that they are in the ball Bðx; rÞ, that is, q ¼ PðxðkÞ 2 Bðx; rÞÞ,

Alternative Methods for Forecasting GDP

177

then qi ¼ q þ oðrd Þ.

(A.3)

If N(r, n), the number of observations falling in the ball Bðx; rÞ, is provided for a given r>0, we characterize r such that k(n) observations fall in Bðx; rÞ. We proceed as follows. We denote Sni all nonordered combinations of the i tuple indices from (nd) indices, then E½Nðr; nÞ ¼

nd X

iPðNðr; nÞ ¼ iÞ ¼

i¼0



¼

nd X

i

i¼0

nd X

X

i

ðj 1 ;...;j i Þ2Sni

nd X

nd i

i

ji Y

ðj 1 ;...;j i Þ2Sni j¼j 1

qj

nY d

ð1  q‘ Þ

‘¼1

‘efj 1 ;...;j i g

qi ð1  qÞndi

i¼0

i¼0

X

! qi ð1  qÞndi

¼ qðn  dÞð1 þ q qÞnd ,

ðA:4Þ

 where q and q are, respectively, the smallest and largest probabilities, qi i ¼ 1; . . . ; n  d. Thus, we obtain a lower bound for E½Nðr; nÞ. If E½Nðr; nÞ ¼ kðnÞ, using Equations (A.2)–(A.4), we obtain  r

kðnÞ ðn  dÞ

1=d (A.5)

DðxÞ,

with DðxÞ ¼ ð1=hðxÞcÞ1=d . Now, using Equation (2), we get X E½wðx X ðiÞ ÞY i , E½mn ðxÞ ¼

(A.6)

i2NðxÞ

R R where Y i ¼ X ðiÞþ1 . We can remark that E½wðx X ðiÞ ÞY i  ¼ Rd R wðx xi Þ R i ; xi Þdxi dyi . Since f ðyi ; xi Þ ¼ f ðyi jxi Þhðxi Þ, we obtain E½wðx X ðiÞ ÞY i  ¼ Ryi f ðy d R R wðx xi Þyi f ðyi jxi Þhðxi Þdxi dyi . Thus, as soon as the weighting function w(  ) vanishes outside the ball Bðx; rÞ, Z Z E½wðx X ðiÞ ÞY i  ¼ wðx xi Þð yi f ðyi jxi Þdyi Þhðxi Þdxi Z

Bðx;rÞ

R

wðx xi Þmðxi Þhðxi Þdxi :

¼ Bðx;rÞ

ðA:7Þ

178

Dominique Gue´gan and Patrick Rakotomarolahy

To compute the bias we need to evaluate E½mn ðxÞ  mðxÞ. We begin to evaluate: X X Z wðx xi ÞmðxÞhðxi Þdxi ¼ mðxÞE½ wðx X ðiÞ Þ ¼ mðxÞ. i2NðxÞ

Bðx;rÞ

i2NðxÞ

(A.8) Then, E½mn ðxÞ  mðxÞ ¼

X Z i2NðxÞ

wðx xi Þðmðxi Þ  mðxÞÞhðxi Þdxi .

Equation (A.9) holds because (iv) in Theorem 1). Then, X Z jE½mn ðxÞ  mðxÞj

i2NðxÞ

(A.9)

Bðx;rÞ

R

i2NðxÞ Bðx;rÞ

wðx xi Þhðxi Þdxi ¼ 1 (assumption

wðx xi Þajjxi  x jjp hðxi Þdxi .

(A.10)

Bðx;rÞ

We get this last expression since the constant a is known and m(  ) is p-continuously differentiable. The inequality (A.10) implies that X wðx X ðiÞ Þ. (A.11) jE½mn ðxÞ  mðxÞj arp E½ i2NðxÞ

The relationship in Equation (A.11) holds because jjxi  x jjp orp , as soon as xi 2 Bðx; rÞ. Now, both cases can be considered:  When r is very small, then the bias is negligible and E½mn ðxÞ ¼ mðxÞ.  If the bias is not negligible, using Equations (A.5) and (A.11), we get 

kðnÞ jE½mn ðxÞ  mðxÞj a ðn  dÞ

p=d

Dðx Þp .

(A.12)

If we choose k(n) as in integer part of nQ, and knowing that k=ðn  dÞ ðk=nÞ, then jE½mn ðxÞ  mðxÞj ¼ Oðnb Þ with b ¼ ðð1  QÞpÞ=d. Proof of Theorem 1 (1) We begin to establish the relationship Equation (5). In the following, we denote Y i ¼ X ðiÞþ1 . We rewrite the left part of Equation (5) as follows: E½ðmn ðxÞ  mðxÞÞ2  ¼ Varðmn ðxÞÞ þ ðE½mn ðxÞ  mðxÞÞ2 .

(A.13)

We first compute the variance of mn ðxÞ, considering two cases: (a) First case: The weights wi, i ¼ 1; :::; k, are independent of (Xn). In that case the variance of mn(x) is equal to

Alternative Methods for Forecasting GDP

179

(A.14)

Varðmn ðxÞÞ ¼ A þ B,

kðnÞ 2 kðnÞ wi VarðY i Þ and B ¼ Si¼1 Sjai wi wj covðY i ; Y j Þ. Using assumpwhere A¼i¼1 kðnÞ Sjai jcovðY i ; Y j Þj. This last term is tion (ii) of Theorem 1, we get jBj Si¼1 negligible due to the result obtained by Yakowitz (1987) on the sum kðnÞ ðkðnÞwi Þ2 ðvðxÞ þ ðE½Y i   mðxÞÞ2 Þ. of covariances. Now, A ¼ ð1=kðnÞ2 ÞSi¼1 Using the fact that the weights are decreasing with respect to the chosen distance, wk    w1 , we get

1 X ðkðnÞwk Þ2 ðvðxÞ þ ðE½Y i   mðxÞÞ2 Þ A 2 kðnÞ i¼1 kðnÞ

1 X ðkðnÞw1 Þ2 ðvðxÞ þ ðE½Y i   mðxÞÞ2 Þ.

kðnÞ2 i¼1 kðnÞ

(A.15)

As soon as kðnÞ ! 1; the product kðnÞwi converges to 1 in case of uniform weights, and can be bounded for exponential weights for all i and for all n; thus, there exist two positive constants c0 and c1 such that Equation (A.15) becomes c21 X c20 X 2 ðvðxÞ þ ðE½Y   mðxÞÞ Þ

A

ðvðxÞ þ ðE½Y i   mðxÞÞ2 Þ. i kðnÞ2 i¼1 kðnÞ2 i¼1 kðnÞ

kðnÞ

(A.16) where vðxÞ ¼ VarðX nþ1 jX n ¼ xÞ. Using assumption (iv) of Theorem 1, we remark that E½Y i  ¼ E½mn ðxÞ. If kðnÞ ¼ ½nQ  where [  ] corresponds to the integer part of a real number, then A ¼ OðnQ Þ thanks to Lemma 1 when n ! 1, and it follows that the relationship (Equation A.14) becomes Varðmn ðxÞÞ ¼ OðnQ Þ,

(A.17)

and ðE½mn ðxÞ  mðxÞÞ2 ¼ Oðn2b Þ.

(A.18)

Substituting Equations (A.17) and (A.18) in Equation (A.13), we get 2b ¼ Q or Q ¼ 2p=ð2p þ dÞ and the proof is complete. (b) Second case: the weights wi, i ¼ 1,y,k, depend on (Xn). We use kðnÞ Varðwðx  X ðiÞ Þ again the relationship (Equation A.14) with A ¼ Si¼1 PkðnÞ Y i Þ and B¼ i¼1 Sjai covðwðx  X ðiÞ ÞY i ; wðx  X ðjÞ ÞY j Þ. Remarking that ðwðx  X ðjÞ ÞY j Þ are f-mixing since (Xj) and (Yj) are f-mixing (Pagan and Ullah, 1999), then B is negligible from the result obtained by Yakowitz (1987). We also remark that

180

Dominique Gue´gan and Patrick Rakotomarolahy kðnÞ A ¼ Si¼1 ðE½ðwðx  X ðiÞ ÞY i Þ2   ðE½wðx  X ðiÞ ÞY i Þ2 Þ, then Z kðnÞ Z X wðx xi Þ2 y2i f ðyi ; xi Þdxi dyi A¼ i¼1

Rd

Z

R

wðx xi Þyi f ðyi ; xi Þdxi dyi

 Rd

2 #

Z

(A.19) .

R

When k increases, the weights wi decrease, and kðnÞwi g where g is a real constant, then  Z Z Z kðnÞ  Z g2 X 2 2 y f ðy ; x Þdx dy  ð y f ðy ; x Þdx dy Þ A¼ i i i i i i i i i i kðnÞ2 i¼1 Rd R Rd R g2 X ðE½Y 2i   E½Y i 2 Þ. kðnÞ2 i¼1 kðnÞ

¼

ðA:20Þ

Under stationary conditions for (Xn) and recalling that Y i ¼ X ðiÞþ1 , thus, Equation (A.20) becomes A ¼ ðg2 =kðnÞÞðE½X 21   E½X 1 2 Þ; 2 A ¼ ðg =kðnÞÞVarðX 1 Þ. Finally, expression (A.14) becomes Varðmn ðxÞÞ ¼

g2 VarðX 1 Þ. kðnÞ

(A.21)

Moreover, when we take kðnÞ ¼ nQ , Equation (A.21) becomes Varðmn ðxÞÞ ¼ OðnQ Þ.

(A.22)

Using Equations (A.22) and (A.18) in Equation (A.13) gives 2b ¼ Q, and Q ¼ ð2pÞ=ð2p þ dÞ, and the proof is complete. (2) We prove now the asymptotic normality of mn ðxÞ. We assume that the variance sn ¼ var½mn ðxÞ exists and is not null, thus mn ðxÞ  Emn ðxÞ X wi Y i  Ewi Y i ¼ . sn sn i¼1 kðnÞ

(A.23)

To establish the asymptotic normality of mn ðxÞ, we distinguish three cases corresponding to the choice of the weighting functions. (i) The weights are uniform, wi ¼ 1=ðkðnÞÞ, then Equation (A.23) becomes mn ðxÞ  Emn ðxÞ X 1 Zi , ¼ sn kðnÞ i¼1 kðnÞ

(A.24)

where Z i ¼ ðY i  EY i Þ=sn . The asymptotic normality of Equation (A.24) is obtained using Theorem 2.2 given in Peligrad and Utev (1997). To compute

Alternative Methods for Forecasting GDP

181

the variance,Pwe follow the work by Yakowitz (1987): varðmn ðxÞÞ ¼ kðnÞ Y i Þ ¼ 1=ðkðnÞÞ½varðYj X ¼ xÞ þ Oðnð2ð1QÞpÞ=d Þ, then 1=ðkðnÞ2 Þvarð i¼1 Equation (A.24) becomes ffi X wi Y i  Ewi Y i mn ðxÞ  Emn ðxÞ pffiffiffiffiffi , ¼ nQ sn s i¼1 kðnÞ

(A.25)

when kðnÞ ¼ ½nQ  and s2 ¼ varðYj X ¼ xÞ, and the proof is complete. (ii) The weights wi are real numbers and do not depend on (Xn)n, then mn ðxÞ  Emn ðxÞ X ¼ wi Z i , sn i¼1 kðnÞ

(A.26)

where Z i ¼ ðY i  EY i Þ=sn . Now, applying again Theorem 2.2 given in Peligrad the asymptotic normality remarking that Pget PkðnÞ and Utev (1997), we kðnÞ wi Z i  ¼ 0 and Var½ i¼1 wi Z i  ¼ 1. To compute s2n ¼ Var½mn ðxÞ, E½ i¼1 we use the stationary condition of the time series (Xn)n, thus Var½mn ðxÞ ¼

kðnÞ X

w2i Var½Y i  ¼

i¼1

kðnÞ X

w2i ½Var½Y nþ1 jX n ¼ x þ B2 ,

i¼1

wherePB is given in Lemma 1. Remarking that ð1=kðnÞ2 Þ kðnÞ 2 wi o1 and then i¼1 Var½mn ðxÞ ¼ ½Var½Y i jX i ¼ x þ B2 

kðnÞ X

w2i .

PkðnÞ

(A.27)

i¼1 ðkðnÞwi Þ

2

o1,

(A.28)

i¼1

PkðnÞ

g2 =kðnÞ, and kðnÞ ¼ ½nQ , we get the result. P (iii) Finally, we assume that wi ¼ ðwðx X ðiÞ ÞÞ=ð K i¼1 wðx X ðiÞ ÞÞ; where w(  ) is a given function. In that latter case, the weights depend on the process (Xn)n. In the following, we denote by N(i) the order of the ith neighbor. We rewrite the neighbor indices in an increasing order such that Mð1Þ ¼ minfNðiÞ; 1 i Kg and MðkÞ ¼ minfNðiÞ efMðjÞ; 8jokg; 1 i Kg for 2 k K, and K ¼ k(n) is the number of neighbors. We introduce a real triangular sequence faKi ; 1 i K and aKi a08ig such that As soon as

Sup K

K X i¼1

a2Ki o1

2 i¼1 wi

and

max jaKi j ! 0.

1 i K

n!1

(A.29)

Now using the sequences MðjÞ; j ¼ 1; . . . ; K and ðaKi Þ; 1 i K, we can rewrite expression (A.23) as K mn ðxÞ  Emn ðxÞ X ¼ aKi Si , sn i¼1

(A.30)

182

Dominique Gue´gan and Patrick Rakotomarolahy

with Si ¼ ðwMðiÞ X MðiÞþ1  EwMðiÞ X MðiÞþ1 Þ=aKi sn . The sequence ðS 2i Þ is uniformly integrable and Si is function only of ðX j ; j MðiÞ þ 1Þ; thus, if we denote Fi , Gi , Fji , and Gji , the sigma algebras generated are fX r gr i , fSr gr i , fX r gjr¼i , and fSr gjr¼i , respectively, then Si 2 FMðiÞþ1 and Gi  FMðiÞþ1 . For a 1 given integer ‘, we have also G1 nþ‘ FnþMð‘Þþ1 ; since Mð1ÞoMð1Þþ 1 Mð2Þo    Mðn þ ‘ÞoMðn þ ‘Þ þ 1 Mðn þ ‘ þ 1Þ. Then sup

Sup

‘

A2G‘1 ;B2G1 nþ‘ ;PðAÞa0

sup ‘

jPðBjAÞ  PðBÞj Sup

jPðBjAÞ  PðBÞj.

(A.31)

A2FMð‘Þþ1 ;B2F1 nþMð‘Þþ1 ;PðAÞa0 1

Under the f-mixing assumption on (Xn)n, the right-hand part of the expression (A.31) tends to zero as n-N and then the left-hand part of Equation (A.31) converges to zero, hence the sequence (Si)i is f-f-mixing. Moreover, for all i: !   K X mn ðxÞ ¼ 1. (A.32) aKi S i ¼ var S i is centered and var sn i¼1 Then, using expressions (A.29)–(A.32), and the Theorem 2.2 given in Peligrad and Utev (1997), we get mn ðxÞ  Emn ðxÞ !D Nð0; 1Þ sn

(A.33)

The variance of mx(x) is given by Equation (A.21). The proof of Theorem 1 is complete. We provide now the proof of Corollary 1. Proof of Corollary 1 From Theorem 1, a confidence interval for a given a can be computed, and has the expression: z1ða=2Þ

mn ðxÞ  Emn ðxÞ

z1ða=2Þ , s^ n

(A.34)

where z1ða=2Þ is the (1a/2) quantile of Student law. Previously, we have established that the estimate mn ðxÞ can be biased; thus, the relationship (Equation A.34) becomes: mn ðxÞ þ B  s^ n z1ða=2Þ mðxÞ mn ðxÞ þ B þ s^ n z1ða=2Þ .

(A.35)

When the bias is negligible, the corollary is established. If the bias is not negligible, we can bound it. The bound is obtained using expressions (A.5)

Alternative Methods for Forecasting GDP

183

and (A.36):

B¼O

kðnÞ ^ ðn  dÞhðxÞc

!!p=d ,

(A.36)

^ with c ¼ pd=2 =ðGððd þ 2Þ=2ÞÞ, hðxÞ being an estimate of the density hðxÞ. Introducing this bound in expression (A.35) completes the proof.

References Akaike, H. (1974). A new look at the statistical model identification. IEEE Transaction on Automatic Control 19, 716–723. Baffigi, A.R., Golinelli, R. and Parigi, G. (2004). Bridge models to forecast the euro area GDP. International Journal of Forecasting 20, 447–460. Becker, R.A., Chambers, J.M. and Wilks, A.R. (1988). The New S Language. New York: Chapman and Hall. Bernanke, B.S. and Boivin, J. (2003). Monetary policy in a data-rich environment. Journal of Monetary Economics 50, 525–546. Box, G.E.P. and Jenkins, G.M. (1970). Time Series Analysis: Forecasting and Control. New York: Holden Day. Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39. Clements, M.P. and Krolzig, H.M. (1998). A comparison of the forecast performance of Markov-switching and threshold autoregressive models of US GNP. Econometrics Journal 1, C47–C75. Collomb, G. (1984). Nonparametric time series analysis and prediction: Uniform almost sure convergence of the window and k-NN autoregression estimates. Mathematic Operational Statistics, Series Statistics 16, 297–307. Darne, O. (2008). Using business survey in industrial and services sector to nowcast GDP growth: The French case. Economics Bulletin 3 (32), 1–8. Devroye, L.P. (1982). Necessary and sufficient conditions for the pointwise convergence of nearest neighbor regression function estimates. Z. Wahrscheinlichkeneitstheorie Verw Gebiete 61, 467–481. Devroye, L.P. and Gyo¨rfi, L. (1985). Nonparametric Density Estimation: The L1 View. New York: Wiley. Diron, M. (2008). Short-term forecasts of euro area real GDP growth: An assessment of real-time performance based on vintage data. Journal of Forecasting 27, 371–390. Donoho, D.L. and Johnstone, I.M. (1992). Minimax estimation via wavelet shrinkage. Department of Statistics, Stanford University, Technical Report 402. Engle, R.F. and Granger, C.W.J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica 55, 251–276. Ferrara, L., Gue´gan, D. and Rakotomarolahy, P. (2010). GDP nowcasting with ragged-edge data: A semi-parametric modeling. Journal of Forecasting 29, 186–199.

184

Dominique Gue´gan and Patrick Rakotomarolahy

Forni, M., Giannone, D., Lippi, M. and Reichlin, L. (2005). Opening the black box: Structural factor models with large cross-sections. European Central Bank Working Paper No. 571. Friedman, J.H. (1988). Multivariate adaptive regression splines (with discussion). Annals of Statistics 19, 1–141. Gue´gan, D. (1983). Une condition d’ergodicite´ pour des mode`les iline´aires a` temps discret. CRAS Se´rie 1, 297–301. Gue´gan, D. and Huck, N. (2005). On the use of nearest neighbors in finance. Revue de Finance 26, 67–86. Gue´gan, D. and Rakotomarolahy, P. (2010). A short note on the nowcasting and the forecasting of euro-area GDP using non-parametric techniques. Economics Bulletin 30 (1), 508–518. Gupta, R. (2006). Forecasting the South African economy with VARs and VECMs. South African Journal of Economics 74, 611–628. Hamilton, J.D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384. Han, J., Fu, Y., Wang, W., Koperski, K. and Zaine, O. (1997). DMQL: A Data Mining Query Language for Relational Databases. Vancouver: WP Simon Fraser University. Hardle, W., Muller, M., Sperlich, S. and Werwatz, A. (2004). Non-Parametric and Semi-Parametric Models. New York: Springer Verlag. Hart, J.D. (1994). Nonparametric Smoothing and Lack-of-Fit Tests. New York: Springer-Verlag. Hoover, K.D. and Perez, S.J. (1999). Data mining reconsidered: Encompassing and the general-to-specific approach to specification search. Econometrics Journal 2, 167–191. Kapetanios, G. and Marcellino, M. (2006). A parametric estimation method for dynamic factors models of large dimensions. IGIER Working Paper No. 305. Koenig, E.F., Dolmas, S. and Piger, J. (2003). The use and abuse of real-time data in economic forecasting. The Review of Economics and Statistics 85, 618–628. Kuan, C.M. and White, H. (1994). Artificial neural networks: An econometric perspective. Econometric Reviews 13, 1–91. Litterman, R.B. (1986). Forecasting with Bayesian vector autoregressions five years of experience. Journal of Business Economic Statistics 4, 25–38. Mack, Y.P. (1981). Local properties of k-NN regression estimates. SIAM Journal on Algebraic and Discrete Methods 2, 311–323. Mizrach, B. (1992). Multivariate nearest-neighbor forecasts of EMS exchange rates. Journal of Applied Econometrics 7, S151–S163. Supplement: Special Issue on Nonlinear Dynamics and Econometrics. Ouyang, D., Li, D. and Li, Q. (2006). Cross-validation and nonparametric k nearest neighbor estimation. Econometrics Journal 9, 448–471. Nowman, B. and Saltoglu, B. (2003). Continuous time and nonparametric modeling of U.S. interest rate models. International Review of Financial Analysis 12, 25–34. Pagan, A. and Ullah, A. (1999). Nonparametric Econometrics. Cambridge, UK: Cambridge University Press. Peligrad, M. and Utev, S.A. (1997). Central limit theorem for linear processes. The Annals of Probability 25, 443–456.

Alternative Methods for Forecasting GDP

185

Pena, D., Tiao, G.C. and Tsay, R.S. (2003). A course in time series analysis. New York: Wiley Series in Probability and Statistics. Prakasa Rao, B.L.S. (1983). Nonparametric Functional Estimation. Orlando, FL: Academic press. Ru¨nstler, G. and Sedillot, F. (2003). Short-term estimates of euro area real GDP by means of monthly data. European Central Bank Working Paper No. 276. Schumacher, C. and Breitung, J. (2008). Real-time forecasting of German GDP based on a large factor model with monthly and quarterly data. International Journal of Forecasting 24, 368–398. Schwartz, G. (1978). Estimating the dimension of ARIMA models. Annals of Statistics 6, 461–464. Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. London: Chapman and Hall. Sims, C.A. (1980). Macroeconomics and reality. Econometrica 48, 1–48. Smets, F. and Wouters, R. (2004). Forecasting with a Bayesian DSGE model: An application to the euro area. Journal of Common Market Studies 42, 841–867. Stock, J.H. and Watson, M.W. (2002). Macroeconomic forecasting using diffusion indexes. Journal of Business & Economic Statistics 20, 147–162. Stone, C. (1977). Consistent nonparametric regression. Annals of Statistics 5, 595–645. Stute, W. (1984). Asymptotic normality of nearest neighbor regression function estimates. Annals of Statistics 12, 917–926. Tkacz, G. and Hu, S. (1999). Forecasting GDP growth using artificial neural networks. Bank of Canada Working Paper No. 99. Tong, H. (1990). Non-Linear Time Series: A Dynamical System Approach. Oxford, UK: Oxford University Press. Yakowitz, S. (1987). Nearest neighbors method for time series analysis. Journal of Time Series Analysis 8, 235–247. Yatchew, A.J. (1998). Nonparametric regression techniques in economics. Journal of Economic Literature 36, 669–721.

Chapter 9

GARCH Models with CPPI Application Hachmi Ben Ameur Amiens School of Management, 18, place Saint Michel, Amiens 80090, e-mail: [email protected]

Abstract Purpose – The aim of this chapter is to examine the constant proportion portfolio insurance (CPPI) method when the multiple is allowed to vary over. Methodology/approach – A quantile approach is introduced under the dependent return hypothesis. We use for example ARCH-type models. Findings – In this framework, we provide explicit values of the multiple as function of the past asset returns and other state variables. We show how the multiple can be chosen to satisfy the guarantee condition, at a given level of probability and for particular market conditions. Originality/value of paper – We show in this chapter that it is possible to choose variable multiples for the CPPI method if quantile hedging is used and in the case of dependent log returns. Upper bounds can be calculated for each level of probability and according to state variables. This new multiple can be determined according to the distributions of the risky asset log return and volatility. Keywords: CPPI, ARCH, value-at-risk, stock returns, crises JEL Classification: C10, G11

1. Introduction The various examples of financial crash (Black Thursday in 1929, the crash of October 1987, the subprime 2007–2008) have shown that the International Symposia in Economic Theory and Econometrics, Vol. 20 F. Jawadi and W.A. Barnett (Editors) Copyright r 2010 by Emerald Group Publishing Limited. All rights reserved ISSN: 1571-0386/DOI: 10.1108/S1571-0386(2010)0000020014

188

Hachmi Ben Ameur

design and management of financial funds should be based on solid foundations, allowing better understanding of the function and structure, and further mastering of the risks inherent to this type of activity (not only profit-seeking but also control of risks in the context of this management). The ARCH models are introduced by Engle (1982) for a study of macroeconomic data on inflation in the United Kingdom. The observation of time series shows that the volatility is not constant as assumed previously for linear estimation models. This is the phenomenon of conditional heteroscedasticity. To overcome these shortcomings, Engel proposed a new class of autoregressive conditional heteroscedasticity that can figure out and analyze the behavior of volatility over time. Then, several models derived from the ARCH model have emerged. The models that estimate the conditional volatility are numerous and have always great importance in studying financial markets. Many extentions from the ARCH model have given several models of a new autoregressive class and conditionally heteroscedastic. Among the very known are the GARCH model of Bollerslev (1986), the EGARCH model of Nelson (1990), the model IGARCH proposed by Engle and Bollerslev (1986) with an effect of persistent volatility, the TGARCH model proposed by Zakoı¨ an (1994) also called the ‘‘threshold model,’’ and the GJR-GARCH model (Glosten–Jagannathan–Runkle GARCH). Campbell, Lo, and MacKinlay, after comparing the linear and nonlinear time series, argued that ‘‘It is both logically inconsistent and statistically inefficient to use volatility measures that are based on the Assumption of constant Volatility over some period when the resulting series moves through time.’’ These models allow considering essential characteristics of the markets: Firstly, the volatility is not constant in time. The consideration of this phenomenon increases the potential predictive models. Secondly, the volatility is not stable; the market has periods of high activity and other periods of relative calm. We observe in the market more and more financial institutions offering the portfolio insurance products. The aim is to reassure savers and investors, with propositions comporting minimum risk and multiple choices that depend on three factors: risk aversion, the expected return, and investment horizon. These models can be very attractive for relatively risk-averse investors; especially during periods of crisis, they guarantee a predetermined proportion of the amount originally invested. In this chapter we examine the GARCH models and we propose an application for the constant proportion portfolio insurance (CPPI) model.

GARCH Models with CPPI Application

189

2. Dependent Returns (ARCH-Type Models) Many previous studies have shown the importance of the asymmetric GARCH model to estimate the conditional volatility. The model implies that negative shocks induce greater volatility than positive shocks. As examples to what has been said, Poon and Granger (2003) compared several studies that concern the GARCH models and they concluded that in general the asymmetric volatility performs better than GARCH. Ronald Heynen, Angelien Kemna, and Ton Vorst (March 1994) concluded that exponential GARCH gives the best description of asset prices according to the Akaike information criterion. Chen and Kuan (2002) have tested several models to determine the conditional volatility according to the modified CCK test of Chen (2001). This test is capable of detecting asymmetric volatility. They only accept the EGARCH for several index prices. Engel and Ng (1993) have shown that the EGARCH model can capture most of the asymmetry of the time series but it presents high conditional variance. Finally, Awartani and Corradi (2005) have studied the daily observation of S&P 500 (composite), and they concluded that the asymmetric GARCH model gives the best estimation to capture the leverage effect.

3. Several GARCH Models The GARCH model is based on the principle of stationarity. It is therefore important to determine whether the series has a unit root or not. We present the tables of three tests that allow studying the unit root of price returns logarithms of S&P500. 1. According to the augmented Dickey–Fuller (ADF) test (see Table 1) that can take into account the differentiated autocorrelation series, we reject the null hypothesis H0: presence of unit root. 2. According to the Elliott–Rothenberg–Stock (ERS) DF-GLS test we also reject the null hypothesis H0: presence of unit root. Several studies have shown that this test is more robust than classical studies (Table 2). 3. According to the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test we accept the null hypothesis H0: series are stationary (Table 3). The three tests that we used give similar results. We can therefore model the volatility of log returns of S&P500 with the appropriate GARCH model. We realized a comparison between the GARCH, GJR-GARCH, and EGARCH on weekly returns of the S & P500. According to information criteria and the maximum of the log likelihood, it appears that the EGARCH model (1,1) provides the best estimation of the conditional volatility of the used data (Tables 4–6).

190

Hachmi Ben Ameur

Table 1: Statistical Test of Augmented Dickey–Fuller: We Test the Null Hypothesis H0: Presence of Unit Root Null hypothesis: RD has a unit root Exogenous: Constant Lag length: 1 (fixed)

Augmented Dickey–Fuller test statistic Test critical values 1% level 5% level 10% level

t-Statistic

Probability

38.15291 3.432271 2.862274 2.567205

0.0000

Augmented Dickey–Fuller test equation Dependent variable: D(RD) Method: Least squares Date: 06/30/09; Time: 23:43 Sample (adjusted): 1/24/1950 6/02/2009 Included observations: 3098 after adjustments Variable

Coefficient

RD(1) D(RD(1)) C

0.979192 0.040997 0.001363

R2 Adjusted R2 SE of regression Sum squared resid Log likelihood Durbin–Watson statistic

0.511382 0.511066 0.019525 1.179947 7799.471 2.000150

Standard error

t-Statistic

Probability

0.025665 0.017958 0.000353

38.15291 2.282989 3.864200

0.0000 0.0225 0.0001

Mean dependent var SD dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

5.17E-06 0.027924 5.033228 5.027381 1619.594 0.000000

From the above statistical table, we reject the null hypothesis H0: presence of unit root.

3.1. The EGARCH(1,1) Model For this we consider the EGARCH(1,1) model with parameter values such as in Nelson (1990). yt ¼ X 0t b þ t pffiffiffiffi t ¼ Z t  ht Z t N:id ð0; 1Þ

ð1Þ

where ht is the conditional volatility. logðht Þ ¼ a0 þ

q X i¼1

ai gðZti Þ þ

p X i¼1

yi logðhti Þ

(2)

GARCH Models with CPPI Application

Table 2:

191

Elliott–Rothenberg–Stock DF-GLS Statistical Test

Stationarity test of Elliott–Rothenberg–Stock Null hypothesis: RD has a unit root Exogenous: Constant Lag length: 15 (automatic based on modified AIC, MAXLAG ¼ 15) t-Statistic Elliott–Rothenberg–Stock DF-GLS test statistic Test critical values 1% level 5% level 10% level

3.326945 2.565712 1.940927 1.616630

DF-GLS test equation on GLS detrended residuals Dependent variable: D(GLSRESID) Method: Least squares Date: 06/30/09 Time: 23:36 Sample (adjusted): 5/02/1950 6/02/2009 Included observations: 3084 after adjustments Variable GLSRESID(1) D(GLSRESID(1)) D(GLSRESID(2)) D(GLSRESID(3)) D(GLSRESID(4)) D(GLSRESID(5)) D(GLSRESID(6)) D(GLSRESID(7)) D(GLSRESID(8)) D(GLSRESID(9)) D(GLSRESID(10)) D(GLSRESID(11)) D(GLSRESID(12)) D(GLSRESID(13)) D(GLSRESID(14)) D(GLSRESID(15)) R2 Adjusted R2 SE of regression Sum squared resid Log likelihood

Coefficient 0.059902 0.898689 0.792578 0.734260 0.698395 0.669589 0.562613 0.506349 0.484749 0.414962 0.367690 0.304846 0.239571 0.177623 0.150863 0.075584 0.486033 0.483520 0.020085 1.237704 7683.551

Standard error

t-Statistic

Probability

0.018005 0.024623 0.029244 0.032277 0.034482 0.0361 35 0.037339 0.037755 0.037609 0.037252 0.036321 0.034500 0.032141 0.029089 0.024858 0.018039

3.326945 36.49765 27.10209 22.74849 20.25405 18.53001 15.06767 13.41147 12.88921 11.13939 10.12324 8.836150 7.453842 6.106111 6.069076 4.190024

0.0009 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Mean dependent var SD dependent var Akaike info criterion Schwarz criterion Durbin–Watson statistic

7.52E-07 0.027948 4.972472 4.941167 2.010598

We also test the nil hypotheses H0: presence of unit root. From the above statistical table we reject the nil hypotheses H0: presence of unit root. Several studies have shown that this test is more robust than classical studies.

192

Table 3:

Hachmi Ben Ameur

Kwiatkowski–Phillips–Schmidt–Shin (KPSS) Statistical Test

KPSS stationarity test Null hypothesis: RD is stationary Exogenous: Constant Bandwidth: 8.45 (Newey–West using Quadratic Spectral kernel) LM-statistic Kwiatkowski–Phillips–Schmidt–Shin test statistic Asymptotic critical values* 1% level 5% level 10% level

0.092764 0.739000 0.463000 0.347000

Residual variance (no correction) HAC corrected variance (Quadratic Spectral kernel)

0.000382 0.000386

KPSS test equation Dependent variable: RD Method: Least squares Date: 06/30/09; Time: 23:41 Sample (adjusted): 1/10/1950 6/02/2009 Included observations: 3100 after adjustments Variable C R2 Adjusted R2 SE of regression Sum squared resid Log likelihood

Coefficient 0.001389 0.000000 0.000000 0.019538 1.183003 7801.497

Standard error

t-Statistic

Probability

0.000351 3.957449 Mean dependent var SD dependent var Akaike info criterion Schwarz criterion Durbin–Watson stat

0.0001 0.001389 0.019538 5.032579 5.030631 2.042173

We test the null hypotheses H0: the series is stationary. From the above statistical table we accept the nil hypotheses H0. The three tests show the stationarity of S&P500 series returns. Estimation of the parameters for several conditional volatility models. *Kwiatkowski–Phillips–Schmidt–Shin (1992, Table 1).

With gðZti Þ ¼ yZ ti þ gðjZ ti j  EjZti jÞ

(3)

We use this model to estimate the conditional volatility. 3.2. Estimation of the Model Parameters We determine the parameters of the EGARCH(1,1) model from the method pseudo-maximum likelihood realized on the software MATLAB. In what follows we detail the methodology of this model.

GARCH Models with CPPI Application

193

Table 4: EGARCH(1,1) Model: Estimation of the Volatility with EGARCH Model Mean: ARMAX(0,1,0); Variance: EGARCH(1,1) Conditional probability distribution: Gaussian number of model parameters estimated: 6 Parameter C MA(1) K GARCH(1) ARCH(1) Leverage(1)

Value

Standard error

t-Statistic

0.0014441 0.0091286 0.32478 0.95906 0.20667 0.10689

0.00030189 0.018169 0.046943 0.0058137 0.016681 0.0095432

4.7836 0.5024 6.9186 164.9653 12.3901 11.2003

Log-likelihood value: 8200.55 Akaike and Bayesian information criteria. AIC: 1.6265e004. BIC: 1.6229e004.

Table 5: Model

GARCH(1,1) Model: Estimation of the Volatility with GARCH

Mean: ARMAX(0,0,0); Variance: GARCH(1,1) Conditional probability distribution: Gaussian number of model parameters estimated: 4 Parameter C K GARCH(1) ARCH(1)

Value

Standard error

t-Statistic

0.0020237 1.0136e005 0.86003 0.11954

0.00029097 1.9356e006 0.010874 0.0089855

6.9551 5.2365 79.0889 13.3038

Log-likelihood value: 8153.96 Akaike and Bayesian information criteria. AIC: 1.6179e304. BIC: 1.6155e304.

We present the descriptive statistics of the S&P500 historical weekly log return and study the variations of the S&P500 during the period 01/1965– 01/2009. We have 3125 observations. We divide our sample into two periods for the estimation of parameters until 01/2005 and the rest of the sample to test the CPPI model. Next graph shows the empirical distribution function of the S&P500 weekly log returns.

194

Hachmi Ben Ameur

Table 6: GJR-GARCH(1,1): Estimation of the Volatility Parameters with GJR-GARCH(1,1) Model Mean: ARMAX(0,0,0); Variance: GJR(1,1) Conditional probability distribution: Gaussian number of model parameters estimated: 5 Parameter C K GARCH(1) ARCH(1) Leverage(1)

Value

Standard error

t-Statistic

0.0015582 1.3502e005 0.85596 0.039595 0.1406

0.00030466 2.0142e006 0.012583 0.01055 0.015617

5.1145 6.7035 68.0236 3.7530 9.0035

Log-likelihood value: 8185.12 Akaike and Bayesian information criteria. AIC: 1.6237e304. BIC: 1.6206e304. The comparison of log likelihood and Akaike and Bayesian information criteria of these latest models show that the EGARCH(1,1) model allows to have the best estimations of the conditional volatility parameters.

Table 7:

Descriptive Statistics

Mean

Maximum

Minimum

Standard deviation

Skewness

Kurtosis

0.0015

0.1412

0.1820

0.0199

0.5018

8.2826

In Table 7 we present the descriptive statistics of the S&P500 log returns. The S&P500 yields present an excess kurtosis compared to the Gaussian distribution (8.2827W3), which is a measure of the ‘‘peakedness’’ of the probability distribution. This shows a strong probability of extreme yields. We have negative value for the skewness (–0.5018). Therefore, the yields of S&P500 do not follow a Gaussian distribution. We estimate the parameters of the extreme quantile Z tk1 , we use the method of pseudo-maximum log likelihood for fitting a mathematical model to S&P500 data. We are interested in estimators of pseudo-maximum likelihood calculated under the fictitious assumption on the perturbation distribution, hence the term pseudo. It is shown that in this case the estimators a are converging and asymptotically normal (for more details see Gourrie´roux and Montfort, 1989). This method is based on the normality assumption of conditional distribution: yt ðyt =yt1 Þ ! Nð0; ht Þ

(4)

GARCH Models with CPPI Application

195

The pseudo log-likelihood function associated in case with hetroscedastic errors is defined in the following: lðaÞ ¼

n X

log f ðyi ; aÞ

(5)

i¼1

We obtain the truncated log-likelihood function: log L ¼ 

T T T 1X ½Y t  mt ðaÞ2 log h^t ðaÞ  log 2p  2 2 2 i¼1 h^t ðaÞ

(6)

The method of pseudo-maximum likelihood estimates a by finding the value of this parameters that maximize l(a). We use an algorithm of optimization.2 We use the EGARCH(1,1) model to estimate the conditional volatility. In Section 4.2 we have developed the criteria of choice of this model. We find that   G K A jt j jtk1 j k1 2 2 log stk1 þ þ E log stk ¼ stk1 0:9573 0:33924 0:20894 stk1 L

þ

tk1 s 0:1066 tk1

pffiffiffiffiffiffiffiffi where Efjtk1 j=stk1 g ¼ 2=p, for the Gaussian law. Then the conditional extreme quantile Z tk1 is Z tk1 ¼

a0 a1 þ t þ stk F 1 ð1  ð1  Þ1=T Þ 0:0014461 0:0097166 ki

All terms are statistically significant, in particular the leverage parameter L is negative and significant. Hence the importance of asymmetrical model (Figure 1). Figure 1 clearly shows that the volatility is not constant in time. The market has periods of high activity and other periods where the market is relatively calm. Next graph shows the evolution of Ztk1 and the conditional volatility, in periods of high volatility the value of Z tk1 is adjusted upward (Figure 2). 2

We use MATLAB with numerical procedures to determine the parameters that minimize the function –l(a).

196

Hachmi Ben Ameur

Standard Deviation

Innovation

Innovations 0.2 0 -0.2 01/1965

10/1987

10/2008

Conditional Standard Deviations 0.1 0.05 0 01/1965

10/1987

10/2008

19/1987

10/2008

Returns Return

0.2 0 -0.2 01/1965

Figure 1: Innovations Standard Deviation and Returns of the S&P500 from 01/1965 to 10/2008. 0

0.08

-0.05

0.06 Ztk

-0.1

0.04 Conditional volatility

-0.15

-0.2 01/2005

0.02

01/2009

0

Figure 2: Trajectories of the Conditional Volatility and Z(tk) for the Period from 01/2005 until 01/2009. We estimate the conditional extreme quantile from the weekly log return of S&P500, we observe that we have a maximum values equal to –16%, in crises periods this last value seems underestimated compared to the maximum negative returns observed, as evidenced by the crisis at the end

GARCH Models with CPPI Application

197

of 2008. Moreover, sequence of negative returns can have an effect more serious than a relatively high yield negative.

4. CPPI Application Recent developments in financial econometrics suggest the use of nonlinear time-series structures to model the attitude of investors toward risk and expected return. This chapter examines the CPPI method when the multiple is allowed to vary over time. A quantile approach is introduced. In this chapter, we provide explicit values of the multiple as function of the past asset returns. These bounds can be statistically estimated from asset return rates, using, for example, ARCH-type models. We show how the multiple can be chosen to satisfy the guarantee condition, at a given level of probability and for particular market conditions. The goal of portfolio insurance is first to limit downside risk in bearish financial market, second to allow some participation in bullish markets. There exist several methods of portfolio insurance: option-based portfolio insurance (OBPI), CPPI, stop-loss, and so on (see, e.g., Poncet and Portait, 1997 for a review about these methods). Here, we examine the CPPI method introduced by Black and Jones (1987) for equity instruments, and Perold (1986) and Perold and Sharpe (1988) for fixed-income instruments (see also Black and Rouhani, 1989; Roman et al., 1989; Black and Perold, 1992). The CPPI method uses a simplified strategy to allocate assets dynamically over time. Basically, two assets are involved: the riskless asset, B, with a constant interest rate r (usually Treasury bills or other liquid money market instruments) and the risky one, S (usually a market index). Usually, the investor, with initial amount to invest V0, wants to recover a fixed percentage a of his initial investment at a given date in the future T. To obtain a terminal portfolio value VT greater than the insured amount pV0, the portfolio manager keeps the portfolio value Vt above the floor F t ¼ pV 0 expðrðT  tÞÞ at any time t in the period [0,T]. For this purpose  The amount et invested in the risky asset is a fixed proportion m of the excess Ct of the portfolio value over the floor.  The constant m is usually called the multiple, et the exposure, and Ct the cushion.  Since Ct ¼ V t  F t , this insurance method consists in keeping Ct positive at any time t in the period.  The remaining funds are invested in the riskless asset Bt. Figure 3 shows a numerical example CPPI where we choose the following value parameters V(0) ¼ 100, P(T) ¼ 100% V(0), and the multiple m ¼ 3.

198

Hachmi Ben Ameur

110 108 106 104

V(t):Portfolio Value

102

Ct: The cushion=V(t)-P(t)

100 98 P(t):The floor

96

0

1 t

Figure 3:

Standard CPPI Example.

Both the floor and the multiple are functions of the investor’s risk tolerance. The higher the multiple, the more the investor will benefit from increases in stock prices. Nevertheless, the higher the multiple, the faster the portfolio will approach the floor when there is a sustained decrease in stock prices. As the cushion approaches zero, exposure approaches zero, too. In continuous time, when asset dynamics have is no jump, this keeps the portfolio value from falling below the floor. Nevertheless, during financial crises a very sharp drop in the market may occur before the manager has a chance to trade. This implies that m must not be too high (e.g., if a fall of 10% occurs, m must not be greater than 10 to keep the cushion positive). Advantages of this strategy over other approaches to portfolio insurance are its simplicity and its flexibility (see, e.g., Black and Rouhani, 1989; Boulier and Sikorav, 1992; De Vitry and Moulin, 1994). Initial cushion, floor, and tolerance can be chosen according to the investor’s own objective (see, e.g., Poncet and Portait, 1997; Prigent, 2001). Banks may bear market risks on the insured portfolios. In that case, banks can use, for example, stress testing since they may bear consequences of sudden large market decreases. For instance, in the case of the CPPI method, banks must, at least, provision the difference on their own capital if the value of the portfolio drops below the floor. Thus, one crucial question for the bank that promotes such funds is what exposure to the risky asset or, equivalently, what level of the multiple to accept? On one hand, as portfolio expectation return is increasing with respect to the multiple, customers want the multiple as high as possible.

GARCH Models with CPPI Application

199

On the other hand, due to market imperfections,3 portfolio managers must impose an upper bound on the multiple.  First, if the portfolio manager anticipates that the maximal daily historical drop (e.g., –20%) will happen during the period, he chooses mr5, which leads to low return expectation. Alternatively, he may consider that the maximal daily drop during the management period will never be greater than a given value (e.g., –10%). A straightforward implication is to choose m according to this new extreme value (e.g., mr10). Another possibility is to take account of the occurrence probabilities of extreme events in the risky asset returns.  Second, he can adopt a quantile hedging strategy. In this case, he determines the multiple as high as possible so that the portfolio value will always be above the floor at a given probability level (typically 99%).

5. Determination of the Multiple When the multiple is assumed to be constant, quantile conditions have been introduced to control the probability that the portfolio value is smaller than the floor4: for a given probability threshold e, P½8t 2 ½0; T; C t 4041   Hamidi et al. (2006) consider a modified quantile hedging strategy where the multiple is conditional. Ben Ameur and Prigent (2007) consider a parametric approach to better determine the relation between state variables and upper bounds on the multiple based on quantile conditions. The higher the multiple m, the higher the amount etk invested on the risky asset. Therefore, an ‘‘aggressive’’ investor would choose high values for m. Nevertheless, in that case, his portfolio is riskier and, as shown in what follows, his guarantee may no longer hold. We assume that the risky asset log return Y follows a GARCH(p,q) model. As it is well known, this kind of dynamics is quite suitable to describe asset fluctuations in a discrete-time setting.5 3 For example, portfolio managers cannot actually rebalance portfolios in continuous time. Additionally, problems of asset liquidity may occur, especially during stock markets crashes. 4 See Prigent (2001) in the Le´vy process case and Bertrand and Prigent (2002) using extreme value theory. 5 The autoregressive conditionally heteroscedastic (ARCH) models, introduced by Engle (1982), are specific nonlinear time-series models. They can describe quite exhaustive set of the underlying dynamics. They have been largely applied on macroeconomics and statistical theory.

200

Hachmi Ben Ameur

The GARCH is defined as follows. Consider the log return Y:   S tk S t  S tk1 ) k ¼ expðY tk Þ  1 Y tk ¼¼ log Stk1 Stk1 Consider the system of autoregressive equations: 8 p > < Y t ¼ a0 þ P ai  Y t þ sk  k; k ki > :

i¼1

Lðstk Þ ¼ b þ C 0 ðk1 Þ þ C1 ðtk1 Þ  Lðstk1 Þ where sk denotes the volatility, the sequence ðtk Þk is i.i.d. with common pdf fW0 and L, C0(.), and C1(.) are deterministic functions. The function L:Rþ ! R is assumed to be strictly increasing. The information delivered by the observation of risk asset returns is generated by ðt1 ; . . . ; tk1 Þk . We have F tk1 ¼ s  algebraðt1 ; . . . ; tk1 Þ. Therefore, we have to search for a multiple mtk that has the following form: mtk1 ¼ gðtk1 ; Y t1 ; . . . ; Y tk1 ; st1 ; . . . ; stk ÞI Ctk1 4L þ hðtk1 ; Y t1 ; . . . ; Y tk1 ; st1 ; . . . ; stk ÞI Ctk1 oL Remark. Note also that the random variables Y tk1 are deterministic functions ðt1 ; . . . ; tk1 Þk . But, for a better financial interpretation of the multiple, we explicitly introduce functions of ðY t1 ; . . . ; Y tk1 Þk itself. Indeed, we examine how the conditional multiple depends on the volatility levels on the past log returns. These kinds of variables are here the state variables. To determine the variable multiple we consider the Var measure, which is the maximum loss not exceeded with a given probability, defined as the confidence level a. We use this condition to determinate the conditional multiple. The previous condition, which is rather strong, can be modified if a quantile hedging approach is adopted, like for the value-at-risk (see Fo¨llmer and Leukert, 1999 for recent application of this notion in financial modeling). The quantile condition is the following:   DS tk L F tk1  ð1  Þ1=T 1 þ mtk1  4 P S tk1 C tk1

GARCH Models with CPPI Application

201

which is equivalent to   L F tk1 mtk1 ðexpðY tk Þ  1Þ4  1  ð1  Þ1=T P C tk1 At time tk, we have two cases: ( expðY tk Þo1 : S tk oStk1 ðS decreasesÞ expðY tk Þ41 : S tk 4S tk1 ðS increasesÞ We assume that the multiple mtk1 is higher than 1 (thus, it is nonnegative). It is the standard assumption. It corresponds to a portfolio profile which is a convex function of the risky asset. Then mtk1 must satisfy the following constraint (for more details see Ben Ameur and Prigent, 2007): mtk1

ðL=C tk1 Þ  1 exp½ðF F tk1 Þ1 ½1  ð1  Þ1=T   1

They are based on the sign of the term: ðF F tk1 Þ1 ð½1  ð1  Þ1=T Þ Lemma. For any real numbers a and b (bW0) and for any random variable e with pdf fe, we have Z ðxaÞ=b h x  ai P½a þ b x ¼ P o f  ðuÞ du ¼ b 1 Thus the pdf of a þ b is given by

x  a 1 f ðxÞ ¼  f  b b We apply this lemma by setting 8 a þ b ¼ Y tk > > > > > > <  ¼  tk ; p P a ¼ aa þ ai  Y tki ; > > > i¼1 > > > : b ¼ st 40: k

202

Hachmi Ben Ameur

We get f FY t ðyÞ k

P   y  ða þ pi¼1 Y tki Þ 1 ¼  f stk stk

F FYk1 ðyÞ tk

P  y  ða þ pi¼1 Y tki Þ , ¼F stk 

in which case, the conditional multiple must satisfy mtk1

exp½ða0 þ

Pp

i¼1 ai

ðL=C tk1 Þ  1  Y tki Þ þ stk  F 1 ð1  ð1  Þ1=T Þ  1

The conditional multiple evolve according to the conditional quantile of the extreme losses and from the conditional market volatility. We estimate the conditional quantile from the weekly log return; we observe that we have a maximum value equal to –16% in crises periods. This value seems underestimated compared to the maximum negative returns observed, as evidenced by the crisis at the end of 2008. Moreover, sequence of negative returns can have an effect more serious than a relatively high yield negative.

6. Application of the Variable Multiple to the S&P500 We apply our model for the very volatile period from 08/2004 to 08/2009, we adopt a weekly rebalancing, and we compare the result to the original CPPI model for m ¼ 3 and m ¼ 6. During this period the market goes through several phases and particularly toward the end where we see the stock market crash related to the subprime crisis. We suppose that the log returns of the S&P500 are dependent and follow an EGARCH(1,1) model. The selection criteria of this model were outlined in section 3. In Figure 4, we observe the dynamique of these portfolios for PT ¼ 100%V 0 and for PT ¼ 95%V 0 . The model CPPI for m ¼ 6 gives a biggest gain than other models when the market is bullish, but with the financial crash of October 2008 the portfolio is fully monetized and falls below the floor at the end of the period for the two floor values PT ¼ 100%V 0 and PT ¼ 95%V 0 , although this value of multiple is not considered a very high value. On the other hand with m ¼ 3, we were unable to capture large market performance. With the two models of the conditional multiple under Var condition with a variable threshold Ltk ¼ 85%C tk1 and under the measure risk expected shortfall with Ltk ¼ 1:1C tk1 we can have higher performance than with standard multiple m ¼ 3.

GARCH Models with CPPI Application

203

Portfolio values for P(T)=100%V0 130 125

portfolio values

120

For m=3 For m=6 With VaR condition With ES condition The floor

115 110 105 100 95 90 08/2004

10/2008 08/2009

tk Portfolio values for P(T)=95%V0 140

130

For m=3 For m=6 With the VaR condition With the ES condition the Floor PT

portfolio values

120

110

100

90

80 08/2004

10/2008 08/2009

tk

Figure 4: Comparison of CPPI Models with Constant and Variable Multiples.

204

Hachmi Ben Ameur

Then, with the subprime crisis and higher volatility on the market, these two models have reduced the exposure to the risky asset S&P500 by decreasing the multiple values. These models remain above the floor where the other models with more exposure are dropped below.

7. Conclusion The GARCH model presents an interesting characteristic: it can rapidly adapt itself to the volatility level. However, it does not forecast sudden crises, such as subprime crises. The information is not containing in the historical series prices; it is brusquely exposed. To take the dependence of the log return on the market into account, we have estimated an asymmetric EGARCH(1,1) model for the conditional volatility and the weekly log returns. The rebalancing occurs in discrete time. As shown in this chapter, it is possible to choose variable multiples for the CPPI method if quantile hedging is used and in the case of dependent log returns. Upper bounds can be calculated for each level of probability and according to state variables. This new multiple can be determined according to the distributions of the risky asset log return and volatility. Other conditions can be imposed on this multiple while the quantile hedging constraints are satisfied. The difference with the standard multiple is significant. Other state variables can also be considered, such as exogenous macro-economic factors. Also, the impact of transaction costs can also be examined. Other approaches are developed to take into account dynamic volatility, for example, the models with regime changing, the models with stochastic volatility, the levy process, and so on. These models consider other series prices characteristics, for example, the leverage effect and ameliorate the understanding of the gap risk.

References Awartani, B.M.A. and Corradi, V. (2005). Predicting the volatility of the S&P 500 stock index via GARCH models: The role of asymmetries. International Journal of Forecasting 21, 167–183. Ben Ameur, H. and Prigent, J.-L. (2007). Portfolio insurance: Determination of a dynamic CPPI multiple as function of state variables. University of CergyPontoise, France, THEMA Working Paper. Bertrand, P. and Prigent, J.-L. (2002). Portfolio insurance: The extreme value approach to the CPPI method. Finance 23, 69–86. Black, F. and Jones, R. (1987). Simplifying portfolio insurance. The Journal of Portfolio Management 14 (1), 48–51.

GARCH Models with CPPI Application

205

Black, F. and Perold, A.R. (1992). Theory of constant proportion portfolio insurance. The Journal of Economics, Dynamics and Control 16, 403–426. Black, F. and Rouhani, R. (1989). Constant proportion portfolio insurance and the synthetic put option: A comparison. In Fabozzi, F.J. (Ed.), Institutional Investor Focus on Investment Management. Cambridge, MA: Ballinger, pp. 695–708. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31 (3), 307–327. Boulier, J.F. and Sikorav, J. (1992). Portfolio Insurance: The Attraction of Security. Quants, 6, CCF, Paris. Chen, T. (2001). Testing conditional symmetry with an application to financial returns. Working Paper. Institute for Social Sciences and Philosophy, Academia Sinica, Taiwan. Chen, Y.-T. and Kuan, C.-M. (2002). Time irreversibility and EGARCH effects in US stock index returns. Journal of Applied Econometrics 17, 565–578. De Vitry, T. and Moulin, S. (1994). Aspects the´oriques de l’assurance de portefeuille avec plancher. Banque et Marche´s 11, 21–27. Engle, R.F. and Bollerslev, T. (1986). Modelling the persistence of conditional variances. Econometric Reviews 5 (1–50), 81–87. Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008. Engel, R. and Ng, V. (1993). Measuring and testing the impact of news on volatility. The Journal of Finance 5, 1749–1778. Fo¨llmer, H. and Leukert, P. (1999). Quantile hedging. Finance and Stochastics 3, 251–273. Gourrie´roux, C. and Montfort, A. (1989). Statistique et mode`les e´conome´triques. Economica, Vol. I et II. Hamidi, B., Jurczenko, E. and Maillet, B. (2006). D’un multiple conditionnel en assurance de portefeuille. A.A. Advisors-QCG, ABN Amro Group, Working Paper. Heynen, R., Kemna, A. and Vorst, T. (1994). Analysis of the term structure of implied volatility. Journal of Financial and Quantitative Analysis 29 (1), 31–56. Perold, A.R. (1986). Constant proportion portfolio insurance. Harvard Business School, Working Paper. Perold, A.R. and Sharpe, W. (1988). Dynamic strategies for asset allocations. Financial Analysts Journal 16–27. Poncet, P. and Portait, R. (1997). Assurance de Portefeuille. In Y. Simon (Ed.), Encyclope´die des Marche´s Financiers. Economica, pp. 140–141. Poon, S.-H. and Granger, C.W.J. (2003). Forecasting volatility in financial markets: a review. Journal of Economic Literature 41 (2), 478–539. Prigent, J.L. (2001). Assurance du portefeuille: Analyse et extension de la me´thode du coussin. Banque et Marche´s 51, 33–39. Roman, E., Kopprash, R. and Hakanoglu, E. (1989). Constant proportion portfolio insurance for fixed-income investment. Journal of Portfolio Management 15 (4), 58–66. Zakoı¨ an, J.M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.