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15.433 INVESTMENTS Class 1: Introduction
Spring 2003
Outline
• Course Overview • Market Overview • Introduction to the Financial System • Overviews of Equity, Fixed Income, Currencies and Derivatives • Financial Innovations • Summary • Questions for Next Class
Course Mechanics
Course Grade:
Class participation
10%
5 group assignments 20% Mid-tern exam
30%
Final exam
40%
Course Materials: • Lecture Notes • Textbook: Bodie, Kane and Marcus • Articles in course reading packet • Additional reading materials and handouts
Course Overview
Broadly speaking, this course will cover the following give themes: • Financial theories: Portfolio theory, the CAPM and the APT
• Equity and equity options, and empirical evidence
• Fixed income instruments and fixed income derivatives, credit market and credit derivatives
• Market efficiency and active investments
• A brief introduction to behavioral finance
Course Objectives
Through this class I want to help you build the following skills: • Analytical ability: modeling skills that are important in making investment decisions, • Quantitative skills: developing problem solving skills, data analysis, probability evaluation of uncertain events. • Empirical knowledge of the financial markets: equity, fixed income (default-free and defaultable) and their respective derivatives.
The Financial System
The financial system can be viewed from two different angles: • Institutional perspective: the financial system encompasses the markets, intermediaries, instruments, clients etc.; and • Functional perspective: the financial system facilitates the allocation and deployment of economic resources, across time and space and in uncertain environment.
While its institutional composition changes over time and space (different places on the globe), the basic functions of a financial system are essentially the same in all economies. The institutional dimension results from the functional needs.
Functional Perspective
The financial system provides six functions: • Transferring economic resources across time (e.g. student loans, retirement funds) and space (global financial market, e.g. institutional investments) • Clearing and settling payments to facilitate trades: money, check, ATM cards, mortgages and mortgage interest payments, etc. • Pooling of resources to undertake large-scale indivisible enterprise (equities, money market funds, mutual funds etc.) • Information to coordinate decision-making, facilitating information flow, ”price discovery”, the normal indices for equities, fixed income rates, volatilities etc. • Dealing with agency problems, mitigate moral hazard, adverse selection and principal-agent based problems that are caused by asymmetry information or incentive mis-matches (collateralization, CEO incentive for company performance etc.). • Managing risks, bundling, packaging, pooling and tranching of risk (insurance companies, CMO, CDO, exotic derivatives etc.)
Institutional Perspective
The institutional compositions of the financial system includes: • Financial markets: equity, fixed income (debt), credits and derivatives • Financial intermediaries: banks, insurance companies, pension funds, mutual funds, investment banks, venture capital firms, asset management firms and information providers etc. • Financial infrastructure and regulation: trading rules, contract enforcement, account system, capital requirements etc. • Governmental and quasi-governmental organizations: SEC, central banks (Federal Reserve System), the Bank for International Settlement (BIS), the International Monetary Fund (IMF), the World Bank, etc. International markets differ in many ways: • The US market is currently the only market offering Financial Futures on individual stocks; • Germany closed the ”Neue Markt”, eliminating a platform for not very liquid stocks, regarding stock exchanges inexperienced companies, thus rising capital is getting more difficult for young companies;
• Investment attitudes and awareness are subject to national regulations, e.g. pension fund regulation - defined benefit vs. defined contribution plans.
Stocks
Common stocks, also known as Equities, represent ownership shares of a corporation. • Two important characteristics: residual claim and limited liability; • Sources of returns: dividends and capital gains; • Calculating returns: buy at time 0 and pay P0 , sell at time T and
receive PT and dividend DT :
– The percentage return is calculated as: rT =
PT + D T − P 0 PO
– The log-return is calculated as: � � PT + D T rT = ln PO
(1)
(2)
For small rT , the log-return and percentage return are close. Question: Why in some situations should we prefer to use logreturns?
• Some determinants of stock returns: – Firm-specific condition: management, productivity, earnings, growthpotential, market-liquidity,
– Market condition: market indices (volatility, volume etc.), e.g. Nasdaq, SP500, DAX, FTSE etc. – Economic condition: macro-economic variables, e.g. GDP-growth, inflation, employment rate, business cycles, liquidity, interest rates etc.
• Some important empirical evidence: – Patterns in the cross-section of stock return: value (value vs. growth), size (small vs. large), momentum (low vs. high); – Time-series behavior of stock returns: time-varying expected returns, predictability, stochastic volatility etc.
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SBTSY10
Figure 1: Daily returns of SP 500-index, Nasdaq-index and T-Bonds 10 years 9/12/2001
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Stocks and Bonds SPX
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Daily Returns
Current Distribution
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Daily Returns Current Distribution
Normal-Distribution
Normal-Distribution
Figure 2: Return distribution of US 10 Year Bond
Data source for Figures 1, 2, and 3: Bloomberg Professional.
Figure 3: Return distribution of S&P 500 Index
Fixed Income
Fixed income, also known as debt-instruments, promise to pay fixed, pre-determined stream of cash flows in the future: Coupon payments, Principal amount (denominations), Time to maturity, Sinking fund obligations, etc.
Figure 4: Cash Flows, Source: RiskM etricsT M , Technical Document, p. 109 Figure 5: Discounted Cash Flows, Source: RiskM etricsT M , Technical Document, p. 109
Treasury bills, notes and bonds vary in maturity. T-bonds may also be callable during a given period.
The value of a bond can be reported in terms of price or yield to maturity. Yield go and price go.
Market Performance
Monthly returns 1926/7 to 2000/12
Equity Portfolio∗ Treasury Bills Mean
0.99%
0.31%
Standard Deviation
5.50%
0.26%
Total Return
$1’828
$16.27
* Value-Weighted
The Term-Structure of Interest Rates
(Yield-Curve)
9
8
Yield Curves for US Treasury
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Figure 6: Treasury Yield Curve, Source: Bloomberg Professional.
How do we describe the term-structure: • Shift • Twist • Butterfly
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The Federal Funds Target Rates (set by FOMC) and the Discount Rates:
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FOMC
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Figure 7: Short Term Interest Rates FED, Discount rates over the last 4 years, Source Bloomberg Professional.
Corporate Bonds
Corporate bonds are similar in structure to Treasury issues, with one important difference: default risk. • Because of the default risk, corporate bonds are cheaper (than their respective Treasury counterparts), paying higher yields. • The difference in yields is referred to as credit spread. • The probability of default varies across firms of different credit qualities (e.g. countries, industries, guarantees etc.) • The probability of default varies over time!
Figure 8: Standard & Poor’s one year transition matrix, Source: RiskM etrics T M − T echnical Document, p. 69.
Derivatives
Overview of Derivatives by Structure: • Forward and Futures; • Options; • Futures; • Path dependent (e.g. look-back), etc.
Overview of Derivatives by Underlying: • Equity Derivatives: stock options, index futures, futures options etc.; • Fixed-Income Derivatives: caps/floors, swaps, swaptions, etc.; • Credit Derivatives: credit swap, collateralized loan obligations, etc.; • Other derivatives: FX, weather, ”exotics”, etc.
Currencies
Currencies are the most liquid financial instrument. Currency instruments have generally spoken the same type of parameters, however with different characteristics, e.g. currency return volatility, which is not shaped in the same form as the equity return volatility. Currency positions usually have a much shorter maturity proprietary traders on Wall Street take positions up to half an hour!
Financial Innovation
Does financial innovation add value? If taken to excess, any virtue can readily become a vice: The market has seen examples of failed financial products, erroneous assumption and strategies, abusive usage of derivatives, distressed hedge funds etc. Empirical evidence supports the statement, that financial innovation does provide social wealth: • It caters to the investment diversity desired by the investors; • It improves the opportunities for investors to receive efficient riskreturn trade-offs; • It provides risk management tools for all market participants; and • It promotes broad distribution and liquidity to economic resources.
Securitization
Figure 9: Securitization of ”stand-alone” mortgages in liquid standardized mortgage-backed securities.
Securitization of the mortgage market: The national mortgage market and mortgage-backed securities transform the residential house finance from fragmented, local-based sources to a free-flowing, international base of capital with depth and usually a higher credit rating.
Other examples: collateralized debt (loan or bond) obligations, asset backed debt, etc.
Credit Enhancement
FleetBoston plays ”Good Bank/Bad Bank,” unloading $1.35 Billion in troubled Loans Patriarch Partners LLC, a NY fund management boutique, created a CLO to raise about $1 billion to acquire the loans.
See the following figure how FleetBoston’s problem loans made their way from Fleet’s books to a special collateralized-loan obligation, funded by investors.
$ 275 million (face value) in Ark subordinated securities
$ 725 million cash FleetBoston
Patriarch Partner's Ark CLO
$ 1.35 billion in funded loans plus $ 150 million in unfunded loans
Figure 10: Setting-up a special purpose vehicle.
$ 925 million in triple-A rated bonds $ 75.75 million in single-A rated bonds $ 35.5 million of equity
Patriarch Partner's Ark CLO
$ 1.036 billion cash
Summary
What to learn in this course? Analytical modeling skills, quantitative tools, and empirical knowledge about the financial market and the financial instruments. How to think about the financial system? • Functional perspectives: across time and space; • Institutional compositions: results from the functional needs. • Does financial innovation add value? • Sometimes abusive usage of financial innovation causes disruptions.
Overall, however, financial innovation provides diversified in-vestment opportunities for investors, risk management tools and techniques for market participants and liquidity to the overall market.
Focus: BKM Chapters 1 & 2; • p.16/17 (market structure); • p.19 (securitization); • p.28-46 (know the different instruments, e.g. CD, CP, Bankers Acceptances etc.);
• p.48-54 (know the most important market indexes, what kind of different weighting-schema exist etc.); Reader: Sharpe (1995);
Type of potential questions: p. 60ff. questions 1, 5, 11, 17, 23
Preparation for Next Class
Please read: • BKM Chapters 3 & 5, • Fama (1995), and • Kritzman (1993). Questions: • What is a normal distribution? • How do we model uncertainty? • Is variance the only measure of uncertainty? • What is the average daily return? • What about the average variability? • What assumptions did you have to make in order to obtain the estimates? • How accurate are your estimates? • If given more observations (a large N), can you improve the precision of your estimates?
15.433 INVESTMENTS Class 2: Securities, Random Walk on Wall Street Reto R. Gallati MIT Sloan School of Management Spring 2003 February 5th 2003
Outline Probability Theory
• A brief review of probability distributions • Evaluating random events with normals. • Large surprises and normal distributions.
Statistical Data Analysis
• Empirical distributions. Sample statistics. • The precision of sample statistics.
Summary Questions for Next Class
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What makes an event random • Flipping a Coin:
• Forecasting tomorrow’s temperature
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Probability Distributions Mathematical tools for random events. It has two components: outcomes and their likelihood. Examples:
• Binomial Distribution = X
0 with probability p 1 with probability 1-p
• Standard Normal Distribution
Figure 1: Normal distribution, Source: RiskM etricsT M - Technical Document, p. 69.
1 standard deviation 68.26% probability 2 standard deviations 95.54% probability 3 standard deviations 99.74% probability
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Figure 2: Normal distribution, Source: CreditM etrics
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Figure 3: Normal distribution, Source: CreditM etricsT M -Technical Document, p. 37.
-Technical Document, p. 70.
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A Digression to History In his 1900 dissertation on ”The Theory of Speculation,” Louis Bachelier searched for ”a formula which expresses the likelihood of a market fluctuation. ” He ended up with a mathematical formula that describes the Brownian Motion. In the finance world, Brownian Motion came to be called the random walk, once described as the path a drunk might follow at night in the light of a lamp post. Using the geometric Brownian motion to describe the random fluctuations in stock prices, Fisher Black, Myron Scholes, and Bob Merton worked out the Black Scholes option pricing formula. This work was done in the spring of 1970, when both Merton and Scholes were at MIT Sloan!
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Model of the Behavior of Stock Prices Wiener Processes The change ∆z during a small period of time ∆t is: √ ∆z = ε ∆t
(1)
where ε is a random drawing from a standardized normal distribution N(0,1). The values of ∆z for any two different short intervals of time ∆t are independent. It follows from the first property that ∆z itself has a normal distribution with: mean of ∆z = 0 √ standard deviation of ∆z = ∆t variance of ∆z = ∆t
(2) (3) (4)
The second property implies that z follows a Markov process.1 Consider the increase in the value of z during a relatively long period of time T. This can be denoted by z(T ) − z(0). It can be regarded as the sum of the increases in z in N small time intervals of length ∆t, where N= Thus
T ∆t
N √ εi ∆t z(T ) − z(0) =
(5)
(6)
i=1
where the εi (1, 2, . . . , N ) are random drawing from N(0,1). 1 A Markov process is a particular type of stochastic process where only the present value for a variable is relevant for predicting the future. The past history of the variable and the way that the present has emerged from the past are irrelevant. Stock prices are usually assumed to follow a Markov process. Suppose that the price of IBM stock is $ 100 now. If the stock price follows a Markov process, our predictions should be unaffected by the price one week ago, one month ago, or one year ago. The only relevant piece of information is that the price is now $ 100. Predictions for the future are uncertain and must be expressed in terms of probability distributions. The Markov property implies that the probability distribution of the price at any particular future time is not dependent on the particular path followed by the price in the past. The Markov property of stock prices is consistent with the weak form of market efficiency. This sates that the present price of a stock impounds all the information contained in a record of past prices. If the weak form of market efficiency were not true, technical analysts could make above-average returns by interpreting charts of the past history of stock prices. There is very little evidence that they are, in fact, able to do this. Statistical properties of the stock price history of IBM may be useful in determining the characteristics of the stochastic process followed by the stock price (e.g. its volatility). The point being made here is that the particular path followed by the stock in the past is irrelevant.
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relatively large value of delta t
z
value of delta t relatively large value of delta t
Figure 3: Relatively large value of ∆t
small value of delta t
z
value of delta t small value of delta t
Figure 4: Small value of ∆t
true process
z
value of delta t true process
Figure 5: The true process obtained as ∆t → 0
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From the second property of Wiener processes, the εi ’s are independent of each other, if follows from equation 6 that z(T ) − z(0) is normally distributed with mean of [z(T ) − z(0)] = 0 √ standard deviation of [z(T ) − z(0)] = ∆t variance of [z(T ) − z(0)] = ∆t
(7) (8) (9)
This is consistent with the discussion earlier in this section. Generalized Wiener Process The basic Wiener process, dz, that has been developed so far has a drift rate of zero and a variance rate of 1.0. The drift rate of zero means that the expected value of z at any future time is equal to its current value. the variance rate of 1.0 means that the variance of the change in z in a time interval of length T equals T . A generalized Wiener process for a variable x can be defined in terms of dz as follows dx = a dt + b dz
(10)
where a and b are constants. To understand equation 10, it is useful to consider the two components on the righthand side separately. The a dt term implies that x has an expected drift rate of a per unit of time. With the b dz term, the equation is dx = a dt
(11)
dx =a dt
(12)
x = x0 + a t
(13)
which implies that
or
where x0 is the value of x at time zero. In a period of time of length T , x increases by an amount a T . The b dz term on the right-handed side of euqation 10 can be regarded as adding noses or variability to the parth followed by x. The amount of this noise or variability is b times a Wiener process. A Wiener process has a standard deviation of 1.0. It follows that b times a Wiener proces has a standard deviation of b. In a small time interval ∆t, the change in value of x, ∆x, is from equation 1 and 10, given by √ ∆x = a∆t + ε ∆t 15.433
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where, as before, ε is a random drawing from a standardized normal distribution. Thus ∆x has a normal distribution with mean of ∆x = a∆t √ standard deviation of ∆x = b ∆t variance of ∆x = b2 ∆t
(15) (16) (17)
Similar arguments to those given for a Wiener process how that the change in the value of x in any time interval T is normally distributed with mean of ∆x = aT √ standard deviation of ∆x = b T variance of ∆x = b2 T
(18) (19) (20)
value of variable, x
generalized Wiener process: dx = a dt+ b dz
dx = a dt
Wiener process: dz
time
Figure 6: Wiener processes
Thus, the generalize Wiener process given in equation 10 has an expected rift rate (i.e. average rift per unit of time) of a and a variance rate (i.e., variance per unit of time) of b2 . It is illustrated in Figure (6). Process for Stock Prices 15.433
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It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected rift rate and a constant variance rate, However, this model fails to capture a key aspect of stock prices. This is that the expected percentage return required by investors from a stock is independent of the stock price. If investors require a 14% per annum expected return when the stock price is $ 10, then ceteris paribus, they will also require a 14% per annum expected return when it is $ 50. Clearly, the constant expected rift-rate assumption is inappropriate and needs to be replaced by the assumption that the expected return (that is, expected rift divided by the stock price) is constant. If S is the stock price at time t, the expected drift rate in S should be assumed to be µS for some constant parameter, µ. This means that in a short interval of time, ∆t, the expected increase in S is µS∆t. The parameter, µ, is the expected rate of return on the stock, expressed in decimal form. If the volatility of the stock price is always zero, this model implies that ∆S = µS∆t
(21)
dS = µSdt
(22)
dS = µdt S
(23)
ST = S0 eµT
(24)
in the limit as ∆t → 0
or
so that
where S0 and ST are the stock price at time zero and time T . Equation 24 shows that when the variance rate is zero, the stock price grows at a continuously compounded rate of µ per unit of time. In practice, of course, a stock price does exhibit volatility. A reasonable assumption is that the variability of the percentage return in a short period of time, ∆t, is the same regardless of the stock price. In other word, an investor is just as uncertain of the percentage return when the stock price is $ 50 as when it is $ 10. This suggests that the standard deviation of the change in a short period of time ∆t should be proportional to the stock price and leads to the model dS = µSdt + σSdz
(25)
dSS = µdt + σdz.
(26)
or
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Equation 26 is the most widely used model of stock price behavior. The variable σ is the volatility of the stock price. the variable µ is its expected rate of return. The discrete-time version of the model is √ ∆S = µ∆t + σε ∆t S √ ∆S = µS∆t + σSε ∆t
(27) (28)
The variable ∆S is the change on the stock price, S, in a small interval of time, ∆t, and ε is a random drawing from a standardized normal distribution (i.e.g, normal distribution with a mean of zero and standard deviation of 1.0). The parameter, µ, is the expected rate of return per unit of time from the stock and the parameter, σ, is the volatility of the stock price. Both of these parameters are assumed constant. The left-hand side of equation 27 is the return provided by the stock in a short period √ of time, ∆t. The term µ∆t is the expected value of this return, and the term σε ∆t is the stochastic component of the return. The variance of the stochastic component (and therefore, of the whole return) is σ 2 ∆t. This is consistent with the definition of √ the volatility, σ. That is, σ is such that σ ∆t is the standard deviation of the return in a short time period, ∆t. Equation 27 shows that ∆S/S is normally distributed with mean µ∆t and standard √ deviation σ ∆t. In other words, √ ∆S ≈ N µ∆t, σ ∆t (29) S
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Why Normals? Model the random fluctuation of stock prices using geometric Brownian motion. Implication for stock returns: normal distribution (for continuously compounded returns). The annualized stock return is normal with mean (µ) and standard deviation (σ) . For the S&P 500 index returns, µ is roughly 12%, and σ is roughly 15%. The latter is also called volatility. Fix a time horizon, say ∆t. The stock return over ∆ is normally distributed with √ mean µ∆t and standard deviation σ ∆t. What is the distribution of daily returns?
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Events that are not Normal A negative surprise: on October 19, 1987, the S&P 500 index dropped more than 23% on one day. A positive surprise: on January 3, 2001, the Nasdaq composite index gained more than 14% on one day. Suppose we use normal distribution to characterize daily stock returns. What are the probabilities of such surprises?
A Positive Surprise
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Figure 4: Nasdaq Dec 1 to Jan 18 2001, jumps on January 2nd and 3rd
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The Probability of a Crash Let r denote the daily return, which is:
• normally distributed with • mean 0.12/252 = 0.00048, √ • standard deviation 0.15 · 252 = 0.0094
What is the probability of an ’87 crash? Prob (r < 0.23) =?
• First, convert r to a standard normal X=
r − 0.00048 0.0094
(30)
• Second, convert the critical value for r to that for X: −0.23 − 0.00048 ≈ −23 0.0094
(31)
• Finally, knowing that X is a standard normal, P rob(X < −23) = N (−23) = 10−127 !
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What Normal Fails to Capture...
There are large movements (both up and down) in stock prices that cannot be captured at all by the normal distribution. In mathematical terms, the tail distribution of a normal random variable is too thin. Historical stock returns exhibit fat tails. If we make financial decisions based on normal distribution, we will miss out on the large movements. The consequences are catastrophic! This is especially important for leveraged investments over a short time horizon. Tail fatness is also an important issue in risk management.
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6/14/1998
6/14/1997
6/14/1996
6/14/1995
6/14/1994
6/14/1993
6/14/1992
6/14/1991
6/14/1990
6/14/1989
6/14/1988
6/14/1987
6/14/1986
6/14/1985
Index Level
Data Analysis
6,000
5,000
4,000
3,000
2,000
1,000
-
CCMP
Figure 7: SP 500 and Nasdaq Index, index points, source: Bloomberg Professional.
40%
SPX
30%
20%
10%
-10% 0%
-20%
-30%
SPX
Figure 8: SP 500 daily returns, source: Bloomberg Professional
MIT Sloan
30% 25%
Probability
20% 15% 10% 5% 0% -5% -0.080
-0.060
-0.040
-0.020
-
0.020
0.040
Daily Returns Current Distribution
Normal-Distribution
0.060
0.080
Preliminaries for Data Analysis When given the ”raw data,” first look for trends. If there are any, the first step is always to de-trend the data. Why? The i.i.d. assumption for r1 , r2 , . . . , rN : returns are independent and identically distributed. The longer we observe, the more we know about the probability distribution . . . but do not forget structural changes!
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Empirical Distribution 1. Sort the outcomes r1 , r2 , . . . , rN 2. Denote the minimum by x and the maximum by x. Divide [x, x] evenly into K
bins: bin 1: [x, x + ∆x]
bin 2: [x + ∆x, x + 2∆]
... bin 3: [x − x] , x
where ∆x = (x + x) /K
3. Fixing a number K, count the number Nk of the ri ’s that fall to the k-th bin. 4. Repeat this for k = 1, 2, ..., K, we get a list of bins and their respective ”likelihood” Nk . 5. Finally, we need to renormalize the likelihood so that the probability of r i falling into [x, x] is one.
30% 25%
Probability
20% 15% 10% 5% 0% -5% -0.080
-0.060
-0.040
-0.020
-
0.020
0.040
0.060
0.080
Daily Returns Current Distribution
Normal-Distribution
Figure 10: Daily return distribution of S&P 500 index. source: Bloomberg Professional
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Fat Tails 5% 4%
Probability
3% 2% 1% 0% -1% -0.080
-0.070
-0.060
-0.050
-0.040
-0.030
-0.020
-0.010
-
0.010
Daily Returns Current Distribution
Normal-Distribution
Figure 11: Left fat tail of daily Return distribution of S&P 500 index, 5% on left hand side.
5% 4%
Probability
3% 2% 1% 0% -1% -0.010
-
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
Daily Returns Current Distribution
Normal-Distribution
Figure 12: Right fat tail of daily Return distribution of S&P 500 index, 5% on right hand side
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Sample Statistics mean
N 1 ri N i=1
(33)
N 1 (ri − µ)2 N i=1
(34)
µ=
variance: σ2 =
skewness (lack of symmetry): skew =
1 N
kurtosis (peakedness): kurt =
15.433
1 N
N
(ri − µ)3 √ σ3
(35)
(ri − µ)4 σ2
(36)
i=1
N
i=1
22
MIT Sloan
Standard Errors Take the sample mean µ as an example: mean: µ =
1 N
N
i=1 ri
We know that ri ’s are random draws from a stationary (and ergodic2 ) distribution. In fact, to simplify our analysis, we assumed that they are i.i.d. This implies that the sample mean µ is itself a random variable.
• What is its mean?
• What is its standard deviation?
Standard errors: measure the precision of the estimators.
2 ergodic: Of, pertaining to, or possessing the property that in the limit all points in a space are covered with equal frequency, or that each sufficiently large selection of points is equally representative of the whole. [Oxford English Dictionary]
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Conditional Version So far, we assumed that the distribution of stock returns stays the same throughout history. We calculate the sample statistics as if history repeats itself with the same probability law. But we know that this cannot be true. How do we use the data when we have reasons to believe that the probability law changes over time? For example, suppose we believe that the daily returns within each month t are normally distributed with mean µt and standard deviation σt , which vary over time t. The simplest way to take the conditional information into account is to calculate the sample mean and standard deviation month by month.
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Time Series Patterns While the unconditional version of sample statistics gives us a static picture of daily stock returns, the conditional version provides more dynamic information. For example, • The conditional expected returns are time varying, but there is not much persistence. • The conditional volatilities are also time varying. Moreover, they seem to be highly persistent. • There is a negative relationship between returns and volatility: when the market goes down, the volatility goes up. We will revisit these issues in more detail in Class 9.
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Summary We use probability distributions to characterize and evaluate random events. There is a long tradition of using normal distributions to characterize the fluctuations in stock prices. The normal distributions, however, are not adequate to capture large surprises. The empirical distribution and sample moments are useful statistical tools to extract information from the data. Stationarity is one important assumption. The precision of the sample moments can be measured by their standard errors. The sample statistics can be used in both unconditional and conditional versions. The conditional version provides more dynamic information about the data.
Focus: BKM Chapters 3 & 5 (Chapter 3: This is all general stuff, you need to now the basic things like IPO, private placements, secondary market etc.); • p. 137 (probability distribution, standard deviation); • p. 141 (figure 5.4); • p.149 bottom and 150 (continuous compounding); Reader: Fama (1995). Type of potential questions: chapter 3 concept check question 2 3, p. 98 ff. question 2, 5, 11, 17, 22 chapter 5, p. 146 ff. question 10, 12, 15
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Questions for Next Class Please read: • BKM Chapters 6 and 7, • Elton and Gruber (2000), and • Kritzman (1992)
Think about the following questions: • Two important components in making an investment decision: the investment opportunity and the investor. • Admittedly, one investor may differ from another. And the investment opportunity does not stay constant across space or time. • If you were asked to build a model of investments for a generic investor on a generic market environment, what are the basic features you would include in your model?
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15.433 INVESTMENTS
Class 3: Portfolio Theory
Part 1: Setting up the Problem
Spring 2003
A Little History
In March 1952, Harry Markowitz, a 25 year old graduate student from the University of Chicago, published ”Portfolio Selection” in the Journal of Finance. The paper opens with: ”The process of selecting a portfolio may be divided into two stages. The first stage starts with observation and experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of portfolio”. Thirty eight years later, this paper would earn him a Nobel Prize in economic sci ences.
Introduction Two basic elements of investments:
• the investment opportunity;
• the investor.
Our task for this class:
• a model for financial assets; • a model for investors; • optimal portfolio selection.
Modelling Financial Returns
Virtually all real assets are risky. Financial assets, claims on real assets, bear such risk:
• some are designed to minimize risk • some are designed to capture risk
25%
20%
15%
Daily returns (ln)
10%
5% 0%
-5%
-10%
-15%
6
10
/4
/1
99
96
96
19
19 4/
7/
4/
4/
5
96
99
4/ 1/
/1 /4
10
19
95
95
19 4/
19 4/ 4/
7/
4 1/
4/
19
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94 10
/4
/1
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4/
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3
19 4/
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/1
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93
93
19 4/ 7/
19 4/ 4/
1/
4/
19
93
-20%
30%
30%
25%
25%
20%
20% Probability
Probability
Figure 1: Return of Mexican Peso, Source: Bloomberg Professional.
15% 10%
15% 10%
5%
5%
0%
0%
-5% -0.080
-0.060
-0.040
-0.020
-
0.020
0.040
0.060
Daily Returns
Current Distribution
Normal-Distribution
0.080
-5% -0.080
-0.060
-0.040
-0.020
-
0.020
0.040
0.060
Daily Returns
Current Distribution
Normal-Distribution
Figure 2: Return of S & P 500 Index,
Figure 3: Return of 10 Year Treasury Bills,
Source: Bloomberg Professional.
Source: Bloomberg Professional.
0.080
Modelling Investors
Overall, investors are risk averse, although some are more so than the others. ”We next consider the rule that the investor does (or should) consider expected return a desirable thing and variance of return an undesirable thing”. - Markowitz (1952). Heterogeneity of investors: • individual investors vs. corporations • investors with different marginal tax rates • informed vs. uninformed • young vs. old • Behavior issues: loss aversion, mental accounting, over confidence, over reaction, under reaction, etc.
Choose A or B
A:
B:
�
$240� 000 with probability 100%
�
$1� 000� 000 with probability 25% $0
with probability 75%
Choose C or D C:
D:
�
�
−$750� 000 with probability 100%
$0
with probability 25%
−$1 000 000 with probability 75% �
�
Equivalent Choices:
A+D :
�
B+C :
�
$240� 000 with probability 25% −$760� 000 with probability 75%
$250� 000 with probability 25% −$750� 000 with probability 75%
Setting up the Problem What do we need . . . a recipe and some ingredients. The investment opportunity:
• riskfree rf = 7% • risky rp : E (rp ) = 15%, std (rp ) = 22%. BKM,
A mean-variance investor:
157 ff.
1
U (r) = E(r) − 0.005 · A · var (r)
(1)
The optimal portfolio selection:
• invest a portion y of the total wealth in the risky asset, leaving the rest in the riskfree account
• possible portfolios: ry = (1 − y) · rf + y · rp
• the optimal portfolio? max y∈R
U (ry )
where R stands for the space of real numbers.
The coefficient 0.005 is in the literature as well written as calibrate the subjective risk aversion coefficient A.
1
1
2.
It is a calibration coefficient to
p.
Portfolio Construction The opportunity set is fixed: rf and rp
Our only choice variable: y [how much to invest in risk portfolio]
The end product: ry = (1 − y) · rf + y · rp
E (ry ) = E ((1 − y) · rf ) + E (y · rp )
(2)
(3)
= (1 − y) · 0.07 + y · 0.15
(4)
= 0.07 + 0.08 · y
(5)
var (ry ) = var ((1 − y) · rf ) + var (y · rp ) +2 · cov ((1 − y) · rf , y · rp )
(6)
= 0 + y 2 · 0.222 + 0
(7)
= 0.222 · y 2
(8)
std (ry ) =
�
var (ry ) = 0.22|y|
(9)
The Risk Return Combinations
Every choice of y gives rise to one pair of return E and risk std.
• For y � 0, we have: y=
E (ry ) − 0.07 std (ry ) = 0.08 0.22
(10)
• More generally, we have, for any y ¿ 0: y=
E (ry ) − rf std (ry ) =
std (rp ) E (rp ) − rf
(11)
y may vary over the entire positive real line, but this relation holds regardless. A linear relation between E and std: E (ry ) − rf =
E (ry ) − rf std (ry ) std (rp )
(12)
The Capital Allocation Line
Collecting all y ∈ R,we get all of the risk-return (µ, σ) combinations available to in vestors.
E(r)
E(rP )
rf
CAL
P
Line) n o i t a Alloc l a t i (Cap
E(rp) rf
S
σ
σP
Figure 4: Capital Allocation Line
e)
weight y
CA
P
L(
Ca
pit
al A
llo
cat
ion
Lin
wy
E(rP ) σP 0 0
rf
Figure 5: Capital Allocation Line, a different view.
σ
The Sharpe Ratio
One measure of the attractiveness of a portfolio r is its Sharpe Ratio (S): Intuitively, S measures extra return per extra risk. S=
E (r) − rf std (r)
(13)
Recall that the CAL can be re-written as: E (ry ) − rf =
E (ry ) − rf std (ry ) std (rp )
(14)
For the extra risk std (ry ) chosen (through y), the extra reward is: Sp · std(ry ). Moreover, the Sharp Ratio Sy of any portfolio thus constructed from rf and rp is the same: Sy = S p =
E (ry ) − rf std (rp )
for any y � 0. Does that make sense to you?
(15)
Forming the Optimization Problem We are now ready to ”feed” our portfolio to the optimization machine:
max y ∈ R
U (ry )
(16)
where
U (r) = E(r) − 0.005 · A · var (r)
(17)
From our earlier derivation, we know that: E(ry ) = 0.07 + 0.08 · y; var(ry ) = 0.222 · y 2
(18)
Our optimization problem therefore becomes max f (y):
max y∈R
U (ry )
f (y) = 0.07 + 0.08 · y − 0.0005 · A · y 2
(19)
The Optimization Machine Three components of an optimization problem: • the objective function f (y); • the variable y; and • the search space R
Three ways to solve an optimization problem: • analytical; • numerical; and • graphical.
The mathematical foundation: • let y* be the solution of f � (y) = 0; • if f �� (y∗) < 0, then y* is truly the optimal solution.
A Pictorial Solution
f (y) 0
0
f ' (y)
y
optimal portfolio weight y*
0 0
0 optimal portfolio weight y*
Figure 6: Optimal portfolio weight
y
An Analytical Solution
The risk aversion coefficient is set at A = 4 and the optimal weight is y*=0.41. Let’s take some derivatives: f � (y) =
∂f (y) = 0.08 − 0.222 · A · y ∂y
(20)
∂f � (y) = −0.222 · A ∂y
(21)
f �� (y) =
1. look for y∗ that satisfies f � (y∗) = 0: y∗ =
0.08 0.222 · A
2. check the optimality of y∗: f ”(y∗) < 0?
(22)
Determinants of Portfolio Weights
More generally, the optimal solution can be expressed as E (rp − rf ) y∗ = var (rp ) · A
(23)
A more risk-averse investor (with a larger risk-aversion coefficient A) will invest less in the risky asset.
least risk aversion Utility
medium risk aversion
most risk aversion
0
portfolio weight y
y
Figure 7: Utility-function.
If the risk premium, E (rp ) - rf , of the original risky asset decreases, a risk-averse investor will reduce his holdings in the risky asset accordingly. If the original asset is risky (with var(rp ) > 0), but pays zero risk premium, then no risk-averse investor will hold the risky asset. If the risk premium is negative, a risk-averse investor will start shorting the asset. What is the optimal portfolio weight y* of an investor with the following utility func tion? HINT: This investor cares only about the Sharpe Ratio.
Going Beyond
Our setup assumes the following: 1. A mean variance investor; 2. Investment horizon is fixed to one year; 3. No dynamic rebalancing in between.
Of course, this setup is a very rough characterization of the real investment problem. Nevertheless, this example is valuable: • First, it provides a framework for us to think about the portfolio optimization problem. • Second, although simple, it provides rich intuition.
Now we can go beyond to the next step.
Three Extensions
1. The Skewness Extension: allow skewed asset returns and add preference for posi tive skewness and aversion to negative skewness. 2. The Horizon Extension: allow investment horizon to vary. 3. The Dynamic Extension: allow for dynamic rebalancing.
Leisure Readings ”Fourteen Pages to Fame,” Chapter 2 of Capital Ideas by Peter Bernstein.
Focus: Chapters 6 & 7: • p.157 (eq. 6.1) • p. 161 • p. 163 to 166 • p. 188 • p. 191 to 195 (utility function, utility curves, CAL) Reader: Kritzman (1992) type of potential questions: Chapter 6 concept check question 3 & 4, p. 168 ff. ques tions 2, 9, 10 chapter 7 concept check question 2, 3, 4 & 5, p. 200 ff. questions 4, 8, 13.
Questions for the Next Class
Please read BKM Appendix A, B of Chapter 6, Black (1995) and Kritzman (1992). What about market crashes? Can event risks small probability but high impact events ever be ignored in making an investment decision? What do you think of BKM’s defense of mean variance analysis? What is the ma jor assumption in Paul Samuelson’s proof? Is this assumption realistic?
15.433 INVESTMENTS Class 4: Portfolio Theory Part 2: Extensions
Spring 2003
Introduction
• Should someone with longer investment horizon put more in the stock market?
• Is there value in dynamic re-balancing?
• How do market crashes affect investment behavior above and beyond their impact on mean and variance?
One Day in LTCM
On Aug. 17, 1998, Russia defaulted on its local debt. Friday August 21 at LTCM, Greenwich, CT: ”When he saw the quotation for U.S. swap spreads, he stared at his screen in dis belief. On an active day, U. S.’ swap spreads might change by as much as a point. But on this morning, swap spreads were wildly oscillating over a range of 20 points. ”That Friday, Long? Term lost money wherever it looked. Credit spreads simply exploded. ? Though these moves may seem small in absolute terms, the effect on Long? Term was magnified by the fund’s potent level of leverage and its immense position size.” ”LongTerm, which had calculated with such mathematical certainty that it was unlikely to lose more than $35 million on any single day, had just dropped $553 million on that one Friday in August.” ”When Genius Failed” by Roger Lowenstein
Fat Tails
5% 4%
Probability
3% 2% 1% 0% -1% -0.080
-0.070
-0.060
-0.050
-0.040
-0.030
-0.020
-0.010
-
0.010
Daily Returns
Current Distribution
Normal-Distribution
Figure 1: Left fat tail of daily return distribution of S&P 500 index.
5% 4%
Probability
3% 2% 1% 0% -1% -0.010
-
0.010
0.020
0.030
0.040
0.050
0.060
Daily Returns
Current Distribution
Normal-Distribution
Figure 2: Right fat tail of daily return distribution of S&P 500 index.
n
0.070
0.080
Quantify Fat Tails Skewness
Skewness
Kurtosis
E (x − E (x))3 skew (x) = std (x)3
(1)
E (x − E (x))4 kurto (x) = std (x)4
(2)
0.1 0.0 -0.1 -0.5
-0.3
Skewness / Normal
0.2
0.3
0.4
0.5
What is the skewness of a standard normal?
-5
-4
-3
-2
-1
0
1
2
3
4
xx
Figure 3: Skewness of a standard normal distribution, splus.
What is the kurtosis of standard normal?
Caution: skewness and kurtosis might not exist for some random variables.
55
A Model with Event Risk Daily Return r:
r = x + y
x is the ”normal” component:
y is the ”jump” component:
y =
x and y and are independent.
�
J with probability p 0 with probability 1-p
Moments of the Crash Model
mean E (r) = µ + J · p
(3)
var (r) = σ 2 + J 2 · p · (1 − p)
(4)
variance:
skewness: skew (r) =
J 3 · p · (1 − p) · (1 − 2p) � 3 var (r2 )
(5)
The Skewness of the Crash Model
Model
P
J
skew
once a month
1/12
-2%
-1.0
once a year
1/365
-10%
-5.6
1/365/100
-50%
-7.1
once a lifetime
Each jump model is calibrated so that the annualized expected return is 12%, and the annualized volatility is 15%. That is, the returns from the three models are equally attractive to a mean-variance investor!
Skewness Preference A utility function that captures the risk attitude of an investor in three ways:
1 1 U (r) = E (r) − · A · var (r) + · B · E (r − E (r))3 2 6
1. the expected return is desirable;
2. the variance of return is undesirable (A > 0);
3. positive skewness is desirable, negative skewness is not (B > 0).
If we set B = 0, we are back to a mean-variance investor.
(6)
Modifying the Original Problem Two modifications of our original problem:
1. Investor has a preference for skewness; 1 1 U (r) = E (r) − · A · var (r) + · B · E (r − E (r))3 2 6
(7)
2. The risky asset rP is negatively skewed:
skew = -2 The optimization problem: max y ∈ R
U (ry )
(8)
Just as before, rP is the portfolio return: ry = (1 − y) · rf + y · rp
(9)
Formalizing the Problem
As before, we calculate the utility U (ry ) of an investor who invests a fraction y of his wealth in the risky asset rp , leaving the rest in the riskfree account rf : ry = rf + y · (rp − rf )
(10)
E (ry ) = 0.07 + 0.08 · y; var (ry ) = 0.222 · y 2
(11)
Two old pieces of U (ry ):
The new piece of U (ry ): E (ry − E (ry ))3 = y 3 · Ep − E((rp )3
(12)
E (x − E (x))3 = skew (x) · std (x)3
(13)
E (ry − E (ry ))3 = 0.223 · (−2) · (y)3
(14)
Recall that by definition
we therefore have:
The Optimization Machine
Armed with all three pieces of U (ry ), we write our optimization problem in mathemat ical terms: max y ∈ R
U (ry )
1
1 f (y) = 0.07 + 0.08 · y − · A · 0.222 + · B · 0.223 · (−2) · y 3 6 2
In our original problem, f (y) is a linear-quadratic function of y.
What about now?
(15)
(16)
Optimization: Analytical Method The mathematical foundation:
• let y∗ be the solution of f � (y) = 0
• let f �� (y∗) < 0, then y* is truly the optimal solution.
Let’s take some derivatives: f � (y) = 0.08 − 0.222 · A · y +
f �� (y) = −0.222 · A +
1 · 0.223 · (−2) · B · y 2 2
1 · 0.223 · (−2) · B · y 2
(17)
(18)
What do you think? How do we look for the optimal y*? Does it even exist?
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Optimization: A Pictorial Solution
f(y) 0
f ' (y)
0
portfolio weight y
0
portfolio weight y
0
portfolio weight y
0 ?
f '' (y) 0 ?
Figure 4: Risk aversion and optimal weight.
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The variance aversion coefficient is set at A = 4, and the skew preference is set at B = 5. The optimal portfolio weight is y* = 0.37. Recall that, given the same choice set (rp and rf ), a mean-variance investor with A = 4 will invest y* = 0.41.
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Determinants of Optimal Allocation
What we already know:
1. a more variance-averse investor tends to invest less in the risky asset: A ↑ ⇒ y ∗ ↓
2. given a higher risk premium, a risk-averse investor will invest more in the risky asset: (E(rp ) − rf ) ↑ ⇒ y ∗ ↑
3. given the same risk premium, if the risky asset becomes more volatile, a risk-averse investor will reduce his holdings in it: std(rp ) ↑ ⇒ y ∗ ↓
New Results An investor who is more averse to negative skewness tends to invest less in a negatively skewed risky asset: B ↑ ⇒ y ∗ ↓
Given the same risk premium and volatility, if a risky asset becomes more negatively
skewed, an investor with skewness preference will reduce his holdings: skew(r p ) ↓ ⇒ y ∗ ↓
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The Horizon Effect
To study the horizon effect, we need to understand how stock returns aggregate over time. Let’s start with the simplest model. r1 = µ + σ · ε 1 r2 = µ + σ · ε 2 r3 = µ + σ · ε 3 ... The daily shocks t are independent and standard normally distributed.
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Focus: BKM Appendix A, B of Chapter • 6 p. 172 to 177 middle (probability, distribution, skewness [no equation], normal distribution) • p. 178 middle to 181 middle (utility, utility function) Reader: Black (1995) type of potential questions: -
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Questions for Next
Please read: • BKM Chapter 8, • Kritzman (1994), and • Kritzman (1991)
Think about the following questions:
• What do we mean by diversification?
• When does diversification work, when doesn’t it?
• Can you think of other real life examples (non?financial) when the principle of diversification is used?
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15.433 INVESTMENTS Class 5: Portfolio Theory Part 3: Optimal Risky Portfolio
Spring 2003
Introduction
• Having determined the appropriate exposure to risk, the investor’s next task is to build his risky portfolio rp .
• This selection will be made from the whole universe of risky assets that are available for investment.
How Big is the Universe of Risky Assets?
We live in an exciting time, with an expansive investment environment. Here is a list c of the plain-vanilla instruments reported by RiskMetrics , source : RiskM etricsT M −
T echnical Document Equity Indices: 31 countries.
Foreign Exchange Rates: 31 currencies.
Money Markets around the world: 111.
Swaps around the world: 121.
Gov’t Bonds around the world: 153.
Commodities: 33.
Over 400 different instruments! This does not include individual equities, mutual funds, venture capital funds, futures, options, and other derivatives ... .
Two Risky Assets
1. r1
µ1 = 13%, σ1 = 20%
2. r1
µ2 = 8%, σ2 = 12%
3. correlated returns : ρ = corr (r1 , r2 ) = 30%.
Mixing the Risky Assets • a fraction of w in equity
• a fraction of 1 - w in debt
• the risky portfolio rp = w · r1 + (1 − w) · r2
wy
wi 0
P E(rP) σP 0 0
rf
Figure 4: Efficient frontier.
σ
The Risky Portfolios
Let’s first calculate the mean and variance of the risky portfolio:
µP = w · µ1 + (1 − w) · µ2
(1)
σp2 = w2 · σ12 + (1 − w)2 · σ2 2 + 2 · w · (1 − w) · cov (r1 , r2 )
(2)
= w2 · σ12 + (1 − w)2 · σ22 + 2 · w · (1 − w) · σ1 σ2 ρ1,2
(3)
Spanning the mean-std space:
µi
risky P1 o
mean µ (%)
µ1
µ2
risky P 2 o
σ2
0 0
Figure 5: Efficient frontier.
σ1 std σ (%)
σi
The Optimal Portfolio Problem
the investment opportunity • one riskfree rf • two risky assets r1 and r2 a mean-variance investor U (r) = E (r) −
1 1 · A · var (r) + · B · E (r − E (r))3 2 6
(4)
investment decisions:
1. choose the overall exposure to risk:
y in the risky portfolio rp and 1 − y in rf .
2. choose the right risky portfolio rp :
w in r1 , and 1 − w in r2 .
possible portfolios: ry,w = (1 − y) · rf + y · (w · r1 + (1 − w) · r2 )
(5)
the optimal portfolio: max y ∈ R, w ∈ R
U (ry,w )
(6)
The Separation Principle
In addition to the choice variable y, which controls the investor’s overall exposure to risk, our current problem also involves the choice variable w, which controls the right mix of the risky assets. The Complication: Need to solve y and w simultaneously in the optimization problem: max y ∈ R, w ∈ R
U (ry,w )
(7)
Solution: The Separation Principle (James Tobin 1958): • the choice y of the overall risk exposure is investor specific, depending on his degree of risk aversion; • the choice w of the optimal risky portfolio is the same for all investors.
A Brief Review of CAL
mean µ (%)
µi
rf o
0 0
std σ (%)
σi
Figure 6: Efficient frontier.
Pick any risky portfolio rp , invest a fraction y in it, leaving the rest in the riskfree account rf . What are the possible (E-Std) combinations? E (ry ) − rf =
E (ry ) − rf std (ry ) std (rp )
(8)
A Brief Review of the Sharpe Ratio
µi
risky P1
o
mean µ (%)
µ1
rf o
risky P2
o
0 0
std σ (%)
σi
Figure 7: Efficient frontier.
One measure of the attractiveness of a portfolio r is its Sharpe Ratio: S=
What is its graphical interpretation?
E (r) − rf std (r)
(9)
The Optimal Risky Portfolio
µi risky P1 o
mean µ (%)
µ1
rf o µ 2
o
σ2
0 0
risky P2
σ1 std σ (%)
σi
Figure 8: Efficient frontier.
Each CAL is uniquely identified by its slope. The best CAL is the one with the steepest slope, or the highest Sharpe Ratio. The Optimal Risky Portfolio is the Tangency Portfolio: the unique portfolio with the highest Sharpe-Ratio. Conceptually, it is very important to notice that the definition of the Optimal Risky Portfolio does not involve the degree of risk aversion of any individual investor. In such an ideal world, every investor, regardless of his level of risk aversion, will agree on the best CAL, and allocate his wealth between rf and the optimal risky portfolio. The portion y invested in the optimal risky portfolio, however, will depend on each investor’s degree of risk aversion, as discussed in Class 3. In practice, however, different investors might have very different ideas about their Optimal Risky Portfolio. Why?
Multiple Risky Assets
The opportunity set generated by the two risky assets is relatively simple. As we generalize to the case of multiple risky assets, the opportunity set becomes considerably more complicated. Suppose that we have n securities, whose random returns are denoted by r1 , r2 , . . . , rn As usual, we construct portfolios by mixing these n securities: rp = w 1 · r 1 + w 2 · r 2 + · · · + w n · r n
(10)
where the w’s are the portfolio weights, adding up to 1. Each chosen w gives rise to an investment opportunity with
E (rp ) = var (rp ) =
n �
wi · E (ri )
i=1 n � n �
(11)
wi · wj · cov (ri , rj )
(12)
i=1 j=1
covi,j = covi,j =
n � 1 (ri − r¯i ) · (rj − r¯j ) n i=1 n �
pt · (ri − r¯i ) · (rj − r¯j )
equal weighted
(13)
probability weighted
(14)
i=1
This results in an enormous degree of freedom, and a very rich opportunity set. Of course, not all portfolios in the opportunity set are good deals. Constructing the mean?variance frontier: an efficient set of portfolios that achieve the lowest possible risk for any given target rate of return.
The Optimal Portfolio in Practice
Implication of the Separation Principle: A portfolio manager will offer the same risky portfolio to all clients. In practice, different managers focus on different subsets of the whole universe of financial assets, derive different efficient frontiers, and offer different ”optimal” portfolios to their clients.Why? The theory of portfolio selection builds on many simplifying assumptions: • No Market Frictions (tax, transactions costs, limited divisibility of financial assets, market segmentation). • No Heterogeneity in Investors (e.g. rich vs. poor, informed vs. uninformed, young vs. old). • Static expected returns and variance - no forecastability in returns or volatility (e.g. financial analysts, accounting information, macro?economic variables do not play any role in making an investment decision).
Our next step toward reality (in Classes 19-20): link the following two bodies of ideas, • Security Analysis: subjective, judgmental • Portfolio Selection: objective, statistical.
Some important questions to think about: • Can security analysis improve portfolio performance? • How do analysts’ opinions enter in security selection?
A Digression to Diversification
Diversification is a universal concept. Put in simple terms, one should not put all eggs in one basket. In investments, diversifying means holding similar amounts of many risky assets instead of concentrating all of your investment in only one. In social science, it is believed that individuals are of ”bounded rationality”. One way to mitigate such a problem is through decision-making by multiple agents. For example, in gymnastics or figure skating competitions, the score is averaged over multiple judges, after taking away extremes on either end. The mathematical foundation of diversification - the strong law of large numbers!
σε2p
=
n � �2 � 1 i=1
βp
n
BKM,
σε2i
1� = ·βi n i=1
1 = σε2 n
299 f.
(15)
n
1� = ·αi n i=1
(16)
n
αp
1� = ·εi n i=1
(17)
n
εp
2 σp2 = βp2 σM + σε2p
(18) (19)
p.
Leisure Readings
”The Interior Decorator Fallacy” Chapter 3 of Capital Ideas by Peter Bernstein.
Focus: BKM Chapter 8 • p. 210-213 (eq. 8.2, eq. 8.4, table 8.2) • p. 217 middle to 229 (Markowitz, efficient frontier, mean variance) • p. 234-239 (separation property, CAL) Reader: Kritzman (1994) type of potential questions: concept check question 1, 2, 3, p. 286 ff. questions 5, 11, 18
Questions for the Next Class
Please read BKM Chapters 9-11, Roll and Ross (1995) and Kritzman (1991), and think about the following questions: The separation principle implies that every investor, regardless of his degree of risk aversion, will hold the same optimal risky portfolio. What implication does this result have on the entire market? Suppose there are I investors in the market. Each investor i has? a risk aversion coefficient Aj , with optimal exposure to the optimal risky portfolio. yi∗ =
E (rp ) − rf var (rp ) · Ai
(20)
In equilibrium, adding yi∗ across all investors, what do we get?, (Hint: In equilibrium, supply equals demand, e.g., the amount of borrowing equals that of lending.)
15.433 INVESTMENTS Classes 6: The CAPM and APT Part 1: Theory
Spring 2003
Introduction
So far, we took the expected return of risky asset as given. But where does expected return come from? Using the intuition that investors are risk averse, one explanation is that the risk premium - expected return in excess of the riskfree rate - is a reward for bearing risk. Does this make sense? The Capital Asset Pricing Model (CAPM) provides a simple, yet elegant framework for us to think about the question of reward and risk.
”The CAPM ”
In market equilibrium, investors are only rewarded for bearing systematic risk - the type of risk that cannot be diversified away. They should not be rewarded for bearing idiosyncratic risk, since this uncertainty can be mitigated through appropriate diversification.
Sharpe on CAPM Bill Sharpe, one of the originators of the CAPM, in an interview with the Dow Jones Asset Manager:
”But the fundamental idea remains that there’s no reason to expect reward just for bearing risk. Otherwise, you’d make a lot of money in Las Vegas. If there’s reward for risk, it’s got to be special. There’s got to be some economics behind it or else the world is a very crazy place. I don’t think differently about those basic ideas at all”. - Sharpe (1998)
Assumptions 1. Perfect Markets • Perfect competition - each investor assumes he has no effect on security prices • No taxes • No transactions costs • All assets publicly traded, perfectly divisible • No short-sale constraints • Same riskfree rate for borrowing and lending
2. Identical Investors • Myopic1 • Same holding period • Normality or Mean-Variance Utility • Homogeneous expectations
1
myopic: adj. Of, pertaining to, or affected with myopia; short-sighted, near-sighted
The Equilibrium Market Portfolio
Recall that every investor holds some combination of the riskless asset and the tan gency portfolio.
mean µ (%)
µi CM
µΜ
L
Market o
rf o
σΜ
0 0
std σ (%)
σi
Figure 1: Equilibrium market portfolio and efficient frontier.
When we aggregate the portfolios of all individual investors, lending and borrowing will cancel out, and the value of the aggregated risky portfolio will equate the entire wealth of the economy. The tangent portfolio has become the equilibrium market portfolio.
The Market Price of Risk There are N mean-variance investors in the economy, each with $1.
Investor Risk Aversion
Portfolio Weight µM −rf 2 A σM 1 µM −rf 2 A σM 2 µM −rf 2 A σM 3
1
A1
2
A1
3
A1
...
...
...
AN
µM −rf 2 A σM N
N
Aggregating over all Investors, the total wealth invested in the market portfolio is:
µM − r f $1 · σM
�
1 1 1 +
+ . . . +
A1 A2 AN
�
(1)
In equilibrium, the total wealth invested in the market portfolio must be:
$1 · N
(2)
2 ¯ µM − r f = σ M A
(3)
This implies:
where is an overall measure of the risk aversion among the market participants:
� � 1 1 1 1 1 = ·
+
+ . . . +
AN A1 A2 N A¯
(4)
Pricing the Individual Risky Assets
The market portfolio is made of individual risky assets: r M = w 1 r1 + w 2 r2 + . . . + w N rN
(5)
where wi is the fraction of total market wealth invested in asset i. How are the individual risky assets priced in equilibrium? To answer this question, we deviate the equilibrium holding wi of asset i slightly away from its optimal level, and see how such a deviation affect our investor’s maximized utility. Let’s focus on a representative investor with the average risk aversion . 1
U (r) = E (r) − A¯ var (r) 2
(6)
∗ r = y ∗ (w1∗ r1 + w2∗ r2 + . . . + wN rN ) + (1 − y ∗ ) rf
(7)
His portfolio holding:
What is his y ∗ ? In equilibrium, wi∗ is the optimal solution for this investor. This means: ∂U (r) =0 ∂wi
(8)
Keeping everything else fixed, what if we change wi a little? ∂E (r) = E (ri ) − rf ∂wi
(9)
∂var (r) = 2 · cov (rM , ri ) ∂wi
(10)
So it must be that:
E (ri ) − rf = A¯ cov (rM , ri )
(11)
E (rM ) − rf = A¯ var (rM )
(12)
E (ri ) − rf = βi (E (rM ) − rf )
(13)
Recall that:
This takes us to:
where βi =
cov (rM , ri ) var (rM )
(14)
Our derivation uses the quadratic utility function. In general, the proof goes through for any utility function with a preference for mean, and aversion to variance.
Risk and Reward in the CAPM
For a risky asset ri , the right measure of the rewardable risk is not its variance var (ri ), but its covariance cov (ri , rM ) with the market. The exposure to the market risk can be best quantified by: βi =
cov (rM , ri ) var (rM )
(15)
For one unit exposure to the market risk, the reward is the same as the market: E(rM ) − rf
(16)
For β unit of exposure to the market, the reward is: βi · (E (rM ) − rf )
(17)
For zero exposure to the market risk, the reward is zero, no matter how risky the asset is. In summary, the risk and reward relation in the CAPM is a linear relation.
Systematic vs. Idiosyncratic Each investment carries two distinct risks:
• Systematic risk is market-wide and pervasively influences virtually all security prices. Examples are interest rates and the business cycle.
• Idiosyncratic risk involves unexpected events peculiar to a single security or a limited number of securities. Examples are the loss of a key contract or a change in government policy toward a specific industry.
The Security Market Line
Expected Return ( µi)
Sec
Alpha (α)
(µM)
ur
M ity
ar
L ket
ine
Expected Risk Premium
M
(r f)
defensive
aggressive β = 1 βi
Risk free Investment
Investment in Market
Figure 2: Equilibrium market portfolio and efficient frontier.
Beta (β)
Systematic vs. Idiosyncratic Each investment carries two distinct risks:
• Systematic risk is market-wide and pervasively influences virtually all security prices. Examples are interest rates and the business cycle.
• Idiosyncratic risk involves unexpected events peculiar to a single security or a limited number of securities. Examples are the loss of a key contract or a change in government policy toward a specific industry.
A Linear Factor Model
A simple model to capture these two types of risks is the linear factor model: ri = E (ri ) + βi · F + εi
(18)
which carries two risky components:
• systematic F : common to all securities.
• idiosyncratic εi : specific only to security i.
Both the common factor F and the idiosyncratic component are zero mean random variables.
The Arbitrage Pricing Theory A SINGLE-FACTOR VERSION Assume a frictionless market with no taxes or transaction costs. Assets are perfectly divisible. There is no short-sale constraint. Assume a one-factor linear model:
• βi asset i’s sensitivity to the common factor. • F common factor, with E(F ) = 0. • εi firm-specific return, with zero mean, and independent of the common factor or other firms’ idiosyncratic component. Possible common factor: unexpected changes in inflation, industrial production, etc. The Arbitrage Pricing Theory: E (rj − rf ) E (ri − rf ) = βj βi
(19)
The expected return of any asset is determined by its exposure to the common factor, and has nothing to do with its idiosyncratic component. In deriving APT, one need not make any assumption about investors’ preferences, or assume any specific distribution for the asset returns. The APT is not an equilibrium concept. It does not rely on the existence of a market portfolio. It is based purely on no-arbitrage conditions.
Summary
In equilibrium, the tangency portfolio becomes the market portfolio. The expected return of the market portfolio depends on the average risk aversion in the market. The intuition of the CAPM: expected return of any risky asset depends linearly on its exposure to the market risk, measured by β. Diversification is an important concept in finance. It builds on a power mathemat ical machine called Strong Law of Large Number. Like the CAPM, the basic concept of the APT is that differences in expected return must be driven by differences in non-diversifiable risk. The APT is based purely on no-arbitrage condition. It is not an equilibrium con cept, and does not depend on having a market portfolio.
Focus: BKM Chapters 9-11 • p. 263 bottom to 284, (CAPM, assumptions, beta, liquidity, covariance, expecta tions, SML, zero-beta model, alpha) • p. 287, eq. 10.5, eq.10.6 & eq.10.7 • p. 300 to 308 • p. 308 to 313 middle (eq. 10.15, eq. 10.16) • p. 324 to 334 middle (diversification, eq. 11.2, APT and CAPM, Multifactor APT, eq. 11.5, eq. 11.6) Reader: Roll and Ross (1995). type of potential questions: chapter 9 concept check question 1, 2, 3 & 4, p. 286 ff. questions 1, 4, 17, 22, 23, 25 chapter 10 concept check question 1, 2, 3 & 4, p. 314 ff. questions 4, 5, 6, 7, 18, 19 chapter 11 concept check question 2, 3, 4 & 5, p. 335 ff. questions 3, 5, 7, 10, 16
Questions for Next Class Please read: • BKM Chapter 13, • Jagannathan and McGrattan (1995), • Kritzman (1993), and • Kritzman (1994) Think about the following questions:
• What are the predictions of the CAPM?
• Are they testable?
• What is a regression?
• What is t-test?
TECHNICAL NOTES: The Mathematical
Foundation of Diversification
Behind the concept of diversification, there is deep, elegant, and powerful mathemati cal machinery. STRONG LAW OF LARGE NUMBERS: Let x1 , x2 , . . . , be a sequence of identically distributed and independent random variables with mean µ. We have: 1 xi = µ N →∞ N lim
almost surely.
(20)
A Simulation Based ”Proof”
STEP 1: We start with X1 , assume that it is standard normal, and simulate it 10,000 times. Using the 10,000 scenarios, we plot its ”empirical” probability distribution, which is basically the histogram of the 10,000 scenarios normalized so that the total probability is one. For notational purpose, we write y1 = xi .
5
4
Likelihood
3
2
1
n = 10
-4
-2
2
4
Fig ure 3: Likelihood and simulation outcome for y1 at n=1.
STEP 2: We repeat STEP 1 ten times, each time with a new seed in our random number generator, so that xi , x2 , ..., x10 are indeed independent. We add them up, scenario by scenario, and re scale the sum by 10. That is, we have 10,000 scenarios of y10 =
1 (x1 + x2 + · · · + x10 ) 10
We then plot the ”empirical” probability distribution of y10 . Chart:
(21)
5
4
Likelihood
3
2
1 n = 10
-4
-2
2
4
Outcome of Simulation
Figure 4: Likelihood and simulation outcome for y1 at n=10.
STEP 3: Finally, we repeat STEP 1 one hundred times, obtaining 10,000 scenarios of: y100 =
1 (x1 + x2 + · · · + x100 ) 100
(22)
5
4
n = 100
Likelihood
3
2
1
-4
-2
2
4
Outcome of Simulation
Figure 5: Likelihood and simulation outcome for y1 at n=100.
According to Law of Large Number, as N grows infinitely large, N 1 � yN = xi N i−1
(23)
approaches zero with probability one.
For xi normally distributed, it is actually easy to see that yN = N1
proach zero with probability one. Since:
N 1 1 � var (yN ) = 2 var (xi ) = N N i−1
�N
i−1
xi will ap
(24)
5
4
n = 100
Likelihood
3
2
1
n = 10
-4
n = 10
-2
2
Outcome of Simulation
Figure 6: Likelihood and simulation outcome for y1 at n= 1 / 10 / 100.
4
15.433 INVESTMENTS Class 7: The CAPM and APT Part 2: Applications and Tests
Spring 2003
Predictions and Applications
• Predictions: – CAPM: In market equilibrium, investors are only rewarded for bearing the market risk. – APT: In the absence of arbitrage, investors are only rewarded for bearing the factor risk. • Applications: – professional portfolio managers: evaluating security returns and fund performance. – regulatory commissions: cost of capital for regulated firms. – court rulings: evaluating claims of lost future income. – corporate manager: capital budgeting decisions.
Testability of CAPM and APT
The wide acceptance of the CAPM and APT makes it all the more im
portant to test their predictions empirically.
Recall from Class 6, both theories build on assumptions that are un
realistic at times.
How does a product of abstract reasoning hold in reality?
Unfortunately, the predictions of the CAMP and APT are hard to test
empirically:
• Neither the market portfolio in CAPM nor the risk factor in APT is observable. • Expected returns are unobservable, and could be time varying. • Volatility is not directly observable, and is time varying.
An Ideal Test of the CAPM
In an ideal situation, we have the following inputs: 1. Riskfree borrowing/ lending rate rf 2. Expected returns on the market E(rM ) and on the risky asset E(ri ). 3. The exposure to market risk βi =
cov (rM , ri ) var (rM )
(1)
These inputs allow us to examine the relation between reward (E(rf ) − rf ) and risk βi 1. More risk, more reward? 2. Do they line up? 3. What is the reward for a risk exposure of 1? 4. Zero risk, zero reward?
E(r)
A Linear Relation between Risk and
Reward
α
E(rM)-rf
rf
0 0
βM
Figure 1: Beta with and without risk-free interest rate.
Beta (βι )
Some Practical Compromises
the market portfolio rM is unobservable: use a proxy, e.g., the S&P 500 index. expected returns E(rM ) and E(rf ) are unobservable: use sample av erage T 1 � = rM,t N t
µM
T 1 � µi = ri,t N t
(2)
unobservable risk exposure βi =
cov (rM , ri ) var (ri )
(3)
use sample estimates: βi = where
cov � (rM , ri ) � (ri ) var
(4)
T 1 � � (ri ) = var (rt,M − µM ) N t
(5)
T 1 � cov (rt,M − µM ) (rt,i − µi ) � (rM , ri ) = N t
(6)
Testing the Linear Relation
pick a proxy for the market portfolio rM , and record N monthly returns: rt,M : i = 1, . . . , N
(7)
for the same sample period, collect a sample of I firms, each with N monthly returns: rt,1 : i = 1, . . . , I
and t = 1, . . . , N
(8)
construct the sample mean for rM . for the i-th firm, construct the sample mean µi , and the sample es timate βi . for i = 1,. . . ,I test the linear relation: µi − r f = γ 0 + γ 1 β i
(9)
Implications of the CAPM γ0 = 0: zero exposure → zero reward. γ1 = µM − rf : one unit of exposure same reward as the market.
0.7
SML
Historical Return Forecasted Return 0.1 0.2 0.3 0.4
0.5
0.6
QCOM
VRTS
NT AMGN
AA
AOL
HD
0.0
FDC
-0.1
PFE CAH
BAC
UN MO
-0.3
WMI
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Beta
Figure 2: Betas for top 100 market weighted stocks from SP 500 against SP 500.
2.4
2.6
Regression: The Basic Setup
Two variables x and y, N pairs of outcomes: (xi , yj ), i = 1, 2, . . . , N.
(10)
We have reasons to believe that y and x are related. In particular, we would like to use x to explain y: yi = a + bxj + ε.
y: the dependent variable. x: the independent (explanatory) variable. εi : a random disturbance with zero mean. coefficients: intercept a, slope b.
(11)
Regression: Motivation
Some motivating examples:
1. On day i, xi is the temperature at Orlando, yi is the price of the futures contract on frozen concentrated orange juice. 2. For firm i, xi is its leverage ratio, yi is its probability of default. 3. On day i, xi is the Fed fund target rate, yi is the 3 month T bill rate. 4. At the i-th second, xi is the number of packets sent by [email protected] to [email protected], yi is the number of packets received by the latter.
In each case, the outcome of x might contribute to the outcome of y: yi = a + b · x i + ε i
(12)
but there might be other random factors, captured by εi , that have noth ing to do with x. The slope coefficient b is of particular interest as it measures the sensitivity of y to x.
The Regression Coefficients
The objective: find a and b that best capture the linear relation between y and x. How: find a and b that minimize the squared differences: min
a∈R,b∈R
N �
(yi − a − b · xi )2
(13)
i=1
o
E(µi)-rf
o
b / βi
o o o o o
0 0
Figure 3: Regression.
X / βi
Beta (β)
Regression: The Solution Solving for the optimization problem, we get:
• an estimate �b for the slope coefficient b: �N (yi − µ �y ) (xi − µ �x ) �b = i=1� N �x ) i=1 (xi − µ
(14)
• an estimate � a for the intercept a: � a = µy − �bµx Notice the familiar notation for sample means: T 1 � µ xi �x = N t
T 1 � µ yi �y = N t
(15)
Why do we call our solutions estimates? Why put hats on b and a? a hat is close to the Do we always get �b that is close to the real b, � real a?
In the large sample (large N), we are pretty confident that they do, why?
The Standard Error
In Order to guess the real values of b and a, we use the data to help us. Given N pairs of observations (yi , xi ), our regression solutions �b and
a are the best guesses. But we can never be 100% sure. � How to quantify our uncertainty about �b and � a.
We think of �b and � a as random variables. For any estimate, say �b, we can get an estimate of its standard deviation, which is usually referred to as
the Standard Error. The standard error of an estimate is one measure of its precision.
Interpreting the Regression Result
For the purpose of this class, you will use a ”canned” regression package (e.g., Excel): Input: (yi , xi ), i = 1, . . . , N Output: • the estimates �b and � a and • their standard errors: s�b and s�a • their t statistics: t�b = �b/s�b and t�a = � a/s�a • the R squared The standard errors and the t stat’s provide measures of precision of your estimates. The R squared tells you how much of the randomness in the depen dent variable y is explained by the explanatory variable x.
More on the Slope Estimate Recall that our estimate for the slope coefficient b: �N (yi − µ �y ) (xi − µ �x ) �b = i=1� N �x ) i=1 (xi − µ
(16)
Some familiar notation:
T 1 � (rt,M − µM ) var � (ri ) = N t
(17)
T 1 � cov (rt,M − µM ) (rt,i − µi ) � (rM , ri ) = N t
(18)
We have
βi =
cov � (rM , ri ) var � (ri )
(19)
Intuitively, b is a measure of the covariance cov(x,y) between x and y, scaled by the variance var(x) of x.
Back to testing the CAPM
The implication of the CAPM: µ �i − rf = γ0 + γ1 βi
(20)
� on the 43 industrial portfolios tell us that this relation Our data (� µ, β)
does not hold exactly.
One possibility: our measures of the expected returns are contaminated by noises that are unrelated to the β’s. What we still like to know: • On average, is reward related to risk at all? γ1 = 0 or not ? • On average, does zero risk result in zero reward? γ0 = 0 or not? • On average, does one unit of risk exposure pay the market return? γ1 = µM − rf = 5.9% or not?
(21)
Regression in Action Set up a regression: • The dependent variable: yi = µ �i − rf • The independent variable: xi = β�i
• Add noise εi that is unrelated to βi .
Feed the data to the regression package: estimate standard error t-stat γ0
6%
1.8%
3.5
γ1
0.17%
1.7%
0,1
R squared = 2% Recall the implications of the CAPM: 1. The intercept γ0 = 0 2. The slope γ1 = µ �M − rf = rf
0.7
SML
Historical Return Forecasted Return 0.1 0.2 0.3 0.4
0.5
0.6
QCOM
VRTS ORCL
NT AMGN
AA
AOL
CMCSK
HD
DOW
SUNW EMC
0.0
IP
-0.1
BK BAC ONE UN SWY WB
-0.3
LMT MO WMI
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
Beta
Figure 4: Betas (without rf ) for top 100 market weighted stocks from S&P 500 against S&P 500.
Historical Return Forecasted Return 70% 60% 50% 40% 30% 20% 10% 0% -10% -20% -30% 0.00
0.50
1.00
1.50
2.00
2.50
3.00 Beta
Figure 5: Betas (with and without rf ) for top 100 market weighted stocks from S&P 500 against S&P 500.
A Rule of Thumb with t-Stat
To get a sense of how an estimate, say �b, differs significantly from zero, its t-stat � tb is the most telling statistics.
A Rule of Thumb: think of � tb as standard normal (not a bad assumption
for a large sample). The larger the magnitude (absolute value) of b, the more likely it is significantly different from zero. Hypothesis Testing: The null: �b = 0, the alternative: �b = � 0.
1. A t-stat of 1.960 rejects the null with significance level 5%; 2. A t-stat of 2.576 rejects the null with significance level 1%; For example,� tγ1 = 0.1, what can we say about γ �1 ? What about γ �1 ?
A Summary of the CAPM Tests
In general, the test results depend on the sample data, sample peri ods, statistical approaches, proxy for the market portfolio, etc. But the following findings remain robust: • The relation between risk and reward is much flatter than that pre dicted by the CAPM γ �1 = µ �i − rf . • The risk measure β cannot even begin to explain the cross sectional variation in the expected returns. (γ �1 is statistically insignificant, R squared is close to zero.)
• Contrary to the prediction of the CAPM, the intercept γ �0 is signifi cantly different from zero.
Some Possible Explanations
1. Is the stock market index a good proxy for the market portfolio? • only 1/3 non governmental tangible assets are owned by the corpo rate sector. • among the corporate assets, only 1/3 is financed by equity • what about intangible assets, like human capital? • what about international markets? 2. Measurement error in β: • Except for the market portfolio, we never observe the true β. • To test the CAPM, we use estimates for β, which are measured with errors. • The measurement error in β will cause a downward biased estimate for the slope coefficient, and an upward biased estimate for the in tercept. 3. Measurement error in expected returns • we use sample means µ �M and µ �i as proxies for the real, unob-servable expected returns
• it is known that means are hard to estimate, and there are noises in �M and µ �i and our estimates µ • if the noises in µ �M and µ �i are correlated, then we have a statistical problem (errors in variables)
4. Borrowing restrictions
• This class covers only one version of the CAPM, assuming unre stricted borrowing. • In practice, borrowing restrictions are realistic. It includes margin rules, bankruptcy laws that limit lender access to a borrower’s future income, etc. • Fisher Black showed that borrowing restrictions might cause low-β stocks to have higher expected returns than the CAPM predicts.
Going Beyond the CAPM
Is β a good measure of risk exposure? What about the risk associated with negative skewness? Could there be other risk factors? Time varying volatility, time varying expected returns, time varying risk aversion, and time varying β?
Focus: BKM Chapter 13 • p. 383 (13.1) • p. 386 to 392 (beta, CAPM, SML, market index, concept check question 3 & 4), • p. 391 to 393 top (13.2) • p. 399 bottom (13.4 to 13.6) Reader: Kritzman (1993) and Kritzman (1994). type of potential questions: concept check question 1, 2, 3 & 4
Preparation for Next Class Please read: • Fama and French (1992) and • Jegadeesh and Titman (1993).
15.433 INVESTMENTS Classes 8 & 9: The Equity Market Cross Sectional Variation in Stock Returns
Spring 2003
Introduction
Equities are common stocks, representing ownership shares of a corporation. Two important characteristics: • limited liability: non-negative stock prices, • residual claim: equities are inherently more risky than fixed income securities. In most countries, the equity market is perhaps the most popular venue of investments for individual investors. It also remains to be an important component of institutional investments. We will examine the equity market from two perspectives: • cross-sectional (Classes 8 & 9 ), and • time-series (Class 10).
”Cross-Section” vs. ”Time-Series”
These two concepts are empirically motivated. For a publicly traded firm i, the follow ing information can be readily obtained. • The stock price Pi,t at any time t. • The cash dividend Di,t−1 paid between t-1 and t.
At any time t, we can calculate the realized stock return for ri,t for firm i: • percentage returns: ri,t =
Pi,t +Di,t −Pi,t−1 Pi,t−1
• log-returns: ri,t = ln (Pi,t + Di,t − ln (Pi,t−1 ))
• cross-section of stock returns: ri,t ;
• time series of stock returns: ri,t ;
i = 1, 2, . . . , N
i = 1, 2, . . . , T
The Cross-Sectional Distribution of
Returns
0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 -0.150 -0.010
-0.100
-0.050
-
0.050
Normal-Distribution
0.100
0.150
Current Distribution
Figure 1: Distribution of the Nasdaq-index returns for the year 2000, source: Bloomberg Professional
0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 -0.080 -0.010
-0.060
-0.040
-0.020
-
Normal-Distribution
0.020
0.040
0.060
Current Distribution
Figure 2: Distribution of the Nasdaq-index returns for the year 1999, source: Bloomberg Professional
0.140 0.120 0.100 0.080 0.060 0.040 0.020 -0.020 -0.100
-0.050
-
Normal-Distribution
0.050
0.100
Current Distribution
Figure 3: Distribution of the Nasdaq-index returns for the year 1998 (from left to right), source: Bloomberg Professional
What has changed, what has influenced the market?
Multi-Factor Regressions For each asset i, we use a multi-factor time-series regression to quantify the asset’s tendency to move with multiple risk factors: ri,t−1 − rf,t = αi + βi (rM,t − rf,t ) + fi Ft + εi,t
• 1. Systematic Factors: rM,t : risk premium λM = E (rM,t − rf ) Ft : risk premium λF = E (Ft )
• 2. Idiosyncratic Factors: εi,t : no risk premium E (εi,t )
• 3. Factor Loadings: βi,t : firm’s sensitivity to the market risk fi,t : firm’s sensitivity to the factor risk See
BKM
p. 559 - 572 and article from Fama (1992)
The Pricing Relation
Using the intuition of the CAPM, the reward for asset i should be related to its exposure to the market risk, as well as its exposure to the systematic risks. Given the risk premia of the systematic factors, e.g., λm and λF , the determinants of expected returns: E (ri,t−1 − rf,t ) = βi · λM + fi · λF
(1)
Without the systematic factors F, we are back to the CAPM. What are the additional systematic factors? The intuition of the CAPM: these factors should be proxies for the real, macroeco nomic, aggregate, non-diversifiable risk.
Size: Small or Big We can sort the cross-section of stocks by their Market Capitalizations:
”Share Price x Number of Shares Outstanding”
Cap Decile
Market
NYSE
NYSE
AMEX
NASDAQ
Cap(m$)
Ticker
Stocks
Stocks
Stocks
Total
10
511,391
GE
172
5
80
257
9
10,486
NSM
172
3
81
256
8
4,428
GLM
172
5
136
313
7
2,237
BLC
172
5
166
343
6
1,387
GES
172
5
217
394
5
889
SFG
172
11
254
437
4
534
PNK
172
15
251
438
3
353
FFD
172
32
400
604
2
198
SXI
172
73
551
796
1
95
AVS
172
412
1,399
1,983
Size Range as of December 31, 2000. Source: www.dfafunds.com.
Value or Growth
We can also sort the cross-section of stocks by their Book-to-Market (BtM) ratios: • Growth Stocks: Firms with low BtM ratios • Value Stocks: Firms with high BtM ratios.
Decile
BtM
NYSE
NYSE
AMEXS
NASDAQ
Ticker
Stocks
Stocks
Stocks
Total
1
0.01
IN
155
71
824
1,050
2
0.14
SYY
155
31
362
548
3
0.25
TDX
155
36
223
414
4
0.36
STJ
155
24
177
356
5
0.45
FLO
155
37
229
421
6
0.58
DOL
155
43
248
446
7
0.72
HCC
155
56
274
485
8
0.92
TWR
155
51
251
457
9
1.19
MTN
155
71
279
505
10
1.81
ZAP
155
58
195
408
Value and Growth Definitions as of December 31, 2000. Source: www.dfafunds.com
Portfolios Formed on Size and BtM
Sort the stocks traded on the NYSE, NASDAQ, and AMEX by their size and BtM.
Size labels: A (small), B, C, D, and E (big).
BtM labels: 1 (low), 2, 3, 4, and 5 (high).
• low BtM growth stocks • high BtM value stocks The 25 Fama-French Portfolios:
A (s) 1 (l) 2 3 4 5 (h)
B
C
D
E (b)
Explain the Fama-French Portfolios
Start with the one-factor empirical model: ri,t − rf,t = αi + βi (rM,t − rf,t ) + fi Ft + εi,t
(2)
For each portfolio i, we perform the above regression and obtain an estimate βˆi of the factor loading βi . This regression procedure is equivalent to constructing sample estimates for βi , why? Estimate the market risk premium: T 1� ˆ λM = (rM,t − rf,t ) T i=1
The mean excess return: E (ri,t − rf,t )
ˆM • The model predicted: βˆi · λ • Measured from the data:
1 T
�T
i=1
(ri,t − rf,t )
(3)
A One-Factor Model (CAPM)
Figure 4: One-factor model, Source: Jun Pan, Investments 15.433 Spring 2001.
A Three-Factor Model The Fama and French empirical Factors: • SMB rsmb : small minus big - return on the small portfolio minus that on the big portfolio; • HML rhml : high minus low - return on the high BtM portfolio (value) minus that on the low BtM portfolio (growth). A three-factor regression model: ri,t − rf,t = αi + βi (rM,t − rf,t ) + si · rsmb,t + hi · rhml,t + εi , t
(4)
Pricing Relation: E (ri,t ) − rf,t = βi · λM + si · λsmb,t + hi · λhml,t where λsmb and λhml are the risk premiums of the Fama-French factors.
(5)
The Fama-French Three-Factor Model
Figure 5: Fama-French three factor model, Source: Jun Pan, Investments 15.433 Spring 2001.
The Factor Premia
Using monthly returns from 1963 to 2000, the (annualized) premia for the three factors are:
Factor
Estimate
S.E.
t-stat
Market
6%
2.5%
2.5%
SMB
1.9%
1.9%
1%
HML
5%
2%
2.6%
The Market Risk Premium The market risk premium has its foundation in the CAPM. Investors are risk averse. They are worried about holding stocks that do badly at the times when the market does badly. The market risk premium is a reward for holding the market risk.
What are the Size and Value Factors?
Unlike the market portfolio, the Size and Value portfolios are empirically motivated. Where do the size and value premia come from? If we think of them as risk premia, then we need to understand the real, macroeco nomic, aggregate, non-diversifiable risk that is proxied by the SMB and HML portfolios. In particular, why are investors so concerned about holding stocks that do badly at the times that the hml and smb portfolios do badly, even though the market does not fall?
Some Explanations Value: proxies for the ”distress risk”. Size: proxies for the illiquidity of the stock. HML and SMB contain information above and beyond that in the market return for forecasting GDP growth. Proxies for variables that forecast time-varying investment opportunities or time-varying risk aversion. Over-reaction: earnings announcements. Seasonal: the January effect. Survival bias. Data snooping
Other Factors
Empirical Factors: price-to-earning ratios, strategies based on five-year sales growth, etc. Macroeconomic Factors: labor income, industrial production, inflation, investment growth, consumption wealth ratio, etc. The Market Skewness Factor: • If asset returns have systematic skewness, expected returns should include rewards for accepting this risk. • Co-Skewness: the level of exposure to the systematic skewness. • Harvey and Siddique (Journal of Finance 2000) report that systematic skewness is economically important and commands an average risk premium of 3.60 per year.
Long-Term Reversals Firms whose three- and five-year returns are high (low) tend to have low (high) returns in subsequent years. Firms with low (high) BE/ME, E/P, CF/P, D/P, and prior sales growth tend to have low (high) returns in subsequent years. All these patterns seem to be manifestations of the same value vs. growth phenomenon. This ”reversal” effect makes sense given return predictability and mean-reversion, and is explained by the Fama-French three factor model.
Short-Term Momentum Firms with high returns in the prior year tend to have high returns in the next few months. Firms with low short-term returns tend to have low returns in subsequent months.
At the moment, the momentum effect is the most-studied anomaly in Finance. It cannot be explained by the Fama-French three factor model. Risk-based stories: Proxy for systematic skewness: the low expected return momentum portfolios (losers) have higher skewness than high expected return portfolios (winners). Behavioral (non-risk- based) stories: • 1. underreaction: bad news travels slowly; • 2. overreaction: positive feedback; • 3. overconfidence.
Beta Hedging Recall that:
E (ri ) = rf + βi (E (rM ) − rf )
(6)
where βi =
cov (rM , ri ) var (rM )
(7)
Let’s assume we are long-beta, e.g. β = 1.3, and we are not optimistic about the short-term outlook of the market. We anticipate some bade earnings numbers, which will send the market down . . . and our portfolio even faster. Let’s hedge our BetaExposure! ΔP ΔM ≈β P M
(8)
A stock index futures contract worth: ΔP ΔM ≈β P M
(9)
The generated change in portfolio value ΔV due to an adverse change in the market ΔM is:
ΔV
= ΔP + N ΔF ΔM ΔM = β·P +N ·F M M
(10) (11)
The optimal N∗ isN ∗ = − β·P (12)The optimal hedge with a stock index futures is given F by beta of the portfolio times its value divided by the notional of the futures contract. Example: A portfolio manager holds a stock portfolio worth $10 mio., with a beta of 1.5 relative to S&P 500. The current S&P 500 index futures price is 1400, with a multiplier of $250. Compute:
1. The notional of the futures contract 2. the optimal number of contracts to hedge the beta-exposure against adverse mar ket movements. Solution: 1. The notional of the futures contract is: $250 · 1� 400 = $350� 000
(13)
2. the optimal number of contracts to hedge the beta-exposure against adverse mar ket movements is: N ∗ = −
β · P
1.5 · $10� 000� 000 = −
= −42.9 F
1 · $350� 000
(14)
However: A typical US stock has a correlation of 50% with the S&P 500-index. Using the
regression effectiveness we find that the volatility of the hedged portfolio is still about
(1 − 0.52 = 87%) of the unhedged volatility for a typical stock.
If we wish to hedge an industry index with S&P futures, the correlation is about 75%
and the unhedged volatility is 66% of its original level.
The lower number shows that stock market hedging is more effective for diversified
portfolios.
Summary
The cross-sectional variation in stock returns cannot be fully explained by beta. Adding additional factors (size and value) helps. While the risk premium associated with the beta risk has its theoretical foundation in the CAPM, the premia associated with the size and value factors do not. There are many other puzzling patterns in stock returns, some of which are hard to reconcile with market efficiency.
Focus: Patterns in the Cross-Section of Stock Returns • Reader: Fama and French (1992) type of potential questions: how is value / growth style etc. defined? What was the general setup of the style analysis?
Focus: More Patterns in the Cross-Section of Stock Returns • Cochrane (1999), • Kritzman (1991a), and • Kritzman (1991b) type of potential questions: Cochrane: p. 39 to 43, 50 to 51, Kritzman: historical vs. implied volatility, normal assumption of volatility vs. nonlinearity
Questions for Next Class Please Plase read for class 9: • BKM 13.4 and 13.5 • Cochrane (1999), Kritzman (1991a) and Kritzman (1991b) In the next class, we will examine stock returns from a time-series perspective. Our focus is on dynamic models that allow for: • 1. time varying expected return µt • 2. and time-varying volatility σt Why are these issues important? Please read for class 10: BKM Chapters 20.
15.433 INVESTMENTS Class 10: Equity Options Part 1: Pricing
Spring 2003
SPX S&P 500 Index Options
Symbol: SPX Underlying: The Standard & Poor’s 500 Index is a capitalizationweighted index of 500 stocks from a broad range of industries. The component stocks are weighted according to the total market value of their out-standing shares. The impact of a component’s price change is proportional to the issue’s total market value, which is the share price times the number of shares out-standing. These are summed for all 500 stocks and divided by a predetermined base value. The base value for the S&P 500 Index is adjusted to reflect changes in capitalization resulting from mergers, acquisitions, stock rights, substitutions, etc.
Multiplier: $100. Strike Price Intervals: Five points. 25-point intervals for far months. Strike (Exercise) Prices: In-,at- and out-of-the-money strike prices are initially listed. New series are generally added when the underlying trades through the highest or lowest strike price available. Premium Quotation: Stated in decimals. One point equals $100. Minimum tick for options trading below 3.00 is 0.05 ($5.00) and for all other series, 0.10 ($10.00). Expiration Date: Saturday immediately following the third Friday of the expiration month. Expiration Months: Three near-term months followed by three additional months from the March quarterly cycle (March, June, September and December). Exercise Style: European - SPX options generally may be exercised only on the last business day be-fore expiration. Settlement of Option Exercise: The exercise-settlement value, SET, is calculated using the opening (first) reported sales price in the primary market of each component stock on the last business day (usually a Friday) before the expiration date. If a stock in the index does not open on the day on which the exercise & settlement value is determined, the last reported sales price in the primary market will be used in calculating the exercise-settlement value. The exercise-settlement amount is equal to the difference between the exercise-settlement value, SET, and the exercise price of the option, multiplied by $100. Exercise will result in delivery of cash on the business day following expiration. Position and Exercise Limits: No position and exercise limits are in effect. Each member (other than a market-maker) or member organization that maintains an end of day position in excess of 100,000 contracts in SPX (10 SPX LEAPS equals 1 SPX full value contract) for its proprietary account or for the account of a customer, shall report certain information to the Department of Market Regulation. The member must report information as to whether such position is hedged and, if so, a description of the hedge employed. A report must be filed when an account initially meets the aforementioned applicable threshold. Thereafter, a report must be filed for each incremental increase of 25,000 contracts. Reductions in an options position do not need to be reported. However, any significant change to the hedge must be reported. Margin: Purchases of puts or calls with 9 months or less until expiration must be paid for in full. Writers of uncovered puts or calls must deposit / maintain 100CUSIP Number: 648815 Last Trading Day: Trading in SPX options will ordinarily cease on the business day
(usually a Thursday) preceding the day on which the exercise-settlement value is cal culated.
Trading Hours: 8:30 a.m.- 3:15 p.m. Central Time (Chicago time).
Equity Options: The Basics
A European style call option is the right to purchase, on a future date T, one unit of the underlying asset at a pre determined price K: • At time 0 (today), pay C. • At time T (the expiration date), exercise the option and get (ST − K)+.
A put option gives the right to sell: • At time 0 (today), pay P. • At time T, get (K − ST )+. Most of the index options (e.g., SPX, QEX, NDX) are European style. All of the options on the individual stocks are American style, which allow for exercise before the expiration date T.
S&P 500 Options
.SPX (CBOE),1118.49,-1.82, Nov 12 2001 @ 15:27 ET (Data 15 Minutes Delayed), Calls Last Sale Net 02 Mar 750.0 (SPZ CJ-E) 318.40 pc 02 Mar 775.0 (SPZ CO-E) pc 02 Mar 800.0 (SPX CT-E) pc 02 Mar 850.0 (SPX CJ-E) pc 02 Mar 900.0 (SXB CT-E) 189.00 pc
Bid 370.10 346.20 322.30 275.50 229.80
Ask 374.10 350.20 326.30 279.50 233.80
Vol -
02 Mar 925.0 (SXB CE-E) 02 Mar 950.0 (SXB CJ-E) 02 Mar 975.0 (SXB CO-E) 02 Mar 995.0 (SXB CS-E) 02 Mar 1025. (SPQ CE-E) 02 Mar 1050. (SPQ CJ-E) 02 Mar 1075. (SPQ CO-E) 02 Mar 1100. (SPT CT-E) 02 Mar 1125. (SPT CE-E) 02 Mar 1150. (SPT CJ-E) 02 Mar 1175. (SPT CO-E) 02 Mar 1200. (SZP CT-E) 02 Mar 1225. (SZP CE-E) 02 Mar 1250. (SZP CJ-E) 02 Mar 1275. (SZP CO-E) 02 Mar 1280. (SZP CP-E) 02 Mar 1300. (SXY CT-E) 02 Mar 1325. (SXY CE-E) 02 Mar 1350. (SXY CJ-E) 02 Mar 1375. (SXY CO-E) 02 Mar 1400. (SXZ CT-E) 02 Mar 1425. (SXZ CE-E) 02 Mar 1450. (SXZ CJ-E) 02 Mar 1475. (SXZ CO-E) 02 Mar 1500. (SXM CT-E) 02 Mar 1550. (SXM CJ-E) 02 Mar 1600. (SPB CT-E) 02 Apr 1020. (SYU DD-E) 02 Jun 700.0 (SPZ FT-E) 02 Jun 725.0 (SPZ FE-E) 02 Jun 750.0 (SPZ FJ-E) 02 Jun 800.0 (SPX FT-E) 02 Jun 850.0 (SPX FJ-E) 02 Jun 900.0 (SXB FT-E) 02 Jun 950.0 (SXB FJ-E) 02 Jun 995.0 (SXB FS-E) 02 Jun 1025. (SPQ FE-E) 02 Jun 1045. (SPQ FI-E) 02 Jun 1050. (SPQ FJ-E) 02 Jun 1075. (SPQ FO-E) 02 Jun 1100. (SPT FT-E) 02 Jun 1125. (SPT FE-E) 02 Jun 1150. (SPT FJ-E) 02 Jun 1175. (SPT FO-E) 02 Jun 1200. (SZP FT-E) 02 Jun 1225. (SZP FE-E)
144.90 153.00 148.00 94.00 108.00 80.00 72.00 53.00 47.40 29.50 21.00 19.00 16.50 8.50 6.30 5.40 5.00 3.50 1.30 1.00 0.80 0.35 2.00 0.50 0.50 0.55 225.00 161.50 164.00 121.00 115.00 108.50 91.30 79.00 61.00 54.00 46.50 35.50
pc pc 61.00 pc pc 2.50 (14.00) (1.00) (6.00) pc (7.50) (5.30) pc pc 2.50 pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc pc (10.30) pc (3.00) pc (6.00) pc pc pc
207.40 185.70 164.40 148.00 124.80 106.20 88.80 72.30 57.50 44.80 33.70 24.50 17.60 11.80 8.00 7.30 5.00 3.10 1.90 1.00 0.45 0.05 420.80 397.40 374.00 327.70 282.80 239.30 197.80 162.60 140.50 123.10 106.70 91.60 77.60 64.80 53.10 42.40 33.90
211.40 189.70 168.40 152.00 128.80 110.20 92.80 76.30 61.50 48.80 37.70 28.50 20.60 14.80 10.00 9.30 6.40 4.50 2.80 1.90 1.30 0.95 0.90 0.90 0.90 0.90 0.90 424.80 401.40 378.00 331.70 286.80 243.30 201.80 166.60 144.50 127.10 110.70 95.60 81.60 68.80 57.10 46.40 37.90
2.00 4.00 2.00 560.00 1.00 3.00 3.00 1.00 5.00 401.00 5.00 -
1,078 2 2,109 1,818 8,059 3,929 13,293 7,351 8,354 3,148 13,211 3,922 10,727 3,035 3 6,073 2,241 4,957 789 8,718 269 1,986 94 1,173 28 931 121 28 2,670 1,358 1,350 3,393 1,186 5,411 1,885 5,415 673 7,255 1,451
02 Mar 925.0 (SXB OE-E) 02 Mar 950.0 (SXB OJ-E) 02 Mar 975.0 (SXB OO-E) 02 Mar 995.0 (SXB OS-E) 02 Mar 1025. (SPQ OE-E) 02 Mar 1050. (SPQ OJ-E) 02 Mar 1075. (SPQ OO-E) 02 Mar 1100. (SPT OT-E) 02 Mar 1125. (SPT OE-E) 02 Mar 1150. (SPT OJ-E) 02 Mar 1175. (SPT OO-E) 02 Mar 1200. (SZP OT-E) 02 Mar 1225. (SZP OE-E) 02 Mar 1250. (SZP OJ-E) 02 Mar 1275. (SZP OO-E) 02 Mar 1280. (SZP OP-E) 02 Mar 1300. (SXY OT-E) 02 Mar 1325. (SXY OE-E) 02 Mar 1350. (SXY OJ-E) 02 Mar 1375. (SXY OO-E) 02 Mar 1400. (SXZ OT-E) 02 Mar 1425. (SXZ OE-E) 02 Mar 1450. (SXZ OJ-E) 02 Mar 1475. (SXZ OO-E) 02 Mar 1500. (SXM OT-E) 02 Mar 1550. (SXM OJ-E) 02 Mar 1600. (SPB OT-E) 02 Apr 1020. (SYU PD-E) 02 Jun 700.0 (SPZ RT-E) 02 Jun 725.0 (SPZ RE-E) 02 Jun 750.0 (SPZ RJ-E) 02 Jun 800.0 (SPX RT-E) 02 Jun 850.0 (SPX RJ-E) 02 Jun 900.0 (SXB RT-E) 02 Jun 950.0 (SXB RJ-E) 02 Jun 995.0 (SXB RS-E) 02 Jun 1025. (SPQ RE-E) 02 Jun 1045. (SPQ RI-E) 02 Jun 1050. (SPQ RJ-E) 02 Jun 1075. (SPQ RO-E) 02 Jun 1100. (SPT RT-E) 02 Jun 1125. (SPT RE-E) 02 Jun 1150. (SPT RJ-E) 02 Jun 1175. (SPT RO-E) 02 Jun 1200. (SZP RT-E) 02 Jun 1225. (SZP RE-E)
02 Jun 1250. (SZP FJ-E) 02 Jun 1300. (SXY FT-E) 02 Jun 1325. (SXY FE-E) 02 Jun 1350. (SXY FJ-E) 02 Jun 1375. (SXY FO-E) 02 Jun 1400. (SXZ FT-E) 02 Jun 1425. (SXZ FE-E) 02 Jun 1450. (SXZ FJ-E) 02 Jun 1475. (SXZ FO-E) 02 Jun 1500. (SXM FT-E) 02 Jun 1525. (SXM FE-E)
27.50 17.50 11.20 7.00 6.00 4.00 1.80 2.20 1.90 1.20 0.90
pc pc pc (1.00) pc -pc pc pc pc pc
26.20 15.00 10.80 8.10 5.60 4.00 2.90 1.90 1.25 0.60 0.25
30.20 18.00 13.80 10.10 7.60 5.40 3.90 2.80 2.15 1.50 1.15
11.00 10.00 -
6,959 7,065 1,469 5,481 787 12,350 329 8,703 123 8,537 320
02 Jun 1550. (SXM FG-E) 02 Jun 1550. (SXM FJ-E) 02 Jun 1600. (SPB FT-E) 02 Jun 1650. (SPB FJ-E) 02 Jun 1700. (SPV FT-E) 02 Jun 1750. (SPV FJ-E) 02 Jun 1800. (SYV FT-E)
0.30 0.40 2.90 0.10 0.65 0.05
0.10 -
0.95 0.90 0.90 0.90 0.90 0.90
5.00 -
3,458 6,212 3,849 3,439 1,120 4,132
pc -pc pc pc pc pc
Figure 2: Source: www.cboe.com
Open Int Puts 32 02 Mar 750.0 (SPZ OJ-E) 02 Mar 775.0 (SPZ OO-E) 02 Mar 800.0 (SPX OT-E) 02 Mar 850.0 (SPX OJ-E) 51 02 Mar 900.0 (SXB OT-E)
Last Sale 4.80 4.00 5.00 9.00 12.50
Net 1.10 pc pc 1.50 1.10
Bid 3.10 4.00 5.00 7.40 10.90
Ask 4.50 5.30 6.30 9.40 13.80
Vol 1.00 3.00 7.00
Open Int 13,629 12,283 6,600 3,801 13,075
23.00 17.00 25.00 24.00 32.20 40.00 48.80 55.00 70.00 79.00 93.00 97.60 140.00 143.00 193.00 72.50 195.00 328.00 250.50 276.00 290.00 344.00 343.00 408.50 7.40 11.00 7.30 10.00 14.50 19.50 29.00 37.50 44.00 52.00 66.40 76.00 84.50 103.00 86.00 150.00 179.50
pc pc pc 0.60 2.20 3.50 6.50 4.00 8.00 5.00 pc pc 18.00 (15.00) pc pc 16.80 pc pc pc 19.00 pc pc pc pc pc pc pc pc pc pc pc pc pc (1.00) pc pc pc pc 6.40 6.00 3.50 pc pc pc pc
13.20 16.30 19.70 22.80 28.80 35.10 42.40 51.00 62.00 73.00 86.70 102.80 120.20 139.10 159.60 163.70 181.10 203.80 227.20 251.10 275.30 299.70 324.30 349.10 373.90 423.30 472.90 4.50 5.40 6.70 9.70 13.50 19.30 26.60 35.70 43.20 50.40 58.60 68.20 78.80 90.60 103.60 117.50 133.60
16.20 19.30 22.70 26.80 32.80 39.00 46.40 55.00 64.90 77.00 90.70 106.80 124.20 143.10 163.60 167.70 185.10 207.80 231.20 255.10 279.30 303.70 328.30 353.10 377.90 427.30 476.90 5.90 7.40 8.70 11.70 16.50 22.30 30.60 39.70 47.20 54.40 62.60 72.20 82.80 94.60 107.60 121.50 137.60
2.00 110.00 626.00 101.00 103.00 436.00 1.00 5.00 1.00 200.00 200.00 3.00 2.00 10.00 3.00 -
633 7,719 3,991 11,110 9,778 8,300 4,833 5,684 3,239 9,298 4,549 8,920 3,514 4,579 1,080 2 2,007 213 918 266 1,230 679 60 55 734 14 1,064 2,852 2,993 6,983 4,332 9,137 6,127 1,350 12,129 1,144 14,313 1,412 5,536 160 6,462 446
02 Jun 1250. (SZP RJ-E) 02 Jun 1300. (SXY RT-E) 02 Jun 1325. (SXY RE-E) 02 Jun 1350. (SXY RJ-E) 02 Jun 1375. (SXY RO-E) 02 Jun 1400. (SXZ RT-E) 02 Jun 1425. (SXZ RE-E) 02 Jun 1450. (SXZ RJ-E) 02 Jun 1475. (SXZ RO-E) 02 Jun 1500. (SXM RT-E) 02 Jun 1525. (SXM RE-E)
153.00 244.90 227.00 248.00 280.00 290.00 388.00 396.00 347.00
pc pc pc pc pc 1.00 pc pc pc pc pc
150.60 188.20 208.70 230.10 252.20 275.00 298.20 321.90 345.90 369.80 394.10
154.60 192.20 212.70 234.10 256.20 279.00 302.20 325.90 349.90 373.80 398.10
200.00 -
2,280 4,474 152 2,724 5 5,677 3,946 2,882 101
02 Jun 1550. (SXM RG-E) 02 Jun 1550. (SXM RJ-E) 02 Jun 1600. (SPB RT-E) 02 Jun 1650. (SPB RJ-E) 02 Jun 1700. (SPV RT-E) 02 Jun 1750. (SPV RJ-E) 02 Jun 1800. (SYV RT-E)
417.00 302.00 513.00 590.00 724.70 811.00
pc pc pc pc pc pc pc
418.60 467.60 516.80 566.00 615.20 664.50
422.60 471.60 520.80 570.00 619.20 668.50
-
1,622 944 350 248 43 596
The Value of a Call Option
max[0,S(T)-K]
Payoff in $
Payoff in $
K Terminal Spot Price, S(T)
0
0
Terminal Spot Price, S(T)
K
- max [0, S(T) - K]
Figure 3: Value of a Long Call
Figure 4: Value of Short Call
at Expiration
at Expiration
Notice that the short call option payoff is unbounded from below.
Payoff in $
max [0, K - S(T)]
Payoff in $
K
Terminal Spot Price, S(T)
0
0
Terminal Spot Price, S(T)
K -K
Figure 5: Value of a Long Put at Expiration
- max [0, K - S(T)]
Figure 6: Value of a Short Put at Expiration
Hence you can buy a put option if you are a pessimist, i.e., you have a hunch, but are not absolutely certain, about the stock price, S, going below the strike price, K, at maturity (for Euro-pean options). Notice that the long put option payoff is bounded above by K.
Option Pricing
”Suppose the underlying security does not pay dividend, rT = ln (ST ) − ln (SO )
(1)
� � S0 = E e−ri,T · ST
(2)
or equivalently:
A call option paying (ST − K)+ at time T must be worth: � � C0 = E e−ri,T · (ST − K)+
(3)
From the intuition of the CAPM, we know that the above evaluation depends on the risk aversion coefficient A of the representative investor. ”For example, if the underlying asset is the market portfolio, then E (rT ) − rf = A¯ · var (rT )
(4)
The question is: how to evaluate � � C0 = E e−ri,T · (ST − K)+
(5)
Single Period Binominal-Valuation
Problem
Suppose that a European call option struck at $ 50 matures in one year (t = 1). The riskless rate per year is 25%, compounded annually, so that one dollar invested at the riskless rate grows to $ 1.25 over the year. The stock underlying the option pays no dividends over the year and its current price of $ 40 will either double or halve over the year. Assuming frictionless markets and no arbitrage, what must be the current price of the call, C0 ?
• Strike Price of the call = K = $ 50 • Riskless return over the period = R(1) = 1.25
• Unit Bond price = B(0, 1) =
1 1.25
= $0.80
• Stock price at the beginning of the period = S0 = $40 • Up return of the stock = U = 2 • Down return of the stock = D =
1 2
• Call value at the end of the period given an Up return of the stock = CU = max[0, S0 · U − K] = $30 • Call value at the end of the period given a down return of the stock = CD = max[0, S0 · D − K] = $0 • Call value at the beginning of the period = C0 =? (Ans: $ 12. The next section will derive the price).
Spanning the Payoffs
Consider a portfolio of m0 shares of the underlying stock and B0 dollars invested in the riskless asset, Assume you form the portfolio at time 0. The portfolio has two possible values at the end of the period, depending on whether the stock price goes up or down: U p : m0 · 80 + B0 · 1.25
(6)
Down : m0 · 20 + B0 · 1.25
(7)
Similarly, the call value has two possible values: Up: 30 or Down: 0 We can choose the number of shares, m0 , and the amount invested in the riskless asset, B0 , today, so that the value of the stock-bond portfolio equates to the value of the call next year:
1. m0 · 80 + B0 · 1.25 = 30 2. m0 · 20 + B0 · 1.25 = 0 Subtracting (2) from (1) and solving for the number of shares, m0 , we get m0 =
30 − 0 1 = 80 − 20 2
(8)
Plugging for m0 into either (1) or (2), we get B0 = −8$. The minus sign for B0 means we have to borrow $8 at the riskless rate.
Valuation
Since buying
1 2
of a share of the stock and shorting $8 of the risk-less asset duplicates
the payoff of the all, avoiding arbitrage requires that the current price of the traded call
equals the cost of duplicating it. Hence, the current call value is: V (0) = 12 · $40 − $8 =
$12.
Remarks:
• The required number of shares, m0 = 12 , is the difference in next year’s possible call values, expressed as a proportion of the difference in next year’s possible stock prices: m0 =
30 − 0 1 = 80 − 20 2
(9)
• For a European call option, the required number of shares is always between 0 and 1. In a graph of call values against stock prices, the required number of shares is the slope of the graph. For this reason, the required number of shares is often called the delta of the call. Delta is the sensitivity of the call option price to stock price changes. It is a very useful parameter in hedging. The amount invested in the riskless asset is negative (B0 = −$8). Consequently, the riskless bond must be shorted. Short selling bonds is equivalent to borrowing. The amount to be repaid at the end of the period is (B0 · 1.25) = $10, irrespective of being in the up state or down state. • This approach can be used in a binomial framework to value any claim whose payoff is contingent on the price of the stock (e.g., a put). • Notice that the probabilities q and 1−q do not enter the valuation argument at all. Hence, 2 individuals who disagree about the probabilities of outcomes (q, 1 − q), but who agree on the outcomes (i.e., U and D), will agree that the traded option’s value today is 12. • Since we assume that the stock can only take on two values from any node, we only need two assets (the underlyer and the riskless asset) to replicate the payoff of the option. Any additional uncertainty (example, stochastic interest rates) would require additional assets.
Fisher Black on Option Pricing
”I applied the Capital Asset Pricing Model to every moment in a warrant’s life, for every possible stock price and warrant value ... I stared at the differential equation for many, many months. I made hundreds of silly mistakes that led me down blind alleys. Nothing worked . . . [The calculations revealed that] the warrant value did not depend on the stock’s ex pected return, or on any other asset’s expected return. That fascinated me. [He adds:] Then Myron-Scholes and I started working together”.
Risk Neutral Pricing
It turns out that Black was not far from the truth!
• Two hypothetical investors: one risk neutral (A = 0), and one risk averse (A > 0). • Suppose both investors are willing to pay the market price S0 for the underlying asset. But their required rates of return are different: – For the risk neutral investor, it is simply the riskfree rate rf . – For the risk averse, it is the riskfree rate rf plus a positive risk premium. • If they agreed on So, they would also agree on C0 for the call. If this is true, the easiest way to get C0 is to let the risk neutral investor do the pricing. Hence the term ”risk neutral pricing.”
One key assumption: All investors, regardless of their risk attitudes, agree on today’s stock price S0 .
Risk-Neutral Valuation
Risk neutral valuation is a trick for valuing options quickly when investors can be riskaverse. It is not a model which assumes investors are all risk-neutral. Recall that the version of the binomial model we studied assumed frictionless markets, no arbitrage, European options, a constant riskless rate, no payouts, and a multiplicative binomial process for the underlyer’s spot price. We were able to value a European call and put without any knowledge of investor preferences or beliefs regarding the likelihood of the up or down states. This is called Valuation by Duplication. Since the same value results regardless of investor preferences, we can pretend that investors are risk-neutral as an aid in calculating values. In this case, the expected return on the underlyer is the riskless return rf . Letting π denote the risk-neutral probability of an up jump, we have: St =
1 · [π · St · U + (1 − π) · St · D] rf
(10)
implying π · U + (1 − π) · D = rf
(11)
or equivalently rf − D U −D Under risk-neutrality, the expected payoff of a 1-period call is:
π=
− c+ 1 = (1 − π) · c1
(12)
(13)
where recall: � � + c+ 1 = max 0, S1 − K is the call value in the up state � − � c− 1 = max 0, S1 − K is the call value in the down state π=
rf −D U −D
is the risk-neutral probability of an up jump found by equating the expected
return from the underlying to the riskless return. In a risk-neutral world, the expected return on an option is also the riskless return. Thus, the initial call value c0 is given by discounting the above expected payoff at the
riskless return rf :
� � − c0 = rf−1 · π · c+ 1 + (1 − π) · c1
(14)
Valuation by Duplication is the reason why risk-neutral valuation works. The same idea can be used for puts and for multiple periods. Risk-neutral valuation permits very quick calculation of option values. Risk-neutral valuation, hence, simplifies the valuation problem (compared to valuation by duplication which was mighty cumbersome for multiple periods). Furthermore, the composition of the duplicating portfolio is easily calculated once the option values are known. Armed with the risk-neutral probabilities π and 1 − π, we can duplicate and price anything. For example, to duplicate a contingent claim paying XU dollars if the spot price goes up and XD dollars if the spot price goes down, we use risk-neutral valuation to calculate the current price of the contingent claim as: rf−1 · [π · XU + (1 − π) · XD ]
(15)
which is interpreted as the discounted expected payoff under risk-neutrality. Example: Recall (see previous page) for 1-period calls, � � − c0 = rf−1 · π · c+ 1 + (1 − π) · c1
(16)
where XU = C1+ and XD = C1− . If we think of XU and XD as option values in successor nodes, then we have a recipe for valuing in multiple periods as illustrated next. Note: The risk neutral probabilities π and 1 − π are usually referred to also as the equivalent martingale probabilities.
Two Period Example
Given T = 2, S0 = 40, U = 2, D =
1 ,r 2 f
=
5 , 4
K = 50, what is the risk-neutral
probability of an up jump? 5 − 12 rf − D 1 4 π= = 1 = 2 U −D 2− 2
(17)
Use risk-neutral valuation to calculate the values of a call at each node bow. Also give the number of stocks and the amount lent in order to replicate the call value at each node. Compare your answers with those obtained using the method of replication (see 2-period example in Section 3 of Overheads 3).
+
160
80
+ c1
40
40
c2
0
c0
c2 -
20
c1
10
Figure 7 : Two period binomial
Figure 8: Two period binomial
tree, call option
tree, call option
-
c2
− 0 Answer: Note that c+ 2 = max[0, 160 − 50] = 110, while c2 = c2 = 0$.
c+ 1 c− 1 c0
� � 1 1 1 = · · 110 + · 0 2 2 rf = 0 � � 1 1 + 1 − = · · c + · c1 rf 2 1 2
(18) (19) (20) (21)
The call’s Delta and the amount lent at each node is given by:
0 c+ 11 2 − c2 =
+ 0 12 S2 − S 2 + 0 0 + 1 c S − c S
= 2 � 2 + 2 02� = −29 3 rf S 2 − S 2
m + = 1
(22)
B1+
(23)
0 c+ 11 1 − c1 =
+ 0 15 S1 − S 1 + 0 0 + 11
c S − c S
= 1 � 1 + 1 01� = −11 15 rf S1 − S 1
m0 =
(24)
B0
(25)
Use risk-neutral valuation to calculate the values of a put at each node bow. Also give the number of stocks and the amount lent in order to replicate the put value at each node.
+
160
80
40
+ p1
0
p0
40
p2
p2 -
p1
20 10
Figure 9: Two period binomial
Figure 10: Two period binomial
tree, put option
tree, put option
-
p2
0 Answer: Note that p− 2 = max[0, 50 − 10] = $40, while p 2 = max[0, 50 − 40] = $10
and p+ 2 = 0. Therefore,
p+ 1 p− 1 p0
1 = rf 1 = rf 1 = rf
�
� 1 1 · · 10 + · 0 = $4 2 2 � � 1 1 · · 10 + · 40 = $20 2 2 � � 1 + 1 − · · p + · p1 = $9.60 2 1 2
(26) (27) (28)
We can use Put-Call Parity to check whether our answers make sense. Recall p0 + S0 =
c0 + K · B(0, 2).
LHS = p0 + S0 = 9.60 + 40 = $49.60.
RHS = c0 + K · B(0, 2) = $17.60 + $32 = $49.60.
What is the Intuition?
If there is only one source of uncertainty in the underlying stock, then the effect of the random shock is fully reflected in the underlying stock price. In such a setting, option becomes redundant. Investors are risk averse: they are worried about the systematic random fluctuations in the stock price. This fear is fully ex-pressed when investors price the underlying security. When it comes to price options, investors are already at a comfortable level regarding risk and reward. Options, being redundant in this setting, provide no additional information about the risk or the reward. There is no assumption about investors being risk neutral. Risk neutral pricing is a trick to simplify option pricing.
The Assumptions
1. Constant riskfree borrowing and lend rate rf .
2. The underlying asset can be continuously traded with no transactions costs, no short sale constraints, perfectly divisible, and no taxes.
3. We assume no dividends, but this restriction can be relaxed.
4. The price of the underlying asset follows a Geometric Brownian Motion: dSt = µ · dt + σ · dBt St
(29)
This model is the continuous time version of the random walk model we have studied in Class 9. The increment ∆Bt of a Brownian motion is normally distributed with mean zero and variance ∆t.
The Black Scholes Formula
A European call struck at K, expiring on date T. C0 = S0 · N (d1 ) − e−rT KN (d2 )
(30)
where � + rf + K √ d1 = σ· T � � � ln SK0 + rf − √ d2 = σ· T
ln
� S0 �
σ2 2
σ2 2
� �
·T
·T
(31)
(32)
and where
• S0 is the initial stock price, • σ is its volatility • rf is the riskfree rate N (dx ) is the probability that the outcome of a standard normal distribution is less than d.
Put/Call Parity
c − p = Stz − K · B (t, T )
(33)
= max [0, Stz − K · B (t, T )] − max [0, K · B(t, T ) − Stz ] .
(34)
c − p = S − e−r(T −T ) · K
put − call parity
The value of the call minus the value for the put is equal to the value for the stock minus the present value of K. We have two investments: 1. Buy the call option and sell the put option. Value: c − p 2. Go long the stock and sell a riskless, zero-coupon bond maturing at time T to K. Value: S − e−r(T −t) · K
Neither of these instruments incur any costs during their lifetime. Let‘s examine their values at time T, starting with investment one, the long-call, short-put investment. Since the call and the put have the same strike, at expiration either the call will be in the money or the put will be in the money - but never both. Write ST for the value for the stock at time T. If the call is in the money, the payoff is ST − K, since the position is long. On the other hand, if the short put is in the money, the its payoff is −(K − ST ) = ST − K. That is, since the position is short, its payoff is the negative of the usual K − ST , independent of the stock price at Time T. The second investment, the stock-bond portfolio is comparatively easy to value. At time T, the bond will have matured to a value of K, and therefore the long-stock, short-bond position will have a value of ST − K. Both investments have the same value at time T, and moreover, cost nothing to maintain. Therefore, our basic arbitrage argument tells us the investments must have the same initial value, that is: c−p � �� �
value of investment 1
=
−r(T −t) · K� S � − e ��
value of investment 2
(35)
Reflections of Volatility
Option Price (S)
in-the-money
at-the-money
out-of-the-money Volatility (σ)
Figure 11: Volatility-levels for in, at or out-of-the-money options.
100 90 80 70 60 50 40 30 20
vxn
7/2/2001
1/2/2001
7/2/2000
1/2/2000
7/2/1999
1/2/1999
7/2/1998
1/2/1998
7/2/1997
1/2/1997
7/2/1996
1/2/1996
7/2/1995
1/2/1995
7/2/1994
1/2/1994
7/2/1993
1/2/1993
7/2/1992
1/2/1992
7/2/1991
0
1/2/1991
10
vix
Figure 12: Distribution of the Nasdaq-index returns for the year 2000, data-source: Bloomberg.
VXN is based on the implied volatility of the Nasdaq-100 (NDX) options, while the VIX volatility is based on the implied volatilities of the S&P 100-OEX options.
The Options Market and Volatility
A casual inspection of the options market leads one to believe that stock volatility does not stay constant over time. Why? Investors express their view on the future market volatility by trading options. Effectively, the equity options market serves as an information central, collecting updates about the future market volatility: • short dated options: near term volatility; • long dated options: long term volatility.
100
1400
90
1200
80 1000
70
800
60 50
600
40
400
30 20
200
10
0
NDX
7/2/1997
1/2/1997
7/2/1996
1/2/1996
7/2/1995
1/2/1995
7/2/1994
1/2/1994
7/2/1993
1/2/1993
7/2/1992
1/2/1992
7/2/1991
1/2/1991
0
VXN
Figure 13: VXN: Impl. volatility of options on the NDX; NDX: Impl. volatility of Nasdaq-100 index, datasource: Bloomberg Professional.
Important: VXN and NDX can have somewhat negative correlations!
The Option Implied Volatility
At time 0, a call option struck at K and expiring on date T is traded at C0 . At the same time, the underlying stock price is traded at S0 , and the riskfree rate is rf . If we know the market volatility at time 0, we can apply the Black Scholes formula: 0 = BS (S0 , K, T, σ, rf ) CBS
(36)
Volatility is something that we don’t observe directly. But using the market observed price C0 , we can back it out: � � C0 = BS S0 , K, T, σ I , rf
(37)
If the Black Scholes model is the correct model, then the Option Implied Volatility σ I should be exactly the same as the true volatility σ.
Why Options?
Hedging, (speculative) investing, and asset allocation are among the top reasons for option trading. In essence, options and other derivatives provide a tailored service of risk by slicing, reshaping, and re packaging the existing risks in the underlying security. The risks are still the same, but investors can choose to take on different aspects of the existing risks in the underlying asset.
Reasons Institutions, Use Equity Derivatives 0%
10%
20%
30%
40%
50%
Investing
44% 42%
Asset Allocation Indexing
40%
Market Timing
26%
International Access
Riasing Cash (e.g. to cover redemption) Tax Minimization
70%
61%
Hedging
Tailoring Cash Flow
60%
25% 14% 13% 10%
Figure 14: Reasons Institutions, Use Equity Derivatives, Source: Greenwich Associates Survey of 118 Institutional Investors in 1998.
Further Reading
EASY AND ENTERTAINING: ”The Universal Financial Device” Chapter 11 of Capital Ideas by Peter Bernstein. SERIOUS, INTRODUCTORY LEVEL MATERIALS: Options, Futures, and Other Derivative Securities by John Hull. INSTITUTIONAL, FOR PRACTITIONERS: the Risk magazine, published monthly. http://www.riskpublications.com
Focus: BKM Chapters 20. • p. 652-657, know how to read listed option quotations, difference American vs. European option • p. 662-670, option strategies • p. 671-673, put-call parity • p. 674-679, know basic information about optionlike securities • p. 683-684, know basic information about exotic options type of potential questions: concept check question 1, 2, 3 & 4, 6, 8, 9, p. 688, question 14
Questions for the Next Class
Please read:
• BKM Chapters 21, and
• think about the following questions: There are two sources of uncertainty affecting the SP 500 index: 1. marginal movements due to small chunks of information arrival. 2. market crashes Neither type is diversifiable, and investors are averse to both. • If we want to gauge the fear of market crash, where do we look?
• Why would someone purchase a deep out of the money put option on the SP 500 index?
15.433 INVESTMENTS Class 11: Equity Options Part 2: Empirical Evidence
Spring 2003
The Black Scholes Model
Stock prices follow a geometric Brownian motion. Translating into discrete time, stock prices follow a Random Walk: rt+∆t = µ ·
√ � ∆t + σ ∆εt+∆t
(1)
This model implies that, over time interval ∆t stock returns are normally √ distributed with standard deviation σ ∆t. The market is dynamically complete. Options are redundant, and can be replicated by a dynamic hedging strategy that involves the riskfree account r and the underlying stock S.
The Black-Scholes formula for a call option struck at K and expiring at T: C0BS = BS (S0 , K, T, σ, rf )
(2)
which depends on today’s stock price So, the volatility of the underlying stock a, and the riskfree rate rf .
Implications of the Black & Scholes
The Black-Scholes volatilities implied by options with difference strike prices K and maturities T should be the same. They should also remain constant over time. The volatility implied by options should be consistent with the level of volatility observed in the underlying stock market.
0%
Figure 2: Which volatility?
Implied Volatility
7/15/2001
4/15/2001
1/15/2001
10/15/2000
7/15/2000
4/15/2000
1/15/2000
10/15/1999
7/15/1999
Volatility MA(20)
4/15/1999
1/15/1999
10/15/1998
7/15/1998
4/15/1998
1/15/1998
10/15/1997
7/15/1997
4/15/1997
1/15/1997
10/15/1996
7/15/1996
Implied Volatility S&P500 Index
Volatility MA(60)
Figure 1: Which Volatility?
45%
40%
35%
30%
25%
20%
15%
10%
5% 7/15/2001
5/15/2001
3/15/2001
1/15/2001
11/15/2000
9/15/2000
7/15/2000
5/15/2000
3/15/2000
1/15/2000
11/15/1999
9/15/1999
7/15/1999
5/15/1999
3/15/1999
1/15/1999
11/15/1998
9/15/1998
7/15/1998
5/15/1998
3/15/1998
1/15/1998
11/15/1997
9/15/1997
7/15/1997
5/15/1997
3/15/1997
1/15/1997
11/15/1996
9/15/1996
7/15/1996
S&P 500 Index Volatility
Time Series of Stocks and Options 50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
Evidence against the Black & Scholes
The Black-Scholes implied volatility does not stay constant over time.
The volatility implied by the options market is on average higher than that observed directly from the underlying stock market.
On any given day, options (both puts and calls) with different K and T exhibit a pattern of ”smile” or ”smirk”: • OTM puts have higher implied-vol than ATM options and OTM calls. • This ”smile” pattern is more pronounced in short-dated options. • The volatility implied by long-dated options differs from that implied by short-dated options.
The Assumption of Constant Volatility
The fact that volatility is not constant can be seen from both the underlying stock market and the options market.
In fact, volatility is itself a stochastic process. For example, letting Vt = σt2 , a popular stochastic volatility model is: � � � dVt = k V¯ − Vt dt + η Vt dBtv
(3)
Some empirical ”regularities”: 1. volatility is stochastic (random); 2. volatility is persistent; 3. volatility is mean-reverting; 4. the volatility shocks are negatively correlated with the underlying price shocks. For each regularity, can you provide a relevant empirical evidence from the options market?
The Assumption of No Jump
The fact that stock prices experience sudden jumps can be seen from both the underlying stock market and the options market (why?).
Price is continuous in the Black Scholes: rt+∆t = µ ·
√ � ∆t + σ ∆εt+∆t
(4)
√ As we shrink the time interval ∆t to zero, the volatility σ ∆t of the surprise component also shrinks to zero.
In 1976, three years after the Black Scholes, Bob Merton proposed a jump extension. rt+∆t = µ ·
√
∆t + σ
� ∆εt+∆t + Jt+∆t
(5)
Can you provide the relevant empirical evidence from the options market indicating negative price jumps in the underlying?
The Assumption of Complete Market
Although it is not an explicit assumption made by the Black Scholes, market is dynamically complete with respect to the underlying stocks and the riskfree account.
The intuition is that there is only one source of uncertainty (associated with the Brownian motion), which can be fully reflected by the underlying stock prices.
If in addition to the Brownian shocks, there are jump risks in the underlying stock, then the market is no longer complete.
Similarly situation arises if the stock volatility has random shocks of its own.
In both cases, investors’ aversion to the additional sources of uncertain will be reflected in option prices.
Fear of Jumps
The options market, especially that of OTM puts, provides a unique opportunity for investors to express their aversion to negative price jumps.
The market prices of such OTM puts reflect not only the probability and magnitude of such jumps, but also investors’ aversion to such jumps.
This aversion to negative jump risks is closely related to the aversion to negative skewness.
Just like backing out Black-Scholes implied volatility, one can back out the probability and magnitude of negative jumps from OTM puts.
It is found that option-implied jump risks are much exaggerated in comparison with those observed directly from the spot market (Jun Pan 2001), why?
Summary
The Black-Scholes model provides a powerful framework for us to formulate option pricing.
The Black-Schole formula provides a powerful tool for us to measure the sensitivity of option prices with respect to the market condition.
To option traders, such sensitivity measures are as essential, if not more, as breathing air in and out.
There is mounting evidence, from both the underlying stock market and the options market, suggesting that the assumptions of the Black Scholes are overly simplified.
In this class, we examined the effect of stochastic volatility, price jumps, and fear of jumps. Focus: BKM Chapters 21. • p. 697-703, • p. 708-711, • p. 714-716, • p. 718-722 type of potential questions: concept check question 1 to 8, table 21.3/21.4, p. 732 question 7, 10, 13, 14, 34, 22
Questions for the Next Class
Please read: • BKM Chapter 14, • Kritzman (1993) What are the main differences between stocks and bonds?
Why do people believe that bonds are safer than stocks? Look up the Wall Street Journal, get a feeling of the daily percentage changes in stock prices and bond prices.
15.433 INVESTMENTS
Class 13: The Fixed Income Market Part 1: Introduction
Spring 2003
Bonds. 9/6/1993
-7%
-12%
SBTSY10 CCMP
8%
3%
-2%
-7%
-12% 9/6/1993
3/6/1998
1/6/1998
11/6/1997
9/6/1997
7/6/1997
5/6/1997
3/6/1997
1/6/1997
11/6/1996
9/6/1996
7/6/1996
5/6/1996
3/6/1996
1/6/1996
11/6/1995
9/6/1995
7/6/1995
5/6/1995
3/6/1995
1/6/1995
11/6/1994
9/6/1994
7/6/1994
5/6/1994
3/6/1994
1/6/1994
11/6/1993
1/6/1999
3/6/2000
3/6/2000
9/12/2001
7/12/2001
5/12/2001
3/12/2001
1/12/2001
3/6/2001 5/6/2001
3/6/2001 5/6/2001
9/6/2001
9/6/2001
7/6/2001
1/6/2001
1/6/2001
7/6/2001
9/6/2000 11/6/2000
11/6/2000
7/6/2000
11/12/2000
9/6/2000
7/6/2000
5/6/2000
1/6/2000
1/6/2000
5/6/2000
9/6/1999
9/12/2000
7/12/2000
5/12/2000
3/12/2000
7/6/1999
11/6/1999
1/12/2000
11/6/1999
5/6/1999
9/6/1999
7/6/1999
11/12/1999
3/6/1999
1/6/1999
11/6/1998 3/6/1999
9/6/1998 11/6/1998
9/6/1998
5/6/1999
9/12/1999
7/12/1999
5/12/1999
3/12/1999
7/6/1998
-2%
5/6/1998
3%
7/6/1998
8%
1/12/1999
CCMP
5/6/1998
3/6/1998
1/6/1998
11/6/1997
9/6/1997
7/6/1997
5/6/1997
3/6/1997
1/6/1997
11/6/1996
9/6/1996
7/6/1996
5/6/1996
3/6/1996
1/6/1996
11/6/1995
9/6/1995
7/6/1995
5/6/1995
3/6/1995
1/6/1995
11/6/1994
9/6/1994
7/6/1994
5/6/1994
3/6/1994
1/6/1994
11/6/1993
SPX
11/12/1998
9/12/1998
7/12/1998
5/12/1998
3/12/1998
1/12/1998
11/12/1997
9/12/1997
7/12/1997
5/12/1997
3/12/1997
1/12/1997
11/12/1996
9/12/1996
7/12/1996
5/12/1996
3/12/1996
1/12/1996
11/12/1995
9/12/1995
7/12/1995
5/12/1995
3/12/1995
1/12/1995
Stocks and Bonds
SPX
8%
3%
-2%
-7%
-12%
SBTSY10
Figure 1: Returns from July 1985 to October 2001 for the S&P 500 index, Nasdaq-index and 10 year Treasury
12% 10%
Probability
8% 6% 4% 2% 0% -2% -0.080
0.020
0.120
0.220
0.320
0.420
0.520
0.620
0.720
0.820
0.920
Monthly Returns
Current Distribution
Normal-Distribution
Figure 4: Return-distribution of 1-month Libor rates from 1985 to 2001.
7% 6%
Probability
5% 4% 3% 2% 1% 0% -1% -0.080
-0.060
-0.040
-0.020
-
0.020
0.040
Monthly Returns
Current Distribution
Normal-Distribution
Figure 5: Return-distribution of 10-year US treasury bonds from 1985 to 2001. Data source for Figures 1, 4, and 5: Bloomberg Professional.
0.060
0.080
Zero-Coupon Rates
n-year zero rt,t+n : the interest rate, determined at time t, of a deposit that starts at time t and lasts for n years. All the interest and principal is realized at the end of n years. There are no intermediate payments. Suppose the five-year Treasury zero rate is quoted as 5% per annum. Consider a five-year investment of a dollar:
compounding annual
$ 1 grows into (1+0.05)5 = 1.276
semiannual
(1+0.05/2)10 = 1.280
continuous
e0.05·5 = 1.284
Zero Coupon Yield-Curve
For any fixed time t, the zero coupon yield curve is a plot of the zero-coupon rate rt+n , with varying maturities n:
r t,n
5
10
15
20 Maturity n (years)
Figure 6: Zero-coupon yield curve.
Treasury Bills
T-bills are quoted as bank discount percent rBD . For a $10’000 par value T-bill sold at P with n days to maturity: rBD =
10 000 − P 360 · 10 000 n
(1)
Conversely, the market price of the T-bill is � n � P = 10 000 · 1 − rBD · 360
(2)
Treasury Bond and Notes
Figure 8: Cash flow representation of a simple bond, Source: RiskM etrics T M , p. 109.
Maturity at issue date: T-notes are up to 10 years; T-bonds are from 10 to (30)
years.
Coupon payments with rate c%: semiannual (November and May).
Face (par) value: $1,000 or more.
Prices are quoted as a percentage of par value.
If purchased between coupon payments, the buyer must pay, in addition to the quoted
(ask) price, accrued interest (the prorated share of the upcoming semiannual coupon).
Some T-bonds are callable, usually during the last five years of the bond’s life.
U.S. Government Bonds and Notes
Footnote1
1 Footnote: Treasury bond, note and bill quotes are from midafternoon. Colons in bond and note bid-and-asked quotes represent 32nds; 101:01 means 101 1/32. Net change in 32nds. n-Treasury Note. i-Inflation-indexed issue. Treasury bill quotes in hundredths, quoted in terms of a rate of discount. Days to maturity calculated from settlement date. All yields are to maturity and based on the asked quote. For bonds callable prior to maturity, yields are computed to the earliest call date for issues quoted above par and to the maturity date for issues quoted below par.
Bond Pricing with Constant Interest Rate
Assume constant interest rate r with semiannual compounding.
All future cash flows should be discounted using the same interest rate r. (Why?)
1 1 1 0 0
0 1
2
3
4
5
n
5
n
Zero Rate r t,n
12
10
8 6 4 2 0
1
2
3
4
Figure 11: Bond pricing with constant interest rate.
The bond price as a percentage of par value:
P =
10 � i=1
�
c 2
1+
� r t 2
+�
100 1+
� r t 2
(3)
Time-Varying Interest Rates
In practice, interest rates do not stay constant over time. If that is the case, then the short- and long-term cash flows could be discounted at different rates. That is, rt,t+n varies over n.
1
2
3
4
5
n
Zero Rate r t,n
Figure 12: Bond pricing with time varying interest rates.
The time-t bond price as a percentage of par:
P =
10 � i=1
�
1
Where n1 = 0.5, n2 = 1, . . . , n10 = 5
c 2 r i �t + 0,n 2
+�
100 1+
r0,ni �10 2
(4)
Yield to Maturity
The yield to maturity (YTM) is the interest rate that makes the present value of a bond’s payment equal to its price:
P =
10 � i=1
� 1+
c 2 � Y TM t 2
+�
100
1+
� Y T M 10 2
(5)
where bond has T-year to maturity and pays semiannual coupon with rate c%:
If the interest rate is a constant r, then the YTM equals r;
In practice, the interest rate is not a constant; The time-t n-year zero-coupon rate
rt,t+n varies over time t, and across maturity n; Intuitively, the YTM for a T-year bond is a weighted average of all zero-coupon rate
rt,t+n between n = 0 and n = T ;
What is the difference between the YTM and the holding period return for the same
bond?
Duration
Duration =
T � P V (CFt ) t=1
P
·t
(6)
T 1 � t · (CFt ) = P t=1 (1 + r)t � � 1 1 · CF1 2 · CF2 T · CFT + = + + ··· + P (1 + r)1 (1 + r)2 (1 + r)T
�T
Duration =
t·(CFt ) t=1 (1+r)t
P
M odif ied Duration =
(7) (8)
�T
t·CFt t=1 (1+r)t CFt t=1 (1+r)t
= �T
M acaulay Duration 1 + mr
(9)
(10)
where m represents the number of interest payments per year and r the interest rate.
Ef f ective Duration = −
1 dB · B dy
Dollar Duration = D · B
(11)
(12)
where B stands for bond value. Foonote2
2
If nothing else mentioned, we assume that a duration is defined as a modified duration!
Example: An obligation with a redemption price of 100 and an current market-price of 95.27 has a coupon of 6% (annual coupon payments) and has a remaining maturity of 5 years with a yield of 7%. Calculate the Macaulay-Duration.
t
cash flow
pv-factor
pv of cf
cf weight
pv time -weighted with t
#1
#2
#3
1
6.00
0.9346
5.6075
0.05886
0.05886
2
6.00
0.8734
5.2401
0.05501
0.11002
3
6.00
0.8163
4.8978
0.05141
0.15423
4
6.00
0.7629
4.5774
0.04805
0.19219
5
106.00
0.71299
75.5765
0.79329
3.96644
Duration
#4#=#2·#3
#5=#4/price
#6=#1*#5
4.48
Duration Hedge
Recall, that the price change (dP) from a change in yields (dy) is: dP = −D · P · dy
(13)
[if you have to think twice why D is negative → don’t select fixed income portfolio manager as a career option!] ∆S = −DS · S · ∆y
(14)
∆F = −DF · F · ∆y
(15)
Where DS is the duration of the spot position and DF is the duration of the Futuresposition. 2 σS2 = (DS · S)2 · σ∆y
(16)
2 σF2 = (DF · F )2 · σ∆y
(17)
2 σSF = (DF · F ) · (DS · S) · σ∆y
(18)
The number of futures contracts is: N =−
DS · S σSF =− 2 DF · F σF
(19)
If we have a target duration DT , we can get it by using: N =−
σSF DS · S =− DF · F σF2
(20)
Example 1: A portfolio manager has a bond portfolio worth $10 mio. with a modified
duration of 6.8 years, to be hedged for 3 months. The current futures price is 93-02,
with a notional of $100’000. We assume that the duration can be measured by CTD,
which is 9.2 years.
Compute:
1. The notional of the futures contract;
2. The number of contracts to buy/sell for optimal protection. Solution: 1. The notional of the futures contract is: (93 + 2/32)/100 · $100 000 = $93 062.5
(21)
2. The number of contracts to buy/sell for optimal protection. N∗ = −
6.8 · $10 000 000 DS · S =− DF · F 9.2 · $93 062.5
(22)
Note that DVBP of the futures is 9.2 · $93 062.5 · 0.01% = $85 Example 2: On February 2, a corporate treasurer wants to hege a July 17 issue of $ 5 mio. of CP with a maturity of 180 days, leading to anticipated proceeds of $ 4.52 mio. The September Eurodollar futures trades at 92, and has a notional amount of $ 1 mio. Compute: 1. The current dollar value of the of the futures contract; 2. The number of contracts to buy/sell for optimal protection. Solution: 1. The current dollar value is given by: $10 000 · (100 − 0.25 · (100 − 92)) = $980 000
(23)
Note that the duration of futures is 3 months, since this contract refers to 3-month LIBOR. 2. If rates increase, the cost of borrowing will be higher. We need to offset this by a gain, or a short position in the futures. The optimal number of contracts is: N∗ = −
DS · S 180 · $4 520 000 000 =− = −9.2 90 · $980 000 DF · F
Note that DVBP of the futures is 0.25 · $1 000 000 · 0.01% = $25
(24)
Convexity
The duration should not be used for big swings in the term structure. The accuracy of the estimation of the duration-coefficient depends on the convexity. Duration is an approximate estimate of a convex form with a linear function. The stronger the yield-curve is ”curved”, the more the real value deviates from the estimated values. We receive a better estimate applying the first two moments of a Taylor-expansion to estimate the price changes: dP =
dP 1 d2 P · dr + · 2 dr2 + ε dr 2 dr
ε is the residual part of the Taylor expansion and
d2 P dr 2
(25)
is the second derivative of the
bond price relative to the yield. Dividing both parts by the price we obtain: dP dP 1 1 d2 P = · · dr + +ε P dr P 2 dr2
(26)
Replacing the second derivative of the price equation and reformulating the notation of the Taylor-expansion, we get: � � �� T � 1 dP 1 t · CFt 1 dP = · − · t P 2 dr 1 + r t=1 (1 + rt ) � � 1 1 1 1 · 2 · CF1 2 · 3 · CF2 T · (T + 1) · CFt = · · + + ··· + 2 P (1 + r)2 (1 + r) (1 + r)2 (1 + r)T T � 1 1 1 t · (t + 1) · CFt = · · (27) · 2 2 P (1 + r) t=1 (1 + r)t
The notation of the price-convexity results from the Taylor-expansion. The first part is the approximation based on the duration. The second part is an approximation based on the convexity of the price/yield-relationship. The percent-approximation results from the duration and the convexity by summing up the individual components. The convexity is defined as: convexity =
1 d2 P 1 · · 2 dr2 P
(28)
dP dP 1 1 d2 P 1 = · · dr + · 2 · (dr)2 + ε P dr P 2 dr P
(29)
∆P ∆r = −D · + convexity · (∆r)2 1+r P
(30)
Based on duration and convexity the price change for small changes in the market return can be expressed as price change instead a percentage number: ∆P = −Duration ·
∆r + convexity · (∆r)2 1+r
(31)
Building Zero Curves
The coupon-bearing T-notes and bonds can be thought of as packages of zero-coupon securities. For any time t, a zero-curve builder uses all such coupon-bearing securities traded in the market at time t to calculate the zero-coupon rates rt,t+n for all possible maturities n. A sophisticated procedure will take into account of the illiquidity and mis-pricing of bonds, as well as tax-related issues. In recent years coupon-bearing securities have been ”stripped” into simpler packages of zero-coupon securities. For example, a 5-year T-note can be divided up and sold as 10 separate zero-coupon bonds, or as a ”strip” of coupon payments and a five-year zero-coupon bond.
Focus: BKM Chapter 14
• All pages except p. 434 after-tax returns Style of potential questions: Concept checks 1, 2, 3, 4, 6, 8, 9, p. 443 ff question 1, 5,
14, 15
Preparation for Next Class
Please read: • BKM Chapter 15, and • Kao (1993).
15.433 INVESTMENTS
Class 14: The Fixed Income Market
Part 2: Time Varying Interest Rates and Yield Curves
Spring 2003
US0001M
Figure 1: Time varying interest rates, Source: Bloomberg. US0003M
Ju n01
Ju n00
Ju n99
Ju n98
Ju n97
Ju n96
Ju n95
Ju n94
Ju n93
Ju n92
Ju n91
Ju n90
Ju n89
Ju n88
Ju n87
Ju n86
Ju n85
T-Bill Rates (monthyly, %)
Time-Varying Interest Rates
12
10
8
6
4
2
0
A Model for Stochastic Interest Rates
Let rt be the time-t one-period interest rate: rt+1 = rt = k (r¯ − rt ) + σ · εt+1
To be consistent with our earlier notation, rt = rt,1 ! Important:
• εt is the random shock that occurs at time t. • The shocks follow a standard normal distribution. • The shocks are independent across time. • k, r, and σ are constant coefficients.
(1)
•
The Coefficients and the Moments
r is the long-run mean of the interest rate. E (rt ) =?
(2)
σ is related to the volatility of the interest rate. var(rt ) =?
(3)
k captures the rate at which the interest rate reverts to its long-run mean, r¯, 0 < k S0 · e(r−rf )·T . An investor can: • Borrow S0 · e−rf ·T in the domestic currency at rate r for time T . • Use the cash to buy spot e−rf ·T of the foreign currency and invest
this at the foreign risk-free rate. • Short a forward contract on one unit of the foreign currency. The holding in the foreign currency grows to unit at time T because of the interest earned. Under the terms of the forward contract, the hold ing is exchanged for F0 at time T . An amount S0 · er−rf ·T is required to repay the borrowing. A net profit of F0 − S0 · er−rf ·T , is therefore, made at time T . Suppose next that F0 < S0 · er−rf ·T . An investor can: • Borrow e−rf ·T in the foreign currency at rate r for time T . • Use the cash to buy S0 · e−rf ·T of the domestic currency and invest this at the domestic risk-free rate. • Take a long position in a forward contract on one unit of the foreign currency. In this case, the holding in the domestic currency grows to S0 · er−rf ·T at time T because of the interest earned. At time T the investor pays F0 and receives one unit of the foreign currency. The latter is used to repay the borrowings. A net profit of S0 · er−rf ·T − F0 , is therefore, made at time T . Note that equation is identical to equation (??) with q replaced by r f . This is not a coincidence. A foreign currency can be regarded as in investment asset paying a known dividend yield. The ”dividend yield” is the risk-free rate of interest in the foreign currency. To see why this is so, note that interest earned on a foreign currency holding is denomi nated in the foreign currency. Its value when measured in the domestic currency is therefore proportional to the value of the foreign currency.
The value of a forward foreign exchange contract is given by equation ?? with q replaced by rf . It is: f = S0 · e−rf ·T
(26)
Example: Suppose that the six-month interest rates in the United States and Japan are 5% and 1% per annum, respectively. The current yen/dollar exchange rate is quote as 100. This means that there are 100 yen per dollar or 0.01 dollars per yen. For a six-month forward contract on the yen S0 = 0.01, r = 0.05, rf = 0.01, T = 0.5. From equation (??), the forward foreign exchange rate as: 0.01e(0.05−0.01)·0.5 = 0.01020 This would be quoted as 1/0.010202 or 98.02.
(27)
Focus: BKM Chapter 16
• p. 485-498 (duration, convexity, eq. 16.2, 16.3, 16.4) • p. 500-508 (immunization) • p. 509-512
• p. 514-515 (swaps) Style of potential questions: Concept check questions, p. 519 ff. ques
tion 1, 3, 4, 10, 26
BKM Chapter 22
• p. 744-749
• p. 758-759
Style of potential questions: Concept check questions, p. 762 ff., ques
tion 4, 13
BKM Chapter 23
• p. 767-772 (eq. 23.1) • p. 786
• p. 790-794
Style of potential questions: Concept check questions,p. 797 ff. question 1,4,7,11, 13,25
Preparation for Next Class Please read:
• BKM chapter 14, and • Duffie and Gˆarleanu (2001).
15.433 INVESTMENTS Class 16: Risk Management
Spring 2003
Introduction
The recent, notable increase in focus on financial risks can be traced in part to the concerns of regulatory and investors about risk exposure of financial institutions through their large positions in OTC derivatives.
The dramatic increase in the availability and usage of derivative products can be traced to several developments: • 1. Because of the rapid improvement in financial modelling and com puter systems, complex derivatives can be offered at more favorable prices and liquidity. • 2. With the liberalization of financial markets around the world, the demand for more sophisticated hedging instruments with wider coverage range has also increased. There certainly have been periods of high volatility in the financial mar ket, but what distinguishes the recent period from earlier periods is that investors have had lower cost access to derivatives that permit highly leveraged positions and, hence, potentially large changes in value for a given change in the value of the underlying instrument. The recent losses on derivative positions, by both financial and non-financial corporations, are clear manifestations of this effect.
Some Losses on Derivatives Position
Orange County: $ 1.7 billion, leverage (reverse repos) and structured
notes
Showa Shell Sekiyu: $ 1.6 billion, currency derivatives
Metallgesellschaft: $ 1.3 billion, oil futures
Barings: $ 1 billion, equity and interest rate futures
Codelco: $ 200 million, metal derivatives.
Proctor & Gamble: $ 157 million, leveraged currency swaps.
Air Products & Chemicals: $ 113 million, leveraged interest rate and
currency swaps.
Dell Computer: $ 35 million, leveraged interest rate swaps.
Louisiana State Retirees: $ 25 million, IOs/POs.
Arco Employees Savings: $ 22 million, money market derivatives.
Gibson Greetings: $ 20 million, leveraged interest rate swaps.
Mead: $ 12 million, leveraged interest rate swaps.
Figure 1: ”Accidents” of the last two decades, source: Reto Gallati, Risk Management and Capital Adequacy, McGraw-Hill, New York, March 2003.
The Economics of Risk Management
The economics of risk management for financial firms is far from an ex act science.
While rigorous and empirically testable models can be brought to the task of measuring financial risks, some of the benefits and costs of bear ing these risks are difficult to quantify.
In a hypothetical world of perfect capital markets, adding or subtract ing financial risk has no impact on the market value of a publicly traded corporation or on the welfare of its shareholders.
We can certainly agree, however, that capital markets are not perfect, and that market imperfections underly significant benefits to bearing and controlling financial risks.
It is difficult to quantify the costs and benefits in bearing/controlling risk. So, rather than a recipe providing in each case the appropriate amount of each type of risk to be borne in light of the costs and benefits, one should aim for a critical review of the nature of risks, the channels through which they can be measured and mitigated.
An appropriate appetite for risk is ultimately a matter of judgment that is informed by quantitative models for measuring risk and based on a conceptual understanding of the implications of risk.
The Leverage of Financial Firms
Compared with other types of corporations, financial firms have rela tively liquid balance sheets, made up largely of financial positions.
This relative liquidity allows a typical financial firm to operate with a high degree of leverage.
For example, major broker-dealers regulated by SEC frequently have a level of accounting capital that is close to the regulatory minimum of 8% of account assets, implying a leverage ratio on the order of 12 to 1.
Ironically, in light of the relatively high degree of liquidity that fosters high leverage, a significant and sudden financial loss (or reduced access to credit) can cause dramatic illiquidity effects.
The Firm’s Vulnerability to Losses
Figure 2: S&P 500 returns and VaR estimates (1.65σ)
The primary focus of risk-management teams at financial institutions is not on traditional financial risk, but rather on the possibility of ex treme losses.
The benefits of this particular focus of risk management usually come from the presence of some kind of non-linearity in the relationship be tween the market value of the firm and its raw profits from operations.
Such non-linearity is typically associated with events that cause a need for quick access to additional capital or credit.
Capital - A Scarce Resource
If new capital could be obtained in perfect financial markets, we would expect a financial firm to raise capital as necessary to avoid the costs of financial distress.
In such a setting, purely financial risk would have a relatively small impact, and risk management would likewise be less important.
In fact, however, externally raised capital tends to be more costly than retained earnings as a source of funding.
External providers of capital tend to be less well informed about the firm’s earnings prospects , charging the firm a ”lemon’s premium” that reflects their informational disadvantage.
They might also be concerned that the firm’s managers have their own agenda, and may not use the capital efficiently.
A Brief Zoology of Risks
The risks faced by financial institutions fall largely into the following broad categories: • Market Risk - unexpected changes in prices or rates. • Credit Risk - changes in value associated with unexpected changes in credit quality. • Liquidity Risk - the risk of increased costs, or inability to adjust
financial positions (for example through widening of spreads), or of lost access to credit. • Operational Risk - fraud, systems failures, trading errors (such as deal mispricing). • Systemic Risk - breakdown in market-wide liquidity, chain-reaction default.
Risk Management in a Non-Financial
Firm, the Case of Merck
Related Materials: • Lecture Notes: ”Corporate Financial Risk Management,” by Dar rell Duffle, Graduate School of Business, Stanford University, Spring Quarter, 1996.
• Judy Lewent and John Kearney, ”Identifying Measuring and Hedg ing Currency Risk at Merck,” Journal of Applied Corporate Finance, vol 2, 1990, pp. 19-28.
Financial Background
As of 1994, Merck had been an extremely profitable firm with a low debt load (the debt/equity ratio is under 2%).
There is no financial distress on the horizon, essentially eliminating that as a motive for hedging.
For 1994, Merck’s sales were $15 billion, of which 32% were foreign. Sales were enhanced by 1% in dollar terms in 1994 by changes in foreign exchange rates. Fluctuations in currency prices reduced earnings by 2% in 1993. R&D expenditures in 1994 were $ 1.2 billion.
A Strong Dollar Scenario
Consider a scenario in which the U.S. dollar has strengthened dramati cally, say 20%. At current levels of foreign revenue, the 20% strength ening in U.S. dollar translates into an unexpected shortfall of about $1 billion in revenues. Management is not to blame for fluctuations in foreign currency markets. Exchange rates are difficult to predict, af ter correcting for differences between domestic and foreign interest rates.
Managers, however, will be expected to deal with the immense amount of shortfall represented by this sort of scenario:
1. Are dividends to be cut? 2. What R&D program? How will it be funded?
Funding all positive NPV projects and at the same time maintaining stable dividends in this scenario could call for issuing new debt.
Not only are debt underwriting costs considerable, crucial information regarding the profitability of the R&D program is likely to be unknown by potential bond investors. This means that the rate of return demanded by outside bond investors may include an extra risk premium for the information that they do not hold.
In other words, what might have been a positive NPV project, when funded with retained earnings, may now be a negative NPV project, when funded with new debt, and may therefore be dropped.
Finally, shareholders may not be aware that weak earnings are due to
financial market effects beyond management control, and could inappro priately blame management.
A Program of Foreign Exchange
Quoting from the 1994 annual report, Merck claims that: ”A significant portion of the Company’s cash flows are denominated in foreign currencies. The Company relies on sustained cash flows gener ated from foreign sources to support its long-term commitment to U.S. dollar-based research and development. To the extent the dollar value of cash flows is diminished as a result of a strengthening dollar, the Company’s ability to fund research and other dollar based strategic initiatives at a consistent level may be impaired. To protect against the reduction in value of foreign currency cash flows, the Company has instituted balance sheet and revenue hedging programs to partially hedge this risk”.
Some Details: The value of purchased currency options, the largest category of hedging instruments shown in Merck’s disclosure under ”fair value of financial instruments”, was $ 42.5 million as of year-end 1994, on a notional amount of $ 1.79 billion underlying these options. • The carrying value of these options is shown as $ 97.6 million, in dicating a loss of $ 55 million, more than half of the value of the options. This is consistent with the hedging role of these options and the fact that sales were enhanced by approximately 1% (roughly $150 million) due to fluctuations in exchange rates. • On a ”delta” basis, one may therefore draw the conclusion that
Merck has hedged roughly one-third of its exposure to foreign ex change rates.
Accounting Issues
Consider a 750 million Euro put option hedge against a 1 billion Swiss franc receivable on next year’s sales.
Suppose the puts expire in one year, and were purchased for about 7.5 million dollars. Suppose the market value of the options dropped to 2.5 million dollars during the next quarter because of a risk in the value of the Euro.
Since there is roughly a 90% correlation between Swiss franc price changes and Deutsch Mark price changes, it is quite likely that the market value of receivable francs has risen and at least partially offset the loss on the options.
If the put position is marked-to-market for accounting purposes, as would be required by an SEC ruling, then the 5 million dollar decline in value of the put position would show up on the balance sheet or income state ment as a reduction of $ 5 million.
The receivable, however, would not typically be marked to market under current accounting standards.
Before the publication of FAS 133, if the options were written on Swiss francs rather than marks, and a number of other conditions were met, then accounting standards would allow losses or gains on the put options to be deferred until the francs are received and recorded.
This situation is often called hedge accounting. In his 1996 lecture notes, Darrell Duffie wrote: ”The criteria for hedge accounting are governed by a bewildering, complicated, and quickly changing array of different ac counting standards”. Sure enough, we now have a new accounting rule, FAS 133 (amended by FAS 137 and 138), which is based on a non economic separation of option time value and intrinsic value.
Futures and Basis Risk
Basis risk arises when the characteristics of the futures contract differ from those of the underlying. For example quality of agricultural products, types of oil, cheapest to deliver (CTD) bond, etc. Basis = Spot − F utures
(1)
Cross Hedging Hedging with a correlated (but different) asset. • In order to hedge an exposure to Norwegian Krone one can use Euro futures. • Hedging a portfolio of stocks with index futures. The optimal Hedge Ratio ∆S
−
change in $ value of the inventory
(2)
∆F
−
change in $ value of the one f utures
(3)
N
−
number of f utures to buy/sell
(4)
∆V = ∆S + N · ∆F 2 2 2 σ∆V
= σ∆S + N 2 · σ∆F + 2 · σ∆S,∆F 2
∂σ∆V 2 = 2 · N · σ∆F ∂N
Minimum variance hedge ratio: σ∆S σ∆S,∆F = −ρ∆S,∆F · Nopt = − 2 σ∆F σ∆F
(5) (6) (7)
(8)
Hedge Ratio as Regression Coefficient
The optimal amount can also be derived as the slope coefficient of a regression ∆S/S on ∆F :
∆S ∆F +ε = α + βSF · S F βSF =
σSF σS = ρSF · 2 σF σSF
(9)
(10)
Optimal Hedge One can measure the quality of the optimal hedge ratio in terms of the amount by which we have decreased the variance of the original portfolio.
2 R2 = σS2 − σV2 σS2 = ρSF � σV = σS 1 − R2
(11)
where V stands for Value including hedge. If R2 is low the hedge is not effective!
At the optimumum the variance of the hedged portfolio is: σV2 = σS2 −
2 σSF σF2
(12)
Example: An airline company needs to purchase 10’000 tons of jet fuel in 3 months. We can use heating oil futures traded on NYMEX. Notional for each contract is 42’000 gallons. We need to check whether this hedge can be efficient. Spot price of jet fuel is $ 277/ton. Futures price of heating oil is $ 0.6903/gallon. The standard deviation of jet fuel price rate of changes over 3 months is 21.17%, that of futures 18.59%, and the correlation is 0.8243. Compute: • The notional and the standard deviation of the unhedged fuel cost in dollars. • The optimal number of futures contracts to buy/sell, rounded to the closest integer. • the standard deviation of the hedge fuel cost in dollars. Solution: The notional is N = $2 770 000, the standard deviation in dollars is: σ(∆S/S) · S · NS = 0.2117 · 277 · 10 000 = $586 409
(13)
The standard deviation of one futures contract in dollars is: σ(∆F/F ) · F · NF = 0.1859 · 0.6903 · 42 000 = $5 390
(14)
The futures notional in dollars is: F · NF = 0.6903 · 42 000 = $28 993
(15)
The position corresponds to a liability (payment), hence we have to buy futures as a protection. 0.2117 = 0.9387 0.1859 = 0.8243 · 0.2117 · 0.1859 = 0.03244 βSF = 0.8243 ·
σSF
(16) (17)
The optimal hedge ratio is: HRopt = βSF ·
NS · S = 89.7, or 90 contracts NF · F
2 = $586 4092 = 343 875 515 281 σunhedged 2 −σSF /σF2 = −(2 605 268 452/5 390)2 2 = $331 997 σhedged
(18) (19) (20) (21)
The hedge has reduced the standard deviation from $586’409 to $331’997. R2 = 67.95% (= 0.82432 )
Term structure strategies Bullet strategy: Maturities of securities are concentrated at some point on the yield curve.
Barbel strategy: Maturities of securities are concentrated at two extreme maturities.
Ladder strategy: Maturities of securities are distributed uniformly on the yield curve.
Example:
bond coupon maturity yield duration convexity
A
8.5%
5
8.5
4.005
19.81
B
9.5%
20
9.5
8.882
124.17
C
9.25%
10
9.25
6.434
55.45
Portfolios: • Bullet portfolio: 100% bond C • Barbell portfolio: 50.2% bond A, 49.8% bond B Dollar-duration of barbell portfolio: 0.502 · 4.005 + 0.498 · 8.882 = 6.434
(22)
It has the same duration as bullet portfolio.
Dollar-convexity of barbell portfolio: 0.502 · 19.81 + 0.498 · 124.17 = 71.78 The convexity here is higher!
The yield of the bullet portfolio is 9.25%. The yield of the barbell portfolio is 8.998%.
This is the cost of convexity !
(23)
Stock Index Futures
A stock index tracks changes int he value of a hypothetical portfolio of stocks. The weight of a stock in the portfolio equals the proportion of the portfolio invested in the stock. The percentage increase in the stock index over a small interval of time is set equal to the percentage increase in the value of the hypothetical portfolio. Dividends are usually not in cluded in the calculation so that the index tracks the capital gain/loss from investing in the portfolio.1 If the hypothetical portfolio of stocks remains fixed, the weights assigned to individual stocks in the portfolio do not remain fixed. When the price of one particular stock in the portfolio rises more sharply than others, more weight is automatically given to that stock. Some indices are con structed from a hypothetical portfolio consisting of one of each of a num ber of stocks. The weights assigned to the stocks are then proportional to their market prices, with adjustments being made when there are stock splits. Other indices are constructed so that weights are proportion to market capitalization (stock price · number of shares outstanding). The underlying portfolio is then automatically adjusted to reflect stock splits, stock dividends, and new equity issues.
The following summary highlights the key differences between the most important stock indices: • The Down Jones Industrial Average is based on a portfolio consisting of 30 blue chip stocks in the United States. The weights given to the stocks are proportional to their prices. • The Standard & Poor’s 500 (S&P500)Index is based on a portfolio 1 An exception to this is a total return index. This is calculated by assuming that dividends on the hypothetical portfolio are reinvested in the portfolio
of 500 different stocks, 400 industrials, 40 utilities, 20 transportation companies, and 40 financial institutions. The weights in the portfolio at any given time are proportional to their market capitalizations. • The NASDAQ 100 is based on 100 stocks using the National Asso ciation of Securities Dealer Automatic Quotations Service. All futures contracts on stock indices are settled in cash, not by delivery of the underlying asset. All contracts are marked to market on the last trading day, and the positions are then deemed to be closed. For most contracts, the settlement price on the last trading day is set at the clos ing value of the index on that day. For the futures on the S&P 500, the last trading day is the Thursday before the third Friday of the delivery month.
Futures Prices of Stock indices An index can be thought of as an investment asset that pays dividends. The asset is the portfolio of stocks underlying the index, and the div idends are the dividends that would be received by the holder of this portfolio. Often there are many stocks underlying the index providing dividends at different times. To a reasonable approximation, the index can then be considered as an asset providing a continuous dividend yield. if q is the dividend yield rate, equation ?? gives the futures price, F 0 , as:2 F0 = S0 · e(r−q)·T
(24)
Example: Consider a 3-month futures contract on the S&P 500. Sup pose that the stocks underlying the index provide a dividend yield of 2 For a total return index, dividends are assumed to be reinvested in the portfolio underlying the index so that q=0 and F0 = S0 · er·T
3% per annum, that the current value of the index is 900, and that the continuously compounded risk-free interest rate is 8% per annum. in this case, r=0.08, S0 =900, T =0.25, and q=0.03, and the futures price, F0 , is given by: F0 = S0 · e(r−q)·T = 900 · e(0.08−0.03)·0.25 = 911.23
(25) (26)
In practice, the dividend yield on the portfolio underlying an index varies week by week throughout the year.
Hedging using Index Futures Stock index futures can be used to hedge the risk in a (usually welldiversified) portfolio or individual stocks (individual stock-futures work better for some specific stocks). We will use β as the coefficient from the CAPM and the regression. This is the slope of the best-fit line ob tained when the excess return on the portfolio over the risk-free rate is regressed against the excess return on the market over the riskfree rate. When β = 1.0, the return on the portfolio tends to mirror the return of the market; when β = 2.0, the excess return on the portfolio tends to be twice as great as the excess return on the market; when β = 0.5, it tends to be half as great; and so on.
When the β of the portfolio equals 1, the position in futures contracts should be chosen so that the value for the stocks underlying the futures contacts equals the total value of the portfolio being hedge. When β = 2, the portfolio is twice as volatile as the stocks underlying the futures con tract and the position in futures contacts should be twice as great. When
β = 0.5, the portfolio is half as volatile as the stocks underlying the fu
tures contract and the position should be half as great. In general, if we
define:
P portfolio value;
F value of assets underlying one futures contract.
The correct number of contracts to short in order to hedge the risk
in the portfolio is:
P (27) F The formula assumes that the maturity of the futures contract is close to β·
the maturity of the hedge and ignores the daily settlement of the futures contract.
Example: A company wishes to hedge a portfolio worth $ 2’100’000 over the next three months using an S&P 500 index futures contract with four months to maturity. The current level of the S&P 500 is 900 and the β of the portfolio 1.5. The value of the assets underlying one futures contract is 900 · 250 = $225 000. The correct number of futures contracts to short is, therefore: 2 100 000 1.5 · = 14 225 000
(28)
To show that the hedge works, we suppose the risk-free rate is 4% per year and the market provides a total return of -7% in the course of the next three months. This is bad news for the portfolio. The risk-free rate is 1% per three months so that the return on the market is 8% below the risk-free rate. We therefore expect the return (including dividends)
on the portfolio during the three months to be 1.5 · 8 = 12% below the risk-free rate, or -11%. Assume that the dividend yield on the index is 2% per annum, or 0.5% per three months. This means that the index declines by. 7.5% during the three months, from 900 to 832.5. Equation ?? gives the initial futures price as: 4
900 · e(0.04·−0.02)· 12 = 906.02
(29)
and the final futures price as: 1
832.5 · e(0.04·−0.02)· 12 = 833.89.
(30)
The gain on the futures position is: (906.02 − 833.89) · 250 · 14 = 252 455
(31)
The total loss on the portfolio is 0.11 · 2 100 000 = $231 000. The net gain form the hedge position is 252 455 − 231 000 or about 1% of the value of the portfolio. This is as expected. The return on the hedged position during the three months is the risk-free rate. It is easy to verify that roughly the same return is realized regardless of the performance of the market.
Questions for Next Class
Read: • Kritzman (1994a) • Kritzman (1994b) • Ross (1999), and • Perrold (1999) regarding hedge funds.
15.433 INVESTMENTS Class 17: The Credit Market Part 1: Modeling Default Risk
Spring 2003
The Corporate Bond Market
25 20 15 10 5 0
Mortgage Rates (Home Loan Mortgage Corporation)
FED Fund
Apr-01
Apr-99
Apr-97
Apr-95
Apr-93
Apr-91
Apr-87
Apr-85
Apr-83
Apr-81
Apr-79
Apr-77
Apr-75
Apr-73
Apr-71
-10
Apr-89
-5
Spread
Figure 1: Mortgage and FED rates, Source : www.federalreserve.gov/releses/hr
AAA
BAA
Spread
Figure 2: Corporate rating spreads, Source : www.federalreserve.gov/releses/hr
Jan-99
Jan-94
Jan-89
Jan-84
Jan-79
Jan-74
Jan-69
Jan-59
Jan-54
Jan-49
Jan-44
Jan-39
Jan-34
Jan-29
Jan-24
Jan-19
-
Jan-64
20 18 16 14 12 10 8 6 4 2
16% 14% 12% 10% 8% 6% 4%
1 Year Maturity
30 Years Maturity
Figure 3: Corporate rating spreads, Source : Moody’s
100% 90% 80% 70% 60% 50% 40% 30% 20%
BBB
AAA
AA-
Figure 4: Corporate rating migration for industry-sector, Source : Standard Poor’s.
D
CCC
B
B
B+
BB
BB
BB+
BBB
BBB
BBB+
A
A
A+
AA-
AA
AAA
0%
AA+
10%
Caa
B3
B2
B1
Ba3
Ba2
Ba1
Baa3
Baa2
Baa1
A3
A2
A1
Aa3
Aa2
Aa1
0%
Aaa
2%
Bond Valuation with Default Risk
100
Cashflow Conditioning on Survival
c/2 0.5
1
1.5
2
2.5
3
3.5
4
τi
Non-Default Default RandomDefaultTime τ%
Figure 5: Chart cash flow Conditioning on survival.
Assuming no default risk, P0 =
8 �
er·ti + 100 · er·4
i=1
How does the default risk affect the bond price?
(1)
Modelling Default Risk
Modelling default risk is central to the pricing and hedging of credit sen sitive instruments. Two approaches to modelling default risk: • Structural approach, ”first-passage”: default happens when the to tal asset value of the firm falls below a threshold value (for example, the firm’s book liability) for the first time.
• Reduced-form, ”intensity-based”: the random default time τ� is gov erned by an intensity process λ.
For pricing purpose, the reduced-form approach is adequate, and will be the focus of this class.
Modelling Random Default Times
The probability of survival up to time t: (2)
P rob(τ� ≥ t) The probability of default? before time t: P rob (τ� < 0) = 1 − P rob (τ� ≥ t)
(3)
We assume that T� is exponentially distributed with constant default
intensity λ:
1
Survival Probability:
(
)
Prob τ% ≥ t = e- λt
0 0
100 t (year)
Figure 6: Survival Probability.
Default Probability and Credit Quality One-Year default probability = 1 − eλ Default intensity λ =?
D CCC B B B+ BB BB BB+ BBB BBB BBB+ AA A+ AAAA AA+ AAA 0%
10% AAA BB
20% AA+ BB
Figure 7: Survival Probability.
AA B+
30% AAB
40%
50%
A+ B
A CCC
60% A D
70% BBB+
80% BBB
90% BBB
100% BB+
AAA AA+ AA AAA+ A ABBB+ BBB BBBBB+ BB BBB+ B BCCC D
AAA 91.95% 2.31% 0.62% 0.00% 0.00% 0.08% 0.14% 0.00% 0.07% 0.05% 0.17% 0.00% 0.00% 0.00% 0.00% 0.00% 0.19% 0.00%
AA+ 4.11% 84.71% 1.36% 0.15% 0.03% 0.06% 0.04% 0.00% 0.03% 0.00% 0.00% 0.00% 0.00% 0.03% 0.00% 0.00% 0.00% 0.00%
AA 2.86% 8.75% 85.42% 3.44% 0.83% 0.49% 0.11% 0.08% 0.07% 0.11% 0.00% 0.12% 0.00% 0.00% 0.07% 0.00% 0.00% 0.00%
AA0.48% 2.88% 7.24% 83.67% 4.47% 0.66% 0.35% 0.13% 0.17% 0.21% 0.08% 0.06% 0.05% 0.10% 0.00% 0.00% 0.00% 0.00%
A+ 0.16% 0.19% 2.60% 8.61% 82.27% 5.25% 1.13% 0.59% 0.45% 0.11% 0.08% 0.06% 0.09% 0.00% 0.00% 0.18% 0.19% 0.00%
A 0.20% 0.48% 1.49% 3.02% 8.08% 82.50% 8.58% 2.26% 0.93% 0.69% 0.51% 0.37% 0.05% 0.03% 0.14% 0.00% 0.00% 0.00%
A0.12% 0.10% 0.25% 0.50% 2.75% 5.44% 77.39% 8.32% 2.24% 0.59% 0.34% 0.18% 0.28% 0.23% 0.21% 0.00% 0.19% 0.00%
BBB+ 0.04% 0.00% 0.50% 0.23% 0.46% 3.18% 7.21% 75.24% 7.83% 2.67% 0.67% 0.31% 0.33% 0.10% 0.00% 0.36% 0.19% 0.00%
Figure 8: Survival Probability, Migration table, Source: RiskMetrics T M .
16% 14% 12% 10% 8% 6% 4% 2% 0% Aaa Aa1 Aa2 Aa3 A1 A2 A3 Baa1 Baa2 Baa3 Ba1 Ba2 Ba3 B1
B2
Figure 9: One-Year Default Rates by Modified Ratings, 1983-1995, Source: Moodys (1996).
B3
BBB 0.04% 0.38% 0.22% 0.15% 0.40% 1.11% 3.00% 8.36% 77.76% 9.46% 4.21% 1.59% 0.52% 0.13% 0.14% 0.00% 0.56% 0.00%
Pricing A Defaultable Bond
For simplicity, let’s first assume that the riskfree interest rate r is a constant. Consider a τ -year zero-coupon bond issued by a firm with default intensity λ: P0 = $100 · e−r·τ · P rob(τ� ≥ τ )
(4)
P0 = $100 · e−r·τ · e−λ·τ
(5)
P0 = $100 · e−(r+λ)·τ
(6)
where we assume that conditioning on a default, the recovery value of the bond is 0 (we have also assumed risk-neutral pricing). The yield on the defaultable bond is r + λ, resulting in a credit spread of λ.
Time Variation of Default Probability
10 8 6 4 2 0 -2
GDP Growth / Chain QQQ%
Linear (GDP Growth / Chain QQQ%)
Figure 10: Chart Annual GDP Growth Rate, source: Bureau of Economic Analysis Stochastic
Dec-01
Dec-00
Dec-99
Dec-98
Dec-97
Dec-96
Dec-95
Dec-94
Dec-93
Dec-92
Dec-91
Dec-90
Dec-89
Dec-88
Dec-87
Dec-86
-4
Default Intensity
In general, the credit quality of a firm changes over time. A more realistic model is to treat the arrival intensity as a random process. Suppose that intensities are updated with new information at the begin ning of each year, and are constant during the year. Then the probability of survival for t years is
For example,
� � E e−λ0 +λ1 +···+λt−1
(7)
� � ¯ − λt + εt+1 λt+1 − λt = k λ
(8)
Can you calculate the probability of survival for τ years? What is the price of a τ -year zero-coupon bond? What if the riskfree interest rate is also stochastic?
Example: A portfolio consists of two long assets $100 each. The prob ability of default over the next year is 10% for the first asset, 20% for the second asset, and the joint probability of default is 3%. What is the expected loss on this portfolio due to credit risk over the next year assuming 40% recovery rate for both assets. Probabilities: 0.1 · (1 − 0.2)
−
def ault probability of A
(9)
0.2 · (1 − 0.1)
−
def ault probability of B
(10)
0.03
−
joint def ault probability
(11)
Expected losses: 0.1 · (1 − 0.2) · 100 · (1 − 0.4) = 4.8
(12)
0.2 · (1 − 0.1) · 100 · (1 − 0.4) = 10.8
(13)
0.03 · 200 · (1 − 0.4) = 3.6
4.8 + 10.8 + 3.6 = $19.2 mio.
(14)
(15)
Example: Assume a 1-year US Treasury yield is 5.5% and a Eurodollar deposit rate is 6%. What is the probability of the Eurodollar deposit to default assuming zero recovery rate)? 1 1−π = 1.06 1.055 π = 0.5%
(16) (17)
Example: Assume a 1-year US Treasury yield is 5.5% and a and a default probability of a one year CP is 1%. What should be the yield on the CP assuming 50% recovery rate? 1 1−π 0.5π = + 1+x 1.055 1.055 = 6%
(18) (19)
Some Practitioner’s Credit Risk Model
RiskMetrics: CreditM etricsT M
http://riskmetrics.com/research
Credit Suisse Financial Products: CreditRisk+
http://www.csfb.com/creditrisk
KMV Corporation / CreditM onitor T M
http://www.kmv.com
Focus: BKM Chapter 14 • p. 415-422 (definitions of instruments, innovation in the bond mar ket) • p. 434-441 (determinants of bond safety, bond indentures) Style of potential questions: Concept check questions, p. 448 ff. ques tion 31
Questions for Next Class Please read: • Reyfman, • Toft (2001), and • Altman, Caouette, Narayanan (1998).
15.433 INVESTMENTS Class 18: The Credit Market Part 2: Credit Derivatives
Spring 2003
Introduction
Credit derivatives are financial instruments whose payoffs are linked to certain credit events.
A commonly stipulated credit event is default by a named issuer, which could be a corporation or a sovereign issuer.
More generally, credit events can be defined in terms of downgrade, merger, cross default, failure to pay, restructuring, etc.
According to a survey by the British Bankers’ Association, the credit derivatives market accounted for an estimated $ 586 billion at the end of 1999. It is predicted to reach $ 1.58 trillion by 2002.
Some key credit derivative instruments: credit swap, total return swaps, collateralized debt obligations, spread options, etc.
Credit Swap
()
$100 − Y τ%
0
1
2
3
4
5
U
Figure 1: Cash flows of a credit swap, buyer of a credit swap.
T : the random credit event time. τ : the maturity of the credit swap. U : annuity of the credit swap. $ 100: par value of the underlying. Y (t): time t market value of the underlying.
τ%
τ
Use, Credit Swap to Mitigate Credit Risk
Credit risk often emerges whenever a corporation enters into long term business relationships with large customers or suppliers.
For example, a credit exposure arises when a company has a long term product delivery contract with a customer in exchange for a future cashflow stream.
This exposure can be particularly pronounced if a large upfront investment is required.
To mitigate such credit risk, the company can purchase a credit swap, whose payoff is linked to the credit event of its customer.
Credit Swap Valuation
Define the credit event time as the time when firm XYZ defaults. Use the model of constant default intensity:
For simplicity, let’s assume risk neutral pricing, and constant risk free rate r. Let at (λ) be the time O value of receiving $ 1 at the end of year t in the event that default is after that date: Let −bt (λ) be the time O value of receiving $ 1 at the end of year t in the event that default happens in year t:
bt (λ) = e−r· P rob (t − 1 ≤ T < t) � � −r·t −λ(t−1) −λ t = e e · −e
(1) (2)
The time O value of an annuity of $ 1 paid at the end of each year until default or maturity τ , whichever comes first, is A0 = al (λ) + a2 (λ) + · · · + aτ (λ)
(3)
The time O value of a $ 1 payment made at the end of the firm’s default year, provided the default date is before the maturity date τ , is B0 = bl (λ) + b2 (λ) + · · · + bτ (λ)
(4)
Suppose that the loss of face value at default carries no risk premium and, has an expected value of f. The time 0 value of the default swap is V0 = B0 · f − A0 · U for any given annuity U.
(5)
The default swap spread U is obtained by solving V0 = 0, leaving: U=
B0 f A0
(6)
Collateralized Debt Obligations
A CDO is an asset backed security whose underlying collateral is typically a portfolio of bonds (corporate or sovereign) or bank loans.
A CDO cashflow structure allocates interest income and principal repayments from a collateral pool of different debt instruments to a prioritized collection (tranches) of CDO securities.
A Typical CDO Contractual Relationships
Collateral Manager
Underlying Securities (Collateral)
Senior Notes
Figure 2: CDO-relationships.
Trustee
CDO Special Purpose Vehicle (SPV)
Mezzanine Notes
Hedge Provider (if needed)
Equity
Why CDO?
In perfect capital markets, CDOs would serve no purpose; the costs of constructing and marketing a CDO would inhibit its creation.
In practice, CDOs address some important market imperfections.
First, banks and certain other financial institutions have regulatory capital requirements that make it valuable for them to securitize and sell some portion of their assets, reducing the amount of (expensive) regulatory capital that they must hold.
Second, individual bonds or loans my be illiquid, leading to a reduction in their market values. Securitization may improve liquidity, and thereby raise the total valuation to the issuer of the CDO structure.
The Balance Sheet CDO
FleetBoston plays ”Good Bank-Bad Bank,” unloading $ 1.35 Billion in troubled loans.
Patriarch Partners LLC, a NY fund management boutique, created a CLO to raise about $ 1 billion to acquire the loans.
See the following figure how FleetBoston’s problem loans made their way from Fleet’s books to a special collateralized-debt obligation, funded by investors.
C ( xt , λt ) ≈ delta − gamma in ( xt , λt ) e viv sur ,t) p (λ 0
C ( x0 , λ 0 ) 1-
p( λ,
def aul t
0
t)
RC (x 0 , λ0 )
( recovery
fraction R )
C ( x,λ ) =value of defaultable position at underlying x, default intensity λ
Figure 3: Setting-up a special purpose vehicle.
Economic Principles of Credit Risk
Several Economic principles distinguish credit risk from other types of market risk, and might thereby influence the choice of risk measurements and techniques for risk mitigation: • Adverse selection: setting credit exposure limits; • Winner’s curse: setting limits on credit risk concentrations (by industry, by geographic location, by credit rating, etc.); • Moral hazard: limit borrower’s access to credit.
Adverse Selection
There may be a significant amount of private information regarding the credit quality of a junk bond or a bank loan. An investor may be concerned about being ”picked off” when trading such instruments.
Given the risk of being picked off, the buyer offers a price that, on average, is below the price at which the asset would be sold in a setting of symmetric information.
This reduction in price due to adverse selection is sometimes called a ”lemon’s premium”.
Mitigating Adverse Selection
In general, adverse selection cannot be eliminated by securitization of assets in a CDO, but it can mitigated.
The seller achieves a higher total valuation (for what is sold and what is retained) by designing the CDO structure so as to concentrate the ”lemon’s” problem into small subordinate tranches, leaving the large senior tranch relatively immune to the effects of adverse selection.
The large senior tranche can therefore be sold at a small lemon’s premium.
The issuer can retain significant fractions of smaller subordinate tranches that are more subject to adverse selection.
Credit Switches
While both single name credit default swaps and synthetic securitizations are effective in transferring credit risk from one risk taker to another, there is often significant cost associated with these transactions.
Instead of eliminating the credit exposures, a corporation can retain the same amount of credit risk, but use zero cost credit switches to diversify the credit portfolio.
For example, one can use credit switches to diversify credit risks across different industries.
Measuring Credit Risk
For the purpose of measuring features of the distribution of P&L, credit risk should be viewed and computed as part of market risk.
At the same time, the measurement of credit risk provides its own set of challenges because many credit sensitive instruments are relatively illiquid, remain on a firm’s books, and cannot be reliably marked to market without modeling the default process.
Moreover, the problems of moral hazard and adverse selection recommend the use of limits on the bilateral exposures to counterparties.
Specialized Measures of Credit Risk
There are reasons (e.g., adverse selection and moral hazard) to track credit risk, by counterparty, that go beyond the contribution made by credit risk to overall market risk. To better understand the nature of the exposures to credit risk, risk managers have explored several complementary measures of credit risk, including: • Market value of default loss • Exposure: the risk that one’s exposure to a given counterparty (or within a given concentration area) excess a given threshold.
In addition to these, and other, internal measures of credit risk, financial institutions are required to measure and report credit risk to the appropriate regulatory authorities.
Measuring Credit Exposures
Regulators require financial institutions to measure both current and potential future exposure. Guidelines are set forth by both the DPG and BIS. For example, the DPG guidelines provide the following definitions: • Current Exposure: Xt = Yt+ for a position t with market value Yt , where Yt is the combination of all positions subject to bi-lateral netting agreements with one counterparty. • Potential Exposure: the 99% worst case exposure. • Aggregating Credit Risks: adding current and potential exposures, and then multiplying by an applicable default ratio for the counterparty.
Exposure Disclosure
The DPG recommends reporting of: • the top 20 credit exposures, by counterparty; • aggregate credit risk organized by the 3 categories ”net replacement value,” gross replacement value,” and ”new exposure”; • aggregate credit risk organized by the 3 classes ”credit rating of counterparty”, ”counterparty industry sector” and ”counterparty geographic location” Large broker dealers of OTC derivatives commonly report, to the public, the current replacement cost of their portfolios as a measure of credit risk.
Credit Risk Value and VaR
One can extend the methodology for estimation of portfolio market risk measures, such as VaR, so as to include credit risk: • Catalog the sensitivities of all positions to fluctuations in market value due to 1. changes in underlying market wide risk factors (such as prices,rates, market credit spreads, and volatilities) 2. counterparty specific fluctuations in credit quality • Applying a risk model for changes in market wide risk factors and counterparty specific credit quality, estimate the overall risk all positions, allowing for correlation.
Modeling Correlated Defaults
There are N counterparties, each with some default intensity λi . We dial in correlations between default intensity by allowing for common credit events, arriving according to some intensity process λc . At the arrival of a common credit event, all counterparties are exposed (with probability p) to immediate default.
Effectively, there are two components to the default intensity of counterparty i: λi = λIi + pλc where λIi is the default intensity specific to counterparty i.
(7)
Non Financial Firms
Non financial firms can manage and mitigate their exposures to the credit quality of their customer, suppliers, or clients through a variety of credit derivatives. • Credit Swaps; • Total Return Swaps; • Credit Switches.
Focus: • Reyfman and Toft (2001) • Altman, Caouette, Narayanan (1998).
Preparation for Next Class
Please read: • BKM Chapter 12, • Rubinstein (2001) • Daniel and Titman (1999) and • Fama (1998).
15.433 INVESTMENTS Class 19: Security Analysis
Spring 2003
Introduction
What explains the price we observe from the market? In an efficient market, the price reflects all information available to pub lic. Implicitly, there should be a pricing model that links the market information to the market price. Gathering all of the relevant information, processing it through some pricing model, we have something to say about what the price of a se curity should be. If the market is efficient, and our pricing model is right, then we will be in agreement with the market. What if we disagree? Either our pricing model is wrong, or the mar ket price is wrong. What Information to Use?
1. The broad economic environment (a) The global economy (b) The domestic economy: GDP, employment, inflation, interest rates, budget deficit, consumer confidence index, the business cycle, economic indicators. (c) Federal government policy: fiscal policy, monetary policy, supply-side policies.
2. The industry environment (a) Sensitivity to the Business cycle (b) Industry life cycles (c) Industry structure and performance 3. The firm-specific information (a) accounting information, dividend payout (b) growth opportunity (c) ...
The Present Value Model
To determine the price of a treasury bond, we calculate the present value of future cash flow. In principle, one can apply the same approach to stock valuation, think ing of the future dividend as a stream of coupon payments. Additional issues needed to be addressed:
• Unlike the coupon payments, the dividend payouts are uncertain. What are the appropriate discount rates?
• Dividends are known to be sticky, and some firms do not even pay dividends. Where do we get information about the growth compo nent of a firm?
• Unlike fixed-income securities, stocks do not have maturity dates. How do we take care of dividend payments that are postponed into the infinite future?
A Simple Two-Period Model By definition: � 1 − P�0 P�1 + D � R1 = (1)
P�0 Let I0 be the collection of public information available at time 0, it must be
� � � � � 1 |I�0 − P0 � E P�1 |I�0 + E D � �1 |I�0 =
E R (2) P0 where we take our expectation conditioning on the information available in I0 . Define the intrinsic value of the firm � � � � � � � � E P1 |I0 + E D1 |I0 � � V0 = (3)
� � 1 + E R1 |I0 � � � � � � � � If the market is efficient, accurately reflecting E P1 |I0 , E D1 |I0 , and
� � � � E R1 |I0 , then the market price must agree with the intrinsic value of
the firm.
Projecting into the Infinite Horizon We can move into the future by applying the two-period recursively:
V0
� � �1 E D � � = �1 1 + E R
� � �2 E D � �� � � �� + � � �2 1 + E R1 1 + E R � � �3 E D � �� � � �� � � �� + � � � �3 1 + E R1 1 + E R2 1 + E R +...
� � �n E D � �� + . . . +� � n � i=1 1 + E Ri
(4)
where all expectations are taken with respect to the information available in I0 .
The Gordon Model
Using all of the information available at time 0, the market participants agree that: 1. the dividend growth is constant �
�
� n = D0 (1 + g)n E D
2. the expected return is constant � � �n = k E R
(5)
(6)
where g ≥ 0 and k ≥ 0. This implies that
V 0 = D0
�n ∞ � � 1+g n=1
1+k
(7)
Suppose that the expected rate of return is always higher than the ex pected growth rate: k>g
(8)
Letting x = (1 + g)/(1 + k) < 1, we have � � V0 = D0 x + x 2 + x 3 + . . . � � = D0 x 1 + x + x 2 + . . . = D0 x
1 1−x (9)
Now plugging x = (1 + g)/(1 + k) back in:
What if k < g?
V 0 = D0
1+g 1 · 1 + k 1 − (1 + g)) / (1 + k)
= D0
1+g 1+k 1+g · = D0 1 + k (k − g) (k − g)
(10)
The Economic Implication? The Gordon Model starts from an identity: � � � � 1 = P 1 + D1 − P 0 R P�0
(11)
Making additional assumptions about • constant growth • constant expected return It then arrives at another identity: V0 =
� � �1 E D k+g
There is no economic input in the process. Nevertheless, it is widely used by stock market analysts.
(12)
Stock Prices, Dividend Policies, and
Investment Opportunities
Cash Cow: 100% dividend payout ratio. It provides a stream of divi dends E(Dn ) = $5, maintaining a zero dividend growth g = 0. V0cc =
$5 k cc (13)
Growth Prospects: 40% dividend payout ratio. In any given year, it plows back (1 - 40%) of its earnings to a project generating an ex pected return of k∗ per year. That is, E(D1 ) = $5 · 40% and E(D2 ) = ($5 + $5 · (1 − 40%) · k ∗ ) · 40%, maintaining a dividend growth rate of g = (1 − 40%) · k ∗ . What are the expected returns kCC and kGP ? For example, cc
k =E
�
P1cc + D1cc − P0cc P0cc
�
(14)
Suppose we pick kCC and kGP exogenously. For a risk-neutral investor, kCC and kGP = rf . 1. If rf > k∗, then 2. If rf < k∗, then What if kCC and kGP = k∗ ? What makes the valuation of Cash Cow different from that of Growth Prospects? Different dividend policies? Different investment opportuni ties?
Earnings, Earnings Forecast, and Financial
Analysts
One of the main determinants of a firm’s value is its investment oppor tunity, which can be affected by macro-economic variables as well as the condition of the industry. Focusing on each firm, however, its projected earnings are among the most informative in predicting the growth opportunity of a firm. For this reason, the stock market analysts, as well as investors, pay close attention to firms’ quarterly earnings reports. In addition, financial analysts invest a great deal of time and energy in forecasting firms’ future earnings.
Inflation and the Stock Market
Inflation neutral: changes in the rate of inflation, whether expected or unexpected, ought to have no effect on the expected real rate of return on common stocks. This is consistent with the wisdom that common stock, representing ownership of the income generated by real assets, should be a hedge against inflation. Empirically, however there is much evidence that common stock returns and inflation are negatively related during the post-1953 period. Possible explanations: 1. Stagflation: negative correlation between inflation and real activity. 2. Increased uncertainty when inflation is high, inducing a higher re quired rate of return.
Monetary Policy and the Economy
Using the tools of monetary policy, the Fed can affect the volume of money and credit and their price/interest levels. The initial link between monetary policy and the economy occurs in the market for reserves. The Fed policies influence the demand for or supply of reserves at banks and other depository institutions, and through this market, the effects of monetary policy are transmitted to the rest of the economy. A change in the reserves market will trigger a chain of events that affect other short-term interest rates, foreign exchange rates, long-term inter est rates, and the amount of money and credit in the economy, and levels of employment, output, and prices.
Financial Statement Analysis
The major financial statements: • The income statement is a summary of the profitability of the firm over a period of time, such as a year. • The balance sheet provides a list of the firm’s assets and liabilities at a particular time, say the year-end. • The cash-flow statement details the cash flow generated by the firm’s operations, investments, and financial activities.
Financial Leverage of ROE
Let L = Debt / Equity be the leverage ratio of a firm. A firm with L = 0 is all-equity financed. Let be the return on assets, and ROE be the return on equity. Let r be the interest rate on the firm’s debt. Suppose that the corporate tax rate is c. A straightforward derivation shows that: � · (1 + L) − r · L) ROE = (1 − c) · (R
(15)
Preparation for Next Class
Please read: • Read BKM Chapters 26 & 27, • Thomas (2000), Waring, Whitney, Pirone and Castille (2000), and • Strongin, Petsch and Sharenow (2000).
15.433 INVESTMENTS Class 20: Active Portfolio Management
Spring 2003
Financial instruments are increasing in
number and complexity
Supranationals Interest rate-options Currencyforwards
Currencyoptions
Government
Cross-currency hedges Agencies
Futures
Semigovernments Agencies
Proxy hedges Complex domestic markets Index-linked bonds
Swaps Supranationals
Inflation protected government bonds
Volatility options Swaptions
CDO / MBA Exotic currency options
Key question for every investor
What is the goal for the total portfolio?
What is the time frame for achieving that goal?
What is the tolerance for loss/uncertainty within a shorter term (one-,
three-, six-month) period?
Which kinds of risk are acceptable/unacceptable?
What are you willing to pay for active risk management? (e.g. cur rency hedges)
How do you monitor/evaluate your risk management?
The risk-versus-return compass
Increasing compensated risks can increase returns
Two major types of compensated risk:
• Credit • Market
Are these areas of ”skill” ? Optimize the risk exposure Insufficient evidence of ”skill” ? Ignore, hedge or transfer the risk?
Same Risk More Return
Less Risk Same Return
Starting Portfolio
Same Risk Less Return
More Risk Same Return
Higher Moments of Asset
Asset −→ Return −→ Risk
∂(asset) ∂∆
= return
∂(return) ∂∆
∂(risk) ∂∆
= risk
change in value of asset
speed of change
= higher moments of risk profile of speed
Active vs. Passive management
Active management means allocation of resources based on an active strategy. Usually active management is performed against a benchmark, requiring intended over-/ underweights of positions.
Passive management means following an index, benchmark or another portfolio using quantitative techniques, such as principal component analysis to replicate an index.
The discussion of active vs passive management is linked to the efficient market discussion: Can information add value (performance).
Stock Picking
Figure 3: Bottom-up vs. top down approach
Bottom-Up
Tactical Asset Allocation
Top-Down
Strategic Asset Allocation
From Where does Superior Performance
Come?
From Where does Superior Performance Come? Superior performance arises from active investment decisions which differentiate the portfolio from a ”passive” benchmark These decisions include: • Market Timing: Altering market risk exposure through time to make advantage of market fluctuations; • Sectoral emphasis: Weighting the portfolio towards (or away from) company attributes, such as size, leverage, book/price, and yield, and towards (or away from) industries; • Stock selection: Marking bets in the portfolio based on information idiosyncractic to individual securities; • Trading: large funds can earn incremental reward by accommodating hurried buyers and sellers.
Some Definitions
Active management: The pursuit of transactions with the objective of profiting from competitive information - that is, information that would lose its value if it were in the hands of all market participants Active management is characterized by a process of continued research to generate superior judgment, which is then reflected in the portfolio by transactions that are held in order to profit from the judgment and that are liquidated when the profit has been earned.
Alpha: The ”risk adjusted expected return” or the return in excess of what would be expected from a diversified portfolio with the same systematic risk When applied to stocks, alpha is essentially synonymous with misvaluation: a stock with a positive alpha is viewed as undervalued relative to other stocks with the same systematic risk, and a stock with a negative alpha is viewed as overvalued relative to other stocks with the same systematic risk When applied to portfolios, alpha is a description of extraordinary reward obtainable through the portfolio strategy. Here it is synonymous with good active management: a better active manager will have a more positive alpha at a given level of risk.
Alpha, historical: The difference between the historical performance and what would have been earned with a diversified market portfolio at the same level of systematic risk over that period. Under the simplest procedures, historical alpha is estimated as the constant term in a time series regression of the asset or portfolio return upon the market return.
Alpha, judgmental: The final output of a research process, embodying in a single quantitative measure the degree of under or overvaluation of the stocks Judgmental alpha is a product of investment research and unique to the individual or organization that produces it is derived from a ”forecast” of extraordinary return, but it has been adjusted to be the expected value of subsequent extraordinary return. For example, among those stocks that are assigned judgmental alphas of 2 percent, the average performance (when compared to other stocks of the same systematic risk with alphas of zero) should be 2 percent per annum. Thus, average experienced performance for any category of judgmental alpha should equal the alpha itself. A judgmental alpha is a prediction, not retrospective experience.
Alpha, required: The risk adjusted expected return required to cause the portfolio holding to be optimal, in view of the risk/reward tradeoff. The required alpha is found by solving for the contribution of the holding to portfolio risk and by applying a risk/reward tradeoff to find the corresponding alpha. It can be viewed as a translation of portfolio risk exposure into the judgment which warrants that exposure.
Expected Return
CAPM
(rM)
ine tL e k r Ma y t M Expected uri Sec Risk Premium
(rf)
defensive
aggressive β=1
Risk free Investment
Beta (β)
Investment in Market
Figure: CAPM and market-aggressivity
βi =
E (ri ) − rf E (rM ) − rf
(1)
βi =
cov (ri , rM ) 2 σM
(2)
APT
APT says: • Expected excess return for any asset is a weighted combination of the asset’s exposure to factors.
APT does not say: • What the factors are or what the weights are.
So what? • CAPM forecasts can be used for performance measurement, i.e. beat the index; • APT forecasts are difficult to use for performance - remember they are arbitrary; • A good APT forecast can help you to outperform the index; • APT is an active management tool based on a multifactor model.
Factor Models
R2 = 1 −
var (εi ) var (ri )
ri = [bi,1 F1 + bi,2 F2 + · · · + bi,n Fn ]
(3)
(4)
A factor models tries to explain the variation of return, which is a transformation of the original level: asset behavior. Some techniques help to understand what moves the assets and thus determines return and risk. The principal component analysis is frequently used, but . . . first hand interpretation is maybe not intuitive .
”Shall I go long principal component 2 and short principal component 4 ?” Le Penseur, Rodin 1880
The Treynor-Black Model
Mix Security Analysis with Portfolio Theory Suppose that you find several securities appear to be mispriced relative to the pricing model of your choice, say the CAPM.
According to the CAPM, the expected return of any security with β k is: M = rf + βk · (E (rM ) − rf ) µCAP k
(5)
Let A be subset with ”mis-priced” securities. For any security k ∈ A, you find that M rk = αk + µCAP + εk k
(6)
where αk is the perceived abnormal return. You would like to exploit the ”mis-pricing” in the subset A. For this, your form a portfolio A, consisting of the ”mis-priced” securities. At the same time, you believe that the rest of the universe is fairly priced.
The rest of the portfolio allocation problem then becomes a standard one:
• The objective is that of a mean-variance investor. • The choice of assets: 1. The market portfolio with µM and σM 2. The portfolio of ”mis-priced” securities A, 3. with µA and σA
4. The riskfree asset. • The solution: same as the one we considered in Class 5.
The Black-Litterman Model
Mix Beliefs with Portfolio Theory The Black-Litterman asset allocation model, developed when both authors were working for Goldman Sachs, is a significant modification of the traditional mean-variance approach. In the mean-variance approach of Markowitz, the user inputs a complete set of expected returns and the variance-covariance matrix, and the portfolio optimizer generates the optimal portfolio weights. Due to the complex mapping between expected returns and portfolio weights, users of the standard portfolio optimizers often find that their specification of expected returns produces output portfolio weights which may not make sense. These unreasonable results stem from two well recognized problems:
1. Expected returns are very difficult to estimate. Investors typically have knowledgeable views about absolute or relative returns in only a few markets. A standard optimization model, however, requires them to provide expected returns for all assets.
2. The optimal portfolio weights of standard asset allocation models are extremely sensitive to the return assumptions used. These two problem compound each other; the standard model has no way to distinguish strongly held views from auxiliary assumptions, and the optimal portfolio it generates, given its sensitivity to the expected returns, often appears to bear little or no relation to the views the investor wishes to express.
In practice, therefore, despite the obvious attractions of a quantitative approach, few global investment managers regularly allow quantitative models to play a major role in their asset allocation decision. In the Black-Litterman model, the user inputs any number of views or statements about the expected returns of arbitrary portfolios, and the model combines the views with equilibrium, producing both the set of expected returns of assets as well as the optimal portfolio weights. Since publication of 1990, the Black-Litterman asset allocation model has gained wide application in many financial institutions.
How relevant are factors in relation to
different styles?
100%
100%
90%
90%
80%
80%
70%
70%
60%
60%
50%
50%
40%
40%
30%
30%
20%
20%
10%
10%
0%
0% PC 1
PC 2
PC 3
PC 4
PC 5
PC 6
PC 7
PC 9 PC 10
Factors for ”Value” portfolio
PC 1
PC 2
PC 3
PC 4
PC 5
PC 6
PC 7
PC 9 PC 10
Factors for ”Growth” portfolio
Depending on the nature of the investments, the influencing factors are different. Thus, the principal components, reflecting the ”explanatory power” of existing, but ”unknown” factors are different in structure and dimension. What makes a ”good” factor? • Interpretable: It is based on fundamental and market-related characteristics commonly used in security analysis • Incisive: It divides the market into well defined slices • Interesting: It contributes significantly to risk, or it has persistent or cyclical positive or negative exceptional return
Why Factors? • Change in behavior (company restructuring, new business strategy etc),
• reflected in sensitivities to factors; Screening of universe for ”adequate” investments, depending on investment objective; • Handling of information-overflow
Some examples of Style-definitions: • Large Cap Value: Stocks in Standard & Poor’s 500 index with high book-to-price ratios • Large Cap Growth: Stocks in Standard & Poor’s 500 index with low book-to-price ratios • Small Cap Stocks: Stocks in the bottom 20 • Each styles reacts different and thus fits different clients in different ways
Factor Definitions:
Size: Captures differences in stock returns due to differences in the market capitalization of companies This index continues to be a significant determinant of performance as well as risk.
Success: Identifies recently successful stocks using price behavior in the market as measured by relative strength. The relative strength of a stock is significant in explaining its volatility.
Value: Captures the extent to which a stock is priced inexpensively in the market. The descriptors are as follows: • Forecast Earnings to Price; • Actual Earnings to Price; • Actual Earnings to Price; • Yield. Variability in Markets (VIM): Predicts a stock’s volatility, net of the market, based on its historical behavior. Unlike beta, this index measures the stock’s overall volatility.
Growth: Uses historical growth and profitability measures to predict future earnings growth. The descriptors are as follows: • Dividend payout ratio over five years Computed using the last five years of data on dividends and earnings; • Variability in capital structure;
• Growth rate in total assets; • Earnings growth rate over last five years; • Analyst-predicted earnings growth; • Recent earnings change Measure of recent earnings growth.
Return Decomposition
Total Return
Total Excess
Normal
Risk-Free
Active
Active Systematic
Active Specific
Figure: Return decomposition
Risk Decomposition Common Risk
21 x 21 = 441
Specific Risk 4.5 x 4.5 = 20.25
Total Risk 21.48 x 21.48 = 461.25
Figure: Risk decomposition,
Return and Risk, A Two Factor Linear
Model
Return: rp = ap + bp,1 F1 + bp,2 F2 + εp
rBM = aBM + bBM,1 F1 + bBM,2 F2 + εBM
(7)
Excess Return: rp − rBM = ap + (bp,1 − bBM,1 )F1 + (bp,2 − bBM,2 ) F2 + (ap + εp − aBM − εBM )
(8)
Variance of Excess Return : var (rp − rBM ) = (bp,1 − bBM,1 )2 var (F1 ) + (bp,2 − bBM,2 )2 (F2 ) + 2 · (bp,1 − bBM,1 ) (bp,2 − bBM,2 ) · cov (F1 , F2 ) + var (εp ) + var (εBM ) − 2 · cov (εp , εBM ) Tracking Error:
TE =
15.433
� varp − rBM
(9)
� � 2 2 � � (bp,1 − bBM,1 ) var (F1 ) + (bp,2 − bBM,2 ) (F2 ) � = � +2· (bp,1 − bBM,1 ) (bp,2 − bBM,2 ) · cov (F1 , F2 ) (10) � +var(εp ) + var (εBM ) − 2 · cov (εp , εBM )
23
MIT Sloan
Tracking Error
The tracking error is defined as: ”the standard deviation of active return”.
σA = std [rAP ] = σ [rP ] − σ [rBM ]
= σAP = σP − σBM
(11)
The tracking error measures the deviation from the benchmark, as the rp is the sum of the weighted returns of all positions in the portfolio and rBM is the sum of the weighted returns of all positions in the benchmarks. Portfolio and benchmark do not always contain the same positions!
Tracking error is called as well active risk.
Information Ratio
Information Ratio: A measure of a portfolio manager’s ability to deliver, relating the relative return to the benchmark and the relative risk to the benchmark: • Expected Active Return (alpha) • Active Risk IR =
expected active return α = active risk TE
(12)
Implied alpha: Alpha backed-out through reverse engineering; how much has my expected return to be to justify all other parameters ceteris paribus
Forecasts
Some examples: • MCAR: How much does active risk increase if I increase the holding x by 1 % and reduce cash by 1 % • MCTCFR: How much does common factor risk increase if I increase the holding x by 1 % and reduce cash by 1 % • MCASR: How much does specific active risk increase if I increase the holding x by 1 % and reduce cash by 1 %
Performance Attribution
The identification of individual return components can be performed quite easily, subject to the history of the restructuring of the portfolio.
The straight forward approach is based on the definition of a passive benchmark portfolio, which reflects the long-term investment strategy. In the context of the investment strategy (or the strategic asset allocation) the investment management decides which asset categories (equities, fixed income, currencies, etc.) are over-/underweighted relative to the benchmark (strategy). The weights of specific asset categories - as determined in the investment strategy - are called normal weights.
For each asset category of the portfolio exists a corresponding asset category of the benchmark (index), relative to which the performance is calculated. The return of these indices are called normal returns. It is obvious, that the the normal return is a return of a passive investment in the corresponding asset category of the benchmark.
For equities, fixed income and for currencies exist different indices, reflecting different needs.
The normal weight of the asset category i (ws,i ) multiplied with the normal return (rs,i ) is the return of this intended asset category. Summed up over all returns from the different asset categories, the portfolio has the following strategy/-benchmark return:
rstrategy =
�N
i=1 ws,i
· rs,i
Against this benchmark-portfolio we want to know the realized return of the actively managed portfolio. We have a positive excess-return, if the effectively realized portfolio return (rportf olio ) exceeds the strategy return (rstratey ): rexcess
return
= rstrategy − rportf olio
The current portfolio return (rportf olio ) is calculated from the effective breakdown of the portfolio in the different asset categories (wp,i ) as well as the effectively realized returns (rs,i ) of the individual asset categories: rportf olio =
�N
i=1 wp,i
· rp,i
The difference between the strategy return and the realized portfolio return results from the fact, that the portfolio manager restructures the portfolio through market timing strategies based on the on the assumption of predicting the direction of the performance. Overperformance by timing the market can be achieved by adjusting the overall market exposure of the portfolio. Various techniques exist to time the market: • tactical over- and underweights of categories and thus deviates from the normal weights thourghchanges in the asset class mix (especially stock and cash positions), also called rotation (sector rotation, asset class rotation) • timing within an asset class: changing the security mix by shifting the proportions of conservative (low beta) and dynamic (high beta) securities.
• derivatives instruments: especially index futures and the use of options. Security selection is the identification of over/-under priced securities. So a superior valuation process is needed to compare the true value for a security with the current market value.
Overall, the return of a portfolio can be decomposed in four return components, which summed up again result in (rportf olio ):
• rstrategy = • rtiming =
�N
i=1 ws,i
�N
i=1 rs,i
• rselectivity = • rcumulative
�N
· rs,i
· (wp,i − ws,i )
i=1 ws,i
ef f ect
=
· (rp,i − rs,i )
�N
i=1 (wp,i
− ws,i ) · (rp,i − rs,i )
Figure 1 highlights the decomposition of the portfolio return in the individual components and their relationship to a active respectively passive portfolio management. Quadrant (1) is put together from passive selectivity and passive timing. It represents the long-term investment strategy and serves as the benchmark return for the observation period in examination. If the portfolio management performs a passive market timing, we receive the return in quadrant (2). It represents the return from timing and strategy. We understand timing as the deviation in the weight of the individual asset category from the normal weight. Within the individual asset categories we invest in a passive index portfolio. Through subtraction of the strategy return from quadrant (1) we receive the net result from timing.
Quadrant 3 reflects the returns from selectivity and strategy. Selectivity is the active choice of individual securities within an asset strategy. The normal weights are kept equal. The return from selectivity is received through subtraction of the strategy return in quadrant (1) from quadrant (3). In quadrant (4) we finally find the realized return of the portfolio over the observation period in examniation, calculated as the product of the current weights of the individual asset categories with the current returns within the asset categories. Not obvious from the figure is the fourth component, the cumulative effect (also called interaction effect), which is based on cross product of return- and weight differences. The residual term can be derived from the interaction between timing and selectivity. It is based on the fact that the portfolio manager puts more weight on the asset categories with a higher return than in the benchmark index (selectivity).
selectivity
market
timing
active
passive
active
passive
(4)
(2)
realized return �N i=1 wp,i · rp,i
timing & strategy �N i=1 wp,i · rs,i
(3)
(1)
selectivity & strategy �N i=1 ws,i · rp,i
strategy
�N
i=1 ws,i
· rs,i
Timing = (2)-(1)
Selectivity = (3) - (1)
Residual = (4)-(3)-(2)+(1)
Figure 1: Performance components in an active portfolio
Example without Currency Components
All returns are calculated in the domestic currency. All foreign exposures are perfectly hedged back into the domestic portfolio currency. The upper part of the table contains the normal weights and normal returns required to calculate the passive strategy of the individual asset categories. In the second part of the table are the effective weights and the current returns of the individual asset categories in the specific quarters. The current weights and returns are adjusted from quarter to quarter to reflect the restructuring of the tactical asset allocation and the stock picking and result in the active over/-underweights.
In the lower part of the table are the individual performance components resulting in the individual quarters. They are calculated using the equations in the previous equations.
From the results in Table 1 it is obvious that the return from active management varies substantially from quarter to quarter and reflects no constant pattern. The timing-return varies between -0.15% in the 4th quarter x1 and max 0.28% in the 1st quarter x1. Selectivity has even more variation: min is -0.18% in and 1.48% in the 1st quarter. The residual terms have a surprising big impact, with 0.18% of the portfolio return in 1st quarter and 2n quarter and reducing the portfolio return with -0.39%!.
normal weights normal returns current weights current returns
FI $ FI Euro Eq $ Eq Euro FI $ FI Euro Eq $ Eq Euro FI $ FI Euro Eq $ Eq Euro FI $ FI Euro Eq $ Eq Euro
strategy timing selectivity interaction realized
return from active manage
x0/3 57.50 12.50 22.50 7.50 -0.75 -19.37 -28.62 -1.94 57.50 12.50 22.50 7.50 -0.32 -2.06 -26.19 -21.61
x0/4 57.50 12.50 22.50 7.50 1.26 2.50 1.39 4.37 53.70 16.50 22.10 7.70 0.81 4.15 1.99 2.37
x1/1 57.50 12.50 22.50 7.50 4.53 14.52 16.02 6.71 55.70 15.80 22.30 6.20 4.22 7.05 16.69 17.75
x1/2 57.50 12.50 22.50 7.50 2.09 3.31 0.85 4.48 52.30 16.10 23.60 8.00 2.12 4.68 0.97 4.89
x1/3 57.50 12.50 22.50 7.50 0.46 -0.99 -1.12 2.15 54.50 15.90 23.50 6.10 1.15 1.74 0.16 -2.97
x1/4 57.50 12.50 22.50 7.50 0.91 -3.16 -2.18 2.18 53.80 15.70 23.10 7.40 0.78 2.33 -3.89 -3.24
-9.44 0.00 1.48 0.00 -7.96
1.68 0.06 -0.07 0.08 1.74
8.53 0.28 -0.13 -0.39 8.29
2.14 0.04 0.25 0.05 2.48
0.05 -0.09 0.64 0.16 0.76
-0.20 -0.15 -0.18 0.18 -0.35
4.94 0.19 -0.06 0.02 5.09
7.70 0.33 1.93 0.10 10.06
1.48
0.06
-0.24
0.34
0.71
-0.15
0.16
2.36
x2/1 entire period 57.50 12.50 22.50 7.50 3.26 7.07 7.79 5.71 52.40 16.50 23.20 7.90 2.32 5.93 10.09 7.08
Table 1: Example for performance components without currency exposure
The active management contributed in 5 out of 7 quarters positively to the overall return. Over the time period of 7 quarters the active management added 2.36%, with contribution from timing of 0.33%, selectivity contributed 1.93% and the residual term 0.10%. Looking at the realized return of the portfolio (10.06%), the contribution from active management with 2.36% is substantial!
Even more important is the contribution from the strategy, which added 7.70% to the portfolio return, and thus is the most important component. This example shows quite nicely, that the most important contribution to the return is from the strategic asset allocation. The substantial part of the achieved investment performance is based on the strategy and not from the active management through the portfolio manager.
strategy timing selectivity residual total
Brinson/Hood/Beebower (1986) return components variance (%) components (%) 10.11 93.6 -0.66 1.7 -0.36 4.2 -0.07 0.5 9.01 100
Brinson/Singer/Beebower (1991) return components variance (%) components (%) 13.49 91.5 -0.26 1.8 0.26 4.6 -0.07 2.1 13.41 100
comments
91 pension funds, USA 1974-1983, n=43
82 pension funds, USA 1977-1987, n=45
Table 2: Performance components for US pension funds
Table 2 the study has be carried out over a time period of 10 years. In the first analysis the pension funds realized a return of 9.01%. This is 1.10% below the strategy return of 10.11. The active management destroyed value worth 1.10% (0.66% timing, 0.36%). For the second analysis the results look not better. The selectivity added in average 0.26% to the annual return, however the active return does not look better, active management lowered the returns by 8 basis points in average. A comparison between the dimension of strategy return and the ”added value” of active management shows in both studies that the active component is only a small fraction of the total return.
Performance Attribution with a currency
component
We know from previous classes and from own experience that diversification can improve the performance of the portfolio. Diversification can be generated through investments in e.g. different asset categories or individual sectors, industries etc. Especially the diversification across the border lines is important, adding additional low correlations to the portfolio. Looking at the performance attribution of international diversified portfolios, we want to know the impacts of strategy, timing and selectivity and as well the contribution from fx-components from the portfolio allocation. Exposure to foreign currencies can be generated through direct investments in fx (buy, sell), or through investments in foreign securities, without completely hedging the fx exposures. The portfolio return is increases through the fx-return by carrying the fxexposures during the observation period.
We define with fs,j the strategic component in currency j and fp,j the effectively held exposure to currency j. The return of an internationally diversified portfolio including fx-exposures can be calculated as following:
rportf olio =
�N
i=1 wp,i · rp,i +
�N
i=1 fp,i
· rf x,j
wp,i is again the current portfolio weight for asset category i. The current return of an actively managed portfolio yields rp,i is no longer exclusively in the domestic currency (1 to 1), but in the local currency of the particular investment. Thus, the first sum of previous equation includes the weighted return of the investments in different securities,
measured in the local currency of the corresponding market. For investments in American securities it reflects e.g. the local equity return in US$. With the second term in the previous equation we add an additional return components coming from the fx-return for the exposure held in the particular currency. The currency return rf x,j is weighted with the corresponding fx-component in the portfolio. In the example, where e.g. 20% of the equity portfolio are invested in Euro-denominated securities (without hedging the portfolio against the Euro-exposure), we have to add to the local US-equity return a 20%-investment (exposure) in Euro-currency. For a portfolio, which is exclusively invested in the domestic portfolio currency, the second terms is dropped.
The strategy return is calculated analogous to the portfolio return using the passive strategy weights and normal returns in the local currency:
rstrategy =
�N
i=1 ws,i
· rs,i +
�N
i=1 fs,i
· rf x,j
Again, here as well we add a fx-component. The different fx-returns are weighted according their corresponding strategic fx-weights fs,j . The decision, to vary the proportions of individual currency exposures in the portfolio is considered part of the tactical asset allocation. Through conscious deviations from strategic currency weights the active portfolio manager can add (loose) an additional return component to the domestic portfolio return.
Independent from the previous statement is the decision to invest in local markets and asset categories to benefit from the local performance.
Accordingly, the timing-component of an international diversified portfolio is split into two parts: one in a market-component, which reflects the investment decisions regarding the specific market and asset categories, and a second part which reflects the currency component from the allocation decision in different currency exposures:
rmarkets = rcurrency =
�N
i=1 rs,i
· (wp,i − ws,i )
�N
i=1 rf x,j
· (fp,j − fs,j )
The market component is calculated through multiplication of the passive normal returns rs,j , measured in the specific local currency of the investment position, with deviation of the portfolio weights from normal weights. The return, coming from the deviation from the strategic currency allocation, is reflected in the fx-component.
The return component from selectivity is defined as:
rselectivity =
�N
i=1 ws,i
· (rp,i − rs,i )
The portfolio return, rp,i , and as well the normal return, rs,i , are measured in the local portfolio currency. The performance component from selectivity decisions is thus not be affected from the currency of specific investments, but is calculated exclusively through the choice of specific securities within a market or an asset category.
Legend:
ws,i strategy weight (normal weight) for asset class i
rs,i strategy return (normal return) for asset class i
wp,i portfolio weight (effective weight) for asset class i
rp,i portfolio return (effective return) for asset class i
Similar terms:
• Timing: Allocation • Cumulative effect: Interaction effect
Performance Measure
Capital Market oriented View Sharpe’s measure:
rp − r f (13) σp Divides average portfolio excess return over the sample period by the Sharpe =
standard deviation of returns over that period. It measures the reward to (total) volatility trade-off?
Treynor’s measure: rp − rf (14) βp Gives excess return per unit of risk over the sample period by the stanT reynor � s =
dard deviation of returns over that period. It uses systemic risk instead of total risk.
Jensen’s measure: Jensen� s α = rp − [rf + βp (rM − rf )]
(15)
It is the average return on the portfolio over and above that perdicted by the CAPM, given the portfolioo’s beta and the average market return. Jensen’s measure is the portfolios alpha value.
Appraisal ratio:
Appriasal Ratio =
αp σεp
(16)
It divides the alpha of the portfolio by the nonsystematic risk of the portfolio. It measures abnormal return per unit of risk that in principle could be diversified away by holding a market index portfolio.
Summary
The (simple) performance attribution allows to figure out where the portfolio manager manager added value and where he destroyed value.
It is key to learn from past errors and not to repeat them. Performance attribution is an essential element in investment to ensure that exposures are rewarded with the appropriate risk premium.
The capital market oriented performance attribution is another approach to analyze the performance. It allows to calculate the riskpremiums for the factors / styles to which the portfolio is exposed.
Focus: BKM Chapter 26 • p. 874-883, 890-897 (learn general definitions and assumptions) type of potential questions: Concept check questions, p. 899 ff. question 2,3,4,5
BKM Chapter 27 • p. 917-933 (objectives of active portfolios, market timing, security selection, portfolio construction, multifactor models and portfolio management, quality of forecasts) type of potential questions: Concept check questions, p. 899 ff. question 2,3,4,5
• Thomas (2000), p. 26 f. information ratio Strongin, Petsch and Sharenow (2000), p. 18 dealing with stock-specific benchmark, p. 23 f. portfolio manager patterns
Preparation for Next Class
Please read:
• Kritzman (1994a), • Kritzman (1994b), • Ross (1999), and • Perrold (1999). Video (Optional): ”Trillion Dollar Bet”, (a PBS Documentary). I have just one copy of the video tape. It is to be distributed by Joon Chae ([email protected]) on a first come first serve basis. If there is excess demand, some alternative means of distribution will be worked out. Please contact Joon directly.
15.433 INVESTMENTS Class 21: Hedge Funds
Spring 2003
The Beginning of Hedge Funds
In 1949, Alfred Jones established the first hedge fund in the U.S.
At its beginning, the defining characteristic of a hedge fund was that it hedged against the likelihood of a declining market.
Two speculative tools were merged into a conservative form of investing: 1. leverage was used to obtain higher profits. 2. short selling was em ployed to hedge against the downside risk.
By combining long and short positions, Jones exploited the relative pric ing of stocks, while minimizing his exposure to the overall market.
To align the manager/investor incentives, Jones employed performance based fee compensation. He also kept all of his own money in the fund.
A Little History of Hedge Funds
While mutual funds were the darlings of Wall Street in the 60’s, Jones’ hedge fund was outperforming the best mutual funds even after the 20% incentive fee deduction. The news of Jones’ performance created excite ment, and by 1968, approximately 200 hedge funds were in existence.
During the 60s bull market, many of the new hedge fund man-agers stopped hedging the downside risk, and went into the bear market of the early 70s with long, leveraged positions. Many were put out of busi ness.
During the next decade, only a modest number of hedge funds were established.
Over the past 10 years, however, the number of funds has increased at an average rate of 25 74% per year.
Investment Flexibility
Hedge funds are largely unregulated. This lack of regulation permitted a hedge fund manager to employ leverage, to sell short, and to charge performance based fees, practices that normally were not available to regulated investors such as mutual funds.
A second differentiating characteristic was the amount of leverage em ployed. Hedge fund managers were able to leverage their portfolios through the use of futures, options, and repurchase agreements, as well as through more traditional sources of financing such as banks and bro kerage firms.
While hedge funds came in all shapes and sizes, they tended to have one common trait: low correlation with the U.S. equity market.
Some Structural Details
The majority of the funds are organized as limited partnerships, allowing only 99 investors. General partners usually have a significant investment in the partnership. Performance based compensation. Typically, 20% of net profits in ad dition to 1% management fee. Most performance fees are subject to a ”high watermark,” and some require a ”hurdle rate.” Restricted withdrawals. Typically, limited partners are allowed to with draw only annually, and some had a ”lock up” period as long as three years. A culture of secrecy. Rarely are limited partners given a list of portfolio holdings.
Size and Location
According to a report by Tremont & TASS in 1999, there are 5,000 funds in the whole industry. However, over 90% of the $325 billion under management in the industry is managed by some 2,600 funds. Figure 3: Growth in Hedge Funds, Source: www.tassresearch.com
Location of hedge funds:
USA
33.9%
Cayman Islands
18.9%
British Virgin Islands 16.5% Bermuda
11%
Bahamas
7.2%
Others
12.5% Source: Tremont TASS (June 1999)
Domicile of Fund Managers In U.S.: 91% Outside U.S.: 9% Source: Tremont TASS (June 1999)
A Taxonomy of Hedge Fund Strategies
Directional Trading: based on speculation of market direction in mul tiple asset classes. Both model based systems and subjective judgment are used to make trading decisions.
Relative Value: focus on spread relationships between pricing compo nents of financial assets. Market risk is kept to a mini-mum. Many managers use leverage to enhance returns.
Specialist Credit: based on credit sensitive securities. Funds in this strategy conduct a high level of due diligence in order to identify rela tively inexpensive securities.
Stock Selection: combine long and short positions, primary in equities, in order to exploit under and overvalued securities. Market exposure can vary substantially.
Low Correlation Among Hedge Fund
Hedge fund mangers exhibit much lower correlation with one another than traditional active managers: • Between 1990 and 2000, the average correlation among Lipper (mu tual funds) managers has been on the order of 90%, • while hedge fund managers resemble S&P 500 stocks have an average correlation on the order of 10%. • for the universe of stocks, the average correlation is about 20%. The low average correlation among hedge fund managers suggests that pooling funds into portfolios or indices can significantly reduce their total risk, providing distinct advantages relative to traditional active strategies.
Effects of Age
Between 1990 and June 2000 Fund Age Annualized Annualized Sharpe (year)
Return
Volatility
Ratio
≤1
27.2%
6.1%
3.7
1≥x≤2
23.5%
5.9%
3.2
2≥x≤3
18.7%
5.5%
2.5
3≥x≤5
18.1%
5.2%
2.5
x≥5
14.7%
5.9%
1.6
Source: Morgan Stanley Dean Witter
The Rise of LTCM
The fund that LTCM managed had commenced operations with $1 bil lion of capital in early 1994, and had subsequently raised an additional $2 billion.
In September 1997, after three and half years of investment returns that far exceeded even the principals’ expectations, the Fund’s net capital stood at $6.7 billion.
Since inception, the Fund’s returns after fees had been 19.9% from Febru ary 24 through December 31, 19941 42.8% in 1995, 40.8% in 1996, and 11.1% in 1997 through August.
These returns were achieved without exposure to the stock market. In many of its trades, the firm was in effect a seller of liquidity, diversified across many markets.
Some LTCM Trades
Convergence Trades on Swap Spreads
Yield Curve Relative Value Trades Butterfly Trades
Selling Equity Volatility
Risk Arbitrage
Equity Relative Value Trades on Royal Dutch/Shell
Fixed Rate Residential Mortgages
Japanese Government Bond Swap Spread
The Fall of LTCM
Following a successful 1997, LTCM began the year with about $4.8 bil lion in capital, having just returned $2.7 billion to out-side investors. In May and June the Fund experienced its two worst months ever, with gross returns of 6.7% and 10.1% respectively. The losses were distributed across many positions, with no single trade experiencing a large loss. In response to the losses, LTCM reduced the risk of the portfolio. By July 21, the fund was up 7.5% on the month. But the month of August was a continuation of adverse movements, and by mid August, overall posi tions had been cut by an additional 5%. On Monday, August 17, 1998, in an event that stunned the world, Rus sia defaulted on its government debt. LTCM had only a small exposure to Russian government credit, and the Fund suffered a correspondingly small loss in these positions. Friday, August 21, was the worst day in the Fund’s history, as many of the Fund’s trades moved adversely and substantially: 1. During the morning, the U.S. swap spread widened by 19 bps, com pared with a typical daily move of less than a basis point. The U.K. gilt swap spread also widened dramatically that day. The Fund had large short positions in both of these swap spreads. 2. LTCM suffered a significant loss in a risk arbitrage position related to the planned acquisition of telephone equipment maker Ciena Corp by Tellabs, Inc. LTCM estimated that the combination of the risk arb loss and the un precedented widening of the U.S. and U.K. swap spreads and other spreads had resulted in a one day loss of about $553 million - 15% of its
capital.
The Fund was now down to $2.95 billion, with a dangerously high lever
age ratio of 42.
To reduce their risk, the partners would have to sell something. But
what? Investors wanted only the safest bonds, which LTCM didn’t have.
August was the worst month ever recorded in credit spreads. Unlike the
the ballooning spreads traditionally linked with an economic collapse,
this one was caused by a panic on Wall Street.
In August, three quarters of all hedge funds lost money, and LTCM lost
the most: $1.9 billion, which was 45% of its capital.
Focus: • Ross (1999) and Perrold (1999)
Preparation for Next Class
Please read for next class: • BKM Chapter 27, • Chow and Kritzman (2001), • Bernstein (1995), and • Lewent and Kearney (1990).
15.433 INVESTMENTS Class 22: Market Efficiency
Spring 2003
Types of Market Efficiency
The Weak Form of Efficiency: Prices accurately reflect all in-formation that can be derived by examining market trading data such as past prices, trading volume, short interest, etc. The Semi-strong Form of Efficiency: Prices accurately reflect all public available information, including past prices, fundamental data on the firm’s product line, quality of management, balance sheet composition, patents held, accounting practices, earning forecasts, etc. The Strong Form of Efficiency: Prices accurately reflect all information that is known by any one, including inside information.
Strong Form
Semistrong Form Weak Form
Figure 1: Return distribution of US 10 Year Bond
An inefficiency ought to be an exploitable opportunity. If there is nothing investors can properly exploit in a systematic way, time in and time out, then it’s very hard to say that information is not being properly incorporated into stock prices”. Richard Roll
Financial markets are efficient because they don’t allow investors to earn above average returns without taking above average risks”. Burton Malkiel
The efficient markets theory holds that the trading by investors in a free and compet itive market drives security prices to their true ’fundamental’ values. The market can better assess what a stock or a bond is worth than any individual trader.” Andrei Shleifer
Information Arrivals and Price Updates
The efficient market theory states that security prices reflect all currently available information. One interesting empirical question is: how does the market ad-just to the arrival of new information? Event study methodology is one such tool to measure the eco-nomic impact of events.
Paths to Efficient Prices How does information get impounded in prices?
If gathering information is costly, can prices still perfectly reflect information?
If market prices deviate from their fundamental values, what brings them back?
How do prices deviate from their fundamental values in the first place?
Limits of Arbitrage
Arbitrage plays a critical role in the analysis of securities markets, because its effect is to bring prices to fundamental values and to keep market efficient. The textbook example assumes that the arbitrageur has access to infinite amount of capital. In practice, the arbitrageurs are capital constrained, and their effectiveness in bringing prices to fundamental values is limited. Mutual
Fund Performance
Equity funds: on average, active managers underperform index funds when both are measured after expenses, and those that do outperform in one period are not typically the ones who outperform in the next. Fixed Income funds: on average, bond funds underperform passive fixed income in dexes by an amount roughly equal to expense, and there is no evidence that past performance can predict future performance. Peter Lynch and Magellan Fund Let rt be the monthly returns of Magellan Fund:
rt − rf,t = α + β (rM,t − rf,t ) + srsmb,t + hrhml,t + εt
Overall Period Peter Lynch Post-Lynch 76/6-98/12
76/6-90/5
90/6-98/12
α
0.51 (0.11)
0.75 (0.13)
0.07 (0.14)
β
1.12 (0.03)
1.13 (0.03)
1.04 (0.04)
s
0.34 (0.04)
0.55 (0.05)
0.05 (0.05)
h
0.02 (0.05)
-0.01 (0.06)
0.00 (0.06)
R2
0.91
0.94
0.90
(1)
Anomalies? The size effect The value effect
The short term momentum
The long term reversal
The new issues puzzle
The weekend/holiday effect
The January effect
The Bottom Line
The efficient market hypothesis is a useful framework for modelling financial markets. Like any model, the ’efficient market hypothesis is not a perfect description of re ality; some prices are almost certainly ”wrong”. However, it would be naive to think that prices are always wrong or that it is easy to exploit pricing errors. ” Instead of asking whether or not the market is efficient, the more relevant questions are: • how efficient is the market? • how does the market react to new information arrivals? and why? • what are the mechanisms that bring market prices to fundamental values?
Focus: BKM Chapter 12 • p. 343-368 (12.2,12.4) type of potential questions: Concept check questions, p. 375 ff. question 4,5,7,8,16,17,21
• Rubinstein: Rational markets vs. rational investors, anomalies (pp. 18, 20-21,2326) • Fama: critical review of model setup for testing EMH (time, return, bias etc.)
Questions for Next Class Please read: • BKM Chapters 17-19
• Business Week (2001), • Wall Street Analysts: Who Needs ’Em?” (The New Yorker)
15.433 INVESTMENTS Class 23: Commodities
Spring 2003
Introduction
The following table taken from the annual report of the Bank for In ternational Settlements (BIS) reports the spectacular growth in exchange-traded derivatives, such as futures and options. The figures indicate the amount of the underlying controlled by the derivative in billions of US dollars at year end: Type of Exchange Traded Instru 1986
1987
1988
1989
1990
1995
1998
ment
06/199 9
Interest Rate Options and Futures
516
609
1'174
1'588
2'054
15'669
50'015
54'072
Currency Options and Futures
49
74
60
66
72
120
18'011
14'899
Stock Index Options and Futures
18
41
66
108
158
442
1'488
1'511
142
415
441
Commodities Total
583
724
1’300
1’762
2’284
16’373 69’929 70’923
Table 1: Derivatives volumes, Source: Bank for International Settlements. Visit their Web Page http://www.bis.org/ for the latest statistics.
2
Constant Proportional Storage Costs (Known Storage Costs)
• Recall, financial assets may have cash payouts which have to be factored in the cost of carry relation. In the case of stocks, the payouts are dividends, while for Treasury bonds, they are coupons. • Non-financial assets, such as gold, silver, oil, corn, etc., have storage costs associated with the cash-and-carry strategy. The alteration to the cash-and-carry strategy of before is that we have to consider, both, the cost of financing (i.e., the standard shorting of K bonds) as well as the cost of storage when pur chasing the asset spot. • Notice that dividends over the life of the forward contract are a benefit accruing solely to the spot asset holder and not the forward contract party, while storage costs over the life of the forward contract are a disbenefit borne by the spot asset holder and not the forward contract party. • Hence, storage costs can be viewed as a negative dividend payout. Depending on how storage costs are defined, all we 3
have to do is to replace the present value of dividends in the cost of carry relation with the present value of storage costs, with the appropriate adjustment in sign. • Usually, commodities have no „dividends” or cash payouts associated with them, hence, we only have to consider storage costs when determining the cost of carry relation for them. • Similarly, financial assets usually have dividends or cash payouts but do not have storage costs associated with them, hence, we only have to consider dividends when determining the cost of carry relation for them.
Notation: Let: • I(K) be the initial investment required to buy a forward contract maturing at date T with delivery price K. • B(t,T) be the current price of a unit bond paying 1 at maturity. •
a = �k = t+1 B(t, k ) T
be the current price of an annuity paying $1 at the
end of each period until the forward contract matures.
4
• D be the constant Dollar storage payout per period on the risky asset. • D (k) be the time-varying, known Dollar storage payout per pe riod on the risky asset. • y be the gross, continuously compounded, proportional storage cost yield. • S(t) be the current Spot price of the underlying asset. • F(t,T) be the current forward price of the underlying asset.
The cost of carry relation between the forward price, F(t,T), and the spot price, S(t), depends on the storage cost structure: • No Storage Costs: F(t,T) =
F(t, T) =
S(t) B(t, T)
• Constant Dollar Storage Costs: F (t ,T ) =
S (t ) + D � a , where D B (t , T )
is the size of each (constant) storage cost payout and a is the current value of an annuity (a) paying $1 at each ex-storage cost date.
5
• Time-varying,
Known
Dollar
S(t) + � k =t +1 D(k) � B(t, k )
Storage
Payouts:
T
F (t ,T ) =
�
T k = t +1
B (t , T )
,
where
D(k) � B (t , k ) is the present value of the total storage
costs associated with the underlying asset over the remaining life of the forward contract. S (t ) e y ( T- t) • Continuous Proportional Payouts: F (t ,T ) = , where B (t , T ) y is the continuously compounded, proportional storage cost measured as a percentage of the spot price.
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Convenience Value of Commodities
• Convenience is the benefit from storage of the physical com modity, but not from holding the forward contract on the physical commodity. • Storage may benefit from profit opportunities available when there are temporary shortages in the commodity, perhaps avoiding suspension of the production process. • Frequently, due to a shortage, the asset underlying the forward contract cannot be borrowed in order to execute a short sale. The shortage arises mainly due to the convenience arising from owning the asset spot for the production process. • When shortages occur, the spot price for the commodity is high, generating a convenience yield. • When the convenience is high, the spot-forward curve goes into „backwardation” mode, i.e., (loosely speaking) the term structure of current forward prices is lower than the spot price. (Note: The spot price may also drop dramatically, due to a temporary glut or oversupply, causing the spot-forward curve 7
to reverse into „contango”, i.e., (loosely speaking) the term structure of forward prices rises above the spot price.) • Some financial assets, like Treasury bonds and mortgagebacked securities, have convenience values due to their collat eral value available in repo (repurchase) markets. In other words, there are times (i.e., during shortages) when these fi nancial assets behave just like commodities.
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Basis
In a well-functioning market, the forward price at delivery is equal to the prevailing spot price. Mathematically, F(T,T) = S(T).
However, convergence is not always the norm. Frequently, the forward price does not converge to the spot price due to: • the date of spot delivery T, does not match the date of hedge expiration T’ (with T
„
T'),
• the assets underlying spot and forward contracts are not identi cal, and • the date of spot delivery, T, is uncertain.
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Example
Consider the following typical Short Hedge. a)
A farmer has wheat to sell at harvest time, which we pretend to
know to be at date T with certainty. b)
At t = 0: farmer shorts a forward contract at F(0,T') for deliv
ery at T' after (but close to) T. (Assume that's because there isn't a forward contract of identical maturity to the spot delivery date T.) At t = T: farmer sells spot wheat at S(T) and „closes” out forward position at F(T,T'). c)
The farmer's payoff ( Note: He has a short hedge): غ S (T ) - F (T ,T ¢ ) øß + F (T ,T ¢ ) 424 3 144 42444 3 1
locked -invalue
basisrisk
The difference between these two prices, i.e., [S(T) - F(T,T')], is known as forward-spot basis. The risk the hedger (farmer) faces due to this difference is called basis risk. (Important Note: Strictly speaking, basis risk only arises in futures contracts and is known as futures-spot basis. Basis is due to the 10
standardization of (exchange-traded) futures contracts. Forward contracts are usually tailor-made and hence do not carry basis risk. We speak in terms of forward contracts just to illustrate ideas and because we do not cover futures contracts until later. Hence, the term „futures contracts” is more correct usage instead of „forward contracts” in this sub-section. And ignore margins.) • Example of 1: The forward (futures) contract expires on June 15 while spot delivery is set for May 15. The norm is to choose a forward (futures) delivery month T', which falls after the spot delivery date T. This is because the front-end forward (futures) price is usually very volatile when close to expiration. Picking a later-dated forward (futures) contract reduces this risk. The forward (futures) contract is usually unwound just prior to spot delivery. • Example of 2: The underlying contract in the forward (futures) contract is corn while the spot delivery calls for wheat. Usu ally, the forward (futures) with an underlying which is most closely correlated in price to the spot contract is chosen.
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• Example of 3: The wheat processor taking spot delivery from the farmer is not certain when she needs to take delivery. In other words, the timing of buyers' demand is not known with certainty. Strengthening of the basis (i.e., [S(T) - F(T,T')] �ing) benefits the short hedger (farmer). Conversely, weakening of the basis benefits the long hedger.
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Preparation for Next Class
Please read Read Statman (1999) and Nagy and Obenberger (1994).
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