Chapter 11. Return and Risk: The Capital Asset Pricing Model (CAPM) Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

Know how to calculate expected returns Know how to calculate covariances, correlations, and betas Understand the impact of diversification Understand the systematic risk principle Understand the security market line Understand the risk-return tradeoff Be able to use the Capital Asset Pricing Model 11-1

11.1 Individual Securities 11.2 Expected Return, Variance, and Covariance 11.3 The Return and Risk for Portfolios 11.4 The Efficient Set for Two Assets 11.5 The Efficient Set for Many Assets 11.6 Diversification 11.7 Riskless Borrowing and Lending 11.8 Market Equilibrium 11.9 Relationship between Risk and Expected Return (CAPM) 11-2

The characteristics of individual securities that are of interest are the: Expected Return Variance and Standard Deviation Covariance and Correlation (to another security or index) 11-3

m ( ) Pr E R R A j A, j j1 m 2 A A j A, j A j 1 V a r ( R ) P r R E ( R ) S D ( R ) V a r ( R ) A A A m CoRR v(, ) Pr RERRER () () AB AB, j Aj, A Bj, B j 1 AB, Corelation AB, AB Covariance ABABAB,, 2 11-4

Consider the following two risky asset world. There is a 1/3 chance of each state of the economy, and the only assets are a stock fund and a bond fund. Rate of Return Scenario Probability Stock Fund Bond Fund Recession 33.3% -7% 17% Normal 33.3% 12% 7% Boom 33.3% 28% -3% 11-5

Stock Bond Scenario Deviation Deviation Product Weighted Recession -18% 10% -0.0180-0.0060 Normal 1% 0% 0.0000 0.0000 Boom 17% -10% -0.0170-0.0057 Sum -0.0117 Covariance -0.0117 Deviation compares return in each state to the expected return. Weighted takes the product of the deviations multiplied by the probability of that state. 11-11

Cov( a, b) a b.0117 (.143)(.082) 0.998 i.e., Stock fund and bond fund almost perfectly negatively correlated! 11-12

A portfolio is a combination of securities ER ( ) wer ( ) wer ( ) P A A B B 2 2 w w 2 ww P AA BB ABAB, σ 2 2 2 P (waσ A ) (wbσ B ) 2(wAσ A )(wbσb )ρa, B 11-13

Stock Fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 0.0324 17% 0.0100 Normal 12% 0.0001 7% 0.0000 Boom 28% 0.0289-3% 0.0100 Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks. 11-14

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012 Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: r P w B r B w 5% 50% ( 7%) 50% (17%) S r S 11-15

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012 Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio. E r ) w E( r ) w E( r ) ( P B B S S 9% 50% (11%) 50% (7%) 11-16

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012 Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The variance of the rate of return on the two risky assets portfolio is σ 2 2 2 P (wbσ B ) (ws σ S ) 2(wBσ B )(ws σ S )ρbs where BS is the correlation coefficient between the returns on the stock and bond funds. 11-17

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012 Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than either stocks or bonds held in isolation. 11-18

Look at the last equation on p. 11-13: σ 2 2 2 P A A B B A A B B )ρa, B If : σ (w ρ A B 2 P σ, 1 (w A σ A ) w (w B σ B σ ) ) 2 σ P 2(w )(w That is, the standard deviation of the portfolio equals the weighted average of the standard deviations of the securities. Hence, if ρ A, B 1, the variance and the standard deviation of the portfolio is less than the weighted average of the standard deviations of the individual securities. w A σ σ A w B σ σ B 11-19

Conclusion: if the correlation coefficient between two securities equals 1, no risk reduction is possible (the diversification effect does not apply). If the correlation coefficient is smaller than 1, risk reduction is possible (i.e., we could have used an example with positive but not perfect correlation). If the correlation coefficient is equal to -1, complete risk reduction is possible. 11-20

Now let s characterize the efficient set of return-risk possibilities more completely for this example We do this by constructing expected returns and standard deviations of returns for all possible portfolios of the stock fund and the bond fund i.e., can invest 0%, or 5%, or 10%, or 15%, or 20% etc. in stocks 11-21

% in stocks Risk Return 0% 8.2% 7.0% 5% 7.0% 7.2% 10% 5.9% 7.4% 15% 4.8% 7.6% 20% 3.7% 7.8% 25% 2.6% 8.0% 30% 1.4% 8.2% 35% 0.4% 8.4% 40% 0.9% 8.6% 45% 2.0% 8.8% 50.00% 3.08% 9.00% 55% 4.2% 9.2% 60% 5.3% 9.4% 65% 6.4% 9.6% 70% 7.6% 9.8% 75% 8.7% 10.0% 80% 9.8% 10.2% 85% 10.9% 10.4% 90% 12.1% 10.6% 95% 13.2% 10.8% 100% 14.3% 11.0% Portfolio Return 12.0% 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% Portfolo Risk and Return Combinations 100% bonds 0.0% 5.0% 10.0% 15.0% 20.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds. 100% stocks 11-22

% in stocks Risk Return 0% 8.2% 7.0% 5% 7.0% 7.2% 10% 5.9% 7.4% 15% 4.8% 7.6% 20% 3.7% 7.8% 25% 2.6% 8.0% 30% 1.4% 8.2% 35% 0.4% 8.4% 40% 0.9% 8.6% 45% 2.0% 8.8% 50% 3.1% 9.0% 55% 4.2% 9.2% 60% 5.3% 9.4% 65% 6.4% 9.6% 70% 7.6% 9.8% 75% 8.7% 10.0% 80% 9.8% 10.2% 85% 10.9% 10.4% 90% 12.1% 10.6% 95% 13.2% 10.8% 100% 14.3% 11.0% Portfolio Return 12.0% 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% Portfolo Risk and Return Combinations 100% bonds 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) 100% stocks Note that some portfolios are better than others. They have higher returns for the same level of risk or less. 11-23

return = -1.0 100% stocks 100% bonds = 0.2 = 1.0 Relationship depends on correlation coefficient -1.0 < < +1.0 If = +1.0, no risk reduction is possible If = 1.0, complete risk reduction is possible 11-24

return Individual Assets Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios. P 11-25

return minimum variance portfolio Individual Assets The section of the opportunity set above the minimum variance portfolio is the efficient frontier. P 11-26

The return on any security consists of two parts. First, the expected returns Second, the unexpected or risky returns A way to write the return on a stock in the coming month is: R R where U R is the expected part of the return U is the unexpected part of the return 11-27

Any announcement can be broken down into two parts, the anticipated (or expected) part and the surprise (or innovation): Announcement = Expected part + Surprise. The expected part of any announcement is the part of the information the market uses to form the expectation, R, of the return on the stock. The surprise is the news that influences the unanticipated return on the stock, U. 11-28

R R U can be rewritten as R R m where m is systematic or market risk is idiosyncratic or unsystematic risk 11-29

Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns. This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another. However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion. The CAPM is based on this idea that only systematic risk requires a risk premium, because only systematic risk cannot be diversified away. 11-30

In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Diversifiable Risk; Idiosyncratic Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n 11-31

A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree. An unsystematic risk is a risk that specifically affects a single asset or small group of assets. Unsystematic risk can be diversified away. Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation. On the other hand, announcements specific to a single company are examples of unsystematic risk (e.g. labor strikes, part shortages e.t.c.) 11-32

Total risk = systematic risk + unsystematic risk The standard deviation of returns is a measure of total risk. For well-diversified portfolios, unsystematic risk is very small. Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk. 11-33

return 100% stocks r f 100% bonds In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills. Now investors can allocate their money across T-bills and a portfolio. 11-34

return r f P With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope. 11-35

return M r f With the capital allocation line identified, all investors choose a point along the line some combination of the risk-free asset and the market portfolio M. In a world with homogeneous expectations, M is the same for all investors. P 11-36

return Balanced fund 100% stocks r f 100% bonds Where the investor chooses along the Capital Market Line depends on her risk tolerance. The big point is that all investors have the same CML. 11-37

A diversified investor must worry about the systematic risk which cannot be diversified away. Researchers have shown that the best measure of the risk of a security in a large (diversified) portfolio is the beta (b) of the security. Beta is a measure of relative volatility; it measures the responsiveness of a security to movements in the market portfolio (i.e., systematic risk). b i Cov( R 2 ( R Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio. i, R M M ) ) 11-38

Security Returns Slope = b i Return on market % R i = a i + b i R m + e i 11-39

Expected Return on the Market: R M R F Market Risk Premium Expected return on an individual security: R i R F β i ( R M R F ) Market Risk Premium This applies to individual securities held within welldiversified portfolios. 11-40

This formula is called the Capital Asset Pricing Model (CAPM): R i R F β i ( R M R F ) Expected return on a security = Riskfree rate + Beta of the security Market risk premium If b i = 0, then the expected return is R F If b i = 1, then the expected return is RM If b i >1, then the expected return is greater than R F 11-41

Expected return R i R F β i ( R M R F ) RM R F 1.0 b 11-42

Expected return 13.5% 3% 1.5 β i 1.5 R F 3% RM 10% Ri 3% 1.5 (10% 3%) 13.5% b 11-43