Chapter 11 Return and Risk: The Capital Asset Pricing Model (CAPM) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Key Concepts and Skills 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 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline 11.1 Individual Securities 11.2 Expected Return, Variance, and Covariance 11.3 The Return and Risk for Portfolios 11.4 The Efficient Set 11.5 Riskless Borrowing and Lending 11.6 Announcements, Surprises, and Expected Return 11.7 Risk: Systematic and Unsystematic 11.8 Diversification and Portfolio Risk 11.9 Market Equilibrium 11.10 Relationship between Risk and Expected Return (CAPM) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 11.1 Individual Securities 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) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 11.2 Expected Return, Variance, and Covariance 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. Scenario Recession Normal Boom McGraw-Hill/Irwin Rate of Return Probability Stock Fund Bond Fund 33.3% -7% 17% 33.3% 12% 7% 33.3% 28% -3% Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Expected Return Scenario Recession Normal Boom Expected return Variance Standard Deviation McGraw-Hill/Irwin Stock Fund Rate of Squared Return Deviation -7% 0.0324 12% 0.0001 28% 0.0289 11.00% 0.0205 14.3% Bond Rate of Return 17% 7% -3% 7.00% 0.0067 8.2% Fund Squared Deviation 0.0100 0.0000 0.0100 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Expected Return Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock Fund Rate of Squared Return Deviation -7% 0.0324 12% 0.0001 28% 0.0289 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 0.0100 7% 0.0000 -3% 0.0100 7.00% 0.0067 8.2% E (rS ) 1 (7%) 1 (12%) 1 (28%) 3 3 3 E (rS ) 11% McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Variance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock Fund Rate of Squared Return Deviation -7% 0.0324 12% 0.0001 28% 0.0289 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 0.0100 7% 0.0000 -3% 0.0100 7.00% 0.0067 8.2% (7% 11%) .0324 2 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Variance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock Fund Rate of Squared Return Deviation -7% 0.0324 12% 0.0001 28% 0.0289 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 0.0100 7% 0.0000 -3% 0.0100 7.00% 0.0067 8.2% 1 .0205 (.0324 .0001 .0289) 3 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Standard Deviation Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock Fund Rate of Squared Return Deviation -7% 0.0324 12% 0.0001 28% 0.0289 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 0.0100 7% 0.0000 -3% 0.0100 7.00% 0.0067 8.2% 14.3% 0.0205 McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Covariance Scenario Recession Normal Boom Sum Covariance Stock Bond Deviation Deviation -18% 10% 1% 0% 17% -10% Product -0.0180 0.0000 -0.0170 Weighted -0.0060 0.0000 -0.0057 -0.0117 -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. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Correlation Cov(a, b) a b .0117 0.998 (.143)(.082) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 11.3 The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock Fund Rate of Squared Return Deviation -7% 0.0324 12% 0.0001 28% 0.0289 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 0.0100 7% 0.0000 -3% 0.0100 7.00% 0.0067 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. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Portfolios Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% Scenario Recession Normal Boom Expected return Variance Standard Deviation 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.0016 0.0000 0.0012 9.0% 0.0010 3.08% The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: rP wB rB wS rS 5% 50% (7%) 50% (17%) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Portfolios Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% Scenario Recession Normal Boom Expected return Variance Standard Deviation 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.0016 0.0000 0.0012 9.0% 0.0010 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 (rP ) wB E (rB ) wS E (rS ) 9% 50% (11%) 50% (7%) McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.0016 0.0000 0.0012 9.0% 0.0010 3.08% The variance of the rate of return on the two risky assets portfolio is σ P2 (wB σ B )2 (wS σ S )2 2(wB σ B )(wS σ S )ρ BS where BS is the correlation coefficient between the returns on the stock and bond funds. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.0016 0.0000 0.0012 9.0% 0.0010 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. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. % in stocks 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50.00% 55% 60% 65% 70% 75% 80% 85% 90% 95% McGraw-Hill/Irwin 100% Risk Return 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.08% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.00% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolio Return 11.4 The Efficient Set Portfolo Risk and Return Combinations 100% stocks 12.0% 11.0% 10.0% 9.0% 8.0% 100% bonds 7.0% 6.0% 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds … Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. % in stocks 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% McGraw-Hill/Irwin 100% Risk Return 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.1% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolio Return The Efficient Set Portfolo Risk and Return Combinations 12.0% 11.0% 10.0% 100% stocks 9.0% 8.0% 7.0% 6.0% 100% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Portfolios with Various Correlations return 100% stocks = -1.0 100% bonds = 1.0 = 0.2 McGraw-Hill/Irwin 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 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. return The Efficient Set for Many Securities Individual Assets P Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. return The Efficient Set for Many Securities minimum variance portfolio Individual Assets P The section of the opportunity set above the minimum variance portfolio is the efficient frontier. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. return Optimal Portfolio with a Risk-Free Asset 100% stocks rf 100% bonds In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. return 11.5 Riskless Borrowing and Lending 100% stocks Balanced fund rf 100% bonds Now investors can allocate their money across the T-bills and a balanced mutual fund. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. return Riskless Borrowing and Lending rf P With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Expected vs. Unexpected Returns Realized returns are generally not equal to expected returns. There is the expected component and the unexpected component. At any point in time, the unexpected return can be either positive or negative. Over time, the average of the unexpected component is zero. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 11.6 Announcements and News Announcements and news contain both an expected component and a surprise component. It is the surprise component that affects a stock’s price and, therefore, its return. This is very obvious when we watch how stock prices move when an unexpected announcement is made or earnings are different than anticipated McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 11.7 Risk: Systematic Risk factors that affect a large number of assets Also known as non-diversifiable risk or market risk Includes such things as changes in GDP, inflation, interest rates, etc. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Risk: Unsystematic Risk factors that affect a limited number of assets Also known as unique risk and asset-specific risk Includes such things as labor strikes, part shortages, etc. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Returns Total Return = expected return + unexpected return Unexpected return = systematic portion + unsystematic portion Therefore, total return can be expressed as follows: Total Return = expected return + systematic portion + unsystematic portion McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 11.8 Diversification and Portfolio Risk 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. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Portfolio Risk and Number of Stocks McGraw-Hill/Irwin In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Diversifiable Risk The risk that can be eliminated by combining assets into a portfolio Often considered the same as unsystematic, unique, or asset-specific risk If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Total Risk 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. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. return 11.9 Market Equilibrium M rf P 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. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. return Market Equilibrium 100% stocks Balanced fund rf 100% bonds Just where the investor chooses along the Capital Market Line depends on his risk tolerance. The big point is that all investors have the same CML. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Risk When Holding the Market Portfolio Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (b)of the security. Beta measures the responsiveness of a security to movements in the market portfolio (i.e., systematic risk). bi McGraw-Hill/Irwin Cov( Ri , RM ) ( RM ) 2 Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Security Returns Estimating b with Regression Slope = bi Return on market % McGraw-Hill/Irwin Ri = a i + biRm + ei Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. The Formula for Beta bi Cov( Ri , RM ) ( RM ) 2 Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 11.10 Risk and Return (CAPM) Expected Return on the Market: R M RF Market Risk Premium • Expected return on an individual security: Ri RF βi ( R M RF ) Market Risk Premium This applies to individual securities held within welldiversified portfolios. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Expected Return on a Security This formula is called the Capital Asset Pricing Model (CAPM): Ri RF βi ( R M RF ) Expected return on a security RiskBeta of the = + × free rate security Market risk premium • Assume bi = 0, then the expected return is RF. • Assume bi = 1, then Ri R M McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Expected return Relationship Between Risk & Return Ri RF βi ( R M RF ) RM RF 1.0 McGraw-Hill/Irwin b Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Expected return Relationship Between Risk & Return 13.5% 3% βi 1.5 RF 3% 1.5 b R M 10% R i 3% 1.5 (10% 3%) 13.5% McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Quick Quiz How do you compute the expected return and standard deviation for an individual asset? For a portfolio? What is the difference between systematic and unsystematic risk? What type of risk is relevant for determining the expected return? Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%. What is the expected return on the asset? McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.