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The Capital Asset Pricing Model Global Financial Management 1 Overview Utility and risk aversion » Choosing efficient portfolios Investing with a risk-free asset » Borrowing and lending » The markt portfolio » The Capital Market Line (CML) The Capital Asset Pricing Model (CAPM) » The Security Market Line (SML) » Beta » Project analysis 2 Efficient Portfolios with Multiple Assets Investors Efficient E[r] prefer Frontier Portfolios Asset 1 of other Portfolios of assets Asset 2 Asset 1 and Asset 2 Minimum-Variance Portfolio 0 s 3 Utility in Risk-Return Space Indifference curves 25.00% tau=0.5, Ubar=6% tau=0.5, Ubar=8% Investors 20.00% tau=0.5, Ubar=10% prefer tau=0.25, Ubar=6% tau=0.25, Ubar=8% 15.00% tau=0.5, Ubar=10% Return 10.00% 5.00% 0.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 18.00% 20.00% 22.00% 24.00% Risk 4 Individual Asset Allocations 14.00% Point x is the optimal portfolio for the less risk Return 12.00% averse investor (red line) Point y is the optimal 10.00% x portfolio for the more risk 8.00% y averse investor (black line) 6.00% 4.00% 2.00% Risk 0.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00% 11.00% 12.00% 13.00% 14.00% 15.00% 16.00% 17.00% 18.00% 19.00% 20.00% 5 Introducing a Riskfree Asset Suppose we introduce the opportunity to invest in a riskfree asset. » How does this alter investors’ portfolio choices? The riskfree asset has a zero variance, and zero covariance with every other asset (or portfolio). » var(rf) = 0. » cov(rf, rj) = 0 for all j. What is the expected return and variance of a portfolio consisting of a fraction (1-a) of the riskfree asset and a of the risky asset (or portfolio)? 6 Risk and Return with a Riskfree asset Expected Return ErP aE rj 1 a )r f Variance and Standard Deviation VarrP s 2 a 2 s 2 s P as j P j Hence, the risk-return tradeoff is: sP E rP r f sj Erj rj ) 7 Risk and Return with a Riskfree asset Expected The line represents Return all portfolios depending on a Asset j (a=1) E(rj) rf Riskfree asset (a=0) 0 sj Standard Deviation 8 Investing with Borrowing and Lending Expected a =2 Return a = 0.5 M E[rM ] rf a =1 a=0 Lending Borrowing 0 sM Standard Deviation 9 Optimal Investing With Borrowing and Lending 25.00% Y = optimal risk- Return return tradeoff tau=0.5, Ubar=8% for risk-averse 20.00% tau=0.25, Ubar=6% investor Portfolio X = optimal risk- 15.00% return tradeoff for risk-tolerant investor 10.00% X Y 5.00% rf=4% Risk 0.00% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 10 The Capital Market Line Expected Return M E [ rm ] E [ rIBM ] A IBM rf Systematic Diversifiable Standard Risk Risk Deviation 11 The Capital Market Line The CML gives the tradeoff between risk and return for portfolios consisting of the riskfree asset and the tangency portfolio M. » Portfolio M is the market portfolio. The equation of the CML is: E (rM ) rf E (rp ) rf s p sM The expected rate of return on a risky asset can be thought of as composed of two terms. » The return on a riskfree security, like U.S. Treasury bills; compensating investors for the time value of money. » A risk premium to compensate investors for bearing risk. E(r) = rf + Risk x [Market Price of Risk] 12 Everybody holds the Market Everybody holds the tangency portfolio M » If all hold the same portfolio, it must be the market! Nobody can do better than holding the market » If another asset existed which offers a better return for the same risk, buy that! Can’t be an equilibrium Write the weight of asset j in the market portfolio as wj. Then we have: ) ) E rM ) j 1 w j E rj rf rf j N Var rM ) i 1 j 1 wi w j Covrj , ri ) i N j N » Simply use expressions for multi-asset case 13 All Risk-Return Tradeoffs are Equal Hence, if you increase the weight of asset j in your portfolio (relative to the market), » Then expected returns increase by: E rj ) rf » Then the riskiness of the portfolio increases by: i 1 wi Covrj , ri ) Covrj , rM ) N » Hence, the return/risk gain is: E rj ) rf Cov rj , rM ) » This must be the same for all assets – Why? 14 All Assets are Equal Suppose that for two assets A and B: E rA ) rf E rB ) rf CovrA , rM ) CovrB , rM ) » Asset A offers a better return/risk ratio than asset B – Buy A, sell B – What if everybody does this? » Hence, in equilibrium, all return/risk ratios must be equal for all assets E rA ) rf E rB ) rf CovrA , rM ) CovrB , rM ) 15 The Capital Asset Pricing Model If the risk-return tradeoff is the same for all assets, than it is the one of the market: E rA ) rf E rB ) rf E rM ) rf CovrA , rM ) CovrB , rM ) Var rM ) This gives the relationship between risk and expected return for individual stocks and portfolios. » This is called the Security Market Line. Cov rA , rM ) E rA ) rf Var rM ) ) E rM ) rf rf A E rM ) rf ) Cov rA , rM ) A where Var rM ) 16 Capital Asset Pricing Model A Graphical Illustration Expected Return Expected Market Return Expected market risk premium Risk free rate 0 0.5 1.0 Beta Expected Risk free Beta Expected market = + x return rate factor risk premium 17 The Intuitive Argument For the CAPM Everybody holds the same portfolio, hence the market. Portfolio-risk cannot be diversified. Investors demand a premium on non-diversifiable risk only, hence portfolio or market risk. Beta measures the market risk, hence it is the correct measure for non-diversifiable risk. Conclusion: In a market where investors can diversify by holding many assets in their portfolio, they demand a risk premium proportional to beta. 18 The SML and mispriced stocks Suppose for a particular stock: ) Er ) r Cov rj , rM ) E rj r f VarrM ) M f Remember the definition of expected returns: ) E Pj1 D1 Pj0 ) E rj j Pj0 Then P0 falls, so that E(rj) increases until disequilibrium vanishes and the equation holds! 19 The SML and mispriced stocks Expected Return Stock j is overvalued at X: » price drops, E(rM) » expected return rises. Y At Y, stock j would be undervalued! » expected return falls X » price increases rf j =1 20 The CML and SML E(r) CML E(r) SML M M E(rM) E(rIBM) IBM rf rf sIBM,M/sM sM sIBM s IBM 1.0 21 The Capital Asset Pricing Model The appropriate measure of risk for an individual stock is its beta. Beta measures the stock’s sensitivity to market risk factors. » The higher the beta, the more sensitive the stock is to market movements. The average stock has a beta of 1.0. Portfolio betas are weighted averages of the betas for the individual stocks in the portfolio. The market price of risk is [E(rM)-rf]. 22 Using Regression Analysis to Measure Betas Rate of Return on Stock A Slope = Beta x x x x x x x x x x Rate of Return x x on the Market x x x Jan 1995 23 Calculating the beta of BA Return on B A 40 30 20 10 0 -10 -20 B eta -30 -40 -30 -20 -10 0 10 20 Beta is the slope of a regression line which best fits Return on the market index the scatter of monthly returns on the share and on the market index. 24 Betas of Selected Common Stocks Stock Beta Stock Beta AT&T 0.96 Ford Motor 1.03 Boston Ed. 0.49 Home Depot 1.34 BM Squibb 0.92 McDonalds 1.06 Delta Airlines 1.31 Microsoft 1.20 Digital Equip. 1.23 Nynex 0.77 Dow Chem. 1.05 Polaroid 0.96 Exxon 0.46 Tandem 1.73 Merck 1.11 UAL 1.84 Betas based on 5 years of monthly returns through mid-1993. 25 Beta and Standard Deviation Risk of a Market risk Specific risk = + Share (Variance) of the share of the share Beta of Risk of This is the major share x market element of a share's risk Risk of a Market risk of Specific risk of portfolio = the portfolio + the portfolio Beta of Risk of This is negligible Portfolio x market for a diversified portfolio 26 Testing the CAPM Black, Jensen and Scholes Average Monthly Return Theoretical Line • • • Fitted Line • • • • • • Beta 27 Estimating the Expected Rate of Return on Equity The SML gives us a way to estimate the expected (or required) rate of return on equity. ) E rj r f j E rM ) r f We need estimates of three things: » Riskfree interest rate, rf. » Market price of risk, [E(rM)-rf]. » Beta for the stock,j. 28 Estimating the Expected Rate of Return on Equity The riskfree rate can be estimated by the current yield on one-year Treasury bills. » As of early 1997, one-year Treasury bills were yielding about 5.0%. The market price of risk can be estimated by looking at the historical difference between the return on stocks and the return on Treasury bills. » This difference has averaged about 8.6% since 1926. The betas are estimated by regression analysis. 29 Estimating the Expected Rate of Return on Equity E(r) = 5.0% + (8.6%) Stock E(r) Stock E(r) AT&T 13.3% Ford Motor 13.9% Boston Ed. 9.2% Home Depot 16.5% BM Squibb 12.9% McDonalds 14.1% Delta Airlines 16.3% Microsoft 15.3% Digital Equip. 15.6% Nynex 11.6% Dow Chem. 14.0% Polaroid 13.3% Exxon 9.0% Tandem 19.9% Merck 14.5% UAL 20.8% 30 Example of Portfolio Betas and Expected Returns What is the beta and expected rate of return of an equally-weighted portfolio consisting of Exxon and Polaroid? Portfolio Beta p (1 / 2)(.46) (1 / 2)(.96) p 0.71 Expected Rate of Return E (rp ) 5.0% (8.6%)(0.71) 111% . How would you construct a portfolio with the same beta and expected return, but with the lowest possible standard deviation? Use the figure on the following page to locate the equally-weighted portfolio of Exxon and Polaroid. Also locate the minimum variance portfolio with the same expected return. 31 Graphical Illustration E(r) E(r) CML SML M M 13.6% 11.1% 5.0% 5.0% sM s 0.71 1.0 32 Example The S&P500 Index has a standard deviation of about 12% per year. Gold mining stocks have a standard deviation of about 24% per year and a correlation with the S&P500 of about r = 0.15. If the yield on U.S. Treasury bills is 6% and the market risk premium is [E(rM)-rf] = 7.0%, what is the expected rate of return on gold mining stocks? 33 Example The beta for gold mining stocks is calculated as follows: s gM r gM s g s M .15(.24) 2 0.30 sM sM2 .12 The expected rate of return on gold mining stocks is: E(rg ) 6.0% ( 7.0%)(0.30) 7.1% Question: What portfolio has the same expected return as gold mining stocks, but the lowest possible standard deviation? Answer: A portfolio consisting of 70% invested in U.S. Treasury bills and 30% invested in the S&P500 Index. Beta (.7)( 0) (.3)(1.0) 0.30 E ( rp ) 6.0% ( 7.0%)(0.30) 8.1% Sd ( rp ) (.7)( 0) (.3)(12.0%) 3.6% 34 Using the CAPM for Project Evaluation Suppose Microsoft is considering an expansion of its current operations. » The expansion will cost $100 million today » expected to generate a net cash flow of $25 million per year for the next 20 years. » What is the appropriate risk-adjusted discount rate for the expansion project? » What is the NPV of Microsoft’s investment project? 35 Microsoft’s Expansion Project The risk-adjusted discount rate for the project, rp, can be estimated by using Microsoft’s beta and the CAPM. rP r f E rm r f ) Thus, the NPV of the project is: rP 0.05 1.2 * 0.086) $25 NPV t 1 20 $100 $53.92 million 1153) . t 36 Company Risk Versus Project Risk The company-wide discount rate is the appropriate discount rate for evaluating investment projects that have the same risk as the firm as a whole. For investment projects that have different risk from the firm’s existing assets, the company-wide discount rate is not the appropriate discount rate. In these cases, we must rely on industry betas for estimates of project risk. 37 Company Risk versus Project Risk Suppose Microsoft is considering investing in the development of a new airline. » What is the risk of this investment? » What is the appropriate risk-adjusted discount rate for evaluating the project? » Suppose the project offers a 17% rate of return. Is the investment a good one for Microsoft? 38 Industry Asset Betas Industry Beta Industry Beta Airlines 1.80 Agriculture 1.00 Electronics 1.60 Food 1.00 Consumer Durables 1.45 Liquor 0.90 Producer Goods 1.30 Banks 0.85 Chemicals 1.25 International Oils 0.85 Shipping 1.20 Tobacco 0.80 Steel 1.05 Telephone Utilities 0.75 Containers 1.05 Energy Utilities 0.60 Nonferrous Metals 1.00 Gold 0.35 Source: D. Mullins, “Does the Capital Asset Pricing Model Work?,” Havard Business Review, vol. 60, pp. 105-114. 39 Company Risk versus Project Risk The project risk is closer to the risk of other airlines than it is to the risk of Microsoft’s software business. The appropriate risk-adjusted discount rate for the project depends upon the risk of the project. If the average asset beta for airlines is 1.8, then the project’s cost of capital is: rp rf p E rm rf ) rp 0.05 180.086) 20.5% . 40 Company Risk versus Project Risk Required Return SML Project-specific Discount Rate Project IRR A Company-wide Discount Rate Company Beta Project Beta 41 Project Evaluation: Rules The risk of an investment project is given by the project’s beta. » Can be different from company’s beta » Can often use industry as approximation The Security Market Line provides an estimate of an appropriate discount rate for the project based upon the project’s beta. » Same company may use different discount rates for different projects This discount rate is used when computing the project’s net present value. 42 Summary Optimal investments depend on trading off risk and return » Investors with higher risk tolerance invest more in risky assets » Only risk that can’t be diversified counts If investors can borrow and lend, then everybody holds a combination of two portfolios » The market portfolio of all risky assets » The riskless asset – Covariance with the market portfolio counts In equilibrium, all stocks must lie on the security market line » Beta measures the amount of nondiversifiable risk » Expected returns reflect only market risk » Use these as required returns in project evaluation 43

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Capital Asset Pricing Model, expected return, CAPM model, market portfolio, Certified Associate in Project Management, CAPM formula, project management, Project Management Institute, systematic risk, risk premium, study guide, Beta x, Exam Questions, Asset Pricing, risky assets

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posted: | 1/29/2011 |

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