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- 1. Return and Risk: The Capital Asset Pricing Model, Asset pricing Theories 11-1
- 2. 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) 11-2
- 3. 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 shares fund and a bond fund. Rate of ReturnScenario ProbabilityShares Fund Bond FundRecession 33.3% -7% 17%Normal 33.3% 12% 7%Boom 33.3% 28% -3% 11-3
- 4. Expected Return Shares Fund Bond Fund Rate of Squared Rate of SquaredScenario Return Deviation Return DeviationRecession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000Boom 28% 0.0289 -3% 0.0100Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2% 11-4
- 5. Expected Return Stock Fund Bond Fund Rate of Squared Rate of SquaredScenario Return Deviation Return DeviationRecession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000Boom 28% 0.0289 -3% 0.0100Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2%E (rS ) = 1 × (−7%) + 1 × (12%) + 1 × (28%) 3 3 3E (rS ) = 11% 11-5
- 6. Variance Stock Fund Bond Fund Rate of Squared Rate of SquaredScenario Return Deviation Return DeviationRecession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000Boom 28% 0.0289 -3% 0.0100Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2% (−7% − 11%) = .0324 2 11-6
- 7. Variance Stock Fund Bond Fund Rate of Squared Rate of SquaredScenario Return Deviation Return DeviationRecession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000Boom 28% 0.0289 -3% 0.0100Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2% 1 .0205 = (.0324 + .0001 + .0289) 3 11-7
- 8. Standard Deviation Stock Fund Bond Fund Rate of Squared Rate of SquaredScenario Return Deviation Return DeviationRecession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000Boom 28% 0.0289 -3% 0.0100Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2% 14.3% = 0.0205 11-8
- 9. Covariance 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 theprobability of that state. 11-9
- 10. Correlation Cov (a, b)ρ= σ aσ b − .0117ρ= = −0.998 (.143)(.082) 11-10
- 11. 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. 11-11
- 12. 11.3 The Return and Risk for Portfolios 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. 11-12
- 13. Portfolios Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard 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: rP = wB rB + wS rS 5% = 50% × (−7%) + 50% × (17%) 11-13
- 14. Portfolios Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard 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 (rP ) = wB E (rB ) + wS E (rS ) 9% = 50% × (11%) + 50% × (7%) 11-14
- 15. Portfolios Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08% The variance of the rate of return on the two risky assets portfolio is σ P = (wB σ B ) 2 + (wS σ S ) 2 + 2(wB σ B )(wS σ S )ρ BS 2 where ρBS is the correlation coefficient between the returns on the stock and bond funds. 11-15
- 16. Portfolios Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard 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. 11-16
- 17. 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. 11-17
- 18. Risk: Systematic and Unsystematic We can break down the total risk of holding a stock into two components: systematic risk and unsystematic risk:σ2 R = R +U Total risk becomes ε R = R+m+ε where Nonsystematic Risk: ε m is the systematic risk Systematic Risk: m ε is the unsystematic risk n 11-18
- 19. 12.2 Systematic Risk and Betas• The beta coefficient, β, tells us the response of the stock’s return to a systematic risk.• In the CAPM, β measures the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio. Cov( Ri , RM ) βi = σ ( RM ) 2 • We shall now consider other types of systematic risk. 11-19
- 20. Systematic Risk and Betas• For example, suppose we have identified three systematic risks: inflation, GNP growth, and the dollar-euro spot exchange rate, S($,€).• Our model is: R = R+m+ε R = R + β I FI + βGNP FGNP + βS FS + ε β I is the inflation beta βGNP is the GNP beta βS is the spot exchange rate beta ε is the unsystematic risk 11-20
- 21. Systematic Risk and Betas: Example R = R + βI FI + βGNP FGNP + βS FS + ε• Suppose we have made the following estimates: βI = -2.30 βGNP = 1.50 βS = 0.50• Finally, the firm was able to attract a ε = 1% “superstar” CEO, and this unanticipated development contributes 1% .to × FS + 1% R = R − 2.30 × FI + 1.50 × FGNP + 0 50 the return. 11-21
- 22. Systematic Risk and Betas: Example R = R − 2.30 × FI + 1.50 × FGNP + 0.50 × FS + 1%We must decide what surprises took place in the systematic factors.If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then:FI = Surprise in the inflation rate = actual – expected = 8% – 3% = 5% R = R − 2.30 × 5% + 1.50 × FGNP + 0.50 × FS + 1% 11-22
- 23. Systematic Risk and Betas: Example R = R − 2.30 × FI + 1.50 × FGNP + 0.50 × FS + 1%We must decide what surprises took place in the systematic factors.If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then:FI = Surprise in the inflation rate = actual – expected = 8% – 3% = 5% R = R − 2.30 × 5% + 1.50 × FGNP + 0.50 × FS + 1% 11-23
- 24. Systematic Risk and Betas: Example R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1%If it were the case that the dollar-euro spot exchange rate, S($,€), was expected to increase by 10%, but in fact remained stable during the time period, then:FS = Surprise in the exchange rate = actual – expected = 0% – 10% = – 10% R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1% 11-24
- 25. Systematic Risk and Betas: Example R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1%Finally, if it were the case that the expected return on the stock was 8%, then: R = 8% R = 8% − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1% R = −12% 11-25
- 26. Systematic Risk and Betas: Example R = R − 2.30 × 5% + 1.50 × FGNP + 0.50 × FS + 1%If it were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then:FGNP = Surprise in the rate of GNP growth = actual – expected = 1% – 4% = – 3% R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1% 11-26
- 27. Portfolio Risk and Number of Stocks In a large portfolio the variance terms are σ 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 11-27
- 28. Risk: Systematic and Unsystematic• 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. 11-28
- 29. 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. 11-29
- 30. 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 (β)of the security.• Beta measures the responsiveness of a security to movements in the market portfolio (i.e., systematic risk). Cov( Ri , RM ) βi = σ ( RM ) 2 11-30
- 31. The Formula for Beta Cov ( Ri , RM ) σ ( Ri )βi = =ρ σ ( RM ) 2 σ ( RM ) Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio. 11-31
- 32. 11.9 Relationship between Risk and Expected Return (CAPM)• Expected Return on the Market: R M = RF + Market Risk Premium • Expected return on an individual security: R i = RF + β i × ( R M − RF ) Market Risk Premium This applies to individual securities held within well- diversified portfolios. 11-32
- 33. Capital Asset Pricing Model (CAPM)• The pure time value of money : As measured by the risk free rate, Rf, this is the reward for merely waiting for your money, without taking any risk.• The reward for bearing systematic risk: As measured by the market risk premium, E (RM) - Rf, this component is the reward the market offers for bearing an average amount of systematic risk in addition to waiting.• The amount of systematic risk: As measured by βi, this is the amount of systematic risk present in a particular assets or portfolio, relative to that in an average asset. 11-33
- 34. Expected Return on a Security• This formula is called the Capital Asset Pricing Model (CAPM): R i = RF + β i × ( R M − RF ) Expected Risk- Beta of the Market risk return on = + × free rate security premium a security • Assume βi = 0, then the expected return is RF. • Assume βi = 1, then R i = R M 11-34

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