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Chap010

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  • 1. Return and Risk The Capital Asset Pricing Model (CAPM)
  • 2. Chapter Outline <ul><li>10.1 Individual Securities </li></ul><ul><li>10.2 Expected Return, Variance, and Covariance </li></ul><ul><li>10.3 The Return and Risk for Portfolios </li></ul><ul><li>10.4 The Efficient Set for Two Assets </li></ul><ul><li>10.5 The Efficient Set for Many Assets </li></ul><ul><li>10.6 Diversification: An Example </li></ul><ul><li>10.7 Riskless Borrowing and Lending </li></ul><ul><li>10.8 Market Equilibrium </li></ul><ul><li>10.9 Relationship between Risk and Expected Return (CAPM) </li></ul>
  • 3. <ul><li>An individual who holds one security should use expected return as the measure of the security’s return . Standard deviation or variance is the proper measure of the security’s risk . </li></ul><ul><li>An individual who holds a diversified portfolio cares about the contribution of each security to the expected return and the risk of the portfolio . </li></ul>
  • 4. 10.1 Individual Securities <ul><li>The characteristics of individual securities that are of interest are the: </li></ul><ul><ul><li>Expected Return </li></ul></ul><ul><ul><li>Variance and Standard Deviation </li></ul></ul><ul><ul><li>Covariance and Correlation (to another security or index) </li></ul></ul>
  • 5. 10.2 Expected Return, Variance, and Covariance
  • 6. <ul><li>Expected Return </li></ul><ul><li>Variance (Standard Deviation): the variability of individual stocks. </li></ul><ul><li>Variance </li></ul><ul><li>σ A 2 = Expected value of (R A -R A ) 2 </li></ul><ul><li>Standard Deviation = σ A </li></ul>
  • 7. <ul><li>Covariance and Correlation : the relationship between the return on one stock and the return on another. </li></ul><ul><ul><li>σ AB = Cov (R A , R B ) =Expected value of 【 (R A -R A )*(R B -R B ) 】 </li></ul></ul><ul><ul><li>ρ AB = Corr(R A , R B ) = σ AB / (σ A * σ B ) </li></ul></ul>
  • 8. 10.3 The Return and Risk for Portfolios <ul><li>How does an investor choose the best combination or portfolio of securities to hold? </li></ul><ul><li>An investor would like a portfolio with a high expected return and low standard deviation of return. </li></ul>
  • 9. The Expected Return on a Portfolio <ul><li>It is simply a weighted average of the expected returns on the individual securities. </li></ul><ul><li>Expected return on portfolio = X A R A + X B R B </li></ul>
  • 10. The Variance of the Portfolio <ul><li>Var (portfolio) </li></ul><ul><li>= X A 2 σ A 2 + 2X A X B σ AB + X B 2 σ B 2 </li></ul><ul><li>The variance of a portfolio depends on both the variances of the individual securities and the covariance between the two securities. </li></ul>
  • 11. Diversification Effect <ul><li>As long as ρ<1 , the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities . </li></ul>
  • 12. An Extension to Many Assets <ul><li>The diversification effect applies to a portfolio of many assets . </li></ul>
  • 13. 10.4 The Efficient Set for Two Assets <ul><li>There are a few important points concerning Figure 10.3 : </li></ul><ul><ul><li>The diversification effect occurs whenever the correlation between the two securities is below 1 . </li></ul></ul><ul><ul><li>The point MV represents the minimum variance portfolio . </li></ul></ul><ul><ul><li>An individual contemplating an investment in a portfolio faces an opportunity set or feasible set . </li></ul></ul>
  • 14. 10.4 The Efficient Set for Two Assets <ul><ul><li>The curve is backward bending between the Slowpoke point and MV . </li></ul></ul><ul><ul><li>The curve from MV to Supertech is called the efficient set . </li></ul></ul>
  • 15. 10.4 The Efficient Set for Two Assets <ul><li>Figure 10.4 shows that the diversification effect rises as ρ declines. </li></ul><ul><li>Efficient sets can be calculated in the real world. </li></ul><ul><li>In Figure 10.5 , the backward bending curve is important information. Some subjectivity must be used when forecasting future expected returns. </li></ul>
  • 16. 10.5 The Efficient Set for Many Securities <ul><li>Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios. </li></ul>return  P Individual Assets
  • 17. The Efficient Set for Many Securities <ul><li>The section of the opportunity set above the minimum variance portfolio is the efficient frontier or fficient set . </li></ul>return  P minimum variance portfolio efficient frontier Individual Assets
  • 18. 10.6 Diversification: An Example <ul><li>Suppose that we make the following three assumptions: </li></ul><ul><ul><li>All securities posses the same variance. </li></ul></ul><ul><ul><li>All covariances in table 10.4 are the same. </li></ul></ul><ul><ul><li>All securities are equally weighted in the portfolio . </li></ul></ul>
  • 19. <ul><li>The variances of the individual securities are diversified away , but the covariance terms cannot be diversified away . </li></ul><ul><li>There is a cost to diversification, so we need to compare the costs and benefits of diversification. </li></ul>
  • 20. Total Risk <ul><li>Total risk = systematic risk (portfolio risk) + unsystematic risk </li></ul><ul><li>The standard deviation of returns is a measure of total risk . </li></ul><ul><li>For well-diversified portfolios, unsystematic risk is very small . </li></ul><ul><li>Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk . </li></ul>
  • 21. Systematic Risk <ul><li>Risk factors that affect a large number of assets </li></ul><ul><li>Also known as non-diversifiable risk or market risk </li></ul><ul><li>Includes such things as changes in GDP , inflation , interest rates , etc. </li></ul>
  • 22. Unsystematic (Diversifiable) Risk <ul><li>Risk factors that affect a limited number of assets </li></ul><ul><li>Also known as unique risk and asset-specific risk </li></ul><ul><li>Includes such things as labor strikes , part shortages , etc. </li></ul><ul><li>The risk that can be eliminated by combining assets into a portfolio </li></ul>
  • 23. 10.7 Riskless Borrowing and Lending <ul><li>A risky investment and a riskless or risk-free security. </li></ul><ul><li>The Optimal Portfolio </li></ul><ul><ul><li>Capital Market Line(CML ) </li></ul></ul>
  • 24. <ul><ul><li>Separation Principle : The investor makes two separate decisions: </li></ul></ul><ul><ul><ul><li>Calculates the efficient set of risky assets , represented by curve XAY in F.10.9. He then determines point A , which represents the portfolio of risky assets that the investors will hold. </li></ul></ul></ul><ul><ul><ul><li>Determines how he will combine point A , his portfolio of risky assets, with the riskless asset . His choice here is determined by his internal characteristics , such as his ability to tolerate risk. </li></ul></ul></ul>
  • 25. 10.8 Market Equilibrium
  • 26. Definition of the Market-Equilibrium Portfolio <ul><li>In a world with homogeneous expectations , all investors would hold the portfolio of risky assets represented by point A . It is the market portfolio . </li></ul><ul><li>A broad-based index is a good proxy for the highly diversified portfolios of many investors. </li></ul><ul><li>The best measure of the risk of a security in a large portfolio is the beta of the security. </li></ul>
  • 27. The Formula for Beta
  • 28. Definition of Risk When Investors hold the Market Portfolio <ul><li>Beta measures the responsiveness of a security to movements in the market portfolio. </li></ul><ul><li>F.10.10 tells us that the return on Jelco are magnified 1.5 times over those of the market. </li></ul>
  • 29. <ul><li>While very few investors hold the market portfolio exactly, many hold reasonably diversified portfolios . These portfolios are close enough to the market portfolio so that the beta of a security is likely to be a reasonable measure of its risk . </li></ul>
  • 30. 10.9 Relationship between Risk and Expected Return (CAPM) <ul><li>Expected Return on the Market : </li></ul><ul><li>Expected return on an individual security : </li></ul>Market Risk Premium This applies to individual securities held within well-diversified portfolios.
  • 31. Expected Return on a Security <ul><li>This formula is called the Capital Asset Pricing Model (CAPM): </li></ul><ul><li>Assume  i = 0, then the expected return is R F . </li></ul><ul><li>Assume  i = 1, then </li></ul>Expected return on a security = Risk-free rate + Beta of the security × Market risk premium
  • 32. Expected Return on Individual Security (1/2) <ul><li>Capital-Asset-Pricing Model( CAPM ) : </li></ul><ul><ul><li>R = R F + β * (R M - R F ) (10.17) </li></ul></ul><ul><ul><li>Linearity : In equilibrium, all securities would be held only when prices changed so that the SML became straight. </li></ul></ul><ul><ul><li>Portfolios as well as securities . </li></ul></ul><ul><ul><ul><li>The beta of the portfolio is simply a weighted average of the securities in the portfolio. </li></ul></ul></ul>
  • 33. Expected Return on Individual Security (2/2) <ul><ul><li>F.10.11 differs from F.10.9 in at least two ways: </li></ul></ul><ul><ul><ul><li>Beta appears in the horizontal axis of F.10.11, but standard deviation appears in the horizontal axis of F.10.9. </li></ul></ul></ul><ul><ul><ul><li>The SML (F.10.11) holds for all individual securities and for all possible portfolios , whereas CML (F.10.9) holds only for efficient portfolios . </li></ul></ul></ul>
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  • 40. CML Var(R P ) = σ p 2 = a 2 σ m 2 σ p = a*σ m a = σ p / σ m ( 代入 E(R P ) 中 ) E(R P ) = (1-a) R f + a*E(R m ) , a>0 E(R P ) = R f + ( σ p / σ m )*[E(R m )-R f ] = R f + { [E(R m )-R f ] / σ m } * σ p riskless asset market portfolio
  • 41.  
  • 42. <ul><li>Var (portfolio) </li></ul><ul><li>= X A 2 σ A 2 + 2X A X B σ AB + X B 2 σ B 2 </li></ul><ul><li>=X A 2 σ A 2 + 2X A X B ρ AB σ A * σ B + X B 2 σ B 2 </li></ul><ul><li>(ρ AB = σ AB / σ A * σ B ) </li></ul><ul><li>=X A 2 σ A 2 + 2X A X B σ A * σ B + X B 2 σ B 2 </li></ul><ul><li>(when ρ AB =1 ) </li></ul><ul><li>= (X A σ A + X B σ B ) 2 </li></ul><ul><li>Standard Deviation = (X A σ A + X B σ B ) </li></ul>11 10
  • 43. CML Var(R P ) = σ p 2 = a 2 σ m 2 σ p = a*σ m a = σ p / σ m ( 代入 E(R P ) 中 ) E(R P ) = (1-a) R f + a*E(R m ) , a>0 E(R P ) = R f + ( σ p / σ m )*[E(R m )-R f ] = R f + { [E(R m )-R f ] / σ m } * σ p riskless asset market portfolio
  • 44. β p = = = = + = + = W 1 β 1 + W 2 β 2 σ pm σ m 2 E{ [W 1 R 1 +W 2 R 2 –W 1 E(R 1 )–W 2 E(R 2 )]*[R m –E(R m )]} σ m 2 E{ [W 1 (R 1 –E(R 1 ) +W 2 (R 2 –E(R 2 )]*[R m –E(R m )]} σ m 2 E [W 1 (R 1 –E(R 1 )]*[R m –E(R m )] σ m 2 E [W 2 (R 2 –E(R 2 )]*[R m –E(R m )] σ m 2 σ m 2 W 1 σ 1m σ m 2 W 2 σ 2m
  • 45. E(R A ) = 15.0% E(R Z ) = 8.6% E(R P ) = 0.5 *15.0% + 0.5 *8.6% = 11.8% Beta of Portfolio: = 0.5 *1.5 + 0.5 *0.7 = 1.1 Under the CAPM, the E(R P ) is E(R P ) = 3% + 1.1 *8.0% = 11.8%
  • 46. σ p 2 = E [R P –E(R P )] 2 = E [(W 1 R 1 +W 2 R 2 ) – (W 1 E(R 1 ) + W 2 E(R 2 ))] 2 = E [(W 1 (R 1 – E(R 1 )) + W 2 (R 2 – E(R 2 )] 2 = = W 1 2 σ 1 2 + W 2 2 σ 2 2 + 2W 1 W 2 E [(R 1 – E(R 1 ))*(R 2 – E(R 2 )] W 1 2 σ 1 2 + W 2 2 σ 2 2 + 2W 1 W 2 σ 1 2 = W 1 2 σ 1 2 W 2 2 σ 2 2 W 1 W 2 σ 1 2 W 2 W 1 σ 21 設 Portfolio 中只有兩種資產 Σ W i 2 σ i 2 + Σ Σ W i W j σ ij i=1 2 i=1 2 j=1 2 i≠j
  • 47. σ p 2 = = + + = + = = (when N->∞) (10.10) Σ X i 2 σ i 2 + Σ Σ X i X j σ ij i=1 N i=1 N j=1 i≠j N N N 2 1 Var N(N – 1) N 2 1 COV N 1 Var N(N – 1) N 2 COV N 1 Var COV (N – 1) N COV
  • 48. Portfolio Risk and Number of Stocks Nondiversifiable risk; Systematic Risk; Market Risk Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk n  In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Portfolio risk

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