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39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Ch 05 Mini Case 5/3/2003 Chapter 5. Mini Case Situation To begin, briefly review the Chapter 4 Mini Case. Then, extend your knowledge of risk and return by answering the following questions. a. Suppose Asset A has an expected return of 10 percent and a standard deviation of 20 percent. Asset B has an expected return of 16 percent and a standard deviation of 40 percent. If the correlation between A and B is 0.4, what are the expected return and standard deviation for a portfolio comprised of 30 percent Asset A and 70 percent Asset B? Expected return of a portfolio: ^ ^ ^ r p w A r A (1 w A ) r B Standard deviation of a portfolio: s p WAs A (1 WA )2 s B 2WA (1 WA ) AB s A s B 2 2 2 Asset A Asset B Expected return, r hat 10% 16% Standard deviation, s 20% 40% Using the equations above, we can find the expected return and standard deviation of a portfolio with different percents invested in each asset. Correlation = 0.4 Proportion of Proportion of Portfolio in Portfolio in Security A Security B (Value of wA) (Value of 1-w A) rp sp 1.00 0.00 10.00% 20.00% 0.90 0.10 10.60% 19.94% 0.80 0.20 11.20% 20.55% 0.70 0.30 11.80% 21.78% 0.60 0.40 12.40% 23.53% 0.50 0.50 13.00% 25.69% 0.40 0.60 13.60% 28.17% 0.30 0.70 14.20% 30.89% 0.20 0.80 14.80% 33.80% 0.10 0.90 15.40% 36.85% 0.00 1.00 16.00% 40.00% Michael C. Ehrhardt Page 1 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model b. Plot the attainable portfolios for a correlation of 0.4. Now plot the attainable portfolios for correlations of +1.0 and -1.0. AB = +0.4: Attainable Set of Risk/Return Combinations 20% 15% Expected return 10% 5% 0% 0% 10% 20% 30% 40% Risk, sp Correlation = 1 Proportion of Proportion of Portfolio in Portfolio in Security A Security B (Value of wA) (Value of 1-w A) rp sp 1.00 0.00 10.00% 20.00% 0.90 0.10 10.60% 22.00% 0.80 0.20 11.20% 24.00% 0.70 0.30 11.80% 26.00% 0.60 0.40 12.40% 28.00% 0.50 0.50 13.00% 30.00% 0.40 0.60 13.60% 32.00% 0.30 0.70 14.20% 34.00% 0.20 0.80 14.80% 36.00% 0.10 0.90 15.40% 38.00% 0.00 1.00 16.00% 40.00% AB = +1.0: Attainable Set of Risk/Return Combinations 20% Expected return 15% 10% 5% 0% 0% 20% 40% Risk, sp Correlation = -1 Michael C. Ehrhardt Page 2 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Proportion of Proportion of Portfolio in Portfolio in Security A Security B (Value of wA) (Value of 1-w A) rp sp 1.00 0.00 10.00% 20.00% 0.90 0.10 10.60% 14.00% 0.80 0.20 11.20% 8.00% 0.70 0.30 11.80% 2.00% 0.67 0.33 12.00% 0.00% 0.60 0.40 12.40% 4.00% 0.50 0.50 13.00% 10.00% 0.40 0.60 13.60% 16.00% 0.30 0.70 14.20% 22.00% 0.20 0.80 14.80% 28.00% 0.10 0.90 15.40% 34.00% 0.00 1.00 16.00% 40.00% AB = -1.0: Attainable Set of Risk/Return Combinations 20% Expected return 15% 10% 5% 0% 0% 20% 40% Risk, sp c. Suppose a risk-free asset has an expected return of 5 percent. By definition, its standard deviation is zero, and its correlation with any other asset is also zero. Using only Asset A and the risk-free asset, plot the attainable portfolios. Asset A Risk-free Asset Expected return, r hat 10% 5% Standard deviation, s 20% 0% Using the equations above, we can find the expected return and standard deviation of a portfolio with different percents invested in each asset. Correlation = 0 Proportion of Proportion of Portfolio in Risk- Portfolio in Security A free Asset (Value of wA) (Value of 1-w A) rp sp 1.00 0.00 10.00% 20.00% 0.90 0.10 9.50% 18.00% 0.80 0.20 9.00% 16.00% 0.70 0.30 8.50% 14.00% 0.60 0.40 8.00% 12.00% 0.50 0.50 7.50% 10.00% Michael C. Ehrhardt Page 3 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model 0.40 0.60 7.00% 8.00% 0.30 0.70 6.50% 6.00% 0.20 0.80 6.00% 4.00% 0.10 0.90 5.50% 2.00% 0.00 1.00 5.00% 0.00% Attainable Set of Risk/Return Combinations with Risk-Free Asset 15% Expected return 10% 5% 0% 0% 5% 10% 15% 20% Risk, sp d. Construct a reasonable, but hypothetical, graph that shows risk, as measured by portfolio standard deviation, on the X axis and expected rate of return on the Y axis. Now add an illustrative feasible (or attainable) set of portfolios, and show what portion of the feasible set is efficient. What makes a particular portfolio efficient? Don't worry about specific values when constructing the graph-merely illustrate how things look with "reasonable" data. FEASIBLE AND EFFICIENT PORTFOLIOS The feasible set of portfolios represent all portfolios that can be constructed from a given set of stocks. An efficient portfolio is one that offers: the most return for a given amount of risk or the least risk for a given amount of return. Expected Portfolio Efficient Set Return, r p Feasible Set Risk, s p Feasible and Efficient Portfolios . Michael C. Ehrhardt Page 4 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model e. Now add a set of indifference curves to the graph created for part b. What do these curves represent? What is the optimal portfolio for this investor? Finally, add a second set of indifference curves which leads to the selection of a different optimal portfolio. Why do the two investors choose different portfolios? OPTIMAL PORTFOLIOS An investor's optimal portfolio is defined by the tangency point between the efficient set and the investor's indifference curve. The inderference curve reflect an investor's attitude toward risk as reflected in his or her risk/return trade off function. Expected IB 2 I Return, r p B1 Optimal Portfolio IA 2 Investor B IA 1 Optimal Portfolio Investor A Risk s p Optimal Portfolios . f. What is the Capital Asset Pricing Model (CAPM)? What are the assumptions that underlie the model? CAPM The Capital Asset Pricing Model is an equilibrium model that specifies the relationship between risk and required rate of return for assets held in well diversified portfolios. Assumptions Investors all think in terms of a single holding period. All investors have identical expectations. Investors can borrow or lend unlimited amounts at the risk free rate. All assets are perfectly divisible. There are not taxes and transaction costs. All investors are price takers, that is, investors buying and selling will not influence stock prices. Quantities of all assets are given and fixed. g. Now add the risk-free asset. What impact does this have on the efficient frontier? EFFICIENT SET WITH A RISK-FREE ASSET When a risk free asset is added to the feasible set, investors can create portfolios that combine this asset with a portfolio of risky asset. The straight line connecting krf with M, the tangency point between the line and the old efficiency set, becomes the new efficient frontier. Efficient Set with a Risk-Free Asset Michael C. Ehrhardt Page 5 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Efficient Set with a Risk-Free Asset Expected Z Return, r p . B ^ rM M . The Capital Market r RF A . Line (CML): New Efficient Set sM Risk, sp . h. Write out the equation for the Capital Market Line (CML) and draw it on the graph. Interpret the CML. Now add a set of indifference curves, and illustrate how an investor's optimal portfolio is some combination of the risky portfolio and the risk-free asset. What is the composition of the risky portfolio? OPTIMAL PORTFOLIO WITH A RISK-FREE ASSET The optimal portfolio for any investor is the point of tangency between the CML and the investors indifference curve. Expected Return, r p CML I2 I1 . . ^ M rM ^ R rR R = Optimal r RF Portfolio sR sM Risk, sp . Capital Market Line The capital market line is all linear combinations of the risk free asset and portfolio M. rhat= rrf + (rm-rrf)/sm x sp Intercept Slope Risk Measure Michael C. Ehrhardt Page 6 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model The CML gives the risk and return relationship for efficient portfolios The SML , also part of CAPM, gives the risk and return relationship for individual stocks. SML = ri + (RPm) x b i. What is a characteristic line? How is this line used to estimate a stock's beta coefficient? Write out and explain the formula that relates total risk, market risk, and diversifiable risk. Beta Calculation Run a regression line of past returns on Stock I versus returns on the market. The regression line is the characteristic line. Year rm ri 1 15% 18% 2 -5% -10% 3 12% 16% Beta Calculation 25% 20% 15% Stock Return 10% beta calculation 5% Linear (beta 0% calculation) -10% -5% 0% 10% 20% -10% y = 1.4441x - 0.0259 R² = 0.9943 -15% Market Return R2 measures the percent of a stock's variance as explained by the market. Relationship between stand alone, market, and diversifiable risk s2j = b2j *s2m + s2ej s2j = stand alone risk of stock J 2 2 b j *s m= market risk of stock J s2ej = diversifiable risk of stock J j. What are two potential tests that can be conducted to verify the CAPM? What are the results of such tests? What is Roll's critique of CAPM tests? Test to verify CAPM Beta stability test and tests based on the slope of the SML. Test of the SML indicate a more-or-less linear relationship between realized return and market risk. Slope is less than predicted Irrelevance of diversifiable risk specified in the CAPM model can be questioned. Betas of individual securities are not good estimators of future risk. Michael C. Ehrhardt Page 7 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Betas of ten or more randomly selected stocks are reasonably stable. Past betas are good estimates of future portfolio volitility. Conclusions regarding CAPM It is impossible to verify. Recent studies have questioned its validity. Investors seemed to be concerned with both market and stand alone risk. Therefore, the SML may not produce the correct estimate of rj. CAPM/SML concepts are based on expectations, yeta betas are calculated using historical data. k. Briefly explain the difference between the CAPM and the Arbitrage Pricing Theory (APT). CAPM and the Arbitrage Pricing Theory The CAPM is a single factor model. The APT proposes that the relationship between risk and return is more complex and may be due to multiple factos such as GDP, growth, expected inflation, tax rate changes, and dividend yield. l. Suppose you are given the following information. The beta of company, bi, is 0.9, the risk-free rate, rRF, is 6.8 percent, and the expected market premium, rM - rRF, is 6.3 percent. Because your company is larger than average and more successful than average (that is, it has a lower book-to-market ratio), you think the Fama-French three-factor model might be more appropriate than the CAPM. You estimate the additional coefficients from the Fama-French three-factor model: The coefficient for the size effect, ci, is -0.5, and the coefficient for the book-to-market effect, di, is -0.3. If the expected value of the size factor is 4 percent and the expected value of the book-to-market factor is 5 percent, what is the required return using the Fama-French three-factor model? (Assume that ai = 0.0.) What is the required return using CAPM? Required Return for stock I under the Fama-French-3-Factor Model Fama and French propose three factors: The excess market return, rm-rrf. The return on, S, a portfolio of small firms minus the return on B, a portfolio of big firms. This return is called rsmb, for S minus B. The return on, H, a portfolio of firms with high book-to-market ratios minus the return on L, a portfolio of firms with low book-to-market ratios. This return is called rhml, for H minus L. Required return for Stock I ri = rrf + (rm-rrf) b + (rsmb) c + b= Sensitivity of stock I to the market c= Sensitivity of stock I to the size factor c= Sensitivity of stock I to the book-to-market factor b= 0.9 rrf = 6.8% RPm = 6.3% c= -0.5 value for size factor = 4.0% d= -0.3 book-to-market factor= 5.0% ri = 8.97% CAPM = 12.47% Michael C. Ehrhardt Page 8 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 9 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 10 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 11 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 12 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 13 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 14 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 15 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 16 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 17 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Michael C. Ehrhardt Page 18 4/24/2010 39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls Model Excess Returns on Wal-Mart, kS-kRF 120% 110% 100% 90% 80% 70% 60% 50% 40% Excess Returns on the Market, kM-kRF 30% 20% 10% 0% -30% -20% -10% 0% 10% 20% 30% Michael C. Ehrhardt Page 19 4/24/2010