Capital Asset Pricing Model Homework Problems

1 Capital Asset Pricing Model Homework ProblemsPortfolio weights and expected return1. Consider a portfolio of 300 shares of firm A worth $10/share and 50 shares of firm Bworth $40/share. You expect a return of 8% for stock A and a return of 13% for stock B.(a) What is the total value of the portfolio, what are the portfolio weights and what isthe expected return?(b) Suppose firm A s share price goes up to $12 and firm B s share price falls to $ is the new value of the portfolio? What return did it earn? After the pricechange, what are the new portfolio weights?2. Consider a portfolio of 250 shares of firm A worth $30/share and 1500 shares of firm Bworth $20/share. You expect a return of 4% for stock A and a return of 9% for stock B.(a) What is the total value of the portfolio, what are the portfolio weights and what isthe expected return?

2 (b) Suppose firm A s share price falls to $24 and firm B s share price goes up to $ is the new value of the portfolio? What return did it earn? After the pricechange, what are the new portfolio weights?Portfolio volatility3. For the following problem please refer to Table 1 (Table , p. 336 in Corporate Financeby Berk and DeMarzo).(a) What is the covariance between the returns for Alaskan Air and General Mills?(b) What is the volatility of a portfolio withi. equal amounts invested in these two stocks?ii. 20% invested in Alaskan Air and 80% invested in General Mills?iii. 80% invested in Alaskan Air and 20% invested in General Mills?4. Suppose the historical volatility (standard deviation) of the return of a mid-cap stock is50% and the correlation between the returns of mid-cap stocks is 30%.

3 (a) What is the average varianceAvgV arof a mid-cap stock?(b) What is the average covarianceAvgCovof a mid-cap stock?(c) Consider a portfolio ofnmid-cap stocks. What is an estimate of the volatility ofsuch a portfolio whenn= 10?n= 20?n= 40? What is the limiting volatility?1 Table 1:Historical Annual Volatilities and Correlations for Selected Stocks (basedon monthly returns, 1996-2008).Alaskan SouthwestFord General GeneralMicrosoftDellAirAirlines MotorMotorsMillsVolatility (StDev)37% 50%38%31%42%41%18%Correlation Consider a portfolio of two stocks. Data shown in Table 2. Letxdenote the weight onStock A and 1 xdenote the weight on Stock B. Correlation coefficient equals AB.(a) Write down a mathematical expression for the portfolio s mean return and volatility(standard deviation) as a function ofx.

4 (b) What is the portfolio s mean return and volatility whenx= if AB= 0? AB=+1? AB= 1?(c) Suppose AB= 1? Are there portfolio weights that will result in a portfolio withno volatility? If so, what are the weights?Table 2:Stock Expected Return VolatilityStock A15%40%Stock B7%30%Minimum variance portfolio6. Consider the data shown in Table 2. The risk-free rate isrf= 3%.(a) What is the minimum variance portfolio when AB= 0? What is its expected returnand volatility?2(b) What is the minimum variance portfolio when AB= What is its expectedreturn and volatility?(c) What is the minimum variance portfolio when AB= What is its expectedreturn and volatility?7. Consider two stocks,AandB, such that A= , B= , RA= , RB= (a) What is the minimum variance portfolio when AB= 0 and what is its volatility?

5 (b) What is the minimum variance portfolio when AB= and what is its volatility?(c) What is the minimum variance portfolio when AB= and what is its volatility?8. Consider three risky assets whose covariance matrix is = ,(1)and whose expected returns are R1= , R2= , R3= The risk-free rate isrf= The inverse of the covariance matrix is 1= .(2)What is the minimum variance portfolio and what is its volatility?9. Consider three risky assets whose covariance matrix is = 2 1 01 2 10 1 2 .(3)The expected returns are R1= , R2= , R3= The risk-free rate isrf= for the minimum variance portfolio using the first-order optimality conditions, ,without computing the inverse of the covariance matrix. What is the minimum variance?

6 (Suggestion:By symmetryx 1=x 3. )Tangent portfolio10. For the data of problem 6 determine the tangent portfolios and their respective meanreturns and For the data of problem 7 determine the tangent portfolios and their respective meanreturns and For the data of problem 8 determine the tangent portfolio and its mean return For the data of problem 9 determine the tangent portfolio and its mean return Suppose the expected return on the tangent portfolio is 10% and its volatility is 40%.The risk-free rate is 2%.(a) What is the equation of the Capital Market Line (CML)?(b) What is the standard deviation of an efficient portfolio whose expected return of8%? How would you allocate $1,000 to achieve this position?

7 15. Suppose the expected return on the tangent portfolio is 12% and its volatility is 30%.The risk-free rate is 3%.(a) What is the equation of the Capital Market Line (CML)?(b) What is the standard deviation of an efficient portfolio whose expected return How would you allocate $3,000 to achieve this position?Security market line16. Suppose the market premium is 9%, market volatility is 30% and the risk-free rate is 3%.(a) What is the equation of the SML?(b) Suppose a security has a beta of According to the CAPM, what is its expectedreturn?(c) A security has a volatility of 60% and a correlation with the market portfolio of 25%.According to the CAPM, what is its expected return?(d) A security has a volatility of 80% and a correlation with the market portfolio of-25%.

8 According to the CAPM, what is its expected return?17. Stock A has a beta of and Stock B has a beta of Supposerf= 2% and RM= 12%.(a) According to the CAPM, what are the expected returns for each stock?(b) What is the expected return of an equally weighted portfolio of these two stocks?(c) What is the beta of an equally weighted portfolio of these two stocks?(d) How can you use your answer to part (c) to answer part (b)?418. Suppose you estimate that stock A has a volatility of 32% and a beta of , whereasstock B has a volatility of 68% and a beta of (a) Which stock has more total risk?(b) Which stock has more market risk?(c) Suppose the risk-free rate is 2% and you estimate the market s expected return as10%. Which firm has a higher cost of equity Capital ?

9 19. Consider a world with only two risky assets,AandB, and a risk-free Asset . The two riskyassets are in equal supply in the market, , the market portfolioM= + Itis known that RM= 11%, A= 20%, B= 40% and AB= The risk-free rate is2%. Assume CAPM holds.(a) What is the beta for each stock?(b) What are the values for RAand RB?20. Consider a world with only two risky assets,AandB, and a risk-free Asset . Stock Ahas 200 shares outstanding, a price per share of $ , an expected return of 16% anda volatility of 30%. Stock B has 300 shares outstanding, a price per share of $ , anexpected return of 10% and a volatility of 15%. The correlation coefficient AB= CAPM holds.(a) What is expected return of the market portfolio?(b) What is volatility of the market portfolio?

10 (c) What is the beta of each stock?(d) What is the risk-free rate?21. Suppose you group all stocks into two mutually exclusive portfolios of growth or valuestocks. Suppose the growth stock portfolio and value stock portfolio have equal size interms of total value. Furthermore, suppose that the expected return of the value stocksis 13% with a volatility of 12%, whereas the expected return of the growth stocks is 17%with a volatility of 25%. The correlation of the returns of these two portfolios is risk-free rate is 2%.(a) What is the expected return and volatility of the market portfolio (which is a 50-50combination of the two portfolios)?(b) Does CAPM hold in this economy?Improving the Sharpe ratio22. Suppose portfolioP s expected return is 14%, its volatility is 30% and the risk-free rateis 2%.