Capital Asset Pricing Model Homework Problems

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1 Capital Asset Pricing Model Homework Problems Portfolio weights and expected return 1. Consider a portfolio of 300 shares of firm A worth $10/share and 50 shares of firm B worth $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 is the expected return? b Suppose firm A s share price goes up to $12 and firm B s share price falls to $36. What is the new value of the portfolio? What return did it earn? After the price change, what are the new portfolio weights? 2. Consider a portfolio of 250 shares of firm A worth $30/share and 1500 shares of firm B worth $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 is the expected return? b Suppose firm A s share price falls to $24 and firm B s share price goes up to $22. What is the new value of the portfolio? What return did it earn? After the price change, what are the new portfolio weights? Portfolio volatility 3. For the following problem please refer to Table 1 Table 11.3, p. 336 in Corporate Finance by 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 with i. 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 is 50% and the correlation between the returns of mid-cap stocks is 30%. a What is the average variance AvgV ar of a mid-cap stock? b What is the average covariance AvgCov of a mid-cap stock? c Consider a portfolio of n mid-cap stocks. What is an estimate of the volatility of such a portfolio when n 10? n 20? n 40? What is the limiting volatility? 1

2 Table 1: Historical Annual Volatilities and Correlations for Selected Stocks based on monthly returns, Alaskan Southwest Ford General General Microsoft Dell Air Airlines Motor Motors Mills Volatility StDev 37% 50% 38% 31% 42% 41% 18% Correlation with: Microsoft Dell Alaska Air Southwest Airlines Ford Motor General Motors General Mills Consider a portfolio of two stocks. Data shown in Table 2. Let x denote the weight on Stock A and 1 x denote the weight on Stock B. Correlation coefficient equals ρ AB. a Write down a mathematical expression for the portfolio s mean return and volatility standard deviation as a function of x. b What is the portfolio s mean return and volatility when x 0.4 if ρ AB 0? ρ AB +1? ρ AB 1? c Suppose ρ AB 1? Are there portfolio weights that will result in a portfolio with no volatility? If so, what are the weights? Table 2: Stock Expected Return Volatility Stock A 15% 40% Stock B 7% 30% Minimum variance portfolio 6. Consider the data shown in Table 2. The risk-free rate is r f 3%. a What is the minimum variance portfolio when ρ AB 0? What is its expected return and volatility? 2

3 b What is the minimum variance portfolio when ρ AB 0.4? What is its expected return and volatility? c What is the minimum variance portfolio when ρ AB 0.4? What is its expected return and volatility? 7. Consider two stocks, A and B, such that σ A 0.30, σ B 0.80, RA 0.10, RB 0.06 and r f a What is the minimum variance portfolio when ρ AB 0 and what is its volatility? b What is the minimum variance portfolio when ρ AB 0.6 and what is its volatility? c What is the minimum variance portfolio when ρ AB 0.6 and what is its volatility? 8. Consider three risky assets whose covariance matrix Σ is Σ , and whose expected returns are R , R2 0.10, R The risk-free rate is r f The inverse of the covariance matrix is Σ What is the minimum variance portfolio and what is its volatility? 9. Consider three risky assets whose covariance matrix Σ is Σ The expected returns are R , R , R The risk-free rate is r f Solve for the minimum variance portfolio using the first-order optimality conditions, i.e., without computing the inverse of the covariance matrix. What is the minimum variance? Suggestion: By symmetry x 1 x 3. Tangent portfolio 10. For the data of problem 6 determine the tangent portfolios and their respective mean returns and volatilities. 3

4 11. For the data of problem 7 determine the tangent portfolios and their respective mean returns and volatilities. 12. For the data of problem 8 determine the tangent portfolio and its mean return and volatility. 13. For the data of problem 9 determine the tangent portfolio and its mean return and volatility. 14. 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 of 8%? How would you allocate $1,000 to achieve this position? 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 of 16.5%? How would you allocate $3,000 to achieve this position? Security market line 16. 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 0.6. According to the CAPM, what is its expected return? 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%. According to the CAPM, what is its expected return? 17. Stock A has a beta of 1.20 and Stock B has a beta of 0.8. Suppose r f 2% and R M 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? 4

5 18. Suppose you estimate that stock A has a volatility of 32% and a beta of 1.42, whereas stock 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 as 10%. Which firm has a higher cost of equity capital? 19. Consider a world with only two risky assets, A and B, and a risk-free asset. The two risky assets are in equal supply in the market, i.e., the market portfolio M 0.5A + 0.5B. It is known that R M 11%, σ A 20%, σ B 40% and ρ AB The risk-free rate is 2%. Assume CAPM holds. a What is the beta for each stock? b What are the values for R A and R B? 20. Consider a world with only two risky assets, A and B, and a risk-free asset. Stock A has 200 shares outstanding, a price per share of $3.00, an expected return of 16% and a volatility of 30%. Stock B has 300 shares outstanding, a price per share of $4.00, an expected return of 10% and a volatility of 15%. The correlation coefficient ρ AB 0.4. Assume CAPM holds. a What is expected return of the market portfolio? b What is volatility of the market portfolio? 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 value stocks. Suppose the growth stock portfolio and value stock portfolio have equal size in terms of total value. Furthermore, suppose that the expected return of the value stocks is 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 The risk-free rate is 2%. a What is the expected return and volatility of the market portfolio which is a combination of the two portfolios? b Does CAPM hold in this economy? Improving the Sharpe ratio 22. Suppose portfolio P s expected return is 14%, its volatility is 30% and the risk-free rate is 2%. Suppose further that a particular mix of asset i and P yields a portfolio P with 5

6 an expected return of 22% and a volatility of 40%. Will adding asset i to portfolio P be beneficial? Explain how. 23. Assume the risk-free rate is 4%. You are a financial advisor and your client has decided to invest in exactly one of two risky funds, A and B. She comes to you for advice. Whichever fund you recommend she will combine it with the risk-free asset. Expected returns are R A 13% and R B 18%. Volatilities are σ A 20% and σ B 30%. Without knowing your client s tolerance for risk, which fund would you recommend? 24. You are currently invested in Fund F. It has an expected return of 14% with a volatility of 20%. The risk-free rate is 3.8%. Your broker suggests you add Stock B to your portfolio with a positive weight. Stock B has an expected return of 20%, a volatility of 60% and a correlation of 0 with Fund F. a Is your broker right? b You follow your broker s advice and make a substantial investment in Stock B so that now 60% is in Fund F and 40% is in Stock B. You tell your finance professor about your investment and he says you made a mistake and should reduce your investment in Stock B. Is your finance professor right? c You decide to follow your finance professor s advice and reduce your exposure to Stock B. Now Stock B represents only 15% of your risky portfolio with the rest invested in Fund F. Is the correct amount to hold of Stock B? 25. In addition to risk-free securities, you are currently invested in the Jones Fund, a broadbased fund with an expected return of 12% and a volatility of 25%. The risk-free rate is 4%. Your broker suggests you add a venture capital VC fund to your current portfolio. The VC fund has an expected return of 20%, a volatility of 80% and a correlation of 0.2 with the Jones Fund. a Is your broker right? b Suppose you follow your broker s advice and put 50% of your money in the VC fund. You sell 50% of your value of the Jones Fund. What is the Sharpe ratio of your new portfolio? c What is the optimal fraction of your wealth to invest in the VC fund? 6

7 Capital Asset Pricing Model Homework Solutions 1. Portfolio value 300$ $40 $5,000. Portfolio weights are x A 300$10/$5, % and x B 50$40/$5, %. Expected return 0.608% % 10%. New portfolio value 300$ $36 $5,400. Return is $5,400 - $5,000/$5,000 8% or, equivalently, 0.60[$12 - $10/$10] [$36 - $40/$40] % % 8%. New portfolio weights are x A 300$12/$5, % and x B 50$36/$5, %. 2. Portfolio value 250$ $20 $37,500. Portfolio weights are x A 250$30/$37, % and x B 1500$20/$37, %. Expected return 0.204% % 8%. New portfolio value 250$ $22 $39,000. Return is $39,000 - $37,500/$37,500 4% or, equivalently, 0.20[$24 - $30/$30] [$22 - $20/$20] % % 4%. New portfolio weights are x A 250$24/$39, % and x B 1500$22/$39, %. 3. Cov :50: StDev %. 20:80: StDev %. 80:20: StDev %. 4. AvgV ar AvgCov σ n AvgV ar/n + 1 1/n AvgCov 0.25/n + 1 1/n σ %. σ %. σ %. σ 27.39%. 5. E[Rx] 0.15x x x. StDevx x x ρ AB x1 x ρ AB x ρ AB 0.18x When ρ AB 0, σ 24.08%. When ρ AB +1, σ %. When ρ AB 1, σ %, and the portfolio x A σ B /σ A + σ B 3/7 and x B 4/7 will have no volatility. 6. In general, x A σ2 B σ AB σ 2 A +σ2 B 2σ AB. ρ AB 0 σ AB 0 and x A 0.09/ %. Expected return % % 9.88% and σ %. ρ AB 0.4 σ AB and x A /[ ] 27.27%. Expected return % % 9.18% and σ %. 7

8 ρ AB 0.4 σ AB and x A /[ ] 39.88%. Expected return % % 10.19% and σ %. 7. In general, x A σ2 B σ AB σ 2 A +σ2 B 2σ AB. ρ AB 0 σ AB 0 and x A 0.64/ %. σ %. ρ AB 0.6 σ AB and x A /[ ] %. σ %. ρ AB 0.6 σ AB and x A /[ ] 77.01%. σ %. 8. Minimum variance portfolio is proportional to Σ 1 e , , Since e T Σe , minimum variance portfolio , , , minimum variance 1/e T Σ 1 e 1/ and the minimum volatility 1/ %. 9. The first-order optimality conditions are Σx λe and x 1 + x 2 + x 3 1. In equation form, we have: i 2x 1 + x 2 λ, ii x 1 + 2x 2 + x 3 λ and iii x 2 + 2x 3 λ. Equations i and iii imply that x 1 x 3. Using this fact, equations ii and iii imply that x 2 0. Since the sum of the x i 1, it follows that x 1 x 3 0.5, x 2 0 and λ 1. Minimum variance λ ˆR , , When ρ AB 0: Σ, Σ x Σ 1 ˆR 0.75, 0.4 4, e T Σ 1 ˆR , 0.75 x , , E[R P ] , ˆR 0 E[R P ] r f , V arr P ˆR e T Σ 1 ˆR σ P %. 8.

9 When ρ AB 0.4: Σ, Σ x Σ 1 ˆR , , e T Σ 1 ˆR , x , , E[R P ] , ˆR 0 E[R P ] r f , V arr P When ρ AB 0.4: Σ Σ 1 ˆR e T Σ 1 ˆR σ P % x Σ 1 ˆR , ,, e T Σ 1 ˆR , x , , E[R P ] , ˆR 0 E[R P ] r f , V arr P ˆR e T Σ 1 ˆR σ P % ˆR , , When ρ AB 0: Σ, Σ x Σ 1 ˆR 0.8 8, , e T Σ 1 ˆR ,

10 x , , E[R P ] , ˆR 0 E[R P ] r f , V arr P When ρ AB 0.6: Σ, Σ 1 ˆR e T Σ 1 ˆR σ P % x Σ 1 ˆR , , e T Σ 1 ˆR , x , , E[R P ] , ˆR 0 E[R P ] r f , V arr P ˆR e T Σ 1 ˆR σ P %. When ρ AB 0.6: Σ, Σ x Σ 1 ˆR , , e T Σ 1 ˆR , x , , E[R P ] , ˆR 0 E[R P ] r f , V arr P ˆR e T Σ 1 ˆR σ P %. 12. ˆR , , , 0.07, x Σ 1 ˆR 0.50, 0.875, , 10

11 e T Σ 1 ˆR , 0.50 x , , , , E[R P ] , ˆR 0 E[R P ] r f , V arr P ˆR e T Σ 1 ˆR σ P %. 13. ˆR , , , 0.07, Σ x Σ 1 ˆR 0.04, 0.05, 0.01, e T Σ 1 ˆR , 0.04 x 0.10, , , 0.5, E[R P ] , ˆR 0 E[R P ] r f 0.074, V arr P ˆR e T Σ 1 ˆR σ P %. 14. RP σ P σ P. For an efficient portfolio whose expected return is 8%, we have σ P σ P 30%. Allocate $750 to the tangent portfolio and $250 to the risk-free asset. 15. RP σ P σ P. For an efficient portfolio whose expected return is 16%, we have σ P σ P 45%. Allocate $4,500 to the tangent portfolio and borrow $1,500 at the risk-free asset. 16. Ri β i. R i %. β i / R i %. β i /0.3 2/3 R i /3 3%. 17. RA %. RB %. R P % % 14%. β P R P %. 11

12 18. Stock B has more total risk. Stock A has more market risk. R A %. RB %. Firm A has the higher cost of equity capital. 19. CovR A, R M CovR A, 0.5R A + 0.5R B 0.5σ 2 A + 0.5σ AB V arr M β A 0.05/ /3 R A / %. The market s beta of 1 equals 0.5β A + 0.5β B. Since β A 2/3, this implies that β B 4/3. You can verify this quantity via the formula for β B. The market s expected return of 11% must equal 0.5 R A R B. Since R A 8%, this implies that R B 14%. You can verify this quantity via the SML. 20. Value of the market portfolio 200$ $4 $1,800. Portfolio weights are x A 1/3 and x B 2/3. R M 1/316% + 2/310% 12% σ AB σ A σ B ρ AB CovR A, R M CovR A, 1/3R A + 2/3R B 1/3σ 2 A + 2/3σAB 1/ / V arr M 1/ / /32/ β A 0.042/ r f r f r f 4%. The market s beta of 1 equals 1/3β A +2/3β B. Since β A 1.5, this implies that β B You can verify that the SML holds for security B as it should if the market portfolio is efficient. In particular, you could use security B to determine that the risk-free rate is 4%, too. 21. Sharpe ratios of the value and growth portfolios are / and / , respectively. RM 0.513% % 15%. σ M % Thus, the Sharpe ratio for M is / Since the Sharpe ratio for M is less than the Sharpe ratio for the value portfolio, the market portfolio is not efficient. According to CAPM, investors could reallocate their investments to improve the Sharpe ratio so that they could achieve a higher expected return for the same level of volatility or, alternatively, they could reduce their volatility and still achieve the same expected return. 12

13 22. Sharpe ratio of P is / whereas the Sharpe ratio of P is / A portfolio of 25% in the risk-free asset and 75% in portfolio P will have a volatility of % the same as σ P yet have a higher expected return of 17%. 23. Sharpe ratio of A is / whereas the Sharpe ratio of B is / You should recommend fund B. 24. Since β BF 0, the required return for stock B to compensate for its risk to fund F is the risk-free rate of 3.8%. Since stock B s expected return is higher, it will pay to add stock B to fund F with a positive weight. The new portfolio P has an expected return of 0.620% % 16.4%. We also have that σ BP CovR B, 0.4R B + 0.6R F 0.4σ 2 B V arr P β P B R B 3.8% % 3.8% 29%. Since the actual expected return for stock B is 20% < 29%, you can increase the Sharpe ratio by reducing the weight of B in the portfolio. The new portfolio P has an expected return of % % 14.9%. We also have that σ BP CovR B, 0.15R B + 0.6R F 0.15σ 2 B V arr P β P B R B 3.8% % 3.8% 20% Since the actual expected return for stock B is 20%, this is the correct weight. Note: The formula derived in the handout shows that the optimum value for x the dollar amount to invest in F per dollar invested in fund F is x σ2 ˆR F B σ BF ˆRF σb 2 ˆR F σ BF ˆRB The portfolio weight on stock B is x/1 + x / %. 25. We have that R P 0.520% % 16% CovR V C, R J σ V C σ J ρ V CJ βv J C 0.04/ R V C 4% % 4% 9.12%. 13

14 Since the actual expected return of the VC fund is 20% > 9.12%, you can increase the Sharpe ratio by adding the VC fund to the Jones Fund with a positive weight. The Sharpe ratio of the Jones Fund is / With a mix, new Sharpe ratio The 50% weight on the VC Fund is too large; it should be reduced. As a function of x, the weight to allocate to the VC fund, the Sharpe ratio Sx is Sx 0.20x x x x x1 x Enumeration in increments of 1% yields the optimum weight on the VC fund is 13%. Note: The formula derived in the handout shows that the optimum value for x the dollar amount to invest in the VC fund per dollar invested in the Jones Fund is x σ2 ˆR J V C σ V CJ ˆRJ σv 2 ˆR C J σ V CJ ˆRV C The portfolio weight on stock B is x/1 + x / %. 14

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