Capital Asset Pricing Model Homework Problems

[Pages:14]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?

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Table 1: Historical Annual Volatilities and Correlations for Selected Stocks (based on monthly returns, 1996-2008).

Alaskan Southwest Ford General General

Microsoft Dell

Air Airlines Motor Motors Mills

Volatility (StDev)

37% 50%

38%

31% 42%

41%

18%

Correlation with:

Microsoft

1.00 0.62

0.25

0.23 0.26

0.23

0.10

Dell

0.62 1.00

0.19

0.21 0.31

0.28

0.07

Alaska Air

0.25 0.19

1.00

0.30 0.16

0.13

0.11

Southwest Airlines

0.23 0.21

0.30

1.00 0.25

0.22

0.20

Ford Motor

0.26 0.31

0.16

0.25 1.00

0.62

0.07

General Motors

0.23 0.28

0.13

0.22 0.62

1.00

0.02

General Mills

0.10 0.07

0.11

0.20 0.07

0.02

1.00

5. 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 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 AB = 0? AB = +1? AB = -1?

(c) Suppose 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 rf = 3%. (a) What is the minimum variance portfolio when AB = 0? What is its expected return and volatility?

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(b) What is the minimum variance portfolio when AB = 0.4? What is its expected return and volatility?

(c) What is the minimum variance portfolio when AB = -0.4? What is its expected return and volatility?

7. Consider two stocks, A and B, such that A = 0.30, B = 0.80, R?A = 0.10, R?B = 0.06 and rf = 0.02.

(a) What is the minimum variance portfolio when AB = 0 and what is its volatility? (b) What is the minimum variance portfolio when AB = 0.6 and what is its volatility? (c) What is the minimum variance portfolio when AB = -0.6 and what is its volatility?

8. Consider three risky assets whose covariance matrix is

0.09 0.045 0.01

= 0.045 0.25 0.06 ,

(1)

0.01 0.06 0.04

and whose expected returns are R?1 = 0.08, R?2 = 0.10, R?3 = 0.16. The risk-free rate is rf = 0.03. The inverse of the covariance matrix is

12.2137 -2.2901 0.3817

-1 = -2.2901 6.6794 -9.4466 .

(2)

0.3817 -9.4466 39.0744

What is the minimum variance portfolio and what is its volatility?

9. Consider three risky assets whose covariance matrix is

210

= 1 2 1 .

(3)

012

The expected returns are R?1 = 0.11, R?2 = 0.09, R?3 = 0.05. The risk-free rate is rf = 0.02. 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 x1 = x3. )

Tangent portfolio

10. For the data of problem 6 determine the tangent portfolios and their respective mean returns and volatilities.

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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 rf = 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)?

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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 0.75.

(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%, A = 20%, B = 40% and AB = 0.75. 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 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 0.50. The risk-free rate is 2%.

(a) What is the expected return and volatility of the market portfolio (which is a 50-50 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

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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 A = 20% and 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?

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Capital Asset Pricing Model Homework Solutions

1. Portfolio value = 300($10) + 50($40) = $5,000. Portfolio weights are xA = 300($10)/$5, 000 = 60% and xB = 50($40)/$5, 000 = 40%. Expected return = 0.60(8%) + 0.40(13%) = 10%. New portfolio value = 300($12) + 50($36) = $5,400. Return is ($5,400 - $5,000)/$5,000 = 8% or, equivalently, 0.60[($12 - $10)/$10] + 0.40[($36 - $40)/$40] = 0.60(20%) + 0.40(-10%) = 8%. New portfolio weights are xA = 300($12)/$5, 400 = 66.?6% and xB = 50($36)/$5, 400 = 33.?3%.

2. Portfolio value = 250($30) + 1500($20) = $37,500. Portfolio weights are xA = 250($30)/$37, 500 = 20% and xB = 1500($20)/$37, 500 = 80%. Expected return = 0.20(4%) + 0.80(9%) = 8%. New portfolio value = 250($24) + 1500($22) = $39,000. Return is ($39,000 - $37,500)/$37,500 = 4% or, equivalently, 0.20[($24 - $30)/$30] + 0.80[($22 - $20)/$20] = 0.20(-20%) + 0.80(10%) = 4%. New portfolio weights are xA = 250($24)/$39, 000 = 15.38% and xB = 1500($22)/$39, 000 = 84.62%.

3. Cov = (0.11)(0.38)(0.18) = 0.007524. 50:50: StDev = (0.5)2(0.38)2 + (0.5)2(0.18)2 + 2(0.5)(0.5)(0.007524) = 21.90%. 20:80: StDev = (0.2)2(0.38)2 + (0.8)2(0.18)2 + 2(0.2)(0.8)(0.007524) = 17.01%. 80:20: StDev = (0.8)2(0.38)2 + (0.2)2(0.18)2 + 2(0.8)(0.2)(0.007524) = 31.00%.

4. AvgV ar = 0.502 = 0.25. AvgCov = (0.50)(0.50)(0.30) = 0.075. n = AvgV ar/n + (1 - 1/n) AvgCov = 0.25/n + (1 - 1/n)(0.075). 10 = 30.41%. 20 = 28.94%. 40 = 28.17%. = 27.39%.

5. E[R(x)] = 0.15x + 0.07(1 - x) = 0.07 + 0.08x. StDev(x) = (0.40)2x2 + (0.30)2(1 - x)2 + 2(0.40)(0.30)ABx(1 - x) = (0.25 - 0.24AB)x2 + (0.24AB - 0.18)x + 0.09. When AB = 0, = 24.08%. When AB = +1, = 0.4(0.4) + 0.6(0.3) = 34%. When AB = -1, = | 0.4(0.4) - 0.6(0.3) | = 2%, and the portfolio xA = B/(A + B) = 3/7 and xB = 4/7 will have no volatility.

6.

In general, xA =

. B2 -AB

A2 +B2 -2AB

AB = 0 = AB = 0 and xA = 0.09/(0.16 + 0.09) = 36%. Expected return =

0.36(15%) + 0.64(7%) = 9.88% and = (0.36)2(0.40)2 + (0.64)2(0.30)2 = 24%.

AB = 0.4 = AB = (0.4)(0.3)(0.4) = 0.048 and xA = (0.09 - 0.048)/[(0.09 +

0.16) - 2(0.048)] = 27.27%. Expected return = 0.2727(15%) + 0.7273(7%) = 9.18%

and = (0.2727)2(0.40)2 + (0.7273)2(0.30)2 + 2(0.048)(0.2727)(0.7273) = 28.03%.

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AB = -0.4 = AB = (0.4)(0.3)(-0.4) = -0.048 and xA = (0.09 + 0.048)/[(0.09 + 0.16) + 2(0.048)] = 39.88%. Expected return = 0.3988(15%) + 0.6012(7%) = 10.19% and = (0.3988)2(0.40)2 + (0.6012)2(0.30)2 + 2(-0.048)(0.3988)(0.6012) = 18.70%.

7.

In general, xA =

. B2 -AB

A2 +B2 -2AB

AB = 0 = AB = 0 and

xA = 0.64/(0.09 + 0.64) = 87.67%.

= (0.8767)2(0.30)2 + (0.1233)2(0.80)2 = 28.09%.

AB = 0.6 = AB = (0.3)(0.8)(0.6) = 0.144 and xA = (0.64 - 0.144)/[(0.09 + 0.64) - 2(0.144)] = 112.22%. = (1.1222)2(0.30)2 + (-0.1222)2(0.80)2 + 2(0.144)(1.1222)(-0.1222) = 28.88%.

AB = -0.6 = AB = (0.3)(0.8)(-0.6) = -0.144 and xA = (0.64 + 0.144)/[(0.09 + 0.64) + 2(0.144)] = 77.01%. = (0.7701)2(0.30)2 + (0.2299)2(0.80)2 + 2(-0.144)(0.7701)(0.2299) = 19.03%.

8. Minimum variance portfolio is proportional to -1e = (10.3053, -5.0573, 30.0095). Since eT e = 10.3053 - 5.0573 + 30.0095 = 35.2575,

minimum variance portfolio = (0.2923, -0.1434, 0.8511), minimum variance = 1/(eT -1e) = 1/35.2575

and the minimum volatility = 1/35.2575) = 16.84%.

9. The first-order optimality conditions are x = e and x1 + x2 + x3 = 1. In equation form, we have: (i) 2x1 + x2 = , (ii) x1 + 2x2 + x3 = and (iii) x2 + 2x3 = . Equations (i) and (iii) imply that x1 = x3. Using this fact, equations (ii) and (iii) imply that x2 = 0. Since the sum of the xi = 1, it follows that x1 = x3 = 0.5, x2 = 0 and = 1. Minimum variance = = 1.

10. R^ = (0.15 - 0.03, 0.07 - 0.03) = (0.12, 0.04). When AB = 0:

=

0.16 0.00 0.00 0.09

,

-1 =

1

(0.16)(0.09)

0.09 0.00 0.00 0.16

=

6.25 0.00 0.00 11.?1

.

x -1R^ = (0.75, 0.4?4),

eT -1R^ = 0.75 + 0.4?4 = 1.19?4,

x =

0.75 0.4?4 1.19?4 , 1.19?4 = (0.628, 0.372)

E[RP ] = 0.628(0.15) + 0.372(0.07) = 0.1202,

R^0 = E[RP ] - rf = 0.0902,

V ar(RP )

=

R^0 eT -1R^

=

0.0902 1.19?4

=

0.0755

=

P

=

0.0755

=

27.48%.

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