MØA 155 PROBLEM SET: CAPM Exercise 1. [2] The expected return on the market portfolio minus the risk free rate is known as the 1. market risk premium. 2. alpha. 3. beta. 4. systematic risk. 5. I choose not to answer. Exercise 2. [3] Suppose the return on short-term government bonds (perceived to be riskless) is 6%. The expected rate of return required by the market for a portfolio with a beta measure of one is 15%. According to the CAPM, 1. What is the expected rate of return on the market portfolio? 2. What would you consider is the expected rate of return on a stock with a β of 0? 3. Suppose you consider buying a share at $40. The stock is expected to pay $3 in dividends next year and to then sell for $41. The stock risk has been evaluated by β = −0.5. Is the stock overpriced or underpriced? Exercise 3. Arnold [4] After his last holiday with Recall Inc., Arnold the Stock Analyst has suffered a partial loss of memory. He was trying to estimate some stock betas and returns, but he can’t remember the procedure. He only has the following data: Annual Annual Rate of Rate of return return Portfolio (nominal) (real) Common stocks 12.0 8.8 Corporate bonds 5.1 2.1 Treasury bills 3.5 0.4 Average rates of return for the period 1926 to 1985. Stock Digital Eq. Exxon General Mills MCI Comm. Compaq Genentech Mesa Petroleum Holly Sugar 1 Beta 1.21 ? ? ? ? 1.95 0.68 0.62 Betas for selected common stocks. 1981-1986. Stock Digital Eq. Exxon General Mills MCI Comm. Compaq Genentech Mesa Petroleum Holly Sugar E[r̃] 17.66 13.46 12.29 20.27 22.03 ? ? ? The current T-bill rate is 7.5%. 1. Arnold needs your help to fill in the question marks. Also explain him how you found the numbers. Exercise 4. Arnold [4] Arnold the Stock Analyst needs your help. He has the following data: State 1 2 3 4 Probability 0.1 0.3 0.4 0.2 rm -0.15 0.05 0.15 0.2 rj - 0.30 0.0 0.2 0.5 rf = 6% He needs to find. 1. E[r̃m ] 2 2. σm 3. E[r̃j ] 4. cov(r̃j , r̃m ) 5. The required return for stock j. Exercise 5. [4] Assume the CAPM is valid. You are given the following information: Stock A B C D rf 10% 10% 10% 0% E[r̃m ] 20% 20% E[r̃] 10% 20% 30% β 1.5 2.0 2/3 P0 100 E[P̃1 ] 125 200 40 48 The stocks pay no dividends next period. This is enough information to complete the table. 1. Calculate all the missing data. Show all your calculations. Note: Here P0 is the price of the stock in period 0, and P1 the price in period 1. 2 Exercise 6. Portfolio (RWJ 10.27) [6] The expected return on a portfolio that combines the risk-free asset and the asset at the point of tangency to the efficient set is 25%. The expected return was calculated under the following assumptions. • The risk free rate is 5%. • The expected return on the market portfolio of risky assets is 20%. • The standard deviation of the efficient portfolio is 4%. 1. In this environment, what expected rate of return would a security earn if it had a correlation of 0.5 with the market and a standard deviation of 2%? Exercise 7. Portfolio (RWJ 10.27) [6] The expected return on a portfolio that combines the risk-free asset and the asset at the point of tangency to the efficient set is 25%. The expected return was calculated under the following assumptions. • The risk free rate is 5%. • The expected return on the market portfolio of risky assets is 20%. • The standard deviation of the efficient portfolio is 4%. 1. In this environment, what expected rate of return would a security earn if it had a correlation of 0.5 with the market and a standard deviation of 2%? Exercise 8. [3] You are given the following information about three stocks that are in your portfolio. In addition, you know that the market portfolio has an expected return of 13% and a standard deviation of 18%. The risk free rate is 5%. Stock Beta A B C 1.1 0.8 1.0 Weight in portfolio 20% 50% 30% 1. What is the expected return on your portfolio? Exercise 9. Analyst (RWJ 10.17) [3] A stock has a beta of 0.9. A security analyst who specializes in studying this stock expects its return to be 13%. Suppose the risk free rate is 8% and the market risk premium is 6%. 1. Is the analyst pessimistic or optimistic about this stock relative to the markets expectation? Exercise 10. Portfolio (RWJ 10.27) [6] The expected return on a portfolio that combines the risk-free asset and the asset at the point of tangency to the efficient set is 25%. The expected return was calculated under the following assumptions. • The risk free rate is 5%. • The expected return on the market portfolio of risky assets is 20%. 3 • The standard deviation of the efficient portfolio is 4%. 1. In this environment, what expected rate of return would a security earn if it had a correlation of 0.5 with the market and a standard deviation of 2%? Exercise 11. Portfolio (RWJ 10.27) [6] The expected return on a portfolio that combines the risk-free asset and the asset at the point of tangency to the efficient set is 25%. The expected return was calculated under the following assumptions. • The risk free rate is 5%. • The expected return on the market portfolio of risky assets is 20%. • The standard deviation of the efficient portfolio is 4%. 1. In this environment, what expected rate of return would a security earn if it had a correlation of 0.5 with the market and a standard deviation of 2%? Exercise 12. Widgets [6] The following 3 firms are the only firms currently operating in the widget industry. Firm A B C βD 0 0.05 0.10 Debt Market value 100 75 50 βE 1.0 1.5 1.5 Equity Market value 200 125 50 We also know that rf = 8% E[r̃m ] = 16%. Disregard taxes. 1. Find expected returns on debt and equity for A, B and C. 2. Find expected returns for each firm A, B and C. 3. Find the asset beta for the widget industry. 4. Is the expected return for the widget industry higher than the market return? Explain how you can conclude this. Exercise 13. The risk free rate is 5%. The market portfolio of risky assets is an efficient portfolio. The expected return on this market portfolio is 20%. The expected return on an efficient portfolio that combines the risk-free asset and the asset at the point of tangency to the efficient set is 25%. The standard deviation of this efficient portfolio is 4%. What is the expected rate of return of a security with correlation of 0.5 with the market and a standard deviation of 2%? (a) 5.0% (b) 10.0% 4 (c) 12.5% (d) 15.0% (e) I choose not to answer. 5 Empirical Solutions MØA 155 PROBLEM SET: CAPM Exercise 1. [2] Market risk premium, (a) is correct Exercise 2. [3] 1. 15%. 2. 6% 3. Over/Underpriced? First calculate the stock’s expected return next period as E[r] = 41 + 3 − 40 4 = = 0.1 = 10%. 40 40 What is the required return for that stock? r = rf + (E[rm ] − rf )β = 0.06 + (0.15 − 0.06) · (−0.5) = 0.015 = 1.5%. The stock is clearly underpriced, it should have a much lower return according to the CAPM. Exercise 3. Arnold [4] 1. Use E[r̃] = rf + β · (E[r̃m ] − rf ) and β= E[r̃] − rf E[r̃m ] − rf E[rm ] = 15.9% Stock Digital Eq. Exxon General Mills MCI Comm. Compaq Genentech Mesa Petroleum Holly Sugar 6 Beta 1.21 0.71 0.57 1.52 1.73 1.95 0.68 0.62 Stock Digital Eq. Exxon General Mills MCI Comm. Compaq Genentech Mesa Petroleum Holly Sugar E[r] 17.66 13.46 12.29 20.27 22.03 23.88 13.21 12.71 Exercise 4. Arnold [4] State 1 2 3 4 Probability 0.1 0.3 0.4 0.2 E[r̃m ] = rm -0.15 0.05 0.15 0.2 rj - 0.30 0.0 0.2 0.5 rf = 6% 0.1 · (−0.15) + 0.3 · 0.05 +0.4 · 0.15 + 0.2 · 0.2 = 2 σm 10% = h i 2 E (rm − E[rm ]) = 0.1 · (−0.15 − 0.1) + 0.3 · (0.05 − 0.1) +0.4 · (0.15 − 0.1) + 0.2 · (0.2 − 0.1) = 0.01 E[r̃j ] = 0.1 · (−0.3) + 0.3 · 0.0 = 15% +0.4 · 0.2 + 0.2 · 0.5 = E[(r̃j − E[r̃j )](r̃m − E[r̃m ])] = 0.1(−0.15 − 0.10)(−0.30 − 0.15) +0.3(0.05 − 0.10)(0 − 0.15) +0.4(0.15 − 0.10)(0.2 − 0.15) +0.2(0.2 − 0.10)(0.5 − 0.15) = 0.0215 cov(r̃j , r̃m ) The required return for stock j: βj = r 0.0215 cov(r̃j , r̃m ) = = 2.15 var(r̃m ) 0.01 = rf + (E[rm ] − rf )β = 0.06 + (0.10 − 0.06) · 2.15 = 14.6% Exercise 5. [4] 7 1. rf + (E[rm ] − rf )β E[r̃] = E[r̃] = β = E[r̃m ] = E[P̃1 ] = P0 (1 + E[r̃]) P0 = E[P̃1 ] 1 + E[r̃] E[P̃1 ] − P0 P0 E[r̃] − rf E[r̃m ] − rf E[r̃] − rf + rf β Plug in the numbers in the correct formulas, and you get the table below: rf 10% 10% 10% 0% E[r̃m ] 20% 20% 15% 30% E[r̃1 ] 10% 25% 20% 20% β 0 1.5 2.0 2/3 P0 100 100 200 40 E[P̃1 ] 110 125 240 48 Exercise 6. Portfolio (RWJ 10.27) [6] 1. Remember that by combining the risk free rate and the market portfolio m, we can write the expected return and standard deviations in terms of the fraction ω invested in the risky (market) portfolio: E[rp ] = (1 − ω)rf + ωE[rm ] σp = ωσm We can use this to find the requested information. We need to find the beta of security of interest. To do this we need the variance of the market portfolio. Step 1: Find ω E[rp ] = (1 − ω)rf + ωE[rm ] 0.25 = (1 − ω)0.05 + ω0.20 ω= 0.25 − 0.05 0.2 1 = =1 0.20 − 0.05 0.15 3 Use this to find σm : σp = ωσm 1 0.04 = 1 σm 3 0.04 σm = 1 = 0.03 13 Next find the beta of the security, let us call it i. cov(ri , rm ) = ρ(ri , rm )σi σm = 0.5 · 0.02 · 0.03 = 0.004 8 βi = cov(ri , rm ) 1 0.004 = = 2 var(rm ) 0.03 3 Expected rate of return: E[ri ] = rf + (E[rm ] − rf )βi = 0.05 + (0.20 − 0.05) · 1 = 10% 3 Exercise 7. Portfolio (RWJ 10.27) [6] 1. Remember that by combining the risk free rate and the market portfolio m, we can write the expected return and standard deviations in terms of the fraction ω invested in the risky (market) portfolio: E[rp ] = (1 − ω)rf + ωE[rm ] σp = ωσm We can use this to find the requested information. We need to find the beta of security of interest. To do this we need the variance of the market portfolio. Step 1: Find ω E[rp ] = (1 − ω)rf + ωE[rm ] 0.25 = (1 − ω)0.05 + ω0.20 ω= 0.2 1 0.25 − 0.05 = =1 0.20 − 0.05 0.15 3 Use this to find σm : σp = ωσm 1 0.04 = 1 σm 3 0.04 σm = 1 = 0.03 13 Next find the beta of the security, let us call it i. cov(ri , rm ) = ρ(ri , rm )σi σm = 0.5 · 0.02 · 0.03 = 0.004 βi = 0.004 cov(ri , rm ) 1 = = var(rm ) 0.032 3 Expected rate of return: E[ri ] = rf + (E[rm ] − rf )βi = 0.05 + (0.20 − 0.05) · Exercise 8. [3] 1. Calculate the beta of the portfolio as βp = 0.20 · 1.1 + 0.5 · 0.8 + 0.3 · 1.0 = 0.92 Therefore, the return on the portfolio is E[rp ] = rf + (E[rm ] − rf )βp = 0.05 + (0.13 − 0.05) · 0.92 = 12.36% 9 1 = 10% 3 Exercise 9. Analyst (RWJ 10.17) [3] 1. The market expects E[r] = rf + (E[rm ] − rf )β = 0.08 + 0.06 · 0.9 = 13.4% The analyst is pessimistic, since his expectation of 13% is lower than the 13.4% expected return for a stock with β = 0.9. Exercise 10. Portfolio (RWJ 10.27) [6] 1. Remember that by combining the risk free rate and the market portfolio m, we can write the expected return and standard deviations in terms of the fraction ω invested in the risky (market) portfolio: E[rp ] = (1 − ω)rf + ωE[rm ] σp = ωσm We can use this to find the requested information. We need to find the beta of security of interest. To do this we need the variance of the market portfolio. Step 1: Find ω E[rp ] = (1 − ω)rf + ωE[rm ] 0.25 = (1 − ω)0.05 + ω0.20 ω= 0.2 1 0.25 − 0.05 = =1 0.20 − 0.05 0.15 3 Use this to find σm : σp = ωσm 1 0.04 = 1 σm 3 0.04 σm = 1 = 0.03 13 Next find the beta of the security, let us call it i. cov(ri , rm ) = ρ(ri , rm )σi σm = 0.5 · 0.02 · 0.03 = 0.004 βi = cov(ri , rm ) 0.004 1 = = 2 var(rm ) 0.03 3 Expected rate of return: E[ri ] = rf + (E[rm ] − rf )βi = 0.05 + (0.20 − 0.05) · Exercise 11. Portfolio (RWJ 10.27) [6] 10 1 = 10% 3 1. Remember that by combining the risk free rate and the market portfolio m, we can write the expected return and standard deviations in terms of the fraction ω invested in the risky (market) portfolio: E[rp ] = (1 − ω)rf + ωE[rm ] σp = ωσm We can use this to find the requested information. We need to find the beta of security of interest. To do this we need the variance of the market portfolio. Step 1: Find ω E[rp ] = (1 − ω)rf + ωE[rm ] 0.25 = (1 − ω)0.05 + ω0.20 ω= 0.2 1 0.25 − 0.05 = =1 0.20 − 0.05 0.15 3 Use this to find σm : σp = ωσm 1 0.04 = 1 σm 3 0.04 σm = 1 = 0.03 13 Next find the beta of the security, let us call it i. cov(ri , rm ) = ρ(ri , rm )σi σm = 0.5 · 0.02 · 0.03 = 0.004 βi = cov(ri , rm ) 0.004 1 = = 2 var(rm ) 0.03 3 Expected rate of return: E[ri ] = rf + (E[rm ] − rf )βi = 0.05 + (0.20 − 0.05) · Exercise 12. Widgets [6] 1. Expected returns on debt and equity for A, B, and C: E[r̃D ] = rf + (E[rm ] − rf )βD A : rD = 0.08 + (0.16 − 0.08) · 0 = 8% B : rD = 0.08 + (0.16 − 0.08) · 0.05 = 8.4% C : rD = 0.08 + (0.16 − 0.08) · 0.1 = 8.8% and E[r̃E ] = rf + (E[rm ] − rf )βE A : rE = 0.08 + (0.16 − 0.08) · 1.0 = 16% B : rE = 0.08 + (0.16 − 0.08) · 1.5 = 20% C : rE = 0.08 + (0.16 − 0.08) · 1.5 = 20% 11 1 = 10% 3 2. Expected return for each firm A, B and C: E[rA ] = D E rD + rE = 13.3% V V E[rB ] = 15.6% E[rC ] = 14.4% 3. Asset beta for the industry Debt Market value 100 75 50 225 Firm A B C Industry βD 0 0.05 0.10 Firm A B C Industry βE 1.0 1.5 1.5 Firm A B C Industry r∗ 13.3% 15.6% 14.4% Equity Mkt value 200 125 50 375 rD 8% 8.4% 8.8% rE 16% 20% 20% Total β∗ Mkt value 0.67 300 0.96 200 0.8 100 600 β∗ = 300 200 100 0.66 + 0.96 + 0.8 = 0.78 600 600 600 4. Depends on whether the asset beta for industry is smaller or larger than one. In this case β ∗ = 0.78 < 1, the return is lower than the market. Exercise 13. Remember that by combining the risk free rate and the market portfolio m, we can write the expected return and standard deviations in terms of the fraction ω invested in the risky (market) portfolio: E[rp ] = (1 − ω)rf + ωE[rm ] σp = ωσm We can use this to find the requested information. We need to find the beta of security of interest. To do this we need the variance of the market portfolio. Step 1: Find ω E[rp ] = (1 − ω)rf + ωE[rm ] 0.25 = (1 − ω)0.05 + ω0.20 ω= 0.25 − 0.05 0.2 1 = =1 0.20 − 0.05 0.15 3 Use this to find σm : σp = ωσm 12 1 0.04 = 1 σm 3 0.04 σm = 1 = 0.03 13 Next find the beta of the security, let us call it i. cov(ri , rm ) = ρ(ri , rm )σi σm = 0.5 · 0.02 · 0.03 = 0.004 βi = cov(ri , rm ) 1 0.004 = = 2 var(rm ) 0.03 3 Expected rate of return: E[ri ] = rf + (E[rm ] − rf )βi = 0.05 + (0.20 − 0.05) · = 10% (b) is correct. 13 1 3