CHAPTER 6 1. Which of the following choices best completes the following statement? Explain. An investor with a higher degree of risk aversion, compared to one with a lower degree, will prefer investment portfolios (e) None of the above is true. The answer is not (d) “with a higher Sharpe ratio” because: The way the question is phrased, the high-risk-aversion investor will prefer the high Sharpe ratio investment as will the low-risk-aversion investor. You can’t say the high-risk-aversion investor would not prefer it more. The way I read the question at first was that the high-risk-aversion investor will have the risk-free and the high Sharpe ratio investment on their indifference curve. The low-risk-aversion investor will have some lower Sharpe ratio investment in their indifference curve. Therefore the high-riskaversion investor will prefer the investment with the high Sharpe ratio. But that is not what the question says. Since both investors would prefer the high Sharpe ratio investment, the high-risk-aversion investor would not prefer it more compared to the low-riskaversion investor. They would both prefer it. And you can’t determine which would prefer it more. 5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to bills? Utility for T-bills is: U = E(r) – 0.5Aσ2 = 0.07 - 0.5A(0)2 = 0.07 Utility for the portfolio: U = E(r) – 0.5Aσ2 = 0.12 - 0.5A(0.18)2 For what risk aversion level (A) would an investor be indifferent? Set the Utility values equal and solve for A: 0.12 – 0.0162A = 0.07 A = 0.05/0.0162 = 3.09 For A = 3.09, the investor would be indifferent between the risky portfolio with E(r) = 12% and σ = 18% and the risk-free rate of 7%. More risk aversion (A > 3.09) means an investor would derive less utility from the risky portfolio. (Compute the utility for the risky portfolio if A = 3.5.) Less risk aversion (A < 3.09) means an investor would derive more utility from the risky portfolio. (Compute the utility for the risky portfolio if A = 2.5.) Therefore the risk-aversion parameter (A) must be less than 3.09 for the risky portfolio to be preferred to T-Bills.
1
6. Draw the indifference curve in the expected return–standard deviation plane corresponding to a utility level of .05 for an investor with a risk aversion coefficient of 3. (Hint: Choose several possible standard deviations, ranging from 0 to .25, and find the expected rates of return providing a utility level of .05. Then plot the expected return–standard deviation points so derived.) U = E(r) – 0.5Aσ2 E(r) = 0.5Aσ2 + U Set U = 0.05 and A = 3 and use Excel to compute E(r) for values of σ between 0 and 0.25: Table and chart for questions 6 through 9 below. 7. Now draw the indifference curve corresponding to a utility level of .05 for an investor with risk aversion coefficient A = 4. Table and chart for questions 6 through 9 below. 8. Draw an indifference curve for a risk-neutral investor providing utility level .05. Risk neutral means A = 0. Table and chart for questions 6 through 9 below. 9. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the indifference curve for a utility level of .05 for a risk lover Risk lover means A < 0. In this case A = -3. U = E(r) – 0.5Aσ2 E(r) = 0.5Aσ2 + U (for a constant U) Set U = 0.05 and use Excel to compute values of E(r) for the different values of A for σ from 0.00 to 0.25. σ 0.00% 5.00% 10.00% 15.00% 20.00% 25.00%
Question 6 3.00 = A 5.00% 5.38% 6.50% 8.38% 11.00% 14.38%
Question 7 4.00 = A 5.00% 5.50% 7.00% 9.50% 13.00% 17.50%
Question 8 0.00 = A 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%
Question 9 -3.00 = A 5.00% 4.63% 3.50% 1.63% -1.00% -4.38%
Highlight the table in Excel and insert “Scatter” Chart. See Below.
2
20.00%
15.00%
Indifference Curves for Various Risk-Aversions given rf = 5%
E(r)
10.00%
5.00%
0.00% 0.00% -5.00%
-10.00%
5.00%
10.00%
15.00%
3.00 = A
4.00 = A
0.00 = A
-3.00 = A
20.00%
25.00%
σ
Note that the greater the value of A, the more return required for a given level of risk for the investor to be indifferent between a risky asset and the risk-free. In other words, the red line for A = 4 is above the blue line for A = 3. The more risk adverse investor (A = 4) requires more return for an given level of risk. Hint: Another way to produce this chart is it to type the following into Wolfram Alpha: plot y=(3)(.5)x^2 + .05 and y=(4)(.5)x^2 + .05 and y=(0)(.5)x^2 + .05 and y=(-3)(.5)x^2 + .05 from x = 0 to .25 10. For Problems 10 through 12 : Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 80 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 5%. Calculate the expected return, standard deviation and variance of portfolios invested in T-bills and the S&P 500 index with T-Bill weights from 0.0 to 1.0 in 0.2 increments. (Hint: T-Bills earn 5% and the S&P500 earns 8.00% more so E(rS&P500) = 5% + 8% = 13%) Answer for 10 through 12 in the table below: 11. Calculate the utility levels of each portfolio of Problems 10 for an investor with A = 2. What do you conclude? Answer for 10 through 12 in the table below: 3
12. Repeat Problems 11 for an investor with A = 3. What do you conclude? Produce a table that computes E(rC), σC, σ2C, and U for A = 2.0 and A = 3.0 for E(rS&P500) = 13.00%; σS&P500 = 20.00%; rf = 5.00% and Weights in 20% increments. Recall y = allocation to the risky asset (S&P500). y = WS&P500 1.0 0.8 0.6 0.4 0.2 0.0
1 – y = WT-Bills 0.0 0.2 0.4 0.6 0.8 1.0
E(rC) 13.00% 11.40% 9.80% 8.20% 6.60% 5.00%
σC 20.00% 16.00% 12.00% 8.00% 4.00% 0.00%
σ2C 0.0400 0.0256 0.0144 0.0064 0.0016 0.0000
U for A = 2.0 0.0900 0.0884 0.0836 0.0756 0.0644 0.0500
U for A = 3.0 0.0700 0.0756 0.0764 0.0724 0.0636 0.0500
For the lower risk aversion (A = 2), Utility is maximized with 100% in the S&P500. For the higher risk aversion (A = 3), Utility is maximized with 40% in T-Bills and 60% in the S&P500. Extra Questions: Solve for the Utility maximizing allocation between the risky portfolio and T-bills for each level of risk aversion. Then solve for the E(rC)* and σC* and the U* for these “optimal” portfolios. A = 2.0: y* = [E(rP) – rf]/Aσ2P = (0.13 – 0.05)/[2.00(0.20)^2] = 1.00 E(rC)* = rf + y*[E(rP) – rf] = 0.05 + 1.00[0.13 – 0.05] = 13.00% σC* = y*σP = 1.00(0.20) = 20.00% U* = E(rC*) – A(0.5)(σC*)2 = 0.13 – (2)(0.5)(0.2)2 = 0.0900 A = 3.0: y* = [E(rP) – rf]/Aσ2P = (0.13 – 0.05)/[2.00(0.20)^2] = 1.00 E(rC)* = rf + y*[E(rP) – rf] = 0.05 + 0.67[0.13 – 0.05] = 10.33% σC* = y*σP = 0.67(0.20) = 13.33% U* = E(rC*) – A(0.5)(σC*)2 = 0.13 – (2)(0.5)(0.2)2 = 0.0900 Note that U* = 0.0767 for the 67-33 portfolio exceeds U = 0.0764 for the 60-40 portfolio in the table above. Use these inputs for Problems 13 through 19: You manage a risky portfolio with expected rate of return of 18% and standard deviation of 28%. The T-bill rate is 8% 13. Your client chooses to invest 70% of a “combined” portfolio in your risky portfolio (E(rP) = 18.00% and σP = 28.00%) and 30% in a T-Bill money market fund. What is the expected return and standard deviation on the client’s combined portfolio? The client’s 70-30 Combined Portfolio: 4
E(rC) = (0.7)(0.18) + (0.3)(0.08) = 15.00% σC = (0.7)(0.28) = 19.60% 14. Suppose that your risky portfolio includes the following investments in the given proportions: Stock A = 25%, Stock B = 32% and Stock C = 43%. What are the investment proportions of your client's overall portfolio, including the position in T-bills? Stock A = Stock B = Stock C =
25% 32% 43% 100%
of 70% of 70% of 70%
= 18% = 22% = 30% 70%
15. What is the reward-to-volatility ratio (S) of your risky portfolio (P) and your client's (70-30) combined portfolio? S = [E(r) - rf]/σ Your Risky Portfolio: SP = [0.18 – 0.08]/0.28 = 0.3571 Client’s Combined Portfolio: SC = [0.15 – 0.08]/0.196 = 0.3571 Note that both portfolios lay on the Capital Allocation Line (CAL) defined by rf = 8% and E(rP) = 18.00% and σP = 28.00%. The Slope of the CAL is 0.3571 at any point on the line. 16. Draw the CAL of your portfolio on an expected return–standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL. E(rC) = rf + y[E(rP) – rf] σC = yσP Substitute for y in E(rC) to get return as a function of risk for a given risk portfolio and risk-free: In other words, we need E(rC) as a function of σC for a given E(rP) and σP: σC = yσP y = σC/σP E(rC) = rf + y[E(rP) – rf] = rf + σC/σP [E(rP) – rf] = rf + σC [E(rP) – rf]/σP The return of the combined portfolio (E(rC)) is a linear function of the risk of the combined portfolio (σC). The intercept (rf = 0.08) and the slope ([E(rP) – rf]/σP = 0.3571) are constants. Produce this table in Excel: σC 0.00% 19.60% 28.00%
E(rC) 8.00% 15.00% 18.00%
Highlight the table and insert a “Scatter” chart. 5
20% 28.00%, 18.00% 18% 16% 14%
19.60%, 15.00%
E(rC)
12% 10% 8%
0.00%, 8.00%
6%
E(rC) = rf + σC[E(rP) - rf]/σP
4%
E(rC) = 0.08 + σC[0.3571]
2% 0% 0%
5%
10%
15%
20%
25%
30%
σC
17. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall combined portfolio will have an expected rate of return of 16%. (a) What is the proportion in the risky portfolio (y)? E(rC) = rf + y[E(rP) – rf] y = [E(rC) - rf]/[E(rP) – rf] y = [0.16 – 0.08]/[0.18 – 0.08] = 0.80 80% in the risky, 20% in the risk-free (b) What are your client's investment proportions in your three stocks and the T-bill fund? Stock A = Stock B = Stock C =
25% 32% 43% 100%
of 80% of 80% of 80%
= 20% = 26% = 34% 80%
(c) What is the standard deviation of the rate of return on your client's portfolio? σC = yσP = 0.80(0.28) = 22.4%
6
18. Suppose that your client prefers to invest in your risky fund a proportion y that maximizes the expected return on the combined portfolio subject to the constraint that the combined portfolio's standard deviation (σC) will not exceed 18%. (a) What proportion of the combined portfolio will be invested in the risky portfolio (y)? σC = yσP y = σC/σP = 0.18/0.28 = 64.29% (b) What is the expected return on the combined portfolio? E(rC) = rf + y[E(rP) – rf] = 0.08 + (0.6429)(0.18 – 0.08) = 14.43% 19. Your client's degree of risk aversion is A = 3.5. (a) What proportion of the combined portfolio will be invested in the risky portfolio (y) in order to maximize utility? y* = [E(rP) – rf]/Aσ2P = (0.18 – 0.08)/[3.5(0.28) 2] = 36.44%
(b) What is the expected return and standard deviation of the return on your client's optimized portfolio? E(rC)* = rf + y*[E(rP) – rf] = 0.08 + 0.3644[0.18 – 0.08] = 11.64% σC* = y*σP = 0.3644(0.28) = 10.20% Extra Question: Calculate your clients utility for y = 1, y = 0.70, y = 0.80. y = 0.6429 and y* = 0.3644 given a riskaversion level of 3.5. U* = E(rC*) – A(0.5)(σC*)2 y = WP 100.00% 80.00%
1 -y = WT-Bills 0.00% 20.00%
E(rC) 18.00% 16.00%
σC 28.00% 22.40%
U for A = 3.5 0.0428 0.0722
70.00% 64.29% 36.44% 0.00%
30.00% 35.71% 63.56% 100.00%
15.00% 14.43% 11.64% 8.00%
19.60% 18.00% 10.20% 0.00%
0.0828 0.0876 0.0982 0.0800
Note the greatest utility for y* = 36.44 20. Look at the data in Table 6.7 on the average risk premium of the S&P 500 over T-bills, and the standard deviation of that risk premium. Suppose that the S&P 500 is your risky portfolio.
7
(a) If your risk-aversion coefficient is A = 4 and you believe that the entire 1926–2009 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity? A = 4, E(rM) − rf = 7.93%, σM = 20.81% y* = [E(rP) – rf]/Aσ2P = (0.0793)/[4(0.2081) 2] = 45.78% 45.78% allocated to the S&P500 and 54.22% allocated to T-bills
(b) What if you believe that the 1968–1988 period is representative? A = 4, E(rM) − rf = 3.44%, σM = 16.71% y* = [E(rP) – rf]/Aσ2P = (0.0344)/[4(0.1671)2] = 30.80% 30.80% allocated to the S&P500 and 69.20% allocated to T-bills (c) What do you conclude upon comparing your answers to (a) and (b)? The reward to volatility ratio is lower for 1968 to 1988 than 1926 to 2009. Therefore the allocation to the S&P500 is lower. 21. Consider the following information about a risky portfolio (fund) that you manage, and a riskfree asset: E(rP) = 11%, σP = 15%, rf = 5%. (a) Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her combined (or overall or complete) portfolio equal to 8%. What proportion should she invest in the risky portfolio, P, and what proportion in the risk-free asset? E(rC) = rf + y[E(rP) – rf] y = [E(rC) - rf]/[E(rP) – rf] y = [0.08 – 0.05]/[0.11 – 0.05] = 0.50 50% in the risky, 50% in the risk-free (b) What will be the standard deviation of the rate of return on her portfolio? σC = yσP = 0.50(0.15) = 7.5% (c) Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than 12%. Which client is more risk averse? The first client is more risk averse.
8
Additional Questions: 1. Given the following states of the economy, the probabilities of the states and the returns for assets 1 and 2 given the states, calculate the expected return and standard deviation for Asset 1, Asset 2, and a 40-60 portfolio of Asset 1 and 2. (Note that you need to calculate the covariance between 1 and 2.) State Expansion Contraction Drought W1 = 40%
ps 0.6 0.3 0.1
r1s 0.30 0.05 -0.20
r2s 0.10 0.10 -0.30
W2 = 60%
E(r1) =∑psrs = (0.6)(0.3) + (.3)(0.05) + (.1)(-0.20) = 0.1750 E(r2) = (0.6)(0.1) + (0.3)(0.1) + (0.1)(-0.30) = 0.06 1 =(∑ps[rs - E(r)]2)½ = [(0.6)(.3 – .175)2 + (.3)(.05 – .175)2 + (.1)(-.20 – .175)2]½ = .1677 2 =[(.6)(.1 – .06)2 + (.3)(.1 – .06)2 + (.1)(-.30 – .06)2]½ = .12 12 =.6(.3 – .175)(.1 – .06) + .3(.05 – .175)(.1 – .06) + .1(-.20 – .175)(-.30 – .06) = .015 E(rP) = w1E(r1) + w2E(r2) = 0.4(.175) + .6(0.06) = 0.1060 P = [w1212 + w2222 + 2w1w21,2]½ = [(.42)(.16772) + (.62)(.122) + 2(.4)(.6)(.015)]½ = 12.99% 2. Only two investment choices are available and an investor may allocate between either: A risky portfolio P with E(rP) = 10% and σP = 20% and the risk-free asset with rf = 5%. a) Calculate the allocation between the risky portfolio and the risk-free asset for an investor with a risk aversion level of 6. y* = [E(rP) – rf]/Aσ2P = (0.10 – 0.05)/[6(0.20)2] = 0.2083 = 20.83% 20.83% in the risky portfolio and 79.17% in the risk-free b) Calculate the return and standard deviation to the optimal combined portfolio for an investor with a risk aversion level of 6. E(rC) = rf + y[E(rP) – rf] = 0.05 + .2083[0.10 – 0.05] = 0.0604 = 6.04% σC = yσP = (.2083)(0.20) = 0.0417 = 4.17% c) Calculate the risk-free rate at which the investor with a risk-aversion level of 6 is indifferent between the risk free and the optimal combined portfolio. The certainty equivalent rate is equal to the utility derived from the optimal combined portfolio. So calculate the investor’s utility from holding the optimal combined portfolio: 9
U = E(rc) – A σC 2/2 = 0.0604 – 6(0.0417)2/2 = 0.0552 = 5.52% d) Calculate the utility derived the investor with a risk-aversion level of 6 from holding only the risk-free asset and holding only the risky portfolio Only the risk-free: U = E(r) – Aσ 2/2 = 0.05 – 6(0)2/2 = 0.05 Only the risky portfolio: U = E(r) – Aσ 2/2 = 0.10 – 6(0.20)2/2 = -0.02 So the Utility from holding the optimal combined portfolio (20.83% risky and 79.17% risk-free) exceed the utility from holding only either the risky or the risk-free. Now assume the investor has a risk aversion level of 2. e) Calculate the allocation between the risky portfolio and the risk-free asset. y* = [E(rP) – rf]/Aσ2P = (0.20 – 0.05)/[2(0.20)2] = 0.6250 = 62.50% 62.50% in the risky portfolio and 37.50% in the risk-free f) Calculate the return and standard deviation to the optimal combined portfolio. E(rC) = rf + y[E(rP) – rf] = 0.05 + .6250[0.10 – 0.05] = 0.0813 = 8.13% σC = yσP = (.6250)(0.20) = 0.1250 = 12.50% g) Calculate the risk-free rate at which the investor with a risk-aversion level of 2 is indifferent between the risk free and the optimal combined portfolio. U = E(rc) – A σC 2/2 = 0.0813 – 6(0.1250)2/2 = 0.0656 = 6.56% h) Calculate the utility derived the investor with a risk-aversion level of 2 from holding only the risk-free asset and holding only the risky portfolio Only the risk-free: U = E(r) – Aσ 2/2 = 0.05 – 2(0)2/2 = 0.05 Only the risky portfolio: U = E(r) – Aσ 2/2 = 0.10 – 2(0.20)2/2 = 0.06 So the Utility from holding the optimal combined portfolio (62.50% risky and 37.50% risk-free) exceed the utility from holding only either the risky or the risk-free. i) Compare the holdings of the A = 6 investor with the A = 2 investor. Which holds more of the risky asset? Portion in risky Portion in risk-free
A=2 62.50% 37.50%
A=6 20.83% 79.17%
E(rC)
8.13%
6.04%
σC Certainty Equiv (U)
12.50% 6.56%
4.17% 5.52%
The investor with A = 2 holds more of the risky asset.
10
j) Compare the utility level of each investor at their optimal combined portfolio. Which investor is better off? THIS IS A TRICK QUESTION! You can’t compare utility levels across individuals. Each investor is at his or her optimum so each is as well off as they can be. CFA PROBLEMS 1.
Utility for each investment = E(r) – 0.5 4 We choose the investment with the highest utility value. Expected Standard Utility deviation Investment return U E(r) 1 0.12 0.30 -0.0600 2 0.15 0.50 -0.3500 3 0.21 0.16 0.1588 4 0.24 0.21 0.1518
2.
When investors are risk neutral, then A = 0; the investment with the highest utility is Investment 4 because it has the highest expected return.
3.
(b)
4.
Indifference curve 2
5.
Point E
6.
(0.6 $50,000) + [0.4 ($30,000)] $5,000 = $13,000
7.
(b)
8.
Expected return for equity fund = T-bill rate + risk premium = 6% + 10% = 16% Expected return of client’s overall portfolio = (0.6 16%) + (0.4 6%) = 12% Standard deviation of client’s overall portfolio = 0.6 14% = 8.4%
9.
Reward-to-volatility ratio =
10 0.71 14
11
