CHAPTER 6: CAPITAL ALLOCATION TO RISKY ASSETS
PROBLEM SETS
1. (e) The first two answer choices are incorrect because a highly
risk averse investor would avoid portfolios with higher risk premiums and higher standard deviations. In addition, higher or lower Sharpe ratios are not an indication of an investor's tolerance for risk. The Sharpe ratio is simply a tool to
absolutely measure the return premium earned per unit of risk.
2. (b) A higher borrowing rate is a consequence of the risk of the
borrowers’ default. In perfect markets with no additional cost of default, this increment would equal the value of the borrower’s option to default, and the Sharpe measure, with appropriate treatment of the default option, would be the same. However, in reality there are costs to default so that this part of the
increment lowers the Sharpe ratio. Also, notice that answer (c) is not correct because doubling the expected return with a fixed risk-free rate will more than double the risk premium and the Sharpe ratio.
3. Assuming no change in risk tolerance, that is, an unchanged risk-aversion coefficient (A), higher perceived volatility increases the denominator of the equation for the optimal investment in the risky portfolio (Equation 6.7). The proportion invested in the risky portfolio will therefore decrease.
4. a. The expected cash flow is: (0.5 × $70,000) + (0.5 × 200,000)
= $135,000.
With a risk premium of 8% over the risk-free rate of 6%, the required rate of return is 14%. Therefore, the present value of the portfolio is:
$135,000/1.14 = $118,421
b. If the portfolio is purchased for $118,421 and provides an
expected cash inflow of $135,000, then the expected rate of
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return [E(r)] is as follows:
$118,421 × [1 + E(r)] = $135,000
Therefore, E(r) = 14%. The portfolio price is set to equate the expected rate of return with the required rate of return.
c. If the risk premium over T-bills is now 12%, then the required
return is:
6% + 12% = 18%
The present value of the portfolio is now:
$135,000/1.18 = $114,407
d. For a given expected cash flow, portfolios that command
greater risk premiums must sell at lower prices. The extra discount from expected value is a penalty for risk.
5. When we specify utility by U = E(r) – 0.5Aσ2, the utility level
for T-bills is: 0.07
The utility level for the risky portfolio is:
U = 0.12 – 0.5 × A × (0.18)2 = 0.12 – 0.0162 × A In order for the risky portfolio to be preferred to bills, the following must hold:
0.12 – 0.0162A > 0.07 bills.
6. Points on the curve are derived by solving for E(r) in the
following equation:
A < 0.05/0.0162 = 3.09
A must be less than 3.09 for the risky portfolio to be preferred to
U = 0.05 = E(r) – 0.5Aσ2 = E(r) – 1.5σ2
The values of E(r), given the values of σ2, are therefore: 0.00 0.05 0.10 0.15
0.0000 0.0025 0.0100 0.0225
2
E(r) 0.05000 0.05375 0.06500 0.08375
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0.20 0.25
0.0400 0.0625
0.11000 0.14375
The bold line in the graph on the next page (labeled Q6, for Question 6) depicts the indifference curve.
7. Repeating the analysis in Problem 6, utility is now:
U = E(r) – 0.5Aσ2 = E(r) – 2.0σ2 = 0.05
The equal-utility combinations of expected return and standard
deviation are presented in the table below. The indifference curve is the upward sloping line in the graph on the next page, labeled Q7 (for Question 7). 0.00 0.05 0.10 0.15 0.20 0.25
The indifference curve in Problem 7 differs from that in Problem 6 in slope. When A increases from 3 to 4, the increased risk aversion results in a greater slope for the indifference curve since more expected return is needed in order to compensate for additional σ.
0.0000 0.0025 0.0100 0.0225 0.0400 0.0625
2
E(r) 0.0500 0.0550 0.0700 0.0950 0.1300 0.1750
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投资学第10版习题答案06
