CHAPTER 6: RISK AVERSION AND
CAPITAL ALLOCATION TO RISKY ASSETS
PROBLEM SETS
1. (e)
2. (b) A higher borrowing rate is a conquence 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 becau doubling the expected return with a fixed risk-free rate will more than double the risk premium and the Sharpe ratio.
怎么建群
3. Assuming no change in risk tolerance, that is, an unchanged risk aversion coefficient (A), then higher perceived volatility increas the denominator of the equation for the optim
al investment in the risky portfolio (Equation 6.7). The proportion invested in the risky portfolio will therefore decrea.
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 prent value of the portfolio is:
$135,000/1.14 = $118,421
b. If the portfolio is purchad for $118,421, and provides an expected cash inflow of $135,000, then the expected rate of return [E(r)] is as follows:
$118,421 × [1 + E(r)] = $135,000
描写眼神的句子Therefore, E(r) = 14%. The portfolio price is t 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 prent value of the portfolio is now:
$135,000/1.18 = $114,407
d. For a given expected cash flow, portfolios that command greater risk premia must ll at lower prices. The extra discount from expected value is a penalty for risk.
5.When we specify utility by U = E(r) – 0.5Aσ 2, 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
掰玉米In order for the risky portfolio to be preferred to bills, the following must hold:
0.12 – 0.0162A > 0.07 A < 0.05/0.0162 = 3.09
A must be less than 3.09 for the risky portfolio to be preferred to bills.
6. Points on the curve are derived by solving for E(r) in the following equation:
U = 0.05 = E(r) – 0.5Aσ 2 = E(r) – 1.5σ 2
The values of E(r), given the values of σ 2, are therefore:
| 2 | E(r) |
0.00 | 0.0000 | 0.05000 |
0.05 | 0.0025 | 0.05375 |
0.10 | 0.0100 | 0.06500 |
0.15 | 0.0225 | 0.08375 |
0.20 | 0.0400 | 0.11000 |
0.25 | 0.0625 | 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) – 0.5Aσ 2 = E(r) – 2.0σ 2 = 0.05
The equal-utility combinations of expected return and standard deviation are prented in the table below. The indifference curve is the upward sloping line in the graph on the next page, labeled Q7 (for Question 7).
| 2 | E(r)笑死病 |
0.00 | 0.0000 | 0.0500 |
0.05 | 0.0025 | 0.0550 |
0.10 | 0.0100 | 0.0700 |
0.15 | 0.0225 | 0.0950 |
| | |
0.20 | 0.0400 | 0.1300 |
0.25 | 0.0625 | 0.1750 |
| | |
The indifference curve in Problem 7 differs from that in Problem 6 in slope. When A increas from 3 to 4, the incread risk aversion results in a greater slope for the indifference curve since more expected return is needed in order to compensate for additional σ.
8. The coefficient of risk aversion for a risk neutral investor is zero. Therefore, the corresponding utility is equal to the portfolio’s expected return. The corresponding indifference curve in the expected return-standard deviation plane is a horizontal line, labeled Q8 in the graph above (e Problem 6).
9. A risk lover, rather than penalizing portfolio utility to account for risk, derives greater utility as variance increas. This amounts to a negative coefficient of risk aversion. The corresponding indifference curve is downward sloping in the graph above (e Problem 6), and is labeled Q9.
南靖土楼门票
歌唱祖国歌曲10. The portfolio expected return and variance are computed as follows:
(1) WBills | (2) rBills | (3) WIndex | (4) rIndex | rPortfolio (1)×(2)+(3)×(4) | Portfolio (3) × 20% | 2 Portfolio |
0.0 | 5% | 1.0 | 13.0% | 13.0% = 0.130 | 20% = 0.20 | 0.0400 |
0.2 | 5% | 0.8 | 13.0% | 11.4% = 0.114 | 16% = 0.16 | 国民收入决定理论0.0256 |
0.4 | 5% | 0.6 | 13.0% | 9.8% = 0.098 | jdtab 12% = 0.12 | 0.0144 |
0.6 | 5% | 0.4 | 13.0% | 8.2% = 0.082 | 8% = 0.08 | 0.0064 |
0.8 | 5% | 0.2 | 13.0% | 6.6% = 0.066 | 4% = 0.04 | 0.0016 |
1.0 | 5% | 0.0 | 13.0% | 5.0% = 0.050 | 0% = 0.00 | 0.0000 |
| | | | | | |
11. Computing utility from U = E(r) – 0.5 × Aσ 2 = E(r) – σ 2, we arrive at the values in the column labeled U(A = 2) in the following table:
