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Chapter 06  Risk Aversion and Capital Allocation to Risky Assets
CHAPTER 6: RISK AVERSION AND
CAPITAL ALLOCATION TO RISKY ASSETS
PROBLEM SETS
1.
(e)
2.
(b) A higher borrowing 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 riskfree 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), then higher perceived volatility increases the denominator of the equation for the
optimal investment in the risky portfolio (Equation 6.12). 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 riskfree 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 return [E(r)] is derived as follows:
$118,421
×
[1 + E(r)] = $135,000
Therefore, E(r) =
14%. The portfolio price is set to equate the expected rate or
return with the required rate of return.
c.
If the risk premium over Tbills 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
61
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View Full DocumentChapter 06  Risk Aversion and Capital Allocation to Risky Assets
d.
For a given expected cash flow, portfolios that command greater risk premia
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) – 0.5A
σ
2
, the utility level for Tbills is: 0.07
The utility level for the risky portfolio is: U = 0.12 – 0.5A(0.18)
2
= 0.12 – 0.0162A
In order for the risky portfolio to be preferred to bills, the following inequality 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) – 0.5A
σ
2
= E(r) – 1.5
σ
2
The values of E(r), given the values of
σ
2
, are therefore:
σ
σ
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 following graph (labeled Q6, for Question 6) depicts the
indifference curve.
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 Spring '10
 AttilaOdabaşı

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