CHAPTER 6: RISK AVERSION AND
CAPITAL ALLOCATION TO RISKY ASSETS
PROBLEM SETS
1.
(e)
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 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 6.7).
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 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 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 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
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.
61
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When we specify utility by U =
E(r) – 0.5Aσ
. 2
, the utility level for Tbills 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
⇒
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:
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 graph on the next page (labeled Q6, for Question 6) depicts the
indifference curve.
7.
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 Three '11
 3213
 Standard Deviation, Utility, CML, Risk premium

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