CHAPTER 6: RISK AND RISK AVERSION
1. a.
The expected cash flow is: .5
×
70,000 + .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 at $118,421, and provides an expected payoff of $135,000,
then the expected rate of return, E(r), is derived as follows:
$118,421
×
[1 + E(r)] = $135,000
so that E(r) =
14%.
The portfolio price is set to equate the expected return with the
required rate of return.
c.
If the risk premium over bills is now 12%, 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.
2.
When we specify utility by U =
E(r) – .005A
σ
2
, the utility from bills is 7%, while that
from the risky portfolio is U = 12 – .005A
×
18
2
= 12 – 1.62A.
For the portfolio to be
preferred to bills, the following inequality must hold: 12 – 1.62A > 7, or,
A < 5/1.62 = 3.09.
A must be less than 3.09 for the risky portfolio to be preferred to bills.
3.
Points on the curve are derived as follows:
U = 5 = E(r) – .005A
σ
2
=
E(r) – .015
σ
2
The necessary value of E(r), given the value of
σ
2
, is therefore:
σ
σ
2
E(r)
0%
0
5.0%
5
25
5.375
10
100
6.5
15
225
8.375
20
400
11.0
25
625
14.375
61
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The indifference curve is depicted by the bold line in the following graph (labeled Q3, for
Question 3).
E(r)
σ
5
4
U(Q3,A=3)
U(Q4,A=4)
U(Q5,A=0)
U(Q6,A<0)
4.
Repeating the analysis in Problem 3, utility is:
U = E(r) – .005A
σ
2
=
E(r) – .02
σ
2
= 4
leading to the equalutility combinations of expected return and standard deviation
presented in the table below.
The indifference curve is the upward sloping line appearing
in the graph of Problem 3, labeled Q4 (for Question 4).
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 Spring '08
 Lazrak
 Standard Deviation

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