8/3/2019 Ch06 Revised

1/3

CHAPTER 6: RISK AND RISK AVERSION

1. 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 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 rateor 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 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) 0.005A 2 , the utility level for T-bills is 7%.The utility level for the risky portfolio is: U = 12 0.005A 182 = 12 1.62A

In order for the risky portfolio to be preferred to bills, the following inequality must

hold:

12 1.62A > 7 A < 5/1.62 = 3.09

A must be less than 3.09 for the risky portfolio to be preferred to bills.

6-1

8/3/2019 Ch06 Revised

2/3

3. Points on the curve are derived by solving for E(r) in the following equation:

U = 5 = E(r) 0.005A2 = E(r) 0.0152

The values of E(r), given the values of2

, are therefore: 2 E(r)

0% 0 5.000%5% 25 5.375%

10% 100 6.500%15% 225 8.375%20% 400 11.000%25% 625 14.375%

The bold line in the following graph (labeled Q3, for Question 3) depicts the

indifference curve.

E(r)

5

4

U(Q3,A=3)U(Q4,A=4)

U(Q5,A=0)

U(Q6,A

8/3/2019 Ch06 Revised

3/3

4. Repeating the analysis in Problem 3, utility is now:

U = E(r) 0.005A2 = E(r) 0.0202 = 4

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 above, labeled Q4 (for Question 4).

2 E(r)

0% 0 4.000%5% 25 4.500%

10% 100 6.000%15% 225 8.500%20% 400 12.000%25% 625 16.500%

The indifference curve in Problem 4 differs from that in Problem 3 in both slope andintercept. When A increases from 3 to 4, the increased risk aversion results in a greaterslope for the indifference curve since more expected return is needed in order tocompensate for additional . The lower level of utility assumed for Problem 4 (4%rather than 5%) shifts the vertical intercept down by 1%.

7. c [Utility for each portfolio = E(r) 0.005 4 2

We choose the portfolio with the highest utility value.]

8. d [When investors are risk neutral, then A = 0; the portfolio with the highest

utility is the one with the highest expected return.]

6-3