problem set 4

PROBLEM SET 4

1) Consider the following utility functions, where W is wealth:

(a) U (W ) = W 2
1
(b) U (W ) =
W
(c) U (W ) = −W
(d) U (W ) = W
(e) U (W ) = ln(W )
(f) U (W ) =

W 1−γ
, with γ = 2
1−γ

How likely are each of these functions to represent actual investor preferences? Why?
2) Suppose investors have preference described by the following utility function with A > 0:
U = E(r) − 1 Aσ 2
2
Each investor has to choose between three portfolios with the following characteristics:

E(rA ) = 20%

σA = 20%

E(rB ) = 12%

σB = 22%

E(rC ) = 15%

σC = 28%

(a) Which portfolio would every investor pick and why?
(b) What utility would an investor with a risk aversion parameter, A, of 1 get from the three portfolios?
(c) What must be the risk aversion of an investor that is indifferent between picking portfolio B and portfolio C?

1

3) Consider an investment universe consisting of three assets with the following characteristics: E(r1 ) = 12%
E(r2 ) = 17%
E(r3 ) = 7%

σ1 = 25%

ρ1,2 = 0.5

σ2 = 30%

ρ1,3 = 0.25

ρ2,3 = 0.35

σ3 = 20%

(a) What is the expected return and standard deviation of an equally weighted portfolio investing in all three assets?
(b) What would the diversification benefit be for an investor that shifted her investment to the equally weighted portfolio from an investment consisting only of asset 1?
(c) If choosing between investing all her capital in asset 2 or in the equally weighted portfolio, what would an investor with a risk aversion parameter, A, of 3 choose?
(d) What about an investor with a risk aversion parameter, A, of 1?
(e) What is the covariance between the return on the equally weighted portfolio investing in all three assets and the return of an equally weighted portfolio investing only in assets 1 and 3?
4) Consider investors with preferences represented by the utility function U =
E(r) − 1 Aσ 2 .
2
(a) Draw the indifference curve representing a utility level