Consider an investor having the utility function U = E(r) – 0.5 A 2 . A. On a st
ID: 2801415 • Letter: C
Question
Consider an investor having the utility function U = E(r) – 0.5 A 2 .
A. On a stand-alone basis, which investment would they select if they are risk-averse with A = 3?
B. If they are risk-neutral, which investment would they pick?
C. If the investor with A=3 allocates their wealth between a risky portfolio P having expected return of 12% and standard deviation of 16% and the risk-free asset which returns 6%, what fraction (y) of their wealth will they allocate to the risky portfolio?
Investment Expected Return Standard Deviation 1 20% 38% 2 24% 34% 3 33% 28% 4 34% 27%Explanation / Answer
A. U = E(r) – 0.5 * A * 2
Investment 1: U = 0.2 - 0.5*3*0.382 = 0.2 - 0.216 = -0.016 = -1.6%
Investment 2: U = 0.24 - 0.5*3*0.342 = 0.24 - 0.1734 = 0.0666 = 6.66%
Investment 3: U = 0.33 - 0.5*3*0.282 = 0.33 - 0.1176 = 0.2124 = 21.24%
Investment 4: U = 0.34 - 0.5*3*0.272 = 0.34 - 0.1094 = 0.2306 = 23.06%
with A=3 risk aversion an investor would select Investment 4 with highest utility.
B. For Risk neutral investors A=0
So they would select investment with highest E(r) i,e. Investment 4
C. U = 0.12 - 0.5*3*0.162 = .12 - .0384 = 0.0816 = 8.16%
This Utility function is greater than risk free rate of 6%. So the investor will be better off investing his entire portfolio in risky asset to maximaise his utility.