Question

You manage a mutual fund with 17% expected return and 27% volatility. The risk-free rate is 7%. Assume a new client’s utility function is U = E(r) − 1 2Aσ2 .

a. What is his optimal allocation y, if his risk aversion, A, is 2, 5, or 10? How does the optimal allocation change with A? Explain the intuition.

b. What happens to his optimal allocation if the expected return on your fund goes up to 20% (for A = 2 only; other assumptions stay the same)? Explain the intuition for the change.

c. What happens to his optimal allocation if the T-bill rate drops to 5% (for A = 2 only; other assumptions stay the same)? Explain the intuition for the change.

d.What happens to his optimal allocation if the return standard deviation of your fund goes up to 35% (for A = 2 only; other assumptions stay the same)? Explain the intuition for the change

Answer 1

Solution for a)

The formula for finding out the optimal allocation, y, is

y = (E(R) -R_f)/Aσ², where E(R) is the expected return on the risky asset, R_f is the risk-free return, A is the risk aversion of the investor and σ² is the variance of the risky asset.

Hence, if the investor's risk aversion, A=2, the optimal weightage of the risky asset should be (0.17-0.07)/[2*(0.27²)] = 0.6859 or 68.59%. Hence, the allocation towards the risk-free asset will be 31.41%

if the investor's risk aversion, A=5, the optimal weightage of the risky asset should be (0.17-0.07)/[5*(0.27²)] = 27.44%. Hence, the allocation towards the risk-free asset will be 72.56%

if the investor's risk aversion, A=10, the optimal weightage of the risky asset should be (0.17-0.07)/[10*(0.27²)] = 13.72%. Hence, the allocation towards the risk-free asset will be 86.28%

A is the investors risk-aversion towards the risky asset. Hence higher the risk-aversion or A, the lower will be his tolerance towards risky assets and hence, lower will be his allocation towards such assets. If A is negative it means that the investor is a risk taker and loves taking on risk. If A=0 he is risk-neutral.

Solution for B

Expected return is now 20% and A=2. The optimal portfolio now shifts. Plug the new values in the aforementioned formula

the optimal weightage of the risky asset should be (0.20-0.07)/[2*(0.27²)] = 0.8916 or 89.16%. His allocation towards risk-free asset will be only 1-0.8916= 0.1084 or 10.84%

The allocation to the risky asset increased from 68.59% to 89.16% as the expected return of the mutual fund jumped from 17% to 20%. Since the investors risk-aversion or A was relatively low at 2, he is willing to take on more risk, and hence, and increase in expected return of the mutual fund caused his allocation towards the mutual fund to jump by quite a bit.

Solution for C

The T-Bill rate has dropped from 7% to 5%. Use the formula to calculate new allocation. the optimal weightage of the risky asset is (0.17-0.05)/[2*(0.27²)] = .8230 or 82.3%. The allocation towards the risk-free asset fell from 31.4% to (1-0.8230) or 17.70% after the T-Bill rate has dropped from 7% to 5%.

Again, since the investors risk-averseness or A is low at 2, he would shift more of his portfolio towards the mutual fund, because the risk-free asset is not yielding as much. If A was higher, the allocation shift from the T-Bill to the mutual fund would have been lower.

Solution for D

Now, the standard deviation of the mutual fund has increased from 27% to 35%. The new allocation towards the mutual fund would be (0.17-0.07)/[2*(0.35²)] = 0.4082 or 40.82%. The allocation towards the risk-free asset, hence, is as high as 59.18%

The allocation towards the mutual fund fell from 68.59% to 40.82%, as the standard deviation rose from 27% to 35%. Since A=2, the investor is still slightly risk averse. Hence, an increase in volatility in the mutual fund will cause his allocation to the mutual fund to fall.

If A was negative, the allocation to mutual fund would have actually increased because the investor is a risk-taker