2. Consider a risky portfolio that offers an expected rate of return of 12% and

Question

2. Consider a risky portfolio that offers an expected rate of return of 12% and a standard deviation of 20%. T-bills offer a risk-free 7% return. (1) What is the certainty equivalent rate of return of the risky portfolio? What is the risk premium of the risky portfolio? (2) What is the reward-to-volatility ratio (Sharpe ratio) of the risky portfolio? What's the meaning of the Sharpe ratio? (3) Suppose a security investor's utility function is U = E(r)-12×A× a. If A-2, would the investor choose the risky portfolio or the T-bills? b. If A-4, would the investor choose the risky portfolio or the T-bills? c. What is the maximum level of risk aversion for which the risky portfolio is still preferred to bills?

Explanation / Answer

1. Certainty equivalent rate of return = expected return/(1 + Risk premium%)

Risk premium = 12-7 = 5%

Certainty equivalent rate = 12/(1+5%)= 12/1.05= 11.43

2) sharpe ratio is the excess return earned per unit volatility of the portfolio

Sharpe ratio = (Mean return portfolio- Risk free return)/ portfolio volatility

= (12-7)/20= 5/20= 0.25

3)

a. For A=2

Sigma^2= 0.2*0.2= 0.04= 4%

U = 12-(0.5*2*4)= 8

U for risk frew security = 7-(0.5*2*0)= 7

Thus investor will chose risky asset.

b. A=4

U = 12-(0.5*4*4)=4

For risk free asset

U = 7

Thus investor will chose risk free asset.

c. For max level of risk aversion

U ( Risk free) = U (risky asset)

7 = 12-(0.5*A*4)

=> A = 2.5

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