Question

Question 7. For a mean-variance optimizer with A = 1, which of the following is most preferred? Compute the utility of each: • A certain return of 2 percent • A gamble that gives a return of 5 percent half of the time and 0 percent half of the time • A gamble that gives a return of 4 percent half of the time and 1 percent half of the time • A gamble that gives a return of 3 percent three-quarters of the time and 1 percent one-quarter of the time

Answer 1

Q-7) The formula for utility theory is equal to U = E(r ) - 1/2*A*Standard deviation^2 1) When the return is certain at 2% then Stnadard deviation would be 0, A = 1 U = 2-(1/2)*1*0 2) probability (P) Return (%) Expected return P* Return^2 0. 55 2.5 12.5 5 0. 0 0 0 Sum 2.5 12.5 Standard deviation = (12.5-2.5^2)^(1/2) 2.5 2.5-(1/2)*1*2.542 -0.625

3) Return (%) Expected return P* Return^2 probability (P) 0.5 0.5 0.5 0.5 Sum 2.5 Standard deviation = (8.5-2.5^2)^(1/2) 1.5 U = 2.5-(1/2)*1*1.542 1.375 4) probability (P) Return (%) Expected return p* Return^2 0.25 0.75 2.25 0.25 3 0.75 2.25 0. 253 0.75 2.25 0.25 1 0.25 0.25 Sum 2.5 Standard deviation = (7-2.5^2)^(1/2) 0.866 2.5-(1/2)*1*(0.866^2) 2.125

The payout will be either 0 or 200 with equal probability Expected value = (0*0.5 + 200*0.5) 100 required rate to discount the cash flows = (5 + 20) 25 % the price of the asset should be = 100/(1+0.25) 80