Question

Question 10. You want to figure out the risk aversion for your client. After showing him a number of investment alternatives you find that the two alternatives below, A and B, give your client the same Utility. Using the standard definition for Utility: a) What is your clients risk aversion? b) What is your client’s utility for these two investments?

Stdev A: 18%

Stdev B: 15%

E(r) A: 10%

E(r) B: 7%

Answer 1

Answer:-

U = E(r) – 0,5 x A x sigma squared
U = Utility
A = risk aversion coefficient
sigma square = square of volatility
Standard deviation is a measure of volatility
Given
Both alternatives A and B have same utility
U (A) = U (B)

Stdev A: 18% = 0.18
Stdev B: 15% = 0.15
E(r) A: 10%
E(r) B: 7%
U(A) = 10 % - 0.5 x A x (0.18)²
U (B) = 7 % - 0.5 x A x (0.15)²
U (A) = U (B)

10 % - 0.5 x A x (0.18)²  = 7 % - 0.5 x A x (0.15)²

3 % = 0.5 x A x (0.18)² - 0.5 x A x (0.15)²
3 % = 0.0162 A - 0.01125 A
3 % = 0.00495 A
0.03 = 0.00495 A
A = 0.03 / 0.00495 = 6.06

a) Therefore the clients risk aversion A = 6.06

For A
U (A) = 10 % - 0.5 x A x (0.18)²
U (A) = 0.10 - 0.5 x 6.06 x (0.18)²
U (A) = 0.10 - 0.09817
U (A) = 0.00183
U(A) = 0.183 %

For B
U (B) = 7 % - 0.5 x A x (0.15)²
U (B) = .07 - 0.5 x A x (0.15)²
U (B) = 0.07 - 0.5 x 6.06 x (0.15)²  
U (B) = 0.07 - 0.06817
U (B) = 0.00183
U (B) = 0.183 %