Question

If Adam's utility function is U = W^0.5, and he invests in a business that can yield $6000 with probability 2/5, and $4000 with probability 3/5, then his risk premium to avoid bearing this risk is

Select one:

a. $4800.

b. $5200.

c. $49.

d. $72.

Answer 1

Answer

The correct answer is (c) $49.

Expected Utility(EU) is given by :

EU = P₁U(W₁) + P₂U(W₂)

where P₁ = Probability that W = W₁ = 6000. Thus P₁ = 2/5 and P₂ = Probability that W = W₂ = 4000. Thus P₂ = 3/5

Hence EU = P₁U(W₁) + P₂U(W₂) = (2/5)*U(6000) + (3/5)*U(4000)

=> EU = (2/5)*6000^0.5 + (3/5)*4000^0.5

Thus, If he invests, His expected wealth = P₁W₁ + P₂W₂ = (2/5)*6000 + (3/5)*4000 = 4800.

Amount he wants to have with certainty to avoid this investment risk is that amount such that Utility from that amount is equal to expected utility if he invest(EU).

As EU = (2/5)*6000^0.5 + (3/5)*4000^0.5 and Let X be the wealth he wants to have with certainty.

Hence We have :

X^0.5 = (2/5)*6000^0.5 + (3/5)*4000^0.5

=> X = ((2/5)*6000^0.5 + (3/5)*4000^0.5)²

=> X = 4751(approx)

Risk premium = Expected wealth from Gamble - Amount he wants to have with certainty

Here, Expected wealth from Gamble = 800 and Amount he wants to have with certainty(Certainty equivalent) = 4751

=> Risk premium = 4800 - 4751 = 49

Thus, risk premium = 49

Hence, the correct answer is (c) $49.