Question

Questions 1 – 4 refer to prospects X and Y below, as well as the following information ?? = ($0, 0.50; $50, 0.40; $100, 0.10) ?? = ($0, 0.5; $30, 0.20; $80, 0.30) Mark has utility of wealth given by ??(??) = ??0.4 1. What is the expected value of prospect X (????(??))? 2. What is the standard deviation of prospect X (????(??))? (Round your answer to the nearest cent, and don’t worry, I’ll include a healthy margin of error so you won’t get this wrong due to rounding). 3. What is the value of the expected utility of Y for Mark (EU(Y))? 4. How much is Mark willing to spend to acquire Y (so, what is the value of ????(??))? (Hint, if ??(??) = ??0.4,??ℎ???? ??−1(??) = ??2.5).

Answer 1

1. The Expected Value of the prospect X is the weighted average of the return the weights being the probability.

E(X) = 0*0.5 + 50*0.4 + 100*0.1

= 0 + 20 +10 = 30

Hence Expected Value of prospect X is 30

2. The standard deviation of the prospect X is the squared root of the weighted average of the squared difference of the return from the expected value. The weights being the probabilities.

Variance(X) = 0.5*(0-30)^2 + 0.4*(50-30)^2 + 0.1*(100-30)^2

= 450 + 160 + 490

= 1100

So SD(X) = (1100)^1/2 = 33.16624 ~ 33

3. The Expected Utility of Y is the weighted average of the utility received from the return of the prospect.

EU(Y) = 0.5*0.4*(0) + 0.2*0.4*(30) + 0.3*0.4*(80)

= 0 + 2.4 + 9.6

= 12 is the Expected Utility of the prospect Y

4. The amount of money spent to acquire the prospect Y should depend on the expected utility from the prospect. The expected utility from the prospect Y is 12. Hence Mark will spend that much amount of money where his utility is equal to the expected utility.

So. 0.4Y = 12

Y = 30