a. A utility function is given as U=3W, W>=0. At the wealth level of $1500, a person is faced with a game that she has a probability of 0.5 to win $200 and a probability of 0.5 to lose $200. Find out the utility of expected wealth and the expected utility.
b. Is he risk averse, risk netral, or risk loving ?
c. Calculate the risk premium of the game (How much does he/she is willing to pay to avoid the game ?
- Anonymous1 decade agoFavorite Answer
W1=wealth if gain $200
W2=wealth if lose $200
U1=utility if gain $200
U2=utility if lose $200
U=utility of expected wealth
p1=probability of gaining $200
p2=probability of losing $200
W(e)=p1*W1+p2*W2 = 0.5*$1700+0.5*$1300 = $1500
U=3*W(e)=3*1500=4500 (Utility of expected wealth)
U(e)=p1*U1+p2*U2=0.5*5100+0.5*3900=4500 (Expected Utility)
The utility of the game is same as utility of his original weatlh. So he is risk neutral.
(c) Risk premium = W(e) – W0 = 1500 – 1500 = 0
Since he is risk neutral, there is no need for him to pay anything to avoid the game.
不過以前學的utility function不如這一題這樣簡單用linear function (U=3W),以前學的是quadratic 或 natural log function.
U = r(p) – CE
U= r(p)– 0.005Aσ^2
[r(p) = return of portfolio, CE = certainty equivalent, U=expected utility]