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# investment 問題

請回答

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 ?

### 1 Answer

- Anonymous1 decade agoFavorite Answer
Let:

W0=original wealth

W1=wealth if gain $200

W2=wealth if lose $200

W(e)=expected wealth

U0=original utility

U1=utility if gain $200

U2=utility if lose $200

U=utility of expected wealth

U(e)=expected utility

p1=probability of gaining $200

p2=probability of losing $200

(a)

p1=0.5

p2=0.5

W0=$1500

W1=$1500+$200=$1700

W2=$1500-$200=$1300

W(e)=p1*W1+p2*W2 = 0.5*$1700+0.5*$1300 = $1500

U0=3*W0=3*1500=4500

U1=3*W1=3*1700=5100

U2=3*W2=3*1300=3900

Ans:

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)

(b)

U0=4500=U(e)

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.

Finance 已丟下四年多了,希望沒弄錯吧.

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=ln(W)

U = r(p) – CE

U= r(p)– 0.005Aσ^2

[r(p) = return of portfolio, CE = certainty equivalent, U=expected utility]

好像是這樣之類的...不過已忘得一乾二淨了.

以前我覺得CE,utility function是很難明的,到現在亦如此認為

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