Question

1. Suppose that an individual has a wealth of $50,000 and the utility of U(W) = vW. This individual has the option of investing all wealth in risky stock, which is worth $100 per share, which will be worth $105 per share in a good state with probability 1/2 and $95 share in a bad state with probability 1/2. Assume, the interest rate is zero. per (a) Find the expected value and the expected utility of investing all wealth in the risky asset. Does this individual prefer investing or not investing? Explain. (b) Find the general expression of absolute risk aversion for this investor. How does it depend on wealth? Explain.

Please write clearly and a detailed answer step by step.

Answer 1

1)

a)

Probability of favorable state=p=1/2=0.5

Wealth in case of favorable state=W₁=50000-100+105=$50005

Utility in case of favorable state=U(50005)=50005^1/2=223.61797781 utils

Probability of unfavorable state=1-p=1/2=0.5

Wealth in case of unfavorable state=W₂=50000-100+95=$49995

Utility in case of unfavorable state=U(49995)=49995^1/2=223.59561713 utils

Expected value =p*W₁+(1-p)*W₂=0.5*50005+0.5*49995=$50000

Expected utility=p*U(50005)+(1-p)*U(49995)=0.5*223.61797781+0.5*223.59561713=223.60679747 utils

Let us check calculate the utility of not investing i.e. wealth is $50000,

Utility in case no investment is made=U(50000)=50000^1/2=223.60679775 utils

We can see that utility in case of not investing is higher than the expected utility of investing in risky asset. So, individual will prefer not to invest.

b)

Given U(W)=W^0.5

U'(W)=dU(W)/dW=0.5W^-0.5

U''(W)=-0.25W^-1.5

Absolute Risk aversion=-U''(W)/U'(W)=-(-0.25W^-1.5)/(0.5W^-0.5)=0.5/W

We can see that absolute risk aversion is decreasing in the given case. It means that absolute risk aversion decreases as wealth increases. It means that if wealth increases, investor keeps more dollars in risky asset.