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 per share in a bad state with probability 1/2. Assume, the interest rate is zero. (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.

Answer 1

A) total number of shares bought = 50,000/100

= 500

So in good state, Wg = 105*500 = 52,500, Pg = .5

g : good state, b: bad state

Wb= 95*500 = 47500

So,

expected value EV = Pg*Wg + Pb*Wb

= .5*52,500 + .5*47,500

= 50,000

EU = Pg*U(Wg) + Pb*U(Wb)

= .5*√52,500 + .5*√47,500

= 223.54

.

If no investment is done , then w = 50,000

U(50,000) = √50,000

= 223.606 > EU

Thus individual doesn't prefer to invest

b) absolute risk aversion coefficient = -U"(w)/U'(w)

U' = ​​​​​.5*W^{​​​​​-.5}

U" = -.25*W^{​​​​​​-1.5}

Coefficient = +.25*W^{​​​​​​-1.5}/.5*W^{​​​​​​-.5}

= .5/W

As wealth rises , this coefficient falls,

So as individual gets wealthy, individual gets less risk averse