Risky investment

An individual with wealth w is deciding how much to invest in the stock market.

Denote the investment by z. With probability the price of this stock will go up by

r X 100%, and with probability (1) the price of this stock will go down by r X 100%.

The current price of the stock is $1.

(a) What is her total wealth when she makes a profit on her investment? What is

her total wealth when she makes a loss on her investment?

Her Bernoulli utility function is given by

u(x) = x^a , a (0, 1)

(b) What is her expected utility from the investment level z? [Hint: Each lottery in

this case is denoted by a different level of z because nothing else varies across lotteries.

So we can think of the expected utility as a function of z.]

(c) What is her expected utility maximizing level of investment in terms of , w

and r ?

(d) If = 0.5, how much does she invest? What if = 1?

(e) Suppose the parameter values are = 0.5, = 4/5 and r = 1. How much does

she invest?

(f) What is the certainty equivalent and risk premium for the lottery corresponding

to z ? What is the risk premium?

(g) What happens to the risk premium as a-> 1 ? Provide an intuition for this

result?