Question

Suppose an investor has exponential utility function U(x) = -e^(-a*x) and an inital wealth of W....

Suppose an investor has exponential utility function U(x) = -e^(-a*x) and an inital wealth of W. The investor is faced with an opportunity to invest an amount w <= W and obtain a random payoff x. Show that his evalution of this incremental investment is independent of W.

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Answer #1

Here for the given utility function we can see that the payoff "x" is in exponent.

Now if we want to calculate the returns on the increamental investment, say we increase the investment amount marginally by say p, then corresponding marginal utility which is dervived from this investment is dependent completely on x which is a random payoff generated and hence is independent of W. Whatever be the amount invested whether it be w < W or w = W the return is x only and hence the marginal increment in return or utility also does not depend on x and is = a*e^(-a*x) which is positive but decreasing as well and is independent of W.

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