Question

7. Consider an investor who has initial wealth w and has to decide how to invest it. There is a riskless asset with rate of return r. The risky asset has return xi with probabilityπi, i = 1, . . . , n. Denote by α the fraction of wealth that the investor puts into the risky asset. Write the investor’s problem. Show that if the investor has constant relative risk aversion, then the fraction of wealth invested in the risky asset α, does not change with w(that is, dα*/dw = 0, where α∗ denotes the solution to the investor’s problem).

Answer 1

The investor's problem is,

if the investor has constant risk averse, then

the first order condition for this problem is,

Here,

Now we find that , that is,

Suppose the investor is risk averse then,

If is some constant then,

But it must zero when,

using the first order condition,then