Suppose an individual has $100 to invest. Two assets are available. One asset will yield a return of 10%, while the other risky asset will yield 0% with probability .5 and 21% with probability .5
Suppose the investor's utility function is given by U(x) = ln(x) where x is the wealth after investing (assume she is investing for just one period). How much will she invest in the risky asset?
Her EU as a function of the amount of investment in the risky asset (X) is:
EU(inv) = .5 ln((100 - X)1.1 + X) + .5 ln((100 - X)1.1 + 1.21X)
Simplify this to
EU = 0.5 ln(110 - 0.1X) + 0.5 ln(110 + 0.11X)
Take the derivative and set = 0:
-0.05/(110 - 0.1X) + 0.055/(110 + 0.11X) = 0
Solve for X = 50.
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