Question

Show that the optimal capital allocation decision of an investor with a utility function of the form
Where ? is the proportion of wealth allocated to the optimal risky portfolio and 1 − ? is the optimal

U = E(η- Ασή Ε(rp) - Yr y* =- Ασβ

proportion of wealth allocated to the risk free asset.

Answer 1

Let the complete portfolio have y proportion in optimal risky portfolio and 1-y proportion in risk free asset

As risk free asset have zero volatility, standard deviation of complete portfolio=proportion in risky portfolio*standard deviation of risky portfolio=y*s

Expected return of complete portfolio=proportion in risky portfolio*returns of risky portfolio+proportion in risk free asset*returns of risk free asset=y*rp+(1-y)*rf

U=Expected returns-0.5*A*standard deviation(or volatility of the portfolio)^2=y*rp+(1-y)*rf-0.5*A*(y*s)^2

Optimal allocation will be when utility is maximized

Differentiating utility w.r.t. y, we get

dU/dy=r-rf-0.5*A*s^2*2*y

Setting dU/dy=0, we get y=(rp-rf)/(A*s^2)

To confirm that utility has been maximized, we get d2U/dy2=-0.5*A*s^2*2

As A is positive, d2U/dy2 is negative hence Utility is maximized