Expected Utility: Optimal Asset Investment

amount invested q

10

return

r

1

-0.8

return

r

2

0.8

r

1

's probability p

0.3

An investor begins with 10 units of wealth that can be invested in a risky asset, or maintained as cash with no return. Assume the risky asset yields a rate of return of

r

1

or

r

2

with probabilities

p

and

1-p

, respectively, and let

q

be the number of units of wealth that the investor decides to invest in the asset. The value of the investor's portfolio at the end of the period will be

10-q+(1+r)q

. Let the two possible end-of-period values of the portfolio be

x

1

and

x

2

, shown above along the horizontal axis. The Bernoulli logarithmic utility function of wealth (constant relative risk aversion, CRRA) is plotted in blue. The orange line is a plot of the expected value and the corresponding expected utility of the portfolio for different values of

p

. The optimal portfolio is the one for which the expected utility is a maximum, as shown by the green dot.