A student asked me whether it is possible for the value of in problem set 4 to be negative. Here, represents the optimal level of exposure to the risky asset; 1- represents the optimal level of exposure to the safe bond. While it is certainly *theoretically *possible for to have a negative value, for this particular problem it turns out that > 0. The reason is positive in this case is that the 60/40 probabilities imply a positive expected return on the risky asset which exceeds the expected return on the bond. Thus, even though the stock is risky, a square root utility investor is willing to invest some of her money in the stock because, in an expected utility sense, some *positive *exposure to risk is worthwhile.

If you obtain a negative value for , this means that you must have made a math error somewhere. In the case of investor A, *E*(*U*(*W*)) = .6(105 + 25 )^{.5} + .4(105 – 35 )^{.5}, then one can find the optimal value for investor A’s by applying the chain rule individually to both the .6(105 + 25 )^{.5} and the .4(105 – 35 )^{.5} terms, setting the resulting equation equal to zero (this is the so-called “first order condition”) and solving for . Obviously the same logic applies to solving investor B’s problem; the only difference is that *E*(*U*(*W*)) = .6ln(105 + 25 ) + .4ln(105 – 35 ) for investor B.

This raises an interesting question; specifically, what would have to be different about this problem in order to obtain a negative value for ? If this problem had been originally parameterized such that the expected return on the stock was less than the expected return on the bond, this would guarantee a negative value for . For example, suppose that everything stayed the same, but that the state probabilities for the stock were 55/45 rather than 60/40. Note that with 55/45 state probabilities, the stock has an expected return of .55(.3) + .45 (-.3) = 3%, which is less than the guaranteed 5% return on the bond. Under this scenario, investor A’s optimal is -48.33%, which implies that she would optimally sell short $48.33 of stock and invest her initial wealth of $100 plus the $48.33 in proceeds from the short sale in the bond. From date 0 to date 1, she would earn 5% or $7.42 on her $148.33 bond investment. At date 1, she would close out her short position by buying the stock back at either $48.33 x (1.30) = $62.83 (in which case she would lose $7.08 on her $100 net investment) or at $48.33 x (.7) = $33.83 (in which case she would gain $21.92 on her $100 net investment). Thus the expected return on her portfolio is .55(-7.08%) + .45(21.92%) = 5.97%, and the standard deviation is 14.43% (short selling is risky because you might get stuck having to close out the short position at a high price; this is why the standard deviation is so high). On the other hand, investor B’s optimal is -24% when the state probabilities for the stock are 55/45 rather than 60/40; I will leave it as an exercise for the reader to determine the expected return and risk for investor B’s optimal portfolio under this alternative scenario.

As I noted in my previous Problem Set #4 hint, if you get stuck on the math at all, you might consider inputting the data into an Excel spreadsheet and use Solver to find the optimal value for . If you do this, be sure to email your spreadsheet model to risk@garven.com sometime prior to the beginning of next Thursday’s class meeting, at which time I’ll also collect your completed problem sets.