A student asked me whether it is possible for the value of α in problem set 4 to be negative. Here, α represents the optimal level of exposure to the risky asset; 1-α represents the optimal level of exposure to the safe bond. While it is certainly theoretically possible for α to have a negative value, for this particular problem it turns out that α > 0. The reason α is positive in this case is because 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 α, this means that you must have made a math error somewhere. Since E(U(W)) = .6(105 + 25α )^.5 + .4(105 – 35α )^.5, then one can find the optimal value for a by applying the chain rule individually to both the .6(105 + 25α )^.5 and the .4(105 – 35α )^.5 terms, setting the resulting equation equal to zero (this is the so-called “first order condition”) and solving for α .

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 α . For what it’s worth, I just did this a few moments ago and confirmed that the (positive) α value which I obtained using the calculus is identical to the α value indicated by Excel (obviously I was not surprised, since in both cases I knew a priori that my calculus and Excel code were correct :-)).

This raises an interesting question; specifically, what would have to be different about this problem in order to obtain a negative value for α ? If this problem were reparameterized such that the risk/return characteristics of the stock vis-a-vis the bond were sufficiently worsened, and/or if the investor was inclined to act in a more risk averse fashion (e.g., if initial wealth declined and/or the investor’s utility function were different), then a negative value for α is a possibility. For example, suppose that everything stayed the same, but that the state probabilities for the stock were 55/45 rather than 60/40. If this occurred, then you would find that the investor’s optimal α is -48.33%. Note that with 55/45 state probabilities, the stock has an expected return of .55(.3) + .45 (-.3) = 3% and a standard deviation of 14.92% (check this calculation for yourself). If short selling were prohibited, then this investor would optimally invest all of her initial wealth in the bond. However, if short selling were allowed, then at date 0, she would 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).

I have posted my spreadsheet for this problem at http://risk.garven.com/wp-content/uploads/2017/09/assetallocation.xls. You are welcome to use this spreadsheet if you are interested in numerically validating your calculus-based solution procedure.

Problem set #4 involves determining how to (optimally) allocate your initial wealth W₀ = $100 to (risky) stock and (safe) bond investments. Let α represent the allocation to stock; then the plan is to invest $100α in the stock and $100(1-α) in the bond. The key here is to find the value for a which maximizes expected utility. The problem is based on the following facts:

• U(W) = W^.5;
• W₀ = $100;
• current bond and stock prices are B₀ and S₀ respectively;
• end-of-period bond price is B₁ = B₀(1.05) with probability 1.0; and
• end-of-period stock price is S₁ = S₀(1.3) with probability .6 and S₁ = S₀(.7) with probability .4.

In order to compute expected utility of wealth, you must first determine state-contingent wealth (W_s). Since there is a 60% chance that the stock increases in value by 30%, a 40% chance that the stock decreases 30%, and a 100% chance that the bond increases in value by 5%, this implies the following:

• 60% of the time, W_s = αW₀(1.3) + (1-α)W₀(1.05) = α100(1.30) + (1-α)100(1.05) = α130 + (1-α)105 = 105 + 25α.
• 40% of the time, W_s = αW₀(.7) + (1-α)W₀(1.05) = α100(.7) + (1-α)100(1.05)] = α70 + (1-α)105 = 105 – 35α.

Therefore, expected utility is: E(U(W)) = .6(105 + 25α)^.5 + .4(105 – 35α)^.5. It is up to you to solve for the optimal value of α. This requires solving the first order condition, which involves differentiating E(U(W)) with respect to α, setting the result equal to 0 and solving for α.