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 (Ws). 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, Ws = 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, Ws = 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.