Problem set #4 involves determining how to (optimally) allocate your initial wealth *W*_{0 }= $100 to (risky) stock and (safe) bond investments. Let *α* represent the allocation to stock; then the plan is to invest $100*α* 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};*W*_{0 }= $100;- current bond and stock prices are
*B*_{0 }and*S*_{0}respectively; - end-of-period bond price is
*B*_{1 }=*B*_{0}(1.05) with probability 1.0; and - end-of-period stock price is
*S*_{1}=*S*_{0}(1.3) with probability .6 and*S*_{1}=*S*_{0}(.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 }=*α**W*_{0}(1.3) + (1-*α*)*W*_{0}(1.05) =*α*100(1.30) + (1-*α*)100(1.05) =*α*130 + (1-*α*)105 = 105 + 25*α*. - 40% of the time,
*W*_{s }=*α**W*_{0}(.7) + (1-*α*)*W*_{0}(1.05) =*α*100(.7) + (1-*α*)100(1.05)] =*α*70 + (1-*α*)105 = 105 – 35*α*.

Therefore, expected utility is: *E*(*U*(*W*)) = .6(105 + 25*α*)^{.5} + .4(105 – 35*α*)^{.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*.