Hints for solving problem set #4 (Hint #1)

Problem set #4 consists of two problems: 1) an optimal (expected utility maximizing) portfolio problem, and 2) a stochastic dominance problem.  We’ll discuss stochastic dominance tomorrow (and also (hopefully) work a class problem in connection with that concept), but in the meantime allow me to provide you with some hints for setting up  the first problem.

The first problem involves determining how to (optimally) allocate  initial wealth W0 = $100 to (risky) stock and (safe) bond investments for two investors who are identical in all respects except utility. Let \alpha represent the allocation to stock; then the plan is to invest $100\alpha in the stock and $100(1-\alpha) in the bond. The key here is to find the value for \alpha which maximizes expected utility. The problem is based on the following facts:

  • U(W) = W.5; for Investor A and U(W) = ln W for Investor B;
  • W0 = $100 for both investors;
  • 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
  • eEnd-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 for either investor, 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 = \alphaW0(1.3) + (1-\alpha)W0(1.05) = \alpha100(1.30) + (1-\alpha)100(1.05) = \alpha130 + (1-\alpha)105 = 105 + 25\alpha.
  • 40% of the time, Ws = \alphaW0(.7) + (1-\alpha)W0(1.05) = \alpha100(.7) + (1-\alpha)100(1.05) = \alpha70 + (1-\alpha)105 = 105 – 35\alpha.

Therefore, expected utility for Investor A is: E(U(W)) = .6(105 + 25\alpha).5 + .4(105 – 35\alpha).5, and expected utility for Investor B is E(U(W)) = .6ln (105 + 25\alpha) + .4ln(105 – 35\alpha). It is up to you to solve for the optimal value of \alpha for each investor.  There are two ways to do this – via calculus or a spreadsheet model.  Actually, I would encourage y’all to work this problem both ways if time permits because doing so will help you develop an even better grasp of the underlying principles and concepts in Finance 4335.

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