Hints for solving problem set #4

Problem set #4 (which is due on Thursday, February 13) consists of two problems: 1) an optimal (expected utility-maximizing) portfolio problem, and 2) a stochastic dominance problem. We’ll discuss stochastic dominance on Tuesday, but in the meantime here are some hints for setting up the first problem.

The first problem involves determining how to (optimally) allocate initial wealth W0 = $1,000 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 $1,000\alpha in the stock and $1,000(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 = $1,000 for both investors;
  • Current bond and stock prices are B0 and S0 respectively;
  • End-of-period bond price is B1 = B0(1.03) 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 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 in value by 30%, and a 100% chance that the bond increases in value by 3%, this implies the following:

  • 60% of the time, Ws = \alphaW0(1.3) + (1-\alpha)W0(1.03) = \alpha1,000(1.30) + (1-\alpha)1,000(1.03) = \alpha1,300 + (1-\alpha)1,030 = 1,030 + 270\alpha.
  • 40% of the time, Ws = \alphaW0(.7) + (1-\alpha)W0(1.03) = \alpha1,000(.7) + (1-\alpha)1,000(1.03) = \alpha700 + (1-\alpha)1,030 = 1,030 – 330\alpha.

Therefore, expected utility for Investor A is: E(U(W)) = .6(1,030 + 270\alpha).5 + .4(1,030 – 330\alpha).5, and expected utility for Investor B is E(U(W)) = .6ln (1,030 + 270\alpha) + .4ln(1,030 – 330\alpha). There are two ways to solve for the optimal value of \alpha for each investor – via calculus or a spreadsheet model; either approach suffices.  If you decide to provide a spreadsheet-based instead of a calculus-based solution,  then in addition to turning in a completed problem set, also email your spreadsheet model to fin4335@gmail.com.

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