Optimal Weight of Risky Portfolio

Question

"Suppose that the investor has a quadratic utility function. That is,

$$U \left[ W \right] = W - \frac{1}{250}W^2.$$

Assume the investor is maximizing its expected utility and is considering in investing $100 either in the risk-free asset that yields 3% per year or to a risky asset that yields 10% per year with probability 0.5 and -2% with probability 0.5

What is the optimal weight on the risky portfolio?"

Answer 1

Let y be the % invested in risky portfolio. \begin{eqnarray*} E(W) &=& p_1W_1 + p_2W_2\\ &=&0.5(y\cdot1.1\cdot100 + (1-y)\cdot1.03\cdot100) + 0.5(y\cdot0.98\cdot100 + (1-y)\cdot1.03\cdot100)\\ &=&103+y \end{eqnarray*}

\begin{eqnarray*} E(W^2) &=& p_1W_1^2 + p_2W_2^2\\ &=&0.5(y\cdot1.1\cdot100 + (1-y)\cdot1.03\cdot100)^2 + 0.5(y\cdot0.98\cdot100 + (1-y)\cdot1.03\cdot100)^2\\ &=&0.5[(103+7y)^2 + (103-5y)^2] \end{eqnarray*}

\begin{eqnarray*} E(U[W]) &=& E(W) - \frac{1}{250}\ E(W^2)\\ &=&103+y-\frac{1}{500}\ [(103+7y)^2 + (103-5y)^2]\\ \frac{dEU}{dy}\ &=& 0\\ .\\ .\\ .\\ y &=& \frac{22}{37}\ \end{eqnarray*}