Choosing Risky-Asset Weight with Quadratic Utility

Summary

The document solves a two-asset allocation problem for an investor with quadratic utility over terminal wealth. The investor begins with $100 and chooses a share to place in a risky asset, with the remainder earning the risk-free rate. The risky asset has two equally likely returns, one positive and one negative. The solution writes terminal wealth in each outcome as a function of the risky allocation, then calculates expected wealth and expected squared wealth.

Substituting those quantities into expected utility produces a quadratic objective in the allocation. Setting its derivative to zero gives the stated optimal risky weight of 22/37. This result is specific to the supplied utility function, return outcomes, probabilities, initial wealth, and risk-free return. The worked example does not discuss constraints such as prohibiting borrowing or short selling, nor does it examine whether quadratic utility remains sensible outside this small setup.

Key ideas

• The risky allocation determines terminal wealth separately in each return outcome.
• Expected quadratic utility depends on both expected wealth and expected squared wealth.
• The optimal allocation is found by differentiating expected utility with respect to the risky weight and setting the derivative to zero.
• For the stated assumptions, the solution gives a risky weight of 22/37.
• The result depends on the specified utility and return distribution and does not address additional allocation constraints.

Tags

# Optimal Weight of Risky Portfolio


# Optimal Weight of Risky Portfolio












"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 by SMLJKNN (score 2, accepted)

https://quant.stackexchange.com/a/47411

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*}

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.