How Wealth and Risk Aversion Affect Quadratic-Utility Portfolio Weights

Summary

The document frames portfolio selection as maximizing expected utility over asset weights subject to a fully invested constraint. It presents an expected-utility expression for quadratic utility in terms of initial wealth, portfolio mean, and portfolio variance, then asks why the resulting weights vary with starting wealth and whether higher risk aversion should lead toward the minimum-variance portfolio.

This is a conceptual question rather than a resolved analysis: it supplies no derivation of optimal weights, numerical example, or answer explaining the wealth dependence. The displayed objective includes both variance and squared portfolio mean, so conclusions about the effect of risk aversion depend on the precise utility specification and assumptions. The text is useful for identifying what must be examined in a utility-based allocation model, but it does not establish the expected limiting behavior or validate an implementation.

Key ideas

• The allocation problem maximizes expected utility subject to weights summing to one.
• The stated quadratic-utility expression depends on initial wealth as well as portfolio mean and variance.
• The author observes that computed weights change with initial wealth and questions this behavior.
• The document does not provide a derivation or answer establishing a minimum-variance limit as risk aversion rises.

Tags

# Portfolio Theory - Maximizing Expected Utility Function


# Portfolio Theory - Maximizing Expected Utility Function












I am trying to implement a portfolio selection tool based on utility functions. So, I should maximize the expected utility of a given utility function: $$ \begin{align} &\max_{w}\ E[u(W_0(1+w^TR))]\\\\ & s.t.\\ &w^Te = 1, \ e^T=[1,...,1] \end{align} $$ Using the quadratic utility function we get: \begin{align} &E[u(W_0(1+w^TR))] = u(W_0) + W_0\mu_p(1-bW_0)-\frac{b}{2}W_0^2(\sigma_p^2 + \mu_p^2)\\\\ &\mu_p \ \text{is the portfolio mean and} \ \sigma_p \ \text{is the portfolio standard deviation} \end{align}

I already implemented this, but I get different weights for different initial wealth values, is this normal? One thing that I find odd is that for a large enough initial wealth, the model doesn't tend to the minimum variance portfolio as risk aversion goes up.

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