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Why Mean-Variance Indifference Implies Quadratic Utility

Article Quant Q&A · Author: Alkibiades
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Summary

The document raises a result from expected-utility theory: if an agent ranks contingent claims only by their mean and variance, then utility must be quadratic, and asks for a mathematical explanation. It does not provide a derivation, proof, or examples; it is a question rather than a complete instructional treatment.

The topic connects utility functions to mean-variance preferences, but the claim needs careful assumptions about the set of distributions and the domain of utility. In particular, concavity alone does not establish the result, and unrestricted quadratic utility is not globally increasing and concave. The document offers no evidence or discussion of these limits, so its value is primarily as a pointer to a theoretical question rather than a demonstrated method.

Key ideas

  • The document asks why mean and variance would be sufficient to determine preferences over contingent claims.
  • It states that indifference among claims with matching means and variances would imply quadratic utility.
  • No mathematical proof or supporting examples are included.
  • The claim depends on assumptions about the available distributions and utility domain.

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# Quadratic utility function


# Quadratic utility function












May you can help me undertanding the following conclusion: Suppose we have an agent who has preferences over contingent claims, represented by a concave function $U$. This simply means that $\mathbb{E}U(X)\le\mathbb{E}U(Y)$ where $X,Y$ represent two claims. Now suppose that if two contingent claims have the same mean and the same variance they are indifferent for the agent.

If we consider distributions concentrated on three points than the function $U$ must be quadratic.

I do not understand the last part, may you could show me that mathematically?

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