Why Mean-Variance Utility Selects Efficient Portfolios
Summary
The document explains why maximizing mean-variance utility, expressed as expected return minus a risk-aversion-weighted variance term, selects an efficient portfolio. The efficient frontier consists of portfolios that offer the highest return at each fixed level of risk. Since variance is a one-to-one transformation of nonnegative volatility, the same efficiency criterion applies when risk is measured by variance.
The argument is that a portfolio below the frontier could be replaced by another with the same risk and higher return, which would also raise the utility objective for a positive risk-aversion parameter. The explanation is conceptual and gives no numerical example. It assumes valid portfolios and a positive risk penalty; it does not discuss constraints, estimation error, or cases where the optimizer may fail to attain a maximum.
Key ideas
- The efficient frontier contains portfolios with the highest return at each risk level.
- Using variance instead of volatility preserves the ordering of risk levels.
- A dominated portfolio cannot maximize mean-variance utility when risk aversion is positive.
- The argument assumes a valid portfolio set and an attained optimum.
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Full text
# Prove that the portfolio that maximizes utility lies on the efficient frontier
# Prove that the portfolio that maximizes utility lies on the efficient frontier
When maximizing mean-variance utility in a portfolio optimization framework
$max \{R - \lambda \sigma ^2\}$
where R is portfolio return, $\lambda$ is a risk aversion parameter, and $\sigma^2$ is portfolio volatility, how can I be sure that the result lies on the efficient frontier? I can show that $\lambda$ is effectively equals $\frac{R}{\sigma^2}$ but I don't quite see how this problem develops the efficient frontier
## Answer by Attack68 (score 4, accepted)
https://quant.stackexchange.com/a/49399
The efficient frontier is defined as the set of portfolios which have the highest return for a given measure of volatility, i.e. $\{S: s \in P \; s.t. \nexists \; t \in P \; \text{where} \;R(s) < R(t) \; \text{and} \; \sigma(s)=\sigma(t) \}$, where $P$ is the set of all validly constructed portfolios.
Therefore this also holds for the efficient frontier when the risk is squared, i.e. a one-to-one mapping for risk to variance.
The optimisation framework $max \{R-\lambda\sigma^2 \}$ must therefore return the efficient frontier since by definition there does not exist a valid portfolio for a given risk or variance where $R$ is greater and therefore increases the objective function above that of the efficient frontier.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.