Why Portfolio Utility Uses Half the Variance Penalty

Summary

The document explains the factor of one half in a mean-variance portfolio objective that subtracts a risk penalty proportional to portfolio variance from expected return. The factor is a mathematical convenience: when differentiating the quadratic variance term with respect to portfolio weights, the factor of two from the derivative cancels the one-half multiplier. The resulting first-order condition is simpler to write.

Multiplying the variance penalty by one-half does not change the optimization structure when the coefficient on risk is interpreted consistently. The discussion addresses the notation in a utility expression using expected returns, a covariance matrix, portfolio weights, and a risk-aversion parameter. It does not derive the efficient frontier or discuss constraints, estimation error, or how to select the risk-aversion coefficient; its scope is limited to explaining the normalization in the objective.

Key ideas

• A one-half multiplier on portfolio variance is used for algebraic convenience.
• Differentiating a quadratic variance term produces a factor of two that cancels the multiplier.
• The objective expresses a tradeoff between expected return and a variance-based risk penalty.
• The explanation does not address portfolio constraints or choosing the risk-aversion parameter.

Tags

# Efficient Frontier Derivation: why minimize half the portfolio variance instead of just the variance?


# Efficient Frontier Derivation: why minimize half the portfolio variance instead of just the variance?












In Robert Merton's derivation of the efficient frontier of a portfolio, he minimizes $\frac{1}{2}\sigma^2 $ over the investment weights in each asset, where $\sigma^2$ represents portfolio variance. I am confused why the function he minimizes is half the variance, instead of just the variance. It doesn't make a difference in calculations, but I cannot figure out why he (and all other derivations) do this.

## Answer by Bob Jansen (score 2)

https://quant.stackexchange.com/a/14913

I assume you're talking about this formula:

$$U(w) = w'\mu - \frac{1}{2} \lambda w' \Sigma w = w'\mu - \frac{1}{2} \lambda \sigma_\omega^2$$

where $\sigma_\omega^2$ denotes the portfolio variance for a portfolio with weights $\omega$.

Dividing by two is purely done for convenience, optimizing this formula requires taking the derivative with respect to $\omega$ and setting it to $0$. When the derivative is taken the factor $\frac{1}{2}$ is canceled by the square.

See this question on more information on setting $\lambda$.

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