Inferring Mean-Variance Risk Aversion from Portfolio Weights

Summary

The note explains how to infer an investor’s risk-aversion coefficient for a portfolio on an efficient frontier, provided the investor follows mean-variance preferences. It starts with utility equal to expected portfolio return minus a risk penalty proportional to portfolio variance. Applying the first-order condition for optimal weights and multiplying by those weights gives an implied coefficient equal to expected portfolio return divided by portfolio variance.

This provides a way to label frontier portfolios with a preference parameter after an optimization, but the result depends on the assumed utility model. The question concerns portfolios generated by a multi-objective evolutionary algorithm; the answer does not establish that every such algorithm’s output corresponds to mean-variance utility maximization. The formula is therefore an interpretation under a specific preference assumption, not a general method for recovering an investor’s preferences from any efficient frontier.

Key ideas

• Under mean-variance utility, risk aversion penalizes portfolio variance relative to expected return.
• The implied coefficient for a portfolio is its expected return divided by its variance.
• The derivation assumes the portfolio satisfies the mean-variance first-order condition.
• Portfolios from other optimization methods may not map directly to this preference parameter.

Tags

# Is it possible to derive the "risk tolerance" from the portfolio efficient frontier?


# Is it possible to derive the "risk tolerance" from the portfolio efficient frontier?












I am trying to solve the Portfolio Optimization Problem using a "Multi-objective Evolutionary Algorithm". After obtaining the efficient frontier, I would like to know if we can infer for each point of the efficient frontier the corresponding risk tolerance parameter (which is related to the investor's preference). Thanks.

## Answer by John (score 4, accepted)

https://quant.stackexchange.com/a/4527

Strictly speaking the risk aversion coefficient depends on the form of investor preferences. Your "multi-objective evolutionary algorithm" may or may not be easy to place in this format. However, it becomes easy if you think about the risk aversion coefficient in mean/variance space if you were a mean-variance variance investor.

In this case you would have utility $$U\left(w\right)\equiv w'\mu-\frac{1}{2}\lambda w'\Sigma w $$ with the first order condition $$\mu-\lambda\Sigma w=0$$ multiplying both sides by $w'$ and solving for $\lambda$ gives $$\lambda=\frac{w'\mu}{w'\Sigma w}=\frac{\mu_{p}}{\sigma_{p}^{2}} $$ or that the implied risk aversion coefficient given portfolio holdings if you were a mean-variance investor is the ratio between the portfolio mean to portfolio variance.

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