How Risk Aversion Selects a Portfolio Along the Capital Allocation Line

Summary

The document explains how the risk-aversion coefficient in mean-variance utility relates to the maximum-Sharpe portfolio. Its central point is that the coefficient does not determine the maximum Sharpe ratio. With a risk-free asset available, the maximum-Sharpe portfolio is the tangency portfolio, and an investor combines it with the risk-free asset according to their risk tolerance: higher risk aversion favors more risk-free investment, while lower risk aversion favors more exposure to the tangency portfolio.

The explanation is conceptual and refers to an efficient-frontier diagram, distinguishing the unconstrained frontier, the frontier without short sales, and the line combining the risk-free asset with the tangency portfolio. It does not derive an explicit formula for the optimal weights or discuss estimation error, transaction costs, or constraints beyond short selling. The interpretation therefore depends on the stated setup, especially the availability of a risk-free asset and the assumptions behind mean-variance optimization.

Key ideas

• The risk-aversion coefficient does not set the maximum Sharpe ratio.
• The tangency portfolio is the risky portfolio with the maximum Sharpe ratio.
• Risk aversion determines how an investor allocates between the tangency portfolio and the risk-free asset.
• Higher risk aversion implies a larger allocation to the risk-free asset in this setup.

Tags

# how to relate risk aversion and sharpe ratio in optimisation


# how to relate risk aversion and sharpe ratio in optimisation












I am trying to optimise the following: U(w)=w′μ−λ/2w′Σw which is the typical risk aversion problem. I would like to set lambda in order to have the max sharpe but I cannot find in literature what is the relation between them. Can anybody help?

Thanks Luigi

## Answer by phdstudent (score 3, accepted)

https://quant.stackexchange.com/a/54956

$\lambda$ is independent of the maximum sharpe ratio. The maximum sharpe ratio portfolio will give you a combination of the risk free asset and the tangency portfolio. Then your risk aversion just makes you choose the combination between these two assets. See picture below.

The blue line is the efficient frontier with short-sales allowed. The red-curve is the efficient frontier without short sales. The purple line is the combination between the $r_f$ and the tangency portfolio. Then if your $\lambda$ is high you are investor 1, and tilt more towards risk free. If $\lambda$ is low you are investor 2 and tilt more towards tangency.

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