# 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.

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You mean the $\lambda$ in $\mu - \lambda \sigma$? – Bob Jansen Nov 12 '12 at 11:16
Exactly, known also as the "risk aversion coefficient". – omar Nov 14 '12 at 4:22

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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