# 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

$$\lambda$$ 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.
• The Sharpe ratio formula has $r_f$ in it. So how can you say "I never mentioned I have a risk free rate"? – noob2 Jun 16 '20 at 19:09