# 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

## 1 Answer

$$\lambda$$ 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.

• What is the red curve in your plot? Btw I understood what you mean, however if I increase lambda I just start to walk down on the efficient frontier towards the minimum variance..I do not move along the straight line..for lambda equal zero I optimise the max return and lambda tending to infinity it becomes a minimum variance problem..so I do not think that increasing lambda I move on the line..thanks – Luigi87 Jun 16 '20 at 16:56
• When you have a risk-free you are on the straight line of the plot. Not on the efficient frontier. I clarified what the red curve is. – phdstudent Jun 16 '20 at 17:33
• At the end i made a code which loops through lambda and for each computes the sharpe (actually I never mentioned I have a risk free rate) what I have got is a function like a bell shape, so there exists a relation between them. Then I choose the value of lambda that provided me with an allocation such that the Sharpe was maximum (the top of the bell curve). – Luigi87 Jun 16 '20 at 17:48
• 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
• I assume it zero – Luigi87 Jun 16 '20 at 19:16