mean variance optimization vs max sharpe ratio

Question

I keep reading/hearing that the results from mean-var optimization is max Sharpe ratio. It seems making sense if you fix either target return or target risk, but in general, it doesn't seems right, for example, $J1$ and $J2$ are target function:

$J1 = \mu\prime w - \lambda w\prime\Sigma w.$

$J2 = (\mu\prime w)/\sqrt{w\prime\sigma w}$

The optimal solution of $J1$ and $J2$ should be very different, because $J1$ depends on lambda, $J2$ does not, not to mention the derivatives respect to w are very different.

what am I missing here?

Answer 1

In theory in the case of a constrained optimisation and in practice they are not.

However... A lot of practitioner wants to achieve the best Sharpe Ratio for their portfolio. But as you describe it in J2 the term is not linear nor quadratic and is much harder to optimise especially in the context of the multitude of constraints that would occur in a typical portfolio optimisation framework

J1 is nicely quadratic so it is a lot easier to optimise. And it has this nice property that you would want to maximise u'w and minimise wSw which aligns in terms of conceptual goals with getting the best possible Sharpe Ratio

But in reality they are not equivalent and J2 is highly unpractical and rarely used. Also with J2 a passive portfolio with 0 tracking error would be always the best solution in the absence of other constraints... So the vast majority of practitioner would use a variant of J1

Answer 2

You do not find a correct solution because, mathematically speaking, the problem are not well posed.

Firstly, in $J1$ seems that you have in mind the pure risky ptf while in the $J2$ case this is not possible. For simplification you can assume that $r_f =0$ but it exist, otherwise Sharpe ratio don’t have any sense.

Moreover, probably you have in mind the unconstrained version but also in this case you have to note that in $J1$ like in $J2$ case the the minimal constraint $w’1=1$ holds.

Third and, and maybe most important, in $J1$ the optimization strategy return you entirely the efficient frontier (through lambda) while $J2$ return you only one point. In this point $w$= tangent ptf.

The problem that, probably, you have in mind is well posed if in $J1$ you add the riskless asset. In this case $w$ is interpretable as the weight of risky ptf and $(1-w)$ as riskless asset weight. Then $w$ become unique and equal to tangent ptf as in $J2$ case. I got the proof but it is not short and now it is in some notebook.