# Many quants optimize sharpe ratios, sortino ratios, or anything of the form A/B. What about maximizing something of the form (AB)/(CD)?

The Sharpe ratio is defined as return/risk, generally as mean(ret)/sd(ret), where ret represents the data set of returns of an investment. However, I have seen other ratios that I also like. What I tend to do is filter stocks by Sharpe ratio, and then filter the top 100 stocks through another filter for the sortino ratio, or the omega ratio. This seems to prefer one ratio over another though.

What if I mixed the Sharpe ratio and the Sortino ratio together? Why not maximize the Sharpe(ret)*Sortino(ret)? What does this mean?

If the answer is yes, then why not just take a bunch of ways of understanding return, such as mean, an average growth factor, and multiply them. Then risk could be the multiplication of standard deviation, maximum drawdown, average drawdown, downside deviation, etc? It seems like maximizing (ABCD)/(EF*G) is a method of optimizing a portfolio while attending to many ratios and factors instead of one at a time.

• 1. I think it is not usefull mix the two index simply because the are almost the same thing...so that you obtain some redundance. In fact Sortino index is just the same of Sharpe but a asymmetric weight wrt the return-index you have choosen. Morally you should apply it if you want to be more sensitive about the index you are expecting. 2. I'm not sure about your philosophy to multiply indexes and maximise them... Infact the multiplication should have some theoretical justfication. Other wise you could maximise with also funxctions like $x+y$ or $\sqrt{x^2 + y^2}$ Mar 29, 2018 at 7:22

## 1 Answer

There are many reasons why you should not do this, which can be summed up by:

You are constructing an objective function which is difficult to reason about, and might be doing something you don't want.

The measures you mention, Sharpe, Sortino, etc are usually chosen for theoretical reasons. (The Sharpe is a good measure of the Signal-Noise Ratio, which is related to probability of a loss; the Sortino is basically the same but with a tinfoil hat; and so on.) By jumbling them together you probably lose the theoretical advantages of all of them unless you understand their scale, how they correlate to each other, and so on.

On the other hand, many of these measures are used merely heuristically. For that use, it is fine to multiply them together. Add them, take them to powers. It's all heuristics. But again, unless you understand what can happen when you combine them in this way (and it gets more complicated the more you add), you can wind up with a displeasing heuristic that optimizes something you hadn't intended.