Combining Sharpe, Sortino, and Other Portfolio Objectives
Summary
The document considers whether a portfolio objective should combine several performance and risk measures, such as multiplying Sharpe and Sortino ratios or dividing multiple return measures by multiple risk measures. The response cautions that an objective assembled this way can become difficult to interpret and may reward behavior the optimizer's designer did not intend.
Sharpe and Sortino ratios are often chosen for particular theoretical motivations; combining them without understanding their scales and relationships can obscure those motivations. If the measures are used only as heuristics, combining them may still be reasonable, but the resulting objective should be examined carefully. The discussion offers conceptual guidance rather than a tested construction: it supplies no portfolio results, mathematical analysis of a specific combined score, or rules for choosing weights. Its central lesson is to understand what an objective rewards before optimizing against it.
Key ideas
- Combining performance and risk ratios can create an objective that is hard to interpret.
- Mixing measures may weaken the theoretical rationale for choosing each measure separately.
- Heuristic combinations are possible, but their behavior should be understood before use.
- The discussion does not evaluate a specific combined objective empirically.
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# Answer by steveo'america (score 2, accepted) # 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. ## Answer by steveo'america (score 2, accepted) https://quant.stackexchange.com/a/40017 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.
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