# Alternative relative performance measure to Sharpe ratio for non-IID return

The Sharpe ratio is often used to compare the relative performance of portfolios despite its IID-assumption for the returns being violated.

I can find ample warnings about the consequences of breaching its assumptions.

What I am having difficulty to find, however, are alternatives to the Sharpe ratio as a relative performance measure. Has a standard-solution crystalized in the community? Can somebody point me to literature?

• The response of the Sharpe ratio to violations of the i.i.d. assumption are covered in great detail in chapter 4 of Short Sharpe Course. The bottom line is that violations of i.i.d. normality you are likely to encounter in practice have very little effect on the distribution of the Sharpe ratio. – steveo'america Dec 13 '19 at 0:11
• @steveo'america can you not envision a dominant portfolio having a lesser sharpe due to distributions not being normalized? – amdopt Dec 13 '19 at 1:02
• well, it's been some four years since I wrote a paper on the topic, so I'm a little rusty. But by asking about iid'' I assume the OP is asking about the statistical properties of Sharpe ratio, not about use of Signal-noise ratio as a utility function. – steveo'america Dec 13 '19 at 22:59
• @steveo'america Your amount of rust should have been considered before commenting. The OP asked about options. They were given options. Feel free to provide a better answer. – amdopt Dec 14 '19 at 0:41
• @amdopt touche. However, none of the proposed alternatives can be characterized as 'standard-solutions' with much practical currency. In ~10 years of pitching quant strategies, discussion of the Sharpe (realized and backtested) was routine, but I only recall one instance of clients asking about any of these others (and in that case, they were filling out a TPS report, and couldn't even define the measure they were asking us to quote). – steveo'america Dec 17 '19 at 1:11

I don't know that there is a "standard-solution crystalized in the community," but there are alternatives. The ones that I prefer are Omega, Sortino, and Kappa. All three of these ratios, unlike Sharpe, do not assume normally distributed returns.

Omega Ratio: This is the probability-weighted ratio of gains versus losses for a given minimum acceptable return. Omega looks at all moments instead of just volatility like Sharpe. The advantage of using the integral is the entire distribution may be considered. So instead of considering volatility alone, Omega considers, amongst others, kurtosis and skewness. This is important when returns are asymmetric.

$$Omega(r)={{\int_r^\infty(1-F(x))dx}\over\int_{-\infty}^r F(x)dx}$$

where $$F$$ is the cumulative distribution function of returns and $$r$$ is the minimum acceptable return that defines our gain or loss -- $$r$$ does not have to be zero!

Sortino Ratio: Sortino puts more emphasis on downside risk than Sharpe. Sortino is a performance measure that penalized returns that fall below a user-specified target return. Therefore, Sortino does not punish upside volatility as Sharpe does.

$$Sortino = {{r_p - t}\over {DD}}$$

where $$r_p$$ is the average portfolio return, $$t$$ is the target return, and $$DD$$ is the downside deviation: $$DD = \sqrt{\frac {1}{N} \cdot \sum_i^N min(0,r_i-t)^2}$$

Kappa-3 Ratio: Though the higher Kappa, the better, interpretation can be tricky, and this ratio is best used to rank investments versus one another.

$$K_n(\tau)={{\mu-\tau}\over{^n\sqrt{LPM_n(\tau)}}}$$

where $$\mu$$ is the mean return, $$\tau$$ is the return threshold, and $$LPM_n$$ is the n-th order lower partial moment:

$$LPM_n(\tau)=\int_{-\infty}^t (\tau-R)^ndF(R)$$

Please note that I have listed these three ratios in this order for a specific reason: setting the Kappa $$n$$ parameter to 1 gives you Omega, setting it to 2 gives you Sortino. The most common setting is three hence the Kappa-3 name. Kappa is a way of "unifying" Omega and Sortino.

There are many other performance measurements--I have only listed the three that I prefer as a substitute for Sharpe. I use metrics that compare returns to drawdowns in conjunction with the metrics above to gain a larger, more well-rounded picture when optimizing a portfolio of investments or trading models.