# Risk-adjusted returns ratio that does not reward high risk for negative returns

Think of Sharpe ratio, Treynor ratio, or anything where (excess) returns $r$ are divided by something that represents risk, $\sigma$:

$$\mathrm{performance} = \frac{r}{\sigma}$$

If the returns are negative (let's say for a short period), a performance indicator based on such a ratio is better (=less negative), the higher the risk (e.g. volatility) is.

First question: Does this make sense? I would not say the true performance is better (less poor), if the risk increases – even when the returns are negative.

Second question: Are there established (or even proposed) risk-adjusted return measures that penalize (or at least don't reward) high risk even for negative returns? I'd be quite happy to derive one, but I would expect that someone has already figured this out.

Yes, you are correct on both terms - it doesn't make much sense, and there exists a well-cited solution by C. Israelsen: "A refinement to the Sharpe ratio and information ratio." Journal of Asset Management 5.6 (2005): 423-427.

The adjustment he gives is to define $$SR_{adj} = \frac{r}{\sigma^{\frac{r}{abs(r)}}},$$

which solves the ranking problem during periods of negative (excess) return.

• Interesting. This amounts to using $\frac{r}{\sigma}$ if $r>=0$ and $r \sigma$ otherwise. – noob2 Feb 15 '17 at 19:18
• Exactly what I was looking for – thanks @Forgottenscience! Also kudos to @noob2 for the clarification. I guess you'd have to be an academic not to prefer the piecewise definition. – Reunanen Feb 16 '17 at 8:51
• One more problem with the Israelson refinement is left: U would prefer a slight positive performance over a slight negative performance everytime - even when the vol is extremely high for the positive r and low for the negative one! Example: r in % vol in % r/vol Rank r/vol^(r/abs(r)) Rank 0,01 100,00 0,0001 1 0,0001 1 -0,01 1,00 -0,0100 2 -0,0100 2 But is the positive performance with 100 % vol really better than the negative one...? – Dom Sep 7 '17 at 10:37

1) In a certain, theoretical sense, it does make sense: suppose two portfolio managers delivered negative returns (-1%, say), and one had a higher volatility ("risk") than the other. Then the high-risk fund did better, in a way: despite higher risk, the portfolio manager succeeded in providing the same small loss as the low-risk manager.

2) I agree that this implies a perverse effect: maximise risk for a given negative return. In numerical portfolio optimisation, one may rather prefer a linear combination of risk and return instead of a ratio, or simply put a 'safeguard' into the objective function: when return turns negative, change the selection criterion (e.g. select only based on risk).

1. Does this make sense? Consider this: You are an investor. You have 2 investments. 1 high risk (hr) and the other low risk (lr). You expect the hr to be volatile and expect the opposite from lr. If the hr has a small loss and the lr has an equal small loss shouldn't the hr have a better ratio? It should. It performed better based on it's volatility and return potential.

2. Yes. You might consider something like an Omega ratio. Omega does not assume a normalized distribution of returns and allows you to set a minimum acceptable return.

Using Omega to optimize a portfolio (i.e. assign weightings to each investment) is far more effective than other metrics in helping a manager achieve an expected return of a group of assets.