The Sharpe ratio is a frequently utilized statistic in asset management. As a measure of excess return per unit risk, it’s most frequent usage is to compare how ‘efficient’ one set of returns is to another. However, under a few mathematical assumptions, the Sharpe ratio has another interesting application:
If we assume portfolio returns are normally distributed and a risk-free rate of 0%, then our portfolio’s Sharpe ratio can actually tell us the probability of a loss occurring.
Proof: Let us assume that our asset returns, r, follow a multivariate normal return distribution, r ~ N(μ, Σ), and we have a portfolio weights w. Therefore, our portfolio return, rp = wtr, is distributed rp ~ N(μp=wtμ, σp = wtΣw).
If we assume a risk free rate of zero, the probability of a negative return, P(rp < 0), is equivalent to P(Z < -μp / σp), where Z~N(0,1).
Note: If you want to try this exercise to calculate the probability of loss on your own portfolio, it is likely easier to just take historical portfolio values and use them to estimate the expected return, μp, and volatility, σp, of the portfolio instead of trying to estimate the expected return, volatility, and correlations for all the underlying assets. The formula in Excel for P(Z < -μp / σp) is: “normdist(-μp / σp, 0, 1, true)”.
We can plot a curve to show our probability of loss for a given Sharpe ratio:
We can extend this math to construct a surface that tells us the probability of a loss below a given standard deviation level for a given Sharpe ratio. For an nσ loss, the formula is P(Z < -(n + μp / σp)). We can now plot a surface that tells us the probability of loss for a given Sharpe ratio and standard deviation level:
Like the Richter-Gutenberg law in seismology, this model does nothing to predict losses (telling us when a loss will occur), but it may help us forecast them (telling us the probability with which they occur).
Consider that from 2/1993 to 12/2006, the monthly Sharpe ratio of the SPY ETF was 0.21, indicating a loss probability of 41.63%. From 1/2007 to 11/2012, 42.85% of months were losses. Returns may not be perfectly normal – but this model does a pretty good job of identifying the frequency of losses.
So it would seem that to manage risk, all we have to do is increase our Sharpe ratio! Unfortunately, the math is a little trickier.
Consider two assets whose returns are distributed normally. The first has expected return 4% and volatility 10% and the second has expected return 6% and volatility 12%. The former Sharpe ratio is 0.4, the latter 0.5 indicating a 34.46% chance of a loss and 30.85% chance of loss respectively. So this is a victory in risk management, right?
If we consider the average magnitude of a loss for each of these distributions, the first is -6.69% and the second is -7.69%. But surely the lower probability of loss in the second portfolio cancels out this larger average magnitude loss? Unfortunately this is not the case. If we were to consider a situation where positive returns are clamped to 0, the expected return of the portfolios would be the probability of a loss times the average loss: -2.30% and -2.37% for the portfolios respectively.
Now, of course, we can’t have a higher expected return with a larger average loss unless we are compensated on the right tail. Some quick math and we find that we are: first portfolio has an expected positive return of 9.62% and the second an expected positive return of 12.11% – their probability adjusted expected positive returns are 6.30% and 8.37%. You’ll note that the left tail and right tail probability adjusted expected returns sum to the expected returns for each portfolio.
Nevertheless, in volatile times when risk management is on the forefront of your mind, it is not enough to rely on a higher Sharpe ratio and a lower probability of loss: it is always important to consider the magnitude of your losses as well.