# Normality assumption in Sharpe ratio

I have read that the Sharpe ratio imposes a normality assumption, but I fail to see how. Standard deviation is statistic for any type of distribution. Anyone have any ideas?

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This is a basic statistics question. The Sharpe ratio is calculated using the mean and variance of a distribution, therefore it is a less descriptive measure the more the mean and variance do not completely describe the distribution. – Joshua Ulrich Jan 16 '14 at 6:32
@JoshuaUlrich, I disagree with that notion. SR calculations for any return distributions are 100% accurate. SR of a bond portfolio can be fairly compared with an emerging market stock portfolio. After all the returns are scaled by their own volatility measure. The problem arises due to a problem with the definition of SR itself, it penalizes out-sized returns to the upside, which are desirable yet not properly accounted for in SR computations. – Matt Wolf Jan 17 '14 at 2:32
@MattWolf: so you would argue that there's no normality assumption? Even if you account for upside/downside deviation, the Sharpe Ratio still has issues if the distributions are non-normal. – Joshua Ulrich Jan 17 '14 at 15:03
@JoshuaUlrich, that is not what I said, of course are the moments different for different distributions. My point was, and sorry if that was not clear, that it is impossible to attach an accurate distributional assumption to any financial asset returns, hence the most often used normal and log-normal assumptions for most asset returns, which validates the usage of mean and stdev under normal distributional assumptions. In fact most all financial asset returns exhibit dynamic distributional features. – Matt Wolf Jan 18 '14 at 11:07

You are correct that you can compute Sharpe ratios on portfolios with any return distribution. The issue is comparing Sharpe ratio's of non-normally distributed portfolios (which in reality is almost any portfolio). To take an extreme example. Consider two portfolios, with returns in excess of benchmark.

1. 50% chance of 10% return, 50% chance of a 20% return
2. 50% chance of 10% return, 50% chance of a 100% return

The Sharpe ratios are $$1. \frac{0.5 \cdot 0.1 + 0.5 \cdot 0.2}{\sqrt{0.5 (0.1 - 0.15)^2 + 0.5 (0.2 - 0.15)^2}} = 3 \\ 2. \frac{0.5 \cdot 0.1 + 0.5 \cdot 1}{\sqrt{0.5 (0.1 - 0.55)^2 + 0.5 (1 - 0.55)^2}} \approx 1.22$$

Portfolio 2 clearly dominates portfolio 1, but its Sharpe ratio is much lower.

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Very good example! – Richard Jan 16 '14 at 8:18
Many thanks to both David Nehme and Joshua Ulrich for answering my question. – user6997 Jan 16 '14 at 18:27
@user6997, you should mark this as correct answer if you think it properly addresses your question – Matt Wolf Jan 17 '14 at 2:25
@David, I would argue that this particular observation is not really because of different distributional assumptions but because of the intrinsic flaws of SR itself which is that it penalizes large upside returns because of an increase in volatility such upside returns cause. It is absolutely fair to compare SRs of portfolios of entirely different asset classes with intrinsically different distributional assumptions. – Matt Wolf Jan 17 '14 at 2:34