# How Sharp Is the Sharpe Ratio? An Analysis of Global Stock Indices

Investors across the globe use the Sharpe Ratio, among other risk-adjusted metrics, to compare the performance of mutual fund and hedge fund managers as well as asset classes and individual securities. The Sharpe Ratio attempts to describe the excess return relative to the risk of the strategy or investment — that is, return minus risk-free rate divided by volatility — and is among the primary gauges of fund manager performance.

But hidden within the Sharpe Ratio is the assumption that volatility — the denominator of the equation — captures “risk” in its entirety. Of course, if volatility fails to entirely reflect the investment’s risk profile, then the Sharpe Ratio and similar risk-adjusted measures may be flawed and unreliable.

What are the implications of such a conclusion? A common one is that the distribution of returns must be normal, or Gaussian. If there is significant skewness in the returns of the security, strategy, or asset class, then the Sharpe Ratio may not accurately describe “risk-adjusted returns.”

To test the metric’s effectiveness, we constructed monthly return distributions for 15 global stock market indices to determine if any had such exacerbated skewness as to call into question the measure’s applicability. The distribution of returns went as far back as 1970 and were calculated on both a monthly and annual basis. The monthly return distributions are presented blow. Annual return results were qualitatively similar across the various indices studied.

We ranked all 15 indices by their skewness. The S&P 500 came in close to the middle of the pack on this measure, with an average return of 0.72% and a median return of 1% per month. So, the S&P distribution skews just a bit to the left.

**S&P 500 Monthly Return Distributions, Since 1970**

The complete list of indices ranked by their skewness is presented in the chart below. Ten of the 15 indices exhibit left skewness, or crash risk: They are more prone to pronounced nose-dives than they are to steep upward climbs. The least skewed distributions were those of France’s CAC 40 and the Heng Seng, in Hong Kong, SAR.

**Monthly Returns by Global Index**

Index | Mean | Median | Min. | Max. | STD | Skewness |

ASX 200 | 0.58% | 1.01% | -42.3% | 22.4% | 0.048 | -1.3 |

TSX | 0.60% | 0.88% | -22.6% | 16% | 0.044 | -0.77 |

FTSE | 0.53% | 0.91% | -27.6% | 13.7% | 0.045 | -0.73 |

Russell 2000 | 0.84% | 1.60% | -21.9% | 18.3% | 0.055 | -0.55 |

S&P 500 | 0.72% | 1.00% | -21.8% | 16.3% | 0.044 | -0.45 |

DAX | 0.67% | 0.74% | -25.4% | 21.4% | 0.056 | -0.39 |

Nikkei | 0.54% | 0.91% | -23.8% | 20.1% | 0.055 | -0.37 |

MXX | 1.23% | 1.16% | -29.5% | 20.4% | 0.066 | -0.34 |

MOEX | 1.29% | 1.63% | -30% | 33% | 0.079 | -0.29 |

CAC 40 | 0.64% | 0.98% | -22.3% | 24.5% | 0.056 | -0.11 |

Hang Seng | 1.17% | 1.23% | -44.1% | 67.3% | 0.090 | 0.33 |

NSE | 1.50% | 1.05% | -24% | 42% | 0.076 | 0.53 |

KRX | 0.90% | 0.49% | -27.3% | 50.7% | 0.074 | 0.80 |

BVSP | 5.63% | 1.94% | -58.8% | 128.6% | 0.184 | 2.51 |

SSE | 1.65% | 0.63% | -31.2% | 177.2% | 0.151 | 6.26 |

The Shanghai Composite has exhibited the greatest degree of right skewness over time, tending to crash up more than down, and otherwise generating average returns of 1.65% per month and median returns of 0.63% per month.

**Shanghai Composite (SSE) Monthly Return Distribution, Since 1990**

On the opposite end of the spectrum is the Australian ASX. The ASX has the most left skewness of all the indices, with an average monthly return of 0.58% and median monthly return of 1.01% since 1970.

**Australian Stock Exchange (ASX) Monthly Return Distributions, Since 1970**

In the end, the BSVA in Brazil, the Shanghai Composite in China, and, to a lesser extent the ASX in Australia just have too much skewness in their returns to validate the Sharpe Ratio as an appropriate measure for their risk-adjusted performance. As a consequence, metrics that account for skewness in returns may be better gauges in these markets.

Of the other indices, seven had fairly symmetrical distributions and five had moderately skewed ones. All told, this suggests that the Sharpe Ratio still has value as a performance metric and that it may not be as obsolete or ineffective as its critics contend.

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*All posts are the opinion of the author. As such, they should not be construed as investment advice, nor do the opinions expressed necessarily reflect the views of CFA Institute or the author’s employer.*

Image credit: ©Getty Images/NPHOTOS

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I enjoyed reading this – thank you for taking the time to put it together. I have also found the ratio useful for portfolio optimization in US markets, like you say, critics notwithstanding. Best wishes and thanks again.

Interesting read. Thank you. I notice that volatiliy and sharpe ratios are used while there is clearly no presence of a normal return distribution. For instance if put options are used to hedge tail risk. How do you look at the use of volatility and Sharpe ratio in these circumstances? And what alternatives are available?

I notice for the S&P Index you utilized a data series starting in 1970 but not why this period was chosen. What would be the result in terms of distribution skewness if you used the longer history of a similarly constructed index?

[I understand that one can be constructed as least as far back as 1926 and even to the 1870s.]

Thank you for the post. A couple of comments which I hope you don’t mind receiving.

“What are the implications of such a conclusion? A common one is that the distribution of returns must be normal, or Gaussian.” Actually no, just that the distribution must be symtetrical, not skew. There are many symmetrical distributions besides the normal distribution. You go on to talk about skewness, but this point is still important, because you could also address kurtosis.

“Annual return results were qualitatively similar across the various indices studied.” These should be shown so that the reader can make this judgment call themselves.

“…we constructed monthly return distributions for 15 global stock market indices…” Did you use log returns? You should for geometric brownian motion processes. Although this will accentuate the negative skew of the distributions that are already negatively skew according to your calculations, it will reduce the positive skew of those with positive skew. This is even more important when considering longer periods like a year.

“Ten of the 15 indices exhibit left skewness, or crash risk: They are more prone to pronounced nose-dives than they are to steep upward climbs. ” Are you intentionally implying that you can say something about the future based on the past performance of these indices? If so, you should make this assumption explicit.

Presumably, if you considered sub-periods, you may find that the distribution actually had positive skewness over certain periods. Skewness (ex-post) is in fact a random variable that has its own distribution, so you should expect to see positive skewness and negative skewness in a sample from a population where none exists. This is really easy to demonstrate through modelling.

“…just have too much skewness in their returns to validate the Sharpe Ratio as an appropriate measure for their risk-adjusted performance…” What is the issue with skewness in a market if you are comparing funds within that market, that all exhibit the same skewness i.e. all of the returns are biased in the same way. You could simply favour the lowest Sharpe Ratio funds, if you use Sharpe Ratio for fund selection, which seems crazy on its face but I accept some people may actually do this.