# Bibliography

A biased and incomplete list of references relating to the Sharpe Ratio:

• Gosset, W. S., (as "A. Student"), The Probable Error of a Mean, 1908, introduced Student's $$t$$ statistic. Except it wasn't called $$t$$, and it was not normalized in the way it is now computed. In fact, it was exactly equal to the Sharpe Ratio, but for the fact that the standard deviation was commonly computed with a denominator of $$n$$, not $$n-1$$. It was R. A. Fisher, in Applications of "Student's" Distribution that described the $$t$$ statistic in its modern form.
• Johnson, N. L. and Welch, B. L., Applications of the non-central t-distribution, 1940, describes the standard error of the $$t$$ statistic as an estimator of the parameter analogue (the true mean divided by the true standard deviation, which I call the 'signal to noise ratio').
• Roy, A. D., Safety First and the Holding of Assets, 1952, advocates use of what would later be called the Sharpe Ratio to allocate among assets. The justification is that so doing will, via Chebyshev's Inequality, minimize the probability of a loss. This remarkable paper also presents what is essentially the efficient frontier plot, and Markowitz-style portfolio optimization!
• Sharpe, W. F. Mutual Fund Performance, 1965, gives us the 'reward to variability ratio', a name so awful it could only survive as the Sharpe Ratio. Note that Sharpe acknowledges that the mean and volatility of a return series can only be estimated, and not known, and thus the Sharpe ratio is a statistic, not a parameter.
• Miller, R. E. and Gehr, A. K., Sample Size Bias and Sharpe's Performance Measure: A Note, 1978, note that the Sharpe ratio is slightly biased, making the connection to the distribution of the $$t$$ statistic.
• Jobson, J. D. and Korkie, B. M., Performance Hypothesis Testing with the Sharpe and Treynor Measures, 1981, describes the asymptotic distribution of the Sharpe Ratio of $$k$$ assets, allowing multiple comparison tests for equality of Sharpe ratios, even for correlated returns.
• Lo, A. W., The Statistics of Sharpe Ratios, 2002, gives the standard error of the Sharpe Ratio, and describes the effects of autocorrelation. This is the same standard error described by Johnson and Welch.
• Mertens, E. Comments on Variance of the IID estimator in Lo (2002), 2002, describes a more complex standard error estimator for the Sharpe Ratio that takes into account skew and kurtosis of the returns stream.
• Bao, Y. Estimation Risk-Adjusted Sharpe Ratio and Fund Performance Ranking Under a General Return Distribution, 2009, derives the bias and standard error of the Sharpe ratio under general IID returns.
• Pav, S. E. Safety Third: Roy's Criterion and Higher Order Moments, 2015, extends Roy's "Safety First" criterion to include higher order moments of returns.
• Challet, D. Sharper asset ranking from total drawdown durations, 2015, provides a means of computing a Sharpe ratio which is putatively more efficient for fat-tailed distributions. Implementations of this estimator are marred by bias and shrinkage, which clouds our understanding of their performance.
• Paulsen, D. and Soehl, J., Noise Fit, Estimation Error and a Sharpe Information Criterion, 2016, describe a correction for the Sharpe Ratio, which is an unbiased estimate for the signal-noise ratio of the Markowitz Portfolio.
• Pav, S. E. A Short Sharpe Course, 2017, though incomplete, is billed as a complete course for self-study of the statistics of the Sharpe Ratio.
• Pav, S. E. The Sharpe Ratio: Statistics and Applications, 2021. All the material in A Short Sharpe Course, plus chapters on inference on the Markowitz portfolio, overfitting, market timing and backtesting.

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The above references an opinion and is for information purposes only. It is not offered as investment advice.