My post has all the links you need to understand it, really just the Wikipedia definitions of Sharpe Ratio and various test statistics like the standard score.
I spent three hours studying all the various concepts, and I’ve reached the conclusion that there’s a bit of confusion everywhere, due to nobody defining their terms. Wikipedia gives four different definitions of the Sharpe ratio, the original/revised vs. ex-ante/ex-post. Plus, they are defined using momenta of a distribution, but for the examples where it’s used the sample means and sample standard deviation, which are the momenta only for a uniform distribution. Lumifer says that the Sharpe ratio is a descriptive statistics because it’s the excess returns over volatility, which I presume he intends to mean the istantaneous standard deviation. You say that the Sharpe ratio is a test statistics, because it is a measure divided by the standard deviation.
Neither of those assertions are true, though, per se. The Sharpe ratio is a statistics, full stop. How you want to use it determines if it’s a description or a test. Indeed: ex-ante Sharpe over a prior distribution uses the Lumifer definition and still can be used to test a null hypothesis, and your definition can still be used as descriptive statistics if your sample is the entire population (an easy feat for the price of a stock or index).
Besides: multiple hypothesis testing = parameter estimation, so the guy who found the 17th value is best described as a case of overfitting.
I agree though that Lumifer used an unnecessary rude tone.
Wikipedia gives four different definitions of the Sharpe ratio
Sure, but this discussion has a specific context: finance. In finance the words “Sharpe ratio” are well defined, they mean the annualized ratio of the sample mean of excess returns to the sample standard deviation of the same returns.
and still can be used to test a null hypothesis
Can it? Let’s try. I have a series of excess returns for which I know the Sharpe ratio (defined as above), let’s say it is 0.5. Given the null hypothesis that the true mean of these returns is zero, what is the probability that my sample mean is what it is conditional on the null hypothesis being true?
I spent three hours studying all the various concepts, and I’ve reached the conclusion that there’s a bit of confusion everywhere, due to nobody defining their terms.
Wikipedia gives four different definitions of the Sharpe ratio, the original/revised vs. ex-ante/ex-post. Plus, they are defined using momenta of a distribution, but for the examples where it’s used the sample means and sample standard deviation, which are the momenta only for a uniform distribution.
Lumifer says that the Sharpe ratio is a descriptive statistics because it’s the excess returns over volatility, which I presume he intends to mean the istantaneous standard deviation.
You say that the Sharpe ratio is a test statistics, because it is a measure divided by the standard deviation.
Neither of those assertions are true, though, per se. The Sharpe ratio is a statistics, full stop. How you want to use it determines if it’s a description or a test.
Indeed: ex-ante Sharpe over a prior distribution uses the Lumifer definition and still can be used to test a null hypothesis, and your definition can still be used as descriptive statistics if your sample is the entire population (an easy feat for the price of a stock or index). Besides: multiple hypothesis testing = parameter estimation, so the guy who found the 17th value is best described as a case of overfitting.
I agree though that Lumifer used an unnecessary rude tone.
Sure, but this discussion has a specific context: finance. In finance the words “Sharpe ratio” are well defined, they mean the annualized ratio of the sample mean of excess returns to the sample standard deviation of the same returns.
Can it? Let’s try. I have a series of excess returns for which I know the Sharpe ratio (defined as above), let’s say it is 0.5. Given the null hypothesis that the true mean of these returns is zero, what is the probability that my sample mean is what it is conditional on the null hypothesis being true?