Comparing Sharpe Ratios Across Simulated Samples
Summary
The document considers comparing two portfolio strategies when Monte Carlo simulation produces a collection of Sharpe ratio estimates for each strategy. The response uses an approximate normal sampling distribution for an estimated Sharpe ratio, with uncertainty depending on the population Sharpe and the return sample size underlying each estimate. It notes that a collection of Sharpe estimates may therefore have observations with different standard errors.
For unequal sample sizes, the proposed approach forms inverse-standard-error weighted averages for each strategy, then approximates the uncertainty of their difference. The null hypothesis corresponds to a zero difference between the strategy-level Sharpe parameters. This is an approximate framework, not a fully specified test: it relies on the stated normal approximation and does not discuss dependence between simulations, how to estimate unknown population Sharpe values in the standard errors, or alternative bootstrap and paired-simulation methods. The text also distinguishes comparing population Sharpe parameters from simply comparing averages of simulated estimates.
Key ideas
- An estimated Sharpe ratio can be modeled approximately as normal, with standard error tied to sample length and the underlying Sharpe.
- When estimates have different standard errors, inverse-standard-error weighting is proposed for aggregation.
- Compare the aggregated strategy estimates through their difference, with zero as the null value.
- The approximation leaves dependence among simulations and estimation of unknown standard errors unresolved.
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# How to test the difference between samples of sharpe ratios
# How to test the difference between samples of sharpe ratios
I am testing the performance difference between 2 portfolio strategies. I use Monte Carlo simulation in R to generate $N$ simulations of portfolio returns for each strategy. I then compute the Sharpe ratio for each simulation. In the end, each strategy has $N$ observations of Sharpe ratios.
How would I best go about using this data to test whether the true Sharpe ratio of one strategy is greater than that of the other?
All the research I have looked at examines comparisons between samples of returns, and not samples of Sharpe Ratios. The SharpeR package also only seems to have functions that take samples of returns as inputs.
I am also a little unsure of whether my question is even correctly stated - i.e. whether I should instead be asking about whether the true mean Sharpe ratio of one strategy is greater than that of the other.
## Answer by shabbychef (score 1)
https://quant.stackexchange.com/a/64170
Typically we have $$\hat{\zeta}\approx\mathcal{N}\left(\zeta,\frac{1 + \frac{\zeta^2}{2}}{n}\right),$$ where $\hat{\zeta}$ is the observed Sharpe ratio, and $\zeta$ is the unobserved population analogue (the signal-noise ratio). Assuming you observe $\hat{\zeta}_{1,i}$ and $\hat{\zeta}_{2,j}$ for $1 \le i \le M_1$ and $1 \le j \le M_2$, where the Sharpes are observed over samples of size $n_{1,i}$ and $n_{2,j}$, then if the sample sizes are different you should compute weighted averages: $$ \tilde{\zeta}_1 = \frac{\sum_i \frac{\hat{\zeta}_{1,i}}{s_{1,i}}}{ \sum_i \frac{1}{s_{1,i}}}, $$ where $s_{1,i}$ is the estimated standard error $\sqrt{\frac{1 + \frac{\zeta_{1,i}^2}{2}}{n_{1,i}}}$. Similarly compute $\tilde{\zeta}_2$. The claim is that $$\tilde{\zeta}_1 \approx\mathcal{N}\left(\zeta_1, \frac{M_1}{\left(\sum_i \frac{1}{s_{1,i}}\right)^2}\right).$$ From this you can find an approximate normal form for the difference $\tilde{\zeta}_1 - \tilde{\zeta}_2$, which should have zero mean under the null you are testing.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.