Bootstrap standard error from a handful of replicates

Question

You resample your daily returns with replacement, recompute the strategy's Sharpe ratio each time, and (for a tiny illustration) keep just five bootstrap replicates:

$0.8, \quad 1.0, \quad 1.2, \quad 1.4, \quad 1.6.$

Estimate the bootstrap standard error of the Sharpe ratio from these replicates.

Accepted Answer

The whole point of the bootstrap is that the spread of the resampled estimates stands in for the sampling variability of the real estimate. So the bootstrap standard error is simply the standard deviation of the replicate values. Use the sample standard deviation, dividing by B - 1 (here B = 5 replicates). Step 1, the mean of the replicates. \bar{	heta}^ = \frac{0.8 + 1.0 + 1.2 + 1.4 + 1.6}{5} = \frac{6.0}{5} = 1.2. Step 2, the squared deviations from that mean. (-0.4)^2 + (-0.2)^2 + 0^2 + (0.2)^2 + (0.4)^2 = 0.16 + 0.04 + 0 + 0.04 + 0.16 = 0.40. Step 3, divide by B - 1 = 4 and take the squa…

Bootstrap standard error from a handful of replicates

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