The Bootstrap is a technique for simulating the sampling distribution of a statistic invented by Bradley Efron. It is a technique that attempts to solve the following problem: the empirical p.d.f. of a dataset clearly rejects common parametrical representations or the statistic we are computing has a population distribution that is analytically difficult or impossible to compute; however, the statistic is useful and we need to estimate it's sampling distribution to place confidence limits on the observed value.

The method is discussed in many places, such as Efron's excellent little book The Jackknife, the Bootstrap, and Other Resampling Plans, but I will summarize it briefly: we simulate data drawn from the empirical distribution function of the data by resampling with replacement of the actual data. This is clearly not as good as sampling from the population distribution function, but there are strong theorems governing the convgence of the e.d.f. to the p.d.f. and it does allow us to produce monte-carlo simulations of data with all of the measured properties of the sample (although the procedure is a little more complicated in the presence of serially correlated data). It is important to note that the replacement is an important step — it means that the properties of the simulations we create do not exactly match the actual sample and that allows us to estimate quantities such as the bias of an estimator.

The above charts show our use of The Bootstrap to analyze the series of daily returns of the Compact Model Portfolio. The upper chart shows five simulated total return time series (black) and the actual total return time series (red). The returns are accumulated and the dispersion of the final states due to a fortunate run of returns is very evident. The histograms show the distributions of the maximum drawdown and Sharpe Ratio for each simulated series. This are both popular metrics for quant. traders and are examples of statistics with awkward sampling distributions that a traditional analysis only gives use one opportunity to compute from historical data. The maximum drawdown histogram is fitted to the Gamma Distribution, and the Sharpe Ratio histogram to the Student's t Distribution. We learn from these charts that the standard deviation of the Sharpe Ratio is approximately equal to it's sample value and that the probability of a maximum drawdown exceeding 25% of capital is close to unity. It would not be possible to obtain this information via other methods.

Finally, I would like to acknowledge Greg Laughlin for stimulating my interest in using the Bootstrap method.