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Describe bootstrapping and the way in which it improves on Monte Carlo simulation.

FRM® Part I · Quantitative Analysis · Simulation and Bootstrapping · FRM1.QA.13.d

The idea

Bootstrapping resamples the observed data with replacement. Draw T observations from the T actual data points, compute the statistic, repeat thousands of times, and use the spread of the results as the standard error or confidence interval. Its advantage over Monte Carlo is that it needs no assumed distribution or data generating process, so fat tails, skewness and other empirical features carry over automatically. When several assets are involved, resample whole dates so that cross-asset dependence is preserved. Some observations appear more than once, and about 37% are left out of any given resample. The bootstrap standard error is the standard deviation of the replicate estimates, with no division by the square root of the number of replications. Remember that it still assumes the observations are iid and representative.

A practice question

From the question bank, with the reason each wrong option is wrong.

How does bootstrapping draw its samples?

  1. It draws values from a normal distribution fitted to the observed data.Why not: Describes parametric Monte Carlo simulation, not bootstrapping.
  2. It resamples without replacement until every observation has been used once.Why not: Without replacement every resample would just reorder the original data and give the same statistic.
  3. It resamples with replacement from the actual observed data. - correct
  4. It splits the data into two halves and uses one to forecast the other.Why not: Describes out-of-sample validation, which is not bootstrapping.

Bootstrapping draws T observations with replacement from the T actual data points, computes the statistic, and repeats many times. Replacement is what makes the resamples differ from one another and from the original sample.

A harder one

A bootstrap resample of 20 observations is drawn with replacement from an original sample of 20 observations. What is the probability that a particular original observation appears at least once in the resample, closest to?

  1. 36%Why not: Reports the probability that the observation is missing from the resample rather than present.
  2. 95%Why not: Uses the probability of missing the observation on one draw, not the probability it appears across all 20 draws.
  3. 100%Why not: Treats the resample as drawn without replacement, in which case every observation would be used exactly once.
  4. 64% - correct

The probability of missing the observation on all 20 draws is (1 - 1/20)^20 = 0.3585. So it appears at least once with probability 1 - 0.3585 = 0.6415, about 64%. Some observations repeat and others are left out.

The hardest tier

A risk manager bootstraps a statistic and obtains five replicate estimates: 2.0, 2.9, 1.6, 2.4 and 2.1 (a deliberately small example). What is the bootstrap standard error of the statistic, taken as the sample standard deviation of the replicates (divisor B - 1), closest to?

  1. 0.43Why not: Divides the sum of squared deviations by B instead of B - 1.
  2. 0.48 - correct
  3. 0.22Why not: Divides by the square root of the number of replications; that is the error of the average of the replicates, not the standard error of the statistic.
  4. 0.23Why not: Reports the sample variance of the replicates instead of its square root.

The mean of the replicates is 2.20 and the sum of squared deviations is 0.94. The sample variance is 0.94 / 4 = 0.235, so the standard error is the square root, 0.485. The bootstrap standard error is the standard deviation of the replicates themselves.

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