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Outline the circumstances in which bootstrapping does not work well.

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

The idea

The standard bootstrap resamples single observations independently from the sample, so it works well only when the data are iid and representative. It performs poorly in four situations. First, serially dependent data such as autocorrelation or volatility clustering. Resampling points destroys the dependence, and a block bootstrap, which resamples runs of consecutive observations, is the usual fix. Second, structural breaks or regime changes, which mix distributions. Third, tail and extreme-value questions, since a resample cannot contain a loss worse than the worst one observed. A single crash day may be missing from about 37% of resamples when the sample has 100 observations. Fourth, very small samples, where the data may not be representative of the population. The exam point: a bootstrap can only reuse what is in the sample.

A practice question

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

A risk manager has bootstrapped losses from two calm years of data to estimate the risk of a market crash. Why is the result unreliable?

  1. A resample can only contain observed values, so crash-type losses absent from the sample cannot appear. - correct
  2. Bootstrapping always overstates risk because it duplicates the largest observations in the sample.Why not: Duplication of observations does not create losses beyond those in the sample; the usual problem is understatement.
  3. Bootstrapping requires a normal distribution, which calm periods do not provide in the data.Why not: The bootstrap makes no normality assumption.
  4. Bootstrap resamples are too small to contain the most recent observations of the sample.Why not: Resamples are the same size as the original sample and can include any observation.

The bootstrap draws only from the observed data, so it cannot generate outcomes worse than those in the sample. If the sample contains no stressed periods, the resampled distribution understates the possibility of a crash.

A harder one

A risk manager has 250 daily returns showing volatility clustering and uses a moving block bootstrap with blocks of 5 consecutive days (overlapping blocks allowed). How many blocks must be drawn with replacement to build a resample of 250 observations?

  1. 246Why not: Gives the number of distinct overlapping blocks available, not the number to be drawn.
  2. 250Why not: Draws one block per observation, which would produce 1,250 observations in the resample.
  3. 1,250Why not: Multiplies the sample size by the block length instead of dividing.
  4. 50 - correct

Each block supplies 5 observations, so a resample of 250 needs 250 / 5 = 50 blocks. The 246 overlapping blocks are the pool drawn from. Blocks keep the short-range dependence that single-observation resampling destroys.

The hardest tier

A bank's 100 daily returns include one crash day far worse than all others. It runs 1,000 bootstrap resamples of 100 days each. On average, in how many resamples will the crash day be absent, closest to?

  1. 10Why not: Uses 1% as the probability of absence, which is the chance of drawing the crash on a single draw.
  2. 634Why not: Counts the resamples that do include the crash day rather than those that do not.
  3. 990Why not: Uses the probability of missing the crash on one draw and does not compound it over 100 draws.
  4. 366 - correct

The probability of missing the crash day on all 100 draws is 0.99^100 = 0.3660, so about 366 of 1,000 resamples omit it. The tail estimate depends heavily on one observation, so bootstrap risk estimates in the tail are unstable.

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