FRM Exam Part I · Simulation and Bootstrapping

Bootstrapping and Resampling with Replacement for FRM Part I

Updated 11 October 2026 · Fact-checked

Bootstrapping estimates a statistic's sampling distribution by drawing many new samples, with replacement, from your observed data. Each resample is the same size as the original. You compute the statistic on every resample. The spread of those results gives the standard error, and their percentiles give a confidence interval.

Understand Bootstrapping

You usually have one sample of data and want to know how reliable a statistic from it is, such as a mean, a volatility or a VaR. Theory gives formulas for simple cases like the sample mean. For a quantile or a ratio, no simple formula exists. Bootstrapping gets around this by using the data itself.

The idea is to treat your sample as a stand-in for the true population. You draw n observations from your n data points with replacement. Replacement means each draw puts the value back, so one observation can appear several times in a resample and others not at all. This gives one bootstrap sample. You compute your statistic on it. Then you repeat this B times, for example 1,000 times.

The B statistics form an estimate of the sampling distribution. Their standard deviation is the bootstrap standard error. Their percentiles give a percentile confidence interval. You never needed a formula for the statistic's distribution, only a computer.

The basic version is the iid bootstrap. It assumes observations are independent and identically distributed, so every observation is equally likely to be drawn. In risk management the best-known use is in historical simulation. You resample past returns or losses with replacement, compute VaR or ES on each resample, and average the results or study their spread. This also shows how much sampling error sits in a VaR estimate.

Bootstrapping has limits. It can only reuse values you have already seen, so it cannot create a loss worse than your worst observation. If data are serially dependent or volatility clusters, the iid bootstrap loses that structure and can understate risk. If your sample is small or unrepresentative, the bootstrap inherits that weakness.

Key formulas to remember

Bootstrap sample: Draw n values from n observations, with replacement, each draw equally likely (probability 1/n); Each resample has the same size as the original sample. Repeat B times to get B statistics θ*₁, …, θ*_B.
Bootstrap standard error: SE(θ̂) ≈ √[ Σ (θ*_b − mean of θ*)² ÷ (B − 1) ], summed over b = 1 to B; This is the sample standard deviation of the B bootstrap estimates. It is not the standard deviation of the raw data.
Percentile confidence interval: For a (1 − α) interval, use the α/2 and (1 − α/2) percentiles of the B bootstrap estimates; For a 95% interval, use the 2.5th and 97.5th percentiles.
Chance an observation is left out of one resample: (1 − 1/n)ⁿ, which approaches e⁻¹ ≈ 0.368 as n grows; So about 63.2% of the original observations appear at least once in a large resample.
Bootstrap bias estimate: Bias ≈ mean of θ* − θ̂; θ̂ is the statistic from the original sample. Mean of θ* is the average of the bootstrap statistics.

How to solve Bootstrapping questions

Use this order for most bootstrapping questions, whether they ask for a standard error, a confidence interval, a VaR or a conceptual point.

1Identify the statistic you need, such as the mean, standard deviation, VaR or ES, and the original sample size n.
2Check the resampling rule. Draws are with replacement, each resample has size n, and each observation has probability 1/n on every draw.
3Note the number of resamples B and how many bootstrap estimates the question gives you.
4For a standard error, find the mean of the bootstrap estimates, sum the squared deviations from it, divide by B − 1 and take the square root.
5For a confidence interval, sort the bootstrap estimates and read off the α/2 and 1 − α/2 percentiles.
6For bootstrap VaR, compute VaR on each resample, then average the results or look at their spread. Do not take the VaR of the pooled results as a standard error.
7Sanity-check the answer: the standard error should be smaller than the data's standard deviation for a mean, and the interval should contain the original estimate.
8If the question is conceptual, test it against the iid assumption and the fact that no value outside the observed data can be drawn.

Quickest way: Fast routes for bootstrap questions

When to use it: Use these when time is short and the question gives you a list of bootstrap estimates or asks a probability about resampling.

Standard error from a list: enter the B estimates in the calculator's statistics mode and read the sample standard deviation (the s, n−1 key), not the population one.
Confidence interval from a sorted list: the 95% bounds are roughly the 2.5% and 97.5% positions, so with B = 200 use about the 5th smallest and 5th largest values.
Probability a given observation is missing: compute (1 − 1/n)ⁿ. For a probability of being included, take 1 minus that.
Concept questions: eliminate any option that says sampling is without replacement, that new values outside the data can appear, or that bootstrapping needs a normality assumption.

Common mistakes in Bootstrapping

Resampling without replacement
Students link sampling to drawing cards or picking a subset from a population.
Fix: Remember that replacement is what makes each resample different. Without it, a full-size resample would just reproduce the original data.
Using the data's standard deviation as the bootstrap standard error
Both are standard deviations, so they get mixed up.
Fix: The bootstrap standard error is the standard deviation of the statistic across resamples, for example of the B resampled means.
Dividing by B instead of B − 1
Students forget the bootstrap estimates are themselves a sample.
Fix: Use B − 1 unless the question says to use the population formula. Use the sample standard deviation key on your calculator.
Believing bootstrapping can produce extreme values beyond the data
The word 'simulation' suggests new random outcomes.
Fix: Bootstrap draws only reuse observed values. A bootstrap VaR or ES cannot exceed the worst observation in the sample.
Applying the iid bootstrap to dependent data without comment
Students forget that resampling single observations breaks any time ordering.
Fix: If returns show autocorrelation or volatility clustering, the iid bootstrap misses it and can understate risk. Say so when a question hints at dependence.
Confusing the number of observations n with the number of resamples B
Both are just counts in the question stem.
Fix: The resample size n sets the accuracy of the data's information. B sets only how smoothly the bootstrap distribution is approximated. More resamples do not add data.

Worked examples

Example 1

An analyst bootstraps the mean daily return (in %) of a portfolio and obtains five bootstrap means: 1.2, 0.8, 1.0, 1.4 and 0.6. Estimate the bootstrap standard error of the mean. (Five resamples is only for illustration.)

Show the solution

Mean of the bootstrap estimates = (1.2 + 0.8 + 1.0 + 1.4 + 0.6) ÷ 5 = 5.0 ÷ 5 = 1.0.
Deviations from 1.0 are 0.2, −0.2, 0.0, 0.4 and −0.4.
Squared deviations are 0.04, 0.04, 0.00, 0.16 and 0.16, which sum to 0.40.
Divide by B − 1 = 4: 0.40 ÷ 4 = 0.10.
Take the square root: √0.10 = 0.3162.

Answer: The bootstrap standard error of the mean is about 0.316 percentage points.

Example 2

A risk analyst has 10 historical loss observations and draws a bootstrap sample of 10 values, with replacement, each draw equally likely. (a) What is the probability that one specific observation does not appear in the resample? (b) What is the expected number of distinct original observations in the resample? Options for (a): A) 0.100, B) 0.349, C) 0.651, D) 0.900.

Show the solution

Each draw misses the specific observation with probability 1 − 1/10 = 0.9.
Draws are independent, so over 10 draws the probability of missing it every time is 0.9¹⁰.
0.9² = 0.81, 0.9⁴ = 0.6561, 0.9⁸ = 0.43047, and 0.9¹⁰ = 0.43047 × 0.81 = 0.3487.
So (a) is about 0.349, option B. Option C (0.651) is the probability the observation appears at least once.
For (b), each observation appears at least once with probability 1 − 0.3487 = 0.6513.
Expected distinct observations = 10 × 0.6513 = 6.51.

Answer: (a) About 0.349 (option B). (b) About 6.5 distinct observations, so roughly 65% of the original data appear in a resample.

Exam tips

Look for the words 'with replacement' and 'same size as the original sample'. These are the defining features and are often the key to a conceptual question.
When given a list of bootstrap estimates, decide first whether the question wants the standard error (a standard deviation) or an interval (percentiles).
Expect a link to historical simulation VaR. Know that bootstrapping reuses observed losses, so it cannot give values beyond the worst loss in the sample.
Know the iid weakness: it ignores serial dependence and volatility clustering. Options that claim bootstrapping fixes this are wrong.
Keep the (1 − 1/n)ⁿ ≈ 0.368 result ready for quick probability questions, and use your calculator's sample standard deviation key.

Practice questions from Simulation and Bootstrapping

Bootstrapping: frequently asked questions

What is bootstrapping in FRM Part I?

It is a resampling method. You draw new samples of the same size, with replacement, from your observed data and compute a statistic on each. The results estimate the statistic's sampling distribution, standard error and confidence interval.

Why is sampling done with replacement in the bootstrap?

Replacement lets each draw come from the full original data set with equal probability. It makes every resample different and mimics drawing fresh samples from the underlying population. Without replacement, a full-size resample would just copy the original data.

How is bootstrapping used with historical simulation VaR?

You resample past returns or losses with replacement and compute VaR on each resample. The average gives a VaR estimate and the spread shows its sampling error. It cannot produce losses worse than those already in the data.

What are the main limitations of the iid bootstrap?

It assumes observations are independent and identically distributed, so it ignores autocorrelation and volatility clustering. It also depends on the sample being representative and can only reuse observed values. Small samples give unreliable results.