How does bootstrapping work for constructing confidence intervals in risk analysis?

Bootstrapping is a resampling technique that estimates the sampling distribution of a statistic by repeatedly drawing samples (with replacement) from the observed data. It's powerful because it doesn't require assumptions about the underlying distribution — making it ideal for risk metrics that have complex or unknown distributions.

The Bootstrap Procedure

1. Start with your data: You have n observations (e.g., 250 daily P&L values).
2. Draw a bootstrap sample: Randomly select n observations with replacement from the original data. Some observations will appear multiple times, others won't appear at all.
3. Compute the statistic: Calculate whatever you're interested in (mean, VaR, Expected Shortfall, Sharpe ratio, etc.) from this bootstrap sample.
4. Repeat B times: Typically B = 1,000 to 10,000 iterations.
5. Build the confidence interval: Sort the B bootstrap statistics and take the appropriate percentiles.

Worked Example

Kensingston Capital wants a 95% confidence interval for their portfolio's 99% VaR, estimated from 500 daily returns.

1. Original 99% VaR estimate from the 500 returns = $4.2 million.
2. Draw 5,000 bootstrap samples, each of size 500 (with replacement).
3. For each bootstrap sample, compute the 99% VaR.
4. Sort the 5,000 VaR estimates from smallest to largest.
5. The 95% CI is the 2.5th percentile to the 97.5th percentile:

• Lower bound: 125th sorted value = $3.6 million
• Upper bound: 4,875th sorted value = $5.1 million

1. Result: 95% CI for VaR is [$3.6M, $5.1M].

When Bootstrap Beats Normal Theory:

• When the statistic's distribution is skewed (VaR, ES, max drawdown)
• When sample sizes are small and CLT assumptions are shaky
• When the formula for the standard error is unknown or complex (e.g., for ratios like Sharpe)

Limitations:

• If the original sample is not representative of the population, bootstrap won't fix that
• Extremely heavy-tailed data (infinite variance) can produce unreliable bootstrap intervals
• Computationally intensive for very large datasets

Try our FRM question bank for more practice on resampling methods.