Why Bootstrap Risk-Measure Intervals?
An empirical VaR or TVaR is calculated from observed losses, so it depends on which observations happened to enter the data. A single point estimate does not show how much that estimate might vary across samples. The bootstrap provides a practical way to examine this sampling variability: repeatedly draw samples from the observed losses, recalculate the risk measure in each sample, and use the resulting estimates to form an interval.
For tail measures, the interval must be interpreted carefully. At a high confidence level, only a small fraction of observations directly determine the upper tail. Resampling cannot create new tail experience that was not represented in the original data. Consequently, a numerically precise interval from many bootstrap replicates can still be unreliable if the observed tail is sparse or unrepresentative.
Empirical VaR, TVaR, and the Percentile Interval
For a loss variable \(X\) with distribution function \(F_X\), define the \( \alpha \)-level VaR as the left quantile \( \operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\geq\alpha\} \). For an empirical sample, this is the corresponding order statistic under the empirical distribution. In particular, with the inverse empirical distribution convention, it is the observation at rank \( \lceil n\alpha\rceil \) after sorting the sample in increasing order.
Here TVaR is defined by the quantile-integral convention: \( \operatorname{TVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_{\alpha}^{1}\operatorname{VaR}_{u}(X)\,du \). For an empirical distribution this definition averages the upper tail, allowing a fractional contribution from the observation at the boundary when the tail probability does not correspond to a whole number of observations. This convention matters in finite samples; the phrase “average losses above VaR” can give a different result when there are ties or when the boundary observation has positive probability.
To construct percentile intervals, calculate the point estimate from the original data, generate \(B\) bootstrap samples, and calculate the same statistic in each replicate. Sort the resulting \(B\) bootstrap estimates. For a nominal \(100(1-\gamma)\%\) percentile interval, use the \( \gamma/2 \) and \(1-\gamma/2\) quantiles of the bootstrap estimates. This tutorial uses the nearest-rank convention for finite lists: the quantile at probability \(p\) is the sorted value at rank \( \lceil Bp\rceil \), with rank at least 1.
Find empirical VaR or TVaR from the observed losses, using the stated quantile and tail-average convention.
Draw \(n\) observations with replacement for each of \(B\) replicates. Resample the independent observational units, not individual entries that belong to the same dependent unit.
For each replicate, calculate the same empirical VaR or TVaR at the same confidence level.
Sort the \(B\) results and select the bootstrap quantiles corresponding to the desired interval probabilities.
Why Resampling Approximates Sampling Variation
The method has a simple justification. If the original losses are independent observations from an unknown distribution, the empirical distribution \( \widehat F \) estimates that distribution. Conditional on the observed data, a bootstrap sample consists of independent draws from \( \widehat F \). Recalculating a statistic across these samples therefore measures how that statistic varies under repeated sampling from the estimated distribution.
Here \(T\) is the chosen statistic, such as empirical VaR or empirical TVaR, and \( \widehat\theta^{*(b)} \) is its value in bootstrap replicate \(b\). The percentile interval uses quantiles of the conditional distribution of these replicate estimates. This is an approximation to sampling uncertainty, not an exact finite-sample coverage guarantee. It can be especially fragile for extreme quantiles, where a small change in the largest observations can materially change the estimate.
Worked Example: Exact Bootstrap Distribution of VaR
A hypothetical portfolio has four observed annual losses, in thousands of dollars: \(10,20,30,80\). Construct the exact bootstrap distribution of the empirical \(75\%\) VaR for bootstrap samples of size four, then find its nearest-rank \(95\%\) percentile interval. Use the inverse empirical quantile convention.
Method. In every bootstrap sample the \(75\%\) VaR is the third-smallest observation, since \( \lceil4(0.75)\rceil=3 \). Thus, the VaR is at most a given observed loss when at least three of the four draws are no larger than that loss.
VaR equals \(10\). A bootstrap draw equals \(10\) with probability \(1/4\). The third order statistic is \(10\) if at least three of the four draws are \(10\):
VaR is at most \(20\). Each draw is either \(10\) or \(20\) with probability \(1/2\). At least three draws must be from these two values for the third order statistic to be no larger than \(20\):
Subtracting the probability of a VaR of \(10\) gives \(P(\widehat{\operatorname{VaR}}_{0.75}^{*}=20)=(80-13)/256=67/256\).
VaR is at most \(30\). A draw is in \(\{10,20,30\}\) with probability \(3/4\). The third order statistic is at most \(30\) when at least three draws are in this set:
Therefore the probability of a VaR of \(30\) is \((189-80)/256=109/256\), and the probability of a VaR of \(80\) is \(1-189/256=67/256\). The four masses are \(13/256,67/256,109/256,67/256\), which sum to \(256/256\), as required.
The cumulative probability is \(13/256\) at \(10\), \(80/256\) at \(20\), \(189/256\) at \(30\), and \(256/256\) at \(80\). The \(2.5\%\) percentile is \(10\), because \(13/256\approx0.0508\) already exceeds \(0.025\). The \(97.5\%\) percentile is \(80\), because the cumulative probability at \(30\) is below \(0.975\). Thus the exact percentile interval under the bootstrap distribution is \([10,80]\) thousand dollars.
Check. The masses add to one, and the interval endpoints are observed bootstrap VaR values. The common mistake avoided here is to calculate the event “at least three draws are at most \(30\)” incorrectly: its two disjoint cases have probabilities \(108/256\) and \(81/256\), totaling \(189/256\).
Bootstrap TVaR with a Fractional Tail Boundary
For TVaR, the statistic must be recalculated from the complete resampled empirical distribution in every replicate. One should not simply select observations strictly greater than the replicate VaR and average them. That approach can omit the boundary mass and does not in general implement the quantile-integral definition.
Worked Example: A Percentile Interval for TVaR
A hypothetical insurer records four independent annual losses, in thousands of dollars: \(10,20,40,90\). Estimate \(50\%\) TVaR and use the following eight size-four bootstrap samples to form a nearest-rank \(95\%\) percentile interval. For this example, each replicate is listed as an unordered multiset.
The original empirical distribution assigns probability \(1/4\) to each loss. The upper half of the distribution consists of the two largest losses, \(40\) and \(90\), so the original estimate is \( (40+90)/2=65 \) thousand dollars.
For each resample, \(50\%\) TVaR is the average of its two largest observations. The resamples and calculations are:
- \(\{10,10,20,40\}\): \( (20+40)/2=30\).
- \(\{10,20,20,90\}\): \( (20+90)/2=55\).
- \(\{20,20,40,90\}\): \( (40+90)/2=65\).
- \(\{10,40,40,90\}\): \( (40+90)/2=65\).
- \(\{10,10,10,90\}\): \( (10+90)/2=50\).
- \(\{20,40,90,90\}\): \( (90+90)/2=90\).
- \(\{10,20,40,40\}\): \( (40+40)/2=40\).
- \(\{40,40,90,90\}\): \( (90+90)/2=90\).
Sorting the eight bootstrap estimates gives \(30,40,50,55,65,65,90,90\). For \(B=8\), the nearest-rank lower endpoint is at rank \( \lceil8(0.025)\rceil=1\), and the upper endpoint is at rank \( \lceil8(0.975)\rceil=8\). The resulting illustrative \(95\%\) percentile interval is \([30,90]\) thousand dollars.
Check. Each replicate uses the two largest observations because its size is four and the tail probability is \(1-0.50=0.50\). The interval is broad relative to the original estimate of \(65\), showing how much the estimate can vary even in this small sample. Eight replicates are used only to make the arithmetic visible; they are far too few for a stable practical interval.
Sparse Tail Data and Interval Reliability
The bootstrap samples from the empirical distribution, not from the unknown population directly. If the original data contain only one exceptionally large observation, every bootstrap sample can either omit it or include it multiple times. That resampling behavior can make an interval unstable and may fail to represent losses larger than anything observed.
Worked Example: How Often Does a Bootstrap Miss the Only Extreme Loss?
Suppose a hypothetical insurer has \(20\) independent observed losses, in thousands of dollars, equal to \(1,2,\ldots,19,100\). At the \(95\%\) confidence level, calculate the probability that a size-20 bootstrap sample contains no copy of the observed loss \(100\). Explain what this says about a bootstrap estimate of \(95\%\) TVaR.
In each draw, the probability of selecting one of the other 19 observations is \(19/20=0.95\). Bootstrap draws are made with replacement, so the probability that all 20 draws omit the loss of \(100\) is
Thus, about \(35.85\%\) of bootstrap samples contain no copy of the only loss of \(100\). For a sample of size 20, the quantile-integral \(95\%\) TVaR averages the upper \(5\%\) of the empirical distribution, equivalent here to the largest observation because \(20(0.05)=1\). In replicates that omit \(100\), this bootstrap TVaR is no larger than \(19\); in replicates that include it, the value can be much larger.
Check. The probability is for omitting the single extreme observation on every draw, not merely on the first draw. The calculation makes the limitation concrete: the bootstrap can reveal variability caused by the observed maximum, but it cannot show how much larger an unobserved population loss might be.
Practical Choices and Common Traps
For an actual analysis, use enough bootstrap replicates that the requested endpoint quantiles are not determined by only a handful of simulated estimates. Several thousand replicates are a common practical starting point, with more often needed for very high confidence levels or unusually noisy tail statistics. Increasing \(B\) reduces Monte Carlo noise in the bootstrap quantiles; it does not fix a small or unrepresentative original data set.
| Issue | Why it matters | Appropriate response |
|---|---|---|
| Few observations beyond the target quantile | The estimate and its bootstrap distribution depend heavily on a small number of losses. | Report the tail count and interpret the interval cautiously. |
| Dependence between observations | Resampling individual observations independently can break the dependence structure. | Resample the appropriate independent units or groups. |
| Different quantile conventions | Finite-sample VaR and TVaR can change with the order-statistic or boundary convention. | State the convention and apply it consistently to the data and every replicate. |
| Too few bootstrap replicates | Extreme interval endpoints can be dominated by Monte Carlo noise. | Increase \(B\), then assess whether the endpoints are stable across runs. |
| Interpreting the interval as guaranteed coverage | A percentile interval is an approximation, not an exact finite-sample confidence statement. | Consider the data-generating process and tail support when interpreting it. |
A common exam trap is to treat the number of bootstrap replications as the amount of original tail information. For example, generating 100,000 resamples from a data set with one extreme loss does not turn that one loss into 100,000 independent tail observations. The resamples repeatedly reuse the same empirical information.
Check Your Understanding
Use the stated conventions in each problem. Answers are included for checking after you work through the calculations.
- A hypothetical loss sample is \(5,15,25,45\). Using the inverse empirical quantile convention, what is the original \(75\%\) VaR? (Answer: \(25\))
- For a size-four bootstrap sample from the data \(10,20,30,80\), what is the probability that the \(75\%\) VaR equals \(80\)? (Answer: \(67/256\approx0.2617\))
- A data set has four losses \(12,22,42,82\). Under the quantile-integral definition, calculate empirical \(50\%\) TVaR. (Answer: \(62\))
- With \(B=40\) bootstrap replicates, find the nearest-rank indices for the endpoints of a \(90\%\) percentile interval. (Answer: ranks 2 and 38)
- A sample of \(25\) losses contains exactly one observation above a chosen tail threshold. What is the probability that a size-25 bootstrap sample contains no copy of that observation? Give four decimals. (Answer: \((24/25)^{25}\approx0.3604\))