Why Put an Interval Around an Estimated TVaR?
An empirical TVaR is calculated from observed losses, so its value depends on which losses happened to enter the sample. That matters especially in the tail: a small number of large observations can have substantial influence on the estimate. A point estimate alone does not show how much it might vary across samples drawn under similar conditions.
The bootstrap estimates this sampling variability by repeatedly drawing new samples from the observed losses, with replacement, and recalculating empirical TVaR. The resulting estimates form an approximate sampling distribution. A confidence interval constructed from that distribution communicates uncertainty about the estimate; it is not a forecast range for the next loss.
Empirical TVaR Inside Each Replicate
Let the ordered observed losses be \(x_{(1)}\leq\cdots\leq x_{(n)}\), and let \(p\) be the confidence level. The empirical TVaR is the average of the empirical quantile over confidence levels from \(p\) to 1. This is also the average loss in the upper tail when the loss at the VaR boundary receives the appropriate fractional weight. That convention matters when the tail contains a non-integer number of observations or the losses are tied.
For the particularly simple case \(np=k\), where \(k\) is an integer, the upper tail comprises exactly \(n-k\) observations, and the empirical estimate is their arithmetic mean:
When \(np\) is not an integer, use the fractional VaR-mass treatment rather than rounding the tail count. A bootstrap replicate must use the same rule as the original sample. Otherwise, variation between replicates can reflect an inconsistent calculation rather than sampling uncertainty.
Worked Example: Recalculating TVaR in Each Bootstrap Replicate
A hypothetical analyst observes four annual losses, in thousands of dollars: \(10, 20, 40,\) and \(80\). Estimate TVaR at \(p=0.50\), then calculate the TVaR for each of six specified bootstrap replicates. The replicates are provided to make the arithmetic transparent; six replicates are not enough for a dependable confidence interval.
Original estimate. Here \(n=4\) and \(np=4(0.50)=2\). The upper tail contains \(4-2=2\) observations. Their losses are \(40\) and \(80\), so
Replicate calculations. In every row, order the resampled values and average the largest two. The bootstrap resamples are:
| Replicate | Resampled losses, in thousands | Largest two | Replicate TVaR |
|---|---|---|---|
| 1 | 10, 20, 20, 80 | 20, 80 | \((20+80)/2=50\) |
| 2 | 10, 10, 40, 80 | 40, 80 | \((40+80)/2=60\) |
| 3 | 20, 20, 40, 40 | 40, 40 | \((40+40)/2=40\) |
| 4 | 10, 40, 80, 80 | 80, 80 | \((80+80)/2=80\) |
| 5 | 10, 10, 20, 40 | 20, 40 | \((20+40)/2=30\) |
| 6 | 20, 40, 40, 80 | 40, 80 | \((40+80)/2=60\) |
The six estimates, sorted, are \(30,40,50,60,60,80\) thousand dollars. Their mean is \((30+40+50+60+60+80)/6=53.33\) thousand dollars, to two decimal places. The bootstrap mean need not equal the original estimate of \(60\). The calculation checks out: each replicate retains four draws, and each tail average uses exactly two losses.
Method and mistake avoided. The method is to resample first and then recalculate the statistic. Resampling without replacement would keep all four original losses in every replicate; every replicate TVaR would then remain \(60\), falsely suggesting no sampling variation.
Percentile Bootstrap Confidence Intervals
A common interval is the percentile bootstrap interval. Calculate \(B\) bootstrap TVaR estimates, order them, and take the lower and upper quantiles corresponding to the desired confidence level. For a nominal 95% interval, these are the 2.5th and 97.5th percentiles of the bootstrap estimates.
The quantile convention for a finite set of replicates should be specified. One simple convention is the nearest-rank empirical quantile: for probability \(q\), select the ordered value at rank \(\lceil Bq\rceil\), with ranks starting at 1. The examples here use that convention. In practice, other software may interpolate between ordered values; report the convention so the interval can be reproduced.
where \(\widehat{\operatorname{TVaR}}^{\,*}_{(j)}\) is the \(j\)th ordered bootstrap estimate and the nominal confidence level is \(1-\alpha\). If the resulting rank is below 1, use rank 1. A practical bootstrap generally uses many more than a handful of replicates so the interval endpoints are not determined by only the most extreme simulated estimates.
Choose the loss unit, confidence level \(p\), and empirical TVaR convention. Apply these consistently throughout.
For each of \(B\) replicates, draw \(n\) observations with replacement from the original sample of \(n\) losses.
Order each resample and calculate its empirical TVaR at the same \(p\), including fractional VaR-mass treatment when needed.
Use the specified bootstrap quantile convention to obtain the interval endpoints, and state the nominal confidence level.
Why this works, approximately. If the observed losses are representative independent observations from the process of interest, their empirical distribution estimates that process. Sampling from this empirical distribution creates new samples that mimic repeated sampling from the estimated process. Recalculating TVaR in each sample therefore approximates how the estimator varies under repeated sampling.
This justification is approximate, not a guarantee. In particular, the bootstrap cannot generate a loss value larger than the largest observed loss. If the true tail is poorly represented in the sample, resampling can reproduce the sample’s uncertainty but cannot supply unobserved tail information. The interval also depends on the observations being suitable resampling units; if observations are dependent or systematically selected, naive independent resampling may not represent the sampling process.
Worked Example: Constructing a Percentile Interval
A hypothetical insurer has five observed claim losses, in thousands of dollars: \(5,10,20,40,100\). Estimate TVaR at \(p=0.60\). Then use a specified summary of \(B=100\) bootstrap estimates to form a 95% percentile interval. Assume the ordered bootstrap estimates have values \(20\) at rank 3, \(70\) at rank 50, and \(100\) at rank 98.
Original estimate. Since \(np=5(0.60)=3\), the upper tail contains \(5-3=2\) observations. Thus
Interval ranks. For a 95% interval, \(\alpha=0.05\). Under the nearest-rank convention, the lower rank is \(\lceil100(0.025)\rceil=\lceil2.5\rceil=3\). The upper rank is \(\lceil100(0.975)\rceil=\lceil97.5\rceil=98\).
The stated bootstrap values at those ranks are \(20\) and \(100\), respectively. Therefore, the percentile interval is \([20,100]\) thousand dollars. Its midpoint is not required to equal the original estimate, and its width reflects the variation among the bootstrap estimates. The calculation checks: both selected ranks lie between 1 and 100, and the lower endpoint does not exceed the upper endpoint.
Method and mistake avoided. The endpoints are quantiles of the replicate TVaRs, not quantiles of the original losses. Treating the 95% interval as a range expected to contain 95% of individual claims would confuse uncertainty about the estimated TVaR with the distribution of claim amounts.
Interpretation, Assumptions, and Common Traps
A 95% bootstrap confidence interval describes uncertainty in the estimated TVaR under the resampling model. It is not a statement that 95% of losses fall between the endpoints, nor does it mean that a particular future loss has a 95% chance of landing there. A wide interval signals that the estimate is sensitive to the observed sample under the bootstrap procedure; it does not identify the reason for that sensitivity by itself.
The standard bootstrap described here assumes that the observed losses are an appropriate independent sample from the population or process being studied. Match the resampling unit to the data and the target quantity. For example, if each observation is an annual aggregate loss and the target is TVaR of annual aggregate losses, resample annual totals, not the individual payments that happened to make up those totals.
Common exam traps include:
- Resampling without replacement. That produces the original sample in a different order, not bootstrap variation.
- Changing the tail rule. Use the same \(p\) and the same fractional-boundary convention for the original estimate and every replicate.
- Using the wrong observations. Resampling individual components when the target concerns aggregate outcomes changes the estimand.
- Overreading a narrow interval. A bootstrap cannot reveal tail losses absent from the sample; its interval is conditional on the information the sample contains.
- Ignoring the number of replicates and quantile convention. Both affect the stability and reproducibility of estimated endpoints, especially in the extreme percentiles.
Worked Example: Resampling Annual Losses at the Correct Unit
A hypothetical risk manager has five annual aggregate losses, in thousands of dollars: \(20,30,40,50,160\). The target is TVaR at \(p=0.80\). Five bootstrap replicates of these annual totals are supplied below. Calculate each replicate TVaR and the sample standard deviation of the five replicate estimates. Treat the replicates as a small illustration, not as a stable interval calculation.
Original estimate. Here \(np=5(0.80)=4\), so the empirical upper tail contains one annual loss. The original estimate is the largest observation, \(160\) thousand dollars.
Replicate TVaRs. Because each resample has five annual totals and the upper tail contains one observation, each replicate TVaR is its maximum:
| Replicate annual totals, in thousands | Maximum and replicate TVaR |
|---|---|
| 20, 30, 50, 50, 160 | 160 |
| 20, 30, 30, 40, 50 | 50 |
| 20, 20, 30, 40, 50 | 50 |
| 20, 30, 30, 40, 40 | 40 |
| 30, 40, 50, 160, 160 | 160 |
The five estimates are \(160,50,50,40,160\), with mean \((160+50+50+40+160)/5=92\) thousand dollars. The sum of squared deviations from 92 is
The sample variance is \(15{,}480/(5-1)=3{,}870\) in squared thousands of dollars, and the sample standard deviation is \(\sqrt{3{,}870}=62.21\) thousand dollars, rounded to two decimal places. This measures the spread of these five replicate estimates; with only five replicates it is highly variable. The calculation checks: the replicate estimates are all maxima, as required by the one-observation upper tail.
Method and mistake avoided. Each observed value represents a complete annual outcome, so the bootstrap draws complete annual totals. Resampling individual payments and rebuilding annual totals would require a different model and would not be the stated empirical bootstrap for the observed annual-loss sample.
Check Your Understanding
For each problem, use the stated empirical-tail rule and show the intermediate calculations.
- A sample of four losses, in thousands of dollars, is \(4,9,15,32\). Calculate empirical TVaR at \(p=0.75\). (Answer: \(32\) thousand dollars.)
- Five observed losses, in thousands of dollars, are \(3,7,9,15,31\). Calculate empirical TVaR at \(p=0.60\), using the integer-tail formula. (Answer: \(23\) thousand dollars.)
- Four bootstrap TVaR estimates, in thousands of dollars, are \(12,16,20,24\). Calculate their sample standard deviation, rounding to two decimal places. (Answer: \(5.16\) thousand dollars.)
- A bootstrap uses \(B=40\) replicates and the nearest-rank convention. The ordered estimates at ranks 2 and 38 are \(18\) and \(64\), respectively. What is the 90% percentile interval? (Answer: \([18,64]\), in the same units as the estimates.)
- An analyst wants TVaR of annual aggregate losses. The data contain one annual total per year, built from multiple claim payments. Which observations should the analyst resample in the ordinary empirical bootstrap, and why? (Answer: Resample the annual totals, because they are the observed units for the target distribution of annual aggregate losses.)