Clustered Resampling for Policy-Year Experience

Why the Resampling Unit Matters

A claims file may contain many individual claims, but those records are not necessarily independent. Claims from the same policy-year can share coverage terms, weather conditions, operational changes, or other influences. If those claims are related, resampling them one at a time can break the dependence pattern and give a misleading measure of uncertainty.

Clustered resampling addresses this by treating each independent exposure group as one sampling unit. For policy-year data, the cluster is typically all the records belonging to one policy during one year, or all records for a defined group of policies during one year if that is the independent unit. A bootstrap replicate draws whole clusters with replacement and includes every claim, exposure amount, and other relevant record from each selected cluster.

Definition: A cluster bootstrap resamples independent groups of observations, rather than resampling the observations inside each group separately. If there are $n$ observed clusters, each bootstrap sample consists of $n$ draws with replacement from those clusters. The statistic of interest is recalculated from each sample.

The method requires a defensible choice of independent clusters. Clusters should be reasonably representative of the population or process to which the estimate is intended to apply. Dependence within a cluster may be substantial; dependence between clusters is the concern. If the same policy appears across several years and its records remain related across those years, treating each policy-year as independent may not be appropriate. The cluster definition should preserve the dependence relevant to the uncertainty being measured.

How to Construct a Cluster Bootstrap

Let the observed clusters be $C_1,\ldots,C_n$, and let $\hat{\theta}$ be an estimate calculated from them. For bootstrap replicate $b$, draw $n$ clusters independently with replacement from the observed $n$ clusters, then calculate $\hat{\theta}^{*(b)}$ using the complete selected clusters. With $B$ replicates, the bootstrap standard error is the sample standard deviation of the replicate estimates:

\widehat{\operatorname{SE}}_{\text{boot}}(\hat{\theta}) = \sqrt{\frac{1}{B-1}\sum_{b=1}^{B} \left(\hat{\theta}^{*(b)}-\overline{\hat{\theta}}^*\right)^2}, \qquad \overline{\hat{\theta}}^* = \frac{1}{B}\sum_{b=1}^{B}\hat{\theta}^{*(b)}.

The bootstrap average $\overline{\hat{\theta}}^*$ need not equal the original estimate exactly, especially when $B$ is finite. The standard error measures the spread of the replicate estimates; it is not itself an estimate of the parameter. A bootstrap interval can also be formed from quantiles of the replicate estimates, provided the quantile convention is stated.

Set the target and sampling unit.
State what quantity is being estimated and identify the independent exposure cluster. Do not decide the cluster solely from how the data file is arranged.

Resample complete clusters.
Draw the original number of clusters with replacement. Retain all records and linked quantities within each selected cluster.

Recalculate the estimate.
Apply the original estimator to each bootstrap sample. For a rate, recalculate its numerator and denominator from the sampled clusters.

Summarize the replicates.
Use their sample standard deviation for a bootstrap standard error, or their ordered values for a percentile interval.

Why Resampling Whole Clusters Preserves Dependence

The reason for retaining complete clusters can be seen by considering a mean. Suppose each of $n$ independent clusters contains $m$ claim values, and the target is the average claim value across the observed clusters, with equal cluster sizes. Write $Z_j$ for the mean claim value in cluster $j$. The overall mean is then the mean of $Z_1,\ldots,Z_n$. Resampling clusters resamples these cluster means as units, preserving the variation between clusters and the dependence among claims within each cluster.

In contrast, claim-by-claim resampling treats all $nm$ claim values as if they could be independently selected. That procedure does not preserve which claims occurred together. When claims within a cluster tend to move together, the claim-level resample can understate the variability of an estimate based on future clusters.

Justification: In a cluster bootstrap, each replicate mean is the average of $n$ draws from the empirical distribution that assigns probability $1/n$ to each observed cluster mean $Z_j$. Its conditional variance is therefore the empirical variance of the cluster means, using divisor $n$, divided by $n$. This follows because the $n$ resampled cluster draws are independent conditional on the observed data. Resampling individual claims instead uses the empirical distribution of claim values and generally produces a different variance.

Worked Example: Comparing Claim-Level and Cluster-Level Resampling

A hypothetical insurer observes four independent policy-years. Each year has two claims. The claim amounts, in dollars, are $(0,0)$, $(0,0)$, $(40,40)$, and $(40,40)$. Estimate the average claim amount and compare the exact bootstrap standard errors from resampling policy-years and resampling claims. For this calculation, use the exact bootstrap distributions rather than a finite simulation.

Original estimate. There are eight claims with total amount $4(40)=160$ dollars, so the observed average is $160/8=\$20$. The four policy-year means are $0,0,40,40$.

Policy-year bootstrap. Each of four draws selects a policy-year with replacement. Let $K$ be the number of selected years whose claims are both \$40. Each draw has probability $1/2$ of selecting such a year, so $K\sim\operatorname{Binomial}(4,1/2)$. The bootstrap average claim amount is $40K/4=10K$ dollars. Since $\operatorname{Var}(K)=4(1/2)(1/2)=1$, the variance of the bootstrap estimate is $10^2(1)=100$ dollars squared, and its standard error is $\sqrt{100}=\$10.00$.

Claim-level bootstrap. The eight claim records contain four zeros and four \$40 amounts. In eight draws with replacement, let $J$ be the number of selected \$40 claims. Then $J\sim\operatorname{Binomial}(8,1/2)$, and the replicate average is $40J/8=5J$ dollars. Its variance is $5^2\operatorname{Var}(J)=25[8(1/2)(1/2)]=50$ dollars squared. The claim-level bootstrap standard error is $\sqrt{50}=\$7.07$, rounded to the nearest cent.

The claim-level method gives a smaller standard error because it can select one of the high claims without the other high claim from that policy-year. The cluster method retains the observed within-year pairing. Method and trap: resample the four complete policy-years when those are the independent units; treating the eight related claims as eight independent units ignores the clustering.

Unequal Exposure and Other Cluster-Level Quantities

When exposure differs across clusters, do not automatically average cluster-specific rates. If the target is total loss per unit of exposure, a natural estimate is the ratio of total loss to total exposure. For cluster $j$, let $L_j$ be its loss amount and $e_j$ its exposure. The estimate is $\sum_j L_j/\sum_j e_j$. Each bootstrap replicate must resample the $(L_j,e_j)$ pairs together and recompute this ratio. Resampling losses without their corresponding exposures would alter the relationship between the numerator and denominator.

Worked Example: Resampling a Loss Rate With Unequal Exposure

A hypothetical portfolio has three independent clusters with exposure and loss, respectively, of $(1,\$10{,}000)$, $(2,\$40{,}000)$, and $(1,\$30{,}000)$. Estimate loss per exposure unit. Then calculate the bootstrap standard error from the four specified cluster-bootstrap replicates: $(1,10{,}000),(1,10{,}000),(1,30{,}000)$; all three original clusters once each; $(2,40{,}000),(2,40{,}000),(1,30{,}000)$; and $(1,10{,}000),(1,30{,}000),(1,30{,}000)$. For easier arithmetic, report losses in thousands of dollars.

Original estimate. Total exposure is $1+2+1=4$ units. Total loss is $10+40+30=80$ thousand dollars. The estimated rate is $80/4=20$ thousand dollars per exposure unit.

Replicate estimates. Recalculate both totals for each replicate:

First replicate: loss $10+10+30=50$, exposure $1+1+1=3$, rate $50/3=16.6667$.
Second replicate: loss $80$, exposure $4$, rate $80/4=20$.
Third replicate: loss $40+40+30=110$, exposure $2+2+1=5$, rate $110/5=22$.
Fourth replicate: loss $10+30+30=70$, exposure $1+1+1=3$, rate $70/3=23.3333$.

The four replicate estimates have mean $(16.6667+20+22+23.3333)/4=20.5$. Their sum of squared deviations from 20.5 is $(16.6667-20.5)^2+(20-20.5)^2+(22-20.5)^2+(23.3333-20.5)^2=25.2222$, to the shown precision. The sample variance is $25.2222/(4-1)=8.4074$, and the bootstrap standard error is $\sqrt{8.4074}=2.90$ thousand dollars per exposure unit, rounded to two decimals.

This is a small, specified set of four replicates, so its standard error is only an illustration of the calculation; an applied bootstrap would normally use many more replicates. Method and trap: keep exposure attached to its cluster and recompute the ratio. Averaging the three original cluster rates $10/1$, $40/2$, and $30/1$ would give a different estimand from total loss per total exposure.

Bootstrap Intervals and Small Samples

A percentile interval uses the selected lower and upper quantiles of the bootstrap estimates as interval endpoints. It is straightforward to calculate, but its interpretation depends on the sampling design, the statistic, and the available clusters. With few clusters, a bootstrap distribution can be discrete and coarse. A large number of resamples does not create new independent information; it only describes the resampling distribution implied by the observed clusters more precisely.

Worked Example: An Exact Percentile Interval From Four Years

For a hypothetical portfolio, the aggregate losses in four independent policy-years are $10,10,30,30$ thousand dollars. Estimate the mean annual aggregate loss and find the exact 95% cluster-bootstrap percentile interval. Use the inverse empirical distribution convention: the $p$-quantile is the smallest value whose cumulative probability is at least $p$.

The original mean is $(10+10+30+30)/4=20$ thousand dollars. In a bootstrap sample of four years, let $K$ be the number of selected years with loss 30. Each draw has probability $1/2$ of selecting such a year, so $K\sim\operatorname{Binomial}(4,1/2)$. The replicate mean is $10+5K$ thousand dollars.

For $K=0,1,2,3,4$, the replicate means are $10,15,20,25,30$, with probabilities $1/16,4/16,6/16,4/16,1/16$, respectively. The cumulative probabilities are $1/16,5/16,11/16,15/16,16/16$. The 2.5th percentile is 10 because $1/16=0.0625$ already exceeds 0.025. The 97.5th percentile is 30 because the cumulative probability at 25 is $15/16=0.9375$, less than 0.975. Thus the exact percentile interval is $[10,30]$ thousand dollars.

The interval is wide and has only five possible bootstrap estimates, reflecting the small number of observed years. Method and trap: retain the policy-year aggregate as a unit. Resampling individual claim records would not reproduce this year-level bootstrap distribution.

Practical Checks Before Resampling

Before interpreting a clustered bootstrap result, check that the resampled unit matches the target population and dependence structure. The method quantifies sampling variation under the assumption that the observed clusters provide a useful representation of future independent clusters. It does not automatically correct for a change in coverage, incomplete records, or a systematic difference between observed and target exposures.

State the estimand. A mean per claim, a mean annual aggregate, and loss per exposure unit are different targets.
Keep linked data together. Preserve claims, exposure, and any other quantities whose joint behavior is part of the cluster.
Match the cluster to dependence. If dependence persists across multiple policy-years, resampling each policy-year independently may still split related observations.
Use enough replicates for numerical stability. More replicates reduce Monte Carlo noise in the calculated bootstrap summaries, but cannot compensate for very few observed clusters.
Distinguish resampling uncertainty from data limitations. Bootstrap variation reflects the chosen empirical resampling scheme; it cannot reveal unobserved sources of bias by itself.

Clustered resampling is most useful when the independent sampling units are groups but the estimate is calculated from records inside those groups. The central exam skill is to identify the unit that could independently have been observed again, preserve its internal structure, and recompute the stated estimate without changing its definition.

Check Your Understanding

For each problem, identify the resampling unit and calculate or interpret the requested result.

A hypothetical study has four independent policy-years, each with two claims: $(0,0)$, $(0,0)$, $(20,20)$, and $(20,20)$ dollars. For the average claim amount, what is the exact cluster-bootstrap standard error? (Answer: \$5.00.)
Three independent exposure clusters have $(e,L)=(1,6),(1,12),(2,22)$, with losses in thousands of dollars. What is the total-loss-per-exposure estimate? (Answer: 10 thousand dollars per exposure unit.)
For the three clusters in Question 2, a bootstrap replicate selects the first cluster twice and the third cluster once. What is its loss-per-exposure estimate? (Answer: $34/4=8.5$ thousand dollars per exposure unit.)
A portfolio has four independent years with aggregate losses $5,5,25,25$ thousand dollars. In an exact cluster bootstrap of four years, what is the probability that the replicate mean equals 15 thousand dollars? (Answer: $4/16=0.2500$.)
Each of five policy-years contains claims that share a year-specific catastrophe exposure. An analyst resamples all claim rows independently to estimate uncertainty in annual aggregate loss. What is the key defect, and what should be resampled instead? (Answer: Independent claim resampling breaks the within-year dependence; resample complete policy-year clusters, retaining each year’s claims together.)

What You'll Learn