Bootstrap Intervals For Premium And Loss-Ratio Estimates

Why Bootstrap Premium and Loss-Ratio Estimates?

A portfolio’s observed claim cost per exposure and loss ratio are calculated from recorded experience. If the policies observed are viewed as a sample from a broader population of comparable policies, those estimates vary from sample to sample. A policy-level bootstrap uses the observed records to approximate that sampling variation, without requiring a parametric claim model.

The key is to resample the unit that generated the estimate and then recalculate the estimate in every replicate. For aggregate claim cost per exposure, that means resampling policy records containing both exposure and loss. For a loss ratio, it means resampling records containing both loss and earned premium. Keeping each record’s values together preserves their observed association.

Definition: For $n$ observed policies, let $E_j$ be the exposure, $L_j$ the aggregate loss, and $P_j$ the earned premium for policy $j$. The aggregate claim cost per exposure is $$ \hat{c}=\frac{\sum_{j=1}^{n}L_j}{\sum_{j=1}^{n}E_j}, $$ provided total exposure is positive. The aggregate loss ratio is $$ \hat{R}=\frac{\sum_{j=1}^{n}L_j}{\sum_{j=1}^{n}P_j}, $$ provided total earned premium is positive. A policy-level bootstrap draws $n$ policy records with replacement from the observed $n$ records, recalculates the chosen estimate, and repeats this process $B$ times.

Here, “premium” in the loss-ratio calculation means earned premium on the same basis and for the same experience period as the losses. The claim cost per exposure is often described as loss cost; it is not necessarily a final indicated rate, because expenses, profit provisions, and other adjustments may be outside the stated estimand. The calculations below use nominal dollar amounts, with no discounting, interest, mortality, or fitted probability distribution.

What Is Resampled—and What Is Recalculated?

Suppose an observed policy record is $(E_j,L_j,P_j)$. A bootstrap replicate selects $n$ of the observed records, allowing the same policy record to appear more than once and some records not to appear. If the selected indices are $I_1,\ldots,I_n$, the replicate estimates are

\hat{c}^{*}=\frac{\sum_{b=1}^{n}L_{I_b}}{\sum_{b=1}^{n}E_{I_b}}, \qquad \hat{R}^{*}=\frac{\sum_{b=1}^{n}L_{I_b}}{\sum_{b=1}^{n}P_{I_b}}.

Each replicate must recompute its numerator and denominator from its selected records. Resampling losses independently of exposures or premiums would break the observed policy-level pairing. Resampling the already-calculated portfolio total would not create the variation in policy composition that the bootstrap is intended to represent.

Choose the record and estimand.
For claim cost per exposure, retain each policy’s exposure and loss. For a loss ratio, retain its loss and earned premium. Specify the experience period and units consistently.

Draw a bootstrap sample.
Select $n$ policy records with replacement from the $n$ observed records. Each policy record has probability $1/n$ on each draw.

Recalculate the statistic.
Compute the ratio of the relevant replicate totals. Do not average the record-level ratios unless that average is specifically the estimand.

Summarize the replicates.
Use the spread of the $B$ replicate estimates for a bootstrap standard error, and use their empirical quantiles for a percentile interval.

Why the Bootstrap Recalculates a Ratio

The target is a ratio of population totals, such as expected loss divided by expected exposure. Its sample counterpart is the ratio of the corresponding sample totals. The bootstrap treats the empirical distribution of observed policy records as an estimate of the unknown distribution of policy records, then asks how the same ratio would vary across samples drawn from that empirical distribution.

This gives a direct justification for the replicate calculation. If a bootstrap sample contains records $I_1,\ldots,I_n$, its empirical loss total is $\sum_b L_{I_b}$, and its empirical exposure total is $\sum_b E_{I_b}$. Substituting these replicate totals into the original statistic gives $\hat{c}^{*}$. The same argument gives $\hat{R}^{*}$ using earned premium in the denominator. Since the numerator and denominator are formed from the same selected records, their association is reflected in the replicate ratio.

Bootstrap standard error: If $\hat{\theta}^{*}_1,\ldots,\hat{\theta}^{*}_B$ are the replicate estimates, their estimated bootstrap standard error is the sample standard deviation $$ \widehat{\operatorname{SE}}_{\mathrm{boot}}(\hat{\theta}) = \sqrt{\frac{1}{B-1}\sum_{b=1}^{B} \left(\hat{\theta}^{*}_b-\bar{\hat{\theta}}^{*}\right)^2}, \qquad \bar{\hat{\theta}}^{*}=\frac{1}{B}\sum_{b=1}^{B}\hat{\theta}^{*}_b. $$

A percentile interval is formed from quantiles of the bootstrap estimates. For example, the 95% percentile interval uses the 2.5th and 97.5th percentiles. It is an interval procedure for uncertainty under the sampling setup represented by the bootstrap; it is not a statement that a fixed parameter has a 95% probability of lying in this particular observed interval. The quantile convention used by software can affect endpoints, especially when $B$ is small, so it should be stated when exact reproducibility matters.

Worked Example: Bootstrap Standard Error for Claim Cost per Exposure

A hypothetical insurer observes four policies for one experience period. Exposure is measured in policy-years and losses are in dollars. The records are: policy 1, $E_1=1,L_1=100$; policy 2, $E_2=2,L_2=180$; policy 3, $E_3=1,L_3=160$; and policy 4, $E_4=2,L_4=360$. Find the observed aggregate claim cost per exposure and the bootstrap standard error from the four specified resamples: $(1,1,2,4)$, $(1,2,3,4)$, $(2,2,3,3)$, and $(3,3,4,4)$.

Method. First calculate the ratio of aggregate losses to aggregate exposure. Then calculate that same ratio within each resample. The example uses only four replicates to make the arithmetic visible; it is not enough replicates for a stable practical interval.

The observed totals are $\sum E_j=1+2+1+2=6$ policy-years and $\sum L_j=100+180+160+360=800$ dollars. Thus $\hat{c}=800/6=133.3333$ dollars per policy-year.

For resample $(1,1,2,4)$, exposure is $1+1+2+2=6$, loss is $100+100+180+360=740$, and the estimate is $740/6=123.3333$. For $(1,2,3,4)$, the totals are $6$ and $800$, giving $800/6=133.3333$. For $(2,2,3,3)$, exposure is $2+2+1+1=6$, loss is $180+180+160+160=680$, and the estimate is $680/6=113.3333$. For $(3,3,4,4)$, exposure is $1+1+2+2=6$, loss is $160+160+360+360=1{,}040$, and the estimate is $1{,}040/6=173.3333$.

The mean of the four bootstrap estimates is $(123.3333+133.3333+113.3333+173.3333)/4=135.8333$. Their deviations from that mean are $-12.5,-2.5,-22.5,$ and $37.5$. The sum of squared deviations is $156.25+6.25+506.25+1{,}406.25=2{,}075$. Therefore the sample variance is $2{,}075/(4-1)=691.6667$, and the bootstrap standard error is $\sqrt{691.6667}=26.2996$, or $\$26.30$ per policy-year to the nearest cent.

Check. Every resample contains four records, and every one has total exposure $6$, so each replicate’s denominator is consistent with the selected policies. The common mistake avoided is averaging policy-level loss-per-exposure ratios instead of computing the ratio of aggregate losses to aggregate exposure.

Worked Example: Bootstrap Standard Error for a Loss Ratio

A hypothetical portfolio has four policy records with loss and earned premium, respectively: $(\$80,\$100)$, $(\$120,\$200)$, $(\$240,\$300)$, and $(\$160,\$200)$. Calculate the aggregate loss ratio and the bootstrap standard error from four resamples: $(1,1,2,4)$, $(1,2,3,4)$, $(2,2,3,3)$, and $(3,3,4,4)$.

Method. For each resample, divide its total loss by its total earned premium. The observed loss total is $80+120+240+160=\$600$, and earned premium totals $100+200+300+200=\$800$. The observed aggregate loss ratio is $600/800=0.75$, or 75%.

For $(1,1,2,4)$, loss is $80+80+120+160=\$440$ and premium is $100+100+200+200=\$600$, so the ratio is $440/600=0.7333333$. For $(1,2,3,4)$, it is $600/800=0.75$. For $(2,2,3,3)$, loss is $120+120+240+240=\$720$ and premium is $200+200+300+300=\$1{,}000$, so the ratio is $0.72$. For $(3,3,4,4)$, loss is $240+240+160+160=\$800$ and premium is $300+300+200+200=\$1{,}000$, so the ratio is $0.80$.

The replicate mean is $(0.7333333+0.75+0.72+0.80)/4=0.7508333$. The deviations from this mean are $-0.0175,-0.0008333,-0.0308333,$ and $0.0491667$. The sum of squared deviations is $0.003675$, so the sample variance is $0.003675/3=0.001225$. The bootstrap standard error is $\sqrt{0.001225}=0.035$, or 3.50 percentage points.

Check. Each ratio uses the replicate’s loss and premium totals from the same selected policies. This standard error summarizes the spread of the four stated replicates; a practical analysis would use many more replicates. The common mistake avoided is resampling losses and premiums separately, which would discard their policy-level pairing.

Percentile Intervals and the Weighting Trap

The bootstrap standard error summarizes spread, while a percentile interval uses the shape of the replicate distribution directly. For a chosen confidence level, sort the replicate estimates and take the corresponding lower and upper quantiles. This approach can reflect asymmetry in a ratio’s bootstrap distribution, although its reliability still depends on whether the observed records and resampling scheme support the intended inference.

Worked Example: A Percentile Interval and a Ratio-of-Totals Check

A hypothetical portfolio has four records of loss and earned premium: $(\$80,\$100)$, $(\$90,\$100)$, $(\$420,\$600)$, and $(\$160,\$200)$. Its observed loss ratio is $(80+90+420+160)/(100+100+600+200)=750/1{,}000=0.75$. Calculate a 95% percentile interval using these eight bootstrap resamples: $(1,1,1,1)$, $(2,2,2,2)$, $(3,3,3,3)$, $(4,4,4,4)$, $(1,1,2,2)$, $(3,3,4,4)$, $(1,2,3,4)$, and $(1,2,4,4)$.

Method. Recalculate loss divided by premium for every resample, sort the ratios, and use linear interpolation at position $1+(B-1)p$ for percentile $p$, where $B=8$. This explicitly chosen convention gives positions $1.175$ for the 2.5th percentile and $7.825$ for the 97.5th percentile.

The four all-identical resamples have ratios $80/100=0.80$, $90/100=0.90$, $420/600=0.70$, and $160/200=0.80$. Resample $(1,1,2,2)$ has loss $80+80+90+90=\$340$ and premium $\$400$, giving $0.85$. Resample $(3,3,4,4)$ has loss $420+420+160+160=\$1{,}160$ and premium $600+600+200+200=\$1{,}600$, giving $0.725$. Resample $(1,2,3,4)$ gives $750/1{,}000=0.75$. Resample $(1,2,4,4)$ gives loss $80+90+160+160=\$490$ and premium $100+100+200+200=\$600$, giving $0.8166667$.

In sorted order, the ratios are $0.70,0.725,0.75,0.80,0.80,0.8166667,0.85,0.90$. At position $1.175$, interpolate $0.175$ of the way from $0.70$ to $0.725$: the lower endpoint is $0.70+0.175(0.025)=0.704375$. At position $7.825$, interpolate $0.825$ of the way from $0.85$ to $0.90$: the upper endpoint is $0.85+0.825(0.05)=0.89125$. The resulting illustrative 95% percentile interval is $[0.7044,0.8913]$, or approximately 70.44% to 89.13%, to four decimal places.

The average of the four individual policy loss ratios is $(0.80+0.90+0.70+0.80)/4=0.80$, which differs from the aggregate ratio of 0.75. The aggregate ratio weights policies through their premium amounts; the unweighted average gives each policy equal weight. The average is not a substitute for the ratio of totals when the requested quantity is the portfolio loss ratio. With only eight resamples, this interval is for illustrating the calculation, not for drawing a reliable practical conclusion.

Assumptions, Interpretation, and Exam Traps

A nonparametric policy bootstrap assumes the observed policy records are informative about the target population and that the resampled units can reasonably be treated as exchangeable draws for the purpose at hand. The policy must be the appropriate independent sampling unit. If records are dependent or the observation design requires preserving groups, a simple policy-level resample may not represent the sampling process. Also check that replicate denominators are positive; a replicate with zero total exposure or premium does not yield the corresponding ratio.

The bootstrap quantifies sampling uncertainty under this resampling model. It does not, by itself, correct biased or incomplete experience, account for an unrecorded change in coverage, or capture all uncertainty about future conditions. A large number $B$ reduces Monte Carlo noise in the estimated bootstrap standard error or interval endpoints, but it cannot repair an unrepresentative sample or an inappropriate resampling unit.

Wrong weighting: averaging policy loss costs or policy loss ratios generally does not give the ratio of portfolio totals. Use the ratio of sums when that is the defined estimand.
Broken records: resampling losses separately from exposure or premium changes the observed relationship among policy values. Resample the relevant values together.
Changed denominator: keep units and periods aligned. For example, do not divide losses from one period by earned premium from a different period unless that is explicitly the intended measure.
Overstated precision: a percentile interval based on very few replicates is unstable. More replicates improve numerical precision, but not the quality of the underlying sample or assumptions.

Key takeaway: Define the portfolio measure as a ratio of totals, resample complete policy-level records with replacement, and recalculate that same ratio in every replicate. Use the replicate distribution for uncertainty summaries, while keeping the target population and resampling assumptions in view.

Check Your Understanding

For each question, calculate the requested quantity from the given policy-level information.

Three hypothetical records have exposure and loss $(1,\$50)$, $(2,\$100)$, and $(1,\$150)$. What is aggregate claim cost per exposure? (Answer: $\$75.00$ per exposure unit.)
Three records have loss and earned premium $(\$60,\$100)$, $(\$120,\$200)$, and $(\$80,\$100)$. What is the aggregate loss ratio? (Answer: 0.6500, or 65.00%.)
Two policy records have exposure-loss pairs $(1,\$40)$ and $(2,\$120)$. A bootstrap sample selects policy 1 twice. What is its claim cost per exposure? (Answer: $\$40.00$ per exposure unit.)
Eight sorted bootstrap loss ratios are $0.50,0.55,0.60,0.65,0.70,0.75,0.80,0.90$. Using linear interpolation at position $1+(B-1)p$, what is the 95% percentile interval? (Answer: $[0.5088,0.8825]$, to four decimal places.)
Two hypothetical policies have loss-premium pairs $(\$80,\$100)$ and $(\$540,\$900)$. Compare their aggregate loss ratio with the unweighted average policy ratio. (Answer: aggregate ratio $620/1{,}000=0.6200$; unweighted average $(0.80+0.60)/2=0.7000$.)

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