Empirical Deductible Loss Costs

Why Empirical Deductible Loss Costs Need Careful Denominators

A deductible changes what the insurer pays for each claim, but the loss cost depends on more than the payment calculation. The analyst must also identify the population represented by the data and use the matching denominator. In particular, an average payment per ground-up claim is not the same as an average payment per claim that exceeds the deductible, or a cost per policy exposure.

This tutorial estimates insurer-paid loss costs directly from observed claim severities, then compares that empirical estimate with the result from a fitted severity model. Assume an ordinary per-claim deductible, no policy limit, no coinsurance, and claim amounts adjusted to a consistent basis. The observed claims are ground-up claims: claims are included even when their severity is below the deductible.

Definition: Let $X$ be a ground-up claim severity and $d$ an ordinary per-claim deductible. The insurer's payment per loss is $Y^L=(X-d)_+=\max(X-d,0)$. For $n$ observed ground-up claims with severities $x_1,\ldots,x_n$, the empirical average payment per ground-up claim is $\frac{1}{n}\sum_{j=1}^{n}\max(x_j-d,0)$. It is a loss cost per claim, not a premium rate per exposure.

From Claim Payments to a Loss Cost

Apply the deductible separately to every claim before adding payments. Claims with severity at or below $d$ contribute zero to the insurer's payment total, but they remain in the denominator when the target is average payment per ground-up claim. If the data represent $E$ exposure units, the empirical loss cost per exposure is the total insurer payment divided by $E$.

\widehat{L}_{\text{claim}}=\frac{\sum_{j=1}^{n}(x_j-d)_+}{n}, \qquad \widehat{L}_{\text{exposure}}=\frac{\sum_{j=1}^{n}(x_j-d)_+}{E}.

The first estimate averages over ground-up claims; the second averages over exposures. They are connected by the observed ground-up claim frequency $n/E$: $\widehat{L}_{\text{exposure}}=(n/E)\widehat{L}_{\text{claim}}$. This relationship uses the same claim population and exposure base in both estimates.

Identify the target.
Decide whether the requested result is payment per ground-up claim, per exposure, or per claim that produces a positive payment.

Apply the contract term.
For each severity, calculate $\max(x_j-d,0)$. Do not subtract the deductible from a claim already below it and retain a negative payment.

Use the matching denominator.
Divide by all ground-up claims for a per-claim cost, or by the stated exposure count for a per-exposure cost. Keep zero-payment claims in the ground-up claim denominator.

Why the Empirical Average Is the Natural Estimate

Suppose ground-up claim severities $X_1,\ldots,X_n$ are independent observations from the same severity distribution, with finite expected deductible payment. Each transformed observation $Y_j^L=(X_j-d)_+$ is then an observation of the insurer payment per ground-up claim. By linearity of expectation, the expected value of their average equals the expected payment per claim:

E\left[\frac{1}{n}\sum_{j=1}^{n}(X_j-d)_+\right] =\frac{1}{n}\sum_{j=1}^{n}E[(X_j-d)_+] =E[(X-d)_+].

Thus the sample average is an unbiased empirical estimate of the expected insurer payment per ground-up claim under these sampling assumptions. This justification does not make a small sample precise: random variation can still make the estimate differ from the underlying expected payment. Nor does it justify averaging only claims that exceed the deductible when the desired quantity is per ground-up claim; that would condition the average on a different population.

Worked Example: Claim Payments and Cost per Exposure

A hypothetical insurer observes five ground-up claims of $800, $1,200, $2,000, $3,500, and $6,500. Each claim is subject to an ordinary $1,000 deductible. The claims arise from 10 exposure units. Calculate the empirical payment per ground-up claim and the loss cost per exposure.

Method. Apply the deductible to each severity using $Y^L=(X-d)_+$. Every one of the five ground-up claims belongs in the per-claim denominator, including the claim that produces no insurer payment.

Claim payments. The five payments are $\max(800-1{,}000,0)=0$, $1{,}200-1{,}000=200$, $2{,}000-1{,}000=1{,}000$, $3{,}500-1{,}000=2{,}500$, and $6{,}500-1{,}000=5{,}500$, all in dollars.

Total insurer payment. $0+200+1{,}000+2{,}500+5{,}500=\$9{,}200$.

Per ground-up claim. Divide by all five claims: $\$9{,}200/5=\$1{,}840.00$.

Per exposure. Divide the same total by 10 exposure units: $\$9{,}200/10=\$920.00$ per exposure.

Check and trap avoided. The observed ground-up frequency is $5/10=0.5$ claims per exposure, and $0.5\times \$1{,}840=\$920$, agreeing with the direct exposure calculation. Dividing by only the four claims above the deductible would give $2,300, which is an average per positive-payment claim, not the requested average per ground-up claim.

Frequency Weights and Incomplete Claim Data

A severity table may group claims by amount instead of listing every claim. If record $i$ represents $w_i$ claims, apply the deductible to the record severity and multiply its payment by $w_i$. The total ground-up claim count is $\sum_i w_i$, so the empirical per-claim estimate is $\sum_i w_i(x_i-d)_+/\sum_iw_i$. The weights must represent claim counts, not arbitrary importance scores.

If claims below the deductible are absent from the data, the observed positive-payment claims alone cannot supply the full ground-up claim denominator. A calculation from those records can estimate average payment conditional on a claim exceeding the deductible, but not the average payment per ground-up claim. The missing claim count, or a defensible estimate of it, is needed for that conversion. Likewise, a per-exposure cost requires an exposure denominator on the intended basis.

Worked Example: Grouped Claims and Frequency Weights

A hypothetical portfolio has 200 exposure units. Its complete ground-up claim data are summarized as follows: three claims of $400, two claims of $1,000, four claims of $2,500, and one claim of $5,000. The ordinary deductible is $1,000. Find the empirical insurer loss cost per ground-up claim and per exposure.

Method. Treat each listed count as a frequency weight, apply the deductible at each severity, and use all represented claims in the ground-up claim denominator.

Weighted payments. The three $400 claims pay $3\times 0=\$0$. The two $1,000 claims pay $2\times 0=\$0$. The four $2,500 claims pay $4\times(\$2{,}500-\$1{,}000)=\$6{,}000$. The one $5,000 claim pays $\$5{,}000-\$1{,}000=\$4{,}000$.

Totals and estimates. There are $3+2+4+1=10$ ground-up claims and total insurer payment is $\$0+\$0+\$6{,}000+\$4{,}000=\$10{,}000$. The per-claim cost is $\$10{,}000/10=\$1{,}000.00$; the per-exposure cost is $\$10{,}000/200=\$50.00$.

Check and trap avoided. Five claims exceed the deductible, so dividing by five gives $2,000 per positive-payment claim. That is not the per-ground-up-claim cost. The observed frequency is $10/200=0.05$, and $0.05\times \$1{,}000=\$50$ per exposure, confirming the exposure estimate.

Comparing an Empirical Cost with a Fitted Severity Model

A fitted severity model provides a second estimate of expected payment, based on a specified distribution and estimated parameters. The comparison is meaningful only when the fitted model describes the same ground-up claim severity population and the deductible is applied to the model's ground-up severity. It is not a substitute for checking whether the observed sample is complete or representative.

For one explicit comparison, assume claim severities are independent and identically distributed exponential random variables with rate $\lambda>0$, measured in inverse dollars. The density is $f_X(x)=\lambda e^{-\lambda x}$ for $x\geq 0$. Given a complete sample of exact severities, the maximum-likelihood estimate is $\widehat{\lambda}=n/\sum_jx_j=1/\bar{x}$. This fitted model assumes an exponential severity distribution; it should not be used merely because a deductible is present.

Under this model, calculate the expected payment by integrating over severities above the deductible. With the substitution $y=x-d$, the density factor for $x=d+y$ is $\lambda e^{-\lambda(d+y)}$, so:

E[(X-d)_+] =\int_d^\infty(x-d)\lambda e^{-\lambda x}\,dx =e^{-\lambda d}\int_0^\infty y\lambda e^{-\lambda y}\,dy =\frac{e^{-\lambda d}}{\lambda}.

The final equality follows because the integral is the mean of an exponential distribution with rate $\lambda$, namely $1/\lambda$. The formula assumes an ordinary deductible, no policy limit, and the stated exponential model. It is not an empirical-data formula and does not apply unchanged to other severity distributions.

Worked Example: Comparing Observed Payments with an Exponential Fit

A hypothetical sample of five complete ground-up claim severities is $500, $1,000, $2,000, $4,000, and $5,000. Assume, for model comparison, independent exponential severities and an ordinary $1,000 deductible. Calculate the empirical average payment and the fitted model's expected payment per ground-up claim.

Empirical method. Applying the deductible gives payments of $0, $0, $1,000, $3,000, and $4,000. Their total is $0+0+1{,}000+3{,}000+4{,}000=\$8{,}000$, so the empirical average is $\$8{,}000/5=\$1{,}600.00$ per ground-up claim.

Fit the model. The total ground-up severity is $500+1{,}000+2{,}000+4{,}000+5{,}000=\$12{,}500$. The sample mean is $\$12{,}500/5=\$2{,}500$, hence the exponential maximum-likelihood rate is $\widehat{\lambda}=1/2{,}500=0.0004$ per dollar.

Model-implied payment. Here $\widehat{\lambda}d=0.0004\times1{,}000=0.4$. Therefore the fitted expected payment is $e^{-0.4}/0.0004=\$1{,}675.80$, rounded to the nearest cent.

Comparison and check. The empirical estimate is $\$1{,}600.00-\$1{,}675.80=-\$75.80$ below the fitted estimate, a difference of about 4.5% of the fitted value. This difference is plausible as sample variation; it does not by itself establish that either estimate is wrong. The model fit used all five ground-up severities to estimate $\lambda$, while the empirical payment calculation applied the deductible to each of those same five claims.

Common Mistakes and Practical Checks

Dropping zero-payment claims. Keep claims at or below the deductible in the denominator for a cost per ground-up claim. Excluding them changes the target to a conditional average among claims with positive payments.
Confusing claim cost with exposure cost. A per-claim estimate uses a claim count; a per-exposure estimate uses an exposure count. Check the units in the final answer.
Applying the deductible to a total. Apply it claim by claim. In general, $\sum_j(x_j-d)_+$ is not obtained by subtracting one deductible from the aggregate loss.
Ignoring the meaning of a weight. A frequency weight represents how many claims have a given severity. Multiply both the represented payment and the represented claim count by that frequency.
Treating a fitted model as observed data. The model result depends on its distributional assumption and parameter estimate. Compare it with the empirical result, but state the model and its assumptions.
Using only claims that pierced the deductible without adjustment. If sub-deductible claims are missing, the resulting denominator is not the ground-up claim count. Obtain the missing count or clearly report a conditional estimate instead.

Key takeaway: Estimate an empirical deductible loss cost by applying the payment function to each ground-up claim, summing the resulting payments, and dividing by the population that matches the requested cost basis. A fitted severity model can provide a useful comparison, provided its assumptions and parameter estimation are explicit.

Check Your Understanding

For each problem, identify the target measure before choosing its denominator. Amounts are in dollars unless stated otherwise.

A hypothetical sample contains ground-up severities of $600, $1,400, and $2,600, with a $1,000 deductible and four exposure units. Find the empirical payment per ground-up claim and per exposure. (Answer: $666.67 per claim and $500.00 per exposure.)
A grouped hypothetical data set has two claims of $700, three claims of $1,900, and one claim of $3,900. With a $1,000 deductible and 80 exposure units, find the average payment per ground-up claim and per exposure. (Answer: $933.33 per claim and $70.00 per exposure.)
Assume a fitted exponential severity model with rate $\lambda=1/3{,}000$ per dollar and an ordinary deductible of $1,500. Calculate its expected payment per ground-up claim. (Answer: $3{,}000e^{-0.5}=\$1{,}819.59$, rounded to the nearest cent.)
A complete weighted severity sample has two claims of $900, three claims of $1,900, and one claim of $3,900. The deductible is $1,000 and the portfolio has 80 exposure units. Calculate total payment and both loss costs. (Answer: total payment $5,600; $933.33 per ground-up claim; $70.00 per exposure.)
In a hypothetical book there are 10 ground-up claims, but only the four claims above a $1,000 deductible appear in a payment file; their insurer payments total $12,000. What can be calculated from these data, and what is the average payment per ground-up claim if the stated count of 10 is valid? (Answer: the conditional average for positive-payment claims is $3,000; the ground-up claim average is $1,200.00. Dividing by four does not give the ground-up claim average.)

What You'll Learn