Severity Estimation from Censored Policy Payments

Why a Policy Payment Is Not Necessarily the Loss

A claim file may record the amount paid by an insurer rather than the original, or ground-up, loss. A deductible can make small losses produce no payment, while a policy limit can make large losses produce the same capped payment. The payment record is therefore sometimes an exact observation of the loss and sometimes only an interval containing it.

This distinction matters when describing a portfolio’s severity. The empirical average payment is directly calculable from payment records, but it is not generally the empirical average ground-up loss. Likewise, the fraction of payments below a threshold need not equal the fraction of underlying losses below that threshold. The correct calculation depends on what each record reveals.

Definition: Let $X$ be a nonnegative ground-up loss. Suppose a hypothetical policy has per-loss deductible $d$ and insurer payment limit $u$, where $u$ is the maximum payment above the deductible. The insurer payment is $$ Y^L=\min\bigl((X-d)_+,u\bigr), \qquad (z)_+=\max(z,0). $$ A zero payment implies $0\le X\le d$; a positive payment strictly below $u$ identifies $X=d+Y^L$; and a payment equal to $u$ implies $X\ge d+u$. These conclusions assume all losses are nonnegative and the stated deductible and limit are the only coverage modifications.

The limit convention is important: here $u$ is the maximum insurer payment, not the maximum ground-up loss. Thus the ground-up threshold at which the payment becomes capped is $d+u$. If a contract defines its limit differently, the payment formula and the implied loss intervals must be adjusted to that contract.

Translate Each Record Before Calculating

For a record with observed payment $y$, first identify the set of possible ground-up losses. Under the policy above, the interval is

Record interpretation: If $y=0$, then $X\in[0,d]$. If $0<y<u$, then $X=d+y$ exactly. If $y=u$, then $X\in[d+u,\infty)$. A zero payment is not evidence that the loss was zero, and a capped payment is not evidence that the loss was exactly $d+u$.

These are examples of censored observations: the underlying value is not always known exactly, but the record constrains it. A deductible creates uncertainty at the bottom of the loss range, and a limit creates uncertainty at the top. If every claim, including those with zero payment, is in the data, the intervals can be used to calculate empirical bounds without assuming a parametric severity distribution.

Worked Example: Identify the Loss Information in Each Payment

A hypothetical policy has deductible $d=\$80$ and payment limit $u=\$200$. Four recorded payments are $\$0,\ \$35,\ \$200,$ and $\$120$. Translate each record into information about ground-up severity.

Method. Compare each payment with zero and the cap of $\$200$, then use the corresponding interval or exact-loss rule.

The $\$0$ payment implies $0\le X\le \$80$. It does not identify a particular loss in that range.

The $\$35$ payment is positive and below the cap, so the ground-up loss is exactly $d+y=\$80+\$35=\$115$.

The $\$200$ payment is capped, so the loss is at least $d+u=\$80+\$200=\$280$. It could be $\$280$, but it could also be larger.

The $\$120$ payment is positive and below the cap, so the ground-up loss is exactly $\$80+\$120=\$200$.

The records therefore imply, in order, $[0,80]$, exactly $115$, $[280,\infty)$, and exactly $200$, with amounts in dollars. Check: each positive uncapped payment is below $\$200$, and the capped-payment threshold is $\$280$. The common mistake avoided is treating either the zero or the capped payment as an exact ground-up loss.

Empirical CDF Bounds from Censored Records

For complete ground-up observations $x_1,\ldots,x_n$, the empirical CDF at threshold $t$ is the fraction of observed losses no greater than $t$:

\widehat F_X(t)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}(x_i\le t).

When a record gives only an interval $[L_i,U_i]$, its contribution to this count may be uncertain. A record is definitely no greater than $t$ if its largest possible loss is at most $t$. It is possibly no greater than $t$ if its smallest possible loss is at most $t$. Counting these two groups gives lower and upper bounds.

Empirical CDF bounds: If record $i$ implies $X_i\in[L_i,U_i]$, with $U_i=\infty$ allowed, then $$ \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}(U_i\le t) \ \le\ \widehat F_X(t)\ \le\ \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}(L_i\le t). $$ The lower bound counts records definitely at or below $t$; the upper bound counts records that could be at or below $t$. An exactly observed loss is represented by $L_i=U_i$.

The bounds are justified record by record. If $U_i\le t$, every loss compatible with record $i$ is at most $t$, so that record must contribute one to the empirical count. If $L_i>t$, no compatible loss is at most $t$, so it must contribute zero. Only records whose intervals include values on both sides of $t$ are uncertain. The lower bound assigns zero to each such uncertain record; the upper bound assigns one. Dividing the resulting counts by $n$ gives the stated bounds.

Worked Example: CDF Bounds with a Deductible and Limit

A hypothetical policy has $d=\$80$, $u=\$200$, and three observed payments: $\$0,\ \$20,$ and $\$200$. Find the empirical ground-up CDF bounds at $t=\$30$, and find a lower bound for the empirical mean ground-up loss.

Method. Convert payments to loss intervals first. Then count losses that are definitely or possibly at most $\$30$, and use the interval lower endpoints for the mean bound.

The zero payment implies $X_1\in[0,80]$. The $\$20$ uncapped payment identifies $X_2=80+20=\$100$. The capped $\$200$ payment implies $X_3\in[80+200,\infty)=[280,\infty)$.

At $t=\$30$, none of the records is definitely at or below $\$30$: the first could be as high as $\$80$, and the other two losses are definitely above $\$30$. Thus the lower CDF bound is $0/3=0$.

The first record could be at or below $\$30$, because its interval includes losses from $\$0$ through $\$30$. The second and third cannot be at or below $\$30$. Thus the upper CDF bound is $1/3=0.3333$, to four decimal places.

For the mean, the smallest compatible losses are $\$0,\ \$100,$ and $\$280$. Therefore the empirical mean is at least $$ \frac{0+100+280}{3}=\frac{380}{3}=\$126.67 $$ to the nearest cent. There is no finite upper bound because the third loss has no finite upper endpoint. Check: only the zero-payment record could fall below $\$30$; the exact $\$100$ loss and capped loss of at least $\$280$ cannot. The common mistake avoided is counting the zero payment as definitely below the threshold rather than possibly below it.

Bounds for the Empirical Mean

The empirical ground-up mean is $\overline X=n^{-1}\sum_i X_i$. If each record implies $X_i\in[L_i,U_i]$, then adding the record-level bounds and dividing by $n$ gives a lower and an upper bound:

\frac{1}{n}\sum_{i=1}^{n}L_i \ \le\ \overline X\ \le\ \frac{1}{n}\sum_{i=1}^{n}U_i.

If one or more upper endpoints are infinite, this method gives no finite upper bound for the mean. That does not mean the actual losses must have an infinite mean; it means the payment records alone do not provide a finite upper bound. Additional information or assumptions would be needed to make a stronger statement.

Worked Example: Deductible Censoring Without a Payment Limit

A hypothetical policy has a deductible of $\$100$ and no payment limit. Three complete claim records show payments of $\$0,\ \$40,$ and $\$150$. Find bounds for the empirical mean ground-up loss and determine the empirical CDF at $t=\$120$.

Method. With no limit, a positive payment identifies the loss as deductible plus payment. A zero payment only places the loss between zero and the deductible.

The first loss is in $[0,100]$. The second is exactly $100+40=\$140$, and the third is exactly $100+150=\$250$.

The smallest compatible losses sum to $0+140+250=\$390$, so the mean is at least $\$390/3=\$130.00$. The largest compatible losses sum to $100+140+250=\$490$, so the mean is at most $\$490/3=\$163.33$, rounded to the nearest cent. Thus the empirical mean lies between $\$130.00$ and $\$163.33$.

At $t=\$120$, the first loss is certainly at most $\$120$, since its largest possible value is $\$100$. The other two losses, $\$140$ and $\$250$, exceed the threshold. Therefore the empirical CDF is exactly $1/3=0.3333$, to four decimal places.

Check: the CDF is exact here even though the first loss is not, because every value in its interval lies below the chosen threshold. The common mistake avoided is treating the $\$40$ payment itself as the ground-up loss.

Payment Summaries and Data Limitations

The empirical average payment answers a different question from the empirical average loss. For the three payments in the preceding example, the average payment is $(0+40+150)/3=\$63.33$, to the nearest cent. That value is exact for the recorded payments, but it is not a ground-up severity estimate: the deductible has reduced payments, and the zero-payment record could still represent a positive loss.

A policy limit has the opposite information problem at the high end. A payment at the cap establishes a lower bound on the corresponding loss, not an exact value. Consequently, a sample containing capped losses may determine the empirical CDF at some thresholds but leave it uncertain at others, and may yield a finite lower bound but no finite upper bound for the empirical mean.

These calculations also depend on the records being complete in a relevant sense. If the data include all claims, including claims with zero payment, then $n$ is the number of claims represented in the sample. If only paid claims are recorded, zero-payment claims are absent. The observed data then do not directly describe the same set of losses: the count and severity of omitted claims cannot be recovered from the paid records alone. Do not insert hypothetical zero payments or interpret a paid-claim average as the average over all claims without additional information.

Write the coverage rule.
Confirm what the deductible and limit apply to. Under the convention here, $Y^L=\min((X-d)_+,u)$, so the cap threshold for ground-up loss is $d+u$.

Translate every payment.
Use an interval for zero or capped payments and an exact loss for a positive uncapped payment.

Match the quantity to the question.
Use recorded payments for payment summaries; use compatible loss intervals for ground-up CDF or mean bounds.

Check the data scope.
Verify whether zero-payment claims are included and whether the sample represents all claims or only paid claims.

Key takeaway: A censored payment is information about a loss, not automatically the loss itself. Convert each payment into the set of compatible ground-up amounts before calculating empirical CDFs or means; report bounds when the records do not identify a unique answer.

Check Your Understanding

For each question, treat losses as nonnegative and use the stated coverage convention. CDF bounds are for the empirical ground-up distribution.

A policy has $d=\$40$ and payment limit $u=\$60$. Payments are $\$0,\ \$30,\ \$60$. Find the lower bound for the empirical mean loss and the CDF bounds at $t=\$50$. (Answer: Mean lower bound $\$56.67$, with no finite upper bound; CDF bounds $1/3$ and $1/3$.)
A policy has deductible $d=\$25$ and no payment limit. Payments are $\$0,\ \$15,\ \$75$. Find the empirical mean bounds and the empirical CDF at $t=\$30$. (Answer: Mean between $\$46.67$ and $\$55.00$; CDF $1/3=0.3333$.)
A policy has $d=\$100$, payment limit $u=\$300$, and payments $\$50,\ \$300,\ \$0$. Find the empirical mean lower bound and the CDF bounds at $t=\$200$. (Answer: Mean lower bound $\$183.33$, with no finite upper bound; CDF bounds $2/3$ and $2/3$.)
A policy has $d=\$60$, $u=\$100$, and payments $\$0,\ \$40,\ \$100,\ \$80$. Find the empirical payment mean, the ground-up mean lower bound, and the CDF bounds at $t=\$120$. (Answer: Payment mean $\$55.00$; ground-up mean lower bound $\$100.00$, with no finite upper bound; CDF bounds $1/2$ and $1/2$.)
A policy has $d=\$30$, $u=\$70$, and records payments only for claims that received a positive payment. Can the empirical ground-up CDF across all claims be determined from these records alone? State why. (Answer: No. Zero-payment claims are omitted, so the records do not establish the all-claim denominator or the number and severities of losses at or below the deductible.)

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