Policy Limits and Right-Censored Claim Amounts

Why a Policy Limit Changes What the Data Show

A claim record showing a payment equal to the policy limit does not necessarily tell you the full amount of the loss. The insurer may have paid the maximum covered amount even though the ground-up loss was larger. Treating that payment as the exact loss can understate claim severity, especially when several claims reach the limit.

The key distinction is between the ground-up loss, meaning the loss amount before the policy's payment cap, and the observed payment after the cap is applied. This tutorial considers a simple hypothetical policy with no deductible, coinsurance, expenses, or offsets, and a fixed per-claim limit. These assumptions isolate the effect of the limit.

Definition: Let $X$ be a ground-up claim loss and let $u$ be the maximum insurer payment per claim. Under the stated policy assumptions, the observed payment is $Z=\min(X,u)$. If $Z<u$, the ground-up loss is observed exactly: $X=Z$. If $Z=u$, the record is right-censored at $u$: it establishes $X\ge u$, but does not reveal the exact value of $X$.

A payment at the limit may correspond to a ground-up loss exactly equal to the limit or to one greater than it. The record alone cannot distinguish those possibilities. In continuous severity models, the probability of a loss being exactly equal to a particular amount is often zero, but that modeling observation does not make the individual censored claim amount known. If claims are recorded in rounded units or losses have discrete values, equality can also have positive probability.

Right censoring is different from truncation. A capped claim remains in the observed data, but its amount above the cap is unknown. Under truncation, some losses are not present in the data at all. Here, assume every claim in the study is recorded, including claims whose payment reaches the limit.

What an Empirical Payment Summary Measures

Suppose $n$ claims have observed payments $z_1,\ldots,z_n$, all subject to the same limit $u$. Their empirical mean payment is

\bar{z}=\frac{1}{n}\sum_{j=1}^{n}z_j

This is the average amount paid under the capped policy for those claims. It is a valid summary of the observed payments, but it is not generally the average ground-up loss. Payments below the limit identify the corresponding losses; payments at the limit identify only lower bounds.

For each claim, $Z\le X$. More precisely, $X-Z=0$ when $X\le u$, and $X-Z=X-u$ when $X>u$. Therefore, for a fully observed set of ground-up losses,

\bar{X}=\bar{Z}+\frac{1}{n}\sum_{j=1}^{n}(x_j-u)_+ \qquad\text{where}\qquad (a)_+=\max(a,0).

The second term is the average amount above the limit. Capped records do not provide that amount for claims at the limit. They do, however, provide a lower bound: the ground-up loss for each such claim is at least the recorded payment. Thus $\bar{X}\ge\bar{Z}$, but the exact ground-up mean need not be identifiable from the capped records.

Key result: With complete claim counts and a fixed per-claim limit, the empirical mean of recorded payments is an exact mean for the capped payment amounts and a lower bound for the ground-up sample mean. Without additional information or assumptions about losses above the limit, the capped records do not give a finite upper bound on that ground-up mean.

Worked Examples

Worked Example: Separating Payments From Audited Losses

A hypothetical insurer applies an \$8,000 per-claim payment limit and no deductible. Six observed payments, in thousands of dollars, are $2,5,8,8,6,8$. A separate audit has also established that the corresponding ground-up losses were $2,5,11,9,6,14$ thousand dollars. Calculate the mean payment and mean ground-up loss, and identify which claims were censored in the ordinary payment records.

Method. First calculate the mean of the actual recorded payments. Then use the audited amounts to calculate the ground-up mean and compare the two. This comparison is possible only because the audit supplies the amounts hidden by the cap.

The payments at the limit are the third, fourth, and sixth observations. Those three records indicate ground-up losses of at least \$8,000, but do not show the audited amounts of \$11,000, \$9,000, and \$14,000.

The payment total is $2+5+8+8+6+8=37$ thousand dollars. Therefore, the mean payment is $37/6=6.1667$ thousand dollars, or \$6,166.67 to the nearest cent.

The audited ground-up total is $2+5+11+9+6+14=47$ thousand dollars. The mean ground-up loss is $47/6=7.8333$ thousand dollars, or \$7,833.33 to the nearest cent.

The difference between the means is $(47-37)/6=10/6=1.6667$ thousand dollars, or \$1,666.67 per claim to the nearest cent. This is the average amount by which the ground-up losses exceed the recorded payments. The check $7.8333\ge6.1667$ confirms that the uncapped mean is larger.

Common mistake avoided: Treating all three \$8,000 payments as exact \$8,000 losses would produce a mean ground-up loss of \$6,166.67. That is the mean payment, not the audited ground-up mean.

Worked Example: Finding What Capped Records Can Establish

A hypothetical portfolio has five claims subject to a \$12,000 per-claim payment limit. The recorded payments, in thousands of dollars, are $3,12,5,12,9$. Calculate the mean payment, the proportion of claims censored at the limit, and the strongest lower bound on the ground-up sample mean available from these records.

Method. Calculate summaries directly from the recorded payments, then interpret each payment at the limit as a lower bound on its underlying loss. No severity distribution or upper bound on the losses is supplied.

The payment total is $3+12+5+12+9=41$ thousand dollars. The mean payment is $41/5=8.2$ thousand dollars, or \$8,200. Two of the five records equal the limit, so the proportion censored is $2/5=0.4$, or $40\%$.

The three below-limit losses are known exactly: their total is $3+5+9=17$ thousand dollars. Each of the two claims recorded at \$12,000 has a ground-up loss of at least \$12,000. The ground-up total is therefore at least $17+12+12=41$ thousand dollars, and the ground-up mean is at least $41/5=8.2$ thousand dollars.

There is no finite upper bound from these records alone: either or both censored losses could be larger than \$12,000, with no stated maximum. The bound is attained if both ground-up losses equal the limit, but the records do not establish that they do.

Common mistake avoided: Calling \$8,200 the exact mean ground-up loss confuses a payment summary with a loss summary. The data establish a lower bound of \$8,200, not an exact value.

Worked Example: Comparing Expected Payment and Expected Loss Under a Specified Model

For a hypothetical product, suppose ground-up loss $X$, measured in thousands of dollars, has the discrete distribution $P(X=4)=0.40$, $P(X=9)=0.35$, and $P(X=15)=0.25$. Every claim has a \$10,000 payment limit and no other payment adjustment. Find the expected payment, expected ground-up loss, and probability that a payment record is censored.

Method. Apply $Z=\min(X,u)$ to each possible loss before calculating the expected payment. Calculate the expected ground-up loss from the original loss distribution. This is a model-based comparison: the tail amount is available here because the distribution is specified, not because a capped record reveals it.

The possible payments are $4$, $9$, and $10$ thousand dollars, respectively. In particular, the \$15,000 loss produces a \$10,000 payment. Thus,

$$ E[Z]=(0.40)(4)+(0.35)(9)+(0.25)(10) =1.60+3.15+2.50=7.25 $$

The expected payment is \$7,250. The expected ground-up loss is

$$ E[X]=(0.40)(4)+(0.35)(9)+(0.25)(15) =1.60+3.15+3.75=8.50 $$

So the expected ground-up loss is \$8,500. Only the \$15,000 loss exceeds the limit, so the probability of a censored payment is $0.25$. The difference in expectations is $8.50-7.25=1.25$ thousand dollars. This agrees with the expected amount above the limit: $(0.25)(15-10)=1.25$ thousand dollars.

Common mistake avoided: Applying the limit to the loss distribution itself would incorrectly replace the \$15,000 loss by \$10,000 when calculating $E[X]$. The cap changes the payment, not the underlying loss.

Derivation and Interpretation

The relation between the capped mean and the ground-up mean follows claim by claim. Under the stated policy, $z_j=\min(x_j,u)$, so $x_j-z_j$ is zero for a loss at or below the limit and equals $x_j-u$ for a loss above the limit. In either case, $x_j-z_j=(x_j-u)_+$. Rearranging gives $x_j=z_j+(x_j-u)_+$. Summing across claims and dividing by $n$ yields

\bar{x}=\frac{1}{n}\sum_{j=1}^{n}z_j+ \frac{1}{n}\sum_{j=1}^{n}(x_j-u)_+ =\bar{z}+\frac{1}{n}\sum_{j=1}^{n}(x_j-u)_+.

Every term in the second sum is nonnegative, which proves $\bar{x}\ge\bar{z}$. But for a censored record, the value of $(x_j-u)_+$ is unknown. The same payment of $u$ is consistent with many ground-up losses, so the sum cannot be recovered from the capped records alone.

Empirically, the distribution of recorded payments has all its observations at or below $u$. A collection of payments equal to $u$ therefore combines claims that may have different ground-up amounts. That collection does not reveal how the ground-up losses are distributed above the limit. A specified severity model, uncapped audit data, or another justified source of information could add what the payment records lack; without such information, assigning exact losses to the capped claims would be unsupported.

Common Traps

Equating “at the limit” with “equal to the loss.” A payment of $u$ establishes $X\ge u$, not $X=u$.
Using the payment mean as the ground-up mean. The payment mean is exactly calculable from the records, but it is generally only a lower bound for the ground-up mean.
Assuming the tail can be reconstructed from the capped values. Several different sets of losses above $u$ can produce the same observed payments. The cap alone does not identify those losses.
Ignoring the policy assumptions. The formula $Z=\min(X,u)$ applies to a payment cap with no other adjustments. A deductible, coinsurance, or a differently defined limit changes the payment mapping.
Counting payments at the cap as missing claims. Under the setting here, those claims are observed; their amounts are right-censored. The claims still belong in summaries of the observed payment data.

Key takeaway: Summarize capped payments as payments. Mark records at the limit as right-censored ground-up losses, and do not interpret them as exact losses. From capped records alone, the empirical ground-up mean is at least the mean payment, but its exact value is not generally determined.

Check Your Understanding

For each problem, distinguish the observed payment summary from what can be concluded about ground-up losses.

A hypothetical policy has a \$6,000 per-claim limit and no deductible. Six payments, in thousands of dollars, are $1,4,6,2,6,5$. Find the mean payment and the proportion censored. What lower bound does the data give for the ground-up mean? (Answer: Mean payment $=\$4,000$; censored proportion $=2/6=0.3333$; ground-up mean is at least $\$4,000$.)
A hypothetical portfolio has a \$15,000 limit and five recorded payments, in thousands, of $7,11,15,15,4$. Find the mean payment and state whether the exact ground-up mean can be recovered. (Answer: Mean payment $=\$10,400$; two claims are censored; the ground-up mean is at least $\$10,400$, but its exact value cannot be recovered from these records alone.)
Suppose $P(X=2)=0.25$, $P(X=8)=0.50$, and $P(X=20)=0.25$, with amounts in thousands of dollars. A hypothetical policy pays $\min(X,10)$. Calculate the expected payment, expected ground-up loss, and probability of a censored payment. (Answer: Expected payment $=\$7,000$; expected ground-up loss $=\$9,500$; censoring probability $=0.25$.)
Eight hypothetical claims have observed payments, in thousands of dollars, of $1,4,9,7,9,5,9,3$, under a \$9,000 limit. Calculate the mean payment, the censored proportion, and the lower bound on the ground-up total. (Answer: Mean payment $=\$5,875$; censored proportion $=3/8=0.3750$; ground-up total is at least $\$47,000$.)
Four hypothetical claims are recorded under a \$20,000 limit, with payments $6,12,20,20$ thousand dollars. Find the ground-up mean lower bound and explain whether any finite upper bound follows from the records. (Answer: Ground-up mean is at least $(6+12+20+20)/4=\$14,500$; no finite upper bound follows from the records alone.)

What You'll Learn