Censored Claims And Unsupported Limit Extensions

Why a Policy Limit Hides Severity

An insurer’s payment is not always the same as the ground-up loss. When a policy pays up to a per-claim limit, every sufficiently large loss produces the same capped payment. The payment tells the actuary that the loss reached the limit, but not how far above it the loss went. That distinction matters when estimating an increased limits factor (ILF): a higher limit can pay amounts that the existing records never observed.

The discussion below assumes a covered ground-up loss $X$, no deductible, and a policy that pays the covered loss up to its per-claim limit. Other contract terms can change the relationship between the loss and recorded payment, so the terms must be known before interpreting a capped amount.

Definition: A claim is right-censored at limit $L_0$ when the data establish that its ground-up loss reached or exceeded $L_0$, but record only the payment capped at $L_0$. Under the stated contract assumptions, the payment per loss is $Y^L=\min(X,L_0)$. If $Y^L<L_0$, the observed payment identifies $X$; if $Y^L=L_0$, it establishes $X\ge L_0$, not $X=L_0$.

This is censoring, not truncation. A censored record remains in the data: the claim is known to have occurred and its payment reached the cap. With truncation, some claims are absent from the observed data altogether. Also, if losses are discrete and can equal the limit exactly, a payment of $L_0$ does not by itself distinguish $X=L_0$ from $X>L_0$.

What Capped Data Can and Cannot Establish

For an uncapped observation below $L_0$, the payment provides a ground-up value under the assumed coverage terms. For a payment at $L_0$, it provides a threshold statement. If the proposed higher limit is $u>L_0$, the payment for a censored claim at that new limit could be anywhere from $L_0$ up to $u$. Without further information, the old payment cannot say where in that interval it belongs.

This gives a simple bound for a fixed set of claims. Keep each uncensored claim’s known loss unchanged. For each claim censored at $L_0$, its payment at $u$ is at least $L_0$ and at most $u$. These bounds can describe what the observed claims might have paid at the higher limit, but they do not determine an expected loss or a sound ILF for future claims.

Worked Example: Bounding a Higher-Limit Total

A hypothetical book has ten covered claims with no deductible and a per-claim limit of $L_0=\$100{,}000$. The recorded payments are $\$20{,}000,\ \$35{,}000,\ \$50{,}000,\ \$75{,}000,\ \$90{,}000,\ \$60{,}000$, and four payments of $\$100{,}000$. Find the observed total at the current limit and bound the total payment for these same claims at $u=\$200{,}000$. No severity distribution is assumed; interest and mortality are not applicable.

Method. Sum the observed payments, then bound the contribution from the censored claims. The first six payments are below the limit, so their known losses sum to

20{,}000+35{,}000+50{,}000+75{,}000+90{,}000+60{,}000 =\$330{,}000.

The four capped payments contribute $4(\$100{,}000)=\$400{,}000$ at the current limit. Thus the current-limit total is $\$330{,}000+\$400{,}000=\$730{,}000$.

At the higher limit, each of the four censored claims pays at least $\$100{,}000$ and at most $\$200{,}000$. Their combined contribution is therefore between $\$400{,}000$ and $\$800{,}000$. Adding the six known losses gives a higher-limit total between $\$730{,}000$ and $\$1{,}130{,}000$. Relative to the observed current-limit total, the same-claim total ratio is bounded by $1.000$ and $\$1{,}130{,}000/\$730{,}000=1.548$, rounded to three decimals.

Check and trap avoided. The lower bound equals the current-limit total, as a larger limit cannot reduce a claim payment. Treating every $\$100{,}000$ payment as an exact ground-up loss would incorrectly select the lower bound and imply a ratio of 1.000. The records support a range, not that point estimate. No distributional assumption or future-frequency adjustment has been used.

Why the Tail Is Not Identified

The identification problem can be stated directly. For claims censored at $L_0$, the records distinguish losses below the limit from losses that reach it. They do not reveal how the losses above the limit are distributed. Two severity models can agree on all observable information at the current cap while assigning very different amounts to the unobserved excess. Their expected payments at a higher limit, and therefore their ILFs, can differ.

For a continuous severity distribution, the probability contribution of a claim censored at $L_0$ is $P(X\ge L_0)$. An uncensored observation at $x<L_0$ contributes its density $f_X(x)$. Thus, for independent claims under a specified severity model, the likelihood is built from density contributions for uncensored values and tail-probability contributions for capped values. This is useful for fitting a chosen model, but it does not make an arbitrary tail shape observable.

The reason a higher limit depends on the hidden part of the distribution can be seen by separating claims below and above the old limit. For $u>L_0$, under the same no-deductible payment assumptions,

E[\min(X,u)] = E[X\,\mathbf{1}_{\{X<L_0\}}] + P(X\ge L_0)\,E[\min(X,u)\mid X\ge L_0].

The first term concerns losses below the old limit, and the second term requires the conditional severity distribution among losses that reached it. Capped records can inform the proportion reaching $L_0$, but do not reveal the conditional amounts above $L_0$. Therefore the second term cannot generally be calculated from those records alone.

Key result: A higher-limit ILF is not determined by capped payments alone when the ground-up severity distribution above the cap is unknown. A point estimate requires additional severity information or an explicit, defensible assumption about the unobserved tail. The assumption is part of the result, not a fact established by the capped payments.

Model-Based Extension Requires a Stated Tail Assumption

A parametric severity model can supply the missing tail behavior, but its implications should be presented as conditional on that model. The exponential distribution is a convenient illustration. If $X$ has an exponential distribution with mean $\theta$, then its density is $f_X(x)=\theta^{-1}e^{-x/\theta}$ for $x\ge0$. Under this assumption,

E[\min(X,a)] =\int_0^a x\frac{1}{\theta}e^{-x/\theta}\,dx +aP(X>a) =\theta(1-e^{-a/\theta}).

The equality follows by integrating the density over losses below $a$ and adding the capped payment $a$ for losses above $a$. This formula is valid for the stated exponential model and payment terms; it is not a model-free conclusion from capped observations.

Worked Example: Alternative Exponential Tails, Different ILFs

A hypothetical portfolio has four claims, each paid $\$100{,}000$ under a no-deductible per-claim limit $L_0=\$100{,}000$. Compare the indicated ILF from $\$100{,}000$ to $u=\$200{,}000$ under two alternative assumptions: $X$ is exponential with mean $\theta=\$100{,}000$, or $X$ is exponential with mean $\theta=\$200{,}000$. These are assumed model parameters, not estimates claimed to be identified by the four capped payments. Interest and mortality are not applicable.

Method. For each assumed mean, calculate the model’s expected limited payment at each limit using $E[\min(X,a)]=\theta(1-e^{-a/\theta})$, then divide the higher-limit amount by the base-limit amount. The claim count cancels from this severity ratio.

For $\theta=\$100{,}000$, the expected payment at the base limit is

100{,}000(1-e^{-1})=\$63{,}212.06,

and at the higher limit it is

100{,}000(1-e^{-2})=\$86{,}466.47.

The indicated factor is $\$86{,}466.47/\$63{,}212.06=1.368$, rounded to three decimals.

For $\theta=\$200{,}000$, the base-limit expected payment is

200{,}000(1-e^{-0.5})=\$78{,}693.87,

and the higher-limit expected payment is

200{,}000(1-e^{-1})=\$126{,}424.11.

The indicated factor is $\$126{,}424.11/\$78{,}693.87=1.606$, rounded to three decimals. Both models assign positive probability to a claim reaching the existing limit: the probability is $e^{-1}$ for the first model and $e^{-0.5}$ for the second. Consequently, four capped claims are possible under either assumption, although their likelihoods differ.

Check and trap avoided. The results differ because the assumed tail scales differ. The four payments at the cap do not establish either exponential mean. Reporting 1.368 or 1.606 without identifying its model assumption would make a model-dependent indication appear to be an empirical fact.

A fitted model can use both uncensored and censored observations through their respective likelihood contributions. However, a limited sample, a narrow range of observed limits, or very few uncensored large claims may leave the tail estimate highly dependent on the selected model. Parameter estimation does not remove the need to assess whether the model and the data support the proposed extension.

What Additional Evidence Can Support an Extension?

Useful evidence is information that distinguishes ground-up severities above the existing cap, or that gives a credible basis for specifying their distribution. Examples include:

Linked excess-layer records. Payments from a layer attaching above the primary limit can reveal excess amounts, provided the layer’s attachment, limit, and claim linkage are known.
Uncapped or higher-limit observations. Claims from comparable policies with limits above $L_0$ can expose some of the severity range hidden in the capped portfolio.
Detailed claim information. Settlements, valuations, or other documented ground-up amounts may provide severity information, subject to coverage and data-quality review.
A stated severity model. A parametric assumption may support a projection, but the chosen distribution, parameter basis, and uncertainty should be made explicit.

Evidence must match the loss basis used for the proposed ILF. A limit on a different coverage, a different deductible, or a different treatment of expenses can change the recorded payment. Before combining records, check that claim identifiers link the layers correctly and that the attachment and exhaustion terms are understood. A few observed excess payments can establish that some claims exceeded the old cap; they do not, by themselves, establish a reliable shape for the far tail.

Worked Example: Using Linked Excess-Layer Information

A hypothetical primary policy pays covered losses up to $\$100{,}000$, with no deductible. Ten claims each receive a primary payment of $\$100{,}000$. Linked records from an excess layer attaching at $\$100{,}000$ establish exact excess amounts of $\$40{,}000,\ \$80{,}000,\ \$120{,}000,$ and $\$200{,}000$ for four claims, with no upper layer cap over these amounts. For the other six claims, complete records verify that no excess is payable and that the ground-up loss is exactly $\$100{,}000$. Calculate the total payment for these claims at limits of $\$100{,}000$ and $\$200{,}000$, and the ratio. No severity distribution is assumed; interest and mortality are not applicable.

Method. Add the attachment point to each positive excess amount to recover the four ground-up losses: $\$140{,}000,\ \$180{,}000,\ \$220{,}000,$ and $\$300{,}000$. The other six losses are $\$100{,}000$ each, as established by the complete linked records.

At the current limit, the ten payments total $10(\$100{,}000)=\$1{,}000{,}000$. At the $\$200{,}000$ limit, the four larger losses pay $\$140{,}000,\ \$180{,}000,\ \$200{,}000,$ and $\$200{,}000$; the last two are capped at $\$200{,}000$. The remaining six each pay $\$100{,}000$. Thus the higher-limit total is

140{,}000+180{,}000+200{,}000+200{,}000+6(100{,}000) =\$1{,}320{,}000.

The same-claim ratio is $\$1{,}320{,}000/\$1{,}000{,}000=1.320$. As a check, the increase is $\$320{,}000$, equal to the sum of the four excess amounts limited to $\$100{,}000$ each: $\$40{,}000+\$80{,}000+\$100{,}000+\$100{,}000$.

Check and trap avoided. The calculation uses linked, exact excess information and applies the higher limit to each claim. It does not treat an excess payment as a ground-up loss: the attachment point must be added. This sample supports the calculation for these ten claims, but the four observed excess amounts alone do not establish the tail of a broader future portfolio.

Common Mistakes and Practical Interpretation

Equating the cap payment with the loss. Under the stated terms, a payment equal to the limit means $X\ge L_0$; it does not generally mean $X=L_0$.
Extrapolating a point factor from a bound. A possible range of higher-limit payments is not itself an estimate of the expected payment or the ILF.
Presenting a model output as data-only. A parametric factor depends on the distribution and parameter assumptions, especially in an unobserved tail.
Combining unmatched payment bases. Deductibles, coverage definitions, attachment points, and layer limits must be reconciled before using excess records to infer ground-up loss.
Assuming a few large claims define the far tail. Observations can support the range they cover without validating behavior beyond that range.

For an ILF extension, state what is directly observed, what remains censored, and which additional evidence or model supplies the missing tail information. This separation makes the indication interpretable and helps prevent unsupported precision. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, identify what the data support and state any assumptions used in the calculation.

A no-deductible policy has a per-claim limit of $\$80{,}000$. The recorded payments are $\$25{,}000,\ \$45{,}000,\ \$80{,}000,$ and $\$80{,}000$. What is the current-limit total, and what range is possible for the same four claims at a $\$120{,}000$ limit? (Answer: Current total $\$230{,}000$; higher-limit total from $\$230{,}000$ to $\$310{,}000$.)
Eight claims have payments $\$50{,}000,\ \$60{,}000,\ \$70{,}000,\ \$80{,}000,\ \$90{,}000$, and three payments of $\$100{,}000$ at a $\$100{,}000$ limit. Bound the same-claim ratio at a $\$150{,}000$ limit. (Answer: The current total is $\$650{,}000$, the higher-limit total is $\$650{,}000$ to $\$800{,}000$, and the ratio is 1.000 to 1.231.)
Assume $X$ is exponential with mean $\theta=\$150{,}000$, no deductible, and limits $L_0=\$100{,}000$ and $u=\$250{,}000$. Calculate the model-based ILF. (Answer: $150{,}000(1-e^{-5/3})/[150{,}000(1-e^{-2/3})]=1.667$, rounded to three decimals.)
Five claims each pay $\$100{,}000$ at the current limit. Complete linked excess records show excess payments of $\$25{,}000$ and $\$75{,}000$ for two claims and verify zero excess for the other three. With no deductible and a higher limit of $\$200{,}000$, find the same-claim ratio. (Answer: Higher-limit total $=\$500{,}000+\$25{,}000+\$75{,}000=\$600{,}000$; ratio $=1.200$.)
Under an exponential severity model with mean $\theta=\$100{,}000$, two uncensored claims are observed at $\$20{,}000$ and $\$50{,}000$, and three claims are censored at $\$80{,}000$. Write the likelihood contribution for these five observations, ignoring any frequency model. (Answer: $(1/100{,}000)^2 e^{-20{,}000/100{,}000}e^{-50{,}000/100{,}000}(e^{-80{,}000/100{,}000})^3=(1/100{,}000)^2e^{-3.1}$.)

What You'll Learn