Why a Capped Payment Is Not an Exact Severity
A policy limit caps the amount paid for a sufficiently large covered loss. If the payment reaches the limit, the underlying loss is not known to equal that amount: it may be larger. For severity modeling, that observation carries different information from a claim whose uncapped severity is observed exactly. Treating both observations as exact can distort the likelihood and, in turn, the fitted severity distribution.
This tutorial considers claim-level severities under a continuous severity model. The focus is the likelihood contribution from an exact observation and from a severity censored above a known policy limit. The likelihood depends on the severity distribution and its parameters; it is not generally a probability for a particular exact value.
For a continuous distribution, \(P(X=u)=0\), so \(P(X\ge u)=P(X>u)\). A policy payment equal to \(u\) therefore represents the event \(X\ge u\), not an exact observation \(X=u\), when the payment is capped at \(u\). This interpretation assumes the recorded limit is the cap and the underlying severity is otherwise unobserved above it. If the model permits a point mass at the limit, or the record cannot distinguish a capped claim from an exact loss at the limit, the observation mechanism must be specified more carefully.
Likelihood Contributions and Assumptions
Suppose claim severities \(X_1,\ldots,X_n\) are conditionally independent given their recorded risk characteristics. For claim \(i\), let \(y_i\) be the observed amount and let \(\delta_i=1\) mean that the severity is observed exactly; let \(\delta_i=0\) mean it is right-censored at the known threshold \(u_i\). The likelihood contribution for that claim is
Here \(\theta\) denotes the severity-model parameter or parameters. For an exact claim, \(y_i\) is the observed severity. For a censored claim, \(u_i\) is the censoring limit, and the capped payment tells us the event that the severity reached that threshold. If risk characteristics differ, use the corresponding conditional density and survival function for each claim.
This expression assumes the severity model is appropriate for the population being modeled, the claim observations are independent conditional on their modeled characteristics, and the censoring threshold is known. It also assumes the censoring mechanism does not carry additional unknown severity-model information beyond the recorded threshold and censoring status. A fixed policy limit meets that condition when the limit is specified independently of the realized severity. This is not a likelihood for deductible-selected claims, where observation may require crossing a lower threshold.
Use the exact severity if the underlying amount is observed; otherwise record the censoring threshold and mark the claim as censored.
An exact value contributes a density. A censored value contributes the probability of exceeding its threshold.
Under conditional independence, the joint likelihood is the product of the individual contributions, evaluated using each claim’s risk characteristics.
Why the Survival Probability Appears
The distinction follows directly from what the data reveal. An exact continuous value has probability zero of occurring at one precise point, so its likelihood contribution is represented by a density. More formally, for a small interval of width \(h\) around an observed \(x\), the probability is approximately \(f_X(x)h\). After accounting for the interval width, the contribution per unit of severity is \(f_X(x)\). A capped observation does not identify a small interval around the limit; it identifies the entire event that the severity is at least the limit.
Thus replacing a censored observation by a density at the limit would use information the data do not contain. The density \(f_X(u)\) describes local probability per unit of severity near \(u\); the survival probability \(S_X(u)\) describes the total probability of all severities at or above \(u\).
Worked Example: Exact and Capped Claims Under an Exponential Model
A hypothetical portfolio is modeled with independent claim severities \(X\) following an exponential distribution with rate \(\lambda=0.01\) per thousand dollars. Two claims are observed exactly, at 40 and 120 thousand dollars. A third claim is capped at 200 thousand dollars. Construct the joint likelihood contribution and state its units.
Method and assumptions. For this exponential model, \(f_X(x)=\lambda e^{-\lambda x}\) and \(S_X(u)=e^{-\lambda u}\), for nonnegative amounts in thousands of dollars. The first two observations are exact, while the third contributes the probability that its severity reaches or exceeds 200.
The density factors are \(0.01e^{-0.4}=0.006703200\) and \(0.01e^{-1.2}=0.003011942\), each per thousand dollars. The survival factor is \(e^{-2}=0.1353353\), a probability. Multiplying the factors gives
There are two density contributions and one probability contribution, so the joint likelihood has units of per (thousand dollars)\(^2\). The result is approximately \(2.732\times10^{-6}\) per (thousand dollars)\(^2\). As a check, the exponent combines all three severities or thresholds, \(40+120+200=360\), and \(0.01(360)=3.6\), matching the likelihood above.
Common mistake avoided: The capped claim contributes \(e^{-2}\), not \(0.01e^{-2}\). Multiplying by the exponential density factor at 200 would incorrectly treat the capped payment as an exact severity.
Different Risk Classes Use Different Conditional Contributions
If the severity distribution varies by risk characteristics, the same contribution rule applies claim by claim, but the parameters must match each claim’s class or modeled characteristics. The likelihood remains a product under conditional independence. The following example uses a simple exponential model with different means by class; it does not require a common mean across the portfolio.
Worked Example: A Capped Claim in One Class and an Exact Claim in Another
In a hypothetical two-class portfolio, severity is exponential in each class. Class A has mean severity 100 thousand dollars; one claim in that class is observed exactly at 50 and another is capped at 300. Class B has mean severity 200 thousand dollars; one claim is observed exactly at 100. Find their joint likelihood contribution.
Method and assumptions. An exponential distribution with mean \(\mu\) has density \(f_X(x)=\mu^{-1}e^{-x/\mu}\) and survival function \(S_X(u)=e^{-u/\mu}\). Use \(\mu_A=100\) for both Class A claims and \(\mu_B=200\) for the Class B claim. Amounts are in thousands of dollars.
The exact Class A claim contributes \(100^{-1}e^{-50/100}=0.01e^{-0.5}=0.006065307\) per thousand dollars. The Class A capped claim contributes \(e^{-300/100}=e^{-3}=0.04978707\). The exact Class B claim contributes \(200^{-1}e^{-100/200}=0.005e^{-0.5}=0.003032653\) per thousand dollars.
There are two exact observations, so the joint likelihood has units of per (thousand dollars)\(^2\); the capped claim contributes a dimensionless probability. The rounded result is \(9.157819\times10^{-7}\) per (thousand dollars)\(^2\). Checking the exponent, the exact and censored amounts contribute \(0.5+3+0.5=4\), and the two density constants multiply to \(0.00005\), as used above.
Common mistake avoided: Applying Class B’s mean to the Class A capped claim, or applying one portfolio-wide mean to all three claims, would ignore the stated class-specific severity models.
A Gamma Example and the Density-at-the-Limit Trap
The density-versus-survival distinction is not specific to exponential models. In any continuous severity model, an exact value contributes the model’s density and a right-censored value contributes its survival probability. For a Gamma distribution with shape \(k\) and scale \(\theta\), the density is \(f_X(x)=x^{k-1}e^{-x/\theta}/(\Gamma(k)\theta^k)\) for \(x>0\). For the special case \(k=2\), this simplifies to \(f_X(x)=xe^{-x/\theta}/\theta^2\), and its survival function is \(S_X(u)=e^{-u/\theta}(1+u/\theta)\).
Worked Example: Gamma Severity With One Exact and One Censored Claim
A hypothetical severity model is Gamma with shape \(k=2\) and scale \(\theta=100\) thousand dollars. One claim is observed exactly at 100 thousand dollars, and another is capped at 200 thousand dollars. Calculate the joint likelihood contribution and compare it with the incorrect treatment of the cap as an exact value.
Method and assumptions. The Gamma model is continuous, claims are independent, and the cap identifies the event \(X\ge200\). The exact claim contributes the shape-2 Gamma density at 100; the capped claim contributes the shape-2 survival probability at 200.
For the exact claim, \(f_X(100)=100e^{-100/100}/100^2=0.01e^{-1}=0.003678794\) per thousand dollars. For the capped claim, \(S_X(200)=e^{-200/100}(1+200/100)=3e^{-2}=0.40600585\).
The joint contribution has units per thousand dollars because there is one density factor and one dimensionless probability factor. The result checks against direct multiplication: \(0.003678794\times0.40600585\approx0.001493612\).
Common mistake and correction. The wrong calculation uses \(f_X(200)=200e^{-2}/100^2=0.002706706\) for the capped claim. Multiplying that by the exact-claim density gives approximately \(0.000009958\) per (thousand dollars)\(^2\). This is wrong for a capped record: it substitutes a density at 200 for the probability of every severity at least 200. The correct contribution is \(S_X(200)=0.40600585\), giving \(0.001493612\) per thousand dollars.
Practical Checks Before Maximizing a Likelihood
When constructing a likelihood for estimation, write the contribution for each record before simplifying the product. This makes it easier to detect whether a claim is exact or censored, whether the correct limit was used, and whether the relevant class parameters were applied. The likelihood is then maximized over the model parameters; the rule for a capped observation does not change just because the parameters are estimated rather than specified.
- Check the observation mechanism. A recorded payment equal to a limit is censored only when it represents an underlying severity at or above that limit. Do not infer censoring solely from a rounded amount without considering how the record was generated.
- Check the argument of the survival function. Use the threshold at which severity becomes unobserved, not an unrelated payment amount or an assumed value above the cap.
- Check likelihood units. Each exact continuous observation contributes a density, while each censored observation contributes a probability. With \(r\) exact observations, the joint likelihood has \(r\) inverse-severity units when all amounts use the same scale.
- Check independence and conditioning. Multiplication is justified for conditionally independent claims. If dependence is part of the model, a joint distribution is required instead.
Check Your Understanding
For each problem, treat claims as independent and amounts as measured in thousands of dollars. Assume the stated censoring thresholds are known and the severity models are continuous.
- An exponential severity has rate \(0.02\) per thousand dollars. One claim is observed exactly at 25, and a second is capped at 100. What is their joint likelihood contribution, including its units? (Answer: approximately \(0.0016417\) per thousand dollars.)
- An exponential severity has mean 250. One claim is observed exactly at 50 and another is capped at 500. Find the joint likelihood contribution and its units. (Answer: approximately \(0.0004432\) per thousand dollars.)
- Under an exponential model with rate \(0.01\), two severities are observed exactly at 20 and 80, and a third is capped at 150. Find the joint likelihood contribution. (Answer: approximately \(8.2085\times10^{-6}\) per (thousand dollars)\(^2\).)
- A Gamma severity has shape 2 and scale 50. One claim is observed exactly at 50, and a second is capped at 100. Calculate the joint likelihood contribution. (Answer: approximately \(0.0029872\) per thousand dollars.)
- Class A has exponential mean 100, and Class B has exponential mean 200. A Class A claim is observed exactly at 60, a second Class A claim is capped at 200, and a Class B claim is observed exactly at 100. Find the joint likelihood contribution. (Answer: approximately \(2.2525\times10^{-6}\) per (thousand dollars)\(^2\).)