Why the Severity Response Must Match the Target
A severity model is only interpretable when its response represents the quantity the analysis intends to predict. For a closed claim, the final amount paid may be a useful measure of claim cost. For an open claim, however, the amount paid so far is generally only part of the eventual payment, while the case reserve is an estimate of future payment rather than an observed payment. Combining these values without defining the target can make a model’s fitted severity difficult to interpret.
The distinctions matter in pricing and reserving work. A model of amounts paid during a particular development period answers a different question from a model of ultimate claim costs. Likewise, an incurred amount that includes a reserve is not automatically an observed ultimate amount. Before fitting a Gamma or Inverse Gaussian severity GLM, specify whether the response is paid, incurred, or ultimate and make sure open claims are handled consistently with that choice.
The values must be defined on a consistent basis. For example, do not compare gross-of-deductible payments with net-of-deductible ultimate amounts, or combine amounts subject to different policy limits without defining the corresponding severity target. The earlier material on policy limits and severity targets distinguishes a policy limit as part of the target from a limit that only censors an underlying loss. Claim development adds a separate issue: even with the coverage basis fixed, an open claim may not yet reveal its ultimate amount.
Paid, Incurred, and Ultimate Responses
A paid response is an amount actually paid over a specified period or cumulatively through a specified valuation date. It can be appropriate when the modeling objective concerns cash flow or payments at that development age. It is not automatically appropriate as the response for ultimate severity: a small cumulative payment on an open claim may reflect early development, not a small ultimate claim.
An incurred response is the sum of cumulative payments and the case reserve at a valuation date. It includes an estimate of future payments. Its value may change when the reserve is revised, even if no additional cash has been paid. It can be modeled as an incurred-at-valuation response if that is the intended target, but it should not be labeled an observed ultimate amount merely because it includes a reserve.
An ultimate response is the final claim amount on the chosen basis. It is observable only once the claim is sufficiently mature or closed under the relevant definition. Restricting a model to settled claims can create a different problem: claims that settle quickly may not be representative of all claims. Whether that creates bias depends on how settlement timing relates to severity and risk characteristics.
| Response | What is observed? | Appropriate interpretation | Key caution |
|---|---|---|---|
| Cumulative paid | Payments made through a stated valuation date | Paid amount at that development age | Open claims may have additional payments to come |
| Incurred | Cumulative paid plus case reserve | Current incurred estimate | The reserve is an estimate and may be revised |
| Ultimate | Final amount under a defined coverage basis | Ultimate severity | Usually unavailable for open claims without an explicit method or assumption |
Worked Example: Calculating Paid and Incurred Responses
A hypothetical portfolio contains two closed claims with final payments of $9,200 and $14,800, and one open claim with $7,500 paid and a $5,500 case reserve at the valuation date. Calculate the three-claim average paid response and the three-claim average incurred response. Treat all amounts as being on the same coverage and expense basis. This is a deterministic accounting calculation; no interest rate, mortality assumption, or probability distribution is assumed.
Method. Use cumulative paid as the paid response for each claim. For incurred response, add the case reserve to cumulative paid for the open claim. The two closed claims have no case reserve under the stated closure convention.
The paid responses are $9,200, $14,800, and $7,500. Their total is \(9{,}200+14{,}800+7{,}500=31{,}500\), so the average paid response is \(31{,}500/3=\$10{,}500.00\).
The open claim’s incurred amount is \(7{,}500+5{,}500=\$13{,}000\). Thus the incurred responses are $9,200, $14,800, and $13,000, with total \(9{,}200+14{,}800+13{,}000=37{,}000\). The average incurred response is \(37{,}000/3=\$12{,}333.33\), rounded to the nearest cent.
Check and common mistake. The incurred average exceeds the paid average by \((37{,}000-31{,}500)/3=\$1{,}833.33\), which is exactly the open claim’s reserve divided by three. A common mistake is to call $13,000 the open claim’s ultimate amount. The calculation only establishes its current incurred amount; it does not establish that the reserve will equal the final additional payment.
When an Open Claim Can Be Treated as Censored
A right-censored observation provides a known lower bound on an event amount but not the exact amount. In a simplified claim model, suppose the target is ultimate severity \(U\), payments are nonnegative, there are no recoveries or payment reversals, and a claim that remains open has a known cumulative paid amount \(c\). If remaining payment is known to be strictly positive, then \(U>c\). The exact ultimate amount is not observed, but the event contributes information that the ultimate amount exceeds \(c\).
For a continuous severity distribution with density \(f_U\) and survival function \(S_U(u)=P(U>u)\), an exact observed ultimate amount \(u\) contributes \(f_U(u)\) to the likelihood. A right-censored observation at \(c\) contributes \(S_U(c)\). This is the standard likelihood treatment for right censoring when the censoring mechanism is noninformative given the model covariates: the chance of being censored at \(c\) must not depend on the unobserved ultimate amount in a way the likelihood omits.
The conditions deserve attention. Paid-to-date is not automatically an independent censoring threshold. Payment timing can depend on claim complexity, litigation, injury severity, coverage disputes, or the amount itself. Further, an open claim’s paid amount may be zero or may already equal the eventual amount. Thus, the assertion \(U>c\) must be justified by the data definition, and the censoring mechanism must be addressed or reasonably assumed noninformative. If those requirements do not hold, inserting \(S_U(c)\) as though it were a standard censoring contribution can produce a misspecified likelihood.
An incurred amount is usually not a censoring threshold. Because it includes a reserve estimate, it can be above or below the eventual amount. Knowing \(I_t\) does not, in general, establish \(U>I_t\), \(U=I_t\), or \(U<I_t\). A reserve may be used in a separately specified model for incurred amounts or reserve development, but it is not a substitute for an observed ultimate amount or a valid censoring bound.
Worked Example: Likelihood Contributions for Open Claims
Suppose, hypothetically, that ultimate severity \(U\) follows an exponential distribution with mean \(\theta=\$10{,}000\), so \(f_U(u)=\theta^{-1}e^{-u/\theta}\) and \(S_U(u)=e^{-u/\theta}\) for \(u\geq 0\). Assume two open claims are known to have ultimate amounts exceeding their cumulative paid thresholds of $6,000 and $10,000, with the right-censoring assumptions just described. A third claim has an exact ultimate amount of $8,000. Calculate the joint likelihood contribution, treating the claims as conditionally independent. No interest or mortality assumption is involved; the severity distribution and parameter are specified above.
Method. The exact $8,000 observation contributes a density. Each open claim contributes a survival probability at its threshold:
The exact observation contributes \(f_U(8{,}000)=0.0001e^{-0.8}=0.0000449328964\) per dollar. The first open claim contributes \(S_U(6{,}000)=e^{-0.6}=0.548811636\). The second contributes \(S_U(10{,}000)=e^{-1}=0.367879441\).
Multiplying the contributions gives \(L=(0.0000449328964)(0.548811636)(0.367879441)\approx0.0000090717953\) per dollar. The result is rounded to eight significant figures. The units are per dollar because there is one exact density contribution and the two survival contributions are probabilities.
Check and common mistake. The two open claims contribute survival probabilities, not density values at their paid thresholds: their ultimate amounts are not known exactly. Also, \(0.0000090717953\) has eight significant figures; reporting \(0.0000090718\) while calling it eight significant figures would be incorrect because that displayed number has five.
Adjusting an Open Claim to an Ultimate Target
If the target is ultimate severity, an adjustment for an open claim requires a model for the unobserved portion. One route is a conditional expectation: estimate \(E[U\mid\text{information observed at the valuation date}]\) under a stated severity and development model. The appropriate conditioning information may include paid amounts, claim age, coverage, and other covariates. A generic development factor or reserve addition is not automatically valid for every risk class or valuation date.
A short derivation illustrates how an adjustment can arise under a particular distribution. Suppose \(U\) has an exponential distribution with mean \(\theta\), and the only observed information is \(U>c\). Its density is \(f_U(u)=\theta^{-1}e^{-u/\theta}\). The conditional density above \(c\) is \(f_U(u\mid U>c)=f_U(u)/S_U(c)\), for \(u>c\). Substituting the exponential density and survival function gives \(f_U(u\mid U>c)=\theta^{-1}e^{-(u-c)/\theta}\). Therefore the excess \(U-c\), conditional on exceeding \(c\), is exponential with mean \(\theta\), so \(E[U\mid U>c]=c+\theta\). This memoryless result is specific to the exponential distribution; it is not a general rule for claim severity.
Worked Example: Model-Based Ultimate Estimate for an Open Claim
For a hypothetical claim, cumulative paid at the valuation date is $12,000 and the case reserve is $8,000, so current incurred is $20,000. Suppose, for illustration, that ultimate severity \(U\) follows an exponential distribution with mean \(\theta=\$25{,}000\), and that the known open-claim information is exactly \(U>\$12{,}000\). Estimate the conditional expected ultimate amount. Assume no recoveries and that the paid amount is a valid lower bound. No interest or mortality assumption is involved.
Method. Apply the exponential conditional-mean result with \(c=\$12{,}000\) and \(\theta=\$25{,}000\): \(E[U\mid U>12{,}000]=12{,}000+25{,}000=\$37{,}000\). The model-implied expected unpaid amount is therefore \(37{,}000-12{,}000=\$25{,}000\).
The case reserve is $8,000, so current incurred is \(12{,}000+8{,}000=\$20{,}000\). The model-based conditional expected ultimate exceeds current incurred by \(37{,}000-20{,}000=\$17{,}000\).
Check and common mistake. The conditional expected ultimate is at least the paid lower bound, as required, and its expected unpaid component is $25,000 under this exponential model. It is not obtained by adding the $8,000 case reserve to paid and calling the result ultimate. Nor should the $25,000 expected unpaid amount be treated as a universal adjustment: the result depends on the assumed distribution, parameter, and conditioning information.
Practical Specification and Exam Traps
For an ultimate-severity GLM, the cleanest response is an observed ultimate amount on a consistently defined basis. When some claims are still open, defensible approaches depend on available information and the objective: restrict to appropriately mature claims, model the observed incurred target explicitly, estimate ultimate conditionally under a justified development model, or use a censoring likelihood when the censoring conditions hold. Each choice changes what the fitted mean represents. The earlier material on Gamma and Inverse Gaussian severity GLMs supplies candidate response distributions; the response definition and observation mechanism must still be specified separately.
Do not assume that treating current incurred as ultimate is harmless simply because the reserve is intended to estimate future payments. That substitution turns an estimate into a purported observation and can understate or overstate variability. Conversely, do not discard all open claims automatically: a correctly specified censored likelihood can retain their information, but only when the bound and censoring assumptions are credible.
State whether the model predicts paid at a specified age, incurred at a valuation date, or ultimate severity.
Identify exact amounts, open claims with partial payment, and claims represented by reserve-inclusive estimates.
Use a response matching the target, a justified conditional adjustment, or a censoring contribution only when its assumptions are satisfied.
A model of current incurred amounts is not automatically a model of ultimate severity, even if its predictors and link function are the same.
Check Your Understanding
For each problem, identify the target or calculation carefully before using the numbers.
- A hypothetical closed claim has final payments of $18,400. At an earlier valuation date it had $11,000 paid and a $9,000 case reserve. What were its paid and incurred amounts at that earlier date, and what is its final paid amount? (Answer: Earlier paid $11,000; earlier incurred $20,000; final paid $18,400.)
- An open claim has $4,000 cumulative paid and a $3,000 case reserve. Under a no-recovery convention, does the reserve establish that ultimate severity is at least $7,000? Explain. (Answer: No. Incurred is $7,000, but a reserve is an estimate and does not establish \(U\geq7{,}000\); only the stated paid lower bound may be usable, if further payment is known to be due.)
- Assume \(U\) is exponential with mean $20,000 and an open claim is known to satisfy \(U>\$5{,}000\). Find the model-based conditional expected ultimate amount. (Answer: $25,000.)
- Under an exponential model with mean $10,000, an exact ultimate observation is $5,000 and a second claim is right-censored at $5,000. Assuming independent, noninformative censoring, write the likelihood contribution and evaluate it. (Answer: \(f(5{,}000)S(5{,}000)=0.0001e^{-1}\approx0.00003679\) per dollar, rounded to four significant figures.)
- A data extract contains an open claim’s paid-to-date amount but settlement timing is strongly related to claim complexity. Can its paid amount be inserted as an ordinary independent right-censoring threshold without further analysis? (Answer: No. The noninformative-censoring assumption is questionable and the mechanism or an alternative model must be addressed.)