Why the Policy Limit Changes the Severity Target
A policy limit changes what the insurer pays, but it does not reveal the full size of every loss. If a covered ground-up loss exceeds the limit, the insurer observes its own capped payment while the loss itself remains larger and unknown. A severity analysis must therefore decide whether its target is the payment made under the contract or the underlying ground-up loss. Those are different quantities, and the appropriate data interpretation and model depend on the choice.
Assume throughout this tutorial that there is no deductible, coinsurance, or other payment adjustment. Let \(X\) be a nonnegative covered ground-up loss and let \(u\) be the policy limit. The insurer payment is \(Y=\min(X,u)\). With a continuous ground-up severity distribution, a payment equal to \(u\) indicates that the loss reached or exceeded the limit; it does not identify the exact ground-up amount.
Two Modeling Questions, Two Targets
If the business question is the expected insurer payment under a specified limit, the response can be the observed limited payment. The target is then \(E[\min(X,u)\mid Z]\). A fitted model for this mean is directly about the contract being priced; it need not reconstruct the unseen portion of losses above the limit. It does, however, need to represent the capped response appropriately. For example, a continuous Gamma distribution has no point mass at \(u\) and allows values above \(u\), so it is not an exact distribution for a response that has a positive probability of being exactly capped. A mean-modeling approach may still be useful if its assumptions for the mean and variance are reasonable, but the distributional mismatch should not be ignored.
If the question is instead to estimate ground-up severity, the target remains \(X\), even when some observations are capped. An observation below the limit can be used as an exact loss, while an observation at the limit says only that \(X\ge u\). A ground-up model therefore requires a suitable conditional severity distribution, not just a formula for the mean. The distribution determines how much probability lies above each limit and hence how the capped payments relate to the latent severity.
| Modeling choice | Response or information used | Target | Key requirement |
|---|---|---|---|
| Limited-payment severity | Observed payment \(Y=\min(X,u)\), including payments at the cap | \(E[Y\mid Z]\) | A suitable conditional mean model; a full response distribution must account for a possible cap mass |
| Ground-up severity | Exact \(X\) below the limit; information \(X\ge u\) at the limit | The distribution of \(X\), including \(E[X\mid Z]\) | A specified conditional ground-up distribution and correct treatment of capped observations as censored |
A key distinction is that \(E[X\mid Z]\) alone generally does not determine \(E[\min(X,u)\mid Z]\). The capped mean depends on the distribution of losses relative to the limit. Two risks can have the same ground-up mean but different probabilities of exceeding \(u\), and therefore different expected insurer payments. Conversely, a direct model of capped payments answers the limited-payment question but does not, by itself, identify the tail of the ground-up distribution above the cap.
Worked Example: Calculating the Observed Limited-Payment Target
A hypothetical insurer reviews four claims, all with the same policy limit of $9,000. The recorded payments are $2,800, $6,200, $9,000, and $9,000. Under the stated no-deductible assumptions, calculate the sample mean limited payment and state what the two capped observations reveal about ground-up severity.
Method. Use the insurer payments as the response because the requested target is the limited-payment mean. Add all four observed payments:
There are four claims, so the sample mean limited payment is:
The two $9,000 payments are fully observed as insurer payments, so they are valid responses for this target. Each indicates a ground-up loss of at least $9,000, but neither reveals the amount above $9,000. The $6,750.00 average is therefore the average of the limited payments, not an estimate obtained by treating all four ground-up losses as fully known. Common mistake avoided: replacing each capped payment with an assumed exact ground-up loss of $9,000 confuses an observed payment with an unobserved loss.
Why the Ground-Up Distribution Matters
For a nonnegative loss and a fixed limit, expected limited payment can be derived from the ground-up survival function. Write \(S_X(x)=P(X>x)\). For each realized loss, \(\min(X,u)\) is the length of the interval from zero to \(u\) over which the indicator \(1(X>x)\) equals one. Taking expectations gives the tail-integral identity:
This identity shows why a ground-up mean alone is insufficient: the limited expectation uses the survival probabilities across the range up to the limit. The full ground-up mean, when finite, uses the tail across the entire nonnegative range, including amounts above the limit. If the conditional distribution varies by rating variables \(Z\), the calculation is conditional on \(Z\): \(E[\min(X,u)\mid Z]=\int_0^u P(X>x\mid Z)\,dx\).
Worked Example: From an Exponential Ground-Up Model to a Limited Mean
For a hypothetical class of claims, suppose ground-up severity has an exponential distribution with mean \(\theta=\$18{,}000\), so \(S_X(x)=e^{-x/18{,}000}\) for \(x\ge0\). The policy limit is \(u=\$9{,}000\). Calculate the probability that a claim reaches the limit and the expected insurer payment. Report the probability to four decimals and money to the nearest cent.
Method. For the exponential model, the probability of reaching the limit is the survival probability at \(u\):
Rounded to four decimals, the probability of a capped payment is \(0.6065\). To calculate the expected limited payment, integrate the survival function from zero to the limit:
Substitute the evaluated exponential term:
Thus the expected insurer payment is $7,082.45. The ground-up mean remains $18,000; it is not the expected payment under the $9,000 limit. Check: the limited mean is below the $9,000 maximum, as it must be. Common mistake avoided: using the ground-up mean as the limited mean ignores the cap and the probability mass of losses that exceed it.
Reading a Capped Observation Correctly
For ground-up modeling, policy-limit observations are right-censored: their exact values are unavailable, but they are known to be at least as large as the limit. Under a continuous severity distribution, the probability of a loss being exactly equal to the limit is zero, so recording a payment at \(u\) is conventionally interpreted as \(X\ge u\). If a severity model instead allows a genuine atom exactly at the limit, the observation mechanism must distinguish that event from a loss exceeding the limit.
This distinction is about what the data say, not about whether the insurer can record its payment. The payment at the limit is known exactly as a limited payment. It is the ground-up loss that is not known exactly. For a direct limited-payment model, a capped response is an observed response. For a ground-up model, the same record is a censored observation and must not be substituted as though it were an exact ground-up severity.
Worked Example: Separate the Payment Average from the Ground-Up Information
A hypothetical portfolio has four claims, each subject to a $12,000 limit. The insurer records payments of $3,000, $10,000, $12,000, and $12,000. Calculate the observed mean payment, classify the ground-up information for each claim, and explain what additional assumption is needed to predict a different limited mean for this portfolio.
Step 1: Calculate the direct limited-payment average. The total payment is:
Dividing by four claims gives \(37{,}000/4=\$9{,}250.00\). This is the observed mean insurer payment.
Step 2: Classify the ground-up observations. The first two payments are below the $12,000 limit and, under the assumed contract, identify exact ground-up losses of $3,000 and $10,000. The last two payments imply ground-up losses of at least $12,000, but the amounts above the limit are unknown.
Step 3: Identify what is needed for a model-based prediction. To predict the expected limited payment for a different risk mix or limit, one needs either a direct model for the limited-payment mean under the relevant limits and covariates, or a specified ground-up severity distribution conditional on those covariates. The four payments alone do not reveal the sizes of the two losses above $12,000, nor do they determine the ground-up tail. Common mistake avoided: calling $9,250.00 the mean ground-up severity treats both capped losses as if they were exactly $12,000.
Assumptions and Practical Model Checks
The target should follow the business question and the observation process. A limited-payment model is natural when the objective is expected payments under known contract terms. It requires that the recorded payment be measured consistently and that the model’s conditional mean describe the capped response. If the model specifies a full probability distribution, that distribution should accommodate the response’s bounded support and possible probability mass at the limit. A standard continuous Gamma severity distribution does not do so exactly, even if a Gamma-based mean structure is being used as an approximation.
A ground-up severity model is appropriate when the objective requires the latent loss distribution, such as estimating ground-up means or evaluating payments at limits not already represented in the data. It requires a defensible conditional distribution for \(X\), correct covariates, and correct treatment of capped observations as \(X\ge u\). Predictions of limited payments from that model require integrating the fitted distribution up to the applicable limit. With only capped data, the tail above the limits may be weakly identified; extrapolation then depends strongly on the assumed severity distribution.
When limits vary across claims, the same numerical payment can mean different things: a payment of $8,000 may be uncapped under a $15,000 limit but capped under an $8,000 limit. A direct limited-payment model must account for the relevant limit structure, and its fitted mean must be compatible with the payment’s maximum. A ground-up model must use each claim’s actual censoring threshold. In either approach, verify whether the response is per claim, how the limit is applied, and whether any contract terms beyond the stated assumptions change the insurer payment.
Check Your Understanding
For each problem, identify the relevant target and work through the calculation before checking the answer.
- Four claims have a common $8,000 limit and recorded payments of $2,500, $8,000, $6,000, and $8,000. What is the observed mean limited payment, and how many observations are capped? (Answer: Mean limited payment $6,125.00; 2 capped observations.)
- Ground-up severity is exponential with mean $12,000, and the policy limit is $6,000. Find the probability of reaching the limit and the expected limited payment. (Answer: Probability 0.6065; expected payment $4,721.63.)
- At a $15,000 limit, four recorded payments are $3,000, $10,000, $15,000, and $15,000. Find the observed mean payment and state how many ground-up losses are known exactly, assuming a continuous severity distribution. (Answer: Mean payment $10,750.00; 2 ground-up losses known exactly.)
- A proposed ground-up model supplies only \(E[X\mid Z]=\$24{,}000\) for a risk with an $8,000 limit. Can that mean alone determine \(E[\min(X,8{,}000)\mid Z]\)? Explain what is missing. (Answer: No; the conditional ground-up distribution, or equivalent survival probabilities through the limit, is needed.)
- A direct limited-payment model predicts mean payments of $4,200 for each of 40 claims in one group and $7,100 for each of 60 claims in another group. Calculate the combined predicted mean payment per claim. (Answer: \((40(4{,}200)+60(7{,}100))/100=\$5{,}940.00\).)