Parametric Severity Models for Exposure Curves

From a Severity Model to an Exposure Curve

An exposure curve summarizes how expected losses change as the available policy limit changes. When ground-up claim data are sparse at high severities, a fitted parametric severity model can provide a smooth survival function and a basis for calculating limited expected losses. The benefit comes with a responsibility: the resulting curve depends on the chosen distribution, the quality of the data, and the range over which the model is used.

This tutorial fits exponential and Pareto severity models and uses their survival functions to calculate limited expected losses. All claim amounts in the examples are hypothetical and are stated in thousands of dollars. Assume a consistent ground-up loss basis: claims have been adjusted, as needed, to represent the same coverage and valuation basis before fitting.

The curve \(G(u)\) is nondecreasing and lies between zero and one. It approaches one as the limit grows without bound when \(E[X]\) is finite. An increased limits factor is not the same as \(G(u_2)\): it compares limited losses at two limits, while \(G(u_2)\) compares limited loss with unlimited expected loss.

Limited Expected Loss from the Survival Function

For any nonnegative loss, the limited expected loss has a useful integral representation. For a fixed loss amount \(X=x\), the amount \(\min(x,u)\) is the length of the interval from zero to \(u\) over which the indicator \(I(x>t)\) equals one. Therefore,

Taking expectations and interchanging expectation and integration gives the survival-function formula. The interchange is valid here because the integrand is nonnegative.

This result explains why the entire severity distribution below the limit matters: \(L(u)\) is the area under the survival curve from zero to \(u\). It also shows that, wherever the survival function is continuous, the slope of \(L(u)\) is \(S_X(u)\). A model with an excessively heavy tail can materially increase estimates at high limits.

Fitting and Using an Exponential Severity Model

An exponential model with rate \(\lambda>0\) has density \(f_X(x)=\lambda e^{-\lambda x}\) and survival function \(S_X(x)=e^{-\lambda x}\) for \(x\geq 0\). Its mean is \(E[X]=1/\lambda\). For independent, complete observations \(x_1,\ldots,x_n\) from this model, the maximum-likelihood estimate is \(\hat{\lambda}=n/\sum_j x_j=1/\bar{x}\). The model assumes a constant hazard over severity: conditional on exceeding any amount, the remaining severity has the same exponential distribution. That assumption may be unsuitable for claim data with a heavier tail or changing hazard.

Integrating the exponential survival function gives the limited expected loss and, after dividing by the model mean, the exposure curve:

Fitting a Pareto Tail

A Pareto Type I model with threshold \(x_m>0\) and shape parameter \(\alpha>0\) has support \(x\geq x_m\), survival function \(S_X(x)=(x_m/x)^\alpha\) for \(x\geq x_m\), and survival equal to one below \(x_m\). If all observations are complete and independently drawn from this distribution with known threshold \(x_m\), the maximum-likelihood estimate is

This follows from the density \(f_X(x)=\alpha x_m^\alpha x^{-(\alpha+1)}\): the log-likelihood, apart from terms not involving \(\alpha\), is \(n\ln(\alpha)-\alpha\sum_j\ln(x_j/x_m)\). Differentiating with respect to \(\alpha\), setting the derivative to zero, and solving yields the estimate above. For this model \(E[X]=\alpha x_m/(\alpha-1)\) only when \(\alpha>1\); if \(\alpha\leq1\), the unlimited mean is infinite, so the normalized exposure curve \(L(u)/E[X]\) is not defined as an ordinary finite ratio.

When \(u\geq x_m\), integration of the Pareto survival function gives

For \(\alpha>1\), this expression tends to \(x_m+x_m/(\alpha-1)=E[X]\) as \(u\) tends to infinity. For \(u<x_m\), the loss always exceeds the limit, so \(L(u)=u\). At \(\alpha=1\), the integral from \(x_m\) to \(u\) is \(x_m\ln(u/x_m)\); the limited loss remains finite at any finite limit, even though the unlimited mean is infinite.

Right-Censored Claims and Model Fitting

A claim recorded only as exceeding a threshold \(c\) is right-censored: its exact severity is unknown, but the observation establishes \(X>c\). Under a parametric model, an exact observation contributes its density to the likelihood, while a right-censored observation contributes its survival probability. This treatment assumes the censoring mechanism is noninformative for severity, conditional on the model and the information used to set the censoring threshold. A policy limit that suppresses the observed amount is one possible source of censoring.

For independent exponential observations, let \(r\) be the number of exact observations, let their amounts be \(x_1,\ldots,x_r\), and let the remaining observations be censored at amounts \(c_1,\ldots,c_s\). The likelihood is proportional to \(\lambda^r\exp[-\lambda(\sum_jx_j+\sum_kc_k)]\). Differentiating its log with respect to \(\lambda\) gives the estimate

A censored claim contributes its known amount at risk to the denominator but does not count as an exact event in the numerator. This estimator depends on the exponential assumption; other severity distributions require their own likelihood and may not yield such a simple closed form.

Practical Checks Before Using a Parametric Curve

A fitted curve is a model-based estimate, not a direct observation of losses at every possible limit. Check that the fitted distribution is compatible with the data and the intended application before using it for pricing. In particular, a Pareto tail can imply much greater high-limit losses than a thin-tailed exponential model. Sparse observations at large severities make the tail parameter especially influential.

• Check support and units. A Pareto Type I model with threshold \(x_m\) applies only at or above that threshold. Keep the claim amounts, thresholds, and limits in the same units.
• Check whether the unlimited mean exists. For Pareto Type I, the mean is finite only when \(\alpha>1\). Limited expected loss at a finite \(u\) can still be calculated when \(\alpha\leq1\), but a conventional normalized exposure curve cannot use an infinite mean as its denominator.
• Represent censoring correctly. An exact loss contributes a density; a right-censored loss contributes a survival probability. Treating a limit value as an exact ground-up claim generally distorts the fit.
• Check curve behavior. For nonnegative losses, \(L(u)\) must not decrease as \(u\) increases. When the finite-mean exposure curve applies, \(G(u)\) must also be nondecreasing and no greater than one.
• Keep the loss basis consistent. An increased limits factor is meaningful only when the numerator and denominator use compatible severity definitions, claim adjustments, and coverage terms.

These checks are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. The central calculation is compact—fit a model, integrate its survival function, and normalize or take a ratio—but the quality of the answer depends on choosing the right likelihood, retaining consistent units, and stating where the fitted model is being extrapolated.