Nonparametric Exposure Curves and Smoothing

Why Estimate an Exposure Curve Nonparametrically?

A parametric severity model summarizes losses through a specified distribution. A nonparametric exposure curve instead uses observed severities directly, without choosing a named severity distribution. This can be useful when a sample is available but a parametric tail assumption is difficult to justify. The tradeoff is that the empirical curve reflects sampling variation, especially where few observations inform the tail.

Here, let $X$ be a nonnegative ground-up loss with finite, positive mean. At limit $u$, the loss subject to the limit is $\min(X,u)$. The exposure curve expresses expected limited loss as a fraction of expected ground-up loss. It therefore measures the relative expected loss at different limits, rather than the probability that a claim exceeds a limit.

Definition: The exposure curve at limit $u$ is $L(u)=E[\min(X,u)]/E[X]$. Given complete ground-up observations $x_1,\ldots,x_n$, its empirical estimate is $\widehat{L}(u)=\sum_{i=1}^{n}\min(x_i,u)\big/\sum_{i=1}^{n}x_i$, provided the total observed loss is positive. The empirical estimate uses the same observations and denominator at each limit.

This estimator requires ground-up severities: the observed values must not have been capped by policy limits. It also treats the observations as a sample from the severity distribution relevant to the curve being estimated. If losses are censored, the observed values cannot simply be substituted as though they were ground-up. The earlier material on censored claims addresses that distinct data issue.

Shape Properties That Guide Estimation

An exposure curve starts at zero, is nondecreasing, and approaches one as the limit grows without bound. It is also concave: increasing the limit by a fixed amount adds less expected limited loss at higher starting limits. These properties are not merely visual preferences. They follow from the definition and are useful checks on an estimate.

Theorem: For nonnegative $X$ with $0<E[X]<\infty$, $L(0)=0$, $L(u)$ is nondecreasing and concave in $u$, and $L(u)\to1$ as $u\to\infty$. At ordered grid points $0=u_0<u_1<\cdots<u_m$, concavity requires the successive secant slopes $[L(u_j)-L(u_{j-1})]/(u_j-u_{j-1})$ to be nonincreasing.

To see why, fix a possible severity value $x$. As a function of $u$, $\min(x,u)$ increases linearly with slope one until $u=x$, then stays constant. Its slope never increases, so this function is nondecreasing and concave. Taking its expectation preserves both properties, and division by the positive constant $E[X]$ does not change them. Also, $\min(X,u)$ increases to $X$ as $u$ grows, so its expectation approaches $E[X]$; dividing by $E[X]$ gives the limiting value one.

For a single complete sample, the empirical curve inherits these properties. Each term $\min(x_i,u)$ is nondecreasing and concave, and the denominator is fixed and positive. Thus a common-sample empirical curve should not have decreasing ordinates or increasing successive slopes. Apparent violations often arise when ordinates were estimated separately from different samples or bases, rounded, or otherwise processed inconsistently.

Calculating the Empirical Curve

Worked Example: Direct Estimates at Two Limits

A hypothetical insurer has complete ground-up severities of $15,000, $35,000, and $50,000 from a small portfolio. Estimate the exposure curve at limits of $25,000 and $40,000. No parametric severity distribution is assumed; the observations are treated as the available sample from an unknown severity distribution. Interest and mortality assumptions do not apply to this loss-severity calculation.

Method. Apply the empirical limited-loss formula at each limit, using the sum of all ground-up severities as the common denominator. Express the values in thousands of dollars: the observations are 15, 35, and 50, and their total is $15+35+50=100$.

At a limit of 25, the capped losses are $\min(15,25)=15$, $\min(35,25)=25$, and $\min(50,25)=25$. Their sum is $15+25+25=65$, so $\widehat{L}(25)=65/100=0.6500$.

At a limit of 40, the capped losses are $15$, $35$, and $40$. Their sum is $15+35+40=90$, so $\widehat{L}(40)=90/100=0.9000$.

Check and trap avoided. The estimates are nondecreasing, as required. A tempting but incorrect numerator at the first limit is $15+35+25=75$: it fails to cap the $35,000 loss at $25,000. The correct capped total is 65, not 75. The estimates are $0.6500$ and $0.9000$, respectively.

At a limit at least as large as the largest observed severity, every observed loss is uncapped, so the empirical estimate equals one. Between observed severity values, the empirical curve is piecewise linear: as the limit rises, each loss above that limit contributes to the slope, while losses already below it contribute no further increase.

Monotonicity Adjustment for Noisy Ordinates

Sometimes an analyst receives exposure ordinates estimated separately rather than calculated from one shared set of ground-up observations. Sampling variation can then produce a decrease across increasing limits. A standard way to impose nondecreasing order is weighted isotonic regression. It finds nondecreasing fitted values that minimize weighted squared deviations from the original ordinates.

\min_{z_1,\ldots,z_m}\sum_{j=1}^{m} w_j(z_j-y_j)^2 \quad\text{subject to}\quad z_1\le z_2\le\cdots\le z_m,

Here $y_j$ is an observed or separately estimated ordinate at an increasing limit, $w_j>0$ is its weight, and $z_j$ is the adjusted value. A common computational procedure is the pool-adjacent-violators algorithm: whenever adjacent fitted values decrease, pool them and replace them by their weighted mean; repeat until the sequence is nondecreasing. Equal weights give an ordinary arithmetic mean within each pooled block.

Worked Example: Pooling a Decreasing Pair

Suppose separately estimated ordinates at limits 20, 40, and 80 are $0.60$, $0.55$, and $0.53$, with equal reliability weights of one. These are noisy estimates from different subsets of a hypothetical portfolio, not ordinates calculated from one common ground-up sample. No parametric distribution is imposed; interest and mortality assumptions do not apply.

Method. Apply weighted isotonic regression. The first pair violates the required order because $0.60>0.55$, so pool them. Their equal-weight mean is $(0.60+0.55)/2=0.575$. The sequence is now $0.575,0.575,0.53$, and it still decreases at the final pair.

Pool all three values. The equal-weight mean is $(0.60+0.55+0.53)/3=1.68/3=0.56$. The fitted ordinates are therefore $0.56,0.56,0.56$, which satisfy the nondecreasing constraint. Because all weights are equal, the squared-error-minimizing fit over this pooled block is its arithmetic mean.

Check and trap avoided. The fitted values no longer decrease. However, monotonicity alone does not guarantee that a fit is a good representation of a plausible exposure curve: pooling has removed all variation over these limits. Do not treat an adjusted sequence as automatically informative about the tail merely because it is ordered.

The example illustrates an important qualification. Isotonic regression enforces monotonicity, but the exposure curve is also concave. If a set of adjusted ordinates has increasing secant slopes, it still violates the theoretical shape. A full shape-constrained smoother must impose both nondecreasing ordinates and nonincreasing successive slopes. For a grid including $u_0=0$ and $z_0=0$, the slope constraints are

\frac{z_j-z_{j-1}}{u_j-u_{j-1}} \ge \frac{z_{j+1}-z_j}{u_{j+1}-u_j} \quad (j=1,\ldots,m-1).

One may minimize weighted squared error subject to both sets of shape constraints. This is a constrained smoothing problem, not merely a monotonicity adjustment. If the ordinates come from a common complete sample, first investigate why they fail these properties: smoothing inconsistent inputs can hide a data or calculation error rather than correct it.

Smoothing Without Erasing Tail Information

A smoother can reduce the visual impact of sampling noise, but every smoothing choice changes what the data say. In particular, averaging severities and then calculating an exposure curve is not generally equivalent to calculating the curve from the original severities. Because limited loss is a concave function of severity, reducing dispersion while preserving the mean can raise expected limited loss at some limits. A smoothed severity sample can therefore shift exposure ordinates, not just make them look less jagged.

Worked Example: Effect of Grouping Two Tail Observations

A hypothetical ground-up sample, measured in thousands of dollars, is $10,12,18,50,90$. As a simple smoothing illustration, replace the two largest observations by their common mean. The sample is treated as observations from an unknown severity distribution, with no parametric form assumed. Interest and mortality assumptions do not apply.

Method. First calculate the original empirical ordinates. The total is $10+12+18+50+90=180$. At a limit of 40, the capped losses total $10+12+18+40+40=120$, giving $\widehat{L}(40)=120/180=0.6667$. At a limit of 60, they total $10+12+18+50+60=150$, giving $\widehat{L}(60)=150/180=0.8333$.

The mean of the two largest observations is $(50+90)/2=70$, so the grouped sample is $10,12,18,70,70$. Its total remains $180$. At limit 40, the capped total remains $10+12+18+40+40=120$, so the estimate remains $0.6667$. At limit 60, the capped total becomes $10+12+18+60+60=160$, giving $160/180=0.8889$.

Check and trap avoided. Grouping preserves total ground-up loss here, but it raises the estimate at limit 60 from $0.8333$ to $0.8889$. It also changes the sample maximum from 90 to 70, making the grouped empirical curve reach one at a lower limit. Do not assume smoothing preserves the tail or the resulting exposure curve simply because the overall mean is unchanged.

The grouping in this example is a transparent demonstration of an effect, not a universal smoothing recommendation. A practitioner should examine how a chosen smoother changes ordinates at meaningful limits, especially where the tail is supported by only a few observations. A smooth curve that removes a credible high-severity feature may be less useful than a visibly irregular estimate that accurately reflects the available evidence.

Practical Checks and Common Traps

Cap each claim before summing. At limit $u$, calculate $\min(x_i,u)$ for every observation. Do not cap the sample total or substitute uncapped values for claims above the limit.
Keep one denominator. For the direct empirical curve, use the same total ground-up loss at every limit. Changing the denominator across ordinates changes the quantity being estimated.
Check more than monotonicity. An exposure curve should also be concave. On an ordered grid, compare successive slopes, accounting for unequal spacing between limits.
Separate noise adjustment from data repair. A common-sample empirical curve is already nondecreasing and concave. Investigate violations before applying a smoother.
Protect the tail from excessive pooling. Weighting and pooling choices determine how much an isolated tail ordinate influences the fit. A fitted curve should not imply more tail certainty than the data support.

Key takeaway: Direct empirical exposure curves use capped ground-up losses divided by total ground-up loss. Their monotonicity and concavity follow from the limited-loss function. Use constrained smoothing when noisy estimates require it, and check that the adjustment respects both shape properties without erasing credible tail behavior.

Check Your Understanding

Try each problem before reading its answer. Amounts are in thousands of dollars unless stated otherwise.

A hypothetical complete ground-up sample is $8,22,40$. Calculate the empirical exposure ordinates at limits 20 and 30. (Answer: $0.6857$ and $0.8571$)
Three separately estimated ordinates at increasing limits have equal weights and values $0.44,0.41,0.47$. Apply weighted isotonic regression. (Answer: $0.425,0.425,0.47$)
An estimated exposure curve has $L(20)=0.50$, $L(40)=0.72$, and $L(80)=0.84$, with $L(0)=0$. Find the three successive secant slopes and state whether these knots satisfy concavity. (Answer: $0.025,0.011,0.003$; yes, the slopes are nonincreasing.)
For observations $12,28,60$, calculate the empirical exposure ordinate at limit 30. (Answer: $70/100=0.7000$)
For the sample $10,12,18,50,90$, compare the empirical ordinate at limit 60 with the ordinate after replacing 50 and 90 by their mean. (Answer: original $150/180=0.8333$; grouped $160/180=0.8889$)

What You'll Learn