Selection Bias from Coverage Changes and Missing Claims

Why Coverage Changes Can Distort Claims Data

A change in a claims dataset does not necessarily mean that the underlying losses changed. If a deductible rises, smaller losses stop producing insurer payments. If a limit applies, large losses may appear only as capped payments. Reporting rules and incomplete policy participation can omit claims for other reasons. In each case, the observed records may represent a selected part of the losses one wants to study.

The first diagnostic question is therefore not simply “What is the average claim?” It is which losses could enter this dataset, and what does the recorded amount represent? The answer determines whether a comparison is valid, what adjustment might be possible, and what cannot be recovered from the available records.

Definition: Let $X$ be a ground-up loss, meaning the loss amount before applying insurance terms. A claim is observed under a deductible $d$ when $X>d$, assuming the insurer records only positive payments. If the insurer payment is subject to a maximum payment limit $u$, define the payment per loss as $Y^L=\min((X-d)_+,u)$, where $(z)_+=\max(z,0)$. A dataset may record $Y^L$, $X$, or both; these are not interchangeable quantities.

A deductible changes which claims are visible and, if the dataset records ground-up amounts only for visible claims, changes the distribution of recorded loss amounts as well. A payment limit has a different effect: it can leave the claim visible while concealing the exact amount of a large ground-up loss. A reporting threshold may resemble a deductible in the data, but it need not be part of the insurance contract. Its effect still depends on which records are excluded and whether the exclusion rule is known.

Selection by a Deductible

Suppose a claim file includes ground-up loss amounts only when $X>d$. The mean ground-up loss among recorded claims is $E[X\mid X>d]$, not $E[X]$. The two means answer different questions: the first describes losses that exceed the deductible, while the second describes all losses in the target population, including those at or below it.

E[X]=pE[X\mid X\le d]+(1-p)E[X\mid X>d], \qquad p=P(X\le d),\quad 0<p<1.

This identity follows by partitioning the outcomes into the disjoint events $X\le d$ and $X>d$. Their probabilities are $p$ and $1-p$, and the law of total expectation weights each conditional mean by its event probability. The identity requires the relevant expectations to exist. If $p=0$, then $E[X]=E[X\mid X>d]$; if $p=1$, then $E[X]=E[X\mid X\le d]$. The conditional mean for an event of probability zero is not defined.

The identity also shows why the omitted count alone is insufficient to reconstruct the unconditional severity mean: it gives no information about the amounts of the omitted losses. If the omitted count is known, it may help reconstruct a total claim count or frequency. Reconstructing a severity mean generally requires the omitted loss amounts or other defensible information about them.

Worked Example: Comparing Two Deductible Levels

A hypothetical analyst has ten ground-up losses, in dollars: $100, 200, 300, 400, 500, 600, 700, 800, 900, 1{,}000$. One group has a deductible of $200$, and another has a deductible of $500$. Assume the listed losses are the same underlying loss set for both groups and no payment limit applies. For each deductible, find the number of observed claims, the average ground-up loss among observed claims, and the average insurer payment per observed claim.

Method. Select losses strictly greater than the deductible. For each selected loss, subtract the deductible to obtain the insurer payment. This separates the selection effect on ground-up severity from the change in payment amounts.

Deductible $200$. The observed ground-up losses are $300, 400, 500, 600, 700, 800, 900, 1{,}000$, so there are $8$ claims. Their total is $300+400+500+600+700+800+900+1{,}000=5{,}200$. The average ground-up loss is $5{,}200/8=\$650$. The payments are $100,200,300,400,500,600,700,800$, totaling $3{,}600$. The average payment is $3{,}600/8=\$450$.

Deductible $500$. The observed ground-up losses are $600,700,800,900,1{,}000$, so there are $5$ claims. Their total is $4{,}000$, and their average ground-up loss is $4{,}000/5=\$800$. The payments are $100,200,300,400,500$, totaling $1{,}500$. The average payment is $1{,}500/5=\$300$.

Check and trap avoided. Raising the deductible increased the observed ground-up mean from $\$650$ to $\$800$, even though the underlying loss set did not change. The average payment fell from $\$450$ to $\$300$. Comparing either sample mean as if it were an unselected measure of all losses would confuse coverage effects with underlying severity.

Limits, Censoring, and What Payments Reveal

With a payment limit $u$, a payment of exactly $u$ can arise from a ground-up loss of $d+u$ or from any larger loss. Thus, a capped payment does not identify the exact ground-up amount. The insurer payment is known, but the ground-up loss is known only to be at least $d+u$, under the payment definition above.

This matters when the target quantity is the average payment rather than the average ground-up loss. If every payment is present, the payment average can be calculated directly. But replacing each capped payment with its observed value and calling the result a ground-up severity average understates or, at best, equals the true ground-up average. It equals that average only if every capped payment corresponds to a ground-up loss exactly at the cap threshold.

Worked Example: A Limit Hides Ground-Up Amounts

A hypothetical portfolio has eight ground-up losses of $\$300,\ \$700,\ \$900,\ \$1{,}200,\ \$1{,}500,\ \$2{,}000,\ \$2{,}400,$ and $\$3{,}000$. There is no deductible, and the insurer payment limit is $\$2{,}000$ per loss. Find the mean payment and determine how many exact ground-up amounts cannot be recovered from the capped payments alone.

Method. For each loss, record the lesser of the ground-up loss and $\$2{,}000$. Then average the payments, while separately identifying payments at the cap.

The recorded payments are $\$300,\ \$700,\ \$900,\ \$1{,}200,\ \$1{,}500,\ \$2{,}000,\ \$2{,}000,$ and $\$2{,}000$. Their total is $\$300+\$700+\$900+\$1{,}200+\$1{,}500+\$2{,}000+\$2{,}000+\$2{,}000=\$10{,}600$. The mean payment is $\$10{,}600/8=\$1{,}325$.

There are $3$ payments equal to the limit. A payment of $\$2{,}000$ is consistent with a ground-up loss of $\$2{,}000$ or with any larger amount, so none of those three exact ground-up amounts can be identified from capped payments alone. The ground-up total is at least $\$10{,}600$, and its mean is at least $\$1{,}325$, but the capped data provide no finite upper bound.

Check and trap avoided. The payment mean is $\$1{,}325$, not a ground-up mean. All three payments at the limit are ambiguous, including one that could correspond to a ground-up loss exactly equal to the limit.

Reporting Rules and Incomplete Participation

Claims can also be missing because a reporting rule excludes them, a reporting process fails, or only some policies participate in the data collection. The effect depends on how inclusion relates to loss size and other relevant characteristics. If inclusion is more likely for large losses, the observed severity mean may be too high for the target population; if small claims are less likely to be reported, the same direction of bias is possible. Other selection mechanisms can produce bias in the opposite direction.

A useful adjustment is possible when inclusion probabilities are known and the inclusion mechanism is defensible. Suppose each claim $j$ has a known probability $\pi_j>0$ of appearing in the file. Weighting an observed claim by $1/\pi_j$ gives more influence to claims that were less likely to be included. This approach requires the probability model to represent the actual data-collection process. In particular, inclusion must be independent of the unobserved loss amount after conditioning on the factors used to define $\pi_j$. If inclusion depends on unobserved severity even after that conditioning, the weighting may not remove the bias.

\widehat{N}_{\mathrm{HT}}=\sum_{j\ \mathrm{observed}}\frac{1}{\pi_j}, \qquad \widehat{T}_{\mathrm{HT}}=\sum_{j\ \mathrm{observed}}\frac{X_j}{\pi_j}, \qquad \widehat{\bar X}=\frac{\widehat{T}_{\mathrm{HT}}}{\widehat{N}_{\mathrm{HT}}}.

The first two quantities are inverse-probability estimates of the target claim count and total loss; their ratio estimates the target mean. The estimated ratio is not generally exactly unbiased in a finite sample. The method also cannot recover losses that have zero probability of inclusion: if $\pi_j=0$ for a part of the target population, no observed record can represent that part through this weighting scheme.

Worked Example: Adjusting for Unequal Reporting Probabilities

A hypothetical reporting study divides claims into two known strata. In stratum A, each claim has a $0.50$ probability of being included; the observed ground-up losses are $\$100$ and $\$300$. In stratum B, each claim has a $0.25$ probability of being included; the observed loss is $\$600$. Assume inclusion is random within each stratum, independent of loss amount there, and every target claim has a positive inclusion probability. Calculate the unweighted observed mean and the inverse-probability weighted estimate of the target mean.

Method. First calculate the ordinary observed mean. Then give each observed claim weight $1/\pi$, using its own stratum’s inclusion probability, and divide the weighted loss total by the weighted claim count.

The unweighted observed mean is $(\$100+\$300+\$600)/3=\$1{,}000/3=\$333.33$, rounded to the nearest cent.

For stratum A, each weight is $1/0.50=2$. The estimated claim count contribution is $2+2=4$, and the weighted loss total is $2(\$100)+2(\$300)=\$800$. For stratum B, the weight is $1/0.25=4$. Its estimated claim count contribution is $4$, and its weighted loss total is $4(\$600)=\$2{,}400$.

The combined estimated count is $4+4=8$, and the combined estimated loss total is $\$800+\$2{,}400=\$3{,}200$. The weighted mean estimate is $\$3{,}200/8=\$400.00$.

Check and trap avoided. The weighted result is higher because the observed file contains a larger share of stratum B than the assumed target representation: the lower inclusion probability in B is accounted for by its larger weight. The adjustment is justified here by the stated within-stratum random-inclusion assumption; merely knowing that some claims are missing would not justify these weights.

Choosing a Defensible Comparison

Before comparing empirical summaries, identify the target quantity and check whether the datasets measure it on the same basis. A higher average among claims subject to a larger deductible may reflect the exclusion of smaller losses. A lower average payment under a tighter limit may reflect payment terms rather than lower ground-up severity. An observed frequency also depends on whether the denominator is comparable exposure and whether the claims counted are defined consistently.

When a reliable adjustment is not available, restriction is often more defensible than an unsupported correction. Possible restrictions include comparing only policies with the same deductible and limit, comparing losses above a common threshold when ground-up data support it, or limiting the analysis to periods and participants with consistent reporting. Restriction changes the target population or quantity, so state that target clearly rather than implying the restricted result describes all losses.

Define the target.
Specify whether the question concerns ground-up severity, insurer payments, claim frequency, or an aggregate quantity, and name the population and period.

Map the observation rule.
Record how deductibles, limits, reporting thresholds, and participation determine which claims appear and which amounts are recorded.

Assess recoverability.
Determine whether exclusions are known, whether capped amounts conceal severity, and whether any target claims have zero chance of appearing.

Adjust or restrict transparently.
Use a stated adjustment only when its assumptions are supported; otherwise compare a common observable subset and describe the narrower target.

A common exam trap is to treat every recorded claim amount as if it were a complete ground-up loss and every absent claim as if it were a zero. A payment of zero might mean no loss, a loss below a deductible, an unreported claim, or a policy that did not participate. Those possibilities have different implications. Do not impute zeros or compare means until the observation mechanism and the quantity recorded are clear.

Key takeaway: Coverage and data-collection rules can select claims before an empirical summary is calculated. A claim count may sometimes be reconstructed from known omissions, but an unconditional severity mean generally requires the omitted amounts or defensible information about them. Any adjustment or restriction must match the stated target and its assumptions.

The discipline of separating selection from loss experience is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. In practice, it also prevents a change in recorded experience from being mistaken for a change in the underlying risk.

Check Your Understanding

For each problem, identify what the observed data support before interpreting the result.

A hypothetical file contains ground-up losses of $\$80,\ \$120,\ \$180,\ \$260,\ \$400,$ and $\$600$, but records only losses above a $\$200$ deductible. Find the number recorded and their mean ground-up loss. (Answer: 3 claims; mean ground-up loss $\$420$.)
Five hypothetical ground-up losses are $\$250,\ \$700,\ \$1{,}200,\ \$1{,}800,$ and $\$2{,}500$, with no deductible and a payment limit of $\$1{,}500$. Find the mean payment, the number of ground-up amounts not exactly identified by payments alone, and a lower bound for the ground-up mean. (Answer: mean payment $\$1{,}030$; 2 amounts not exactly identified; ground-up mean at least $\$1{,}030$.)
In a hypothetical study, two observed claims in stratum A are $\$120$ and $\$180$, with inclusion probability $0.40$; one observed claim in stratum B is $\$500$, with inclusion probability $0.20$. Assume random inclusion within each stratum, independent of loss amount. Calculate the inverse-probability weighted claim count and mean estimate. (Answer: estimated count $10$; weighted mean $\$325.00$.)
One policy-year group has a $\$100$ deductible and an observed ground-up claim mean of $\$400$; another has a $\$300$ deductible and an observed ground-up claim mean of $\$550$. Can the difference alone establish that underlying severity increased? Name a defensible comparison strategy. (Answer: No; deductible-based selection can raise the observed mean. Restrict to a common coverage basis or make a justified adjustment using adequate loss information.)
A reporting rule excludes every claim above $\$10{,}000$, and the file contains no information about excluded amounts or their number. Can inverse-probability weighting based only on the observed file recover the unconditional mean? (Answer: No; claims above the threshold have zero inclusion probability, so the observed file cannot represent their amounts.)

What You'll Learn