Pure IBNR

Why Claims Can Be Unreported at a Valuation Date

A covered event may occur before a financial statement date even though the insurer has not yet received notice of the resulting claim. At that date, the claim has been incurred but is absent from the reported-claim inventory. Its expected unpaid loss is therefore not represented by a claim-level case reserve. The estimate must reflect both how many claims are still unreported and the expected loss on those claims.

The key timing variable is the reporting delay: the time from claim occurrence to the insurer’s receipt of the report. A claim’s chance of being unreported depends on how long it has had to be reported. Recent claims generally have had less time to emerge than older claims, so applying one reporting percentage to every occurrence period can misstate the liability.

Definition: Pure IBNR is the estimated unpaid loss liability, at a specified valuation date, for covered claims that have occurred by that date but have not yet been reported to the insurer. In this tutorial, “pure” is used strictly for not-yet-reported claims; it does not include additional development on claims already reported. The estimate concerns loss only unless the problem explicitly includes other components.

A reported claim may have an unpaid amount captured in a case reserve, while a claim that has not been reported cannot yet have an individual case reserve. Pure IBNR estimates the latter group in aggregate. It is an estimate, not a list of claims known to exist, and it depends on the valuation date, the definition of occurrence, the reporting-delay assumptions, and the severity assumptions.

From Reporting Delays to an IBNR Estimate

For a claim that occurred $t$ units of time before valuation, let $D$ be its reporting delay, measured in the same time units. The claim is still unreported at valuation exactly when $D>t$. Define the probability of remaining unreported at age $t$ as $q(t)=P(D>t)$. If a reporting-delay cumulative distribution function is available, then $q(t)=1-F_D(t)$, where $F_D(t)=P(D\le t)$.

For a homogeneous occurrence cohort, let $N$ be its number of claims, with expected count $\lambda=E[N]$. Let $X$ denote ultimate loss per claim. If reporting status and severity are independent, and the expected severity is $\mu_X$, the expected pure IBNR for that cohort is:

E[\text{pure IBNR}]=\lambda\,q(t)\,\mu_X.

This formula requires care when severity and reporting are related. For example, if large claims tend to be reported more quickly, the average severity of claims that remain unreported can differ from the average severity of all claims. In that case, use the conditional mean severity among unreported claims, $E[X\mid D>t]$, rather than an overall mean.

E[\text{pure IBNR}]=\lambda\,P(D>t)\,E[X\mid D>t].

For several cohorts, estimate each cohort separately and add the results. Cohorts may differ in age, expected claim count, reporting pattern, or severity. A single average reporting percentage is appropriate only when it correctly represents the mix of cohorts and their relevant severity assumptions.

Why the Expected-Count Formula Works

The formula follows by representing each claim’s contribution with an indicator. For claim $k$, let $I_k=1$ if it is unreported at valuation and $I_k=0$ otherwise. Its pure IBNR contribution is $X_kI_k$, so the cohort total is the sum of these contributions. Under the assumption that each claim has the same distribution and that the claim marks are independent of the Poisson count $N$, linearity of expectation gives:

E\left[\sum_{k=1}^{N}X_kI_k\right] =E[N]\,E[XI] =\lambda\,P(D>t)\,E[X\mid D>t].

The last equality uses $E[XI]=P(D>t)E[X\mid D>t]$. If severity is independent of reporting delay, then $E[X\mid D>t]=E[X]$, which yields the simpler formula. The calculation is an expected liability; it does not assert that the actual number of unreported claims equals its expected value.

Estimating Pure IBNR by Cohort

A practical calculation separates the inputs that can otherwise be confused: expected claims, the probability that a claim remains unreported at its current age, and the mean ultimate loss conditional on remaining unreported. The process is the same whether the expected claim count comes from a frequency model or from an actuarial estimate based on exposure and experience.

Set the valuation date and occurrence cohorts.
Determine which claims have occurred by the valuation date and the elapsed reporting age of each cohort.

Estimate the unreported fraction.
Use a reporting proportion or a delay distribution to calculate $P(D>t)$ for each cohort’s age $t$.

Estimate unreported claim count and severity.
Multiply the cohort’s expected count by its unreported fraction, then by the appropriate conditional mean ultimate loss.

Sum cohort estimates.
Add the cohort amounts to obtain the estimated pure IBNR for the covered claims in scope.

Worked Example: Two Cohorts with Different Reporting Rates

A hypothetical insurer estimates 60 claims in an older occurrence cohort and 40 in a more recent cohort. By the valuation date, 90% of the older cohort and 60% of the recent cohort are reported. The conditional mean ultimate losses for claims that remain unreported are $5,000 and $8,000, respectively. Estimate pure IBNR.

Model assumptions. Treat the cohort claim counts as independent Poisson random variables with parameters 60 and 40, respectively. Treat the stated reporting percentages as applicable to each claim in its cohort. Use deterministic losses of $5,000 and $8,000 per unreported claim, respectively; equivalently, these are the specified conditional means. Assume no discounting, so the effective annual interest rate used is $i=0\%$. Mortality is not applicable to these claim-count and reporting assumptions.

Method. Convert each reported proportion to its complement to obtain the unreported fraction. The older cohort has unreported fraction $1-0.90=0.10$, and the recent cohort has unreported fraction $1-0.60=0.40$.

Older cohort. Its expected unreported count is $60(0.10)=6$. Its pure IBNR is $6($5{,}000)=$30{,}000$.

Recent cohort. Its expected unreported count is $40(0.40)=16$. Its pure IBNR is $16($8{,}000)=$128{,}000$.

Total and check. The estimate is $$30{,}000+$128{,}000=$158{,}000$. As a direct check, $60(0.10)($5{,}000)+40(0.40)($8{,}000)=$158{,}000$. This is the cohort-by-cohort method and avoids applying a reported percentage as though it were an unreported percentage.

Worked Example: Using a Discrete Reporting-Delay Distribution

A hypothetical specialty insurer expects 80 claims at each of three occurrence dates. At the valuation date, the cohorts have reporting ages of 1, 2, and 3 months. Each claim has a deterministic ultimate loss of $1,200. The reporting delay $D$, in months, has probability 0.50 of being 0.5 months, 0.30 of being 1.5 months, and 0.20 of being 2.5 months. Estimate the pure IBNR.

Model assumptions. For each cohort, $N\sim\operatorname{Poisson}(80)$, independently. Claim delays follow the stated discrete distribution, independently across claims. The loss per claim is fixed at $1,200. Assume no discounting, with effective annual interest rate $i=0\%$; mortality is not applicable.

Age-1 cohort. A claim is unreported at age 1 month if its delay is 1.5 or 2.5 months. Thus $P(D>1)=0.30+0.20=0.50$. The expected unreported count is $80(0.50)=40$, giving pure IBNR of $40($1{,}200)=$48{,}000$.

Age-2 cohort. A claim remains unreported at age 2 months only if its delay is 2.5 months. Thus $P(D>2)=0.20$. The expected unreported count is $80(0.20)=16$, giving pure IBNR of $16($1{,}200)=$19{,}200$.

Age-3 cohort. Every possible delay in the model is less than 3 months, so $P(D>3)=0$. Its pure IBNR is $80(0)($1{,}200)=$0$.

Total and check. The estimate is $$48{,}000+$19{,}200+$0=$67{,}200$. The probability at each age follows from the strict condition $D>t$; at age 2, for example, the 1.5-month reports have already arrived and are not counted as unreported.

Worked Example: Severity Depends on Remaining Unreported

A hypothetical insurer estimates 30 claims in group A and 20 claims in group B. At the valuation date, the probability of remaining unreported is 0.20 in group A and 0.50 in group B. Among the unreported claims, ultimate loss is exponentially distributed with mean $6,000 in group A and mean $12,000 in group B. Estimate pure IBNR.

Model assumptions. Claim counts are independent with $N_A\sim\operatorname{Poisson}(30)$ and $N_B\sim\operatorname{Poisson}(20)$. Reporting status has the stated group-specific probabilities. Conditional on being unreported, losses are exponential: for group A the rate is $1/6000$ per dollar, and for group B the rate is $1/12000$ per dollar. Use the stated conditional means; no assumption about the severity of reported claims is needed. Assume no discounting, with $i=0\%$; mortality is not applicable.

Group A. The expected unreported count is $30(0.20)=6$. The expected loss per such claim is $6,000, so the group estimate is $6($6{,}000)=$36{,}000$.

Group B. The expected unreported count is $20(0.50)=10$. The expected loss per such claim is $12,000, so the group estimate is $10($12{,}000)=$120{,}000$.

Total and check. Pure IBNR is $$36{,}000+$120{,}000=$156{,}000$. The direct calculation, $30(0.20)($6{,}000)+20(0.50)($12{,}000)$, gives the same amount. Using one severity mean for both groups would ignore the stated difference in conditional claim severity.

Common Mistakes and Practical Checks

Pure IBNR is sensitive to both reporting maturity and the composition of claims that have not yet been reported. A sound estimate should make those two features visible rather than hiding them in a single blended factor.

Using the reported fraction as the unreported fraction. In the first example, using 90% and 60% directly would give $60(0.90)($5{,}000)+40(0.60)($8{,}000)=$462{,}000$. That is incorrect because those are reported proportions. The correct complements are 10% and 40%, giving $158,000.
Ignoring cohort age. A reporting-delay probability must match the time from occurrence to valuation. A delay distribution evaluated at the wrong age can materially overstate or understate the remaining unreported fraction.
Using an unconditional severity when reporting and severity are related. If the question gives severity specifically for unreported claims, use that conditional mean. Do not replace it with a mean for all claims without an independence assumption.
Adding claims that occurred after valuation. Pure IBNR concerns claims already incurred by the valuation date. Future occurrences are not part of this liability.
Treating expected counts as observed counts. A Poisson parameter of 80 means an expected count of 80, not a certainty that exactly 80 claims occurred. The formula estimates expected aggregate loss.

Key takeaway: For each occurrence cohort, estimate expected unreported claims and multiply by their appropriate conditional mean ultimate loss. The unreported probability is the probability that reporting delay exceeds the cohort’s elapsed age at valuation.

Check Your Understanding

For each problem, estimate pure IBNR. Assume no discounting at $i=0\%$ and no mortality model applies unless stated otherwise.

A portfolio has an expected 120 claims in one cohort. By valuation, 75% are reported, and the mean ultimate loss for an unreported claim is $3,000. Claim count is Poisson with parameter 120. (Answer: $90,000)
Two cohorts have expected claim counts of 40 and 25. Their unreported probabilities are 0.30 and 0.60, and their respective conditional mean losses are $5,000 and $9,000. Counts are independent Poisson with the stated parameters. (Answer: $195,000)
Two occurrence cohorts each have an expected 70 claims, with ages of 1 and 2 months at valuation. Reporting delay $D$ is 0.5, 1.5, or 2.5 months with probabilities 0.40, 0.40, and 0.20. Each claim has deterministic loss of $4,000; counts are independent Poisson with parameter 70. (Answer: $224,000)
Group A has an expected 30 claims and an unreported probability of 0.20; group B has an expected 20 claims and an unreported probability of 0.50. Conditional severity among unreported claims is exponential with mean $6,000 in A and $12,000 in B. Counts are independent Poisson with the stated parameters. (Answer: $156,000)

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