Claim Settlement Delays And Profit Timing

Why Settlement Timing Matters

A claim can be incurred or reported in one policy year and paid in a later one. For an annual cash-flow profit vector, the benefit payment belongs to the year in which the insurer pays it—not automatically to the year in which the claim occurred or was reported. Correct timing matters because a delayed claim shifts outgo between profit-vector components, even if it does not change the total benefit ultimately paid.

This tutorial isolates claim payments. Assume the annual profit vector records cash received and paid during each policy year, with claim payments shown as negative amounts. We do not include claim reserves, investment income, or other cash flows in the calculations. Those items can affect a broader profit test, but they do not change the basic allocation of a claim payment to its settlement year. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A claim cohort is the set of claims assigned to a particular occurrence or reporting year. The settlement year is the policy year in which the insurer actually pays a claim. In a cash-flow profit vector, claim outgo is assigned to the settlement year. A claim’s occurrence or reporting year identifies the cohort; it does not by itself determine the profit-vector component containing the payment.

For clarity, the examples below assume that a claim is reported in the same policy year as the covered event. The payment can occur in that year or a later year. If reporting itself occurs after the covered event, the event year and report year should be tracked separately; the cash payment still belongs to the year it is made. The examples use expected claims per original policy, so policy lapses or deaths do not require a separate adjustment. When counts are given directly, they are expected counts for the stated original cohort.

Allocating Outgo to Settlement Years

Suppose there are $N$ original policies. For claims reported in year $t$, let $q_t$ be the expected number of claims per original policy, $B_t$ the fixed amount paid per claim, and $p_{t,j}$ the proportion of those claims paid $j$ policy years after reporting. Thus, $p_{t,0}$ is the proportion paid in the reporting year, $p_{t,1}$ the proportion paid one year later, and so on. If all claims in the cohort are eventually settled, the proportions sum to 1.

\sum_{j\geq 0} p_{t,j}=1

The claim-payment component for policy year $k$, per original policy, is

\pi_k^{\text{claim}} = -\sum_{t\leq k} q_t B_t p_{t,k-t}.

Each term represents one claim cohort that contributes payments in year $k$. The condition $t\leq k$ rules out payments before the claims are reported. The negative sign reflects the insurer’s cash outgo. If the expected number of claims is supplied as a count $C_t$, rather than per original policy, divide the expected payment amount by $N$ to obtain the component per original policy:

\pi_k^{\text{claim}} = -\frac{1}{N}\sum_{t\leq k} C_t B_t p_{t,k-t}.

When claims have varying amounts, replace $B_t$ by the appropriate expected amount per claim for the relevant payments, provided the assumed payment proportions and amounts are defined consistently. For the fixed-amount examples here, the formulas above apply directly.

Why the Settlement Formula Holds

The formula follows from counting the claims expected to be paid in the year of interest and multiplying by the payment per claim. For a cohort reported in year $t$, the expected number of claims paid $j$ years later is $Nq_t p_{t,j}$. Each such claim generates an outgo of $B_t$, so the expected total outgo from this cohort in year $t+j$ is $Nq_t p_{t,j}B_t$. Dividing by $N$ gives the amount per original policy, and the insurer’s outgo sign makes it negative.

-\frac{Nq_t p_{t,j}B_t}{N} = -q_t B_t p_{t,j}

If more than one cohort settles in a given policy year, their expected payments add. This is valid by linearity of expectation; the claim counts from different cohorts do not need to be independent for their expected cash flows to be added. Setting $j=k-t$ identifies the portion of cohort $t$ paid in policy year $k$, which gives the stated formula.

As a useful check, if every claim is eventually paid and the profit vector includes all settlement years, adding the undiscounted claim components across years gives the negative expected total benefit per original policy. The payment schedule changes when the outgo appears, not the total amount, under the assumptions of fixed benefit amounts and no interest adjustment.

Worked Applications

Worked Example: A Single Cohort Paid in Two Later Years

A hypothetical block contains 800 original policies. For each policy, the probability of a claim being reported in year 1 is 0.05. Each claim pays $12,000. Of the year 1 claims, 60% are paid in year 2 and 40% in year 3. Find the year 2 and year 3 claim components per original policy, and check the total.

Method. Find the expected number of year 1 claims, allocate that count by payment year, and divide each year’s expected payment by the 800 original policies. The claim is reported in year 1, but no payment is made in year 1 under the given settlement pattern.

The expected year 1 claim count is

800(0.05)=40 \text{ claims}.

In year 2, $40(0.60)=24$ claims are paid. The year 2 outgo is $24($12{,}000)=$288{,}000$, so the claim component per original policy is

\pi_2^{\text{claim}} = -\frac{$288{,}000}{800} = -$360.

In year 3, $40(0.40)=16$ claims are paid. The year 3 outgo is $16($12{,}000)=$192{,}000$, giving

\pi_3^{\text{claim}} = -\frac{$192{,}000}{800} = -$240.

Check. The two components total $-$360-$240=-$600$ per original policy. The expected benefit per original policy is $0.05($12{,}000)=$600$, so the total agrees. The common mistake avoided is assigning all $600 to year 1 simply because that is the claim year.

Worked Example: Two Claim Cohorts Settling in the Same Year

A hypothetical block is measured per original policy. The expected number of claims per original policy is 0.03 for claims reported in year 1 and 0.02 for claims reported in year 2. Each claim pays $10,000. Of the year 1 claims, 40% are paid in year 1 and 60% in year 2. Of the year 2 claims, 25% are paid in year 2 and 75% in year 3. Calculate the claim components for years 1, 2, and 3.

Method. Calculate payments separately by reporting-year cohort, then add the contributions that share a settlement year. The year 2 component includes both delayed year 1 claims and year 2 claims paid immediately.

For year 1, only 40% of the year 1 cohort is paid:

\pi_1^{\text{claim}} = -(0.03)(0.40)($10{,}000) = -$120.

For year 2, the delayed year 1 claims contribute $(0.03)(0.60)=0.018$ claims per original policy. The immediately paid year 2 claims contribute $(0.02)(0.25)=0.005$. Together, $0.018+0.005=0.023$ claims per original policy are paid in year 2. Thus,

\pi_2^{\text{claim}} = -(0.023)($10{,}000) = -$230.

For year 3, the remaining 75% of the year 2 cohort contributes $(0.02)(0.75)=0.015$ claims per original policy:

\pi_3^{\text{claim}} = -(0.015)($10{,}000) = -$150.

Check. The components total $-$120-$230-$150=-$500$. The total expected number of claims is $0.03+0.02=0.05$ per original policy, so the expected total benefit is $0.05($10{,}000)=$500$. The common mistake avoided is counting only year 1’s delayed claims in year 2 and omitting the claims from the year 2 cohort that are also paid that year.

Worked Example: Expected Counts, Different Benefit Amounts, and a Timing Trap

A hypothetical group has 500 original policies. The expected number of claims reported in year 1 is 15, with a benefit of $6,000 per claim. Half of those claims are paid in year 2 and half in year 3. The expected number of claims reported in year 2 is 10, with a benefit of $9,000 per claim; all are paid in year 3. Find the year 2 and year 3 claim components per original policy.

Method. Use expected claim counts directly, allocate the year 1 cohort by its settlement proportions, and add all payments made in year 3. Do not divide by the expected claims or by the number of policies remaining; the requested basis is the 500 original policies.

In year 2, half of the 15 year 1 claims are paid: $15(0.50)=7.5$ expected claims. Their expected payment is $7.5($6{,}000)=$45{,}000$. Per original policy,

\pi_2^{\text{claim}} = -\frac{$45{,}000}{500} = -$90.

In year 3, the remaining $15(0.50)=7.5$ year 1 claims generate $7.5($6{,}000)=$45{,}000$ of outgo. The 10 year 2 claims generate $10($9{,}000)=$90{,}000$ of outgo in the same year. Total year 3 outgo is $$45{,}000+$90{,}000=$135{,}000$, giving

\pi_3^{\text{claim}} = -\frac{$135{,}000}{500} = -$270.

Check. The total is $-$90-$270=-$360$ per original policy. The expected total benefit is $(15($6{,}000)+10($9{,}000))/500=$180{,}000/500=$360$ per original policy. A common error is to take the 15 year 1 claims, multiply by one-half, and then divide by 500 to get the year 2 amount; that calculation leaves out the $6,000 benefit per claim. The correct numerator is $7.5($6{,}000)$, not just 7.5.

Common Mistakes and Scope of the Calculation

Putting outgo in the report year by default. In a cash-flow profit vector, use the settlement year. A year 1 claim paid in year 2 contributes to year 2 cash outgo.
Forgetting a cohort that also settles in that year. A year’s component can include payments from several reporting-year cohorts. Add every cohort’s settlement-year contribution.
Confusing an expected count with a per-policy amount. If $C$ is an expected count in a block of $N$ policies, the per-original-policy amount is $CB/N$, not $CB$. If $q$ is already the expected count per original policy, do not divide by $N$ again.
Ignoring the benefit amount. A claim count is not itself a dollar outgo. Multiply the claims expected to be paid by the payment per claim.
Mixing cash-flow profit with incurred-claim accounting. A measure that recognizes the cost when claims are incurred or reported may include a claims reserve to account for unpaid claims. The cash-flow component calculated here records the actual payment; it does not, by itself, describe that reserve-based measure.

Key takeaway: For a cash-flow profit vector, identify each claim cohort, determine how many claims from it are paid in each policy year, multiply by the amount paid per claim, and apply the negative outgo sign. Check that the undiscounted components over all settlement years add to the expected total benefit per original policy.

Check Your Understanding

For each problem, calculate the requested claim-payment components on the stated basis. Treat all amounts as fixed benefits and ignore interest.

A hypothetical block has 400 original policies and 12 expected claims reported in year 1. Each claim pays $4,000; half are paid in year 2 and half in year 3. Find the year 2 and year 3 components per original policy. (Answer: $-$60$ in year 2 and $-$60$ in year 3.)
Per original policy, the expected number of claims reported in year 1 is 0.01 and in year 2 is 0.015. Each claim pays $20,000 if reported in year 1 and $8,000 if reported in year 2. Year 1 claims are all paid in year 2; year 2 claims are all paid in year 2. Find the year 2 claim component. (Answer: $-$320$ per original policy.)
A block begins with 1,000 original policies. There are 20 expected year 1 claims and 18 expected year 2 claims, each paying $10,000. Half of the year 1 claims are paid in year 2 and half in year 3; all year 2 claims are paid in year 2. Find the year 2 and year 3 components per original policy. (Answer: $-$280$ in year 2 and $-$100$ in year 3.)
Per original policy, expected claim counts are 0.02 in year 1 and 0.01 in year 2. Each claim pays $15,000. Of year 1 claims, 30% are paid in year 1 and 70% in year 2. Of year 2 claims, 40% are paid in year 2 and 60% in year 3. Find all three annual components and check their total. (Answer: year 1 $-$90$, year 2 $-$270$, year 3 $-$90$; total $-$450$ per original policy.)
A hypothetical block has 600 original policies. There is a 0.04 probability per original policy of a year 1 claim, paid in full in year 2. Each claim pays $5,000. A separate year 2 cohort has 9 expected claims, each paying $7,000 in year 3. Find the year 2 and year 3 components per original policy. (Answer: year 2 $-$200$ and year 3 $-$105$ per original policy.)

What You'll Learn