Claims, Death Benefits, and Claim-Year Outgo

Why Claim Timing Matters in an Asset-Share Year

A death benefit is a cash outflow from the insurer, but its effect on an annual asset-share ledger depends on when it is paid. A claim settled at midyear does not earn a full year of investment return after payment; a claim expense paid at year-end earns no return during that year. Treating both amounts as if paid at the same date can materially misstate the year-end value of claim outgo.

This tutorial considers the claim side of an annual asset-share calculation. It distinguishes actual claims from expected claims, and death payments from funds that remain for policies still in force. All products and figures below are hypothetical. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Let \(i\) be the effective annual investment return, and let \(\tau\) be the fraction of the policy year elapsed when a payment is made, where \(0\leq \tau\leq 1\). A payment \(C\) made at time \(\tau\) has year-end accumulated value \(C(1+i)^{1-\tau}\). This is an accumulation convention: it values the cash outflow at year-end, not the date on which the insurer actually pays it.

Expected and Actual Claim Outgo

Suppose \(N\) policies are in force at the start of a year. Under a homogeneous mortality assumption, let \({}_1q_x\) be the probability that a policy in force at age \(x\) experiences death during the year. The expected number of deaths is \(N\,{}_1q_x\). This is an expectation, not a forecast that the actual number of claims must equal that value.

If each death benefit \(B\) and claim expense \(E\) are both paid at the same time \(\tau\), the expected year-end claim outgo for the starting group is \(N\,{}_1q_x(B+E)(1+i)^{1-\tau}\). If the benefit and expense are paid at different times, accumulate them separately. For an observed group with \(D\) actual claims, use \(D\), rather than \(N\,{}_1q_x\), to calculate the actual claim outgo.

The expected group outgo is \(N\) times this amount. This formula assumes the stated benefit, expense and payment times apply to every death in the group, and that the same annual effective return is used over the relevant part of the year. It does not apply unchanged if claim amounts or timing vary by claim; in that case, calculate and sum the appropriate expected payment values by claim type or time.

Why the Accumulation Formula Holds

The time from a payment at \(\tau\) to the end of the year is \(1-\tau\) years. Under a constant effective annual return \(i\), an amount invested for \(t\) years accumulates by the factor \((1+i)^t\). To express an outflow at its year-end equivalent, accumulate the payment over exactly the time it is absent from the fund. Therefore, a payment \(C\) at time \(\tau\) has year-end value \(C(1+i)^{1-\tau}\). The exponent is the remaining fraction of the year, not the elapsed fraction.

If there are several payments \(C_j\) at times \(\tau_j\), accumulation is applied payment by payment, because each amount is absent from the fund for a different length of time:

This result assumes the same effective annual return applies over the year. If the return changes during the year, each payment must instead be accumulated using the return factors for the intervals remaining after its payment date. If the asset-share convention records cash flows at their actual dates without accumulating each to year-end, use those actual-date cash flows consistently rather than adding an extra accumulation.

Claim Outgo in the Asset-Share Ledger

An annual asset-share recurrence records premiums and expenses, investment accumulation, and benefits according to the chosen cash-flow timing convention. For the claim component, the key operation is to subtract death benefits and claim expenses when they are paid, or subtract their consistently accumulated equivalents when expressing the ledger at year-end. A claim already paid during the year must not be subtracted again as a second year-end payment.

The cohort perspective and the individual-policy perspective answer different questions. At cohort level, death claims are insurer outgo on policies that terminate because of death. For a policy that continues in force, no death benefit was paid on that policy during the year; its own continuing-policy ledger does not acquire a death payment merely because another policy in the cohort claimed. An aggregate asset-share calculation may include all cohort claims as outgo from the aggregate fund, but that does not make the death benefit an amount retained by continuing policies. The allocation of aggregate experience among survivors is a distinct calculation.

A useful way to keep the accounting straight is to label each cash flow by its event and payment time before combining it with the rest of the ledger:

Common Exam Trap: Using a Full-Year Accumulation Factor

A frequent timing error is to multiply every claim paid during the year by \(1+i\), regardless of its payment date. For the midyear claim in the first worked example, the incorrect calculation would be

That treats the $30,750 as if it were paid at the beginning of the year and were absent from the fund for a full year. The correct year-end value is \(30{,}750(1.21)^{0.5}=\$33{,}825.00\). The overstatement is \(\$37{,}207.50-\$33{,}825.00=\$3{,}382.50\). Conversely, leaving the amount at its nominal payment value understates its year-end equivalent when the payment occurs before year-end and \(i>0\).

Before using any claim amount in an annual recurrence, check three things: whether it is actual or expected, whether the expense is included, and what fraction of the year remains after payment. Also check the denominator: expected outgo per starting policy is divided by the number exposed at the start, not automatically by the number still in force at year-end.