Death Claims In Annual Profit Calculations

Why Death-Claim Timing Matters

A death benefit is an insurer cash outflow, but its annual profit impact depends on more than the stated benefit amount. The calculation must use the probability that a policy in force at the start of the year produces a claim during that year, apply the correct sign, and follow the timing convention used to measure profit. In particular, the expected cash amount paid and the year-end equivalent of that payment are not necessarily the same.

This tutorial isolates the death-claim component of annual profit. Assume a hypothetical policy is in force at the beginning of the policy year, and let $B$ be the death benefit payable if the insured dies during that year. Unless stated otherwise, assume there are no lapses, other exits, settlement delays, or changes in the benefit amount. These assumptions make the death-claim calculation transparent; additional cash flows can be added to the annual profit vector separately.

Definition: Let $q_x$ be the probability that a life aged $x$, alive at the start of the policy year, dies during that year. For a fixed death benefit $B$, the expected nominal claim amount per policy in force at the start of the year is $Bq_x$. Because the claim is an insurer outflow, its contribution to profit is negative: $-Bq_x$, when claims are recorded as nominal cash outgo without an adjustment for payment timing.

The probability is conditional on being alive at the start of the year. It is not the probability of surviving the year, $p_x=1-q_x$, and it is not a probability conditional on death having occurred. For a portfolio of $N$ identical policies in force at the start of the year, the expected nominal claim amount is $NBq_x$. The per-policy calculation is the same as the single-policy calculation.

Nominal Outgo and Year-End-Equivalent Outgo

A timing convention specifies when the claim is treated as paid for the annual profit calculation. If a claim is recorded at the end of the policy year, its amount is already at the year-end measurement date. If the claim is paid earlier, a calculation that expresses all cash flows at year-end may accumulate that outflow from its payment date to the end of the year. This is a date-conversion convention, not a change to the benefit promised by the policy.

Let $U$ be the time of death, measured in years from the start of the policy year, conditional on death during that year. A claim paid immediately at death is paid at time $U$. With effective annual interest rate $i$, the amount accumulated from time $U$ to year-end is multiplied by $(1+i)^{1-U}$. Thus, when a year-end-equivalent basis is specified, the expected claim outgo per policy in force is:

Bq_x E\left[(1+i)^{1-U}\mid T\leq 1\right].

The corresponding profit impact is the negative of this amount. The expression assumes a fixed benefit $B$, payment at death, and accumulation at the effective annual rate $i$ for the remaining fraction of the year. It does not apply unchanged if the benefit varies with the date of death, if a claim is settled later, or if the annual profit basis does not accumulate cash flows to year-end.

Timing distinction: On a nominal cash-outgo basis, expected claim outgo is $Bq_x$, regardless of when during the year claims are paid, provided every claim is paid within that year. On a year-end-equivalent basis, earlier payments are accumulated to year-end according to the stated convention. Do not apply the accumulation factor unless the basis calls for it.

A common approximation assumes that death claims are paid at the midpoint of the year. Under that convention, $U=0.5$ for timing purposes, so the year-end-equivalent claim outgo is $Bq_x(1+i)^{0.5}$. This midpoint convention approximates timing; it does not assert that all actual deaths occur exactly six months after the policy year begins.

Deriving the Expected Claim Contribution

The basic expected-claim formula follows directly from an indicator variable. Define $I=1$ if death occurs during the policy year and $I=0$ otherwise. The claim amount is $IB$: it is $B$ on death and zero on survival. Since $E[I]=P(I=1)=q_x$, linearity of expectation gives:

$$ E[IB]=B E[I]=Bq_x. $$

The insurer pays the claim, so the claim enters the profit identity with a minus sign. Therefore the nominal claim contribution is $-Bq_x$, not $+Bq_x$. If the annual profit is measured at year-end and the payment time is random, the same argument can be applied to the accumulated claim $IB(1+i)^{1-U}$. Conditional on death, the timing factor is averaged over the distribution of $U$, giving the year-end-equivalent formula above.

Under the uniform distribution of deaths assumption (UDD) within the year, $U$, conditional on death in the year, is uniform on $[0,1]$. For a fixed benefit, its expected year-end accumulation factor is:

\begin{aligned} E\left[(1+i)^{1-U}\mid T\leq 1\right] &=\int_0^1 (1+i)^{1-u}\,du\\ &=\frac{i}{\ln(1+i)}. \end{aligned}

For the last equality, write $(1+i)^{1-u}=\exp((1-u)\ln(1+i))$. Integrating this exponential from $u=0$ to $u=1$ gives $\bigl(\exp(\ln(1+i))-1\bigr)/\ln(1+i)$, and $\exp(\ln(1+i))-1=i$. This factor is greater than one when $i>0$: payments made before year-end are accumulated for a positive period. UDD is an assumption about the distribution of death times within the year, not a replacement for the annual probability $q_x$.

Worked Examples

Worked Example: Claim Paid at the End of the Year

A hypothetical insurer has a one-year policy in force at age $x$ at the start of the year. The death benefit is \$150,000, and the annual probability of death is $q_x=0.008$. For this calculation, assume claims are recorded at the end of the year and all other profit components together contribute $+\$3,400$ per policy. Find the expected claim outgo and annual profit after the claim.

Method. Use the annual death probability for a policy in force at the start of the year. Because the claim is recorded at year-end, no timing adjustment is required.

Step 1: Calculate the expected nominal claim.

E[\text{claim}]=\$150{,}000(0.008)=\$1{,}200.

Step 2: Apply the insurer-outgo sign. The profit contribution from claims is $-\$1,200$.

Step 3: Include the other profit components.

\text{Annual profit}=\$3{,}400-\$1{,}200=+\$2{,}200.

Check. The expected claim is $0.8\%$ of the benefit: $0.008\times \$150,000=\$1,200$. Subtracting that outgo reduces the other profit components by exactly \$1,200. The common mistake avoided here is using the survival probability $0.992$ instead of the death probability $0.008$.

Worked Example: Midpoint Claim on a Year-End Basis

A hypothetical policy pays a fixed death benefit of \$80,000. The probability of death during the policy year, conditional on being alive and in force at the start, is $q_x=0.012$. Claims are approximated as paid at midyear. The effective annual interest rate is $i=0.06$, and the other components of profit, already expressed at year-end, total $+\$2,500$. Calculate both the expected nominal claim and the year-end-equivalent claim outgo, then calculate profit on the year-end basis.

Method. First calculate the expected cash amount paid. Then, because this problem specifies a year-end profit basis and midpoint payment, accumulate that expected outgo for half a year.

Step 1: Calculate nominal expected claim outgo.

\$80{,}000(0.012)=\$960.

Step 2: Calculate the midpoint-to-year-end accumulation factor.

(1.06)^{0.5}=\sqrt{1.06}=1.029563\ldots

Step 3: Convert the expected outgo to a year-end equivalent.

\$960(1.029563\ldots)=\$988.380\ldots

Rounded to the nearest cent, the year-end-equivalent claim outgo is \$988.38. Its profit contribution is $-\$988.38$, so:

\text{Year-end profit}=\$2{,}500-\$988.38=+\$1{,}511.62.

Check. The nominal expected claim remains \$960; the additional \$28.38 is solely the accumulation from midpoint to year-end. The common mistake avoided is treating \$988.38 as the nominal benefit expected to be paid. It is a year-end equivalent under the specified interest and timing convention.

Worked Example: Claims Paid at Death Under UDD

A hypothetical one-year policy pays \$40,000 on death during the year. The annual death probability for a life in force at the start is $q_x=0.025$. Assume death times, conditional on death during the year, follow UDD. The effective annual interest rate is $i=0.05$, and other profit components total $+\$1,300$ on a year-end basis. Find the expected claim outgo on that basis and the resulting annual profit.

Method. Calculate the nominal expected claim first. Then use the average accumulation factor for death during the year under UDD.

Step 1: Calculate the nominal expected claim.

\$40{,}000(0.025)=\$1{,}000.

Step 2: Calculate the UDD timing factor.

\frac{i}{\ln(1+i)} =\frac{0.05}{\ln(1.05)} =\frac{0.05}{0.048790164\ldots} =1.024796\ldots

Step 3: Calculate year-end-equivalent claim outgo.

\$1{,}000(1.024796\ldots)=\$1{,}024.796\ldots

Rounded to the nearest cent, expected claim outgo on the year-end basis is \$1,024.80. Therefore:

\text{Year-end profit}=\$1{,}300-\$1{,}024.80=+\$275.20.

Check. The year-end equivalent exceeds the \$1,000 nominal expected claim because payments occur before year-end and $i$ is positive. The common mistake avoided is applying the UDD timing factor to the benefit amount without first weighting by the probability of death; the timing factor applies to the expected claims, which are $Bq_x$.

Common Errors and Checks

A reliable calculation keeps three separate decisions visible: the amount payable on a claim, the probability that a claim occurs, and the date at which the profit measure values the payment. Combining these prematurely is a frequent source of errors.

Using survival probability. The claim occurs on death, so use $q_x$, not $p_x=1-q_x$. A benefit multiplied by $p_x$ would be a survival-weighted amount, not expected death-claim outgo.
Reversing the sign. A death claim is paid by the insurer. Its contribution to profit is negative even though the expected claim amount itself is stated as a positive dollar amount.
Adding a timing factor without a timing basis. On a nominal cash-outgo basis, use $Bq_x$. Accumulate to year-end only when the profit calculation calls for year-end-equivalent values.
Confusing a conditional distribution with the annual death probability. UDD describes when deaths occur, conditional on death during the year. The probability of a claim still comes from $q_x$.
Changing the benefit or exposure unintentionally. Use the benefit applicable to a death during the specified year and calculate per policy in force at the start. If the benefit varies with time or the exposure basis differs, the fixed-benefit formula may need to be modified.

Key takeaway: For a fixed death benefit $B$, the nominal expected annual claim contribution per policy in force is $-Bq_x$. On a year-end-equivalent basis, multiply expected outgo by the accumulation factor implied by the payment timing: $(1+i)^{0.5}$ for midpoint timing, or $i/\ln(1+i)$ for payments at death under UDD. State the basis before calculating.

The treatment of death claims is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In a full annual profit vector, this claim contribution is combined with the other cash-flow components using their own amounts and timing conventions; it should not be counted again through a separate adjustment unless the stated profit definition requires one.

Check Your Understanding

For each problem, identify whether the requested result is nominal claim outgo or a year-end equivalent. Amounts are hypothetical, and all probabilities are conditional on a policy being in force at the start of the year.

A policy pays \$90,000 on death during the year. The annual death probability is $0.006$, and claims are recorded at year-end. Find the expected claim outgo and its profit contribution. (Answer: expected outgo \$540; profit contribution $-\$540$)
A \$60,000 death benefit has annual death probability $0.020$. Claims are assumed paid at midyear, $i=0.04$, and other profit components equal $+\$2,000$ at year-end. Find the year-end-equivalent claim and resulting profit, rounded to cents. (Answer: claim outgo \$1,223.76; profit $+\$776.24$)
A policy pays \$25,000 on death, with $q_x=0.040$. Assume UDD and $i=0.03$. Find the year-end-equivalent expected claim outgo, rounded to cents. (Answer: \$1,014.93)
A fixed \$100,000 death benefit has $q_x=0.010$ and is recorded at year-end. Other annual profit components are $+\$1,800$. Find annual profit after claims. (Answer: $+\$800$)
A policy pays \$75,000 on death. The annual death probability is $0.015$, claims are assumed paid at midyear, and $i=0.08$. Find the year-end-equivalent claim outgo, rounded to cents. (Answer: \$1,169.13)

What You'll Learn