Claim-Related Expenses in Premium Calculations

Why Claim-Related Expenses Need Their Own Timing

A claim-related expense is incurred because an insured event produces a claim, or because an annuity payment is made. Examples in a hypothetical contract include the cost of processing a death claim and an administrative cost incurred each time an annuity installment is sent. These costs differ from expenses triggered by issuing a policy or collecting a premium: a claim expense is payable only when its associated claim or payment occurs.

The key calculation is therefore not simply “add the expense to the benefit.” First identify the event that triggers the expense; then use the probability of that event and discount from the expense’s payment time. If the expense and benefit are paid together, they have the same expected present value factor. If their payment times differ, they generally do not.

Definition: A claim-related expense is an expense whose occurrence is contingent on a covered claim or benefit payment. Its expected present value is found by multiplying the expense amount at each possible payment time by the probability of the triggering event and the discount factor for that time, then summing over possible payment times.

A gross premium under the equivalence principle is set so that the expected present value of gross premiums equals the expected present value of benefits plus expenses. The examples below isolate claim-related expenses to show how they enter that equation. They assume no profit provision, risk margin, or other expense unless explicitly stated.

Match the Expense to the Claim Event

For a discrete annual term insurance, suppose a benefit $B$ is paid at the end of the year of death. Assume a fixed expense $C$ is also paid at that time whenever a death claim occurs. If $q_{x+k-1}$ is the probability of death during policy year $k$, conditional on being alive at its start, the unconditional probability of a claim in year $k$ is ${}_{k-1}p_x q_{x+k-1}$. The expected present value of benefit and expense for an $n$-year term is therefore

\sum_{k=1}^{n} (B+C)v^k\,{}_{k-1}p_x q_{x+k-1}.

This formula uses the end-of-year-of-death payment assumption for both amounts. If the expense is paid immediately when death occurs while the insurance benefit is paid at year-end, the expense must instead be discounted to the time of death. In a continuous model, this difference is represented by a different payment-time integral, as shown later.

When level premiums $P$ are paid annually in advance while the insured is alive for an $n$-year term, their expected present value is $P\sum_{j=0}^{n-1}v^j\,{}_jp_x$. The premium at time zero is certain, which is why the $j=0$ term is one.

Worked Example: A Two-Year Term with a Death-Claim Expense

A hypothetical two-year term policy pays $10,000 at the end of the year of death. A $200 claim-processing expense is paid at the same time as the benefit. The annual effective interest rate is $i=5\%$. The first-year death probability is $q_x=0.10$, and the second-year conditional death probability is $q_{x+1}=0.20$. Level premiums are paid at times 0 and 1 if the insured is alive at the payment time. Find the annual gross premium by the equivalence principle.

Method. Value the combined death benefit and expense using the probability of death in each year. Then divide by the expected present value factor for the premiums. This avoids charging the claim expense in years when no claim occurs.

The discount factor is $v=1/1.05=20/21$. The probability of being alive at time 1 is ${}_1p_x=1-0.10=0.90$. The probability of death in year 1 is 0.10, and the unconditional probability of death in year 2 is ${}_1p_x q_{x+1}=0.90(0.20)=0.18$.

The combined amount paid on a death claim is $B+C=\$10{,}000+\$200=\$10{,}200$. Its expected present value is

\$10{,}200\left[0.10\left(\frac{20}{21}\right)+0.18\left(\frac{20}{21}\right)^2\right] =\$10{,}200\left(\frac{38}{147}\right) =\$2{,}636.7347.

For the premiums, the time-0 payment is certain, while the time-1 payment is made only if the insured survives the first year. Thus the premium present value factor is

1+0.90v=1+0.90\left(\frac{20}{21}\right) =1+\frac{6}{7}=\frac{13}{7}.

Equating the two expected present values gives

P\left(\frac{13}{7}\right)=\$2{,}636.7347, \qquad P=\$2{,}636.7347\left(\frac{7}{13}\right) =\$1{,}419.7802.

The gross annual premium is $1,419.78, rounded to the nearest cent. Check: the premium factor $13/7$ multiplied by the unrounded premium gives an expected premium present value of $2,636.73, equal to the expected present value of the $10,200 claim amount. The common mistake is to use a claim probability of 1 for each year; the expense is paid only when death occurs.

Why the Formula Works

The expected present value is an expectation of discounted cash flows. For a payment $C$ at time $k$ that occurs only if event $D_k$ occurs, its present value is $Cv^k$ when $D_k$ occurs and zero otherwise. Its expectation is therefore

$$ E[Cv^k I(D_k)]=Cv^kP(D_k), $$

where $I(D_k)$ equals 1 if the event occurs and 0 otherwise. Summing this expression across mutually exclusive possible claim years gives the term-insurance expense present value. If a benefit $B$ is paid at exactly the same time on exactly the same event, the claim cash flow is $(B+C)v^k I(D_k)$, so $B$ and $C$ share the same event probability and discount factor. This is why they can be combined in the formula.

If the expense has a different trigger or payment time, that combination is invalid. For example, an expense payable on claim approval could be delayed beyond the benefit payment; the probability of approval and the discount period must then reflect that separate cash flow.

Worked Example: Expense Paid at the Moment of Death

A hypothetical continuous whole-life policy pays $50,000 immediately on death and incurs an additional $500 claim expense at that same instant. Assume a constant force of mortality $\mu_x=0.04$ at all future durations, a force of interest $\delta=0.06$, and a single gross premium paid at issue. Find the premium.

Method. Because both amounts are paid at death, combine them and discount the resulting cash flow from the actual time of death. Under the constant-force mortality assumption, the density of the future lifetime $T_x$ is $0.04e^{-0.04t}$ for $t\geq0$. The discount factor to time $t$ is $e^{-0.06t}$.

The probability density accounts for the chance death occurs around time $t$, and the discount factor accounts for the time value of money. Consequently, the expected present value factor for a unit paid immediately on death is

\int_0^\infty e^{-0.06t}(0.04e^{-0.04t})\,dt =0.04\int_0^\infty e^{-0.10t}\,dt =\frac{0.04}{0.10}=0.4.

The amount paid at death is $\$50{,}000+\$500=\$50{,}500$. Its expected present value, and hence the single gross premium, is

\$50{,}500(0.4)=\$20{,}200.

The single gross premium is $20,200.00. Check: the benefit alone has expected present value $\$50{,}000(0.4)=\$20{,}000$, and the claim expense has expected present value $\$500(0.4)=\$200$; together they total $20,200. The calculation depends on the expense being paid immediately on death. It would not use this same factor if the expense were paid at a later fixed date.

Expenses Triggered by Annuity Payments

An expense charged each time an annuity installment is paid follows the payment pattern of that annuity. For an $n$-year temporary life annuity-immediate with annual payments, the payment at time $k$ is made only if the annuitant survives to time $k$. If the benefit per installment is $R$, and a fixed expense $C$ is incurred at the same time as each installment, the expected present value of the payments and associated expenses is

\sum_{k=1}^{n}(R+C)v^k\,{}_kp_x.

Here ${}_kp_x$ is the probability of survival from age $x$ to time $k$. For an annuity-due, a payment at time zero is certain, so that payment and its associated expense have no discounting and no survival contingency at issue. In either payment pattern, the expense is counted only on installments that are actually made.

Worked Example: A Temporary Annuity with Per-Payment Expenses

A hypothetical contract provides a three-year temporary life annuity-immediate of $500 per year, payable at the end of each year if the annuitant is alive. A $10 administration expense is incurred with each installment. Assume an annual effective interest rate of $i=25\%$, and conditional survival probability 0.50 in each of the three years. Find the single gross premium.

Method. Add the expense to each installment, then weight each combined payment by the probability of survival to its payment date and discount to issue. The annual survival probability is deliberately specified as a simple hypothetical assumption so each duration can be checked directly.

The discount factor is $v=1/1.25=0.8$. Conditional survival is 0.50 in each year, so the survival probabilities to times 1, 2, and 3 are $0.50$, $0.50^2=0.25$, and $0.50^3=0.125$. Each installment together with its expense is $\$500+\$10=\$510$.

\begin{aligned} \text{Expected present value} &=\$510\left[(0.50)(0.8)+(0.25)(0.8)^2+(0.125)(0.8)^3\right]\\ &=\$510(0.400+0.160+0.064)\\ &=\$510(0.624)=\$318.24. \end{aligned}

The single gross premium is $318.24. Check: the annuity benefits alone have expected present value $\$500(0.624)=\$312.00$; the three contingent expenses have expected present value $\$10(0.624)=\$6.24$; the total is $318.24. A common mistake is to multiply the installment expense by three without survival weighting or discounting. The third expense, for example, is incurred only with probability 0.125 and is discounted by $v^3=0.512$, so its expected present value is $\$10(0.125)(0.512)=\$0.64$, not $10.

Common Exam Traps

The most reliable way to avoid timing errors is to write each cash flow with four pieces: amount, trigger, payment time, and discount factor. Then verify that the probability refers to the event that actually causes that payment.

Using survival probability for a death claim. A death benefit is paid on death, so the year-$k$ claim probability is ${}_{k-1}p_xq_{x+k-1}$, not ${}_kp_x$.
Using death probability for an annuity installment. An installment paid at time $k$ requires survival to time $k$, so it is weighted by ${}_kp_x$.
Discounting to the wrong date. An expense paid immediately at death is not automatically discounted to the end of the year of death. Use the time at which the expense is actually paid.
Counting every possible expense as certain. A per-claim expense is not charged in the expected value unless a claim occurs; a per-annuity-payment expense is not incurred if that installment is not paid.
Combining amounts with different triggers. Benefit and expense amounts can be combined into one amount only when they share both the relevant event and payment time.

Key takeaway: A claim-related expense belongs in the gross premium equation with the probability of its own trigger and the discount factor to its own payment time. Sharing a trigger and payment time with a benefit allows the amounts to be combined; otherwise, value the cash flows separately.

Check Your Understanding

For each problem, identify the event and timing associated with the expense before calculating. Answers are rounded to the nearest cent.

A hypothetical one-year term policy pays $8,000 at the end of the year of death and incurs a $120 claim expense at that same time. The single gross premium is paid at issue. If $i=5\%$ and $q_x=0.10$, find the premium. (Answer: $773.33)
A hypothetical two-year temporary life annuity-immediate pays $300 at the end of each year if the annuitant is alive, with a $6 expense on each installment. The annual effective interest rate is $25\%$, and the conditional survival probabilities are 0.90 in year 1 and 0.80 in year 2. Find the single gross premium. (Answer: $361.32)
A hypothetical two-year term policy pays $20,000 at the end of the year of death and incurs a $400 expense at that same time. Annual premiums are paid in advance at times 0 and 1 while the insured is alive. Let $i=0$, $q_x=0.05$, and $q_{x+1}=0.10$. Find the level gross annual premium. (Answer: $1,516.92)
A hypothetical continuous whole-life policy pays $40,000 immediately on death, with a $300 claim expense paid at the same instant. Assume constant force of mortality $\mu_x=0.03$, force of interest $\delta=0.05$, and a single premium at issue. Find the single gross premium. (Answer: $15,112.50)

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