Accumulated Premiums Under Annual Life Contingencies

Why Survival-Weighted Premium Accumulation Matters

When annual premiums are paid only while a policy is in force, not every policy in an original group contributes every premium. A premium due at the start of a policy year is received only if the insured is alive and the contract is still active at that time. To measure the accumulated premium income from the original group at a later date, each premium must therefore be weighted by the probability it was paid, then accumulated with interest.

This tutorial considers the premium component of a retrospective cohort calculation. It does not include accumulated death benefits or expenses. The central task is to keep three things distinct: when a premium is due, how likely it is to be paid, and how long it earns interest.

Definition: Suppose a policy issued at age $x$ has annual premiums $P_k$ due at times $k=0,1,\ldots,t-1$, conditional on the policy being in force at the start of the year. At an effective annual interest rate $i$, the expected accumulated premium income at time $t$, per policy originally issued, is $$ \sum_{k=0}^{t-1} P_k\,{}_kp_x(1+i)^{t-k}. $$ Here ${}_kp_x$ is the probability that a life aged $x$ survives $k$ years. This formula assumes no lapses, premiums due at the start of each year, and accumulation at the same effective annual rate for all cash flows.

Timing and Survival Weights

At time $k$, a premium is paid if the insured is alive at that time and the policy remains in force. Under the no-lapse assumption, the probability of being alive then is ${}_kp_x$, with ${}_0p_x=1$. A premium paid at time $k$ earns interest for $t-k$ years, so its value at time $t$ is $P_k(1+i)^{t-k}$. The factor ${}_kp_x$ weights that value by the chance the payment is received from an original policy.

The time index matters. If valuation is immediately before the premium due at time $t$, that premium is not included: it has not yet been paid. If valuation is immediately after that premium, an additional term $P_t{}_tp_x$ is included, with no further accumulation. The valuation convention must be stated before setting the sum's limits.

\text{Expected accumulated premium income per original policy} =\sum_{k=0}^{t-1}P_k\,{}_kp_x(1+i)^{t-k}.

For level premiums $P_k=P$, the same expression can be written as $$ P\sum_{k=0}^{t-1}{}_kp_x(1+i)^{t-k}. $$ Unlike an ordinary certain annuity accumulation, the terms are not all weighted by one: premiums after issue may fail to be paid because the insured may have died before they are due.

Why the Formula Holds

For a direct justification, define $I_k=1$ if the premium due at time $k$ is paid and $I_k=0$ otherwise. Its accumulated value at time $t$ is $P_k I_k(1+i)^{t-k}$. The total accumulated income from one original policy is the sum of these amounts. Since $E[I_k]={}_kp_x$, linearity of expectation gives the survival-weighted sum. No assumption of independence between the indicators is needed.

E\!\left[\sum_{k=0}^{t-1}P_kI_k(1+i)^{t-k}\right] =\sum_{k=0}^{t-1}P_k(1+i)^{t-k}E[I_k] =\sum_{k=0}^{t-1}P_k\,{}_kp_x(1+i)^{t-k}.

The sum is an expectation per policy originally issued. In a cohort of $N$ identical policies, the expected accumulated premium income is $N$ times this amount. If the cohort method allocates that income across expected survivors at time $t$, the expected number of survivors is $N{}_tp_x$. Thus the accumulated cohort premium income per expected survivor is

\frac{\displaystyle\sum_{k=0}^{t-1}P_k\,{}_kp_x(1+i)^{t-k}} {{}_tp_x}, \qquad {}_tp_x>0.

This allocation is not the same as the accumulated premiums paid by a particular policyholder known to survive to time $t$. Such a policyholder was alive at every earlier premium date and, under the assumptions here, paid each scheduled premium. The cohort allocation instead includes premiums paid by members who later died, then spreads that cohort income across the expected survivors. Keeping these interpretations separate is essential.

Worked Calculations

Worked Example: Level Premiums With Annual Mortality

A hypothetical insurer issues policies at age $x$ with an annual premium of $1,000, due at times 0, 1, and 2. The effective annual interest rate is $i=5\%$. The one-year death probabilities at ages $x$, $x+1$, and $x+2$ are 0.10, 0.20, and 0.25. Find (a) expected accumulated premium income at time 3 per policy issued and (b) the cohort premium allocation per expected survivor at time 3.

Method. First calculate survival probabilities from issue. The probability of surviving one, two, and three years is $$ {}_1p_x=1-0.10=0.90,\qquad {}_2p_x=0.90(1-0.20)=0.72,\qquad {}_3p_x=0.72(1-0.25)=0.54. $$

The premium at time 0 accumulates for three years, the time-1 premium for two years, and the time-2 premium for one year. Weight each by the probability it is paid: $$ \begin{aligned} \text{Time-0 premium: }&1{,}000(1)(1.05)^3=\$1{,}157.625,\\ \text{Time-1 premium: }&1{,}000(0.90)(1.05)^2=\$992.25,\\ \text{Time-2 premium: }&1{,}000(0.72)(1.05)=\$756.00. \end{aligned} $$

Adding the unrounded amounts gives expected accumulated premium income per policy issued: $$ 1{,}157.625+992.25+756.00=\$2{,}905.875, $$ which is $2,905.88 to the nearest cent. The cohort allocation per expected survivor is $$ \frac{2{,}905.875}{{}_3p_x} =\frac{2{,}905.875}{0.54} =\$5{,}381.25. $$

Check. Multiplying the allocation by the survivor probability recovers the per-policy-issue amount: $0.54(5{,}381.25)=2{,}905.875$. The common mistake avoided here is dividing by the survival probability when the requested quantity is per policy originally issued; division is appropriate only for the stated per-survivor cohort allocation.

Worked Example: Varying Premiums and a Shorter Premium History

A hypothetical contract has premiums of $500, $600, and $700 due at times 0, 1, and 2, respectively. The effective annual interest rate is 5%. The one-year death probabilities at the successive ages are 0.20, 0.25, and 0.50. Calculate accumulated premium income at time 3 per original policy and the cohort allocation per expected survivor.

The survival probabilities needed for premiums due before time 3 are $$ {}_0p_x=1,\qquad {}_1p_x=1-0.20=0.80,\qquad {}_2p_x=0.80(1-0.25)=0.60. $$ The probability of surviving to time 3, needed for the survivor allocation, is $$ {}_3p_x=0.60(1-0.50)=0.30. $$

Accumulate each premium to time 3 and apply its payment probability: $$ \begin{aligned} \text{Time-0 premium: }&500(1)(1.05)^3=\$578.8125,\\ \text{Time-1 premium: }&600(0.80)(1.05)^2=\$529.20,\\ \text{Time-2 premium: }&700(0.60)(1.05)=\$441.00. \end{aligned} $$

Therefore, the expected accumulated income per original policy is $$ 578.8125+529.20+441.00=\$1{,}549.0125, $$ or $1,549.01 to the nearest cent. Dividing by the time-3 survival probability gives $$ \frac{1{,}549.0125}{0.30}=\$5{,}163.375, $$ or $5,163.38 per expected survivor. The calculation uses the varying premium amounts individually; replacing them with one level premium would not represent this contract.

Worked Example: Using Life-Table Counts and Solving for a Premium

A hypothetical cohort has $l_x=5{,}000$, $l_{x+1}=4{,}500$, and $l_{x+2}=4{,}000$. Level premiums are due at times 0 and 1, and valuation is at time 2, immediately before any time-2 premium. The effective annual interest rate is 4%. Find the level premium $P$ that makes the expected accumulated premium income per original policy $1,210.56, then find the corresponding cohort allocation per expected survivor.

Life-table counts give the required survival probabilities by taking ratios to $l_x$: $$ {}_1p_x=\frac{4{,}500}{5{,}000}=0.90,\qquad {}_2p_x=\frac{4{,}000}{5{,}000}=0.80. $$

Only the premiums at times 0 and 1 are included. Their accumulated expected values are $P(1.04)^2$ and $P(0.90)(1.04)$. Hence $$ 1{,}210.56=P\left((1.04)^2+0.90(1.04)\right) =P(1.0816+0.936) =2.0176P. $$ Solving gives $$ P=\frac{1{,}210.56}{2.0176}=\$600. $$

The cohort allocation per expected survivor is the accumulated income per original policy divided by ${}_2p_x$: $$ \frac{1{,}210.56}{0.80}=\$1{,}513.20. $$ As a check, the original-policy calculation with $P=\$600$ is $600(1.0816)+600(0.90)(1.04)=648.96+561.60=\$1,210.56$. The time-2 premium is excluded because valuation is immediately before it is due.

Common Timing and Interpretation Traps

The formula is compact, but most errors come from using the wrong survival weight or the wrong accumulation period. A premium due at time $k$ is weighted by ${}_kp_x$, not ${}_{k+1}p_x$, because survival to the start of year $k$ is the condition for paying it. It earns interest for $t-k$ years, not $t-k-1$.

For example, at time 3 the time-2 premium has one year to accumulate. An incorrect calculation might use ${}_3p_x$ as its payment probability and accumulate it for zero years, producing $P_2{}_3p_x$. The correction is $P_2{}_2p_x(1+i)$: the premium is paid at time 2 if the insured survives to time 2, and it then earns one year of interest.

Do not confuse per-issue and per-survivor amounts. The survival-weighted sum is per original policy. Divide by ${}_tp_x$ only when allocating cohort income across expected survivors.
Do not use survival to valuation for every premium. A premium at time $k$ is paid based on survival to $k$, even though some payers may die before time $t$.
Do not include an unpaid premium. State whether valuation is before or after a premium due at the valuation time.
Do not omit interest accumulation. Earlier premiums earn more years of interest than later ones.

Key takeaway: For annual premiums paid at the start of each year, multiply each premium by its probability of being paid and by its accumulation factor to the valuation date. The resulting sum is per original policy; a cohort allocation per survivor requires division by the probability of survival to the valuation date.

Check Your Understanding

For each problem, identify the relevant premium dates before calculating. Assume no lapses and that valuation is immediately before any premium due at that time unless stated otherwise.

A level annual premium of $400 is due at times 0 and 1. At $i=3\%$, ${}_1p_x=0.75$, and ${}_2p_x=0.60$, find expected accumulated premium income per original policy at time 2. (Answer: $733.36)
Premiums of $100, $200, and $300 are due at times 0, 1, and 2. At $i=4\%$, the survival probabilities to those dates are $1$, $0.90$, and $0.70$. Find expected accumulated income per original policy at time 3. (Answer: $525.57)
A level premium of $250 is due at times 0, 1, and 2. At $i=2\%$, ${}_1p_x=0.80$ and ${}_2p_x=0.50$. Find expected accumulated income at time 3, immediately before a premium due at time 3. (Answer: $600.88)
At $i=5\%$, premiums $P$ are due at times 0 and 1. Given ${}_1p_x=0.80$, find $P$ if expected accumulated premium income per original policy at time 2 is $1,942.50. (Answer: $1,000)
A life table has $l_x=5{,}000$, $l_{x+1}=4{,}500$, and $l_{x+2}=4{,}000$. A level premium of $600 is due at times 0 and 1, and $i=4\%$. Find the cohort premium allocation per expected survivor at time 2. (Answer: $1,513.20)

What You'll Learn