Allocating Premium Income to Policy Cohorts

Why Premium Timing Matters

An asset-share projection records the cash flows attributed to a policy or group of similar policies and accumulates them to a specified date. For the premium-income component, the amount received and the date it is received both matter: two cohorts with the same total scheduled premiums can contribute different amounts at the valuation date if their payment dates differ.

The calculation is not simply “annual premium times years.” The premium-paying term determines which payments are scheduled, the payment frequency determines the size and timing of each instalment, and the projection date determines how long each receipt accumulates. The treatment below isolates premium income; claims, expenses, lapses, and other cash flows are outside this calculation.

Definition: For a cohort, premium income included in an asset-share projection is the sum of the actual scheduled premium instalments attributed to that cohort that are due and assumed received by the valuation date. Each instalment is accumulated from its payment date to the valuation date using the applicable investment accumulation factors. Unless stated otherwise, assume policies remain in force, every scheduled premium is paid, and there are no premium waivers or refunds.

A cohort is a group of policies treated alike for the projection. Let $N_c$ be the number of policies in cohort $c$, and let $P_{cj}$ be the premium per policy for scheduled payment $j$, paid at time $t_{cj}$. Let $T$ be the valuation time, with $t_{cj}\leq T$. If $A(t,T)$ is the accumulation factor from time $t$ to time $T$, the cohort’s accumulated premium income is

\text{Premium income for cohort }c\text{ at }T = N_c\sum_{j:\,t_{cj}\leq T}P_{cj}A(t_{cj},T).

This is an aggregate amount for the cohort. The per-policy amount is the same sum without $N_c$. Keep the two bases separate: multiply by the policy count once, not once for each step in the accumulation.

Build the Payment Schedule Before Accumulating

For level premiums paid $m$ times per year in advance over a premium-paying term of $n$ years, the usual schedule from issue is at times $0,1/m,2/m,\ldots,n-1/m$, provided the contract specifies payments at the start of each payment period. There are $mn$ instalments, each normally equal to the annual premium divided by $m$. A payment in advance at time $0$ is included if the projection starts at issue.

For premiums paid in arrears, payments are instead due at the ends of their payment periods. For example, annual arrears premiums over a three-year term are due at times $1,2,3$, not at $0,1,2$. The contract’s payment convention controls; do not infer it just from the phrase “annual premium.”

A premium-paying term is not necessarily the same as the policy term. Once the premium-paying term has ended, do not include further premiums merely because the policy remains in force. Conversely, an instalment due on the valuation date is included only if the projection’s timing convention says it is received by that date. In the examples below, premiums due at the valuation time are included only when explicitly scheduled as a payment at that time.

Specify the cohort.
Record the policy count and the premium amount per policy per instalment. Confirm whether a quoted premium is an annual amount or an instalment amount.

List payment dates.
Use the stated frequency and whether payments are in advance or arrears. Stop the schedule at the end of the premium-paying term.

Accumulate each receipt.
For every included payment, apply the accumulation factor from that payment’s own date to the valuation date.

Aggregate and check.
Sum the per-policy values, multiply by the policy count, and check that later premiums receive less accumulation than earlier premiums.

Accumulation of Premium Receipts

With a constant effective annual investment return $i$, the factor for a fractional period $s$ years is $(1+i)^s$ when the model uses annual effective compounding over fractional years. Alternatively, a problem may directly provide accumulation factors for quarters or half-years; use those stated factors. With changing annual returns $i_1,\ldots,i_k$, a payment made before those years accumulate by the product of the applicable annual factors, such as $\prod_{r=1}^{k}(1+i_r)$. This is consistent with the earlier material on accumulating cash flows at effective investment returns.

Justification: An accumulation factor takes one cash flow at its payment date to its value at the projection date. If a premium $P$ is paid at time $t$, its value at $T$ is $P A(t,T)$. Accumulation is linear, so the value of several receipts is the sum of their separately accumulated values. If each of $N_c$ identical policies pays the same schedule, the aggregate is $N_c$ times the per-policy sum. This gives the cohort formula above; it also explains why different payment dates cannot generally be replaced by one total premium received at the start of the year.

Worked Example: Annual and Semiannual Cohorts at One Year

A hypothetical block contains two cohorts. Cohort A has 4 policies, each paying an annual premium of $1,000 in advance at time 0. Cohort B has 6 policies, each paying $500 at times 0 and 0.5. Both schedules apply over the first year. The annual effective investment return is 21%, and the half-year accumulation factor is 1.10, consistent because $1.10^2=1.21$. Find the combined accumulated premium income at time 1.

Method. List each receipt date, accumulate it separately to time 1, and then multiply each per-policy amount by the cohort size.

For Cohort A, each policy pays $1,000 at time 0. The receipt accumulates for one year:

$$ 4(1{,}000)(1.21)=4{,}840. $$

For Cohort B, each policy pays $500 at time 0 and another $500 at time 0.5. The first instalment accumulates for one year, while the second accumulates for one half-year:

6\left[500(1.21)+500(1.10)\right] =6(605+550) =6(1{,}155) =6{,}930.

Thus the combined amount is $4{,}840+6{,}930=\$11{,}770$. The check is that each cohort’s receipts are accumulated for their actual remaining times; the later semiannual instalment receives only one half-year of accumulation.

Common mistake avoided. The two Cohort B instalments should not both receive the one-year factor. Doing so would overstate their value. Nor should the aggregate include only one policy’s premium schedule: both cohort calculations include the stated policy counts.

Worked Example: Quarterly Premiums Paid in Advance

A hypothetical cohort has 10 policies. Each policy pays a quarterly instalment of $250 in advance at times 0, 0.25, 0.5, and 0.75. The accumulation factor for each quarter is 1.02. Find the aggregate accumulated premium income at time 1, to the nearest cent.

Method. Work from the payment dates. The four instalments accumulate for four, three, two, and one quarters, respectively. The quarterly factor implies an annual accumulation factor of $1.02^4$, but it does not make all four payments time-0 receipts.

\begin{aligned} \text{Per-policy value at time 1} &=250(1.02^4+1.02^3+1.02^2+1.02)\\ &=250(1.08243216+1.061208+1.0404+1.02)\\ &=250(4.20404016)\\ &=1{,}051.01004. \end{aligned}

For 10 policies, the aggregate is $10(1{,}051.01004)=\$10{,}510.1004$, which rounds to $10,510.10 to the nearest cent.

Common mistake avoided. Treating the annual total of $1,000 per policy as if it were all paid at time 0 would give $10(1{,}000)(1.02^4)=\$10{,}824.32$, which is too large. The instalments arrive throughout the year, so the correct calculation gives each its own accumulation period.

Premium-Paying Terms and Valuation Boundaries

A payment schedule should be written with its last due date explicit. For premiums in advance, a two-year term with annual payments from issue generally has payments at times 0 and 1; it does not have a third payment at time 2. For premiums in arrears, a two-year term with annual payments generally has payments at times 1 and 2. These schedules differ even though both premiums are described as annual and the terms have the same length.

The end date requires care. A projection “at time 2” does not by itself establish whether a payment due exactly at time 2 is included. The contract timing and the projection convention must resolve whether that receipt is due and received by the valuation instant. Once that is specified, include it or exclude it consistently; do not silently shift it to just before or after the valuation date.

Worked Example: Different Frequencies and Premium-Paying Terms

A hypothetical group has two cohorts, both issued at time 0, and the accumulated premium income is required at time 2. Cohort A has 3 policies, each paying $700 annually in advance for a two-year premium-paying term. Cohort B has 4 policies, each paying $300 semiannually in advance for a one-year premium-paying term. The annual accumulation factor is 1.21 and the half-year factor is 1.10. No premium is due at time 2 under either schedule.

Method. First list only payments within each premium-paying term; then accumulate each payment to time 2. Cohort A pays at times 0 and 1. Cohort B pays at times 0 and 0.5. Its one-year term ends before time 2, so it has no later payments.

For Cohort A, the time-0 premium accumulates for two years and the time-1 premium for one year:

3(700)(1.21^2+1.21) =2{,}100(1.4641+1.21) =2{,}100(2.6741) =\$5{,}615.61.

For Cohort B, each payment at time 0 accumulates for four half-years, and each payment at time 0.5 accumulates for three half-years:

\begin{aligned} 4(300)(1.10^4+1.10^3) &=1{,}200(1.4641+1.331)\\ &=1{,}200(2.7951)\\ &=\$3{,}354.12. \end{aligned}

The combined accumulated premium income is $\$5{,}615.61+\$3{,}354.12=\$8{,}969.73$. As a check, Cohort A has two payments per policy, while Cohort B has two—not four—payments per policy because its premium-paying term is only one year. Each payment is accumulated for the time remaining to time 2.

Common mistake avoided. Extending Cohort B’s semiannual schedule through time 1.5 would incorrectly charge premiums after its one-year premium-paying term. A policy may continue beyond its premium-paying term, but that does not create additional premiums.

Checks Before Using the Premium Component

Before inserting premium income into an asset-share projection, verify the schedule and the basis of every amount. The following checks catch many errors:

Frequency: confirm whether the quoted amount is per instalment or per year. If it is an annual premium split into $m$ level instalments, each instalment is normally the annual amount divided by $m$.
Timing: identify advance or arrears payment dates explicitly. A premium in advance at time 0 receives the full period’s accumulation; a premium in arrears at the period end does not.
Term: count only dates within the premium-paying term. Do not continue premiums after that term ends.
Valuation date: settle whether a payment due exactly at the valuation time is included under the stated convention.
Aggregation: calculate per policy and then multiply by the cohort size, or calculate aggregate instalments consistently. Do not multiply by the policy count twice.
Accumulation: earlier receipts should generally have at least as much accumulation time as later receipts when the returns are nonnegative.

Premium income is one component of an asset-share projection, not the asset share by itself. The same date-by-date discipline will be needed when other cash flows are incorporated. This tutorial’s premium scheduling and accumulation methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: Build the cohort’s contractual premium schedule first. Include only instalments due under the payment frequency and premium-paying term, accumulate each from its actual payment date, and then aggregate across policies and cohorts.

Check Your Understanding

For each problem, identify the scheduled instalments before accumulating them. Unless stated otherwise, assume every scheduled premium is paid.

A cohort of 2 policies pays $600 per policy annually in advance at times 0 and 1. The annual accumulation factor is 1.05. Find the aggregate value at time 2. (Answer: $2,583.00)
Three policies each pay $80 quarterly in advance at times 0, 0.25, 0.5, and 0.75. The quarter-year factor is 1.01. Find the aggregate value at time 1, to the nearest cent. (Answer: $984.24)
Two policies each pay $500 annually in arrears for a three-year premium-paying term, at times 1, 2, and 3. The annual accumulation factor is 1.04. Find the aggregate value at time 3, including the payment due at time 3. (Answer: $3,121.60)
Four policies each pay $900 annually in advance for a two-year premium-paying term, at times 0 and 1. The annual accumulation factor is 1.10. Find the aggregate value at time 2, with no payment due at time 2. (Answer: $8,316.00)
Cohort A has 2 policies paying $400 annually in advance at times 0 and 1 for a two-year term. Cohort B has 3 policies paying $200 semiannually in advance at times 0, 0.5, 1, and 1.5 for a two-year term. The annual factor is 1.1025 and the half-year factor is 1.05. Find the combined value at time 2, with no premium due at time 2. (Answer: $4,569.78)

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