Why Track the Fund for the Whole Cohort?
A retrospective reserve can be understood as an accumulated fund built from past premiums and reduced by past claims. One way to organize that history is to track the fund for all policies still in force at each year-end. The resulting cohort balance is an aggregate dollar amount; dividing it by the number of surviving policies gives the average fund per survivor.
The order of cash flows matters. In the annual model used here, premiums are collected at the beginning of a policy year, the fund earns interest during that year, and death benefits are paid at the end of the year. A different payment schedule requires a different balance equation. This account-based approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
For a cohort based on a life table, write \(N_t=\ell_{x+t}\), where \(\ell_{x+t}\) is the number alive at age \(x+t\). The number dying during the next year is \(D_t=\ell_{x+t}-\ell_{x+t+1}\). These counts may be actual counts from an observed cohort, or life-table counts used to represent an expected cohort. The interpretation should be stated: expected counts need not be whole numbers, and an expected cohort fund is not necessarily the cash balance of a particular real group.
The Annual Cohort-Fund Balance
Suppose each survivor at time \(t\) pays a premium \(P_t\) at the beginning of the year. Each death during the year generates a benefit \(B_t\), paid at the end of the year. Assume a common effective annual interest rate \(i\), no expenses, no lapses, and no other cash flows. The fund immediately after the time-\(t\) claims is \(F_t\).
The balance \(F_t+N_tP_t\) is the fund after collecting premiums. It earns interest over the year. Then the end-of-year claims are subtracted, leaving the balance for those alive at time \(t+1\). Thus, under this timing convention, claims do not earn interest during the year.
Because \(F_t=N_tV_t\), the same balance can be written directly in terms of the per-survivor amount at time \(t\):
This form makes the mortality adjustment visible. The numerator begins with the amount per policy at time \(t\), adds that year's premium, accumulates interest, and subtracts the benefit cost spread across the policies alive at the start of the year. The denominator then converts the remaining amount per starting policy into an amount per ending survivor. It requires \(N_t>0\) as well as \(N_{t+1}>0\).
Why the Balance Equation Holds
The cohort equation follows from a dollar-by-dollar account balance, rather than from a separate assumption about how the reserve changes. At time \(t\), the cohort has \(F_t\) dollars. The \(N_t\) policies each contribute \(P_t\), so the total just after premiums is \(F_t+N_tP_t\). Under an effective annual interest rate \(i\), every dollar in that account becomes \(1+i\) dollars by the end of the year. The fund before claims is therefore \((F_t+N_tP_t)(1+i)\). There are \(D_t\) deaths, each with a benefit \(B_t\); subtracting their total claims \(D_tB_t\) gives \(F_{t+1}\). Finally, the \(N_{t+1}\) survivors share that remaining fund on a per-policy basis, so division by \(N_{t+1}\) gives \(V_{t+1}\).
Use the aggregate fund after time-\(t\) claims: \(F_t\).
There are \(N_t\) payers, so the premium inflow is \(N_tP_t\).
Multiply the post-premium fund by \(1+i\), then subtract \(D_tB_t\).
Divide the ending fund by \(N_{t+1}\), not by the starting cohort size.
Worked Cohort-Fund Calculations
Worked Example: One Year With an Opening Fund
A hypothetical cohort has 1,000 policies in force at time 0 and an aggregate fund of $20,000 immediately before premiums are collected. Each survivor pays $80 at the beginning of the year. The annual effective interest rate is 5.0%. During the year, 40 policies end in a death claim of $1,000 each, paid at year-end. Find the time-1 aggregate fund and the per-survivor fund. Assume no other cash flows.
Method. Apply the annual cohort balance in the stated order: collect premiums, accumulate interest, pay claims, and divide by the ending survivor count.
The beginning-of-year premium inflow is \(1{,}000(80)=\$80{,}000\). The fund after premiums is \(\$20{,}000+\$80{,}000=\$100{,}000\). After interest, it is \(\$100{,}000(1.05)=\$105{,}000\). The 40 claims total \(40(\$1{,}000)=\$40{,}000\), so the ending fund is \(\$105{,}000-\$40{,}000=\$65{,}000\).
There are \(1{,}000-40=960\) survivors. Thus the per-survivor fund is \(\$65{,}000/960=\$67.7083\ldots\), or $67.71 to the nearest cent. The calculation checks: \(960(\$67.7083\ldots)=\$65{,}000\), up to rounding.
Common mistake avoided: using 1,000 as the divisor at time 1. The ending fund belongs to the 960 survivors, so the correct divisor is 960.
Worked Example: Starting From a Per-Survivor Fund
A hypothetical group has 500 policies alive at time \(t\), with an aggregate fund of $25,000. Each survivor pays a beginning-of-year premium of $120. The annual effective interest rate is 4%. Fifteen policies die during the year, and each death claim of $2,000 is paid at year-end. Calculate the ending aggregate fund and fund per survivor.
Method. First apply the aggregate balance. Then divide by the 485 ending survivors. This also checks that starting from the per-survivor value gives the same result.
The opening amount per survivor is \(\$25{,}000/500=\$50\). The premium inflow is \(500(\$120)=\$60{,}000\), making the post-premium fund \(\$25{,}000+\$60{,}000=\$85{,}000\). Interest adds \(\$85{,}000(0.04)=\$3{,}400\), so the pre-claim fund is \(\$88{,}400\). The claims total \(15(\$2{,}000)=\$30{,}000\). Therefore, the ending aggregate fund is \(\$88{,}400-\$30{,}000=\$58{,}400\).
The number surviving is \(500-15=485\), giving \(\$58{,}400/485=\$120.41237\ldots\), or $120.41 per survivor to the nearest cent.
As a check using the per-survivor form, the death proportion is \(15/500=0.03\), and the survivor proportion is \(485/500=0.97\). Thus \( \big[(\$50+\$120)(1.04)-0.03(\$2{,}000)\big]/0.97 =(\$176.80-\$60)/0.97 =\$120.41237\ldots \). The two calculations agree.
Common mistake avoided: subtracting the full $30,000 claim total from a per-survivor amount. Aggregate dollars must be balanced with aggregate dollars; a per-survivor calculation must first express the claims per starting policy.
Worked Example: Two Years With Changing Cash Flows
A hypothetical cohort follows these life-table counts: \(N_0=200\), \(N_1=190\), and \(N_2=178\). The initial fund is zero. The annual effective interest rate is 4%. In year 0, each survivor pays $500 at the beginning of the year, and each of the 10 deaths receives $3,000 at year-end. In year 1, each of the 190 survivors pays $400 at the beginning of the year, and each of the 12 deaths receives $2,500 at year-end. Find the aggregate and per-survivor funds at times 1 and 2.
Method. Recur the aggregate fund one year at a time, using each year's own premium and benefit. The death counts are \(200-190=10\) and \(190-178=12\).
For year 0, premiums total \(200(\$500)=\$100{,}000\). Accumulating the post-premium fund gives \(\$100{,}000(1.04)=\$104{,}000\). Claims total \(10(\$3{,}000)=\$30{,}000\), leaving \(F_1=\$104{,}000-\$30{,}000=\$74{,}000\). Therefore, \(V_1=\$74{,}000/190=\$389.47368\ldots\), or $389.47 per survivor.
For year 1, the premium inflow is \(190(\$400)=\$76{,}000\). Add this to \(F_1\): \(\$74{,}000+\$76{,}000=\$150{,}000\). After interest, the fund is \(\$150{,}000(1.04)=\$156{,}000\). Claims total \(12(\$2{,}500)=\$30{,}000\), leaving \(F_2=\$126{,}000\). Hence \(V_2=\$126{,}000/178=\$707.86516\ldots\), or $707.87 per survivor.
The check at each duration is the defining relation \(F_t=N_tV_t\): \(190(\$389.47368\ldots)=\$74{,}000\), and \(178(\$707.86516\ldots)=\$126{,}000\). The rounded amounts are reported only after the calculations.
Common mistake avoided: carrying forward the time-0 population of 200 into year 1. Year-1 premiums are paid by 190 survivors, and the time-2 fund is divided among 178 survivors.
Timing, Mortality Counts, and a Frequent Trap
A claim's payment time determines whether it earns interest in the fund calculation. Here, death benefits are paid at the end of the year, so the premium-and-opening-fund balance earns a full year's interest before claims are subtracted. For the first worked example, subtracting the $40,000 of claims before accumulating interest would give \((\$100{,}000-\$40{,}000)(1.05)=\$63{,}000\), which is wrong for end-of-year claims. The correct amount is \(\$100{,}000(1.05)-\$40{,}000=\$65{,}000\). The wrong calculation effectively makes the claims leave the fund at the start of the year and denies the fund interest on dollars that remain invested until year-end.
Mortality counts also need a consistent interpretation. With observed counts, \(D_t=N_t-N_{t+1}\) is the actual number of deaths in that interval. With life-table counts, the same difference represents the expected number of deaths for the life-table cohort. Do not combine an actual claim count with an unrelated expected survivor count unless the problem explicitly defines that hybrid calculation.
| Quantity | Meaning at the stated timing | Use in the annual balance |
|---|---|---|
| \(F_t\) | Aggregate fund after claims at time \(t\), before premiums at time \(t\) | Opening fund |
| \(N_tP_t\) | Premiums collected at the beginning of the year | Add before interest |
| \(D_tB_t\) | Claims for deaths during the year, paid at year-end | Subtract after interest |
| \(N_{t+1}\) | Policies alive after the year's deaths | Divide the ending fund by this count |
The equations assume a common annual rate, a single beginning-of-year premium per survivor, and end-of-year death claims. If premiums are paid at another time, claims are paid on another schedule, or expenses and other benefits enter the account, the cash flows must be placed at their actual times and accumulated only over the period for which they earn interest. A cohort balance is useful precisely because it forces each cash flow to be included once, at the correct time.
Check Your Understanding
For each problem, use the stated timing: premiums at the beginning of the year, interest during the year, and death claims at year-end. Round per-survivor dollar amounts to the nearest cent.
- A hypothetical cohort has 400 survivors, an opening fund of $8,000, a premium of $50 per survivor, and an annual effective interest rate of 3%. Eight deaths each generate a $500 claim. Find the ending fund per survivor. (Answer: $63.37)
- A group has 600 policies, no opening fund, and a beginning-of-year premium of $90 per policy. At 4% annual effective interest, 12 deaths generate claims of $1,500 each. Find the aggregate ending fund and the per-survivor fund. (Answer: $38,160 aggregate; $64.90 per survivor)
- A cohort has 300 survivors and an opening fund of $12,000. Each pays $100 at the beginning of the year; the annual effective interest rate is 5%. Nine deaths generate $2,000 claims at year-end. Find the ending fund per survivor. (Answer: $89.69)
- A hypothetical cohort has counts \(N_0=1{,}000\), \(N_1=970\), and \(N_2=930\), with no initial fund and a 3% annual effective interest rate. Year 0 has a $60 premium per survivor and a $1,000 claim per death. Year 1 has an $80 premium per survivor and a $1,500 claim per death. Find the time-2 aggregate fund and per-survivor fund. (Answer: $52,682 aggregate; $56.65 per survivor)
- A cohort has 200 policies, an opening fund of $5,000, and a $50 premium per survivor. The annual effective interest rate is 4%; five deaths receive $1,000 each at year-end. A calculation subtracts the claims before interest and reports an ending fund of $10,600. Identify the correct ending aggregate fund and per-survivor fund. (Answer: $10,600 aggregate; $54.36 per survivor)