Why an Endowment Reserve Needs Careful Cash-Flow Tracking
An endowment insurance contract combines two possible benefits: a death benefit if the insured dies during the term, and a maturity benefit if the insured survives to the end. A retrospective reserve measures the accumulated premiums remaining after accounting for death claims already paid. For an endowment, the maturity benefit is also important: at the end of the term, it is paid to the surviving policyholders.
The calculation is a cohort cash-flow balance. Premiums are collected from policies in force when each premium is due; death claims are paid for policies that end in death during the following year. At a selected duration, the cohort fund is divided by the number of policies still in force. The maturity claim must be treated separately from earlier death claims when calculating a reserve immediately before the contract ends.
With this timing convention, the premium at time \(j-1\) is paid by \(N_{j-1}\) policies and accumulates for \(k-j+1\) years. The \(D_j\) death claims are paid at time \(j\) and accumulate for \(k-j\) years. Thus the cohort fund at time \(k\), before any maturity payment at that time, is
Here \(C_k\) is the total accumulated fund, not the reserve for one policy. At an intermediate duration, no maturity benefit has yet been paid. At time \(n\), the formula gives the reserve immediately before the maturity payment. The payment \(N_nM\) is then made to survivors. A retrospective reserve based on actual cohort experience need not equal \(M\): the observed deaths and the premium may leave a surplus or a shortfall relative to the maturity claims.
Why the Accumulation Formula Holds
Each premium and claim contributes to the time-\(k\) fund according to how long it remains invested. A payment made at time \(t\) is multiplied by \((1+i)^{k-t}\) to express its value at time \(k\). Premiums are positive inflows; claims are outflows. The following recursion provides a second way to track exactly the same balance.
To justify the recursion, begin at time \(j-1\) with the accumulated fund \(C_{j-1}\). Collect \(N_{j-1}P\) in premiums at that time. The combined amount earns interest for one year, becoming \((C_{j-1}+N_{j-1}P)(1+i)\). Then pay \(D_jB_j\) in death claims at time \(j\). Subtracting those claims gives \(C_j\). Repeatedly substituting this expression accumulates each premium and claim for precisely the number of years in the direct formula.
State whether the reserve is before or after a premium or benefit due at that time. For the formula here, it is after death claims and before a premium at time \(k\).
Use \(N_{j-1}\) for premiums due at the start of year \(j\), and \(D_j\) for claims paid at its end.
Apply the annual interest factor for the number of years from each cash-flow date to time \(k\), or use the fund recursion year by year.
Calculate \(C_k/N_k\). At maturity, do this before subtracting the maturity benefit; then separately account for the payment to survivors.
Worked Endowment Calculations
Worked Example: Reserve at an Intermediate Duration
A hypothetical three-year endowment has 100 policies in force initially. Annual premiums of $200 are paid at the beginning of each year by those then alive. The death benefit is $1,000, payable at each year-end, and the annual effective interest rate is 5%. The observed cohort has 10 deaths in year 1 and 9 in year 2, leaving 90 and then 81 policies in force. Calculate the retrospective reserve at time 2, after year-2 death claims and before any time-2 premium.
Method. Track the aggregate cohort fund with the annual recursion. The maturity benefit is not due at time 2, so it is not subtracted in this calculation.
At time 0, the fund is zero. The first premium collection is \(100(200)=\$20{,}000\). After one year of interest, the fund is \(\$20{,}000(1.05)=\$21{,}000\). The 10 year-1 death claims total \(10(1{,}000)=\$10{,}000\), so
At time 1, 90 policies pay the second premium, for \(90(200)=\$18{,}000\). The combined fund earns one year of interest, and 9 claims are then paid:
There are 81 survivors at time 2. Therefore,
Check. The accumulated premiums are \(20{,}000(1.05)^2+18{,}000(1.05)=22{,}050+18{,}900=\$40{,}950\). Accumulated death claims are \(10{,}000(1.05)+9{,}000=\$19{,}500\). Their difference is \(\$21{,}450\), matching the recursion. The common mistake avoided here is using 81, the number surviving to time 2, as the count paying the time-1 premium; that premium was paid by the 90 policies in force at time 1.
Worked Example: Reserve Immediately Before the Maturity Claim
A hypothetical three-year endowment has 50 policies initially, with annual premiums of $300, a death benefit of $500, and a maturity benefit of $700 per survivor. The annual effective interest rate is 4%. Observed deaths are 5 in year 1, 4 in year 2, and 3 in year 3, leaving 38 survivors at maturity. Calculate the retrospective reserve per survivor immediately before the maturity benefit is paid, and state the cohort position immediately after that payment.
Method. Accumulate premiums and death claims to time 3. The time-3 reserve is first calculated before maturity claims; those claims are then subtracted from the cohort fund.
Premiums of \(50(300)=\$15{,}000\) at time 0 accumulate for three years to \(\$15{,}000(1.04)^3=\$16{,}872.96\). At time 1, 45 policies pay \(\$13{,}500\), which accumulates for two years to \(\$14{,}601.60\). At time 2, 41 policies pay \(\$12{,}300\), which accumulates for one year to \(\$12{,}792\). Total accumulated premiums are \(\$44{,}266.56\).
The year-1 claims total \(\$2{,}500\), accumulated for two years to \(\$2{,}704\). Year-2 claims total \(\$2{,}000\), accumulated for one year to \(\$2{,}080\). Year-3 claims total \(\$1{,}500\), paid at time 3. Total accumulated death claims are \(\$6{,}284\). Thus,
The 38 maturity payments total \(38(700)=\$26{,}600\). The cohort fund after these payments is \(\$37{,}982.56-\$26{,}600=\$11{,}382.56\), or \(\$299.54\) per survivor. The reserve before payment is not the same as the amount left after payment. The remaining amount is an experience surplus in this cohort calculation; the contracts have ended after the maturity claims are settled.
Check. The recursion gives \(C_1=(15{,}000)(1.04)-2{,}500=\$13{,}100\), \(C_2=(13{,}100+13{,}500)(1.04)-2{,}000=\$25{,}664\), and \(C_3=(25{,}664+12{,}300)(1.04)-1{,}500=\$37{,}982.56\). This confirms the direct accumulation.
Worked Example: Separating Death Claims From Maturity Benefits
A hypothetical two-year endowment has 20 policies initially. The annual premium is $50, the annual effective interest rate is 10%, and the death benefit is $200. There are 4 deaths in year 1 and 2 deaths in year 2, leaving 14 survivors at maturity. The maturity benefit is $300 per survivor. Find the reserve at time 1 and the reserve immediately before the maturity payment at time 2.
Method. Use the fund recursion, paying death claims at each year-end. Do not subtract the maturity benefit when calculating the time-1 reserve.
For year 1, the 20 premiums total \(\$1{,}000\). Interest gives \(\$1{,}000(1.10)=\$1{,}100\), and 4 death claims total \(\$800\). Therefore \(C_1=\$300\). With 16 survivors,
At time 1, those 16 policies pay \(16(50)=\$800\). The fund earns interest, and the 2 year-2 claims total \(\$400\):
There are 14 survivors, so immediately before maturity payments the reserve is \(810/14=\$57.8571\ldots\), or $57.86 per survivor. The maturity payments total \(14(300)=\$4{,}200\); the cohort fund is short by \(\$4{,}200-\$810=\$3{,}390\), or \(\$242.14\) per survivor. This shortfall is consistent with the stated premiums and observed experience; the retrospective calculation does not replace the contractually promised maturity benefit.
Check. The year-1 premium fund and claims give \(1{,}000(1.10)-800=\$300\). Adding the next premiums, accumulating, and subtracting year-2 claims gives \(\$810\), as above. The common mistake avoided is subtracting the \(\$4{,}200\) maturity claims at time 1, when they are not yet payable.
Common Errors and Interpretation
| Cash flow or count | Correct treatment | Frequent error |
|---|---|---|
| Premium at time \(j-1\) | Collect \(N_{j-1}P\) | Use the smaller number surviving the year, \(N_j\) |
| Death claims at time \(j\) | Subtract \(D_jB_j\) | Subtract claims for all policies initially in force |
| Reserve at time \(k\) | Divide the cohort fund by \(N_k\) | Report the total cohort fund as a per-policy reserve |
| Maturity at time \(n\) | Calculate the reserve before the maturity payment, then subtract \(N_nM\) to find the post-payment cohort balance | Include the maturity payment in an earlier duration's claims |
A negative cohort balance after maturity payments indicates that the accumulated premiums and investment income, net of death claims, did not cover the maturity claims for the observed cohort. It is not a negative contractual maturity benefit. Conversely, a positive balance after paying all claims is a surplus relative to the actual cash flows used in the retrospective calculation. The reserve is a per-survivor measure at the stated valuation time; the underlying cohort fund and the benefit payable at maturity are distinct quantities.
For an endowment retrospective reserve, the essential discipline is to label each cash flow by its date and eligible policy count. Interest accumulation then follows from the time between that cash flow and the valuation date. This cohort method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, use the stated annual interest rate and the same timing convention: premiums are paid at the beginning of each year by policies then in force, and death claims are paid at year-end.
- A one-year endowment cohort starts with 30 policies. There are 3 deaths during the year. The premium is $80, the death benefit is $400, and \(i=5\%\). Find the reserve at time 1 per survivor, before any maturity payment. (Answer: $48.89.)
- A two-year endowment starts with 40 policies. There are 4 deaths in year 1 and 3 in year 2. The annual premium is $120, death benefits are $600, and \(i=4\%\). Find the time-2 reserve per survivor before maturity. (Answer: $163.29.)
- A two-year endowment starts with 20 policies; 2 die in year 1 and 3 die in year 2. The annual premium is $100, death benefits are $300, \(i=5\%\), and the maturity benefit is $500. Find the reserve per survivor before maturity and the post-maturity balance per survivor. (Answer: $171.00 before maturity; a shortfall of $329.00 per survivor after the maturity payment.)
- A two-year endowment starts with 25 policies; 5 die in year 1 and 4 die in year 2. The annual premium is $150, the year-1 death benefit is $400, the year-2 death benefit is $500, \(i=4\%\), and the maturity benefit is $800. Find the reserve per survivor immediately before maturity and the post-payment balance per survivor. (Answer: $193.50 before maturity; a shortfall of $606.50 per survivor after the maturity payment.)