Common Errors in Retrospective Reserve Calculations

Why Retrospective Reserve Errors Occur

A retrospective reserve is built from the policy cash flows that have already occurred. That sounds direct, but a small timing or unit error can materially change the result. A premium paid at the start of a year earns interest for that year; a claim paid at year-end does not. A cohort fund belongs to all policies in a group, while the reserve usually requested is an amount per policy in force at the valuation date.

The reliable approach is to identify the valuation instant, list each premium and claim with its payment date and number of policies involved, accumulate every amount to that instant, and only then convert the cohort fund to a per-survivor amount. This tutorial focuses on the errors that most often disrupt that sequence. Its methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Consider a closed cohort with $n_0$ policies initially in force. Let $n_j$ be the number in force at time $j$, and let $D_j=n_{j-1}-n_j$ be the number that leave through death during year $j$. If the cohort fund at time $t$ is $F_t$, the retrospective reserve per survivor at time $t$ is $F_t/n_t$, provided $n_t>0$. The fund and reserve are different quantities: $F_t$ is an aggregate amount, while $F_t/n_t$ is an amount per policy then in force.

A Cash-Flow Formula That Makes Timing Visible

Suppose the effective annual interest rate is $i$. In year $j$, a premium of $\pi_j$ is paid at the beginning of the year by each policy then in force, and a benefit of $B_j$ is paid at the end of the year for each death during that year. Assume there are no expenses, lapses, or other cash flows. The fund immediately after the year-$t$ claims is

F_t=\sum_{j=1}^{t}\pi_j n_{j-1}(1+i)^{t-j+1} -\sum_{j=1}^{t}B_jD_j(1+i)^{t-j}.

The exponents follow from the payment dates. A beginning-of-year premium in year $j$ is paid at time $j-1$, so it earns interest for $t-(j-1)=t-j+1$ years. A year-$j$ claim is paid at time $j$, so it earns interest for $t-j$ years before the valuation date. In particular, the final year’s claims have exponent zero.

Why the formula holds: A cash flow of amount $C$ paid at time $s$ has value $C(1+i)^{t-s}$ at time $t$, because it is accumulated for $t-s$ years. Add the accumulated premiums, subtract the accumulated claims, and use the number of policies exposed to each cash flow. The resulting $F_t$ is an aggregate cohort amount. Dividing by $n_t$ allocates that fund across the policies that survived to time $t$.

If the calculation starts with $n_0$ policies and uses life-table survival probabilities, then $n_t/n_0={}_tp_x$ when the cohort counts are consistent with the life table. If $F_t/n_0$ is the fund per original entrant, the reserve per survivor is

\frac{F_t/n_0}{{}_tp_x}=\frac{F_t}{n_0\,{}_tp_x}=\frac{F_t}{n_t}.

This conversion is a common source of error: dividing by $n_0$ gives an amount per original entrant, not an amount per survivor at time $t$.

Worked Calculations and Their Error Checks

Worked Example: Using the Survivor Denominator

A hypothetical two-year term policy cohort has $n_0=600$, $n_1=570$, and $n_2=540$. Each policy in force at the start of a year pays a premium of $120 then; each death generates a $400 claim at that year’s end. The effective annual interest rate is $i=0.05$. Assume the stated cohort counts are the mortality experience for this calculation, with no other decrements or cash flows. Find the reserve per survivor immediately after the time-2 claims.

Method. First calculate the cohort fund at time 2. There are $600-570=30$ deaths in year 1 and $570-540=30$ in year 2. The premiums paid at times 0 and 1 are accumulated for two and one years, respectively:

120(600)(1.05)^2=79{,}380,\qquad 120(570)(1.05)=71{,}820.

Thus, accumulated premiums total $79{,}380+71{,}820=151{,}200$. The year-1 claims are paid at time 1 and accumulate one year; the year-2 claims are paid at the valuation date:

400(30)(1.05)=12{,}600,\qquad 400(30)=12{,}000.

Accumulated claims total $12{,}600+12{,}000=24{,}600$, so the aggregate fund is $151{,}200-24{,}600=126{,}600$. There are 540 survivors at time 2. Therefore,

\frac{126{,}600}{540}=\$234.44.

The reserve is $234.44 per survivor, rounded to the nearest cent. As a check, dividing by the original 600 policies would give $126{,}600/600=\$211.00$ per original entrant, which is not the requested per-survivor reserve. The common mistake avoided here is using the cohort’s starting size as the denominator.

Worked Example: Converting a Per-Entrant Fund to a Per-Survivor Reserve

A hypothetical cohort begins with 1,000 policies. There are 950 policies in force at time 1 and 900 at time 2. Each policy in force at the start of either year pays a $100 premium, and each death produces a $600 claim at year-end. The effective annual interest rate is $i=0.05$. Treat the cohort counts as fixed mortality experience; assume no other cash flows. Find the time-2 retrospective reserve per survivor, and show how the survival-probability denominator gives the same result.

The year-1 and year-2 death counts are $1{,}000-950=50$ and $950-900=50$. Accumulated premiums are

100(1{,}000)(1.05)^2=110{,}250,\qquad 100(950)(1.05)=99{,}750.

Their total is $210{,}000$. Accumulated claims are $600(50)(1.05)=31{,}500$ for year 1 and $600(50)=30{,}000$ for year 2, totalling $61{,}500$. Hence the aggregate fund at time 2 is

F_2=210{,}000-61{,}500=148{,}500.

The survivor count is 900, so the reserve per survivor is $148{,}500/900=\$165.00$. Equivalently, the fund per original entrant is $148{,}500/1{,}000=\$148.50$, and the two-year survival proportion is $900/1{,}000=0.90$. Thus, the per-survivor value is $148.50/0.90=\$165.00$. The common mistake avoided is treating the fund per original entrant as if it were already a reserve per survivor.

This calculation uses the observed cohort proportions: one-year survival is $950/1{,}000=0.95$ in year 1 and $900/950$ in year 2. The cumulative survival proportion is their product, $0.95(900/950)=0.90$, consistent with the cohort counts.

Worked Example: Including Past Claims and Dating Premiums Correctly

A hypothetical two-year policy cohort has $n_0=500$, $n_1=475$, and $n_2=440$. A $150 premium is collected at the beginning of each year from each policy in force then, and a $600 death claim is paid at the end of the year of death. The effective annual interest rate is $i=0.03$. Assume the listed counts describe all mortality, with no other decrements or cash flows. Find the reserve per survivor immediately after the time-2 claims.

There are $500-475=25$ deaths in year 1 and $475-440=35$ deaths in year 2. The time-0 premiums earn interest for two years, while time-1 premiums earn interest for one year:

150(500)(1.03)^2=79{,}567.50,\qquad 150(475)(1.03)=73{,}387.50.

Accumulated premiums total $152{,}955.00$. The year-1 claims earn interest from time 1 to time 2; the year-2 claims are paid at time 2:

600(25)(1.03)=15{,}450,\qquad 600(35)=21{,}000.

The accumulated claims total $36{,}450$, giving an aggregate fund of $152{,}955-36{,}450=116{,}505$. Dividing by the 440 survivors gives $116{,}505/440=\$264.7840\ldots$, or $264.78 per survivor to the nearest cent.

A tempting but incorrect calculation omits the year-1 claims, leaving $152{,}955-21{,}000=131{,}955$ and $131{,}955/440=\$299.90$ per survivor. That is too high because the first-year deaths generated claims already paid into the past. Another timing check is that the year-2 premiums are not accumulated for two years: they were paid at time 1, so they earn one year of interest.

A Practical Audit for Common Errors

Before accepting a retrospective reserve, audit both the cash-flow timeline and the units. The following sequence makes the most frequent errors visible without relying on a memorized formula alone.

Fix the valuation instant.
State whether the reserve is immediately before or after any premium or claim at time $t$. A cash flow at the valuation instant is either included or excluded according to that convention.

Count each cash flow’s policies.
Use $n_{j-1}$ for a beginning-of-year premium under the stated timing, and $D_j=n_{j-1}-n_j$ for year-$j$ death claims.

Accumulate by payment date.
For a payment at time $s$, use $(1+i)^{t-s}$. Do not assign every cash flow the same accumulation period.

Subtract all past outflows.
Include claims paid before the valuation date, not only claims in the final year. Apply the same sign convention to every year.

Convert to the requested unit.
Divide the aggregate fund by $n_t$ for a per-survivor reserve. If starting from a per-original-entrant amount, divide by the cumulative survival proportion ${}_tp_x$.

A useful final check is dimensional: premiums times policies and claims times deaths are aggregate dollars, so their accumulated difference is also aggregate dollars. Only after dividing by a survivor count does the result become dollars per survivor. Also check the direction of likely errors: omitting claims generally overstates the fund, while dividing by too many lives generally understates the per-policy reserve.

Key takeaway: A correct retrospective reserve requires three consistent choices: the cash-flow dates, the cohort counts attached to each cash flow, and the population represented by the final denominator. If any one of these is inconsistent, the arithmetic may be flawless while the reserve is wrong.

Check Your Understanding

For each problem, use a cohort fund first, then divide by the number of policies in force at the valuation time. Assume no expenses or decrements other than those stated.

A two-year cohort has $n_0=400$, $n_1=380$, and $n_2=350$. A $50 premium is paid at the beginning of each year per policy then in force; a $200 claim is paid at year-end per death. At $i=0.04$, find the reserve per survivor immediately after time-2 claims. (Answer: $89.23 per survivor)
A cohort has $n_0=800$, $n_1=760$, and $n_2=720$. Annual beginning-of-year premiums are $75 per policy in force and year-end death claims are $300 per death. At $i=0.05$, calculate the time-2 reserve per survivor. (Answer: $140.83 per survivor)
A cohort starts with 1,000 policies; conditional survival is $0.96$ in year 1 and $0.95$ in year 2, so the counts are $n_1=960$ and $n_2=912$. Each beginning-of-year premium is $40, each year-end death claim is $250, and $i=0.04$. What is the time-2 reserve per survivor? (Answer: $66.67 per survivor)
A cohort has $n_0=300$, $n_1=285$, and $n_2=270$. Each year’s beginning premium is $120 per policy in force and each year-end death claim is $500. At $i=0.03$, find the time-2 reserve per survivor, including both years’ claims. (Answer: $215.53 per survivor)
A three-year cohort has $n_0=200$, $n_1=190$, $n_2=170$, and $n_3=160$. Each year, a $20 premium is paid at the beginning of the year per policy then in force, and a $100 claim is paid at year-end per death. With $i=0.05$, find the time-3 reserve per survivor after the final claims. (Answer: $51.17 per survivor)

What You'll Learn