A Cohort Cash-Flow Table for Reserve Calculations

Why Use a Cohort Cash-Flow Table?

A retrospective reserve calculation combines several items that occur at different times: premiums are received, death claims are paid, interest accumulates, and the number of policies in force changes. A cohort cash-flow table puts those items on the same timeline. It helps ensure that each cash flow receives the correct accumulation factor and that the resulting fund is allocated to the policies still in force at the valuation date.

The method here concerns a discrete annual model. Premiums are paid at the beginning of each policy year by policies then in force; death benefits are paid at the end of the year of death. Interest is earned at an effective annual rate $i$, and there are no expenses, lapses, or other cash flows unless they are explicitly included. The cohort counts may be supplied directly or derived from life-table counts.

Definition: A cohort cash-flow table records, for each policy year, the number of policies in force at its start, the number of deaths during the year, premiums received at the start, death benefits paid at the end, and each cash flow accumulated to a common valuation date. For a cohort that begins with $l_x$ policies, the retrospective reserve per surviving policy at duration $n$ is the accumulated net fund at time $n$, divided by the number $l_{x+n}$ surviving then.

Use a consistent time line. If year $j$ runs from time $j-1$ to time $j$, the beginning-of-year premium is received at time $j-1$, while death benefits are paid at time $j$. Thus, when the valuation date is time $n$, that year's premium accumulates for $n-j+1$ years, but that year's death benefit accumulates for only $n-j$ years. An end-of-year cash flow in the final year is already at the valuation date and receives no interest accumulation.

Building the Table and Formula

Let $l_{x+j-1}$ denote the number of policies in force at the start of year $j$, and let $d_{x+j-1}$ denote the number of deaths during that year. These counts satisfy $l_{x+j}=l_{x+j-1}-d_{x+j-1}$. If the premium per in-force policy in year $j$ is $P_j$, and the benefit per death is $b_j$, aggregate premiums and claims for that year are $l_{x+j-1}P_j$ and $d_{x+j-1}b_j$, respectively.

\begin{array}{c|c|c|c|c} \text{Year }j & \text{In force at start} & \text{Deaths} & \text{Premium at }j-1 & \text{Death claims at }j\\ \hline j & l_{x+j-1} & d_{x+j-1} & l_{x+j-1}P_j & d_{x+j-1}b_j \end{array}

To fill the accumulated-value columns, multiply each beginning-of-year premium by $(1+i)^{n-j+1}$ and each end-of-year claim by $(1+i)^{n-j}$. The aggregate accumulated fund is the accumulated premiums less the accumulated claims.

{}_nV_x = \frac{ \displaystyle\sum_{j=1}^{n} l_{x+j-1}P_j(1+i)^{n-j+1} - \displaystyle\sum_{j=1}^{n} d_{x+j-1}b_j(1+i)^{n-j} }{ l_{x+n} }

This expression uses the convention that the reserve is measured at time $n$, after claims due at that time have been paid and before any later cash flow. In particular, it divides by the cohort count remaining after year $n$'s deaths, not by the count at the beginning of that year. If a problem uses a different premium timing, benefit timing, or valuation instant, the table must reflect that timing rather than applying these exponents mechanically.

Why the Accumulation Method Works

The table is an accounting of cash flows at a common date. A dollar received at time $t$ accumulates to time $n$ as $(1+i)^{n-t}$ dollars under a constant effective annual interest rate. Therefore, each premium contributes its accumulated value to the fund, and each claim removes its accumulated value. The balance at time $n$ is the accumulated net amount attributable to the cohort.

If that balance is shared equally among the $l_{x+n}$ policies still in force, the amount per survivor is the retrospective reserve. For the annual timing above, premiums occur at $t=j-1$ and claims at $t=j$; substituting those times into the accumulation factor gives exactly the two exponents in the formula. This reasoning also explains why changing payment timing changes the factor, even if the cash amount itself is unchanged.

Track the cohort.
For each year, record the number in force at the start, deaths during the year, and survivors at its end.

Record aggregate cash flows.
Multiply the applicable per-policy premium by policies paying it; multiply the benefit by deaths receiving it.

Move every amount to the valuation date.
Use the time between the payment date and valuation date, not merely the year number.

Net and allocate.
Subtract accumulated claims from accumulated premiums, then divide by the number surviving at the valuation date.

Worked Examples

Worked Example: Level Premium and Benefit Over Two Years

A hypothetical cohort begins with 100 policies. In year 1, 12 policies die; in year 2, another 8 die. Each in-force policy pays a premium of $150 at the beginning of each year, and each death generates a $1,000 benefit at year-end. The effective annual interest rate is $i=0.04$. Calculate the retrospective reserve per survivor at time 2, after year 2 claims have been paid.

Method. Build one row per year, accumulate each row's premium and claim to time 2, and divide the net fund by the 80 policies surviving. This avoids giving a year-end claim an extra year of interest.

Year	In force	Deaths	Premium received	Premium accumulated to time 2	Claims paid	Claims accumulated to time 2
1	100	12	$15,000	$15,000(1.04)² = $16,224	$12,000	$12,000(1.04) = $12,480
2	88	8	$13,200	$13,200(1.04) = $13,728	$8,000	$8,000

The year 2 premium is $88\times \$150=\$13{,}200$, since 88 policies are in force at its beginning. The year 2 claim total is $8\times \$1{,}000=\$8{,}000$. The accumulated premiums are $\$16{,}224+\$13{,}728=\$29{,}952$. The accumulated claims are $\$12{,}480+\$8{,}000=\$20{,}480$. Thus, the fund at time 2 is $\$29{,}952-\$20{,}480=\$9{,}472$.

There are $100-12-8=80$ survivors. The reserve per survivor is $\$9{,}472/80=\$118.40$. As a check, the end-of-year claims have been subtracted and the final year's premium has been accumulated for exactly one year; the year 2 claim has not been accumulated at all.

Worked Example: Varying Premiums and Benefits

A hypothetical cohort has 120 policies at time 0. During the next three years, the death counts are 18, 12, and 15, respectively. Beginning-of-year premiums per in-force policy are $300, $350, and $400. The death benefits per claim are $1,200, $1,500, and $1,800. Assume $i=0.05$, and value the reserve at time 3 after year 3 claims.

Method. Compute in-force counts successively, then use the year-specific premium and benefit amounts. The key timing check is that the first premium compounds for three years, whereas the first-year claim compounds for two.

Year	In force	Deaths	Aggregate premium and accumulated value	Aggregate claim and accumulated value
1	120	18	$36,000; $36,000(1.05)³ = $41,674.50	$21,600; $21,600(1.05)² = $23,814
2	102	12	$35,700; $35,700(1.05)² = $39,359.25	$18,000; $18,000(1.05) = $18,900
3	90	15	$36,000; $36,000(1.05) = $37,800	$27,000; $27,000 = $27,000

The in-force sequence is $120$, $120-18=102$, $102-12=90$, and $90-15=75$. For example, the year 2 premium total is $102\times \$350=\$35{,}700$, and its claim total is $12\times \$1{,}500=\$18{,}000$. Summing the accumulated premiums gives $\$41{,}674.50+\$39{,}359.25+\$37{,}800=\$118{,}833.75$. Summing accumulated claims gives $\$23{,}814+\$18{,}900+\$27{,}000=\$69{,}714$.

The net fund is $\$118{,}833.75-\$69{,}714=\$49{,}119.75$. Dividing by 75 survivors gives $\$49{,}119.75/75=\$654.93$ per survivor. The check $120-18-12-15=75$ confirms the denominator, and the year 3 claim is at the valuation date, so its accumulated value remains $27,000.

Worked Example: A Table with Changing Cash Flows and Life Counts

A hypothetical group begins with 150 policies. There are 15 deaths in each of three years, leaving 105 policies at time 3. Premiums are paid at the start of each year: $200 per policy in year 1, $220 in year 2, and $240 in year 3. Death benefits paid at year-end are $1,000, $1,100, and $1,200 per death by year. At $i=0.03$, calculate the reserve per survivor at time 3, after the last claims.

Method. Pair each premium with its beginning-of-year in-force count and each benefit with that year's deaths. Accumulate both columns to time 3 before taking their difference.

Year	In force	Deaths	Premium received	Accumulated premium	Claims paid	Accumulated claims
1	150	15	$30,000	$30,000(1.03)³ = $32,781.81	$15,000	$15,000(1.03)² = $15,913.50
2	135	15	$29,700	$29,700(1.03)² = $31,508.73	$16,500	$16,500(1.03) = $16,995
3	120	15	$28,800	$28,800(1.03) = $29,664	$18,000	$18,000

The premium totals follow from $150\times \$200=\$30{,}000$, $135\times \$220=\$29{,}700$, and $120\times \$240=\$28{,}800$. The death claims are $15\times \$1{,}000=\$15{,}000$, $15\times \$1{,}100=\$16{,}500$, and $15\times \$1{,}200=\$18{,}000$. The accumulated premium total is $\$32{,}781.81+\$31{,}508.73+\$29{,}664=\$93{,}954.54$. The accumulated claim total is $\$15{,}913.50+\$16{,}995+\$18{,}000=\$50{,}908.50$.

The net fund is $\$93{,}954.54-\$50{,}908.50=\$43{,}046.04$. There are $150-15-15-15=105$ survivors, so the reserve is $\$43{,}046.04/105=\$409.962285\ldots$, or $409.96 per survivor to the nearest cent. The in-force counts used for premiums are 150, 135, and 120, while the denominator is the 105 remaining after the last year's deaths.

Common Mistakes and Checks

A cash-flow table is most useful when its timing is explicit. Merely listing annual amounts is not enough: the same amount has a different accumulated value if it is paid at the beginning rather than at the end of a year. Likewise, cohort counts belong to specific points in the year and should not be carried forward without subtracting deaths.

Giving a claim an extra year of interest. A claim paid at time $j$ accumulates for $n-j$ years to time $n$, not $n-j+1$.
Using the wrong count for premiums. A beginning-of-year premium is paid by policies in force at that year's start, before that year's deaths.
Dividing by the initial cohort. The per-survivor reserve at time $n$ uses $l_{x+n}$, after deaths through year $n$, as its denominator.
Rounding too early. Retain the unrounded accumulated totals through the division, then round the final dollar amount to cents.

For example, if a $7,500 claim is paid at the end of year 2 and the valuation date is time 2, the incorrect calculation $7{,}500(1.10)$ gives it a year of interest it has not earned. The correct accumulated amount is $\$7{,}500(1.10)^0=\$7{,}500$. The table's payment-date column makes this error visible.

Key takeaway: Keep counts, payment dates, and accumulated values in separate, clearly labelled columns. Accumulate every premium and claim from its actual payment time to the same valuation date; only then net the amounts and divide by the policies surviving at that date.

Check Your Understanding

For each problem, use the stated cash-flow timing and report the reserve per survivor to the nearest cent.

A cohort has aggregate premiums of $3,000 at time 0 and $3,000 at time 1. Claims of $1,000 are paid at time 1 and $1,000 at time 2. There are 80 survivors at time 2, and $i=0$. What is the retrospective reserve per survivor at time 2? (Answer: $50.00)
At $i=0.05$, aggregate premiums of $3,200 are received at times 0 and 1. Aggregate claims of $1,600 are paid at times 1 and 2. There are 64 survivors at time 2. Calculate the reserve per survivor at time 2. (Answer: $56.38)
A cohort starts with 150 policies and has 15 deaths in each of three years, leaving 105 survivors. At the beginning of the three years, aggregate premiums are $9,000, $6,750, and $4,800, respectively. End-of-year aggregate claims are $4,500, $6,000, and $7,500, respectively. If $i=0.04$, calculate the reserve per survivor at time 3. (Answer: $36.28)
At $i=0.10$, aggregate premiums of $7,500 are received at times 0 and 1, and aggregate claims of $7,500 are paid at times 1 and 2. There are 240 survivors at time 2. Calculate the retrospective reserve per survivor at time 2. (Answer: $6.56)

What You'll Learn