The Cohort Definition of a Retrospective Reserve

What a Cohort Retrospective Reserve Measures

A reserve at a valuation date can be described from two directions. A prospective reserve looks forward from that date. A retrospective reserve looks back: it starts with the premiums received from an original group of policies, allows those premiums to accumulate with interest, and subtracts benefits paid to date, also brought to the valuation date. The resulting amount belongs to the cohort as a whole. To express it as an amount per contract still in force, divide by the number of surviving contracts.

That final division is essential. The cohort starts with one number of contracts, but some contracts may have ended through death or another covered exit. Past premiums from the original group helped support the cohort’s accumulated balance, while benefits were paid for contracts that left. The retrospective reserve per survivor allocates the remaining cohort balance across the contracts still in force.

Definition: For a homogeneous original cohort, the cohort retrospective reserve at duration $k$ is the cohort’s accumulated value at time $k$ of past premiums less past benefits. The retrospective reserve per surviving contract is that accumulated cohort balance divided by the number of contracts from the original cohort that are in force at time $k$. In the notation for a policy issued at age $x$, the per-policy reserve at duration $k$ is denoted ${}_kV_x$.

The accumulated cohort balance is a signed amount. It accounts for cash flows that have already occurred, with each cash flow measured at the valuation date. A premium received before the valuation date contributes its accumulated value at that date; a benefit paid before the valuation date reduces the balance by its accumulated value at that date. If a cash flow occurs exactly at the valuation date, it is included or excluded according to the stated valuation timing.

This definition requires a clear cohort and a clear valuation time. The original cohort is the group of contracts being followed from issue or another specified starting point. The surviving count includes only contracts still in force immediately after the events specified to have occurred by the valuation date.

From Cohort Balance to Amount per Survivor

Let $N_k$ be the number of contracts from the original cohort in force at duration $k$, and let $A_k$ be the cohort’s accumulated net balance at that date. If all surviving contracts are treated as having the same retrospective reserve $R_k$, the aggregate reserve assigned to those contracts is $N_kR_k$. Equating this amount to the cohort balance gives the defining relationship.

N_k R_k=A_k \qquad\Longrightarrow\qquad R_k=\frac{A_k}{N_k}, \quad N_k>0.

This also explains why dividing by the original cohort size is generally wrong: the requested per-policy amount is assigned to the contracts still in force, not to contracts that have already left the cohort. Dividing by the original size would instead report an amount per original contract, including those no longer in force.

Fix the cohort and valuation date.
Record how many contracts began in the cohort and how many remain in force at the valuation date.

Accumulate past cash flows to that date.
Apply the stated interest assumption to premiums and benefits according to when each was received or paid.

Find the net cohort balance.
Subtract accumulated benefits and other specified outflows from accumulated premiums and other specified inflows.

Allocate across survivors.
Divide the net cohort balance by the number of contracts in force at that time.

The arithmetic is meaningful only after the cash-flow timing and the survivor count are established. For example, if premiums are paid at the start of a year and benefits at year-end, the premiums earn interest for that year while those year-end benefits do not earn interest before the valuation at the year-end. If both are stated to occur at the valuation date, the problem must specify the order or convention.

Worked Cohort Calculations

Worked Example: A One-Year Cohort with No Interest

A hypothetical cohort has 200 identical policies. Each policy pays a premium of $50 at issue. At the end of the year, 10 death benefits of $500 each have been paid, and 190 contracts remain in force. Assume an effective annual interest rate of $i=0\%$. Find the one-year retrospective reserve per survivor.

Method. Calculate the cohort’s balance at the year-end, then divide by the year-end survivor count. Because $i=0\%$, the issue premiums have no accumulated interest.

The total premiums received at issue are $200(50)=\$10{,}000$. The total benefits paid at year-end are $10(500)=\$5{,}000$. With no interest, these amounts at the year-end are unchanged, so the accumulated cohort balance is $\$10{,}000-\$5{,}000=\$5{,}000$.

R_1=\frac{\$5{,}000}{190} =\$26.315789\ldots

Rounded to the nearest cent, the retrospective reserve is $26.32 per survivor. The check is that $190(\$26.315789\ldots)=\$5{,}000$, the cohort balance before rounding. The common mistake is to report zero because the total benefits are less than, but close to, the total premiums; the correct calculation allocates the remaining $5,000 over 190 survivors.

Worked Example: Premiums at the Start and Benefits at the End

A hypothetical group of 50 contracts pays a premium of $120 per contract at the start of the year. Five benefits of $800 each are paid at the end of the year, leaving 45 contracts in force. Assume an effective annual interest rate of $i=5\%$. Find the reserve per surviving contract at year-end.

Method. Accumulate the premiums for one year. The benefits are paid at the year-end valuation time, so they are subtracted at their paid amount.

The premiums received at the start of the year total $50(120)=\$6{,}000$. Their accumulated value after one year is $\$6{,}000(1.05)=\$6{,}300$. The benefits paid at year-end total $5(800)=\$4{,}000$. Thus the accumulated cohort balance at year-end is $\$6{,}300-\$4{,}000=\$2{,}300$.

R_1=\frac{\$2{,}300}{45} =\$51.111111\ldots

Rounded to the nearest cent, the reserve is $51.11 per survivor. Before rounding, $45(\$51.111111\ldots)=\$2{,}300$, which checks the allocation. The common mistake is to omit the year’s interest on the beginning-of-year premiums; the premium total at the valuation date is $6,300, not $6,000.

Worked Example: Following Survivors Through Two Years

A hypothetical cohort begins year 1 with 80 contracts. Each contract in force at the start of a year pays a $100 premium at that year’s start. A benefit of $500 is paid at the end of each year for each death during that year. There are 8 deaths in year 1 and 4 deaths in year 2. Assume $i=0\%$ and no other cash flows. Find the retrospective reserve per survivor at the end of year 2.

Method. Track the accumulated cohort balance and the in-force count year by year. With zero interest, the balance carries forward without change except for the next premiums and benefits.

At the start of year 1, premiums total $80(100)=\$8{,}000$. Year 1 benefits total $8(500)=\$4{,}000$. The balance at the end of year 1 is $\$8{,}000-\$4{,}000=\$4{,}000$. The number in force after the 8 deaths is $80-8=72$.

At the start of year 2, the 72 in-force contracts pay premiums totaling $72(100)=\$7{,}200$. Year 2 benefits total $4(500)=\$2{,}000$. The end-of-year-2 balance is therefore $\$4{,}000+\$7{,}200-\$2{,}000=\$9{,}200$. There are $72-4=68$ contracts in force at the valuation date.

R_2=\frac{\$9{,}200}{68} =\$135.294117\ldots

Rounded to the nearest cent, the reserve is $135.29 per survivor. As a check, the cohort balance is $9,200 and $68(\$135.294117\ldots)=\$9{,}200$. The common mistake is to use only the original 80 contracts when finding year 2 premiums; only the 72 contracts still in force pay them.

Worked Example: Using Life-Table Counts for a One-Year Cohort

A hypothetical one-year policy cohort is described by life-table values $\ell_x=1{,}200$ and $\ell_{x+1}=1{,}140$. Each policy pays a $100 premium at issue and a $1,500 death benefit at the end of the year if death occurs during the year. Assume an effective annual interest rate of $i=2\%$, and assume there are no other exits or cash flows. Find the retrospective reserve per survivor at the end of the year.

Method. Use the life-table difference to find deaths, accumulate the issue premiums for one year, and divide the resulting balance by the end-of-year in-force count.

The number of deaths during the year is $\ell_x-\ell_{x+1}=1{,}200-1{,}140=60$. The total premiums at issue are $1{,}200(100)=\$120{,}000$, and their year-end accumulated value is $\$120{,}000(1.02)=\$122{,}400$. The benefits paid at year-end total $60(1{,}500)=\$90{,}000$. The accumulated cohort balance is $\$122{,}400-\$90{,}000=\$32{,}400$, and the survivor count is $\ell_{x+1}=1{,}140$.

R_1=\frac{\$32{,}400}{1{,}140} =\$28.421052\ldots

Rounded to the nearest cent, the reserve is $28.42 per survivor. The check is $1{,}140(\$28.421052\ldots)=\$32{,}400$. The common mistake is to subtract the 60 benefits from the undiscounted issue premiums: the premiums have accumulated to $122,400 by the valuation date.

Assumptions, Interpretation, and Common Traps

A single cohort reserve per survivor is easiest to interpret when contracts are homogeneous: they have the same benefit and premium terms, and the same valuation basis is used for all of them. The cohort calculation also needs a defined treatment of exits and expenses. If a contract lapses without a payment, it leaves the in-force count; if a surrender payment is made, that payment is an outflow in the cohort balance. Expenses or other cash flows should be included when the reserve definition or problem specifies them.

Life-table values can provide the cohort counts when the cohort is modeled using that table. In a one-year interval, $\ell_x$ is the number alive at age $x$, $\ell_{x+1}$ is the number alive one year later, and $\ell_x-\ell_{x+1}$ is the number of deaths in that interval under the table. In an actual-experience question, use the actual number of contracts and claims stated instead; do not substitute an expected life-table count unless the problem asks for an expected or model-based calculation.

Wrong denominator: dividing by the original cohort size reports an amount per original contract. For a reserve per surviving contract, use the in-force count at the valuation date.
Wrong timing: a premium received at the beginning of a period accumulates over that period; a benefit paid at the end of the period is already at the year-end value.
Wrong cohort count: contracts that have left do not pay later premiums, and benefits paid for deaths must be counted as outflows.
Wrong interpretation: the cohort balance is an accumulated net cash-flow measure allocated across survivors. It is not necessarily a separate investment account physically held for each policy.

If the accumulated net balance is negative, the signed retrospective amount per survivor is negative under this definition. That result signals that accumulated past outflows exceed accumulated past inflows on the stated basis; it does not by itself describe a benefit payable to each policyholder. If surviving contracts have materially different terms or histories, a single pooled amount may not represent an appropriate reserve for each contract, and the contracts may need to be analyzed in separate homogeneous groups.

Key takeaway: First determine the accumulated balance for the original cohort at the valuation date. Then divide by the number of that cohort’s contracts still in force. The cohort balance and the per-survivor reserve are related, but they are not the same quantity.

Check Your Understanding

For each problem, identify the surviving cohort count and account for cash-flow timing before calculating the reserve.

A hypothetical cohort has 100 contracts. Each pays $80 at issue. Four benefits of $1,000 are paid at year-end, leaving 96 contracts in force. Assume $i=0\%$. Find the one-year reserve per survivor. (Answer: $41.67)
A hypothetical cohort of 60 contracts pays $150 per contract at the start of the year. Six benefits of $1,200 are paid at year-end, leaving 54 contracts in force. Assume $i=4\%$. Find the year-end reserve per survivor. (Answer: $40.00)
A cohort starts year 1 with 100 contracts. In-force contracts pay $40 at the start of each year; benefits of $300 per death are paid at each year-end. There are 10 deaths in year 1 and 5 in year 2. Assume $i=0\%$. Find the reserve per survivor at the end of year 2. (Answer: $36.47)
A hypothetical cohort has $\ell_x=500$ and $\ell_{x+1}=475$. Each contract pays $200 at issue and a $3,000 benefit at year-end if death occurs during the year. Assume $i=3\%$. Find the one-year reserve per survivor. (Answer: $58.95)
A hypothetical cohort begins with 40 contracts. Each in-force contract pays $250 at the start of each year. Two benefits of $1,000 are paid at the end of year 1, and one benefit of $1,000 is paid at the end of year 2. Assume $i=5\%$, and that 38 contracts are in force at the start of year 2. Find the reserve per survivor at the end of year 2. (Answer: $483.78)

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