Worked Term Insurance Reserve From Cohort Data

Why Cohort Data Matter for a Term-Insurance Reserve

A retrospective reserve looks backward: it compares past premiums with past benefits after accumulating those cash flows with interest. When a problem gives life-table counts, the calculation first produces an aggregate balance for the original cohort. To obtain the balance per policy still in force, divide by the number of survivors at the valuation date—not by the original cohort size.

This tutorial uses a discrete annual model. A level premium is paid at the beginning of each policy year by each person then alive. A death benefit is paid at the end of the year of death. The annual effective interest rate is $i$, and all policyholders in the cohort have the same premium and benefit. These timing assumptions matter: a different premium or benefit timing changes the accumulation factors.

Definition: Let $\ell_{x+j}$ be the number of members of an initial cohort alive at duration $j$, and let $d_j=\ell_{x+j}-\ell_{x+j+1}$ be the number dying during policy year $j+1$. For a term insurance with annual premium $P$, death benefit $B$, zero initial reserve, and annual effective interest rate $i$, the retrospective reserve at duration $k$, immediately after claims at time $k$ and before any premium due at that time, is $$ {}_kV_x= \frac{ \displaystyle\sum_{j=0}^{k-1}\ell_{x+j}P(1+i)^{k-j} - \displaystyle\sum_{j=0}^{k-1}d_jB(1+i)^{k-j-1} }{\ell_{x+k}}. $$ The numerator is the accumulated balance of the original cohort at time $k$; the denominator converts that aggregate balance to an amount per survivor. This form assumes level premiums and benefits, no expenses, and no initial fund. If there is an initial per-policy balance $V_0$, add $\ell_xV_0(1+i)^k$ to the numerator.

Building the Cohort Balance

In the formula, $\ell_{x+j}P$ is the total premium income collected at time $j$, because only those alive at the start of the year pay. That income accumulates for $k-j$ years. In contrast, $d_jB$ is paid at the end of year $j+1$, so it accumulates for $k-j-1$ years. The different exponents reflect the different payment dates.

For a term contract of length $n$, this calculation applies at durations $0\leq k\leq n$. For a valuation during the term, include only premiums and death claims paid by that duration. At $k=n$, the calculation includes the final year’s claims and premiums, but not a premium due at time $n$.

Find the deaths in each year.
Subtract adjacent life-table counts: $d_j=\ell_{x+j}-\ell_{x+j+1}$.

Accumulate premiums and claims separately.
Use the time each cash flow occurs to select its accumulation factor to duration $k$.

Subtract claims from premiums.
The result is the accumulated balance for the original cohort.

Divide by survivors.
Divide the aggregate balance by $\ell_{x+k}$ to obtain the retrospective balance per policy in force.

Why the One-Year Recursion Gives the Same Balance

The direct cohort formula is useful when all past counts and cash flows are available. A one-year recursion gives an independent check and can be more convenient when balances are calculated successively.

Suppose ${}_kV_x$ is the retrospective balance per survivor immediately after the claims at time $k$, before the premium due at time $k$. In the next year, each of the $\ell_{x+k}$ survivors contributes the existing balance and the new premium. That amount earns interest for one year; the year’s death claims are then paid. The remaining aggregate balance is shared among $\ell_{x+k+1}$ survivors.

Derivation: At time $k$, the aggregate balance is $\ell_{x+k}{}_kV_x$. Premiums collected at that time add $\ell_{x+k}P$. After one year of interest and payment of $d_kB$ in claims, the aggregate balance is $$ \ell_{x+k+1}{}_{k+1}V_x = \bigl(\ell_{x+k}{}_kV_x+\ell_{x+k}P\bigr)(1+i)-d_kB. $$ Dividing by $\ell_{x+k+1}$ gives $$ {}_{k+1}V_x = \frac{\ell_{x+k}\bigl({}_kV_x+P\bigr)(1+i)-d_kB}{\ell_{x+k+1}}. $$ This holds under the stated timing assumptions. It is the same cohort balance updated one year at a time: starting balances and premiums earn interest, claims are paid at year-end, and the resulting balance is allocated to survivors.

Worked Term-Insurance Calculations

Worked Example: Accumulating Two Years of Cohort Cash Flows

A hypothetical three-year term policy has a level annual premium of $\$6{,}000$ and a death benefit of $\$100{,}000$, paid at the end of the year of death. The annual effective interest rate is $5\%$. The cohort counts are $\ell_x=1{,}000$, $\ell_{x+1}=960$, $\ell_{x+2}=910$, and $\ell_{x+3}=850$. Find the retrospective reserve at duration 2, immediately after the second year’s claims. Use direct cohort accumulation and check by recursion.

Method: direct cohort accumulation. The numbers dying in the first two years are $$ d_0=1{,}000-960=40,\qquad d_1=960-910=50. $$ The first year’s premiums are collected at time 0 and accumulate for two years: $$ 1{,}000(6{,}000)(1.05)^2 =6{,}000{,}000(1.1025) =\$6{,}615{,}000. $$ The second year’s premiums are collected at time 1 and accumulate for one year: $$ 960(6{,}000)(1.05) =5{,}760{,}000(1.05) =\$6{,}048{,}000. $$ Total accumulated premiums are $\$6{,}615{,}000+\$6{,}048{,}000=\$12{,}663{,}000$.

The first year’s claims are paid at time 1 and accumulate for one year: $$ 40(100{,}000)(1.05)=\$4{,}200{,}000. $$ The second year’s claims are paid at time 2, so they do not accumulate further: $$ 50(100{,}000)=\$5{,}000{,}000. $$ Total accumulated claims are $\$4{,}200{,}000+\$5{,}000{,}000=\$9{,}200{,}000$. Thus the cohort balance is $\$12{,}663{,}000-\$9{,}200{,}000=\$3{,}463{,}000$. There are 910 survivors, so $$ {}_2V_x=\frac{\$3{,}463{,}000}{910}=\$3{,}805.4945\ldots $$ Rounded to the nearest cent, the reserve is $\$3{,}805.49$ per survivor.

Independent recursion check. At duration 1, the aggregate balance is \[ 1{,}000(6{,}000)(1.05)-40(100{,}000)=\$2{,}300{,}000, \] so ${}_1V_x=\$2{,}300{,}000/960=\$2{,}395.8333\ldots$. For the next year, the amount per person before interest is $\$2{,}395.8333\ldots+\$6{,}000=\$8{,}395.8333\ldots$. The aggregate amount after interest is \[ 960(\$8{,}395.8333\ldots)(1.05)=\$8{,}463{,}000. \] Subtract the $\$5{,}000{,}000$ second-year claims to get $\$3{,}463{,}000$; dividing by 910 gives $\$3{,}805.49$, as before. The check confirms that premiums are counted from survivors and that claims are deducted at the correct time.

Worked Example: Correctly Dividing by the Surviving Cohort

A hypothetical two-year term policy has a premium of exactly $\$150{,}000/540$ per policy per year and a death benefit of $\$500$, payable at each year-end death. The annual effective interest rate is $0\%$. Of 300 original policyholders, 240 survive the first year. Find the retrospective reserve at duration 1.

Method: one-year cohort balance. There are $300-240=60$ first-year deaths. Total premiums collected are $$ 300\left(\frac{\$150{,}000}{540}\right)=\$83{,}333.3333\ldots $$ and the claims paid are $60(\$500)=\$30{,}000$. With no interest, the remaining cohort balance is $\$83{,}333.3333\ldots-\$30{,}000=\$53{,}333.3333\ldots$. Divide by the 240 survivors: $$ {}_1V_x=\frac{\$53{,}333.3333\ldots}{240} =\$222.2222\ldots $$ The reserve is $\$222.22$ per survivor, rounded to the nearest cent.

The common mistake is to stop at the cohort balance or divide it by the original 300 lives. Neither gives the balance per in-force policy. The required denominator is 240, giving $\$222.22$; dividing by 300 would give $\$177.78$, which answers a different question.

Worked Example: A Nonzero Retrospective Balance at the End of the Term

A hypothetical two-year term policy has a $\$1{,}000$ end-of-year death benefit, an annual premium of $\$100$, and an annual effective interest rate of $10\%$. The cohort counts are $\ell_x=100$, $\ell_{x+1}=90$, and $\ell_{x+2}=80$. Find the retrospective balance per survivor at duration 2.

Method: accumulate all cash flows to the end of the term. The first-year premium income is $100(\$100)=\$10{,}000$, accumulated for two years: $$ \$10{,}000(1.1)^2=\$12{,}100. $$ The second-year premium income is $90(\$100)=\$9{,}000$, accumulated for one year: $$ \$9{,}000(1.1)=\$9{,}900. $$ Accumulated premiums total $\$22{,}000$. There are 10 deaths in each year. The first year’s claims accumulate to $10(\$1{,}000)(1.1)=\$11{,}000$; the second year’s claims are $10(\$1{,}000)=\$10{,}000$. Accumulated claims total $\$21{,}000$. The balance is $\$1{,}000$, and 80 policyholders survive. Therefore, $$ {}_2V_x=\frac{\$22{,}000-\$21{,}000}{80}=\$12.50. $$

The prospective reserve at the end of the two-year term is zero because no future term benefits or premiums remain. That does not imply that this retrospective balance is zero: the premium in this example was not set to satisfy the equivalence condition. For comparison, the equivalence premium $P$ for these cohort counts satisfies $$ 100P+\frac{90P}{1.1}=\frac{10{,}000}{1.1}+\frac{10{,}000}{(1.1)^2}, \qquad P=\frac{1{,}050}{11}=\$95.4545\ldots. $$ With that premium and a zero initial balance, the present values of premiums and claims agree. Consequently, their accumulated values at time 2 agree as well, and the retrospective balance at the end of the term is zero. An arbitrary premium need not produce that result.

Common Errors and a Practical Check

The main discipline is to keep the cohort total, the valuation date, and the per-survivor amount distinct. A useful check is to calculate the balance both by accumulating all cash flows directly and by applying the one-year recursion. The two methods must agree when they use the same payment timing, counts, and interest rate.

Using the original cohort as the denominator. The numerator is an aggregate balance, but the reserve per policy is obtained by dividing by $\ell_{x+k}$, the number alive at the valuation date.
Using the same accumulation factor for every cash flow. A premium paid at time $j$ accumulates for $k-j$ years; a claim paid at time $j+1$ accumulates for $k-j-1$ years.
Counting premiums from people who have already died. A premium at the start of a year is collected only from those alive at that time, giving $\ell_{x+j}P$.
Assuming the terminal retrospective balance must be zero. The prospective reserve at the end of the term is zero, but the retrospective balance can be nonzero unless the initial balance and past cash flows meet the appropriate equivalence conditions.

These cohort calculations connect life-table counts to policy values without requiring an individual death history. The same careful bookkeeping is useful for checking annual reserve calculations and for recognizing when a supplied cohort balance is aggregate rather than per policy. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated annual payment timing and round the final reserve or premium to the nearest cent.

A two-year term policy has $P=\$200$, $B=\$2{,}000$, and $i=5\%$. The cohort counts are 500, 470, and 430 at durations 0, 1, and 2. Find the retrospective reserve at duration 2. (Answer: $\$153.37$)
A two-year term policy has $P=\$300$, $B=\$1{,}500$, and $i=0\%$. The cohort counts are 600, 570, and 520. Find the reserve at duration 2. (Answer: $\$444.23$)
A two-year term policy has a $\$10{,}000$ death benefit, an annual effective interest rate of $4\%$, and cohort counts of 1,000, 950, and 900. Find the level annual premium that makes the retrospective balance at duration 2 equal to zero, assuming zero initial reserve. (Answer: $\$492.85$)
A two-year term policy has $P=\$50$, $B=\$500$, $i=5\%$, and cohort counts 120, 110, and 100. Find the retrospective balance per survivor at duration 2, and state whether it must equal the prospective reserve at that time. (Answer: Retrospective balance $=\$21.40$; prospective reserve $=0$)
A three-year term policy has $P=\$250$, $B=\$3{,}000$, and $i=3\%$. The cohort counts at durations 0 through 3 are 1,000, 940, 880, and 810. Find the retrospective reserve at duration 3. (Answer: $\$200.90$)

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