Integrated Retrospective Reserve Problem Workshop

Integrating the Cash Flows

A retrospective reserve problem becomes more demanding when cash flows do not stay level. Premiums may change, expenses may vary by duration, and benefits may be paid at a different time from premiums. The reliable method is to track the cohort’s fund in chronological order, keeping the number of policies exposed to each cash flow explicit.

The cohort calculation starts with life-table counts. Let $\ell_k$ be the number alive at time $k$, and let $F_k$ be the cohort fund immediately after claims due at time $k$ have been paid and premiums and expenses at that time have been recorded. Suppose the effective annual interest rate is $i$. During year $k$, each death claim is $b_k$, payable at the end of the year. At time $k+1$, each survivor pays premium $p_{k+1}$ and incurs expense $e_{k+1}$.

Definition: Under this timing convention, the retrospective reserve at time $k$ is the accumulated cohort fund allocated to each policy in force, $F_k/\ell_k$, provided there are no claims already incurred but still unpaid at that valuation time. Premiums and expenses at time $k$ are counted only for the $\ell_k$ policies then in force.

The convention matters: the fund is first accumulated through the year, then claims for deaths during that year are deducted, and then premiums and expenses for surviving policies at the next time point are recorded. When the valuation date falls between policy anniversaries, the number in force and any unpaid claim liabilities must also be considered.

The Cohort Fund Recurrence

With the timing just stated, the fund evolves as follows. The term $\ell_k-\ell_{k+1}$ is the number of deaths during year $k$. The surviving policies, $\ell_{k+1}$, are the ones who pay the next premium and generate the next expense.

F_{k+1} = F_k(1+i) - (\ell_k-\ell_{k+1})b_k + \ell_{k+1}(p_{k+1}-e_{k+1}).

This is a cash-balance equation, not a probability approximation. The life-table counts determine how many cash flows occur; interest applies to the fund accumulated during the year; and expenses reduce the premium cash available to the fund.

Record the opening cash.
At the start of the contract, multiply the number of policies by the premium less the expense incurred at that time.

Accumulate to the claim date.
Multiply the opening fund by $1+i$ for a full year, or by the appropriate fractional accumulation factor for part of a year.

Deduct claims and record survivor cash flows.
Subtract the number of deaths times the benefit, then add premiums less expenses for the policies in force after those deaths.

Allocate and check.
Divide the available fund by the number of policies in force. At a nonannual date, first allow for incurred claims that remain unpaid.

The recurrence follows directly from conservation of cash: the opening fund plus interest is the amount available before year-end claims; subtracting the claims and adding the next survivors’ net premiums gives the closing fund. This explains why the counts and the order of transactions cannot be interchanged casually.

Worked Example: Changing Premiums and Expenses

A hypothetical cohort has $\ell_0=100$, $\ell_1=90$, and $\ell_2=72$. The effective annual interest rate is $i=4\%$. At time 0, each policy pays a premium of $500 and incurs an expense of $20. At time 1, each survivor pays a premium of $700 and incurs an expense of $30. A $4,000 claim is paid at the end of each year for every death in that year. Find the retrospective fund per policy in force at time 2, after the second-year claims.

Method. Calculate the cohort fund at each time point, including the appropriate premium and expense only for the policies then in force.

At time 0, 100 policies contribute a net amount of $500-20=480$ each:

F_0=100(500-20)=48{,}000.

The fund accumulates for one year to $48{,}000(1.04)=49{,}920$. There are $100-90=10$ first-year deaths, so claims total $10(4{,}000)=40{,}000$. The 90 survivors then contribute $90(700-30)=60{,}300$. Thus,

F_1=49{,}920-40{,}000+60{,}300=70{,}220.

From time 1 to time 2, the fund accumulates to $70{,}220(1.04)=73{,}028.80$. There are $90-72=18$ deaths in the second year, with claims of $18(4{,}000)=72{,}000$. Therefore,

F_2=73{,}028.80-72{,}000=1{,}028.80, \qquad \frac{F_2}{\ell_2} = \frac{1{,}028.80}{72} = 14.288\ldots.

The retrospective fund is $14.29 per policy in force, rounded to the nearest cent. As a check, the closing cohort fund is positive and equals $72(14.288\ldots)=1{,}028.80$, as required. The method avoids counting the time-1 premium for the 10 people who died during the first year.

Fractional Valuation Dates and Unpaid Claims

A valuation date between anniversaries requires two separate adjustments. First, determine the number in force at that date under the stated fractional-age assumption. Second, distinguish a claim that has occurred from a claim that has been paid. If a death benefit is payable at the end of the policy year, a death before that date creates an unpaid claim liability until the payment date.

For example, under the uniform distribution of deaths (UDD) assumption within the year from $k$ to $k+1$, the life-table count at $k+s$, for $0\leq s\leq1$, is

\ell_{k+s}=(1-s)\ell_k+s\ell_{k+1}.

If a claim of $b$ per death has been incurred by the valuation date but is payable $1-s$ years later, its value at that date is $b(1+i)^{-(1-s)}$, when fractional-year accumulation follows the effective annual rate. The cohort’s available fund for the surviving policies is the accumulated fund less the value of such unpaid claims.

Worked Example: A Reserve Halfway Through a Policy Year

A hypothetical cohort has $\ell_0=200$, $\ell_1=180$, and $\ell_2=150$. Assume UDD within each year. The effective annual interest rate is $21\%$, with fractional-year accumulation using $(1.21)^t$. At time 0, each policy pays $300 and incurs a $15 expense. At time 1, each survivor pays $320 and incurs a $20 expense. A $2,000 death benefit is paid at the end of the policy year of death. Find the retrospective reserve per policy in force at time 1.5.

Method. Build the fund through time 1, accrue it for half a year, then subtract the value at time 1.5 of claims already incurred but payable at time 2.

At time 0, the net contribution per policy is $300-15=285$, giving

F_0=200(285)=57{,}000.

At time 1, before claims, the fund is $57{,}000(1.21)=68{,}970$. There are $200-180=20$ first-year deaths, so first-year claims total $20(2{,}000)=40{,}000$. The remaining fund is $68{,}970-40{,}000=28{,}970$. At time 1, the 180 survivors contribute $180(320-20)=54{,}000$, giving $F_1=28{,}970+54{,}000=82{,}970$.

The half-year accumulation factor is $(1.21)^{0.5}=1.1$, so the fund at time 1.5 before accounting for pending claims is

$$ 82{,}970(1.1)=91{,}267. $$

Under UDD, the number in force at time 1.5 is $\ell_{1.5}=(180+150)/2=165$. Thus 15 deaths have occurred between times 1 and 1.5. Their claims, totaling $15(2{,}000)=30{,}000$, are payable at time 2, half a year after the valuation date. The value of those claims at time 1.5 is $30{,}000/1.1=27{,}272.727\ldots$. The fund available for the 165 surviving policies is therefore

91{,}267-27{,}272.727\ldots=63{,}994.272\ldots, \qquad \frac{63{,}994.272\ldots}{165}=387.844\ldots.

The retrospective reserve is $387.84 per policy in force, rounded to the nearest cent. The check is that $165(387.844\ldots)$, plus the value of the unpaid claims, equals the accumulated fund of $91,267. Omitting the unpaid claims would incorrectly allocate money backing those claims to surviving policies.

Checking a Retrospective Reserve Prospectively

A prospective check values the expected future benefits less expected future premiums and expenses, conditional on being in force at the valuation date. It provides an independent check only when the future assumptions and timing match the basis of the retrospective calculation. A net premium determined by the equivalence principle is especially useful for this check: at issue, the present value of expected premiums equals the present value of expected benefits.

The following example uses a level net premium and an end-of-year death benefit. The reserve is measured immediately after the time-1 premium, so that premium is not part of the future prospective cash flows. In the retrospective calculation it is included in the fund at time 1.

Worked Example: Matching the Prospective and Retrospective Values

A hypothetical two-year term contract covers 100 lives at issue, 80 at time 1, and 60 at time 2. The effective annual interest rate is $25\%$. A $1,000 benefit is paid at the end of each year of death. A level net premium $P$ is paid at times 0 and 1 by each policy in force at the respective premium date. Find $P$, then check the reserve at time 1 immediately after the premium.

Method. First use the equivalence principle at issue to find $P$, then calculate the reserve both from past cohort cash flows and from future expected cash flows.

There are $100-80=20$ first-year deaths and $80-60=20$ second-year deaths. Let $v=1/1.25=0.8$. The present value of expected benefits at issue is $20(1{,}000)(0.8)+20(1{,}000)(0.8^2)=16{,}000+12{,}800=28{,}800$. The present value of premiums is $100P+80P(0.8)=164P$. Equating the two gives

164P=28{,}800, \qquad P=\frac{28{,}800}{164} =\frac{7{,}200}{41} =175.609756\ldots.

For the retrospective calculation, the time-0 fund is $100P$, which accumulates to $125P$ at time 1. The 20 first-year claims are then deducted, and the 80 surviving policies pay the time-1 premium. The time-1 fund immediately after that premium is

$$ 125P-20{,}000+80P=205P-20{,}000. $$

Substituting $P=7{,}200/41$ gives $205P=5(7{,}200)=36{,}000$, so the fund is $36{,}000-20{,}000=16{,}000$. Dividing by the 80 policies in force gives $16{,}000/80=$200$ per policy.

Prospectively, the remaining obligation is the second-year benefit. Among the 80 policies in force at time 1, 20 claims of $1,000 are expected at time 2. Its value per policy at time 1 is

\frac{20(1{,}000)(0.8)}{80}=$200.

Both methods give $200 per policy. This agreement is a useful check, not a license to mix bases: changing the interest, mortality, cash-flow timing, or expense assumptions in only one calculation can make the two values differ.

Common Errors in Integrated Problems

When a problem combines several cash-flow features, a correct formula applied to the wrong count or date can still produce a wrong reserve. Use a small time line or a cohort table before calculating.

Using the original cohort as the denominator. A per-policy reserve at time $k$ is divided by $\ell_k$, not by $\ell_0$. In the first example, $1{,}028.80/100=$10.29$ is a fund per original policy, not per policy in force. The correct calculation is $1{,}028.80/72=$14.29$.
Applying a premium to deaths from the prior year. A premium at time $k+1$ is paid by those surviving to time $k+1$, unless the problem explicitly states another payment rule.
Subtracting claims at the wrong date. A claim paid at the end of a year reduces the fund then, not at the start of that year. If the valuation date precedes payment, account for the claim as an unpaid liability rather than treating it as already paid.
Ignoring expenses or counting them twice. If the fund receives premium $p$ and incurs expense $e$ at the same date, the net contribution is $p-e$. Do not deduct $e$ again in a later interval unless another expense is specified.
Using annual accumulation over a fraction of a year. Under the stated fractional convention, a half-year factor is $(1+i)^{0.5}$, not $1+i$. Use a different fractional accumulation rule only when the problem specifies one.

Key takeaway: Keep the cohort count, transaction date, and fund balance together at every step. At a nonannual valuation date, separate the cash already paid from claims incurred but still payable. A prospective calculation then provides a strong independent check when it uses the same assumptions and timing.

Check Your Understanding

For each problem, state the cash-flow timing you use and keep the fund total separate from the per-policy reserve.

A hypothetical cohort has 50 policies at time 0 and 45 at time 1. The effective annual interest rate is $5\%$. Each policy pays a net premium of $100 at time 0, and a $1,000 claim is paid at time 1 for each death. Find the retrospective fund per policy in force at time 1 after claims. (Answer: $5.56)
A cohort has $\ell_0=80$, $\ell_1=72$, and $\ell_2=60$, with $i=10\%$. The time-0 net premium is $200 per policy; the time-1 net premium is $250 per survivor. Claims of $500 are paid at each year-end for each death. Find the reserve per policy in force at time 2 after claims. (Answer: $479.33)
A two-year contract has 40 policies at issue, 32 at time 1, and 24 at time 2. The interest rate is $0\%$, and a $600 benefit is paid at the end of the year of death. A level net premium is paid at times 0 and 1 by each policy then in force. Find the premium by equivalence and the reserve at time 1 just after the premium. (Answer: premium $133.33; reserve $150.00 per policy)
Under UDD, a cohort has $\ell_1=90$ and $\ell_2=70$. The death benefit is $1,000, payable at time 2 for deaths in the second year. If $i=10\%$, find the number in force at time 1.5 and the time-1.5 value of claims already incurred but unpaid. (Answer: 80 in force; $9,534.63 of unpaid claims)
A cohort has $\ell_0=60$, $\ell_1=48$, and $\ell_2=40$, with $i=4\%$. At time 0, each policy contributes a net premium of $500; at time 1, each survivor contributes $600. Claims of $1,000 are paid at each year-end for each death. Find the reserve per policy in force at time 2 after claims. (Answer: $1,048.00)

What You'll Learn