Solving for an Unknown Premium From a Retrospective Reserve

Why Solve for a Premium Retrospectively?

A retrospective reserve records the accumulated value of past cash flows for a group of policies still in force at a valuation date. If the interest assumption, in-force counts, past benefit and expense payments, and reserve target are specified, the same cash-flow balance can be rearranged to find an unknown premium. This is useful when a premium is not given directly but must be consistent with the fund accumulated to date.

The key is to value every cash flow at the same date. A premium paid earlier accumulates for longer than a premium paid later. Also, a premium is collected only from policies in force on its payment date, while the per-policy reserve is divided by the number in force at the valuation date.

Definition: For a cohort of $N$ policies, let $n_k$ be the number in force at time $k$, where time is measured in years and $n_0=N$. Let $C_k$ denote aggregate benefit outgo paid at the end of year $k$, and let $E_k$ denote aggregate expense outgo paid then. A retrospective reserve at time $n$, per policy in force at that time, is the accumulated value of premiums less accumulated outgo, divided by $n_n$. This definition assumes the stated cash flows and interest accumulation apply to the cohort being valued.

The Accumulated Cash-Flow Equation

Assume premiums are paid at the beginning of each year, at times $0,1,\ldots,n-1$, by the policies then in force. Assume benefit and expense outgo is paid at the end of each year, at times $1,2,\ldots,n$. Let the effective annual interest rate be $i$, and suppose each net cash flow accumulates at this rate to time $n$. For a level annual premium $P$, the aggregate retrospective reserve at time $n$ is

n_n\,{}_nV = P\sum_{k=0}^{n-1} n_k(1+i)^{n-k} - \sum_{k=1}^{n}(C_k+E_k)(1+i)^{n-k}.

Here ${}_nV$ denotes the retrospective reserve per policy in force at time $n$, under the cohort experience and timing just specified. The first premium, paid at time 0, earns interest for $n$ years; the premium paid at time $k$ earns interest for $n-k$ years. Outgo paid at the end of year $k$ earns interest for $n-k$ years. If the premium is unknown and all other quantities are known, rearrangement gives

P= \frac{n_n\,{}_nV+\displaystyle\sum_{k=1}^{n}(C_k+E_k)(1+i)^{n-k}} {\displaystyle\sum_{k=0}^{n-1}n_k(1+i)^{n-k}}.

Why the formula holds: Move each premium and outgo forward to time $n$. A premium of $P$ collected from $n_k$ policies at time $k$ becomes $n_kP(1+i)^{n-k}$ at time $n$; an outgo of $C_k+E_k$ at time $k$ becomes $(C_k+E_k)(1+i)^{n-k}$. Add the accumulated premiums and subtract accumulated outgo. By definition, the remaining aggregate fund is $n_n{}_nV$. Collecting the terms that contain $P$ and dividing by their positive coefficient gives the solution. This rearrangement requires a known interest rate and cash-flow schedule, and at least one premium payment with a positive in-force count.

If premiums vary by duration, replace $P$ in the sum by the premium applicable at each date. For example, with premiums $P_0,\ldots,P_{n-1}$, the accumulated premium total is $\sum_{k=0}^{n-1} n_kP_k(1+i)^{n-k}$. When only one $P_j$ is unknown, its coefficient is $n_j(1+i)^{n-j}$; the other premium terms are known and can be moved to the other side of the equation. This is still a linear equation in the unknown premium.

Solving for a Level Premium

Worked Example: Level Premium From a Two-Year Cohort Reserve

A hypothetical cohort has 500 policies in force at time 0, 450 at time 1, and 420 at time 2. A level premium $P$ is paid at the beginning of each year by policies then in force. The effective annual interest rate is 4%. During year 1, 50 claims of $10,000 each are paid at year-end; during year 2, 30 claims of $10,000 each are paid at year-end. The reserve at time 2 is $3,000 per policy then in force. Find $P$.

Method. Accumulate premiums and claims to time 2, equate the resulting fund to the aggregate target reserve, and solve for the single unknown. The premium payments occur at times 0 and 1, so their respective accumulation factors are $1.04^2$ and $1.04$.

The aggregate target reserve is

$$ 420(3{,}000)=1{,}260{,}000. $$

The claims paid at the end of year 1 accumulate for one year, while the claims paid at the end of year 2 do not accumulate:

$$ 50(10{,}000)(1.04)+30(10{,}000) =520{,}000+300{,}000 =820{,}000. $$

The accumulated premium total is

500P(1.04)^2+450P(1.04) = \bigl(500(1.0816)+450(1.04)\bigr)P = (540.8+468)P = 1{,}008.8P.

Set the accumulated premiums less accumulated claims equal to the target reserve:

1{,}008.8P-820{,}000=1{,}260{,}000, \qquad P=\frac{2{,}080{,}000}{1{,}008.8} =2{,}061.8557\ldots.

Thus the level premium is $2,061.86 per policy, rounded to the nearest cent. As a check, the accumulated premium fund is approximately $2,080,000; after subtracting $820,000 of accumulated claims, approximately $1,260,000 remains, equal to $420$ times the specified $3,000 reserve. The common mistake avoided here is accumulating both years of claims for the same length of time.

Solving for One Premium in a Changing Schedule

Worked Example: Finding the Second-Year Premium

A hypothetical cohort has 200 policies in force at time 0, 180 at time 1, and 160 at time 2. The effective annual interest rate is 5%. A known premium of $1,000 is paid at time 0, and an unknown premium $Q$ is paid at time 1 by each policy then in force. At the end of each year, 20 claims of $5,000 each are paid. The time-2 reserve is $3,000 per policy in force. Find $Q$.

Method. Accumulate the known first premium and both years of claims to time 2. Then isolate $Q$, whose coefficient reflects 180 premiums paid at time 1 and one year of interest.

The aggregate reserve target is $160(3{,}000)=480{,}000$. The accumulated value of the first premium is

200(1{,}000)(1.05)^2=200{,}000(1.1025)=220{,}500.

The accumulated claims are

$$ 20(5{,}000)(1.05)+20(5{,}000) =105{,}000+100{,}000 =205{,}000. $$

The second premium contributes $180Q(1.05)=189Q$ at time 2. Therefore,

220{,}500+189Q-205{,}000=480{,}000, \qquad 189Q=464{,}500, \qquad Q=2{,}457.6719\ldots.

The second-year premium is $2,457.67 per policy, rounded to the nearest cent. Substitution gives accumulated premiums of approximately $220{,}500+464{,}500=685{,}000$; after $205,000 of accumulated claims, $480,000 remains, matching the target. The common mistake avoided is using 200, rather than 180, as the number of policies paying the second premium.

Using a Cash-Flow Ledger

A cash-flow ledger is useful when the accumulated outgo contains several benefit or expense components. The components need not have the same payment date, but each must be accumulated using its own time factor. The following example also illustrates a zero reserve target: the accumulated premiums exactly cover the stated outgo by the valuation date.

Worked Example: Premium From Specified Accumulated Outgo

A hypothetical two-year cohort has 1,000 policies in force at time 0 and 900 at time 1. A level premium $P$ is paid at times 0 and 1 by policies then in force. The effective annual interest rate is 5%. By time 2, the accumulated value of all benefit and expense outgo is specified by this ledger: $100,000 of benefit outgo and $100,000 of expense outgo paid at time 1, and $100,000 of benefit outgo plus $800,000 of maturity outgo paid at time 2. Find the premium that produces a zero reserve at time 2.

Method. First accumulate each ledger entry to time 2. The two time-1 amounts each earn one year of interest; the time-2 amounts earn none. Then equate total accumulated premiums to accumulated outgo.

\text{Accumulated outgo} = 100{,}000(1.05)+100{,}000(1.05)+100{,}000+800{,}000 = 105{,}000+105{,}000+100{,}000+800{,}000 = 1{,}110{,}000.

The premium accumulation coefficient is

1{,}000(1.05)^2+900(1.05) = 1{,}102.5+945 = 2{,}047.5.

A zero reserve means there is no accumulated fund remaining after outgo, so

2{,}047.5P=1{,}110{,}000, \qquad P=\frac{1{,}110{,}000}{2{,}047.5} =542.1245\ldots.

The premium is $542.12 per policy, rounded to the nearest cent (or $542.1 to the nearest tenth). Checking the rounded premium, its accumulated total is approximately $1,109,990, within rounding of the $1,110,000 outgo. The common mistake avoided is either omitting one of the two time-1 outgo entries or multiplying the time-2 entries by an extra factor of 1.05.

Checks, Assumptions, and Common Traps

Before accepting a solved premium, substitute it back into the original accumulated balance. The resulting fund should equal the specified aggregate reserve, not the per-policy reserve. The most reliable checks are to verify the time of each cash flow, confirm the in-force count attached to each premium date, and confirm that the final division by the valuation-date in-force count is applied only when converting an aggregate fund to a per-policy reserve.

Wrong accumulation period: A premium paid at time 1 accumulates for $n-1$ years to time $n$, not $n$ years. An end-of-year claim paid at time $n$ has no further accumulation to time $n$.
Wrong exposure count: Use the number in force when the premium is collected. Do not use the original cohort size for every premium unless every original policy actually pays each one.
Mixing aggregate and per-policy amounts: Multiply the target per-policy reserve by the number in force at time $n$ before equating it to the aggregate accumulated fund.
Changing the cash-flow convention: The displayed formula assumes premiums at the beginning of each year and outgo at the end. If a problem specifies a different timing, the exponents and possibly the in-force counts must change.
Rounding too early: Keep the exact accumulation factors and unrounded premium through the equation; round only the final monetary answer.

The calculation is linear in one unknown premium under a fixed interest rate and fixed cash-flow assumptions. It does not determine a unique premium if multiple unknown premiums are left unconstrained, nor does this rearrangement alone identify an unknown interest rate or benefit amount. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated payment timing and round money to the nearest cent unless otherwise stated.

A hypothetical cohort has 250 policies at time 0 and 230 at time 1. A single premium $P$ is paid at time 0. The effective annual interest rate is 5%; 20 claims of $5,000 are paid at time 1, and the reserve at time 1 is $600 per policy in force. Find $P$. (Answer: $906.67)
A cohort has 400, 360, and 330 policies in force at times 0, 1, and 2, respectively. A level premium is paid at times 0 and 1. At 4% effective annual interest, 40 claims of $8,000 are paid at the end of year 1 and 30 claims of $8,000 at the end of year 2. The time-2 reserve is $1,500 per policy in force. Find the level premium. (Answer: $1,323.11)
A two-year cohort has 1,000 policies in force at time 0 and 900 at time 1. A level premium $P$ is paid at both times, and the effective annual interest rate is 5%. At time 1, benefit outgo is $100,000 and expense outgo is $100,000; at time 2, benefit outgo is $100,000 and maturity outgo is $800,000. Find $P$ if the time-2 reserve is zero; give the answer to the nearest tenth. (Answer: $542.1)
A cohort has 500 policies in force at time 0, 450 at time 1, and 400 at time 2. The effective annual interest rate is 3%. The time-0 premium is $1,200, and the unknown time-1 premium is $Q$. At each year-end, 50 claims of $4,000 are paid. The time-2 reserve is $2,500 per policy in force. Find $Q$. (Answer: $1,660.11)
A cohort has 1,000 policies in force at time 0, 900 at time 1, and 800 at time 2. Premiums are paid at times 0 and 1, with the time-1 premium $500 greater than the time-0 premium. At 4% effective annual interest, 100 claims of $2,000 are paid at each year-end. The time-2 reserve is $1,000 per policy in force. Find both premiums. (Answer: time-0 premium $366.77; time-1 premium $866.77)

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