Deriving the Retrospective Reserve Recursion

Why the Retrospective Reserve Recurses

A retrospective reserve measures the accumulated value, at a valuation date, of past premiums and other inflows less past claims and other outflows, allocated across the policies still in force. The reserve at one duration therefore contains the starting point for the next duration’s fund balance. To move forward one year, accumulate the available funds, add the premium received, subtract claims paid, and allocate what remains to the survivors.

The key is to distinguish the fund for the whole cohort from the reserve per surviving policy. Death claims are paid from funds for all policies exposed to the year, while the reserve at the end of the year is held only for policies that survive. This difference is why the recursion includes both a mortality subtraction and a division by the survival probability.

Definition: Consider a policy issued at age $x$. Let ${}_kV_x$ be the retrospective reserve per policy in force at time $k$, just before the premium due at that time. Assume the annual effective interest rate is $i$. During the year from time $k$ to $k+1$, each policy in force pays premium $P_{k+1}$ at time $k$, and a death benefit $B_{k+1}$ is paid at time $k+1$ if death occurs during that year. Write ${}_1p_{x+k}$ and ${}_1q_{x+k}$ for the conditional probabilities of survival and death during the year, so ${}_1p_{x+k}+{}_1q_{x+k}=1$. The reserve at time $k+1$ is measured for survivors after the death claims have been paid.

These assumptions specify the timing as well as the mortality basis of the recursion. The premium is available to earn interest for the full year, and the death benefit is paid at year-end, so it does not earn interest during that year. The interest rate and mortality probabilities are treated as deterministic assumptions for the reserve calculation.

Deriving the Annual Recursion

Start with a cohort of policies in force at time $k$. For a convenient scale, imagine there are $N$ such policies. At time $k$, their total reserve is $N\,{}_kV_x$, and they pay total premiums of $NP_{k+1}$. The amount available at the start of the year is therefore $N({}_kV_x+P_{k+1})$. After interest, this grows to $N({}_kV_x+P_{k+1})(1+i)$.

The expected number of deaths during the year is $N\,{}_1q_{x+k}$, so expected death claims are $N\,{}_1q_{x+k}B_{k+1}$. The expected number of survivors at time $k+1$ is $N\,{}_1p_{x+k}$. Their total reserve is $N\,{}_1p_{x+k}\,{}_{k+1}V_x$. Equating the funds remaining after claims to the reserve held for survivors gives the cohort balance.

N({}_kV_x+P_{k+1})(1+i) - N\,{}_1q_{x+k}B_{k+1} = N\,{}_1p_{x+k}\,{}_{k+1}V_x.

Divide both sides by $N$, then divide by ${}_1p_{x+k}$, which must be positive for there to be survivors to whom a reserve is allocated. The result is the annual retrospective reserve recursion:

Theorem: Under the stated annual timing assumptions, with a death benefit $B_{k+1}$ payable at the end of the year of death,

{}_{k+1}V_x = \frac{({}_kV_x+P_{k+1})(1+i)-{}_1q_{x+k}B_{k+1}} {{}_1p_{x+k}}.

The equation can also be written as a survivor-weighted balance:

{}_1p_{x+k}\,{}_{k+1}V_x = ({}_kV_x+P_{k+1})(1+i)-{}_1q_{x+k}B_{k+1}.

This second form is often the quickest check on a calculation. Its left side is the end-of-year reserve per initial policy, weighted by the probability that the policy survives. Its right side is the accumulated fund per initial policy less expected death claims per initial policy. The two sides must agree.

Set the opening amount.
Add the reserve at time $k$ to the premium due at time $k$: ${}_kV_x+P_{k+1}$.

Accumulate interest.
Multiply the opening amount by $1+i$, because it is invested for one full year under the stated timing.

Pay expected death claims.
Subtract ${}_1q_{x+k}B_{k+1}$, the expected claim amount per policy in force at time $k$.

Allocate to survivors.
Divide the remaining amount by ${}_1p_{x+k}$ to obtain the reserve per policy still in force at time $k+1$.

Worked Applications

Worked Example: Correcting the Survivor Allocation

At time $k$, a policy has retrospective reserve ${}_kV_x=\$150$. The premium due is $P_{k+1}=\$30$, the annual effective interest rate is $i=0.04$, the one-year death probability is ${}_1q_{x+k}=0.20$, and the year-end death benefit is $B_{k+1}=\$180$. Find ${}_{k+1}V_x$ and check the survivor-weighted balance.

Method. Apply the cohort recursion. The one-year survival probability is ${}_1p_{x+k}=1-0.20=0.80$.

First combine the reserve and premium: $150+30=180$. Accumulate this amount for one year: $180(1.04)=187.20$. Expected claims per policy in force at time $k$ are $0.20(180)=36.00$. The amount available for survivors is therefore $187.20-36.00=151.20$.

Allocate that amount across the surviving 80% of the cohort: \[ {}_{k+1}V_x=\frac{151.20}{0.80}=\$189.00. \] To check, the survivor-weighted reserve is $0.80(189.00)=\$151.20$, exactly the amount remaining after expected claims.

Common mistake avoided. Do not treat the amount remaining after claims as the reserve per survivor. It is the amount per policy initially in force; divide by the survival probability to obtain the reserve per policy that survives.

Worked Example: A Changing Premium and Benefit

A hypothetical annual renewable contract has reserve ${}_kV_x=\$240$ immediately before a premium of $P_{k+1}=\$40$ is paid. The annual effective interest rate is $i=0.05$, the death probability over the year is ${}_1q_{x+k}=0.10$, and the year-end death benefit is $B_{k+1}=\$200$. Calculate the reserve for survivors at time $k+1$, to the nearest cent.

Method. Use the actual premium and benefit specified for this year; the recursion does not require either amount to be level across durations. The survival probability is $1-0.10=0.90$.

The opening funds including premium are $240+40=\$280$. After one year at 5% interest, the funds are $280(1.05)=\$294.00$. Expected death claims per policy exposed at time $k$ are $0.10(200)=\$20.00$. Thus the funds remaining for survivors are $294.00-20.00=\$274.00$, and the reserve per survivor is \[ {}_{k+1}V_x=\frac{274.00}{0.90} =304.444\ldots =\$304.44. \] The check is $0.90(304.444\ldots)=\$274.00$, before rounding.

Common mistake avoided. Subtract the expected claim before dividing by the survival probability. Dividing the accumulated fund by 0.90 first would incorrectly allocate part of the claim money to survivors.

Worked Example: Recursing Across Two Years

A hypothetical two-year contract begins with ${}_0V_x=\$0$. The annual effective interest rate is $i=0.04$. In the first year, the premium is $P_1=\$50$, the death probability is ${}_1q_x=0.20$, and the year-end death benefit is $B_1=\$80$. In the second year, the premium is $P_2=\$25$, the death probability is ${}_1q_{x+1}=0.10$, and the year-end death benefit is $B_2=\$100$. Find ${}_1V_x$ and ${}_2V_x$, rounding to cents at the end.

Method. Apply the one-year recursion separately to each interval. For year 1, survival probability is $1-0.20=0.80$. The accumulated premium is $(0+50)(1.04)=\$52.00$. Expected first-year claims are $0.20(80)=\$16.00$, leaving $52.00-16.00=\$36.00$ per initial policy. Therefore \[ {}_1V_x=\frac{36.00}{0.80}=\$45.00. \]

For year 2, the opening reserve plus premium is $45+25=\$70$, and accumulation gives $70(1.04)=\$72.80$. Expected second-year claims are $0.10(100)=\$10.00$, leaving $72.80-10.00=\$62.80$. The survival probability is $1-0.10=0.90$, so \[ {}_2V_x=\frac{62.80}{0.90} =69.777\ldots =\$69.78. \]

Check. The first-year weighted reserve is $0.80(45)=\$36.00$, equal to the funds after claims. The second-year weighted reserve before rounding is $0.90(69.777\ldots)=\$62.80$, also equal to the funds after claims.

Common mistake avoided. The second year starts with the reserve at time 1, not with the original reserve at time 0. Each interval’s calculation uses its own premium, mortality probability, and benefit.

Timing, Scope, and Exam Traps

The formula is not independent of cash-flow timing. It assumes the premium is paid at time $k$, before the year’s interest accumulation, and the death benefit is paid at time $k+1$. If a premium is instead paid at year-end, or a benefit is paid at a different time, the accumulation factor and cash-flow balance must reflect that timing. Do not use this formula unchanged when the stated payment dates differ.

The recursion also treats the reserve at time $k+1$ as the amount retained for survivors after death claims have been paid. A contract with an additional benefit payable to survivors at that date needs that survivor payment included as an outflow before the remaining funds are allocated. Similarly, the formula as displayed is for the stated death-benefit setting; it does not silently account for expenses or other cash flows.

A frequent error is to subtract the full benefit $B_{k+1}$ rather than the expected claim amount ${}_1q_{x+k}B_{k+1}$. The benefit is paid only for deaths during the year, so the cohort-level expected outgo per policy exposed is weighted by the death probability. Another error is to multiply by survival probability instead of dividing by it: the remaining cohort fund is spread among fewer policies than were in force at the start.

Key takeaway: For the specified annual timing, accumulate the reserve and premium, subtract expected death claims, then divide by the one-year survival probability. Always verify the result by checking that the survivor-weighted reserve equals the funds remaining after claims.

Check Your Understanding

For each problem, use the stated annual timing: premium at the start of the year, death benefit at the end of the year, and reserve for survivors after claims. Round money to the nearest cent.

A policy has ${}_kV_x=\$100$, premium $P_{k+1}=\$20$, annual effective interest rate $i=0.05$, death probability $0.10$, and death benefit $B_{k+1}=\$100$. Find the next reserve. (Answer: $\$128.89$)
A policy has reserve $\$80$, premium $\$20$, annual effective interest rate $0.04$, death probability $0.25$, and death benefit $\$80$. Find the reserve for survivors and state the survivor-weighted balance. (Answer: reserve $\$112.00$; weighted balance $\$84.00$)
A policy has reserve $\$120$, premium $\$30$, annual effective interest rate $0.03$, death probability $0.20$, and death benefit $\$150$. Calculate the next reserve and verify it using the survivor-weighted form. (Answer: reserve $\$155.63$; weighted balance $\$124.50$)
A contract starts with reserve ${}_0V_x=\$0$. The annual effective interest rate is $0.05$. In year 1, the premium is $\$40$, death probability is $0.20$, and death benefit is $\$50$. In year 2, the premium is $\$20$, death probability is $0.10$, and death benefit is $\$100$. Find ${}_1V_x$ and ${}_2V_x$. (Answer: ${}_1V_x=\$40.00$; ${}_2V_x=\$58.89$)
A policy has reserve $\$200$, premium $\$25$, annual effective interest rate $0.04$, and death probability $0.20$. If the next reserve is $\$210$, find the year-end death benefit that makes the cohort balance hold. (Answer: $\$330.00$)

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