Apply Reserve Recursions with Premiums Before and After Benefits

Why Cash-Flow Timing Changes the Recursion

A one-year reserve recursion is a balance equation across a single policy year. It starts with the reserve at an anniversary and accounts for the premium, interest, and what must be paid or held at the end of the year. The mortality probabilities alone do not determine the recursion: the timing and contingency of each cash flow matter too.

In the standard discrete model used here, a death benefit is paid at the end of the policy year of death. A surviving policyholder has a reserve at the next anniversary. We compare two hypothetical premium conventions: a premium paid at the start of the year, and a premium paid at year-end only if the policyholder survives to that time.

Definition: Let ${}_kV_x$ be the prospective reserve at duration $k$, conditional on the life aged $x+k$ being alive. For these recursions, the reserve is measured immediately before any premium due at that anniversary. Write ${}_1q_{x+k}$ for the probability of death during the next year and ${}_1p_{x+k}$ for the probability of survival through it. Thus ${}_1p_{x+k}+{}_1q_{x+k}=1$. Let $i$ be the effective annual interest rate, $P_k$ the premium amount for the year, and $B_{k+1}$ the death benefit payable at the end of the year if death occurs.

All amounts in the equations are measured on the same valuation basis. The formulas below assume annual effective interest, a discrete one-year mortality model, and no other cash flows during the year. If a contract has expenses or other benefits, they must be included consistently as cash flows; if death benefits are paid at the actual time of death rather than at year-end, this discrete timing formula does not directly apply.

The Two One-Year Recursions

If the premium is paid at the start of the year, it joins the reserve before interest is earned. At year-end, the death branch pays $B_{k+1}$, while the survival branch carries the reserve ${}_{k+1}V_x$. The recursion is

({}_kV_x+P_k)(1+i) = {}_1q_{x+k}B_{k+1} + {}_1p_{x+k}{}_{k+1}V_x.

Solving for the reserve gives

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

If instead the premium is due at year-end only when the policyholder survives, it is not part of the amount that earns interest during the year. On survival, both the year-end premium and the next reserve are required. On death, the death benefit is paid and no survival-contingent premium is collected:

{}_kV_x(1+i) = {}_1q_{x+k}B_{k+1} + {}_1p_{x+k}\bigl(P_k+{}_{k+1}V_x\bigr).

In this second convention, the premium is itself on the survival branch, so it is multiplied by ${}_1p_{x+k}$. These equations use the same valuation point at duration $k$; what changes is when the premium is collected and which policy-year outcome triggers it.

Why the Recursions Hold

The recursions follow by matching the value accumulated to year-end with the year-end requirement in each possible outcome. This argument also explains why the death benefit and the survival reserve must occupy separate branches.

Accumulate the start-of-year funds.
When the premium is due at the start, the available amount is ${}_kV_x+P_k$. At annual effective rate $i$, it grows to $({}_kV_x+P_k)(1+i)$ by year-end.

List the year-end requirements.
Death during the year has probability ${}_1q_{x+k}$ and requires $B_{k+1}$. Survival has probability ${}_1p_{x+k}$ and requires the continuation reserve ${}_{k+1}V_x$. The expected requirement is ${}_1q_{x+k}B_{k+1}+{}_1p_{x+k}{}_{k+1}V_x$.

Equate accumulated funds and expected requirements.
Equating the amounts from Steps 1 and 2 gives the start-premium recursion. For a year-end premium due only on survival, the starting reserve alone accumulates, and the survival branch requires both $P_k$ and ${}_{k+1}V_x$, giving the other recursion.

This is a one-year balance identity on the reserve basis. The death benefit is not a reserve carried forward: it is a payment in the death outcome. Conversely, ${}_{k+1}V_x$ is not a death payment: it is the amount needed to continue the contract after survival.

Worked Applications

Worked Example: Premium Paid at the Start of the Year

A hypothetical policy is at duration $k$. Assume $i=0.05$, ${}_1q_{x+k}=0.10$, and ${}_1p_{x+k}=0.90$. The year-end death benefit is $B_{k+1}=\$100{,}000$, the next reserve is ${}_{k+1}V_x=\$20{,}000$, and a premium of $P_k=\$4{,}000$ is paid at the start of the year. Find ${}_kV_x$. Use the start-premium recursion.

Step 1: Calculate the expected year-end requirement. The death branch contributes $0.10(\$100{,}000)=\$10{,}000$. The survival branch contributes $0.90(\$20{,}000)=\$18{,}000$. Their sum is $\$10{,}000+\$18{,}000=\$28{,}000$.

Step 2: Discount that requirement and subtract the premium paid now.

{}_kV_x = \frac{\$28{,}000}{1.05}-\$4{,}000 = \$26{,}666.6667-\$4{,}000 = \$22{,}666.67.

Check. The reserve plus the start-of-year premium accumulates to $(\$22{,}666.6667+\$4{,}000)(1.05)=\$28{,}000$, matching the expected year-end requirement before rounding. The common mistake is to discount the premium as if it were due at year-end; the premium is paid at the start, so it is subtracted after discounting the year-end requirement.

Worked Example: Premium Due at Year-End on Survival

A hypothetical policy has $i=0.04$, ${}_1q_{x+k}=0.08$, and ${}_1p_{x+k}=0.92$. It pays a death benefit of $B_{k+1}=\$50{,}000$ at year-end on death. If the insured survives, a premium of $P_k=\$2{,}500$ is due at year-end, along with the continuation reserve ${}_{k+1}V_x=\$12{,}000$. Find the reserve at duration $k$.

Step 1: Value the death branch. The probability-weighted death benefit is $0.08(\$50{,}000)=\$4{,}000$.

Step 2: Value the survival branch. The year-end requirement on survival is $\$2{,}500+\$12{,}000=\$14{,}500$. Its probability-weighted value at year-end is $0.92(\$14{,}500)=\$13{,}340$.

Step 3: Discount the total year-end requirement.

{}_kV_x = \frac{\$4{,}000+\$13{,}340}{1.04} = \frac{\$17{,}340}{1.04} = \$16{,}673.08.

Check. Before rounding, $\$16{,}673.076923(1.04)=\$17{,}340$, which equals the probability-weighted death and survival requirements. The common mistake is to add the year-end premium to the death branch as well; under the stated contract, that premium is collected only on survival.

Worked Example: Comparing Premium Timings on the Same Basis

For a hypothetical policy, let $i=0.25$, ${}_1q_{x+k}=0.20$, and ${}_1p_{x+k}=0.80$. The year-end death benefit is $\$30{,}000$, the next reserve is $\$10{,}000$, and the premium is $\$2{,}000$. Calculate the reserve under each of two possible conventions: premium at the start of the year, or premium at year-end on survival.

Start-of-year premium. The expected year-end requirement is $0.20(\$30{,}000)+0.80(\$10{,}000)=\$6{,}000+\$8{,}000=\$14{,}000$. Discounting and subtracting the premium due immediately gives

{}_kV_x = \frac{\$14{,}000}{1.25}-\$2{,}000 = \$11{,}200-\$2{,}000 = \$9{,}200.

Year-end premium on survival. The survival branch now requires $\$2{,}000+\$10{,}000=\$12{,}000$. Its probability-weighted amount is $0.80(\$12{,}000)=\$9{,}600$. Including the death branch gives $\$6{,}000+\$9{,}600=\$15{,}600$, so

{}_kV_x = \frac{\$15{,}600}{1.25} = \$12{,}480.

Check and comparison. For the first convention, $(\$9{,}200+\$2{,}000)(1.25)=\$14{,}000$. For the second, $\$12{,}480(1.25)=\$15{,}600$. The resulting reserves differ by $\$12{,}480-\$9{,}200=\$3{,}280$. This difference is not a calculation inconsistency: the contracts collect the same premium at different times and under different conditions. Applying one timing convention's formula to the other convention is the common error.

Timing Traps and Practical Checks

The most reliable way to choose a recursion is to draw the year-end branches before substituting numbers. Put the death benefit on the death branch. Put the continuation reserve on the survival branch. Add a premium to the branch on which it is actually paid, and determine whether it earns interest during the policy year.

Start-of-year premium: add $P_k$ to the reserve before multiplying by $1+i$. Equivalently, discount the expected year-end requirement and then subtract $P_k$.
Year-end premium on survival: include $P_k$ with the next reserve on the survival branch, so the combined amount is weighted by ${}_1p_{x+k}$. Do not accumulate this premium through the year.
Death benefit versus survival reserve: weight $B_{k+1}$ by the death probability and ${}_{k+1}V_x$ by the survival probability. Neither amount belongs in both branches.
Boundary timing: confirm whether the stated reserve is before or after a premium due at the anniversary. The formulas here define it immediately before any premium due at duration $k$.

A useful final check is to multiply the reserve plus any start-of-year premium by $1+i$, then compare it with the probability-weighted year-end requirements. For an end-of-year premium, multiply the reserve alone by $1+i$ and check against the death benefit branch plus the survival branch containing both the premium and next reserve. These checks often reveal a misplaced probability or an incorrect discounting step.

Key takeaway: A reserve recursion is a cash-flow balance, not a memorized arrangement of terms. Accumulate amounts paid at the start of the year; place year-end amounts on the outcome branch that triggers them; and keep death benefits distinct from reserves for survivors.

Check Your Understanding

For each problem, use the specified premium convention and keep the valuation point immediately before any premium due at duration $k$.

A hypothetical policy has $i=0.03$, ${}_1q_{x+k}=0.05$, ${}_1p_{x+k}=0.95$, a year-end death benefit of $\$80{,}000$, a next reserve of $\$10{,}000$, and a start-of-year premium of $\$1{,}500$. Find ${}_kV_x$. (Answer: $\$11{,}606.80$)
For a hypothetical contract, $i=0.05$, ${}_1q_{x+k}=0.12$, ${}_1p_{x+k}=0.88$, $B_{k+1}=\$40{,}000$, ${}_{k+1}V_x=\$15{,}000$, and $P_k=\$2{,}000$ payable at year-end only on survival. Find ${}_kV_x$. (Answer: $\$18{,}819.05$)
Using $i=0.04$, ${}_1q_{x+k}=0.10$, ${}_1p_{x+k}=0.90$, $B_{k+1}=\$60{,}000$, ${}_{k+1}V_x=\$10{,}000$, and $P_k=\$3{,}000$, calculate the reserve once with the premium at the start of the year and once with it due at year-end on survival. State the difference. (Answer: start premium $\$11{,}423.08$; year-end premium $\$17{,}019.23$; difference $\$5{,}596.15$)
A hypothetical policy has a year-end premium payable only on survival. Given ${}_kV_x=\$12{,}000$, $i=0.05$, ${}_1q_{x+k}=0.10$, ${}_1p_{x+k}=0.90$, $P_k=\$1{,}000$, and ${}_{k+1}V_x=\$8{,}000$, find the year-end death benefit $B_{k+1}$. (Answer: $\$45{,}000$)
A hypothetical policy charges a premium at the start of the year. Given ${}_kV_x=\$6{,}000$, $P_k=\$2{,}500$, $i=0.02$, ${}_1q_{x+k}=0.04$, ${}_1p_{x+k}=0.96$, and $B_{k+1}=\$100{,}000$, find ${}_{k+1}V_x$. (Answer: $\$4{,}864.58$)

What You'll Learn