Discrete Reserve Timing Around Premium Dates

Why Premium-Date Timing Matters

A reserve is a value at a specified instant, so the timing of a premium due at that instant matters. Immediately before payment, the premium is still a future cash flow from the insurer’s perspective: it will be received if the policyholder pays. Immediately after payment, it is no longer outstanding. Confusing the two instants can shift a reserve by one full premium, even though no time has passed and no interest has accrued.

The calculations below use a discrete model: premiums are payable at the start of specified policy years while the insured is alive, and death benefits are paid at the end of the year of death. The amounts are hypothetical. The prospective value is the expected present value of future benefits less the expected present value of future premiums, conditional on the insured being alive at the valuation time.

Definition: At integer duration $k$, let ${}_kV_x^-$ be the prospective reserve immediately before a premium due at time $k$ is paid, and let ${}_kV_x^+$ be the reserve immediately after that premium is paid. If the premium due at time $k$ is $P_k$, and is payable by a policy in force at that instant, then $$ {}_kV_x^- = {}_kV_x^+ - P_k. $$ If no premium is due at time $k$, the two values are equal. A premium payable only at a later time is included in both values, with its payment probability and discounting accounted for.

Set the Valuation Instant Before Summing Cash Flows

At either instant, first condition on the insured being alive at duration $k$. Then list the remaining benefits and premiums by payment date. The premium due at time $k$ belongs in the before-payment list, but not in the after-payment list. Later premiums belong in the relevant list only if the contract schedules them and the policy is in force when they fall due.

For example, suppose a policy has an annual premium at times $0,1,\ldots,n-1$. At duration $k<n$, a premium is scheduled at $k$, so the before-payment value includes it. If premiums stop after duration $r$, a premium at $k>r$ must not be subtracted merely because coverage continues. The contract’s premium schedule, not the benefit period, determines whether a premium remains due.

Fix the instant.
State whether the value is immediately before or immediately after the premium due at duration $k$.

List future benefits.
For each possible payment date, use the probability of reaching the relevant claim event and discount the benefit to duration $k$.

List future premiums.
Include a premium due now only for the before-payment value. For later premiums, include the probability the insured is alive at their due dates.

Subtract and check.
Subtract the expected present value of premiums from that of benefits. If a premium $P_k$ is due now, check that the before-payment value is exactly $P_k$ less than the after-payment value.

Why the Difference Is Exactly One Premium

The before- and after-payment values are measured at the same instant. All future benefit cash flows are therefore identical in the two calculations, and all premiums due after time $k$ are identical too. The only cash flow present before payment and absent after payment is the premium $P_k$, payable at time $k$. Its value at that instant is $P_k$, not a discounted amount.

{}_kV_x^- = \operatorname{EPV}_k(\text{future benefits}) -P_k -\operatorname{EPV}_k(\text{premiums after }k) ={}_kV_x^+-P_k.

Here $\operatorname{EPV}_k$ means expected present value conditional on being alive at duration $k$, with values discounted to that duration. The identity assumes the premium is due and payable at $k$. If no premium is due, the term $P_k$ is absent and there is no before-versus-after premium difference.

Worked Example: One Remaining Year of Coverage

A hypothetical two-year term policy pays $1{,}000$ dollars at the end of the year if death occurs during that year. Premiums of $120$ dollars are due at times 0 and 1, while the insured is alive. Find the reserve at duration 1 immediately before and immediately after the second premium. Assume an effective annual interest rate $i=0.05$, and a conditional probability of death of $q=0.20$ during the remaining year. Use the method of expected present values.

Discount factor. The annual discount factor is $v=1/(1+i)=1/1.05=0.95238095$.

After payment. The premium due at time 1 has already been paid, and no later premium is scheduled. The only possible benefit is $1{,}000$ dollars at time 2, with probability $0.20$: $$ {}_1V_x^+ = (0.20)(1{,}000)v =200(0.95238095) =190.47619. $$ Thus the after-payment reserve is $190.48$ dollars to the nearest cent.

Before payment. The same benefit value applies, but the $120$-dollar premium is still due at time 1: $$ {}_1V_x^- = 190.47619-120=70.47619. $$ Thus the before-payment reserve is $70.48$ dollars to the nearest cent. The check is $190.48-70.48=120.00$ dollars, exactly the premium due. The common error avoided here is discounting the premium at time 1: it is payable at the valuation instant, so its value then is $120$, not $120v$.

Later Premiums Depend on Survival

A premium due after the valuation time is not necessarily certain to be paid. If it is payable only while the insured is alive, its expected present value includes the probability of surviving to its due date. For a premium due one year later, that contribution is $P$ times the one-year survival probability times $v$. This survival adjustment is separate from the before-or-after adjustment for a premium due now.

Worked Example: A Due Premium and a Later Premium

A hypothetical three-year term policy pays $2{,}000$ dollars at the end of the year of death. At duration 1, the insured is alive. Premiums of $150$ dollars are due at times 1 and 2 if the insured is alive. Assume $i=0.05$, conditional death probability $q=0.10$ in policy year 2, and conditional death probability $q=0.20$ in policy year 3. Calculate the reserve immediately before and after the duration-1 premium. Use expected present values, conditioning future premiums on survival.

The discount factor is $v=1/1.05=0.95238095$, so $v^2=0.90702948$. From duration 1, the probability of death in year 2 is $0.10$. The probability of surviving year 2 and dying in year 3 is $(1-0.10)(0.20)=0.18$.

Expected present value of benefits. The year-2 death benefit has value $(0.10)(2{,}000)v=200(0.95238095)=190.47619$. The year-3 death benefit has value $(0.18)(2{,}000)v^2=360(0.90702948)=326.53061$. Therefore, $$ \operatorname{EPV}_1(\text{benefits}) =190.47619+326.53061=517.00680. $$

Expected present value of premiums after time 1. The premium due at time 2 is payable only if the insured survives year 2. Its value at duration 1 is $(0.90)(150)v=135(0.95238095)=128.57143$.

After payment. The time-1 premium has been paid, so only the later premium is subtracted: $$ {}_1V_x^+=517.00680-128.57143=388.43537. $$

Before payment. The time-1 premium is also still due: $$ {}_1V_x^-=517.00680-128.57143-150=238.43537. $$ The reserves are $388.44$ dollars after payment and $238.44$ dollars before payment, to the nearest cent. Their difference is $150.00$ dollars, as required. The calculation avoids treating the time-2 premium as certain: it is payable only if the insured survives the second policy year.

Do Not Invent a Premium at the Valuation Time

A valuation date is not automatically a premium date. If a contract’s premiums stop before coverage ends, a later reserve calculation can still reflect future benefits but must exclude premiums that are no longer scheduled. Conversely, if a premium is due at the valuation instant, it belongs in the before-payment value even if the policyholder has not yet paid it.

Worked Example: Coverage Continues After the Last Premium

A hypothetical three-year term policy pays $1{,}000$ dollars at the end of the year of death. Its only premiums are $250$ dollars at times 0 and 1 while the insured is alive. At duration 2, the insured is alive and coverage continues for one final year, but no premium is due then. Assume an effective annual interest rate $i=1/9$, so $v=0.90$, and a conditional probability of death of $0.10$ during the final year. Find the reserve at duration 2.

The expected benefit payable at time 3 is $(0.10)(1{,}000)=100$ dollars. Discounting one year gives $$ {}_2V_x^+=100(0.90)=90. $$ No premium is scheduled at time 2 or later, so there is no premium value to subtract. The value immediately before time 2 is the same: ${}_2V_x^-={}_2V_x^+=90$ dollars. The answer is $90.00$ dollars to the nearest cent. Subtracting $250$ dollars at duration 2 would incorrectly reuse a premium from an earlier date; it would produce $-160$ dollars, but that cash flow has already occurred.

Common Timing Errors

The sign and size of the adjustment are easy to check once the instant is stated explicitly. Before a premium is paid, the insurer still expects to receive it, so the prospective liability is lower by that premium. After it is paid, that receipt is no longer future cash flow, so the reserve is higher by the same amount. This comparison does not mean the policy’s benefits or mortality have changed.

Discounting a premium due now. A premium payable at the valuation instant is not discounted by a full year. Its present value at that instant is its face amount.
Subtracting the current premium twice. Once the after-payment value is calculated, it already excludes the premium paid at time $k$. Do not subtract it again.
Ignoring survival for later premiums. A premium due next year only contributes when the contract says it is payable, usually conditional on survival to that date.
Assuming coverage implies premiums continue. Check the stated premium-paying period. Coverage may continue after the final scheduled premium.
Using a premium date that is not in the contract. If no premium is due at duration $k$, there is no one-premium adjustment between the two labels.

Key takeaway: Fix the valuation instant, then include exactly the cash flows still outstanding. When a premium $P_k$ is due at that instant, ${}_kV_x^-={}_kV_x^+-P_k$. Premiums payable later remain in both calculations and must be weighted for survival and discounted from their own payment dates.

Check Your Understanding

For each problem, identify whether a premium is due at the valuation time before calculating. Round dollar amounts to the nearest cent.

A policy has one year of coverage remaining. It pays $800$ dollars at year-end if death occurs, with probability $0.15$. At $i=0.04$, a $90$-dollar premium is due now. Find the before- and after-payment reserves. (Answer: $25.38$ dollars before; $115.38$ dollars after.)
At duration 1, a one-year policy has a $500$-dollar year-end death benefit, death probability $0.20$, and $i=0.10$. A $80$-dollar premium is due now and no later premium is scheduled. Find both reserves. (Answer: $10.91$ dollars before; $90.91$ dollars after.)
At duration 1, a policy pays $2{,}000$ dollars at the end of the year of death during the next two years. Conditional death probabilities are $0.10$ in year 2 and $0.20$ in year 3. Premiums of $150$ dollars are due now and at duration 2 if alive. Assume $i=0.05$. Find both duration-1 reserves. (Answer: $238.44$ dollars before; $388.44$ dollars after.)
A policy has one year of coverage left at duration 2. It pays $1{,}200$ dollars at year-end if death occurs, with probability $0.05$. Assume $i=0.04$. All premiums ended at duration 1, so none is due at duration 2. Find the before- and after-date values. (Answer: $57.69$ dollars before and after.)
At duration 1, a three-year policy pays $1{,}600$ dollars at the end of the year of death. The conditional death probabilities in years 2 and 3 are $0.25$ and $0.10$, respectively. Premiums of $200$ dollars are due at durations 1 and 2 if alive. Assume $i=0.06$. Find both duration-1 reserves. (Answer: $142.65$ dollars before; $342.65$ dollars after.)

What You'll Learn