Reserve Values Immediately Before and After Premium Payment

Why the Valuation Instant Matters

An annual premium and a reserve can be associated with the same policy anniversary but refer to different instants. The reserve immediately before the premium is received reflects a premium that is due at that instant; the reserve immediately after reflects that the premium has already been received. Confusing these values shifts the reserve by one full premium.

This distinction is important whenever a problem specifies an anniversary reserve, a premium payment date, or a valuation “before” or “after” payment. The benefit and mortality assumptions need not change between the two instants: the immediate cash flow is what changes. The treatment below assumes a premium is paid at the start of the policy year, provided the policy is then in force, and no benefit or expense is paid at that same instant.

Definition: At time $t$, let ${}_tV_x^-$ denote the reserve immediately before the premium due at time $t$, and let ${}_tV_x^+$ denote the reserve immediately after that premium is received. If no premium is due at time $t$, the two values are equal, provided there is no other cash flow between the valuation instants.

The minus and plus signs are timing labels, not indicators that a reserve is negative or positive. A reserve itself can be positive, zero, or negative, depending on the contract and the valuation basis. If the policy has lapsed, the premium is not paid and this before-and-after premium conversion does not apply as stated.

Prospective Meaning of the Two Values

A prospective reserve is the expected present value of future benefits less the expected present value of future premiums, conditional on the policy being in force at the valuation time. The valuation uses the stated interest and mortality assumptions. To distinguish the anniversary instants, be precise about whether the premium due at time $t$ is still a future cash flow or has already been paid.

Immediately before payment, the premium due at time $t$ is included among future premiums. Since the policy is in force at this instant and payment is due, that premium is certain under the stated timing convention. Immediately after payment, it is no longer a future premium; future premiums begin with a later payment date and are payable only if the policy remains in force then.

{}_tV_x^- = \operatorname{EPV}(\text{future benefits}\mid\text{in force at }t) - \operatorname{EPV}(\text{future premiums including the premium at }t\mid\text{in force at }t)

{}_tV_x^+ = \operatorname{EPV}(\text{future benefits}\mid\text{in force at }t) - \operatorname{EPV}(\text{future premiums after the payment at }t\mid\text{in force at }t)

Both expressions use the same conditional status at the anniversary: the policy is in force at time $t$. The difference is whether the premium at that time remains to be paid. A premium scheduled for a later anniversary is not certain at time $t$: it is paid only if the policy is still in force at that later date.

Deriving the Conversion

Let $P_t$ be the premium due at time $t$. Under the assumptions above, the future benefit cash flows have the same present value immediately before and immediately after the premium. Future premiums after time $t$ also have the same expected present value at both instants. The only change is that the due premium $P_t$ is included in the “before” valuation and excluded from the “after” valuation.

Separate the due premium.
Write the present value of premiums before payment as the due premium $P_t$ plus the present value of premiums payable after time $t$.

Subtract premiums from benefits.
The before-payment reserve subtracts both $P_t$ and the later premiums. The after-payment reserve subtracts only the later premiums.

Compare the expressions.
The before-payment reserve is lower by exactly $P_t$, so adding $P_t$ to it gives the after-payment reserve.

{}_tV_x^- = B_t - (P_t + Q_t), \qquad {}_tV_x^+ = B_t - Q_t

Here $B_t$ is the conditional expected present value of future benefits, and $Q_t$ is the conditional expected present value of premiums payable after time $t$. Subtracting the first expression from the second, or rearranging directly, gives the conversion.

Theorem: If a premium $P_t$ is due and received at time $t$, and no other cash flow occurs between the two valuation instants, then $$ {}_tV_x^+ = {}_tV_x^- + P_t, \qquad {}_tV_x^- = {}_tV_x^+ - P_t. $$ For a level annual premium $P$, replace $P_t$ by $P$. If no premium is due at time $t$, the two reserve values are equal under the same no-other-cash-flow condition.

This is a timing identity, not a new mortality formula. It does not depend on the size of the death benefit, the form of the mortality model, or whether the reserve is calculated prospectively or retrospectively, provided both values use the same basis and differ only by receipt of the premium. It does not apply unchanged if an expense, benefit, refund, or other cash flow also occurs between the instants; those cash flows must be included in the conversion.

Worked Calculations

Worked Example: One-Year Term Reserve at an Anniversary

A hypothetical one-year term policy is in force at the start of its final policy year. The annual effective interest rate is $i=5\%$. The probability of death during that year is $q=0.10$, and the benefit of $10,000 is paid at the end of the year if death occurs. A premium of $900 is due immediately. Calculate the reserve immediately before and after payment, using a prospective net cash-flow valuation with no further premiums.

Method and assumptions. Conditional on being in force at the anniversary, the expected present value of the death benefit is $v qB$, where $v=1/(1+i)$. There are no later premiums or other cash flows.

First calculate the discount factor and the benefit value:

v=\frac{1}{1.05}=0.95238095,\qquad v qB=0.95238095(0.10)(10{,}000)=\$952.38095.

Immediately after the premium has been paid, the due premium is no longer a future cash flow, so ${}_tV^+=\$952.38095$. Immediately before payment, subtract the premium that is still due:

{}_tV^- = \$952.38095-\$900=\$52.38095.

Rounded to the nearest cent, the before-payment reserve is $52.38 and the after-payment reserve is $952.38. The check is ${}_tV^+-{}_tV^-=\$900$, exactly the premium. The common mistake is to subtract the $900 from the after-payment reserve as well, even though it has already been received.

Worked Example: A Later Premium Depends on Survival

A hypothetical two-year term policy is in force at time $t=1$, just before its anniversary premium. The annual effective interest rate is $i=10\%$. The death benefit is $5,000 at the end of either remaining policy year. The conditional probability of death in the first remaining year is $q_1=0.20$; conditional on surviving that year, the probability of death in the second remaining year is $q_2=0.10$. A premium of $600 is due at $t=1$, and another $600 is due at $t=2$ only if the policyholder survives in force to that date. Calculate both reserves at $t=1$.

Method and timing. Immediately after the $t=1$ premium, the only possible future premium is at $t=2$. It is payable with probability $1-q_1=0.80$, and is discounted one year. The death benefit in the first remaining year is discounted one year; the second-year benefit is paid only if the policyholder survives the first year and is discounted two years.

With $v=1/1.10=0.90909091$, the expected present value of benefits immediately after the premium is:

v q_1(5{,}000)+v^2(1-q_1)q_2(5{,}000) =0.90909091(0.20)(5{,}000) +0.90909091^2(0.80)(0.10)(5{,}000)

=\$909.09091+\$330.57851=\$1{,}239.66942.

The expected present value of the future premium at $t=2$ is:

v(1-q_1)(600)=0.90909091(0.80)(600)=\$436.36364.

Therefore, immediately after payment at $t=1$:

{}_1V^+=\$1{,}239.66942-\$436.36364=\$803.30578.

Immediately before payment, the $600 due at $t=1$ is also a future premium in the valuation. Thus ${}_1V^-=\$803.30578-\$600=\$203.30578$. Rounded to the nearest cent, the reserves are $803.31 after and $203.31 before payment. The check is a difference of $600. The common mistake is to discount the premium due now or to multiply it by a survival probability: at $t=1$, conditional on the policy being in force, it is due immediately and is certain.

Worked Example: Different Benefits by Policy Year

A hypothetical two-year policy is in force at time $t=1$. The annual effective interest rate is $i=4\%$. If death occurs in the first remaining year, the benefit at year-end is $8,000; if the policyholder survives and dies in the second remaining year, the benefit at that year-end is $12,000. The corresponding conditional death probabilities are $q_1=0.25$ and $q_2=0.10$. A $900 premium is due at $t=1$, and a further $900 premium is due at $t=2$ only if the policyholder survives to that date. Find both reserves.

Method and assumptions. Use a prospective valuation conditional on being in force at $t=1$. The first-year benefit has probability $0.25$; the second-year benefit has probability $(1-0.25)(0.10)=0.075$. The premium at $t=2$ is payable only after survival through the first remaining year, with probability $0.75$.

The discount factor is $v=1/1.04=0.96153846$. The expected present value of benefits is:

v(0.25)(8{,}000)+v^2(0.75)(0.10)(12{,}000) =\$1{,}923.07692+\$832.10059 =\$2{,}755.17751.

After the $t=1$ premium has been paid, the only future premium is the one at $t=2$. Its expected present value is:

v(0.75)(900)=\$649.03846.

It follows that ${}_1V^+=\$2{,}755.17751-\$649.03846=\$2{,}106.13905$. Before payment, subtract the $900 due at $t=1$, giving ${}_1V^-=\$1{,}206.13905$. To the nearest cent, the after-payment reserve is $2,106.14 and the before-payment reserve is $1,206.14. Their difference is $900. The common mistake is to treat the second-year premium as certain; it must be weighted by the 0.75 probability of surviving to its payment date.

Common Timing Errors

On an exam problem, first locate the valuation instant relative to each scheduled cash flow. Then determine whether the policy is in force and whether a premium is due at that instant. These checks prevent the most frequent errors.

Using one value for both instants. When a premium is due, the values differ by the premium amount—not by its discounted value.
Subtracting the due premium twice. The after-payment value excludes the premium just received; do not subtract it again as a future outgo.
Discounting a premium due immediately. A payment at the valuation instant has time-zero discount factor 1, not $v$.
Applying a survival probability to the current premium. Conditional on being in force immediately before the due date, the premium at that date is certain under the stated convention. A later premium is survival-contingent.
Ignoring other simultaneous cash flows. If a benefit or expense also occurs at the anniversary, the change between reserve values must account for that cash flow as well.

Key takeaway: At an anniversary when an annual premium $P$ is due, the reserve immediately after receipt is the reserve immediately before receipt plus $P$. The identity is reliable only when the valuation basis is unchanged and no other cash flow separates the two instants.

Check Your Understanding

For each problem, state whether the premium due at the valuation time is included as a future premium before payment and excluded after payment.

A hypothetical one-year policy is in force immediately before its final annual premium is paid. The annual effective interest rate is $5\%$, the probability of death in the year is $0.08$, the end-of-year benefit is $25,000, and the premium due now is $1,200. There are no later premiums. Find both prospective reserves. (Answer: before $704.76; after $1,904.76.)
At time $t=1$, a two-year term policy has $i=5\%$, death probabilities $q_1=0.20$ and $q_2=0.10$, and a $6,000 end-of-year death benefit in either remaining year. A $500 premium is due now, and a further $500 is due at $t=2$ only if the policyholder survives. Find both reserves. (Answer: before $697.28; after $1,197.28.)
A policy has a reserve of $-$250$ immediately before a $800 annual premium is received. There are no other cash flows between the two valuation instants. Find the reserve immediately after payment. (Answer: $550.)
At $t=1$, a two-year policy has $i=4\%$. Its benefits are $10,000 at the end of the first remaining year and $20,000 at the end of the second remaining year if death occurs then. The conditional death probabilities are $q_1=0.10$ and $q_2=0.05$. A $1,000 premium is due now and another is due at $t=2$ only if the policyholder survives. Find the before- and after-payment reserves. (Answer: before $-$71.75$; after $928.25.)
A policy is valued at an anniversary when no premium is due, and no benefit, expense, or other cash flow occurs between the two valuation instants. If its reserve immediately before that instant is $1,450, what is its reserve immediately after? Explain the timing. (Answer: $1,450; no cash flow separates the valuations.)

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