Terminal Reserves at Maturity or Expiry

Why the Terminal Boundary Matters

A profit test follows cash flows and reserves over a contract’s life. At the final duration, the reserve is not determined by the word “maturity” or “expiry” alone. It depends on what obligations remain at the reserve valuation time. If all benefits and expenses due under the contract have been paid, there is no future liability and the closing reserve is zero. If a final amount is still payable, the reserve must reflect it, even if insurance coverage has ended.

The timing distinction matters because profit is measured after accounting for the closing reserve. A reserve set to zero too early can omit a genuine obligation and overstate profit. A reserve retained after the obligation has already been paid can understate profit or count the same payment twice. The boundary condition must match the cash-flow definition used throughout the test.

Definition: A terminal reserve is the reserve at the final valuation time being considered. Under a prospective definition, it is the present value of benefits and expenses still payable after that time, less the present value of premiums and other income still receivable. If the contract has been fully settled and no future cash flows remain, the terminal reserve is zero. If settlement is outstanding, the terminal reserve includes the value of that outstanding obligation.

Fix the Valuation Time Before Setting the Boundary

Use a clear timing convention. In the examples below, reserves are measured at policy anniversaries. A reserve described as after settlement is measured after all cash flows due at that time have been paid or received. A reserve described as before settlement is measured just before the stated payment. For an amount due immediately at that valuation time, the pre-settlement reserve includes the amount due, while the post-settlement reserve does not.

For a standard annual profit test, let $V_{t-1}$ be the opening reserve before the premium at time $t-1$. Let $P_{t-1}$ be that premium, $i$ the effective annual interest rate, and $E_t$ and $B_t$ the expenses and benefits paid at time $t$. With a closing reserve $V_t$ measured after those end-of-year cash flows, the funds available before profit are

F_t=(V_{t-1}+P_{t-1})(1+i)-E_t-B_t.

The period profit is $C_t=F_t-V_t$. This identity is a balance: funds after interest and end-of-year outgo are divided between the closing reserve and profit. It applies only when the stated cash flows and reserve timing are consistent. A payment made after the valuation time must not be subtracted as though it had already been paid.

Boundary condition: If the final benefit is paid at time $n$ and $V_n$ is measured after that payment, set $V_n=0$ when no other obligations remain. If $V_n$ is measured immediately before the payment, it must include the amount due at time $n$. If settlement is later than time $n$, the reserve at $n$ includes the present value of the later settlement.

How the Prospective Reserve Gives the Boundary

The prospective definition explains the rule directly. At a valuation time $t$, only cash flows after $t$ belong in the reserve. If the last payment has just been made and the contract has no later premiums, benefits, or expenses, both future-outgo and future-income values are zero. Therefore $V_t=0$. If an amount $B$ is payable at that same time but has not yet been paid, it is still an outstanding obligation. Its value at that time is $B$, not zero.

For example, if an amount $B$ is due one year after a valuation time and the annual effective interest rate is $i$, its value at the valuation time is $B/(1+i)$. This follows by accumulating the reserve for one year: a reserve of $B/(1+i)$ grows to exactly $B$. The same reasoning applies when the settlement date is after the end of insurance coverage; coverage expiry does not erase an unsettled payment obligation.

State the valuation instant.
Specify whether the reserve is measured before or after the final payment.

List what remains due.
Include benefits and expenses still payable, and subtract any future premiums or other income still receivable.

Set the boundary and reconcile profit.
Use zero only after all contract obligations are settled, then check the final-period balance equation.

Worked Applications

Worked Example: A Two-Year Maturity Benefit

A hypothetical two-year savings contract pays a benefit of $1,100 at time 2. Premiums of $500 are paid at times 0 and 1, at the beginning of each coverage year. There are no expenses, death benefits, or other cash flows. The annual effective interest rate is $i=5\%$. Assume survival with certainty to maturity; all cash flows are deterministic, with no random distribution or parameters. Find the prospective reserves at times 0 and 1 and the profit in each year. Reserves at times 1 and 2 are measured after the year-end cash flows.

Method. Work backward from the post-settlement maturity boundary, then check the annual balance equation using unrounded reserves. Since the $1,100 maturity benefit is paid at time 2 and nothing remains afterward, $V_2=0$.

At time 1, the benefit due one year later has present value $1{,}100/1.05$. The time-1 premium of $500 is received immediately at that valuation time, so it is deducted from the prospective liability:

V_1=\frac{1{,}100}{1.05}-500 =1{,}047.619047619\ldots-500 =547.619047619\ldots

At time 0, both premiums remain receivable, at times 0 and 1 respectively. The time-0 premium is not discounted, while the time-1 premium and the maturity benefit are discounted:

V_0=\frac{1{,}100}{1.05^2}-500-\frac{500}{1.05} =997.732426304\ldots-500-476.190476190\ldots =21.541950113\ldots

For year 1, the accumulated opening reserve and premium are $(V_0+500)(1.05)=547.619047619\ldots$. There is no year-1 outgo, so $C_1=547.619047619\ldots-V_1=0$. For year 2, the accumulated reserve and premium are $(V_1+500)(1.05)=1{,}100$, so $C_2=1{,}100-1{,}100-V_2=0$. Both checks reconcile exactly when unrounded reserves are used. To the nearest cent, $V_0=\$21.54$ and $V_1=\$547.62$; the zero profits remain zero before rounding.

Common mistake avoided: Setting $V_1=1{,}100/1.05$ would ignore the premium payable at time 1 and overstate the reserve by $500. Also, using a rounded $V_1$ in the final balance may produce a small rounding difference; the exact reconciliation uses the unrounded value.

Worked Example: Reserve Immediately Before and After Settlement

A hypothetical contract has no premiums, expenses, or payments other than a guaranteed $300 amount payable at time 2. The annual effective interest rate is $i=4\%$. Assume no mortality before time 2 and deterministic cash flows; there is no random distribution or parameter. Calculate the reserve at time 1, then state the reserve immediately before and immediately after the time-2 payment. Check the final-year profit.

Method. Value the outstanding payment prospectively at time 1. At time 2, distinguish the instant before settlement from the instant after settlement.

At time 1, the $300 is due one year later, so

V_1=\frac{300}{1.04}=288.461538462\ldots

The amount grows to $V_1(1.04)=300$ at time 2. Immediately before the payment, the outstanding obligation is therefore $300. Once the $300 is paid, no future obligation remains, so the post-settlement reserve is $V_2=0$. The final-year profit balance is

$$ C_2=V_1(1.04)-300-V_2=300-300-0=0. $$

Common mistake avoided: Saying “the reserve at maturity is zero” without identifying the instant is incomplete here. The reserve is $300 just before settlement and zero immediately after it; these are different valuation times, not conflicting answers.

Worked Example: Coverage Expires Before a Final Refund Is Paid

A hypothetical two-year contract provides coverage through time 2 but, under its terms, pays a final $80 refund at time 3. There are no premiums or expenses after time 0 and no other benefits. The annual effective interest rate is $i=5\%$. Assume the policyholder survives with certainty and the refund is deterministic; there is no random distribution or parameter. Find the reserve at the coverage-expiry time 2 and at time 1, then check the year-3 profit when the refund is paid.

Method. Treat time 2 as coverage expiry, not full financial settlement. At time 2 the refund is still due one year later, so the reserve must remain positive.

At time 2, the present value of the refund due at time 3 is

V_2=\frac{80}{1.05}=76.190476190\ldots

At time 1, the same payment is two years away:

V_1=\frac{80}{1.05^2}=72.562358277\ldots

There are no intervening cash flows. The year-2 accumulated reserve is $V_1(1.05)=76.190476190\ldots=V_2$, giving zero year-2 profit. In year 3, the opening reserve accumulates to the payment:

V_2(1.05)=80,\qquad C_3=80-80-0=0.

If one incorrectly set $V_2=0$ merely because coverage ended, the year-2 balance would report profit of $76.190476190\ldots$ despite the unpaid refund. That apparent profit would be reversed when the refund was paid in year 3. The correction is to retain $V_2=80/1.05$ until settlement, then set the post-payment terminal reserve to zero.

Common mistake avoided: Confusing the end of insurance coverage with the end of all contractual cash flows. The correct boundary is determined by outstanding obligations, not by the coverage label.

Final Checks and Exam Traps

A terminal reserve condition is part of the profit-test model, not a cosmetic final entry. It determines how much of the accumulated fund is treated as profit and how much is held for future payments. Before accepting a final-period result, reconcile the cash-flow timing and reserve timing in the same balance equation.

Zero reserve too early: If a final benefit remains payable after the stated expiry date, its present value belongs in the expiry-date reserve.
Reserve retained too long: After the final payment has been made, do not carry the payment as a future liability.
Benefit omitted from the final period: If the benefit is paid in that period, subtract it in the funds balance before calculating profit and the post-settlement reserve.
Double-counting a payment: Do not both deduct a payment from funds and leave the same payment in a post-settlement reserve.
Premature rounding: Carry unrounded reserves through the final balance check. Round reported dollar amounts only at the end; small apparent differences can result from using displayed rounded values.

Key takeaway: A fully settled contract has a zero post-settlement terminal reserve. A pre-settlement reserve includes amounts due at that instant, and a reserve at coverage expiry can be nonzero when payment obligations continue beyond expiry. Always state the valuation instant and verify the final profit balance.

Check Your Understanding

For each problem, state the reserve timing clearly and use unrounded values in any balance check.

A hypothetical policy pays a certain $250 benefit at time 3, with no later obligations. What are the reserves immediately before and immediately after payment at time 3? Assume deterministic cash flows and no mortality. (Answer: $250 immediately before payment; $0 immediately after.)
A deterministic $220 payment is due at time 2. There are no premiums or expenses, no mortality, and $i=5\%$. What reserve is required at time 1, and what is the post-payment reserve at time 2? (Answer: $220/1.05=\$209.52$ at time 1, rounded to cents; $0 after payment.)
A final $400 amount is due one year after coverage expires at time 4. Assume no mortality, no other future cash flows, and $i=4\%$. Find the reserve at expiry and the reserve immediately after settlement at time 5. (Answer: $400/1.04=\$384.62$ at time 4, rounded to cents; $0 after settlement.)
A policy has an opening reserve of $150 at time 1, earns interest at $i=4\%$, and pays its only final benefit of $156 at time 2. Assume no premiums, expenses, or mortality. What is the post-settlement reserve and the year-2 profit? (Answer: $0 reserve and $150(1.04)-156=\$0$ profit.)
A guaranteed $100 refund is due at time 3, while coverage expires at time 2. Assume deterministic payment, no mortality, no other cash flows, and $i=5\%$. What reserve is required at time 2, and what would the incorrect year-2 profit be if the reserve were set to zero despite no year-2 payment? (Answer: $100/1.05=\$95.24$ reserve at time 2, rounded to cents; the incorrect apparent profit would be $95.24.)

What You'll Learn