Boundary Values at Expiry and Maturity

Why Boundary Values Need Care

A reserve is a value at a specified time, based on the benefits and premiums that remain from that time onward. At expiry or maturity, the set of remaining cash flows can change abruptly: coverage may end, a maturity benefit may be paid, or a premium may be due. The relevant question is therefore not only what is the reserve at the boundary?, but also which side of a boundary payment is being valued?

This distinction is especially important for term insurance and endowment insurance. A term policy has no benefit for survival beyond its term, whereas an endowment policy pays a maturity benefit if the insured survives to the end. The calculations below use net prospective reserves: expected present value of future benefits less expected present value of future premiums, conditional on the policy being in force and the insured being alive at the valuation time.

Definition: At a valuation time, a prospective reserve is the conditional expected present value of benefits still payable minus the conditional expected present value of premiums still receivable. Denote the reserve immediately after cash flows due at time $k$ by ${}_kV_x$, and the value immediately before those cash flows by ${}_kV_x^-$. A premium receivable at the instant is included as a future inflow in the before-payment value, so it is subtracted in the reserve; a benefit payable at the instant is included as a liability in the before-payment value.

The convention for the sides of a boundary is tied to the contract’s timing rules. Immediately before a premium is paid, that premium remains to be received; immediately after, it does not. Immediately before a benefit is paid, that benefit remains a liability; immediately after, it has been settled. If a contract ends without a payment at the boundary, there is no future contractual cash flow after termination, so the post-expiry reserve is zero.

Term Insurance at Its Expiry

For an $n$-year term policy, coverage applies only during the specified term. In an annual model, the reserve at duration $n-1$, after any premium due at that time, reflects the final covered policy year. Once that year has ended and the policy has expired, there are no future benefits or premiums, and the reserve is zero.

For example, suppose the death benefit $B$ is paid at the end of the year of death, the effective annual interest rate is $i$, and $q_{x+n-1}$ is the probability of death during the final policy year for a life alive at its start. If no premium remains after duration $n-1$, then the final-year reserve is $q_{x+n-1}vB$, where $v=1/(1+i)$. The benefit is discounted one year because it is paid at the end of that year.

The annual model locates this final-year risk at duration $n-1$; it does not assign the entire final year’s risk to a reserve at the instant the contract expires. With a continuous-time model and no probability mass at the expiry instant, the remaining exposure tends to zero as time approaches expiry from below. In either convention, once the term has ended, the post-expiry reserve is zero.

Worked Example: A Premium Due in the Final Year of Term Insurance

A hypothetical two-year term policy pays $1,000 at the end of the year of death. The effective annual interest rate is 0%. The annual death probabilities are $q_x=0.10$ and $q_{x+1}=0.10$. Level net premiums are payable at times 0 and 1 if the insured is alive. Find the premium, the reserve at time 1 immediately before and after the premium, and the reserve after expiry at time 2.

Method. First use the equivalence principle at issue, then value only the cash flows remaining at each boundary. This avoids treating a premium due at time 1 as though it had already been received.

The probability of surviving the first year is $p_x=1-q_x=0.90$. The expected present value of benefits at issue is

$$ 1000(0.10+0.90(0.10))=1000(0.19)=190. $$

The premium at time 0 is certain. The premium at time 1 is payable only if the insured survives the first year, so the expected present value of premiums is $P(1+0.90)=1.90P$. Equating the two values gives

1.90P=190 \qquad\Longrightarrow\qquad P=100.

At time 1, conditional on being alive, the expected benefit during the final year is $0.10(1000)=100$. Immediately before the premium is paid, the insurer still expects to receive $100 at that instant, so

{}_1V_x^-=100-100=0.

Immediately after that premium is paid, no premium remains. The final-year expected benefit is still $100, hence ${}_1V_x=100$. After the policy has completed its second year and expired, no benefit or premium remains, so ${}_2V_x=0$.

Check. The final-year reserve after the premium equals the expected final-year benefit, and the premium-free post-expiry reserve is zero. A common error would be to report the before-premium reserve as $100; that omits the $100 premium due at time 1. The corrected before-payment value is $100 less $100, or $0.

Endowment Insurance at Maturity

An endowment policy can pay on death during the term and also pay a maturity benefit if the insured survives to the end. In the last policy year, both possible outcomes must be included in the prospective value. If the death benefit is $B_d$, the survival benefit at maturity is $B_m$, and the last-year probabilities are $q_{x+n-1}$ and $p_{x+n-1}=1-q_{x+n-1}$, then, immediately after the final premium,

{}_{n-1}V_x = v\left(q_{x+n-1}B_d+p_{x+n-1}B_m\right).

This formula assumes both benefits are payable at the end of the final year, the insured is alive at its start, and no further premiums are due after the valuation time. At maturity, conditional on survival, the maturity benefit is due immediately. Thus the reserve immediately before payment is $B_m$; immediately after settlement, if the contract has ended and no other cash flow remains, it is zero.

Why the final-year formula holds: At the start of the final year, there are two mutually exclusive outcomes. Death occurs with probability $q_{x+n-1}$, producing a benefit $B_d$; survival occurs with probability $p_{x+n-1}$, producing $B_m$. The expected payment at year-end is therefore $q_{x+n-1}B_d+p_{x+n-1}B_m$. Discounting that amount one year gives the displayed reserve. If a premium is due at the valuation instant, its amount must also be subtracted from the value immediately before payment.

Worked Example: Different Death and Maturity Benefits

A hypothetical endowment policy is in its final year. The effective annual interest rate is 10%, the probability of death during that year is 0.10, and the probability of survival is 0.90. The death benefit is $5,000 and the maturity benefit is $8,000. A premium of $1,200 is due at the start of this final year. Find the reserve immediately before and after that premium, and immediately before and after maturity payment.

Method. Calculate the final-year expected benefit, discount it one year, and then account for the premium according to whether it has been paid.

Immediately after the premium at the start of the final year, the expected benefit payable at year-end is

$$ 0.10(5000)+0.90(8000)=500+7200=7700. $$

The one-year discount factor is $v=1/1.10$. Therefore the reserve after the premium is

{}_{n-1}V_x=\frac{7700}{1.10}=7000.

Immediately before the premium, the insurer still expects to receive $1,200, so the reserve is $7000-1200=\$5,800$. Immediately after payment it is $7,000.

At maturity, conditional on survival, the $8,000 maturity benefit is due. The value immediately before its payment is $8,000, and after payment it is $0 because the contract has ended. The death benefit is not added to the maturity-date value: it applies to the alternative outcome of death during the final year.

Check. The final-year reserve is the discounted weighted average of the two possible benefits, and $0.10+0.90=1$. A common error is to use only the death benefit in the final-year calculation. That would give $0.10(5000)/1.10$, omitting the $8,000 payment on survival. Including both mutually exclusive outcomes gives the correct $7,000 post-premium reserve.

Comparing Term and Endowment Boundaries

The same final-year mortality assumptions can produce different reserves because the contracts promise different cash flows. The term policy has a benefit only if death occurs before expiry. The endowment policy also promises a survival benefit at maturity. The premium timing matters as well: the before-premium reserve is lower than the after-premium reserve by the premium due, while a benefit due immediately increases the before-payment liability and disappears from the reserve after settlement.

Worked Example: Comparing Term and Endowment Values at the Same Duration

Consider hypothetical two-year contracts with effective annual interest rate 0%. The death probabilities are $q_x=0.20$ and $q_{x+1}=0.25$, so $p_x=0.80$ and $p_{x+1}=0.75$. Both contracts pay $900 at the end of the year of death. Premiums are level, payable at times 0 and 1 while the insured is alive. The endowment contract also pays $900 at time 2 if the insured survives. Find each net premium and the relevant duration-1 and expiry reserves.

Method. Apply the equivalence principle to each contract separately, then calculate the final-year reserve conditional on survival to time 1. Do not count a term survival benefit that the contract does not provide.

For the term policy, expected benefits at issue are

$$ 900(0.20)+0.80(0.25)(900)=180+180=360. $$

The expected number of premiums, with no interest discounting, is $1+0.80=1.80$. Thus $1.80P=360$, giving the term premium $P=\$200$. At time 1, immediately after the premium, the final-year benefit has expected value $0.25(900)=\$225$, so the reserve is $225. Immediately before the premium it is $225-200=\$25$. After expiry at time 2, it is $0.

For the endowment policy, every insured either dies during the term or survives to maturity, and each outcome pays $900. The expected benefit at issue is therefore $900. The premium factor remains $1.80$, so $1.80P=900$ and the endowment premium is $P=\$500$.

At time 1 after the premium, the possible final-year payments are $900 on death and $900 on survival. Their expected total is $0.25(900)+0.75(900)=\$900$. The reserve is $900; immediately before the premium it is $900-500=\$400$. At maturity, the reserve is $900 immediately before the survival benefit is paid and $0 immediately after payment.

Check. The term reserve in the final year is only the expected death benefit; the endowment reserve includes both death and survival outcomes. The calculation also confirms that each net premium equates expected premiums and benefits at issue under the stated assumptions.

Boundary Checks to Use in Practice

Before reporting a reserve at expiry or maturity, identify the valuation instant and list what remains payable or receivable from that instant. In particular:

For term insurance, distinguish the reserve for the final covered year from the reserve after the term has ended.
For endowment insurance, include the survival benefit when valuing the final policy year.
If a premium is due at the boundary, subtract it from the before-payment reserve; do not include it after it has been paid.
If a benefit is due at maturity, include it immediately before payment and remove it after settlement.
State the payment timing and whether the valuation is conditional on the insured being alive and the policy remaining in force.

A useful check is to examine the cash-flow set on each side of the boundary. A term contract with no maturity payment has no future contractual cash flows after expiry. An endowment contract has a maturity liability just before the survival benefit is paid, but none after the payment and termination. These checks help keep boundary values consistent with the contract rather than with a memorized pattern.

Check Your Understanding

For each problem, identify the relevant boundary cash flows before calculating. Amounts are hypothetical; use the stated annual model and assumptions.

A one-year term policy pays $12,000 at the end of the year of death. The effective annual interest rate is 0%, the death probability is 0.04, and a premium of $300 is due at the start of the year. Find the reserve just before and just after the premium, and after expiry. (Answer: $180 before the premium; $480 after the premium; $0 after expiry.)
An endowment is in its final year. The effective annual interest rate is 5%, the death probability is 0.15, and the survival probability is 0.85. The death benefit is $4,000, the maturity benefit is $6,000, and a $500 premium is due at the start of the year. Find the reserve after the premium, before the premium, immediately before maturity payment, and after payment. (Answer: $5,428.57 after the premium; $4,928.57 before it; $6,000 immediately before maturity payment; $0 after payment.)
A term policy has one covered year remaining. The effective annual interest rate is 4%, the death probability in that year is 0.12, the death benefit is $25,000, and a $800 premium is due now. Find the reserve just after and just before the premium, then after expiry. (Answer: $2,884.62 after the premium; $2,084.62 before it; $0 after expiry.)
An endowment is in its final year. The effective annual interest rate is 25%, the death probability is 0.20, and the survival probability is 0.80. The death benefit is $8,000, the maturity benefit is $12,000, and a $1,500 premium is due at the start of the year. Find the reserve on both sides of the premium and maturity benefit. (Answer: $8,960 after the premium; $7,460 before the premium; $12,000 before maturity payment; $0 after payment.)
A two-year endowment pays $950 at the end of the year of death or at time 2 on survival. The effective annual interest rate is 0%, $q_x=0.10$, and $q_{x+1}=0.20$. Level premiums are payable at times 0 and 1 if alive. Find the net premium and the reserve at time 1 before and after the premium, then immediately before and after maturity payment. (Answer: net premium $500; time-1 reserve $450 before and $950 after the premium; $950 immediately before maturity payment and $0 after payment.)

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