Reserves at Maturity and Scheduled Expiry

What Happens to a Reserve at the Contract Boundary?

A contract reaching maturity or scheduled expiry does not automatically have a zero reserve at every instant on that date. The key is whether the valuation is made before or after the terminal cashflows. A maturity benefit may still be payable immediately before settlement; a final premium may also be due. Once all contractual payments have been made and coverage has ended, there is no remaining liability under the terminated contract.

In a multi-state model, the amount due at the boundary can depend on the state occupied at that time. For example, a hypothetical policy might pay a maturity amount to a person in the active state, a smaller amount in the disabled state, and nothing if death has occurred. The reserve must be valued conditional on the relevant state, rather than by using one terminal amount for every policy.

Definition: Let $T$ be the maturity or scheduled expiry time, and let $j$ denote the contract state at $T$. Write $V_j(T^-)$ for the reserve immediately before terminal cashflows are settled, conditional on being in state $j$. Write $V_j(T^+)$ for the reserve immediately after those cashflows are settled and the contract has terminated. If the only remaining cashflows are the terminal benefit, terminal expense, and terminal premium, then $$ V_j(T^-)=B_j(T)+E_j(T)-P_j(T). $$ Here $B_j(T)$ is the benefit payable, $E_j(T)$ is any expense payable, and $P_j(T)$ is a premium received at the boundary, all conditional on state $j$. If termination leaves no continuing contractual liability, $V_j(T^+)=0$.

This convention treats a benefit or expense as a liability and a premium as an inflow, so premiums are subtracted. If a premium was already paid before the valuation instant, it is not included again in $P_j(T)$. Similarly, a payment scheduled after expiry is not a terminal cashflow unless the contract explicitly continues to provide it.

A reserve immediately before $T$ may therefore be positive, zero, or negative. A negative amount can occur when a premium is due but no benefit is payable in that state. It represents the net value of the specified cashflows at that instant; it should not be silently replaced by zero unless a separate minimum-floor rule applies.

Valuing Terminal Cashflows From an Earlier Date

At a date $t<T$, a policy’s state at maturity may not yet be known. Its reserve then reflects the probabilities of reaching each terminal state and the corresponding state-specific terminal cashflow. Assume, for illustration, that there are no other benefits, expenses, or premiums between $t$ and $T$, and that the effective annual interest rate is $i$. If $p_{ij}(t,T)$ is the probability of being in state $j$ at $T$, conditional on state $i$ at $t$, then

V_i(t)=v^{T-t}\sum_j p_{ij}(t,T)V_j(T^-), \qquad v=\frac{1}{1+i}.

The boundary values in this expression are net terminal cashflows when premiums or expenses are due at $T$. If there are additional cashflows before $T$, their expected present values must also be included. The formula does not apply unchanged if the contract continues beyond $T$: in that case there may be future liabilities after the date called “maturity,” and the post-boundary reserve need not be zero.

Timing rule: First identify the valuation instant, the state at that instant, and which cashflows remain unpaid. Then calculate the reserve. “At maturity” is not precise enough by itself: it may mean just before payment, just after payment, or a valuation at an earlier date using the maturity cashflows.

Why the Terminal-Value Formula Holds

The earlier-date formula follows directly from conditional expected present value. Conditional on being in state $i$ at time $t$, the contract reaches terminal state $j$ with probability $p_{ij}(t,T)$. In that event, the net amount due at $T$ is $V_j(T^-)$. Its value at time $t$ is $v^{T-t}V_j(T^-)$. Weighting by each state’s probability and adding gives the reserve:

\begin{aligned} V_i(t) &=E\left[v^{T-t}V_{J_T}(T^-)\mid J_t=i\right]\\ &=\sum_j P(J_T=j\mid J_t=i)\,v^{T-t}V_j(T^-)\\ &=v^{T-t}\sum_j p_{ij}(t,T)V_j(T^-). \end{aligned}

The final equality uses the definition $p_{ij}(t,T)=P(J_T=j\mid J_t=i)$ and the fact that the discount factor is the same for each terminal state. This calculation depends on the stated model assumptions: the transition probabilities describe the state at the boundary, the interest rate is effective annually over the relevant period, and the listed terminal cashflows are the only cashflows remaining.

Worked Example: State-Dependent Maturity Benefits and a Final Premium

A hypothetical two-year contract is valued at time $t=1$, with maturity at $T=2$. The policy is active at $t=1$. Over the final year, the probabilities of being active, disabled, or dead at maturity are $0.85$, $0.10$, and $0.05$, respectively. The effective annual interest rate is $4\%$. At maturity, the active state receives a benefit of $9,000 and pays a final premium of $1,000 at the same boundary; the disabled state receives $4,000 and pays no premium; the dead state receives nothing. There are no other cashflows during the final year. Find the reserve at $t=1$ and the state-specific reserves immediately before and after settlement at maturity.

Method. First calculate the net cashflow at maturity in each state. Then take the probability-weighted amount and discount it one year.

In the active state, the net terminal cashflow is $9{,}000-1{,}000=\$8{,}000$. In the disabled state it is $\$4{,}000$, and in the dead state it is $\$0$. These are also the state-specific reserves immediately before terminal settlement:

V_A(2^-)=8{,}000,\qquad V_D(2^-)=4{,}000,\qquad V_{\text{dead}}(2^-)=0.

The expected net amount payable at maturity, conditional on being active at $t=1$, is

$$ 0.85(8{,}000)+0.10(4{,}000)+0.05(0) =6{,}800+400+0 =7{,}200. $$

The one-year discount factor is $v=1/1.04$. Therefore,

V_A(1)=\frac{7{,}200}{1.04} =6{,}923.0769\ldots =\$6{,}923.08.

After all maturity cashflows are settled and the contract terminates, the reserve is zero in each terminal state. As a check, the probabilities sum to $0.85+0.10+0.05=1.00$, and the undiscounted expected terminal amount is $7,200, which is reduced by discounting at a positive interest rate. A common mistake would be to discount the gross active-state benefit of $9,000 and forget the $1,000 final premium; the correction is to form the state-specific net cashflow before taking the expected present value.

Scheduled Expiry Is Not Always a Maturity Benefit

A scheduled expiry may simply end coverage, with no payment to a policyholder who remains in force. Other state-contingent payments may still be due at expiry—for example, a final settlement for a claim already in progress. The correct boundary value comes from the contract’s actual cashflows, not from the word “expiry.”

The following example isolates the effect of a premium due at the boundary. It also illustrates why a reserve before settlement can be negative even though the reserve after expiry is zero.

Worked Example: A Final Premium at Scheduled Expiry

A hypothetical one-year contract expires at time $T=1$. Immediately before the boundary cashflows, a policy may be active, disabled, or dead. At expiry, an active policy has no benefit and owes a final premium of $300. A disabled policy receives a final settlement of $1,500 and owes no premium. No payment is due in the dead state. Coverage ends at $T$, and there are no cashflows after expiry. Find the state-specific reserves at $T^-$ and $T^+$.

Method. Apply the net terminal cashflow definition separately in each state. The final premium is subtracted because it is an inflow to the contract.

For the active state, $B_A(T)=0$, $E_A(T)=0$, and $P_A(T)=300$, so

V_A(T^-)=0+0-300=-\$300.

For the disabled state, the final settlement is a benefit of $1,500 and no premium is due:

V_D(T^-)=1{,}500+0-0=\$1{,}500.

In the dead state, there is no terminal payment:

V_{\text{dead}}(T^-)=0.

After the final premium or settlement has been paid and coverage has ended, there is no remaining contractual liability. Thus $V_A(T^+)=V_D(T^+)=V_{\text{dead}}(T^+)=0$. The negative active-state value is consistent with the stated timing: the premium is still due immediately after the $T^-$ valuation. If the premium had already been collected before the valuation instant, it would not be deducted again, and the active-state pre-settlement reserve would instead be zero. The common mistake is to include a premium without checking whether it is still unpaid at the valuation time.

Boundary Checks for Practical Valuation

A short timing check can prevent errors in calculations at maturity or expiry. Use the following sequence before applying a formula.

Fix the boundary and the valuation side.
State whether the reserve is required at an earlier date, immediately before terminal settlement, or immediately after settlement.

Identify the state used for conditioning.
At $T^-$, condition on the state at the boundary. At $t<T$, use the probabilities of reaching the possible terminal states from the state at $t$.

List only unpaid terminal cashflows.
Include benefits and expenses as outflows and premiums as inflows. Do not include a premium already paid or a payment that is not due under the contract.

Set the post-expiry value from continuing obligations.
If the contract has fully terminated and no liability remains, the reserve after settlement is zero. If coverage continues, that zero boundary does not apply.

A useful final check is to describe the cashflows in words: “At this instant, in this state, what amount remains to be paid or received?” This question is often more reliable than starting with a formula and trying to fit the contract to it. In particular, do not confuse a reserve conditional on a terminal state with an expected reserve before that state is known.

Key takeaway: At maturity or scheduled expiry, calculate the pre-settlement reserve from the net cashflows still due in each state. If all payments are settled and coverage ends, the post-settlement reserve is zero. At an earlier date, probability-weight the state-specific terminal values and discount them to the valuation date.

Check Your Understanding

For each question, use the stated timing convention and treat a premium as an inflow to the contract.

At maturity, a hypothetical policy pays $5,000 in the active state, $2,000 in the disabled state, and nothing in the dead state. No premium or expense is due at maturity. What are the three reserves immediately before settlement? (Answer: Active $5,000; disabled $2,000; dead $0.)
At time $t=2$, a policy is active. It matures one year later. The effective annual interest rate is $5\%$. At maturity, the probabilities of active, disabled, and dead are $0.60$, $0.30$, and $0.10$. Benefits are $9,000 active and $3,000 disabled; no benefit is paid on death. A $1,000 premium is due at maturity only if active. Find the reserve at $t=2$. (Answer: $5,428.57.)
A contract expires today. Immediately before settlement, the active state owes a final premium of $400 and has no benefit; the disabled state receives a final settlement of $1,500 and owes no premium; the dead state has no payment. Find each $T^-$ reserve and, assuming coverage ends, each $T^+$ reserve. (Answer: $T^-$: active $-\$400$, disabled $1,500, dead $0; all $T^+$ reserves are $0.)
A policy is active one year before maturity. At maturity, the probabilities of active, disabled, and dead are $0.70$, $0.20$, and $0.10$. Benefits are $15,000 active and $5,000 disabled; there is no premium at maturity. The effective annual interest rate is $5\%$. Calculate the reserve one year before maturity. (Answer: $(0.70(15{,}000)+0.20(5{,}000))/1.05=\$10{,}952.38$.)
At maturity, a benefit of $7,000 is payable in the active state, and an expense of $200 is payable in that state. A final premium of $500 is also due in the active state. In the disabled state, a $2,000 benefit is payable with no premium or expense. Find the pre-settlement reserve in each of these states. (Answer: Active $6,700; disabled $2,000.)

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