Boundary Conditions for Terminating States

Why Boundary Conditions Matter

A reserve equation describes how a liability changes while a contract is in force, but it does not determine a unique reserve path on its own. The equation must be paired with a value at the end of the projection period. In a multi-state contract, that value depends on which state the policyholder occupies and what is payable when coverage ends.

Death, recovery, and maturity require different treatment. Death commonly terminates the contract, so after any death benefit has been settled there is no further contract liability. Recovery usually does not terminate the contract: it moves a covered person from a disabled state to an active state. Maturity ends coverage, but a payment due at maturity is part of the liability immediately before it is paid. Confusing these situations can produce an incorrect boundary value even when the differential equation is otherwise correct.

Definition: Let $V_i(t)$ be the reserve at time $t$, conditional on the contract being in state $i$. A boundary condition specifies the reserve at the endpoint of the projection, with the payment timing made explicit. If a benefit is paid at maturity time $T$, this tutorial uses $V_i(T-)$ for the reserve immediately before payment and takes it to equal the maturity amount due in state $i$. Immediately after that payment, the terminated contract has reserve zero.

Terminal Values and State Transitions

A state label describes the contract’s status at a given time; it does not by itself determine the reserve. The key question is whether future contract cash flows remain after entry into the state. If death terminates coverage and the death benefit is settled at the transition, the reserve in the post-settlement death state is zero. The death benefit belongs in the cash flow for the transition into death, not in a continuing death-state reserve.

Recovery has a different interpretation. If a disabled policyholder recovers and coverage continues in the active state, the reserve immediately after recovery is the active-state reserve at that time. A recovery payment, if any, is an additional transition benefit. Only if recovery itself ends the contract should the post-recovery reserve be zero after any recovery settlement has been made.

At maturity, specify whether the reserve is measured before or after the maturity payment. Under the convention here, the pre-payment terminal reserve is the amount due at maturity. If no maturity benefit is due in a particular state, that state’s pre-payment terminal reserve is zero. Writing “the reserve at maturity is zero” without stating that it is measured after payment can omit a benefit that is still due.

Event or state at the endpoint	Boundary interpretation
Death terminates coverage; death benefit settled	Post-settlement death-state reserve is zero; include the death benefit on the transition into death.
Recovery from disabled to active; coverage continues	Use the active-state reserve after recovery, plus any recovery transition benefit where applicable.
Recovery terminates coverage	Post-settlement reserve is zero; include any recovery payment on the terminating transition.
Contract maturity with a benefit due	Immediately before payment, the reserve equals the maturity benefit due in that state.

Worked Example: Assigning Terminal Reserves by State

A hypothetical contract matures at time $T$. Immediately before the maturity payment, it pays $8,000 if the insured is active and $13,000 if the insured is disabled. Death terminates the contract, and a $50,000 death benefit is paid when death occurs. Determine the relevant endpoint reserves under the stated timing convention.

Active at maturity. The $8,000 is due at $T$, so the reserve immediately before payment is $V_A(T-)=\$8{,}000$. Immediately after the maturity payment, the contract has ended and its reserve is zero.

Disabled at maturity. The state-specific amount due is $13,000, giving $V_D(T-)=\$13{,}000$. The active-state maturity amount is not substituted just because the policyholder might have been active earlier.

Death during coverage. The $50,000 is included as a benefit on the transition into death. After that payment has been settled, the death-state reserve is zero. It would be incorrect to set the reserve immediately before death to zero: that reserve must account for the death benefit payable at the transition.

Check. The two maturity boundary values match the amounts due in their respective states, and the death benefit is assigned once, to the transition cash flow. This avoids both omitting the death benefit and counting it again as a continuing liability.

Thiele Equations with Transition Benefits

Assume a continuous-time Markov model on the interval $[0,T)$, with force of interest $\delta$. From state $i$, let $q_{ij}(t)$ be the transition intensity to state $j$. Let $b_{ij}(t)$ be the benefit paid immediately on that transition, positive when it is an outgo, and let $c_i(t)$ be the continuous net outgo rate in state $i$. Premium income is negative net outgo. With reserves defined as expected future net outgo, the Thiele equation is

\frac{dV_i}{dt} = \delta V_i+c_i(t)+\sum_{j\ne i}q_{ij}(t)\bigl(b_{ij}(t)+V_j(t)-V_i(t)\bigr).

The destination reserve $V_j(t)$ appears because a transition changes the state in which future benefits and premiums are valued. For a transition into a settled, terminating state, set $V_j(t)=0$. For recovery into a covered active state, use $V_j(t)=V_A(t)$. The transition payment $b_{ij}$ is separate from that destination reserve.

Assumptions for the equation: Intensities and cash-flow rates are defined per year; the model is Markov with respect to the stated contract states; transition benefits are paid at the transition; and $V_i(t)$ is measured on the same payment-timing basis. If a benefit is paid later than the transition, or if the state description omits information needed to determine future cash flows, the model or cash-flow timing must be adjusted before using this equation.

The equation follows from a short-interval balance. Over an interval of length $h$, starting in state $i$, the probability of a transition to $j$ is $q_{ij}(t)h+o(h)$, and the probability of no transition is $1-\sum_{j\ne i}q_{ij}(t)h+o(h)$. The reserve must equal the continuous outgo during the interval plus the discounted expected value at its end. Expanding that balance to first order gives the equation.

V_i(t)=c_i(t)h+e^{-\delta h} \left[ \left(1-\sum_{j\ne i}q_{ij}(t)h\right)V_i(t+h) +\sum_{j\ne i}q_{ij}(t)h\bigl(b_{ij}(t)+V_j(t+h)\bigr) \right]+o(h).

Using $e^{-\delta h}=1-\delta h+o(h)$ and $V_i(t+h)=V_i(t)+V_i'(t)h+o(h)$, then collecting the terms of order $h$, yields the stated Thiele equation. In particular, the term $-q_{ij}V_i$ accounts for leaving the current state; omitting it would incorrectly treat the policy as remaining in state $i$ after the transition.

Worked Example: The Reserve Change When Recovery Is Possible

At a particular time, a hypothetical disabled state has reserve $V_D=\$40{,}000$. Recovery to the active state occurs at intensity $0.20$ per year, and the active-state reserve is $V_A=\$25{,}000$. Each recovery pays $2,000. Death occurs at intensity $0.03$ per year and pays $80,000. The force of interest is $0.04$, there is no continuous net outgo, and death is terminating. Find $dV_D/dt$ at this time.

Method. Apply the Thiele equation in the disabled state. Recovery leads to the active state, so its transition contribution uses $2{,}000+V_A-V_D$. Death leads to a settled terminating state, so its contribution uses $80{,}000+0-V_D$.

\begin{aligned} \frac{dV_D}{dt} &=0.04(40{,}000) +0.20(2{,}000+25{,}000-40{,}000) +0.03(80{,}000-40{,}000)\\ &=1{,}600+0.20(-13{,}000)+0.03(40{,}000)\\ &=1{,}600-2{,}600+1{,}200\\ &=\$200\text{ per year}. \end{aligned}

The positive derivative means the reserve is increasing locally as time advances under these assumptions. Setting the recovery destination reserve to zero would instead treat recovery as contract termination, contrary to the scenario. The calculation also checks the transition signs: recovery transfers the liability from the disabled reserve to the active reserve, while death removes the continuing reserve after the death payment.

Solving Backward from Maturity

In a finite-term contract, the terminal conditions are specified at $T$, and the differential equations are solved backward to earlier durations. The equations may be coupled: for example, the active-state equation can depend on the disabled-state reserve when disability is entered. A useful way to organize the calculation is to solve a state whose equation is independent of the others first, then substitute that result into the remaining equations.

Fix the cash-flow convention.
Identify continuous outgo, transition benefits, and maturity payments, and state whether endpoint reserves are measured before or after a payment.

Assign endpoint values by state.
Use the maturity amount due for each covered state immediately before payment. Use zero after a terminating event has been fully settled.

Write every transition term.
For each transition, include its intensity multiplied by the transition benefit plus destination reserve minus current reserve.

Integrate from the endpoint.
Solve the reserve equations backward, preserving the state-specific boundary conditions.

Worked Example: Solving Active and Disabled Reserves Backward

A hypothetical one-year contract has active state $A$, disabled state $D$, and terminating death state. Rates are constant over $0\le t<1$, with annual force of interest $\delta=0.15$. From active, disability occurs at intensity $0.10$ and death at intensity $0.05$. From disabled, death occurs at intensity $0.15$; there is no recovery. Death pays $100,000 from either covered state. A transition from active to disabled pays $10,000. At maturity, the benefit is $30,000 if active and $20,000 if disabled. There are no continuous net outgo or premiums. Find both reserves at time 0, in thousands of dollars.

Method and boundary values. Work in thousands of dollars. The death state has reserve zero after settlement. Immediately before maturity, the state-specific maturity benefits give $V_A(1-)=30$ and $V_D(1-)=20$.

Disabled-state equation. Since death is the only transition from disabled, Thiele’s equation is

\begin{aligned} V_D'(t) &=0.15V_D(t)+0.15(100-V_D(t))\\ &=15. \end{aligned}

Integrate and apply $V_D(1)=20$:

V_D(t)=20+15(t-1)=5+15t, \qquad V_D(0)=5.

Active-state equation. Disability pays 10 and enters state $D$; death pays 100 and enters a zero-reserve state. Therefore

\begin{aligned} V_A'(t) &=0.15V_A(t) +0.10\bigl(10+V_D(t)-V_A(t)\bigr) +0.05\bigl(100-V_A(t)\bigr)\\ &=0.10(10+V_D(t))+0.05(100)\\ &=6.5+1.5t, \end{aligned}

where the $V_A$ terms cancel because $0.15=0.10+0.05$. Integrate backward from $V_A(1)=30$:

\begin{aligned} V_A(0) &=V_A(1)-\int_0^1(6.5+1.5t)\,dt\\ &=30-\left(6.5+\frac{1.5}{2}\right)\\ &=30-7.25=22.75. \end{aligned}

Thus $V_A(0)=\$22{,}750$ and $V_D(0)=\$5{,}000$. As a check on the endpoint conditions, the solutions give $V_D(1)=20$ and $V_A(1)=30$, exactly the respective maturity amounts in thousands. The transition payment is included once, in the active-to-disabled term.

Common mistake and correction. If the disabled reserve is incorrectly set to zero, the active equation becomes $V_A'=0.10(10)+0.05(100)=6$, giving the incorrect result $V_A(0)=30-6=\$24{,}000$. This understates the correct active reserve by $1,250 because it ignores the liability carried into the disabled state. The transition does not end the contract.

Common Boundary-Condition Traps

Setting every terminating-state reserve to zero before settlement. Zero applies after the termination cash flows have been paid or otherwise included. A death benefit due at the transition remains part of the originating state’s liability.
Treating recovery as death. If coverage continues after recovery, the destination reserve is the reserve in the recovered-to state. Zero is appropriate only when recovery terminates coverage and its settlement has been accounted for.
Using a post-payment value as a pre-payment boundary. If a maturity benefit is due at $T$, then the pre-payment reserve includes it. State clearly whether a stated boundary value is immediately before or after payment.
Omitting the current-state subtraction in a transition term. Each term contains $b_{ij}+V_j-V_i$. The $-V_i$ reflects that the contract leaves state $i$.
Mixing payment units or time units. If reserves are in thousands, transition benefits and terminal amounts must also be in thousands. Intensities and the force of interest must use the same time unit as the derivative.

Key takeaway: A boundary value is determined by the contract cash flows remaining at that point, not merely by the name of the state. Put benefits on the transition where they are paid, carry the destination-state reserve through nonterminating transitions such as recovery, and apply maturity values on a clearly stated pre- or post-payment basis.

Check Your Understanding

Use the stated payment timing and reserve convention in each problem.

A contract matures with a benefit of $9,000 in the active state and $14,000 in the disabled state. Death terminates coverage and pays $60,000 at the transition. Give the pre-payment maturity reserves and the post-settlement death-state reserve. (Answer: $V_A(T-)=\$9{,}000$, $V_D(T-)=\$14{,}000$, and the post-settlement death reserve is $0$.)
In the disabled state, $\delta=0.03$, recovery intensity is $0.25$, and death intensity is $0.02$. At the time considered, $V_D=60$, $V_A=45$, the recovery benefit is 1, the death benefit is 100, and continuous net outgo is zero, all amounts in thousands. Find $V_D'$. (Answer: $-0.9$ thousand dollars per year.)
A state has only a terminating death transition, with $\delta=0.06$, death intensity $0.06$, and death benefit $90,000. The contract matures in two years with a $12,000 benefit due in this state; there are no other cash flows. Find the reserve at time 0. (Answer: $1,200.)
Over one year, $\delta=0.30$. From active, disability occurs at intensity $0.20$ and death at $0.10$; from disabled, death occurs at $0.30$. Death pays $50,000, active-to-disabled transition pays $4,000, and maturity pays $15,000 active or $25,000 disabled. There are no other cash flows. Find both time-0 reserves in thousands. (Answer: $V_D(0)=\$10{,}000$ and $V_A(0)=\$5{,}700$.)

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