Lump-Sum Benefits on Recovery and Other Transitions

Why Transition Benefits Need Their Own Reserve Term

A transition can change an insurer’s liability in two ways at once. It can trigger an immediate lump-sum payment, and it can move the policy into a state with a different reserve requirement. For example, recovery from disability might trigger a rehabilitation payment while the recovered policyholder still has a positive reserve in the healthy state. Counting only the payment, or only the destination-state reserve, gives the wrong reserve dynamics.

The focus here is the contribution of such payments to a multi-state Thiele equation. The model is continuous in time: transition intensities describe the chance of moving between states over a short interval, and reserves are defined immediately before any transition at that time. All products and numerical data below are hypothetical.

Definition: Let $V_i(t)$ be the prospective reserve at time $t$, conditional on being in state $i$. For a transition from state $i$ to state $j$, let $\lambda_{ij}(t)$ be its transition intensity and $B_{ij}(t)$ the lump-sum benefit paid when it occurs. The payment is made at the transition, and $V_j(t)$ is the reserve required immediately after the transition. Thus the insurer’s liability just after that transition is $B_{ij}(t)+V_j(t)$.

Use the following cash-flow convention. In state $i$, let $p_i(t)$ be the premium income rate and $c_i(t)$ the continuous benefit outgo rate. Positive values of both quantities mean income and outgo, respectively. If the model has finitely many possible transitions out of state $i$, then the Thiele equation is

\frac{dV_i}{dt} = \delta V_i+p_i-c_i +\sum_{j\ne i}\lambda_{ij}\bigl(V_i-V_j-B_{ij}\bigr).

Here $\delta$ is the force of interest. A transition to a state with no future liability has $V_j=0$. The lump sum and destination-state reserve must both be included when the transition occurs; neither replaces the other.

Why the Transition Term Has This Form

The signs are easiest to verify by balancing values over a short interval $h$. Starting in state $i$ at time $t$, the continuous net outgo over that interval is approximately $(c_i-p_i)h$. A transition from $i$ to $j$ occurs with probability approximately $\lambda_{ij}h$, in which case the insurer incurs the lump sum and enters state $j$. The reserve $V_i(t)$ therefore satisfies, to first order,

V_i(t) \approx (c_i-p_i)h +e^{-\delta h} \left[ (1-\sum_{j\ne i}\lambda_{ij}h)V_i(t+h) +\sum_{j\ne i}\lambda_{ij}h\bigl(B_{ij}+V_j(t+h)\bigr) \right].

Using $e^{-\delta h}\approx 1-\delta h$ and $V_i(t+h)\approx V_i(t)+hV_i'(t)$, subtract $V_i(t)$, divide by $h$, and let $h$ tend to zero. Rearranging gives

V_i' =\delta V_i+p_i-c_i +\sum_{j\ne i}\lambda_{ij}(V_i-V_j-B_{ij}).

In particular, a positive transition benefit appears with a minus sign inside the transition term: it is an immediate outgo, not income. The $V_i-V_j$ part accounts for the change in the insurer’s reserve requirement when the policy leaves state $i$ and enters state $j$.

Solving a One-State Reserve Equation

When there is just one state whose reserve is unknown and all transition destinations have specified reserves, the equation often reduces to a scalar linear differential equation. Suppose the coefficients are constant, so that

V'(t)=aV(t)-C,\qquad V(T)=V_T.

Solving this equation backward from time $T$ gives

V(t)=\frac{C}{a}+ \left(V_T-\frac{C}{a}\right)e^{-a(T-t)} \qquad (a\ne 0).

For example, if $V_T=0$, then $V(t)=\frac{C}{a}\bigl(1-e^{-a(T-t)}\bigr)$. This form is useful when the transition intensities, interest force, benefits, and destination reserves are constant over the period. It does not directly apply when these quantities vary with time; then the scalar equation must be solved with the appropriate time-varying coefficients.

Worked Example: A Recovery Payment With No Destination Reserve

A hypothetical disability contract has a disabled state and a healthy state. While disabled, there are no premiums and no continuous benefits. Recovery occurs at constant intensity $0.16$ per year and pays a lump sum of $750. The healthy-state reserve is zero throughout the remaining term. The force of interest is $0.04$, and the disabled-state reserve at time $T=2$ is zero. Find the disabled-state reserve at time 0, to the nearest cent.

Method. Let $V(t)$ be the disabled-state reserve. The only transition is recovery, so $\lambda=0.16$, $B=750$, and $V_{\mathrm{healthy}}=0$. The Thiele equation becomes

V'(t)=0.04V(t)+0.16\bigl(V(t)-0-750\bigr) =0.20V(t)-120.

This is $V'=aV-C$ with $a=0.20$, $C=120$, $T=2$, and $V_T=0$. Therefore

V(0)=\frac{120}{0.20}\left(1-e^{-0.20(2)}\right) =600(1-e^{-0.4}).

Since $e^{-0.4}\approx 0.670320046$, the reserve is

V(0)\approx 600(0.329679954)=197.807972.

Rounded only at the final step, $V(0)=$$197.81. Check: the recovery outgo rate is $0.16(750)=120$, while interest and the chance of leaving the disabled state contribute the coefficient $0.04+0.16=0.20$. Both match the differential equation. A common error is to use $0.04V-120$, omitting the $0.16V$ part of the transition term; the reserve must account for the probability of remaining in the disabled state as well as the benefit on recovery.

Including Continuous Benefits and a Nonzero Destination Reserve

A transition benefit is not the only outgo to include. Continuous benefits paid while in the current state enter through $c_i$, while a transition-triggered lump sum enters through $B_{ij}$. Keeping these cash flows separate helps prevent either from being omitted or counted twice. The destination-state reserve also remains part of the transition liability, even when the lump sum is paid immediately.

Worked Example: Recovery Into a State With a Positive Reserve

A hypothetical contract has a disabled state from which recovery occurs at constant intensity $0.12$ per year. Recovery pays a lump sum of $400 and moves the policy into a healthy state whose reserve is $1,200 throughout the period considered. While disabled, the contract pays continuous benefits at $300 per year and receives no premiums. The force of interest is $0.05$, the period is $T=1.5$ years, and the disabled-state terminal reserve is zero. Find the reserve at time 0, to the nearest cent.

Method. In the disabled state, $c=300$, $p=0$, $\lambda=0.12$, $B=400$, and $V_{\mathrm{healthy}}=1{,}200$. Hence

V'(t)=0.05V(t)-300+0.12\bigl(V(t)-1{,}200-400\bigr) =0.17V(t)-492.

The constant outgo term is $300+0.12(1{,}600)=492$, and the coefficient of $V$ is $0.05+0.12=0.17$. With $V(1.5)=0$, the scalar solution gives

V(0)=\frac{492}{0.17}\left(1-e^{-0.17(1.5)}\right) =2{,}894.117647\left(1-e^{-0.255}\right).

Using $e^{-0.255}\approx 0.774916498$,

V(0)\approx 2{,}894.117647(0.225083502)=651.418.

Rounded to the nearest cent, $V(0)=$$651.42. Check: the recovery-related liability is $400 immediately plus a $1,200 destination reserve, and it is weighted by the recovery intensity $0.12$; the separate continuous benefit rate is $300 per year. Omitting the destination reserve would understate the recovery outgo by $0.12(1{,}200)=144$ per year in the differential equation.

Competing Transitions

When several transitions can occur from the same state, each contributes its own intensity multiplied by the change in liability associated with that transition. Intensities are not probabilities over the entire remaining term; in a short interval of length $h$, $\lambda_{ij}h$ approximates the probability of that particular transition. The sum of transition terms belongs in the reserve equation, even if some transitions lead to states with zero reserve.

Worked Example: Recovery and Death Both Trigger Payments

A hypothetical policyholder is disabled for a period of one year. Recovery occurs at constant intensity $0.08$ per year, pays $500, and leads to an active state with reserve $3,000. Death occurs at constant intensity $0.02$ per year, pays $20,000, and leads to a state with reserve zero. There are no premiums or continuous benefits in the disabled state. The force of interest is $0.04$, and the disabled-state reserve at the end of the year is zero. Find the reserve at time 0, to the nearest cent.

Method. Add the two transition contributions. The recovery transition has $V_{\mathrm{active}}+B=3{,}000+500=3{,}500$; the death transition has $V_{\mathrm{death}}+B=0+20{,}000=20{,}000$. Thus

\begin{aligned} V'(t) &=0.04V(t)+0.08\bigl(V(t)-3{,}500\bigr) +0.02\bigl(V(t)-20{,}000\bigr)\\ &=0.14V(t)-680. \end{aligned}

The coefficient is $0.04+0.08+0.02=0.14$. The constant outgo is $0.08(3{,}500)+0.02(20{,}000)=280+400=680$. Since $V(1)=0$,

V(0)=\frac{680}{0.14}\left(1-e^{-0.14}\right) =4{,}857.142857(1-0.869358235).

Therefore $V(0)\approx 4{,}857.142857(0.130641765)=634.5457$, or $634.55 to the nearest cent. Check: the recovery and death contributions are both outgoes, and the destination reserve of $3,000 is included only in the recovery term. Treating the death payment as the only transition outgo would omit the recovery payment and the active-state liability.

Common Sign and Cash-Flow Traps

The transition term can be checked directly by asking what the insurer owes just after the transition. A transition from $i$ to $j$ creates liability $B_{ij}+V_j$, compared with the pre-transition reserve $V_i$. The difference is therefore $V_i-(B_{ij}+V_j)$, which is exactly the quantity multiplied by $\lambda_{ij}$ in the Thiele equation.

Giving a benefit the wrong sign. A positive lump-sum benefit is an outgo. It appears as $-\lambda_{ij}B_{ij}$, not $+\lambda_{ij}B_{ij}$.
Omitting the destination reserve. A recovery payment may be modest while the reserve needed in the recovered state is substantial. Use $B_{ij}+V_j$ for the post-transition liability.
Confusing a continuous benefit with a transition payment. A continuous benefit rate is paid while the policy remains in a state; a transition benefit is paid only when a specified transition occurs. Their terms are $c_i$ and $\lambda_{ij}B_{ij}$, respectively.
Using a transition probability instead of an intensity. In the differential equation, the multiplier is an intensity per unit time. A probability over a full year cannot be substituted without first matching the model and time interval.

Key takeaway: For each transition, first identify the immediate payment and the reserve required in the destination state. The transition term is the intensity multiplied by $V_i-V_j-B_{ij}$. Add the contributions from all possible transitions, then include the current-state premium and continuous-benefit rates separately.

Check Your Understanding

For each problem, use the stated force of interest and constant transition intensities. Round money to the nearest cent where required.

A disabled state has no continuous benefits or premiums. Recovery occurs at intensity $0.10$ per year, pays $400, and leads to a state with zero reserve. The force of interest is $0.02$, and the disabled-state terminal reserve is zero in two years. Find the reserve at time 0. (Answer: $71.12)
A state pays continuous benefits of $50 per year and no premiums. One transition occurs at intensity $0.07$ per year, pays $1,000, and leads to a state with zero reserve. The force of interest is $0.03$, and the initial-state terminal reserve is zero in one year. Find the reserve at time 0. (Answer: $114.20)
From a disabled state, recovery occurs at intensity $0.05$ per year, pays $200, and leads to a state with reserve $500. Death occurs at intensity $0.01$ per year, pays $10,000, and leads to a state with zero reserve. There are no continuous benefits or premiums, the force of interest is $0.04$, and the disabled-state terminal reserve is zero in two years. Find the reserve at time 0. (Answer: $244.71)
At a particular time, a state has reserve $800 and can transition to a state with reserve $300. The transition intensity is $0.10$ per year, the transition payment is $B$, and the force of interest is $0.03$. There are no premiums or continuous benefits. If the reserve derivative at that time is $20 per year, find $B$. (Answer: $540)

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