Combining State Annuities and Transition Lump Sums

Why the Two Payment Types Need Separate Probability Bases

A multi-state benefit package can pay a continuous annuity while a person occupies a particular state and also pay lump sums when specified transitions occur. For example, a hypothetical contract might pay income during disability, a lump sum on entry to disability, and another lump sum on recovery. These benefits are related through the same state history, but they are not valued using the same probability at a given time.

The annuity payment rate is weighted by the probability of being in the payment state. A transition lump sum is weighted by the probability of being in the origin state multiplied by the relevant transition intensity. The lump sum is discounted to the transition time, not to the start of the year or to the end of a payment period unless the contract specifies that different timing.

Definition: In a continuous-time Markov state model, let $p_i(t)$ be the probability of occupying state $i$ at time $t$, given the specified initial state or initial distribution. Let $\mu_{ij}(t)$ be the transition intensity from state $i$ to state $j$. Over a short interval of length $h$, conditional on being in state $i$ at time $t$, the probability of a transition to $j$ is $\mu_{ij}(t)h+o(h)$. A state annuity pays at a specified rate while a specified state is occupied; a transition lump sum pays when a specified transition occurs.

Assume that the force of interest is constant at $\delta$, so a payment at time $t$ has discount factor $e^{-\delta t}$. Write $b_i(t)$ for the continuous payment rate while in state $i$, and $c_{ij}(t)$ for the lump sum paid at an $i$-to-$j$ transition. A transition benefit is included only for transitions on which the contract pays; other transitions have no corresponding lump sum.

Theorem: For a valuation period from time $0$ to time $T$, the expected present value of continuous state payments and transition lump sums is $$ \operatorname{EPV} = \int_0^T e^{-\delta t} \left[ \sum_i p_i(t)b_i(t) + \sum_i\sum_{j\ne i}p_i(t)\mu_{ij}(t)c_{ij}(t) \right]dt. $$ For an infinite valuation period, use $T=\infty$ only when the integrals converge. The formula assumes that payment rates and transition benefits are defined on the stated time basis, that the model specifies the relevant state probabilities and intensities, and that transition lump sums are paid at the transition time.

Why the Valuation Formula Holds

The formula follows by considering expected payments during a short interval around time $t$. The probability of occupying state $i$ at that time is $p_i(t)$. Conditional on that occupancy, a continuous payment rate $b_i(t)$ produces an expected payment of approximately $b_i(t)h$ over an interval of length $h$. Thus its expected payment contribution per unit of time is $p_i(t)b_i(t)$.

For a transition lump sum, being in the origin state is not enough: the specified transition must also occur during the interval. Conditional on occupancy of $i$, the transition probability is $\mu_{ij}(t)h+o(h)$. Therefore the expected lump sum paid in the interval is $p_i(t)\{\mu_{ij}(t)h+o(h)\}c_{ij}(t)$. Dividing by $h$ and taking the limit gives the expected transition payment rate $p_i(t)\mu_{ij}(t)c_{ij}(t)$. Both types of expected payment are discounted at time $t$; summing them and integrating over the valuation period gives the result.

This argument also explains why a transition lump sum must not be weighted merely by the probability of occupying its destination state. A destination-state probability counts occupancy at a time, not the chance of a particular transition occurring at that time. If a lump sum is instead payable at a fixed future date, its probability and timing must be modeled as a scheduled payment, not as an instantaneous transition payment.

Worked Valuations

Worked Example: Entry into a Permanent Benefit State

A hypothetical contract starts in active state $A$. The only transition is from $A$ to a permanent benefit state $B$, at constant intensity $0.20$ per year. The valuation period is one year. While in $B$, the contract pays continuously at $ \$12{,}000$ per year. It also pays $ \$5{,}000$ immediately on transition from $A$ to $B$. The force of interest is $0.05$ per year. Find the combined expected present value.

Method. First find the occupancy probability and transition intensity contribution separately. Starting in $A$, the probability of still being in $A$ at time $t$ is $p_A(t)=e^{-0.20t}$, so $p_B(t)=1-e^{-0.20t}$. The state annuity is weighted by $p_B(t)$; the transition lump sum is weighted by $p_A(t)(0.20)$.

\begin{aligned} \operatorname{EPV}_{\text{annuity}} &=12{,}000\int_0^1 e^{-0.05t}(1-e^{-0.20t})\,dt\\ &=12{,}000\left[ \frac{1-e^{-0.05}}{0.05} - \frac{1-e^{-0.25}}{0.25} \right]\\ &=12{,}000(0.97541151-0.88479687)\\ &=12{,}000(0.09061464)=1{,}087.38. \end{aligned}

For the lump sum, the discounted transition density is $e^{-0.05t}e^{-0.20t}(0.20)$. Hence

\begin{aligned} \operatorname{EPV}_{\text{transition}} &=5{,}000\int_0^1 e^{-0.25t}(0.20)\,dt\\ &=5{,}000\left(\frac{0.20}{0.25}\right)(1-e^{-0.25})\\ &=5{,}000(0.17695937)=884.80. \end{aligned}

The combined value is $1{,}087.38+884.80=\$1{,}972.18$, rounded to the nearest cent. As a check on the transition probability, the probability of entering $B$ by time 1 is $1-e^{-0.20}=0.18126925$. The discounted transition factor $0.17695937$ is lower, as expected, because the payment is discounted from its actual transition time.

Worked Example: Continuous Benefits with Recovery and Re-entry

A hypothetical policyholder starts in active state $A$, with disability state $D$ also possible. The constant transition intensity from $A$ to $D$ is $0.20$ per year, and the recovery intensity from $D$ to $A$ is $0.10$ per year. There is no death or other exit. While in $D$, a continuous benefit is paid at $ \$6{,}000$ per year. Each $A$-to-$D$ transition pays $ \$2{,}000$, and each $D$-to-$A$ transition pays $ \$500$. The force of interest is $0.10$ per year, and payments continue indefinitely. Find the expected present value.

Method. With $\alpha=0.20$ and $\beta=0.10$, the disability occupancy probability at time $t$, starting from $A$, is $p_D(t)=\frac{\alpha}{\alpha+\beta}(1-e^{-(\alpha+\beta)t})$. This accounts for both incidence and recovery. Since $p_A(t)=1-p_D(t)$, integrate each occupancy probability against the discount factor to get the state exposure relevant to benefits and transition payments.

\begin{aligned} \int_0^\infty e^{-0.10t}p_D(t)\,dt &=\frac{0.20}{0.30} \left(\frac{1}{0.10}-\frac{1}{0.40}\right)\\ &=\frac{2}{3}(10-2.5)=5,\\ \int_0^\infty e^{-0.10t}p_A(t)\,dt &=\frac{1}{0.10}-5=5. \end{aligned}

The disability annuity therefore has value $ \$6{,}000(5)=\$30{,}000$. The discounted expected number of $A$-to-$D$ transitions is the intensity $0.20$ times discounted active-state exposure $5$, or $1$. The corresponding value is $1(\$2{,}000)=\$2{,}000$. The discounted expected number of recovery transitions is $0.10(5)=0.5$, giving $0.5(\$500)=\$250$.

Adding the three components gives $ \$30{,}000+\$2{,}000+\$250=\$32{,}250$. Re-entry matters: a person may receive the disability annuity during more than one spell and may generate multiple transition payments. The transition-intensity integrals count those transitions with the correct discounting.

Worked Example: Disability Income and Competing Death Transitions

A hypothetical contract starts in active state $A$. From $A$, disability $D$ occurs at intensity $0.20$ per year and death at intensity $0.10$ per year. From $D$, death occurs at intensity $0.20$ per year; there is no recovery. Death is absorbing. While in $D$, the contract pays $ \$9{,}000$ per year continuously. It pays $ \$4{,}000$ on the transition from $A$ to $D$, $ \$10{,}000$ on death from $A$, and $ \$8{,}000$ on death from $D$. The force of interest is $0.10$ per year. Find the expected present value over the infinite future.

Method. The total exit intensity from $A$ is $0.20+0.10=0.30$, so $p_A(t)=e^{-0.30t}$. The discounted active-state exposure is $\int_0^\infty e^{-0.10t}p_A(t)\,dt=1/(0.10+0.30)=2.5$. The discounted disability-state exposure is found by integrating its occupancy probability. The transient state probability satisfies $p_D'(t)=0.20p_A(t)-0.20p_D(t)$, with $p_D(0)=0$, giving $p_D(t)=2(e^{-0.20t}-e^{-0.30t})$.

\begin{aligned} \int_0^\infty e^{-0.10t}p_D(t)\,dt &=2\left(\frac{1}{0.30}-\frac{1}{0.40}\right)\\ &=2\left(\frac{10}{3}-\frac{5}{2}\right) =\frac{5}{3}. \end{aligned}

The disability annuity is therefore $ \$9{,}000(5/3)=\$15{,}000$. The discounted transition factors are $0.20(2.5)=0.5$ for $A$-to-$D$, $0.10(2.5)=0.25$ for death from $A$, and $0.20(5/3)=1/3$ for death from $D$. Their respective values are $0.5(\$4{,}000)=\$2{,}000$, $0.25(\$10{,}000)=\$2{,}500$, and $(1/3)(\$8{,}000)=\$2{,}666.67$.

The combined expected present value is $ \$15{,}000+\$2{,}000+\$2{,}500+\$2{,}666.67=\$22{,}166.67$, rounded to the nearest cent. The active-to-death and disability-to-death benefits use different origin-state exposures; applying one death probability to both would not respect the model.

Common Exam Trap: Using the Wrong Probability for a Lump Sum

A common error is to multiply a transition lump sum by the probability of being in the destination state, or to multiply the probability of transition by the lump sum without discounting the actual payment time. In the first worked example, the probability of entering $B$ within the year is $0.18126925$. The incorrect calculation $ \$5{,}000(0.18126925)=\$906.35$ treats every payment as though it were made at time zero.

The correct value is the transition-time discounted integral, $ \$5{,}000\int_0^1 e^{-0.25t}(0.20)\,dt=\$884.80$. The distinction is material because transitions occur throughout the year. A destination-state occupancy probability is also not a substitute for a transition density: a person may already have entered the destination state, or may enter and later leave it.

Key takeaway: For every cash flow, identify the event that makes it payable and the date on which it is paid. Weight a continuous state payment by occupancy probability; weight an instantaneous transition lump sum by origin-state occupancy probability times transition intensity; then discount both at their payment times.

Check Your Understanding

For each problem, identify the relevant occupancy or transition exposure before combining the benefit components.

A contract starts in state $A$, with only a transition to permanent state $B$ at constant intensity $0.10$ per year. For one year, $B$ pays $ \$1{,}000$ continuously per year and entry to $B$ pays $ \$2{,}000$. The force of interest is $0.10$. Find the combined expected present value, rounded to cents. (Answer: $\$226.55$)
A person starts in active state $A$. The constant intensities are $0.30$ from $A$ to $D$ and $0.20$ from $D$ to $A$. There is no other exit. A continuous benefit of $ \$4{,}000$ per year is paid in $D$; transition lump sums are $ \$1{,}000$ on entry to $D$ and $ \$600$ on recovery. Payments continue indefinitely and the force of interest is $0.10$. Find the combined expected present value. (Answer: $\$22{,}100.00$)
A person starts in active state $A$, with intensity $0.10$ from $A$ to benefit state $B$, intensity $0.20$ from $A$ to death, and intensity $0.30$ from $B$ to death. Death is absorbing. The continuous benefit in $B$ is $ \$6{,}000$ per year. Lump sums are $ \$3{,}000$ on $A$-to-$B$, $ \$7{,}000$ on death from $A$, and $ \$5{,}000$ on death from $B$. The force of interest is $0.10$, with an infinite valuation period. Find the combined expected present value. (Answer: $\$8{,}937.50$)
A person starts in active state $A$, and the only transition is to permanent benefit state $B$ at constant intensity $0.25$ per year. The two-year benefit rate in $B$ is $ \$3{,}000$ per year, and the transition lump sum is $ \$4{,}000$. The force of interest is $0.05$. Find the combined expected present value, rounded to cents. (Answer: $\$2{,}701.83$)

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