Expected Present Value of Continuous Benefits

Introduction

Some long-term benefits are paid continuously while a person remains in a specified condition. A hypothetical disability contract, for example, might pay a fixed rate per year for each moment the insured is disabled. To value that promise, the actuary must account for both the chance of occupying the eligible state at each time and the discount applied to payments made at that time.

The central calculation is an integral: at each time, take the benefit rate multiplied by the probability of being in an eligible state, discount that expected rate, and integrate over the benefit term. This tutorial develops that result and applies it to finite-state models. It concerns payments for state occupancy, not amounts paid only when a transition occurs.

Definition: Let $Z_t$ denote the state occupied at time $t$, and suppose the policyholder starts in a known state $i$. The state-occupancy probability is $p_{ij}(t)=P(Z_t=j\mid Z_0=i)$. A continuous state benefit rate $b_j(t)$ is the amount paid per unit of time while the policyholder occupies state $j$ at time $t$. If $b_j(t)=0$, state $j$ is not eligible for this benefit at that time.

The Expected Present Value Integral

Assume a finite-state continuous-time Markov model, a known initial state, deterministic benefit rates, and a constant force of interest $\delta$. Payments accrue continuously during occupancy over the term from time $0$ to time $n$. The present value is random because the state path is random. Its expected value is

\operatorname{EPV} = \int_0^n e^{-\delta t} \sum_j p_{ij}(t)b_j(t)\,dt.

The sum ranges over all states; ineligible states simply have benefit rate zero. If the benefit is payable only in a selected set $J$, the same formula can be written with the sum restricted to $j\in J$. The integral is a present value in currency units: $b_j(t)$ is currency per year, while $dt$ is measured in years.

Assumptions and scope: This formula values benefits accrued continuously according to the state occupied at each instant. It does not include a separate payment caused by entering or leaving a state. The force of interest is constant here, so a payment at time $t$ is discounted by $e^{-\delta t}$. If the force varies deterministically with time, replace this factor by $e^{-\int_0^t \delta(s)\,ds}$.

The state probabilities are obtained from the model’s transition intensities. For constant intensities, the generator matrix $Q$ has off-diagonal entry $q_{jk}$ equal to the intensity of moving from state $j$ to state $k$, and diagonal entry $q_{jj}=-\sum_{k\ne j}q_{jk}$. The row vector of state probabilities satisfies the forward Kolmogorov equations $p'(t)=p(t)Q$, with initial probability one in state $i$. For time-varying intensities, use the corresponding time-varying equations. The earlier material on Kolmogorov equations develops those probability calculations.

Why the Integral Gives the Expected Present Value

Let $I_j(t)$ be an indicator: it equals one if $Z_t=j$ and zero otherwise. The instantaneous payment rate at time $t$ is $\sum_j b_j(t)I_j(t)$. Therefore the present value along a particular state path is the integral of that rate after discounting. Taking expectations gives the desired formula.

Write the pathwise present value.
For each possible path, the present value over the term is $\int_0^n e^{-\delta t}\sum_j b_j(t)I_j(t)\,dt$.

Take expectations.
For nonnegative benefit rates, expectation and integration can be interchanged. Linearity of expectation then gives the integral of $e^{-\delta t}\sum_j b_j(t)E[I_j(t)]$.

Use the state probabilities.
Since $E[I_j(t)]=P(Z_t=j\mid Z_0=i)=p_{ij}(t)$, the expected present value is $\int_0^n e^{-\delta t}\sum_j p_{ij}(t)b_j(t)\,dt$.

This derivation also explains why a benefit paid only while disabled is valued using the probability of being disabled at each time, rather than the probability of ever becoming disabled. Someone may enter the state after time zero, leave it before the term ends, or remain in it; the occupancy probability at each instant accounts for all such paths.

Worked Valuations

Worked Example: A Benefit in One Absorbing State

A hypothetical contract starts in healthy state $H$. The only transition is from $H$ to disabled state $D$, at constant intensity $0.20$ per year; $D$ is absorbing. A continuous benefit of $900 per year is paid while the insured is in $D$, for at most three years. The force of interest is $\delta=0.04$. Find the expected present value.

Method. First find the occupancy probability of $D$. The probability of remaining in $H$ to time $t$ is $e^{-0.20t}$, so

p_{HD}(t)=1-e^{-0.20t}.

The benefit rate is $900 per year in $D$ and zero in $H$. Substituting this probability and the continuous discount factor into the occupancy integral gives

\begin{aligned} \operatorname{EPV} &=900\int_0^3 e^{-0.04t}(1-e^{-0.20t})\,dt\\ &=900\left[ \int_0^3 e^{-0.04t}\,dt - \int_0^3 e^{-0.24t}\,dt \right]\\ &=900\left[ \frac{1-e^{-0.12}}{0.04} - \frac{1-e^{-0.72}}{0.24} \right]. \end{aligned}

The first integral is $(1-e^{-0.12})/0.04=2.826989$. The second is $(1-e^{-0.72})/0.24=2.138532$. Their difference is $0.688457$, so the expected present value is $900(0.688457)=\$619.61$, rounded to the nearest cent.

Check. The benefit probability starts at zero and increases toward one, so the expected present value is positive and below the value of a certain $900 annual continuous payment for three years. That certain-payment value is $900(1-e^{-0.12})/0.04=\$2,544.29$, consistent with the result. The method avoids treating the benefit as payable from time zero.

Worked Example: Competing Exits and a One-Year Benefit

A hypothetical policy starts in active state $H$. It exits $H$ to disabled state $D$ at intensity $0.10$ per year and to an absorbing termination state $X$ at intensity $0.10$ per year. Both $D$ and $X$ are absorbing. A continuous benefit of $600 per year is payable only in $D$, for one year. There is no interest. Find the expected present value.

Method. The total intensity of leaving $H$ is $0.10+0.10=0.20$, so the probability of still being in $H$ at time $t$ is $e^{-0.20t}$. Conditional on leaving $H$, the two exits have equal intensities. Equivalently, solving the state-probability equation gives

p_{HD}(t) = \int_0^t e^{-0.20s}(0.10)\,ds = \frac{1}{2}(1-e^{-0.20t}).

With no interest, the discount factor is one. Thus

\begin{aligned} \operatorname{EPV} &=600\int_0^1 \frac{1}{2}(1-e^{-0.20t})\,dt\\ &=300\left[1-\frac{1-e^{-0.20}}{0.20}\right]\\ &=300(1-0.906346)\\ &=300(0.093654) =\$28.10. \end{aligned}

The unrounded occupancy integral is $0.0468269$ years, and multiplying by $600 gives $28.0961, or $28.10 to the nearest cent. As a check, the probability of being in $D$ at time one is only $\frac12(1-e^{-0.20})=0.090635$, and the average occupancy probability during the year is smaller still.

Common mistake avoided. Using $1-e^{-0.20t}$ as the probability of being in $D$ counts every exit from $H$ as a disabled exit. Half the exits instead go to $X$. The correct occupancy probability in $D$ includes the factor $1/2$, so omitting it would double the answer.

Worked Example: Benefits in Two Eligible States

A hypothetical contract starts in state $H$, with transitions to absorbing states $D$ and $X$ at intensities $0.10$ per year each. A continuous rate of $600 per year is paid in $D$, and $300 per year is paid in $X$, for two years. The force of interest is $\delta=0.10$. Find the expected present value of both benefits.

Method. The total exit intensity from $H$ is $0.20$. By symmetry, the probability of occupying either absorbing state at time $t$ is

p_{HD}(t)=p_{HX}(t)=\frac{1}{2}(1-e^{-0.20t}).

The expected benefit rate at time $t$ is the sum of each state’s occupancy probability multiplied by its rate:

600p_{HD}(t)+300p_{HX}(t) = 450(1-e^{-0.20t}).

Discounting and integrating over the two-year term gives

\begin{aligned} \operatorname{EPV} &=450\int_0^2 e^{-0.10t}(1-e^{-0.20t})\,dt\\ &=450\left[ \frac{1-e^{-0.20}}{0.10} - \frac{1-e^{-0.60}}{0.30} \right]\\ &=450(1.812692-1.503961)\\ &=450(0.308731) =\$138.93. \end{aligned}

The two state contributions have been combined only after weighting their probabilities by their different benefit rates. As a check, both rates are nonnegative and the probability of having exited $H$ increases over the term, so the expected present value is positive. It is also below the present value that would result if the larger rate of $600 were paid with certainty throughout the term.

Common mistake avoided. Adding the two benefit rates first and multiplying by the probability of being in either eligible state would incorrectly apply the same combined rate to both states. The correct expected rate is $600p_{HD}(t)+300p_{HX}(t)$; the state-specific rates must remain paired with their own probabilities.

Interpreting the Model and Avoiding Traps

A continuous benefit rate is not the same as a single payment at a particular time. The integral accumulates payments over the entire duration spent in eligible states. A discrete annual payment would instead use probabilities at specified payment dates; replacing that schedule with an occupancy integral changes the cash-flow timing and therefore the value.

Another important distinction is between an effective annual interest rate and a force of interest. If the given rate is an effective annual rate $i$, the equivalent constant force is $\delta=\ln(1+i)$. The continuous discount factor is then $e^{-\delta t}$, or equivalently $(1+i)^{-t}$. Substituting $i$ directly for $\delta$ is not exact. In Example 1, for instance, the stated force $0.04$ corresponds to effective annual interest $e^{0.04}-1$, approximately $4.081\%$, rather than exactly $4\%$.

The model’s state probabilities must also match the benefit definition. When two or more exits compete, the probability of being in one destination state is generally less than the probability of having left the initial state. Finally, only occupancy cash flows belong in this integral: a separate amount payable at a transition requires a different valuation calculation.

Key takeaway: For a continuous benefit paid while the process occupies eligible states, multiply each state’s benefit rate by its occupancy probability, apply the continuous discount factor, sum across states, and integrate over the payment term.

Check Your Understanding

For each problem, use state-occupancy probabilities and continuous discounting as appropriate.

A hypothetical contract starts in active state $H$, moves to absorbing disabled state $D$ at intensity $0.10$, and pays $500 per year continuously in $D$ for one year. There is no interest. Find the expected present value. (Answer: $24.19)
A policy starts in $H$, with transitions to absorbing $D$ and absorbing $X$ at intensities $0.20$ and $0.10$, respectively. It pays $300 per year continuously in $D$ for two years. The force of interest is $0.05$. Find the expected present value. (Answer: $92.98)
A model starts in $H$ and moves to absorbing $D$ or $X$ at intensities $0.08$ and $0.12$. Rates of $1,000 per year in $D$ and $400 per year in $X$ are paid continuously for one year. The force of interest is $0.04$. Find the combined expected present value. (Answer: $58.38)
A hypothetical contract starts in $H$ and moves to absorbing $D$ or $X$ at intensities $0.25$ per year each. In $D$, it pays a time-varying continuous rate of $800e^{0.02t}$ dollars per year; in $X$, it pays nothing. The benefit term is two years and the force of interest is $0.07$. Find the expected present value. (Answer: $276.12)

What You'll Learn