Present Value of Continuous Care Benefits

Introduction

A continuous long-term-care benefit is paid as a stream while an insured occupies an eligible care state. Its expected present value depends on two quantities at every future time: the probability that the insured is then in a state that qualifies for payment, and the benefit rate payable in that state. Valuing the benefit therefore requires integrating discounted state-contingent payment rates over time.

This tutorial develops that integral and shows how to calculate it from occupancy probabilities. The focus is the value of continuous care benefits, not the derivation of premiums or the timing of discrete claim payments. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Suppose an insured follows a continuous-time state model, and let $p_j(t)$ be the probability of occupying state $j$ at time $t$. If a benefit is paid continuously at rate $b_j(t)$ dollars per year while the insured occupies state $j$, its expected present value over $[0,T]$ is $$ \operatorname{EPV}=\int_0^T e^{-\delta t}\sum_{j\in\mathcal{B}}p_j(t)b_j(t)\,dt. $$ Here $\mathcal{B}$ is the set of states in which the benefit is payable, and $\delta$ is the force of interest. For a benefit payable for as long as the insured remains in an eligible state, use $T=\infty$ when the integral converges.

The payment rate $b_j(t)$ is a flow measured in dollars per year, not a lump sum paid at a transition or at the end of a period. The expression inside the integral is the expected payment rate at time $t$; multiplying by $dt$ gives the expected payment over a short interval, and discounting brings it to time 0. If the contract pays in only some care states, states outside $\mathcal{B}$ contribute zero. Death normally contributes no care benefit, but that must follow from the contract definition rather than from the mathematics alone.

From Occupancy Probabilities to Present Value

Assume the force of interest is constant. Then one dollar paid continuously at time $t$ has time-0 value $e^{-\delta t}$ dollars. For each eligible state, the probability of being in that state at time $t$ is multiplied by the rate payable there. Adding across eligible states gives the expected payment rate at that time.

The expected present value can also be understood directly from a sample path. On that path, the amount paid in a small interval $[t,t+dt]$ is approximately $b_j(t)\,dt$ if the insured occupies eligible state $j$. Taking expectations gives $\sum_j p_j(t)b_j(t)\,dt$. Integrating these expected discounted amounts over time gives the stated formula. This reasoning uses linearity of expectation; it does not require the insured’s state at different times to be independent.

Assumptions and applicability: The formula applies when the state probabilities describe the insured under the specified transition model, the benefit rate conditional on state and time is known, and discounting is represented by the stated force of interest. A constant force gives $e^{-\delta t}$; with a deterministic time-varying force $\delta(s)$, replace it by $\exp(-\int_0^t\delta(s)\,ds)$. A formula based on a single constant $\delta$ does not apply unchanged when the force varies with time.

When transition intensities are constant, the state probabilities can be obtained from the continuous-time Markov model. Suppose the model has transient states, such as active and care states, and absorbing states, such as death. Let $Q_{TT}$ be the submatrix of the generator whose rows and columns correspond to transient states. Its diagonal entries retain the full rates of exit from each transient state, including rates of transition to absorbing states. Let $\pi_T(0)$ be the initial probability row vector restricted to transient states, and let $b$ be a column vector of benefit rates in those states. If no benefit is payable in absorbing states, the EPV is

\operatorname{EPV}=\int_0^\infty e^{-\delta t}\pi_T(0)e^{Q_{TT}t}b\,dt =\pi_T(0)(\delta I-Q_{TT})^{-1}b,

provided the integral converges and the inverse exists. This resolvent expression is for discounted occupancy of the transient states. It is not valid to substitute the full generator, including absorbing states, for $Q_{TT}$ when the desired quantity is transient-state occupancy: an absorbing state can have positive occupancy probability indefinitely.

Worked Examples

Worked Example: Benefit in a Single Care State

A hypothetical policyholder starts in the active state. The constant transition intensity from active to care is $\lambda=0.10$ per year. The intensity from care to death is $\mu=0.20$ per year. There is no recovery. A benefit of $10,000 per year is paid continuously while the insured is in care, with no time limit. The force of interest is $\delta=0.10$ per year. Find the EPV.

Method. First obtain the probability of occupying care. Entry into care at time $s$ has density $\lambda e^{-\lambda s}$. Conditional on entry at $s$, the probability of remaining in care until time $t$ is $e^{-\mu(t-s)}$. Therefore, for $t\geq 0$,

p_C(t)=\int_0^t \lambda e^{-\lambda s}e^{-\mu(t-s)}\,ds =\frac{\lambda}{\mu-\lambda}\left(e^{-\lambda t}-e^{-\mu t}\right).

Substituting $\lambda=0.10$ and $\mu=0.20$ gives $p_C(t)=e^{-0.10t}-e^{-0.20t}$. The discounted expected care occupancy is

\int_0^\infty e^{-0.10t}p_C(t)\,dt =\int_0^\infty\left(e^{-0.20t}-e^{-0.30t}\right)dt =\frac{1}{0.20}-\frac{1}{0.30} =5-\frac{10}{3} =\frac{5}{3}\text{ years}.

Thus the EPV is $$10{,}000\times 5/3=$16{,}666.67$, rounded to the nearest cent. Check: the discounted occupancy is positive and finite, and multiplying years by dollars per year gives dollars. The calculation avoids the mistake of discounting only to care entry: payments after entry must also be discounted to time 0.

Worked Example: Two Successive Care Levels

A hypothetical insured starts active. The only transitions are active to moderate care at intensity $0.12$, moderate care to intensive care at intensity $0.08$, moderate care to death at intensity $0.12$, and intensive care to death at intensity $0.20$, all per year. Benefits are paid continuously at $5,000 per year in moderate care and $11,000 per year in intensive care. The force of interest is $\delta=0.10$ per year. Find the EPV.

Method. The transient states are active, moderate care, and intensive care. Their generator submatrix is

Q_{TT}= \begin{pmatrix} -0.12&0.12&0\\ 0&-0.20&0.08\\ 0&0&-0.20 \end{pmatrix}.

The moderate-care diagonal is $-0.20$, because the total exit intensity from moderate care is $0.08+0.12=0.20$. The intensive-care exit intensity is $0.20$. For a direct calculation, discounted active occupancy is $1/(0.10+0.12)=1/0.22$ years. Multiplying by the transition intensity $0.12$ gives discounted moderate-care occupancy:

D_M=\frac{0.12}{(0.10+0.12)(0.10+0.08+0.12)} =\frac{0.12}{0.22(0.30)} =1.8181818\ldots\text{ years}.

The rate of entry into intensive care is $0.08$ times the probability of being in moderate care. Discounting the time spent in intensive care, where the total exit intensity is $0.20$, gives

D_I=\frac{0.08}{0.10+0.20}D_M =\frac{0.08}{0.30}(1.8181818\ldots) =0.4848484\ldots\text{ years}.

The two benefit components are $5{,}000(1.8181818\ldots)=$9{,}090.91$ and $11{,}000(0.4848484\ldots)=$5{,}333.33$, each rounded to cents. Hence the total EPV is $14,424.24. The calculation avoids omitting the moderate-to-intensive transition when finding the moderate-state exit rate, and it values each state’s occupancy at that state’s own benefit rate.

Worked Example: A Benefit Payable for a Fixed Period

A hypothetical insured starts active. The intensity from active to care is $0.20$ per year, and the intensity from care to death is $0.10$ per year. There is no recovery. A benefit of $12,000 per year is paid continuously while the insured is in care, but only during the first four years after issue. The force of interest is $\delta=0.05$ per year. Find the EPV.

Method. The care occupancy probability is

p_C(t)=\frac{0.20}{0.20-0.10}\left(e^{-0.10t}-e^{-0.20t}\right) =2\left(e^{-0.10t}-e^{-0.20t}\right).

Because payments stop after four years, integrate only from $0$ to $4$. Combining discounting with each exponential term gives

\begin{aligned} \operatorname{EPV} &=12{,}000\int_0^4 e^{-0.05t}2\left(e^{-0.10t}-e^{-0.20t}\right)\,dt\\ &=24{,}000\left[ \frac{1-e^{-0.15(4)}}{0.15} -\frac{1-e^{-0.25(4)}}{0.25} \right]\\ &=24{,}000\left[ \frac{1-e^{-0.6}}{0.15} -\frac{1-e^{-1}}{0.25} \right]\\ &=24{,}000(3.0079224-2.5284822)\\ &=24{,}000(0.4794402) =$11{,}506.56. \end{aligned}

The final value is rounded to the nearest cent. Check: the discounted care occupancy is $0.9588804$ years, and $0.9588804\times$12,000=$11,506.56$. The finite payment period is handled by changing the integration limit, not by assuming the insured leaves care at year 4.

General Calculation Procedure and Common Traps

A reliable calculation separates the state model from the benefit design. First determine which states qualify and the payment rate in each. Next obtain the state occupancy probabilities under the stated initial distribution and transition intensities. Finally, discount and integrate each eligible state’s expected payment rate. The following sequence is useful when the model has several care levels.

Specify the eligible states.
Translate the benefit trigger into a set $\mathcal{B}$. Do not include a state merely because it is related to care if the stated benefit does not apply there.

Write the state benefit rates.
Use dollars per year for continuous benefits. If a benefit varies with time, retain that dependence as $b_j(t)$.

Obtain occupancy probabilities.
Use the specified state model, initial distribution, and transition intensities. For constant intensities, the transient-state probabilities follow from the transient submatrix of the generator.

Discount and integrate.
Evaluate $\int e^{-\delta t}\sum_{j\in\mathcal{B}}p_j(t)b_j(t)\,dt$ over the period during which the benefit can be paid.

Check units and scope.
Occupancy integrates to years; multiplying by a rate in dollars per year gives dollars. Confirm that the time horizon and eligible states match the contract description.

A frequent exam trap is to apply an occupancy formula to the full generator as though all states were transient. For a model with an absorbing death state, the probability of being dead may approach one. Discounted occupancy in that state can be positive when $\delta>0$, and undiscounted occupancy over an infinite horizon does not converge. For care-benefit valuation, use only the relevant eligible-state probabilities in the integral, or use $Q_{TT}$ for a transient-state matrix calculation. Do not remove death-transition intensities from the care-state diagonal: death reduces the time spent eligible for care benefits.

Another error is confusing a continuous rate with a lump sum. A rate of $8,000 per year produces an amount near $8{,}000\,dt$ over a short interval $dt$, not an $8,000 payment each time the insured enters care. Also check the time unit: if intensities and benefit rates are annual, the integration variable must be measured in years and $\delta$ must be an annual force.

Key takeaway: A continuous care benefit is valued by integrating its discounted expected payment rate. For each eligible state, that rate is the state occupancy probability multiplied by the benefit rate in that state. Correct state eligibility, exit intensities, discounting, and integration limits are all essential to the result.

Check Your Understanding

For each problem, calculate the EPV of the stated continuous benefit. State occupancies are governed by the given constant transition intensities.

A hypothetical insured starts active, enters care at intensity $0.20$ per year, and exits care only by death at intensity $0.30$ per year. The benefit is $9,000 per year in care, payable without a time limit. The force of interest is $0.10$ per year. (Answer: $15,000.00)
A hypothetical insured has initial probabilities $0.60$ active and $0.40$ in care. Active enters care at intensity $0.15$ per year; care exits only by death at intensity $0.25$ per year. A continuous care benefit is $8,000 per year, and $\delta=0.05$. (Answer: $22,666.67)
A hypothetical insured starts active. Active enters moderate care at intensity $0.10$; moderate care enters intensive care at intensity $0.20$ or death at intensity $0.10$; intensive care exits to death at intensity $0.30$, all per year. Benefits are $4,000 per year in moderate care and $9,000 per year in intensive care. If $\delta=0.10$, find the EPV. (Answer: $10,625.00)
A hypothetical insured starts active, enters care at intensity $0.10$ per year, and exits care only by death at intensity $0.20$ per year. The continuous care benefit rate is $10{,}000e^{0.02t}$ dollars per year, payable without a time limit, where $t$ is years since issue. The force of interest is $0.08$ per year. (Answer: $24,038.46)

What You'll Learn