Recovery Effects on Premium and Benefit Duration

Why Recovery Changes Both Sides of the Premium Calculation

A long-term care benefit is usually payable only while the insured satisfies the contract’s care eligibility conditions. If an insured recovers, benefit payments stop while the insured is active again. When the contract waives premiums during care but resumes them after recovery, recovery also creates additional premium-paying time. A model that treats care as permanent misses both effects.

This tutorial compares those cash flows in a two-state continuous-time Markov model. The model is deliberately limited to active and care states: it excludes death, expenses, lapses, benefit caps, and any maximum payment period. These assumptions make it possible to isolate the effect of recovery. The transition holding times are exponential, with constant transition intensities over the coverage term, and the insured begins in the active state.

Definition: Let $A$ denote the active state and $C$ the care state. The transition intensity from $A$ to $C$ is $\lambda>0$, and the recovery intensity from $C$ to $A$ is $\rho\geq 0$. For an insured active at time 0, write $p_A(t)=P(X_t=A)$ and $p_C(t)=P(X_t=C)$. Thus $p_A(t)+p_C(t)=1$. A value $\rho=0$ means care is absorbing: once entered, it cannot be left in this model.

Suppose a benefit is paid continuously at rate $b$ per year while in care. Suppose premiums are paid continuously at rate $P$ per year while active, are waived in care, and resume after recovery. The coverage term is $T$, and the force of interest is $\delta$. The net premium principle equates the expected present value of benefits and premiums.

State Probabilities and Present Values

The transition equations for this two-state model are $p_C'(t)=\lambda p_A(t)-\rho p_C(t)$ and $p_A'(t)=-\lambda p_A(t)+\rho p_C(t)$, with $p_A(0)=1$ and $p_C(0)=0$. Since $p_A(t)=1-p_C(t)$, the care-state equation becomes $p_C'(t)=\lambda-(\lambda+\rho)p_C(t)$. Solving gives:

p_C(t)=\frac{\lambda}{\lambda+\rho}\left(1-e^{-(\lambda+\rho)t}\right), \qquad p_A(t)=1-p_C(t).

The expected discounted benefit is the benefit rate multiplied by the discounted expected time spent in care. The expected discounted premiums are the premium rate multiplied by the discounted expected time spent active. In particular, define $B$ as the expected present value of benefits per unit of benefit rate and $E_A$ as the expected present value of premium-paying exposure per unit of premium rate:

B=\int_0^T e^{-\delta t}p_C(t)\,dt, \qquad E_A=\int_0^T e^{-\delta t}p_A(t)\,dt.

These expressions follow by considering a small interval from $t$ to $t+dt$. The probability of being in care at time $t$ is $p_C(t)$, so expected benefit paid in that interval is $b\,p_C(t)\,dt$; discounting and integrating over the term gives $bB$. The same reasoning for the active state gives $PE_A$. Therefore, when $E_A>0$, the continuous net premium rate is $P=bB/E_A$.

Result: For constant intensities and a finite term, let $q=\lambda+\rho$. If $\delta\geq 0$, then $B=\dfrac{\lambda}{q}\left[\dfrac{1-e^{-\delta T}}{\delta}-\dfrac{1-e^{-(\delta+q)T}}{\delta+q}\right]$ and $E_A=\dfrac{1-e^{-\delta T}}{\delta}-B$. When $\delta=0$, interpret $(1-e^{-\delta T})/\delta$ by its limit $T$. The net premium rate is $P=bB/E_A$.

The key point is that $B$ measures expected discounted care time, not the probability of ever entering care. An insured may enter care, recover, and later return to care; the state probability at each time accounts for the resulting time spent in each state. Likewise, $E_A$ includes all active time during which premiums are due, including active time after a recovery.

Worked Example: Calculating a Net Premium With Recovery

A hypothetical contract lasts 2 years. The insured begins active. The constant care-entry intensity is $\lambda=0.20$ per year and the recovery intensity is $\rho=0.30$ per year. The contract pays a continuous benefit of $b=\$1$ per year in care and charges a continuous premium only while active. The force of interest is $\delta=0.05$. Ignore death and all other cash flows. Find the net premium rate.

Method. Use the two-state occupancy probabilities, integrate discounted care and active exposure, and equate expected present values. Here $q=\lambda+\rho=0.50$, and the state probabilities are $p_C(t)=0.4(1-e^{-0.5t})$ and $p_A(t)=1-p_C(t)$.

First calculate the discounted annuity factor over 2 years: $(1-e^{-0.05(2)})/0.05=(1-e^{-0.1})/0.05=(1-0.9048374)/0.05=1.9032516$. The second integral needed for the care-state value is $(1-e^{-(0.05+0.50)(2)})/0.55=(1-e^{-1.1})/0.55=(1-0.3328711)/0.55=1.2129617$.

Therefore, $B=0.4(1.9032516-1.2129617)=0.4(0.6902899)=0.2761160$. The active premium-paying exposure is $E_A=1.9032516-0.2761160=1.6271356$. Thus the premium rate is $P=\$1(0.2761160)/1.6271356=\$0.1697$ per year, to four significant figures.

Check. The discounted total time in the two states is $B+E_A=0.2761160+1.6271356=1.9032516$, the discounted annuity factor over the term. The premium rate is below the benefit rate because the insured is expected to spend more discounted time active than in care. A common mistake is to use the care exposure $B$ as the premium exposure as well; that would ignore the contract’s rule that premiums are due while active.

What Changes When Recovery Is Removed?

If $\rho=0$, the care state is absorbing, and $p_C(t)=1-e^{-\lambda t}$. For the same incidence intensity, this gives a higher or equal care-state probability at every time than the recovery model: once care is entered, the insured remains there. Consequently, expected benefit exposure is higher or equal. Under the specified waiver rule, active premium-paying exposure is lower or equal because it ends permanently at care entry. Both changes tend to raise the net premium.

The comparison depends on the premium rule. If a contract instead charges premiums in every state, the premium exposure would be the discounted term annuity, not $E_A$. The recovery model still changes expected benefits, but it does not change that particular premium exposure. State the contract’s premium rule before calculating the denominator of a net premium.

Worked Example: Comparing Recovery With Absorbing Care

Use a hypothetical 2-year contract with $\lambda=0.20$ per year, $\delta=0.05$, a continuous benefit of \$1 per year while in care, and premiums payable continuously only while active. Compare the net premium rates when (a) $\rho=0.30$ per year and (b) $\rho=0$. The insured begins active; assume no death or other cash flows.

Method. Calculate $B$ and $E_A$ under each transition model. For case (a), the previous state-probability calculation gives $B=0.2761160$ and $E_A=1.6271356$. Hence $P=0.2761160/1.6271356=\$0.1697$ per year, to four significant figures.

For case (b), care is absorbing. The discounted term annuity remains $1.9032516$. The integral of $e^{-(\delta+\lambda)t}$ over the term is $(1-e^{-(0.05+0.20)(2)})/0.25=(1-e^{-0.5})/0.25=(1-0.6065307)/0.25=1.5738772$. Thus $B=1.9032516-1.5738772=0.3293744$, and $E_A=1.5738772$. The net premium rate is $P=\$1(0.3293744)/1.5738772=\$0.2093$ per year, to four significant figures.

Comparison and check. Allowing recovery reduces expected discounted care time from $0.3293744$ to $0.2761160$, while increasing premium-paying exposure from $1.5738772$ to $1.6271356$. The net premium consequently falls from \$0.2093 to \$0.1697 per year. The absorbing-care model has both more benefit exposure and less premium exposure, as expected. The common mistake avoided here is comparing only benefit values while leaving the premium denominator unchanged.

Derivation and Interpretation of the Occupancy Formula

The probability of being in care increases through transitions from active to care and decreases through recoveries. Over a short interval of length $h$, conditional on being active, the probability of entering care is $\lambda h+o(h)$; conditional on being in care, the probability of recovery is $\rho h+o(h)$. Dividing the resulting net probability change by $h$ and taking the limit gives $p_C'(t)=\lambda p_A(t)-\rho p_C(t)$. Using $p_A(t)=1-p_C(t)$ gives the scalar differential equation above.

To solve it, set $q=\lambda+\rho$. Then $p_C'(t)+q p_C(t)=\lambda$. Multiplying by the integrating factor $e^{qt}$ gives $(e^{qt}p_C(t))'=\lambda e^{qt}$. Integrating from 0 to $t$, and using $p_C(0)=0$, yields $e^{qt}p_C(t)=\lambda(e^{qt}-1)/q$, or $p_C(t)=(\lambda/q)(1-e^{-qt})$. This derivation shows why recovery appears in the rate $q$: it drains probability from care as well as incidence adding probability to it.

The long-run care probability in this two-state model is $\lambda/(\lambda+\rho)$, but that limiting value should not be substituted for $p_C(t)$ over a finite contract term. Nor should it be interpreted as the probability of ever entering care. It is the limiting proportion of time in care under the specified transitions.

Worked Example: Expected Care Time and Premium-Paying Time

A hypothetical 3-year contract begins with the insured active. The care-entry intensity is $\lambda=0.10$ per year, recovery intensity is $\rho=0.20$ per year, and the force of interest is $\delta=0$. A continuous benefit of \$12,000 per year is paid in care, and a continuous premium is payable only while active. Ignore death and other cash flows. Find expected total benefit and the net premium rate.

Method. With $q=0.10+0.20=0.30$, the care probability is $p_C(t)=(1/3)(1-e^{-0.30t})$. Expected care time over the term is $B=(1/3)[3-(1-e^{-0.30(3)})/0.30]$. Since $e^{-0.9}=0.4065697$, the bracketed expression is $3-(1-0.4065697)/0.30=3-0.5934303/0.30=3-1.9781010=1.0218990$. Therefore $B=1.0218990/3=0.3406330$ years.

Expected total benefit is $\$12{,}000(0.3406330)=\$4{,}087.60$, rounded to the nearest cent. With zero interest, total exposure over the term is 3 years, so active premium-paying time is $E_A=3-0.3406330=2.6593670$ years. The net premium rate is $P=\$4{,}087.60/2.6593670=\$1{,}537.06$ per year, rounded to the nearest cent.

Check. Expected care and active time sum to $0.3406330+2.6593670=3.0000000$ years. The premium rate is below the benefit rate of \$12,000 per year because the active exposure is much larger than the care exposure. The common mistake avoided is charging premiums for all 3 years despite the stated waiver during care.

Common Traps and Contract Interpretation

Recovery changes the timing and duration of both benefits and premiums; it does not simply “subtract a recovery probability” from a one-time care-entry probability. In the continuous-time model, the state can change repeatedly. Keep the following distinctions clear:

Care probability versus ever-entered probability. $p_C(t)$ is the probability of being in care at time $t$, not the probability of having entered care by then.
Benefit duration versus premium duration. Benefits are valued using care-state exposure; premiums are valued using the states in which the contract requires them to be paid.
Temporary waiver versus permanent cessation. If premiums resume after recovery, active-state exposure includes time after recovery. Treating the first care entry as permanently ending premiums understates expected premiums.
Units and timing. Intensities are rates per year, $T$ must be in years, and continuous payment rates are per year. A discrete annual premium requires its own payment-timing calculation.

This occupancy approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: the same state probabilities support both expected benefit valuation and premium valuation, but each cash flow must be attached to the correct state. It does not apply unchanged if death, expenses, caps, or other states and payment rules are added; those require the corresponding states and cash flows to be represented in the model.

Check Your Understanding

For each problem, assume constant transition intensities, exponential holding times, and an insured initially active. Use the stated benefit and premium rules.

There is no interest, $\lambda=0.10$ per year, $\rho=0.10$ per year, and $T=2$ years. Find expected total time in care. (Answer: 0.1758 years)
A one-year contract has no interest, $\lambda=0.20$ per year, and $\rho=0.20$ per year. It pays \$5,000 continuously per year in care. Find the expected benefit amount. (Answer: \$439.50)
For $T=2$ years, $\lambda=0.10$ per year, $\rho=0.30$ per year, and $\delta=0$, find the expected active premium-paying time when premiums are payable only while active. (Answer: 1.8442 years)
A one-year contract has $\lambda=0.20$, $\rho=0.20$, $\delta=0.10$, and a continuous benefit rate of \$1,000 per year in care. Premiums are payable only while active. Find the net continuous premium rate. (Answer: \$94.73 per year)
For $T=2$ years, $\lambda=0.15$ per year, $\rho=0.35$ per year, and $\delta=0$, find expected care time and expected active premium-paying time. (Answer: 0.2207 years in care and 1.7793 years active)

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