Sensitivity to Recovery and Relapse Assumptions

Why Recovery and Relapse Assumptions Matter

In a long-term care valuation, recovery changes how quickly a person leaves a care claim, while relapse determines how often a recovered person may return to care. These assumptions affect different parts of the contract cash flow: benefits are incurred in care, and, under some premium designs, premiums are collected only while the insured is recovered. A change in one transition intensity can therefore alter both expected benefit outgo and the premium income available to fund it.

This tutorial studies a hypothetical two-state model. It isolates recovery and relapse so their effects can be identified without introducing other transitions. In particular, mortality is excluded from this model; the formulas below are not mortality-adjusted. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model definition: The insured is in either care, state $C$, or recovery, state $R$. The transition intensity from care to recovery is the constant $\rho$, and the transition intensity from recovery to care is the constant $\lambda$, both measured per year. Conditional on the current state, transition times follow the continuous-time Markov model with these intensities. There are no other exits, the force of interest is constant at $\delta>0$, and the valuation begins in care unless stated otherwise.

With constant transition intensity $\rho$, a care spell has an exponential duration with mean $1/\rho$ years. A recovery spell has an exponential duration with mean $1/\lambda$ years when $\lambda>0$. Thus, a higher recovery intensity shortens each care spell on average; a higher relapse intensity shortens each recovery spell and can produce more recurrent claims. These spell means alone do not give the present value of all future payments, since the number and timing of later transitions also matter.

Discounted Occupancy and Premium Equivalence

Let $U_C$ be the expected present value of one dollar per year paid continuously while in care, starting in care. Let $U_R$ be the expected present value of one dollar per year paid continuously while in recovery, also starting in care. These are discounted state-occupancy times, measured in years. They include all future spells in each state, not only the first spell.

U_C=E\left[\int_0^\infty e^{-\delta t}\mathbf{1}_{\{X_t=C\}}\,dt\right], \qquad U_R=E\left[\int_0^\infty e^{-\delta t}\mathbf{1}_{\{X_t=R\}}\,dt\right].

Here $X_t$ denotes the state at time $t$, and the indicator is one when the insured occupies the named state. Suppose a hypothetical policy pays a continuous benefit at rate $B$ dollars per year in care and charges a continuous net premium at rate $P$ dollars per year in recovery. Under the equivalence principle, the expected present value of premiums equals the expected present value of benefits. Therefore $P U_R=B U_C$, provided $U_R>0$.

P=B\frac{U_C}{U_R} =B\frac{\delta+\lambda}{\rho}, \qquad \rho>0.

This is a net premium rate under the stated assumptions, not a gross premium including expenses, profit, or risk margins. It also assumes premiums continue in every recovery period, including after a relapse and subsequent recovery. If premiums are payable only during an initial recovery period, or cease after a specified number of claims, this equivalence formula does not apply without modification.

Deriving the Occupancy Formulas

Let $p_C(t)$ and $p_R(t)$ be the probabilities of occupying care and recovery at time $t$, conditional on starting in care. Over a short interval of length $h$, a person in care moves to recovery with probability approximately $\rho h$, and a person in recovery moves to care with probability approximately $\lambda h$. The forward equation for care occupancy is therefore

\frac{d}{dt}p_C(t)=-\rho p_C(t)+\lambda p_R(t), \qquad p_R(t)=1-p_C(t), \qquad p_C(0)=1.

Substituting $p_R(t)=1-p_C(t)$ gives $p_C'(t)=\lambda-(\rho+\lambda)p_C(t)$. Solving this first-order equation gives $p_C(t)=\lambda/(\rho+\lambda)+\rho e^{-(\rho+\lambda)t}/(\rho+\lambda)$. The recovery probability is $p_R(t)=\rho[1-e^{-(\rho+\lambda)t}]/(\rho+\lambda)$. Discounting and integrating these occupancy probabilities yields the formulas below.

Integrate care occupancy.
$U_C=\int_0^\infty e^{-\delta t}p_C(t)\,dt=\frac{\lambda}{\rho+\lambda}\frac{1}{\delta}+\frac{\rho}{\rho+\lambda}\frac{1}{\delta+\rho+\lambda}$.

Combine the terms.
Putting the terms over a common denominator gives $U_C=\frac{\delta+\lambda}{\delta(\delta+\rho+\lambda)}$.

Integrate recovery occupancy.
$U_R=\frac{\rho}{\rho+\lambda}\left(\frac{1}{\delta}-\frac{1}{\delta+\rho+\lambda}\right)=\frac{\rho}{\delta(\delta+\rho+\lambda)}$.

U_C=\frac{\delta+\lambda}{\delta(\delta+\rho+\lambda)}, \qquad U_R=\frac{\rho}{\delta(\delta+\rho+\lambda)}.

A useful check is that $U_C+U_R=1/\delta$. With no exit from the two-state system, the insured occupies one of the two states at every time, so the total discounted occupancy must equal $\int_0^\infty e^{-\delta t}dt=1/\delta$. The check also helps detect a state-label or transition-rate error.

Worked Example: Pricing with a Higher Recovery Intensity

A hypothetical policy pays $B=\$12{,}000$ per year continuously in care and charges a continuous premium only in recovery. Starting in care, use $\delta=0.05$, $\rho=0.45$, and $\lambda=0.15$, all per year. Find discounted occupancy in each state and the net premium rate. Then recalculate the premium if recovery intensity increases to $0.60$, with all else unchanged.

Method. Calculate the common denominator $D=\delta(\delta+\rho+\lambda)$, use the occupancy formulas, and equate expected present values. This avoids treating the mean of the first care spell as though it were the total discounted care occupancy.

For the original assumptions, $D=0.05(0.05+0.45+0.15)=0.05(0.65)=0.0325$. The discounted care occupancy is $U_C=(0.05+0.15)/0.0325=0.20/0.0325=6.153846$ years. The discounted recovery occupancy is $U_R=0.45/0.0325=13.846154$ years. Their sum is $20.000000=1/0.05$, as required.

The benefit present value is $\$12{,}000(6.153846)=\$73{,}846.15$ to the nearest cent. The premium rate is $P=\$12{,}000(6.153846/13.846154)=\$12{,}000(0.444444)=\$5{,}333.33$ per year. Equivalently, $P=B(\delta+\lambda)/\rho=\$12{,}000(0.20/0.45)=\$5{,}333.33$.

When $\rho=0.60$, $D=0.05(0.05+0.60+0.15)=0.04$. Thus $U_C=0.20/0.04=5$ years and $U_R=0.60/0.04=15$ years. The premium becomes $P=\$12{,}000(5/15)=\$4{,}000.00$ per year. The check $5+15=20=1/\delta$ still holds. Faster recovery reduces discounted care occupancy and increases recovery occupancy, lowering the rate needed to balance the expected values.

How Recovery and Relapse Change Valuation

The premium formula makes the relative effects particularly clear: $P=B(\delta+\lambda)/\rho$. Holding the benefit and discount force fixed, increasing $\rho$ lowers the required premium rate, while increasing $\lambda$ raises it. The reason is not that relapse changes the length of an individual care spell; that spell’s mean remains $1/\rho$. Instead, relapse shortens recovery spells and increases the amount of time, in discounted expectation, spent receiving benefits.

The direction of change in the two occupancy values can also be checked by differentiating. Write $D=\delta(\delta+\rho+\lambda)$. Then

\frac{\partial U_C}{\partial \rho} =-\frac{\delta+\lambda}{\delta(\delta+\rho+\lambda)^2}<0, \qquad \frac{\partial U_C}{\partial \lambda} =\frac{\rho}{\delta(\delta+\rho+\lambda)^2}>0,

\frac{\partial U_R}{\partial \rho} =\frac{\delta+\lambda}{\delta(\delta+\rho+\lambda)^2}>0, \qquad \frac{\partial U_R}{\partial \lambda} =-\frac{\rho}{\delta(\delta+\rho+\lambda)^2}<0.

Thus, a higher recovery intensity shifts discounted occupancy toward recovery, while a higher relapse intensity shifts it toward care. These comparative statics describe a change in one assumption at a time, holding the other model inputs fixed. If multiple assumptions change together, the net effect must be calculated rather than inferred from either direction alone.

Worked Example: Comparing Alternative Relapse Assumptions

A hypothetical contract pays $\$10{,}000$ per year continuously in care and collects premiums continuously only in recovery. Take $\delta=0.04$ and $\rho=0.50$. Compare $\lambda=0.10$ with $\lambda=0.30$. Find discounted occupancy and the net premium rate in each case, rounding money to the nearest cent.

Method. Keep recovery and discount assumptions fixed, vary relapse, and recompute $D$, both occupancy values, and the premium. The trap is to assume that a relapse change affects only the frequency of claims but not the total time in care.

For $\lambda=0.10$, $D=0.04(0.04+0.50+0.10)=0.04(0.64)=0.0256$. Hence $U_C=(0.04+0.10)/0.0256=0.14/0.0256=5.46875$ years and $U_R=0.50/0.0256=19.53125$ years. The total is $25=1/0.04$ years. The net premium is $P=\$10{,}000(0.14/0.50)=\$2{,}800.00$ per year.

For $\lambda=0.30$, $D=0.04(0.04+0.50+0.30)=0.04(0.84)=0.0336$. Thus $U_C=(0.04+0.30)/0.0336=0.34/0.0336=10.119048$ years and $U_R=0.50/0.0336=14.880952$ years. Their sum is again $25$. The premium is $P=\$10{,}000(0.34/0.50)=\$6{,}800.00$ per year. Higher relapse intensity increases care occupancy and more than doubles the rate in this comparison, while recovery occupancy decreases.

As a check, the premium present value in the first case is $\$2{,}800(19.53125)=\$54{,}687.50$, equal to $\$10{,}000(5.46875)=\$54{,}687.50$. In the second case, both benefit and premium present values are $\$101{,}190.48$, subject only to rounding.

Recurrent Claims Over a Finite Period

Discounted occupancy summarizes the value of continuous state-based payments. A separate question is how many new care spells are expected during a finite observation period. Starting in care, the probability of being in recovery at time $t$ is $p_R(t)=\rho[1-e^{-(\rho+\lambda)t}]/(\rho+\lambda)$. Each person in recovery has instantaneous relapse intensity $\lambda$. Therefore, the expected number of relapse transitions during $[0,T]$, not counting the initial care spell, is

E[N_{\mathrm{relapse}}(T)] =\int_0^T \lambda p_R(t)\,dt =\frac{\lambda\rho}{\rho+\lambda} \left[T-\frac{1-e^{-(\rho+\lambda)T}}{\rho+\lambda}\right].

This is an expected transition count, so it need not be an integer. If each relapse initiates a new care spell, the expected number of care spells begun by time $T$ is one plus this value, because the model starts with one spell already in progress. The formula counts relapse entries by $T$; it does not add care spells that begin after $T$.

Worked Example: Expected Recurrent Claims Within Five Years

A hypothetical insured starts in care. Let $\rho=0.40$ and $\lambda=0.20$ per year. Find the expected number of relapses during the first five years and the expected number of care spells begun by that time. Interest is not needed for this transition-count calculation.

Method. Integrate the relapse intensity over the probability of occupying recovery. This accounts for the fact that relapse can occur only while in recovery; multiplying $\lambda$ by five years alone would incorrectly assume recovery occupancy throughout the period.

Here $\rho+\lambda=0.60$, and $e^{-0.60(5)}=e^{-3}=0.0497871$. The expected number of relapses is

\frac{(0.20)(0.40)}{0.60} \left[5-\frac{1-0.0497871}{0.60}\right] =0.1333333[5-1.5836882] =0.1333333(3.4163118) =0.4555082.

Thus the expected number of relapses is approximately $0.4555$, and the expected number of care spells begun by year five, including the initial spell, is $1+0.4555082=1.4555082$. The expectation is fractional because it averages across hypothetical insureds; an individual’s actual count is an integer.

Common Mistakes and Interpretation

Confusing state occupancy with a first-spell mean. The first care spell has mean $1/\rho$, but $U_C$ includes all future care spells, with discounting. In general, $U_C\ne1/\rho$.
Using the wrong transition intensity in the numerator. Starting in care, discounted recovery occupancy is $U_R=\rho/D$, because recovery is entered from care at rate $\rho$. Discounted care occupancy is $U_C=(\delta+\lambda)/D$. The formulas also satisfy $U_C+U_R=1/\delta$.
Reversing the premium ratio. Equivalence is $P U_R=B U_C$, so $P=B U_C/U_R$, not $B U_R/U_C$. Premiums are paid in recovery and benefits in care.
Calling the rate gross. The equivalence calculation includes only the specified benefit and premium streams. It does not include expenses, margins, caps, or mortality.
Interpreting relapse as changing care-spell length. Under this model, the mean of a care spell is $1/\rho$. Relapse changes the duration of recovery spells and the recurrence of care spells.

Key takeaway: For this two-state model, starting in care, discounted care and recovery occupancy are $(\delta+\lambda)/[\delta(\delta+\rho+\lambda)]$ and $\rho/[\delta(\delta+\rho+\lambda)]$, respectively. A continuous recovery-only net premium balancing a care benefit rate $B$ is $B(\delta+\lambda)/\rho$. Recovery reduces expected care exposure; relapse increases it.

Check Your Understanding

Use the two-state assumptions stated in each question. Give money answers to the nearest cent where applicable.

Starting in care, let $\delta=0.05$, $\rho=0.25$, and $\lambda=0.10$. Find $U_C$ and $U_R$. (Answer: $U_C=7.5$ years and $U_R=12.5$ years.)
A policy pays $\$9{,}000$ per year in care and charges continuous premiums only in recovery. Let $\delta=0.03$, $\rho=0.30$, and $\lambda=0.12$. Find the net premium rate. (Answer: $\$4{,}500.00$ per year.)
For $B=\$10{,}000$, $\delta=0.04$, and $\rho=0.50$, compare the net premium rates when $\lambda=0.10$ and $\lambda=0.30$. (Answer: $\$2{,}800.00$ and $\$6{,}800.00$ per year, respectively.)
An insured starts in care with $\rho=0.50$, $\lambda=0.25$, and no mortality. Find the expected number of relapse transitions during the first four years. (Answer: approximately $0.4555$.)
A hypothetical policy pays $\$15{,}000$ per year in care and charges premiums only in recovery. With $\delta=0.04$, $\rho=0.40$, and $\lambda=0.20$, find the net premium rate. (Answer: $\$9{,}000.00$ per year.)

What You'll Learn