Mortality Selection and Competing Death During Care

Why Mortality During Care Must Be Modeled Separately

A long-term care benefit usually stops when the insured recovers or dies. Those are different events: recovery returns the insured to a non-care state, while death terminates the policyholder’s life. Treating both as a single generic “exit from care” may be enough to calculate total time in care, but it is not enough to answer questions about the probability of death during care or the separate probabilities of recovery and death.

Mortality can also differ between people who are healthy and people who have entered care. A model that uses one mortality rate in every state can therefore misstate both the probability of reaching care and the duration of benefits after care begins. The calculations below use a simple continuous-time model to make these effects explicit. The focus is state-specific mortality and its effect on care benefits, rather than a general treatment of multiple decrements.

Definition: Consider four states: healthy $H$, receiving care $C$, recovered $R$, and dead $D$. In the basic model, the possible transitions are $H\to C$ at incidence intensity $\lambda$, $H\to D$ at mortality intensity $\mu_H$, $C\to R$ at recovery intensity $\rho$, and $C\to D$ at mortality intensity $\mu_C$. Rates are measured per year. Unless stated otherwise, they are constant, and the process has the Markov property: given the current state, the future transition probabilities do not depend on the earlier path.

In this model, $\mu_H$ and $\mu_C$ are not interchangeable. The former governs death before care entry; the latter governs death while receiving care. Recovery at rate $\rho$ ends a care episode, but it does not count as death. The model assumes no relapse after recovery and no direct transition from healthy to recovery. These assumptions are appropriate for the calculations here, but a contract with relapse, multiple care levels, or time-varying intensities needs a richer model.

Competing Ways for a Care Episode to End

Suppose a person is in state $C$ at time zero. With constant recovery and mortality intensities, the total rate of leaving care is $k=\rho+\mu_C$. The probability of remaining in care for at least $t$ years is $e^{-kt}$. This probability describes not yet recovering or dying; it does not mean that the person survives in the healthy state.

The probability of recovery by time $T$ is found by accumulating the probability of being in care at time $t$ and recovering in the next small interval. The corresponding calculation for death uses $\mu_C$ instead of $\rho$. Thus, each cause has its own probability, even though both causes contribute to the total rate of leaving care.

\begin{aligned} P(\text{still in care at }T\mid C\text{ at }0) &= e^{-(\rho+\mu_C)T},\\ P(\text{recovery by }T\mid C\text{ at }0) &= \int_0^T \rho e^{-(\rho+\mu_C)t}\,dt = \frac{\rho}{\rho+\mu_C} \left(1-e^{-(\rho+\mu_C)T}\right),\\ P(\text{death in care by }T\mid C\text{ at }0) &= \int_0^T \mu_C e^{-(\rho+\mu_C)t}\,dt = \frac{\mu_C}{\rho+\mu_C} \left(1-e^{-(\rho+\mu_C)T}\right). \end{aligned}

These probabilities sum to one. The expected time in care before recovery or death, with no benefit-period cap, is $1/(\rho+\mu_C)$ years. With a cap of $T$ years, the expected paid duration is the expected time spent in care up to the cap: $\int_0^T e^{-(\rho+\mu_C)t}\,dt$. This duration calculation counts neither time after recovery nor time after death as payable.

Worked Example: Separating Recovery From Death During a Capped Benefit

A hypothetical insured is already receiving care. Recovery has constant intensity $\rho=0.25$ per year, and death during care has constant intensity $\mu_C=0.15$ per year. Benefits are payable continuously while in care, subject to a two-year maximum. Find the probability of remaining in care for two years, the probabilities of recovery and death within two years, and the expected paid duration.

Method. Use the total care-exit intensity $k=\rho+\mu_C$ for remaining in care and expected duration, then allocate exits using the separate recovery and death intensities. The model has no relapse and assumes constant rates.

First, $k=0.25+0.15=0.40$ per year. The probability of still being in care after two years is $e^{-0.40(2)}=e^{-0.8}=0.4493$.

The probability of recovery within two years is \[ \frac{0.25}{0.40}\left(1-e^{-0.8}\right) =0.625(1-0.4493) =0.3442. \] The probability of death during care within two years is \[ \frac{0.15}{0.40}\left(1-e^{-0.8}\right) =0.375(1-0.4493) =0.2065. \]

The expected paid duration, limited to two years, is \[ \int_0^2 e^{-0.40t}\,dt =\frac{1-e^{-0.8}}{0.40} =\frac{0.5507}{0.40} =1.3767\text{ years}. \]

Check. The probabilities add to $0.4493+0.3442+0.2065=1.0000$. The two-year benefit cap is respected because the expected paid duration is less than two years. The common mistake avoided here is assigning all care exits to recovery or treating death as if it were a recovery.

Mortality Before Care Entry and Occupancy of the Care State

For an insured initially healthy, two events compete to determine whether care is reached: entry into care at intensity $\lambda$, and death while healthy at intensity $\mu_H$. Write $a=\lambda+\mu_H$. The probability of remaining healthy through time $s$ is $e^{-as}$, and the probability density of entering care at time $s$ is $\lambda e^{-as}$.

If care begins at time $s$, the probability of still being in care at a later time $t$ is $e^{-(\rho+\mu_C)(t-s)}$. Multiplying these terms and integrating over every possible entry time gives the probability of occupying the care state at time $t$. This distinction matters: someone can enter care but recover or die before time $t$, so “ever entered care” and “in care at time $t$” are not the same probability.

\begin{aligned} P(C\text{ at }t\mid H\text{ at }0) &=\int_0^t \lambda e^{-as} e^{-(\rho+\mu_C)(t-s)}\,ds\\ &=\frac{\lambda}{b-a}\left(e^{-at}-e^{-bt}\right), \qquad b=\rho+\mu_C,\quad a\ne b. \end{aligned}

The second line follows by taking $e^{-bt}$ outside the integral and integrating $e^{(b-a)s}$ from zero to $t$. If $a=b$, the integral instead equals $\lambda t e^{-at}$. The formula applies to the specified constant-intensity, no-relapse model; it is not a general formula for models with multiple care states or transitions back into care.

Worked Example: From Healthy Status to Care Occupancy and Benefit Value

A hypothetical policyholder starts healthy. The intensity of entering care is $\lambda=0.08$ per year, healthy-state mortality is $\mu_H=0.02$, recovery intensity is $\rho=0.20$, and care-state mortality is $\mu_C=0.10$. All rates are constant. Find (1) the probability of entering care before healthy-state death within four years, (2) the probability of being in care at year four, and (3) the expected present value of a continuous benefit of $18,000 per year, payable from care entry until recovery or death, with no maximum duration. The force of interest is $\delta=0.04$ per year.

Method. Use the healthy-state transition rates for care-entry probability, integrate over possible entry times for year-four care occupancy, and discount the expected care-state payment stream for present value. Benefits stop at either recovery or death.

For the first calculation, $a=\lambda+\mu_H=0.08+0.02=0.10$. The probability of entering care by year four before dying in the healthy state is \[ \int_0^4 0.08e^{-0.10s}\,ds =\frac{0.08}{0.10}(1-e^{-0.40}) =0.8(1-0.6703) =0.2637. \]

For the second calculation, $b=\rho+\mu_C=0.20+0.10=0.30$. Since $b-a=0.20$, the probability of occupying care at year four is \[ \frac{0.08}{0.30-0.10}(e^{-0.10(4)}-e^{-0.30(4)}) =\frac{0.08}{0.20}(0.6703-0.3010) =0.1477. \] This is smaller than the probability of having entered care by year four because some entrants have already recovered or died.

For the present value, the probability of being in care at time $t$, starting healthy, is $\frac{\lambda}{b-a}(e^{-at}-e^{-bt})$. Discounting the continuous benefit at force $\delta$ gives \[ \begin{aligned} \text{EPV} &=18{,}000\int_0^\infty e^{-\delta t} \frac{0.08}{0.20}(e^{-0.10t}-e^{-0.30t})\,dt\\ &=18{,}000\frac{0.08}{0.20} \left(\frac{1}{0.04+0.10}-\frac{1}{0.04+0.30}\right)\\ &=18{,}000\frac{0.08}{(0.14)(0.34)} =\$30{,}252.10. \end{aligned} \] The benefit rate is per year, so integrating over time produces dollars. The result is rounded to the nearest cent.

Check. The probability of being in care at year four, $0.1477$, is below the probability of entering care by that time, $0.2637$. The present value is positive and finite because discounting and care exits both reduce future payments. The common mistake avoided is using the care-entry probability as though every entrant remained in care through year four and thereafter.

Mortality Selection Among Care Entrants

Care entrants may have a different mortality assumption from healthy lives. For example, a hypothetical portfolio may classify entrants by an underwriting or clinical risk factor, with different care-state mortality intensities for each class. The class-specific intensity should be used for benefits after care entry. Applying healthy-state mortality to someone in care ignores the state change; using a single average care mortality can also conceal meaningful differences in expected duration and value.

For a person already in care, a continuous benefit paid at rate $c$ dollars per year until recovery or death has expected present value $c/(\delta+\rho+\mu_C)$, assuming constant rates, constant force of interest, and no benefit cap. This follows by multiplying the benefit rate by the probability of remaining in care, $e^{-(\rho+\mu_C)t}$, and the discount factor $e^{-\delta t}$, then integrating over $t\geq 0$.

Worked Example: Comparing Care Entrants With Different Mortality

Two hypothetical groups enter the same care benefit. Each receives a continuous benefit of $24,000 per year. Recovery intensity is $\rho=0.15$ per year for both groups, and the force of interest is $\delta=0.04$. The first group has care-state mortality $\mu_C=0.05$; the second has $\mu_C=0.20$. Find each group’s expected care duration, probability of still being in care after three years, and expected present value. For comparison, healthy-state mortality is $\mu_H=0.03$ in both groups.

Method. For each group, add recovery and care-state mortality to obtain the care-exit intensity. Use that intensity for duration and care-state occupancy, then include interest in the benefit present value.

For the first group, the total care-exit intensity is $0.15+0.05=0.20$. Its expected care duration is $1/0.20=5.0000$ years. The probability of still being in care after three years is $e^{-0.20(3)}=e^{-0.60}=0.5488$. Its expected present value is \[ \frac{\$24{,}000}{0.04+0.15+0.05} =\frac{\$24{,}000}{0.24} =\$100{,}000.00. \]

For the second group, the total care-exit intensity is $0.15+0.20=0.35$. Its expected care duration is $1/0.35=2.8571$ years. The probability of still being in care after three years is $e^{-0.35(3)}=e^{-1.05}=0.3499$. Its expected present value is \[ \frac{\$24{,}000}{0.04+0.15+0.20} =\frac{\$24{,}000}{0.39} =\$61{,}538.46. \]

The first group’s longer expected care duration and higher probability of remaining in care lead to a larger benefit value, all else equal. A mistaken calculation might use healthy-state mortality $0.03$ for the second group, giving $\$24{,}000/(0.04+0.15+0.03)=\$109{,}090.91$. The correction is to use the specified care-state mortality $0.20$, giving $\$61{,}538.46$. The distinction between $\mu_H$ and $\mu_C$ is essential.

Derivation and Practical Checks

The key care-state formulas follow directly from the constant-intensity assumptions. Over a short interval of length $dt$, a person currently in care recovers with probability approximately $\rho\,dt$, dies with probability approximately $\mu_C\,dt$, and remains in care with probability approximately $1-(\rho+\mu_C)dt$. The chance of two transitions in the same short interval is negligible relative to $dt$. If $S_C(t)$ is the probability of remaining in care through time $t$, this gives the differential relation \[ S_C(t+dt)-S_C(t)\approx-(\rho+\mu_C)S_C(t)\,dt. \] Taking the limit yields $S_C'(t)=-(\rho+\mu_C)S_C(t)$, with $S_C(0)=1$, and hence $S_C(t)=e^{-(\rho+\mu_C)t}$. The recovery probability by $T$ is the integral of the recovery rate times the probability still exposed to recovery at each time, $\int_0^T \rho S_C(t)\,dt$; the death probability is obtained in the same way with $\mu_C$. This justifies both the total-exit and cause-specific formulas.

When rates vary with time, the same idea uses cumulative intensities: care-state survival to time $t$ becomes $\exp\left(-\int_0^t(\rho(s)+\mu_C(s))\,ds\right)$. Cause-specific probabilities then integrate the appropriate intensity against this survival probability. This extension requires rates defined on the relevant time scale and is not obtained by simply inserting an average rate into the constant-rate formula.

Identify the state and the endpoint.
Decide whether the calculation starts healthy or already in care, and whether the question asks about care entry, continued care, recovery, death, or benefit value.

Use the rates for that state.
Healthy-state mortality applies before care entry; care-state mortality applies during care. Add recovery to care mortality only when calculating total care exits.

Match the payment rule.
Stop benefits at recovery or death, apply any maximum payment period, and include interest when calculating present value.

Check probability and unit consistency.
Cause-specific exit probabilities plus the probability of remaining in care must sum to one. Intensities, time periods, and the force of interest must use consistent annual units.

The distinction is useful in pricing and valuation because mortality can materially shorten the period for which benefits are paid. Higher mortality during care generally reduces expected care duration and benefit value, but mortality before care entry has a different effect: it reduces the opportunity to reach care in the first place. The two effects should not be conflated. These state distinctions are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: Recovery and death both terminate care benefits, but they are separate transitions with separate intensities. Use healthy-state mortality before care entry and care-state mortality during care; then calculate care occupancy and payment value from the probability of still being in the care state.

Check Your Understanding

Use constant annual intensities and the stated payment assumptions. Probabilities are rounded to four decimals and dollar values to the nearest cent.

A person starts in care with recovery intensity $0.10$ and care mortality $0.20$. For a three-year benefit maximum, find the probability of remaining in care for three years, the probabilities of recovery and death within three years, and the expected paid duration. (Answer: $0.4066$ remaining; $0.1978$ recovery; $0.3956$ death; $1.9781$ expected years.)
A healthy person has care-entry intensity $0.06$ and healthy mortality $0.04$. Find the probability of entering care before healthy-state death within five years. (Answer: $0.2361$.)
A person already in care receives a continuous benefit of $21,600 per year. Recovery intensity is $0.10$, care mortality is $0.05$, and the force of interest is $0.03$. There is no benefit cap. Find the expected present value. (Answer: $\$120{,}000.00$.)
A healthy person has care-entry intensity $0.12$ and healthy mortality $0.03$. After entry, recovery intensity is $0.20$ and care mortality is $0.10$. Find the probability of being in care at time three. (Answer: $0.1848$.)
Two care-entry groups have the same recovery intensity $0.12$, but care mortality is $0.04$ for one group and $0.16$ for the other. Find each group’s expected care duration and the reduction in duration for the higher-mortality group. (Answer: $6.2500$ years and $3.5714$ years; reduction $2.6786$ years.)

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