Mortality Intensities by Health and Care State

Why Mortality Depends on Health and Care State

A long-term care model may distinguish a healthy life from one recovering after an episode and from lives needing different levels of care. Mortality assumptions should reflect those distinctions: the instantaneous risk of death can differ materially by state. Assigning a single rate to every state can therefore misstate both survival and the expected duration of care.

The key is to interpret a mortality intensity as a rate conditional on being alive and in a specified state at the instant under consideration. It is not itself a probability of dying over a year. To obtain a probability, the intensity must be accumulated over time, and the time unit must match the unit used for the rate.

Definition: The state-specific mortality intensity in state $j$, denoted here by $\mu_j(t)$, is the instantaneous rate of death at time $t$, conditional on being alive and occupying state $j$ immediately before $t$. Formally, for small $h>0$, $$ P(\text{death in }[t,t+h)\mid \text{alive and in state }j\text{ at }t) =\mu_j(t)h+o(h). $$ The intensity is measured per unit of time. It is nonnegative, but it is not a probability and may exceed 1 when expressed per year.

The state labels must be defined before mortality assumptions are assigned. For example, a model might distinguish healthy, recovering, moderate-care, and severe-care states. “Recovering” should refer to the model’s specified clinical or contractual condition, not simply to a person who is expected to improve. A care-needing state should likewise have a precise definition. The rates are conditional on these definitions, so changing the criteria for a state can change the appropriate assumptions.

In this tutorial, rates are treated as constant within each named state unless stated otherwise. That is a simplifying assumption for calculation, not a claim that real mortality is constant by age or duration. The focus is how to interpret and use state-specific mortality rates; later refinements can allow rates to vary with age or time spent in a state.

From an Intensity to a Death Probability

Suppose a life is known to remain in a particular state throughout a period of length $t$, and the mortality intensity in that state is constant at $\mu$ per year. If $S(t)$ is the probability of surviving the period under that condition, the probability of death in the period is $1-S(t)$. The intensity determines how quickly survival declines.

Result: With constant mortality intensity $\mu$ per year, conditional on remaining in the state for $t$ years, survival is $e^{-\mu t}$ and the probability of death is $1-e^{-\mu t}$. This formula applies to mortality conditional on that state path; it does not by itself account for uncertainty about transitions between states.

A short derivation explains the exponential form. Over a small interval $h$, conditional survival is approximately $1-\mu h$. Thus the survival function satisfies $S(t+h)\approx S(t)(1-\mu h)$. Subtracting $S(t)$, dividing by $h$, and taking the limit gives the differential equation below. The initial condition is $S(0)=1$, since survival over zero time is certain.

\frac{dS(t)}{dt}=-\mu S(t),\qquad S(0)=1

Separating variables and integrating gives $\ln S(t)=-\mu t$, so $S(t)=e^{-\mu t}$. Therefore the probability of death by time $t$, under the stated condition, is $1-e^{-\mu t}$. For an interval of $t$ years, the relevant accumulated hazard is $\mu t$, not $\mu$ alone.

Worked Example: Comparing Mortality Across Care States

A hypothetical long-term care model assigns constant annual mortality intensities of 0.015 to healthy lives, 0.045 to recovering lives, 0.11 to moderate-care lives, and 0.22 to severe-care lives. Assume in each calculation that the person remains in the stated state for the full year. Find the one-year death probability for each state, then find the six-month death probability for a severe-care life.

Method. Apply $1-e^{-\mu t}$, using $t=1$ year for the first four calculations and $t=0.5$ year for the last. These are conditional probabilities: the assumed state is held fixed during the period.

For a healthy life, the accumulated hazard is $0.015(1)=0.015$. Hence the one-year death probability is $1-e^{-0.015}=0.014888$, or 0.0149 to four decimal places.

For a recovering life, it is $1-e^{-0.045}=0.044002$, or 0.0440 to four decimal places. For a moderate-care life, it is $1-e^{-0.11}=0.104166$, or 0.1042 to four decimal places. For a severe-care life, it is $1-e^{-0.22}=0.197481$, or 0.1975 to four decimal places.

For six months in severe care, the accumulated hazard is $0.22(0.5)=0.11$. The death probability is therefore $1-e^{-0.11}=0.104166$, or 0.1042 to four decimal places.

The ordering of the calculated probabilities matches the ordering of the intensities: the severe-care assumption produces the largest death probability. The check is that each probability is between 0 and 1 and increases when either the intensity or the duration increases. A common mistake is to report 0.22 as the one-year death probability; 0.22 is the intensity, while the corresponding probability is approximately 0.1975.

Mortality Along a Specified State Path

A life may spend part of a period in one state and then move to another. If the state path and time spent in each state are specified, calculate the accumulated hazard by adding each state’s intensity multiplied by its duration. Survival is the exponential of the negative accumulated hazard. This follows by multiplying the conditional survival factors for successive intervals.

Result: For a specified path that spends durations $t_1,\ldots,t_n$ in states with constant mortality intensities $\mu_1,\ldots,\mu_n$, the conditional survival probability along that path is $$ \exp\left(-\sum_{k=1}^{n}\mu_k t_k\right), $$ and the conditional death probability is one minus this quantity. The durations must use the same time unit as the intensities.

Worked Example: Mortality During Recovery and Care

Consider a hypothetical life whose specified path is 0.5 years healthy, followed by 0.6 years recovering, followed by 0.4 years in severe care. Assume constant annual mortality intensities of 0.04, 0.12, and 0.20 in those respective states. Calculate survival and death probabilities over the 1.5-year path, conditional on following it.

Method. Calculate the integrated hazard for each segment. The healthy segment contributes $0.04(0.5)=0.020$. The recovering segment contributes $0.12(0.6)=0.072$. The severe-care segment contributes $0.20(0.4)=0.080$.

The total accumulated hazard is $0.020+0.072+0.080=0.172$. Therefore, conditional survival along the specified path is $e^{-0.172}\approx0.841979$. The conditional death probability is $1-0.841979\approx0.158021$.

To four decimal places, survival is 0.8420 and the death probability is 0.1580. The check is that the two probabilities sum to 1 before rounding, and survival is lower than it would be for 1.5 years spent entirely in the healthy state, since both later segments have higher mortality intensities.

A common error is to use the final state’s intensity for the entire 1.5 years, giving $1-e^{-0.20(1.5)}$. That calculation assumes severe care throughout, which is not the stated path. Another error is to add the intensities without weighting by duration; the correct accumulated hazard is $\sum_k\mu_k t_k$.

Why an Average Intensity Can Mislead

When the state is uncertain, it can be tempting to average the state-specific mortality intensities and then apply the constant-intensity formula. In general, that does not give the exact unconditional death probability. The reason is that survival is an exponential function of the accumulated intensity, and the average of exponentials is not generally the exponential of the average.

Worked Example: A Life Initially in One of Two States

At time 0, a hypothetical insured is in the recovering state with probability 0.40 and in severe care with probability 0.60. Assume that the state does not change during the next year. The annual mortality intensity is 0.08 in recovery and 0.20 in severe care. Calculate the one-year death probability.

Method. Calculate survival conditional on each state, then average those survival probabilities using the stated probabilities of the initial states. In recovery, survival is $e^{-0.08}=0.923116$. In severe care, survival is $e^{-0.20}=0.818731$.

Unconditional survival is $0.40(0.923116)+0.60(0.818731)=0.369246+0.491239=0.860485$. Thus the one-year death probability is $1-0.860485=0.139515$, or 0.1395 to four decimal places.

Common mistake and correction. The weighted average intensity is $0.40(0.08)+0.60(0.20)=0.152$. Treating 0.152 as a constant intensity would give $1-e^{-0.152}\approx0.1410$, which is not the exact probability for this mixture. The correct calculation averages conditional survival first and then subtracts from 1. The check is that the exact result lies between the two conditional death probabilities, $1-e^{-0.08}\approx0.0769$ and $1-e^{-0.20}\approx0.1813$, as it should.

In a model where state transitions can occur during the period, state occupancy is also uncertain after time 0. The path calculation remains valid conditional on a specified path, but an unconditional probability must account for the possible paths and their probabilities. A single weighted average of state intensities is not generally enough. This distinction matters in valuation: mortality determines when a benefit ceases or a payment becomes due, while the state also determines which benefits may be payable.

Assigning and Checking Assumptions

A practical state-by-state assumption table should make both the state definitions and the time basis explicit. The values below are illustrative only; they are not empirical estimates.

State	Illustrative annual mortality intensity	Interpretation
Healthy	0.015	Instantaneous death rate conditional on being alive and healthy
Recovering	0.045	Instantaneous death rate conditional on being alive and recovering
Moderate care	0.11	Instantaneous death rate conditional on being alive and in moderate care
Severe care	0.22	Instantaneous death rate conditional on being alive and in severe care

Before using any assigned intensity, check the following:

State meaning: Confirm that the health and care definitions are mutually interpretable and match the model being valued.
Conditioning: Read the rate as applying to a life alive and currently in that state, not to the original population regardless of later state changes.
Time unit: Convert durations to years when the intensity is annual. For example, three months is 0.25 years.
Probability conversion: Use $1-e^{-\mu t}$ for constant intensity over $t$ years, rather than treating $\mu t$ as an exact probability.
Path dependence: If state changes are specified, calculate each segment’s contribution to accumulated hazard before combining them.

Key takeaway: State-specific mortality is conditional on being alive in the stated health or care state. For a fixed state path, accumulate intensity times duration across its segments, exponentiate to obtain survival, and take the complement to obtain death probability. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, keep the intensity’s time unit and the assumed state path explicit. Probabilities in the answers are rounded to four decimal places.

A hypothetical life remains healthy for two years with constant annual mortality intensity 0.03. Find the conditional probability of death during the two years. (Answer: 0.0582)
A life follows a specified path of one year recovering with annual mortality intensity 0.04, then half a year in care with annual mortality intensity 0.16. Find the conditional probability of death along the path. (Answer: 0.1131)
A life spends 0.4 years in a state with annual mortality intensity 0.05, then 0.8 years in a state with annual mortality intensity 0.15. Find survival along the specified path. (Answer: 0.8694)
At time 0, a life has probability 0.25 of being in recovery, where annual mortality intensity is 0.06, and probability 0.75 of being in care, where it is 0.18. Assume no state changes for one year. Find the unconditional one-year death probability. (Answer: 0.1381)
A hypothetical insured remains in severe care for three months. The annual mortality intensity in that state is 0.24. Find the conditional death probability, converting the duration to years before calculating. (Answer: 0.0582)

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