Long-Term Care Severity and Duration Stresses

Why Long-Term Care Stresses Need a Multi-State View

Long-term care costs depend not only on how many policyholders enter care, but also on which care level they reach, how long they remain there, whether they recover, and how mortality differs by care state. A single “morbidity increase” can hide these distinct effects. A useful stress test therefore changes specified transition assumptions and follows the resulting state probabilities into benefit payments and financial measures.

This tutorial uses an annual, discrete-time model. The calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The transition probabilities and benefits below are hypothetical, not industry data.

Definition: Let the living states be $H$ (healthy), $M$ (moderate care), and $S$ (severe care), with $D$ (dead) absorbing. The one-year transition probability from state $i$ to state $j$ is $p_{ij}$. For each starting state $i$, the probabilities across all possible end states sum to 1. A care-entry stress changes transitions into care; a severity stress changes transitions toward more intensive care; a duration stress changes persistence or recovery; and a care-state mortality stress changes transitions into $D$ from care states.

Set the Projection Basis Before Applying a Stress

For a transparent comparison, first specify the reference model, then change only the assumptions targeted by the stress. In the examples, transitions are made at each year-end, the transition probabilities are constant from year to year, and the state occupied at year-end determines that year’s benefit. Benefits are paid at year-end if the policyholder is then in a care state; no benefit is paid in $H$ or $D$. There are no lapses, expenses, premiums, or other cash flows unless an example explicitly introduces them.

Let $\boldsymbol{\alpha}_0$ be a row vector of starting-state probabilities, $P$ the transition matrix, and $\boldsymbol{b}$ a column vector of end-of-year benefits by state. The death state remains absorbing. For a policyholder starting in $H$, for example, $\boldsymbol{\alpha}_0=(1,0,0,0)$. The distribution after one year is $\boldsymbol{\alpha}_0P$, and after two years it is $\boldsymbol{\alpha}_0P^2$.

Key result: If $i$ is the effective annual interest rate and benefits are paid at each year-end, the expected present value of benefits through year $n$ is the sum of each year’s expected benefit discounted to time 0. For year $t$, the expected benefit is the probability-weighted sum of the benefits in the states reached at that year-end.

E[B_t]=\boldsymbol{\alpha}_0P^t\boldsymbol{b}, \qquad \operatorname{EPV}(B)=\sum_{t=1}^{n}v^t\boldsymbol{\alpha}_0P^t\boldsymbol{b}, \qquad v=\frac{1}{1+i}.

Here, $\boldsymbol{b}$ assigns zero benefit to $H$ and $D$. A care-state benefit is included only when the policyholder is alive in that state at the payment date. This timing convention matters: a policyholder who dies during the year is in $D$ at year-end and receives no year-end care benefit under this model.

Why the Expected-Benefit Formula Holds

The formula follows from conditioning on the end-of-year state. Let $J_t$ denote the state at the end of year $t$, and let $b_j$ be the benefit for state $j$. The benefit is $b_{J_t}$. Its expectation is the sum of each possible benefit multiplied by the probability of the corresponding state:

E[b_{J_t}] =\sum_j P(J_t=j)b_j =\boldsymbol{\alpha}_0P^t\boldsymbol{b}.

A payment at time $t$ has present value $v^t b_{J_t}$, so taking expectations gives $v^tE[b_{J_t}]$. Adding the expected present values for all payment dates gives the stated total. This argument relies on the specified state-contingent payment and timing rules; a benefit paid immediately on care entry, or on a different date, requires a different cash-flow calculation.

Care Entry and Severity Stresses

Care-entry and severity stresses alter different transitions. Increasing $p_{HM}$ or $p_{HS}$ increases entry from healthy status. Increasing $p_{MS}$, with other probabilities adjusted consistently, shifts more moderate-care policyholders to severe care. Each row must still sum to 1. A stress is not applied by simply adding probability to one transition while leaving the remaining row unchanged.

Worked Example: Entry and Severity Stress Over Two Years

A hypothetical policy pays $8,000 for a year-end moderate-care state and $18,000 for a year-end severe-care state. It pays nothing in healthy or dead status. The policyholder starts healthy. The effective annual interest rate is 5%. Under the reference assumptions, the relevant transition probabilities are: from $H$, $p_{HH}=0.80$, $p_{HM}=0.15$, $p_{HD}=0.05$; from $M$, $p_{MH}=0.10$, $p_{MM}=0.55$, $p_{MS}=0.25$, $p_{MD}=0.10$. All other transitions from $H$ and $M$ are zero. Calculate the two-year expected present value. Then apply a combined entry and severity stress: from $H$, set the probabilities to $0.75$ for $H$, $0.20$ for $M$, and $0.05$ for $D$; from $M$, set them to $0.10$ for $H$, $0.45$ for $M$, $0.35$ for $S$, and $0.10$ for $D$. Compare the stressed present value with the reference value.

Method. Find the year-end state probabilities in each projection, multiply care-state probabilities by the corresponding benefit, and discount each year’s expected benefit. The death state is absorbing, so probability already in $D$ cannot return to a living state.

Reference, year 1. Starting in $H$, the year-end probabilities in $M$ and $S$ are $0.15$ and $0$. Thus the expected benefit is $0.15($8,000)= $1,200$.

Reference, year 2. The probability of being in $M$ is $0.80(0.15)+0.15(0.55)=0.2025$. The probability of being in $S$ is $0.15(0.25)=0.0375$. The expected benefit is $0.2025($8,000)+0.0375($18,000)=$1,620+$675=$2,295$.

The reference present value is $1,200/1.05+2,295/(1.05)^2$. The two discounted amounts are $1,142.8571 and $2,081.6327, so the total is $3,224.49, rounded to the nearest cent.

Stressed projection, year 1. The probability of moderate care is $0.20$, giving an expected benefit of $0.20($8,000)=$1,600$.

Stressed projection, year 2. The probability of $M$ is $0.75(0.20)+0.20(0.45)=0.24$. The probability of $S$ is $0.20(0.35)=0.07$. The expected benefit is $0.24($8,000)+0.07($18,000)=$1,920+$1,260=$3,180$.

The stressed present value is $1,600/1.05+3,180/(1.05)^2$. The discounted amounts are $1,523.8095 and $2,884.3537, giving a total of $4,408.16. The increase is $4,408.16−$3,224.49=$1,183.67, rounded to the nearest cent. Higher entry and a greater shift from moderate to severe care both contribute to the increase.

Common mistake avoided: Do not calculate year-two costs using only year-two transitions out of the starting state. The year-two distribution includes everyone who reached each living state in year one and then made another transition.

Duration and Recovery Stresses

Duration risk concerns how long policyholders remain in care. In a discrete-time model, persistence in a care state and recovery out of that state are explicit transition probabilities. Reducing recovery can increase expected payment duration even if the annual benefit amount and the transition into care do not change. A stress to persistence must be accompanied by a consistent adjustment to the other probabilities in that row.

Worked Example: Lower Recovery Increases a Moderate-Care Reserve

A hypothetical policyholder is currently in $M$. The policy pays $8,000 at the end of the next year if the policyholder is in $M$, and $18,000 if in $S$; it pays nothing if the policyholder has recovered to $H$ or died. The effective annual interest rate is 5%. Under the reference assumptions, the next-year probabilities from $M$ are $0.10$ to $H$, $0.55$ to $M$, $0.25$ to $S$, and $0.10$ to $D$. Under the duration stress, recovery to $H$ falls to $0.05$, persistence in $M$ remains $0.55$, the transition to $S$ rises to $0.30$, and mortality remains $0.10$. Calculate the one-year prospective reserve on each basis.

Method. With no premiums or expenses in this example, the prospective reserve is the expected present value of the next year’s benefit. First calculate the expected payment at year-end from the state probabilities.

On the reference basis, the expected year-end payment is $0.55($8,000)+0.25($18,000)=$4,400+$4,500=$8,900$. The one-year reserve is $8,900/1.05=$$8,476.19.

On the stressed basis, the expected year-end payment is $0.55($8,000)+0.30($18,000)=$4,400+$5,400=$9,800$. The stressed reserve is $9,800/1.05=$$9,333.33. The increase is $9,333.33−$8,476.19=$857.14, rounded to the nearest cent.

Both transition rows sum to 1. The stress reallocates 0.05 probability from recovery to severe care, rather than increasing the care probabilities without reducing another transition. In this example, the reserve rises because probability shifts to a state with a higher benefit.

Common mistake avoided: A lower recovery probability does not automatically mean that all of the removed probability becomes time spent in the same care state. The destination of the reallocated probability must be stated; here it moves to severe care.

Care-State Mortality and the Link to Profit

Care-state mortality has a different interpretation from recovery or severity. Higher mortality may reduce the time during which benefits are payable, while lower mortality can extend benefit duration. The net effect depends on the product’s benefits and the transitions among care states. For a benefit paid only if the policyholder is alive in care at a future payment date, a lower probability of death can increase expected benefit costs.

Worked Example: A Mortality Stress Changes Reserve and Projected Profit

A hypothetical policyholder is currently in severe care. At the end of one year, the policy pays $8,000 if the policyholder is in $M$, or $18,000 if in $S$, and nothing if dead. The effective annual interest rate is 4%. On the reference basis, the probabilities are $0.05$ to $M$, $0.75$ to $S$, and $0.20$ to $D$. On the mortality stress basis, the probabilities are $0.05$ to $M$, $0.85$ to $S$, and $0.10$ to $D$. In addition, suppose a single premium of $16,000 is received at time 0 and a $500 expense is paid at time 0. Calculate the reserve for the benefit and the one-year projected profit on both bases, in present-value terms.

Method. Calculate the probability-weighted benefit, discount it one year to obtain the benefit reserve, and subtract that reserve and the time-0 expense from the time-0 premium to obtain projected profit present value.

On the reference basis, the expected year-end benefit is $0.05($8,000)+0.75($18,000)=$400+$13,500=$13,900$. The reserve is $13,900/1.04=$$13,365.38. The projected profit present value is $16,000−$500−$13,365.3846=$2,134.6154, or $2,134.62 to the nearest cent.

On the stressed basis, the expected year-end benefit is $0.05($8,000)+0.85($18,000)=$400+$15,300=$15,700$. The reserve is $15,700/1.04=$$15,096.15. The projected profit present value is $16,000−$500−$15,096.1538=$403.8462, or $403.85 to the nearest cent.

The reserve increases by $15,096.15−$13,365.38=$1,730.77, and projected profit decreases by the same amount. This equality follows because the premium and expense are unchanged between the two bases.

Common mistake avoided: Do not treat the increase in mortality as automatically adverse for benefit costs. In this specified stress, mortality falls from 0.20 to 0.10, so more probability remains in severe care and the expected benefit increases.

Common Stress-Testing Traps

A sound result depends on consistent transitions, benefit timing, and comparisons on a clearly specified basis. Before interpreting a stress result, check each of the following:

Check every transition row. Probabilities from a given starting state, including death, must sum to 1. A stress that changes one probability requires the remaining probabilities to be adjusted if necessary.
Track the full state distribution. Later-year care probabilities depend on all paths into the state, not only on the most recent transition from the initial state.
Apply benefits to the defined payment state and date. A year-end benefit based on being alive in care is not the same as a benefit triggered immediately upon entering care.
Separate severity from duration. A shift from moderate to severe care changes benefit level; a shift from recovery to continued care changes time spent receiving benefits. A stress can affect both, but the changed transitions should be explicit.
Interpret mortality in context. Higher care-state mortality can reduce future care payments, but it may also interact with any death benefit or other contract feature. Do not infer the direction of total profit without listing the cash flows being valued.
Keep the comparison basis consistent. When attributing a change to one stress, hold interest, benefits, expenses, and all unshocked transitions fixed unless the scenario explicitly changes them.

Key takeaway: A long-term care stress is a change to a specified part of the multi-state projection. Recalculate the state distribution, value the resulting state-contingent cash flows, and compare the stressed and reference results using the same timing and discount basis.

Check Your Understanding

For each problem, state the calculation basis clearly. Unless a problem says otherwise, benefits are paid at year-end according to the state then occupied, death is absorbing, and there are no lapses or expenses.

A policyholder starts healthy. The year-one probabilities are $0.82$ to $H$, $0.13$ to $M$, and $0.05$ to $D$. The year-end benefit is $9,000 in $M$, with no other benefit. What is the expected year-one benefit? (Answer: $1,170)
A policyholder starts in $M$. The next-year probabilities are $0.08$ to $H$, $0.62$ to $M$, $0.20$ to $S$, and $0.10$ to $D$. Benefits are $7,000 in $M$ and $16,000 in $S$. At an effective annual interest rate of 4%, calculate the one-year prospective reserve. (Answer: $7,250.00)
A policyholder starts in $S$. Benefits are $6,000 in $M$ and $15,000 in $S$. The reference probabilities are $0.04$ to $M$, $0.76$ to $S$, and $0.20$ to $D$; the stressed probabilities are $0.04$ to $M$, $0.86$ to $S$, and $0.10$ to $D$. At 5% interest, find both one-year reserves and their increase. (Answer: Reference $11,085.71; stressed $12,514.29; increase $1,428.57)
Starting from $H$, the reference transitions are $H$ to $H$ with probability $0.80$, $H$ to $M$ with probability $0.15$, and $H$ to $D$ with probability $0.05$. From $M$, the probabilities to $H,M,S,D$ are $0.10,0.55,0.25,0.10$. With benefits of $8,000 in $M$ and $18,000 in $S$, what is the expected year-two benefit? (Answer: $2,295)
A policyholder starts in $M$. Next-year probabilities are $0.10$ to $H$, $0.50$ to $M$, $0.30$ to $S$, and $0.10$ to $D$. A benefit of $8,000 is paid in $M$, and $20,000 in $S$. At 5% interest, a $12,000 premium is received at time 0 and a $300 expense is paid at time 0. Calculate the one-year projected profit present value. (Answer: $2,176.19)

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