Worked Long-Term Care Case with Changing Care Levels

Introduction

Long-term care benefits can depend on both the level of care and how the insured's health changes over time. An insured may move from healthy status to moderate care, recover, require severe care, or die. A valuation must allow for these possibilities without counting a death benefit more than once or paying a care benefit in the wrong state.

This tutorial develops a hypothetical, two-year contract using an annual Markov model. It combines state-dependent care benefits with a death benefit and premiums that vary by health status. The calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and Cash-Flow Timing

Use four states, observed at integer times: healthy $H$, moderate care $M$, severe care $S$, and dead $D$. During each year, the insured moves from the state at the start of the year to a state at its end. The transition probabilities are constant from year to year. In each row of the following matrix, the entries give the probabilities of the destination states $H,M,S,D$, in that order.

\mathbf{P}= \begin{pmatrix} 0.85&0.10&0.02&0.03\\ 0.20&0.50&0.20&0.10\\ 0&0.10&0.75&0.15\\ 0&0&0&1 \end{pmatrix}.

The rows sum to 1. Recovery is represented by $M\to H$ and $S\to M$; escalation from moderate to severe care is $M\to S$. Death probabilities in a year are 0.03 from $H$, 0.10 from $M$, and 0.15 from $S$. Death is absorbing. This is an annual discrete-time model: its transition probabilities are not annual force-of-mortality rates.

Definition: A Markov state model specifies the states an insured can occupy and the probabilities of moving between them. The Markov assumption is that, conditional on the current state, the next-state distribution does not depend on the earlier path. Here, care and death benefits are paid at year-end: care benefits depend on the state reached, and the death benefit is paid once if the transition during that year is into $D$. A premium is paid at the start of a year if the insured is alive then, at an amount determined by the state at that time.

For the worked case, the year-end care benefits are 12,000 dollars in state $M$ and 30,000 dollars in state $S$; there is no care benefit in $H$. A 4,000-dollar death benefit is paid at year-end on a transition into $D$, and only on that transition. The effective annual interest rate is $i=5\%$, so $v=1/1.05=20/21$. Unless stated otherwise, the insured starts in $H$, and the transition distribution is the matrix above.

Expected Payments and State Probabilities

It is useful to calculate the expected payment during a year conditional on the state at its beginning. Let $r_i$ denote that expected payment when the initial state is $i$. The payment depends on the transition, so each possible destination must be weighted by its transition probability.

r_H=0.10(12{,}000)+0.02(30{,}000)+0.03(4{,}000)=1{,}920.

The 0.03 death probability is included because death during the year produces a 4,000-dollar payment. The same calculation from $M$ or $S$ gives

\begin{aligned} r_M&=0.50(12{,}000)+0.20(30{,}000)+0.10(4{,}000)=12{,}400,\\ r_S&=0.10(12{,}000)+0.75(30{,}000)+0.15(4{,}000)=24{,}300. \end{aligned}

The death-state row has no further payments: once dead, the insured receives neither a care benefit nor another death benefit. Thus $r_D=0$.

Key distinction: A care benefit is determined by the care state reached at year-end. The death benefit is determined by whether the transition is into $D$ during that year. It is not a recurring payment while the insured remains in the absorbing death state.

Worked Example: First-Year Expected Benefit

Assumptions. The insured starts in $H$; the annual transition probabilities are those in the matrix above, including death probabilities 0.03 from $H$, 0.10 from $M$, and 0.15 from $S$. The distribution is Markov, the year-end benefits are 12,000 dollars in $M$, 30,000 dollars in $S$, and 4,000 dollars on a transition into death, and $i=5\%$.

Method. Condition on the state reached at the end of the first year. Starting in $H$, the probabilities of reaching $M$, $S$, and $D$ are 0.10, 0.02, and 0.03. The probability of remaining healthy is 0.85, but that outcome produces no payment.

\begin{aligned} E[B_1\mid X_0=H] &=0.85(0)+0.10(12{,}000)+0.02(30{,}000)+0.03(4{,}000)\\ &=0+1{,}200+600+120\\ &=1{,}920. \end{aligned}

The expected first-year payment is 1,920 dollars at year-end. Its present value at time 0 is $1{,}920v=1{,}920(20/21)=1{,}828.571$ dollars, or 1,828.57 dollars to the nearest cent. The weighted state probabilities sum to 1, and the payment calculation includes all possible payment-producing outcomes.

Common mistake avoided: Do not omit the death transition just because the main benefit is for long-term care. Conversely, do not treat the death benefit as payable in every later year after death.

Valuing Benefits Over Two Years

Let $X_k$ denote the state at time $k$. If the initial state distribution is the row vector $\boldsymbol{\alpha}$, then the distribution at time $k$ is $\boldsymbol{\alpha}\mathbf{P}^k$. To find the expected payment in a later year, weight the conditional expected payment $r_i$ by the probability of being in each state at that year's start.

Justification: In year $k$, the state at the start of the year is $X_{k-1}$. Conditional on $X_{k-1}=i$, the expected payment is $r_i=\sum_j p_{ij}c_{ij}$, where $c_{ij}$ is the payment for the transition from $i$ to $j$. By conditioning on the mutually exclusive possible states at time $k-1$, $$ E[B_k]=\sum_i P(X_{k-1}=i)r_i. $$ Discounting a payment at the end of year $k$ by $v^k$ and adding its expected value gives the expected present value. This conditioning step is why the calculation must use the state distribution at the start of the payment year, not the initial distribution in every year.

Worked Example: Two-Year Benefit Present Value

Assumptions. The insured starts in $H$; annual transitions follow the stated Markov matrix, with death probabilities 0.03 from $H$, 0.10 from $M$, and 0.15 from $S$. Care benefits are 12,000 dollars in $M$ and 30,000 dollars in $S$, and the 4,000-dollar death benefit is paid only on entry into $D$. Payments are at year-end, and $i=5\%$, so $v=20/21$.

After the first year, the state distribution is $(0.85,0.10,0.02,0.03)$. The probability of being in each state at time 2 is found by multiplying this distribution by $\mathbf{P}$:

\begin{aligned} P(X_2=H)&=0.85(0.85)+0.10(0.20)+0.02(0)=0.7425,\\ P(X_2=M)&=0.85(0.10)+0.10(0.50)+0.02(0.10)=0.1370,\\ P(X_2=S)&=0.85(0.02)+0.10(0.20)+0.02(0.75)=0.0520,\\ P(X_2=D)&=0.03+0.85(0.03)+0.10(0.10)+0.02(0.15)=0.0685. \end{aligned}

These probabilities sum to 1.0000. The probability of newly entering $D$ during year 2 is $0.0685-0.0300=0.0385$: the 0.0300 already dead at time 1 do not generate another payment. Equivalently, the year-2 death probability is $0.85(0.03)+0.10(0.10)+0.02(0.15)=0.0385$, conditional on being alive at the start of that year.

The expected year-2 care payment is $0.1370(12{,}000)+0.0520(30{,}000)=1{,}644+1{,}560=3{,}204$ dollars. The expected year-2 death payment is $0.0385(4{,}000)=154$ dollars. Therefore the total expected year-2 payment is $3{,}204+154=3{,}358$ dollars. The present value of the two-year benefit stream is

\begin{aligned} \operatorname{EPV}(B) &=1{,}920v+3{,}358v^2\\ &=1{,}920\left(\frac{20}{21}\right) +3{,}358\left(\frac{400}{441}\right)\\ &=1{,}828.5714+3{,}045.8050\\ &=4{,}874.3764. \end{aligned}

The expected present value is 4,874.38 dollars to the nearest cent. The check on the year-2 calculation is that it includes both the care benefit associated with the year-end care state and the death benefit for transitions into death during year 2, but excludes deaths from year 1.

Common mistake avoided: The time-2 probability in $D$ is cumulative. Multiplying it by the death benefit would incorrectly pay again for people who died in year 1. Use the probability of transition into $D$ during the year being valued.

Premiums Contingent on Health Status

A premium payable only while alive, or at different amounts in different health states, is valued using the state distribution at each premium date. Premiums paid at the beginning of a year are discounted to that beginning-of-year time. In particular, a time-1 premium is multiplied by $v$, not $v^2$.

Worked Example: Equivalence Premium by Health State

Assumptions. Use the same two-year Markov model, starting in $H$, with annual death probabilities 0.03 from $H$, 0.10 from $M$, and 0.15 from $S$. The year-end care and death benefits are unchanged, and $i=5\%$. A premium is due at time 0 and again at time 1 if the insured is alive. The premium is $P$ in $H$, $0.40P$ in $M$, and zero in $S$. No premium is due after death.

Method. Set the expected present value of premiums equal to the 4,874.3764-dollar expected present value of benefits found above. At time 0, the insured is certainly in $H$, so the premium is $P$. At time 1, the state probabilities are $0.85$ in $H$, $0.10$ in $M$, $0.02$ in $S$, and $0.03$ in $D$. Thus the expected time-1 premium is $0.85P+0.10(0.40P)=0.89P$.

\begin{aligned} \operatorname{EPV}(\text{premiums}) &=P+v(0.89P)\\ &=P\left(1+\frac{20}{21}(0.89)\right)\\ &=P\left(1+\frac{89}{105}\right) =\frac{194}{105}P. \end{aligned}

Equating benefit and premium present values gives

\begin{aligned} \frac{194}{105}P&=4{,}874.3764,\\ P&=4{,}874.3764\left(\frac{105}{194}\right)\\ &=2{,}638.1934. \end{aligned}

The equivalence premium is $P=2{,}638.19$ dollars to the nearest cent. The moderate-care premium is $0.40P=1{,}055.28$ dollars to the nearest cent. As a check, the expected premium present value at the unrounded $P$ is $P(194/105)=4{,}874.3764$, matching the benefit present value.

Common mistake avoided: The time-1 premium is not simply $P$, and it is not paid in every state. Weight each state-specific premium by its probability at time 1, then discount the expected payment for one year.

Checks That Prevent Valuation Errors

A sound calculation keeps the transition model, benefit trigger, and payment timing consistent. Before finalizing a valuation, check that each transition row sums to 1, that death is absorbing, and that recovery and escalation are represented by the intended off-diagonal probabilities. Then distinguish probability of being dead by a date from probability of dying during a particular period. The former is cumulative; a death benefit payable on death uses the latter.

Also keep the premium and benefit timelines separate. In this case, premiums occur at the start of each year and benefits at its end. Changing either timing changes the discount factor. The annual Markov formulation is appropriate when the contract and supplied assumptions specify annual state transitions and payment timing. It should not be substituted for a continuous-time model with transition intensities when the problem specifies continuous payment or continuous-time rates.

Key takeaway: For a multi-level long-term care valuation, track state probabilities period by period, attach each payment to its stated transition or state, and discount it from its actual payment date. In particular, count recovery and care-level changes through transitions, and count a death benefit only on entry into death.

Check Your Understanding

Use the annual transition model and payment timing defined above unless a question gives different benefit amounts. Round monetary answers to the nearest cent where needed.

During one year, a hypothetical insured has probabilities 0.30 of a 7,000-dollar care payment, 0.20 of an 18,000-dollar care payment, and 0.03 of a 4,000-dollar death payment; the outcomes are mutually exclusive and all other outcomes pay zero. Find the expected year-end payment. (Answer: 5,820 dollars)
An insured starts in $H$. With the stated transition probabilities and benefits, find the expected payment at the end of year 1, including any death payment. (Answer: 1,920 dollars)
An insured starts in $H$. Find the probability of being in $M$ and in $S$ at time 2, then find the expected year-2 care payment, excluding any death payment. (Answer: $P(X_2=M)=0.1370$, $P(X_2=S)=0.0520$, expected care payment $=3,204$ dollars)
For an insured starting in $H$, find the probability of a new transition into death during year 2 and the expected death payment for that year. (Answer: probability $0.0385$; expected payment $154$ dollars)
Keep the two-year benefit contract, but let premiums be $P$ in $H$, $0.25P$ in $M$, and zero in $S$, payable at times 0 and 1 while alive. Find the equivalence value of $P$ at $i=5\%$. (Answer: $P=2,658.75$ dollars)

What You'll Learn