Integrated Long-Term Care Profit and Reserve Case

Building a Multi-State Long-Term Care Projection

A long-term care contract can move through several living states before death. A policyholder may be healthy, need one of two levels of care, or recover from care and later need care again. These paths affect both the amount of claim paid and the reserve needed for future benefits. A useful integrated projection must therefore track which state the policyholder occupies, not just whether the policyholder is alive.

This tutorial develops a hypothetical annual model and applies it to claims, prospective reserves, and profit by state. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: The states are $H$ (healthy), $C_1$ (needing care at level 1), $C_2$ (needing care at level 2), $R$ (recovered), and $D$ (dead). State $D$ is absorbing: once reached, it cannot be left. A transition probability $p_{sj}$ is the probability that a policyholder in state $s$ at the beginning of a year is in state $j$ at the end of that year. Each row of transition probabilities sums to 1.

Use the following annual timing convention throughout the case. A premium and an expense are paid at the beginning of the year if the policyholder is alive then, including in state $R$. Care benefits and a death benefit are paid at the end of the year, according to the state reached. The annual effective interest rate is $i=5\%$, so $v=1/1.05$. The premium is $1,000 and the expense is $100. End-of-year care benefits are $6,000 in $C_1$ and $12,000 in $C_2$; the death benefit is $2,000 in $D$. There is no claim payment in $H$ or $R$.

For the baseline model, the transition probabilities are:

Beginning state	$H$	$C_1$	$C_2$	$R$	$D$
$H$	0.80	0.10	0.04	0.04	0.02
$C_1$	0.05	0.55	0.15	0.20	0.05
$C_2$	0.02	0.08	0.55	0.25	0.10
$R$	0.35	0.10	0.05	0.45	0.05
$D$	0	0	0	0	1.00

For example, the $C_1$ row sums to $0.05+0.55+0.15+0.20+0.05=1$. The $R$ state is not a second kind of healthy state: the model allows a recovered policyholder to return to care, as shown by the positive probabilities of transitions from $R$ to $C_1$ and $C_2$.

Worked Example: Expected Claim from Each Care Path

Find the probabilities of each end-of-year state and the expected claim for a policyholder who begins the year healthy. Identify the method and check that the probabilities form a complete distribution.

Method. Use the $H$ row of the transition table and multiply each end-state probability by the claim payable in that state. The claims are $0 in $H$ and $R$, $6,000 in $C_1$, $12,000 in $C_2$, and $2,000 in $D$.

The end-state probabilities are $0.80, 0.10, 0.04, 0.04,$ and $0.02$, respectively. Their sum is $0.80+0.10+0.04+0.04+0.02=1.00$.

The expected claim is

0.80(0)+0.10(6{,}000)+0.04(12{,}000)+0.04(0)+0.02(2{,}000) =600+480+40 =\$1{,}120.

Thus the probability of entering either care state is $0.10+0.04=0.14$, but the expected claim is not $0.14$ times one common benefit: the two care levels have different benefit amounts, and death also produces a claim.

Common mistake avoided. Counting only transitions into care omits the death benefit. The correct expected claim sums the payment in every possible end state, weighted by that state's probability.

Prospective Reserves and Profit Timing

Let $V_{t,s}$ be the prospective reserve at the beginning of year $t$, conditional on being in state $s$, immediately before the premium and expense for that year. In this two-year case, $V_{2,s}=0$ for every state because no contract cash flows remain after year 2. Let $C_j$ denote the end-of-year claim in state $j$, with values specified above. A living end state receives a reserve for the next year; the dead state has no future reserve.

Reserve recursion: For a living state $s$ with one year remaining, $$ V_{1,s}= - (1{,}000-100)+v\sum_j p_{sj}C_j. $$ For two years remaining, $$ V_{0,s}= -(1{,}000-100)+v\sum_j p_{sj}\bigl(C_j+V_{1,j}\bigr), $$ where $V_{1,D}=0$. These are net prospective reserves: the present value of future claims and expenses less the present value of future premiums, using the stated cash-flow timing.

The minus sign on the net premium-and-expense amount is important: a premium reduces the liability, while an expense increases it. Here their net effect at the start of each year is a $900 reduction in the reserve. The reserve may be negative in some products or states; that is not, by itself, an arithmetic error.

Why the profit formula works: At the start of a year, the insurer has reserve $V_{t,s}$, receives the premium, and pays the expense. Its accumulated amount at year-end is $(V_{t,s}+1{,}000-100)(1+i)$. Subtract the end-of-year claim and the reserve for the state reached. Hence the state-contingent profit is $$ \Pi_{t,s\to j}=(V_{t,s}+900)(1.05)-C_j-V_{t+1,j}. $$ If the reserve recursion and the transition probabilities use the same assumptions, the conditional expected profit is zero: the recursion makes the accumulated starting reserve and net cash flow equal to the expected claim plus next reserve.

Worked Example: Reserves and Profit by End State

Calculate the one-year reserves in all living states, then calculate the two-year reserve for a policyholder who begins healthy. Finally, find the year-0 profit for each end state and verify its expected value. This method applies the prospective recursion first, then uses the resulting reserves in the profit formula.

Step 1: Expected claims from each beginning state. From the transition table, the expected claims are:

\begin{aligned} E[C\mid H]&=0.10(6{,}000)+0.04(12{,}000)+0.02(2{,}000)=1{,}120,\\ E[C\mid C_1]&=0.55(6{,}000)+0.15(12{,}000)+0.05(2{,}000)=5{,}200,\\ E[C\mid C_2]&=0.08(6{,}000)+0.55(12{,}000)+0.10(2{,}000)=7{,}280,\\ E[C\mid R]&=0.10(6{,}000)+0.05(12{,}000)+0.05(2{,}000)=1{,}300. \end{aligned}

Step 2: One-year reserves. With $v=1/1.05$, the reserve is $-900+vE[C\mid s]$. Therefore:

\begin{aligned} V_{1,H}&=-900+\frac{1{,}120}{1.05}=\$166.67,\\ V_{1,C_1}&=-900+\frac{5{,}200}{1.05}=\$4{,}052.38,\\ V_{1,C_2}&=-900+\frac{7{,}280}{1.05}=\$6{,}033.33,\\ V_{1,R}&=-900+\frac{1{,}300}{1.05}=\$338.10. \end{aligned}

For state $D$, the reserve is zero because there are no future premiums, expenses, or claims after death.

Step 3: Two-year reserve from $H$. The expected next reserve, conditional on starting in $H$, is

0.80(166.6667)+0.10(4{,}052.3810)+0.04(6{,}033.3333)+0.04(338.0952) =\$793.43.

The death-state reserve contributes zero. Thus

V_{0,H}=-900+\frac{1{,}120+793.4286}{1.05} =\$922.31.

Step 4: Year-0 profit by end state. The accumulated starting reserve and net premium less expense are

(V_{0,H}+900)(1.05)=(922.3129+900)(1.05)=\$1{,}913.43.

Subtract the end-state claim and the end-state reserve. The resulting profits are $1,746.76 for $H$; $-\$8,138.95$ for $C_1$; $-\$16,119.90$ for $C_2$; $1,575.33 for $R$; and $-\$86.57$ for $D$. For example, the $C_1$ result is $1{,}913.4286-6{,}000-4{,}052.3810=-\$8{,}138.95$.

Check. Weighting those profits by the $H$-row probabilities gives

0.80(1{,}746.7619)+0.10(-8{,}138.9524)+0.04(-16{,}119.9048) +0.04(1{,}575.3333)+0.02(-86.5714)=\$0.00.

The zero expected profit is the check that the reserve recursion is consistent with the baseline projection assumptions. Individual outcomes can still produce substantial positive or negative profit.

Common mistake avoided. The transition probabilities weight both the claim and the reserve for the destination state. Using only the claim in the year-end profit calculation would omit the liability carried forward for living policyholders.

Changing the Recovery Assumption

An assumption change must preserve a valid transition row. For example, reducing the probability of recovery from care does not simply delete probability mass: that mass must be reassigned to other possible destinations. The resulting claims and reserves depend on where the reassigned probability goes.

Worked Example: Less Recovery from Care Level 1

Suppose the probability of moving from $C_1$ to $R$ falls from 0.20 to 0.10. For this hypothetical change, reassign the removed 0.10 entirely to remaining in $C_1$, so the $C_1$ row becomes $(0.05,0.65,0.15,0.10,0.05)$. Calculate the new one-year reserve in $C_1$, the new two-year reserve from $H$, and the expected profit for a $C_1$ policyholder if the old reserves are retained but the changed transition assumption occurs.

Step 1: Check the changed row. Its probabilities sum to $0.05+0.65+0.15+0.10+0.05=1.00$.

Step 2: Recalculate the expected claim from $C_1$.

0.65(6{,}000)+0.15(12{,}000)+0.05(2{,}000) =3{,}900+1{,}800+100 =\$5{,}800.

The new one-year $C_1$ reserve is

V^{\text{new}}_{1,C_1}=-900+\frac{5{,}800}{1.05}=\$4{,}623.81.

This is $571.43 higher than the old $C_1$ reserve of $4,052.38. The reserves in the other living states at year 1 are unchanged because their transition rows have not changed.

Step 3: Recalculate the two-year reserve from $H$. The first-year transition row from $H$ is unchanged, but its 0.10 probability of reaching $C_1$ now leads to the higher $C_1$ reserve at year 1. The expected next reserve increases by $0.10(571.4286)=\$57.14$. Discounting that change gives

V^{\text{new}}_{0,H} =V^{\text{old}}_{0,H}+\frac{57.1429}{1.05} =922.3129+54.4218 =\$976.73.

Step 4: Measure the experience profit if old reserves remain in force. Conditional on starting in $C_1$, the changed assumption shifts probability 0.10 from ending in $R$ to ending in $C_1$. Under the old reserves, the end-of-year claim plus reserve is $10,052.38 in $C_1$ and $338.10 in $R$. The expected end-of-year outgo therefore increases by

0.10(10{,}052.3810-338.0952)=\$971.43.

The old reserve was calibrated to the baseline transition probabilities, so this additional expected outgo produces an expected profit of $-\$971.43$ for a policyholder starting in $C_1$. The result is not a general penalty for lower recovery: it follows specifically from shifting probability into $C_1$, whose claim and old next reserve together exceed those for $R$.

Check. The change in expected claim alone is $0.10(6{,}000-0)=\$600$. The total expected outgo increase is larger because the shifted transition also adds $0.10(4{,}052.38-338.10)=\$371.43$ of next reserve. The two components sum to $971.43.

Common mistake avoided. A recovery assumption change affects more than the immediate care benefit. It can change the probability of continuing care and the reserve attached to the destination state. Recalculate claims and next-state reserves together.

Checks for an Integrated Case

Fix the state and timing definitions.
Specify when premiums, expenses, benefits, and reserves are measured, and whether a benefit depends on the state entered or the state left.

Validate every transition row.
Check that probabilities are nonnegative and sum to 1, including after any assumption change.

Calculate state-weighted claims.
Apply the benefit attached to each possible destination state, including any death payment.

Carry forward living-state reserves.
Include the destination-state reserve in the prospective recursion and set the reserve after death to zero.

Check expected profit under the valuation basis.
When the same cash flows, transition assumptions, and reserves are used, the conditional expected profit should be zero.

A state-contingent profit result is useful precisely because aggregate expected profit can conceal large gains and losses on different paths. In this case, a healthy policyholder has zero expected profit under the baseline basis, even though a transition into severe care creates a large negative result for that year. When assumptions change, distinguish the experience result using the old reserve basis from a newly calculated prospective reserve using the changed basis.

Key takeaway: In a multi-state long-term care projection, calculate each claim and reserve conditional on the relevant state, then weight by transition probabilities. A recovery change must be traced through both immediate benefits and the future reserve in each destination state.

Check Your Understanding

Use the timing and benefit conventions in this tutorial unless a question states otherwise.

Starting in $R$, calculate the expected end-of-year claim using the baseline transition row. (Answer: $1,300)
Using the baseline assumptions, calculate the one-year reserve in $C_2$, rounded to the nearest cent. (Answer: $6,033.33)
Suppose the baseline $C_1$-to-$R$ probability falls from 0.20 to 0.15, with the removed probability assigned to remaining in $C_1$. What is the new expected claim from $C_1$? (Answer: $5,500)
Under the changed $C_1$ row in Worked Example: Less Recovery from Care Level 1, what is the expected profit conditional on starting in $C_1$ if the old reserves are retained? (Answer: $-\$971.43$)
Starting in $H$ under the baseline assumptions, what is the probability of ending the year in either care state? (Answer: 0.1400)

Beginning state	\(H\)	\(C_1\)	\(C_2\)	\(R\)	\(D\)
\(H\)	0.80	0.10	0.04	0.04	0.02
\(C_1\)	0.05	0.55	0.15	0.20	0.05
\(C_2\)	0.02	0.08	0.55	0.25	0.10
\(R\)	0.35	0.10	0.05	0.45	0.05
\(D\)	0	0	0	0	1.00

What You'll Learn