Long-Term Care Reserve Worked Problems

Why Long-Term Care Reserves Depend on State

A long-term care contract may pay different benefits depending on whether the insured is healthy, needs moderate care, or needs severe care. An insured may recover to a healthier state, progress to a more intensive level of care, or die. Premiums may also stop when care begins. A useful reserve calculation must therefore distinguish the insured’s current state and account for each possible next state.

This tutorial uses an annual, discrete-time model. It is a practical framework for worked reserve problems: transition probabilities describe changes in state over each year, and benefits are paid at the year-end according to the state reached. This timing is an assumption of the model, not a claim that real care needs begin only at an anniversary. For contracts with continuous payments or transitions, use the cash-flow timing specified for that model.

Definition: Let $H$ denote healthy, $C_1$ moderate care, $C_2$ severe care, and $D$ dead. At duration $k$, $V_{k,j}$ is the prospective reserve immediately before any premium due at that time, conditional on the insured being in state $j$. Let $p_{k,jl}$ be the probability of moving from state $j$ at duration $k$ to state $l$ by duration $k+1$. Let $b_{k,jl}$ be the benefit paid at the end of that year for that transition, and let $P_{k,j}$ be the premium received at duration $k$ in state $j$. A dead state is absorbing, and its future reserve is zero unless the contract explicitly provides otherwise.

The Prospective Reserve Recursion

The reserve is the expected present value of future benefits minus the expected present value of future premiums, conditional on the current state. For one year of projection, the transition-based recursion is

V_{k,j}=-P_{k,j}+v\sum_l p_{k,jl}\left(b_{k,jl}+V_{k+1,l}\right), \qquad v=\frac{1}{1+i}.

Here $i$ is the effective annual interest rate. The premium is deducted at the valuation time, and the transition benefit and next reserve are discounted one year. A premium waiver is represented by setting $P_{k,j}=0$ in the states where no premium is payable. If no benefit is paid on a transition, its $b_{k,jl}$ is zero. At the end of a fully specified contract, the terminal reserve is usually zero, unless a maturity or other terminal payment is due.

Derivation: Conditional on being in state $j$ at duration $k$, the insured is in state $l$ at the next anniversary with probability $p_{k,jl}$. Given that transition, the end-of-year benefit is $b_{k,jl}$, and the value at that anniversary of the remaining contract is $V_{k+1,l}$. The conditional expected year-end value is therefore $\sum_l p_{k,jl}(b_{k,jl}+V_{k+1,l})$. Discounting this value by $v$ and subtracting the premium paid immediately at duration $k$ gives the recursion. The transition probabilities in each row must sum to 1.

For a projection across several years, apply the recursion backward from the terminal duration. At each step, use the transition probabilities and cash flows for that year, along with the next year’s reserves. The reserve is conditional on the state occupied at the valuation time; it is not an average over all insureds unless a population-average reserve is specifically required.

Worked Problems

Worked Example: Recovery From Care With a Death Payment

A hypothetical long-term care contract is valued at the start of a year for an insured currently in care state $C$. Over the year, the insured recovers to $H$ with probability 0.20, remains in care with probability 0.70, or dies with probability 0.10. The annual effective interest rate is 5%. A year-end benefit of $12,000 is paid if the insured is in care at year-end, and a $5,000 death benefit is paid if death occurs. There is no premium while in care. At the next anniversary, the reserves immediately before any premium are $3,000 in $H$, $8,000 in $C$, and $0 in $D$. Calculate the current reserve.

Method. Apply the one-year prospective recursion conditional on the current care state. The transition probabilities sum to $0.20+0.70+0.10=1.00$. The transition cash flows and next reserves, in dollars, are:

Recovery: benefit $0$, next reserve $3,000, total $3,000. Remaining in care: benefit $12,000, next reserve $8,000, total $20,000. Death: benefit $5,000, next reserve $0$, total $5,000.

\begin{aligned} V_C &=0+\frac{0.20(3{,}000)+0.70(20{,}000)+0.10(5{,}000)}{1.05}\\ &=\frac{600+14{,}000+500}{1.05}\\ &=\frac{15{,}100}{1.05}\\ &=\$14{,}380.95. \end{aligned}

The result is positive, as expected given the substantial care benefit and the supplied future care reserve. Common mistake avoided: recovery does not imply a zero year-end value; the recovered state has a $3,000 continuation reserve, which must be included in the recovery branch.

Worked Example: Different Care Levels and a Healthy-State Premium

A hypothetical contract has states $H,C_1,C_2,D$. In one year, the transition probabilities are as follows: from $H$, the probabilities of $H,C_1,D$ are 0.80, 0.15, and 0.05; from $C_1$, the probabilities of $H,C_1,C_2,D$ are 0.20, 0.50, 0.20, and 0.10; from $C_2$, the probabilities of $H,C_1,C_2,D$ are 0.05, 0.15, 0.60, and 0.20. Each row sums to 1.00. The annual effective interest rate is 5%. Year-end benefits are $6,000 in $C_1$, $14,000 in $C_2$, and $2,000 on death; there is no benefit in $H$. Next-anniversary reserves are $4,000 in $H$, $9,000 in $C_1$, $18,000 in $C_2$, and zero in $D$. A premium of $1,000 is due immediately in $H$ only; premiums are waived in care. Calculate the reserve in each living state.

Method. For each current state, add the transition benefit to the reserve in the destination state, weight by the transition probability, discount the expected total, and deduct any premium due now. All amounts below are in thousands of dollars.

From $H$, the expected year-end total is $0.80(4)+0.15(6+9)+0.05(2+0)=3.20+2.25+0.10=5.55$. Thus:

V_H=-1+\frac{5.55}{1.05}=-1+5.285714\ldots=4.285714\ldots

From $C_1$, the expected year-end total is $0.20(4)+0.50(6+9)+0.20(14+18)+0.10(2+0)=0.80+7.50+6.40+0.20=14.90$. No premium is due in care, so:

V_{C_1}=\frac{14.90}{1.05}=14.190476\ldots

From $C_2$, the expected year-end total is $0.05(4)+0.15(6+9)+0.60(14+18)+0.20(2+0)=0.20+2.25+19.20+0.40=22.05$. Therefore:

V_{C_2}=\frac{22.05}{1.05}=21.00.

Converting from thousands and rounding to the nearest cent gives $V_H=\$4,285.71$, $V_{C_1}=\$14,190.48$, and $V_{C_2}=\$21,000.00$. The severe-care reserve is highest here because its benefit and continuation reserve are largest. Common mistake avoided: the $1,000 premium is deducted only in $H$, not in either care state.

Worked Example: Two-Year Reserve With Recovery and Premium Waiver

A hypothetical two-year contract has states $H,C,D$, with $D$ absorbing. In each year, the transition probabilities from $H$ to $H,C,D$ are 0.80, 0.15, and 0.05. From $C$ to $H,C,D$, they are 0.25, 0.65, and 0.10. The annual effective interest rate is 5%. At each year-end, the benefit is $10,000 if the destination state is $C$, $3,000 if it is $D$, and zero if it is $H$. A $1,000 premium is due at the start of each year in $H$ only. The reserve at the end of year 2 is zero in all states. Find the reserve at time 0 for an insured in $H$.

Method. First calculate the reserves one year before maturity, then use those values to calculate the time-0 reserve. Work in thousands of dollars. At duration 1, the terminal continuation reserves are zero. In $H$, the expected year-end benefit is $0.15(10)+0.05(3)=1.65$, so:

V_{1,H}=-1+\frac{1.65}{1.05} =-1+1.571428\ldots =0.571428\ldots

In $C$, there is no premium, and the expected year-end benefit is $0.65(10)+0.10(3)=6.80$. Hence:

V_{1,C}=\frac{6.80}{1.05}=6.476190\ldots

At time 0, an insured in $H$ can remain healthy, enter care, or die. The expected year-end total before discounting is:

\begin{aligned} &0.80V_{1,H}+0.15(10+V_{1,C})+0.05(3+0)\\ &=0.80(0.571428\ldots)+0.15(16.476190\ldots)+0.15\\ &=0.457142\ldots+2.471428\ldots+0.15\\ &=3.078571\ldots \end{aligned}

Deduct the time-0 premium and discount the expected year-end total:

V_{0,H}=-1+\frac{3.078571\ldots}{1.05} =-1+2.931972\ldots =1.931972\ldots\text{ thousand}.

Rounded to the nearest cent, the time-0 reserve is $1,931.97. As a check, both year-1 living-state reserves are positive, and the time-0 calculation includes the care benefit as well as the year-1 continuation reserve in the care branch. Common mistake avoided: do not replace the year-1 care reserve with zero; it represents obligations after the first year and is part of the time-0 prospective value.

Checks That Matter in Long-Term Care Problems

State-dependent benefits and premium rules make long-term care reserves particularly sensitive to how a problem is read. Before calculating, write down the states, the transition probabilities from each state, the benefit attached to each possible transition or destination state, and the premium payable in each current state. Then apply the recursion consistently.

A useful check is to verify that each transition row sums to 1. A second is to inspect the cash-flow timing: a benefit paid at year-end is discounted, whereas a premium paid immediately is not. A third is to check that death is absorbing and that no post-death reserve is carried forward unless a specific contractual payment requires it. Finally, when recovery is possible, include the reserve in the recovered state; recovery changes the future benefit and premium pattern rather than automatically ending the contract.

Key takeaway: A long-term care reserve is conditional on the current health or care state. Each transition contributes its probability-weighted benefit and destination-state reserve; state-dependent premiums are then included at the time they are payable.

Check Your Understanding

For each problem, use the stated annual transition model and reserve timing. Amounts are in dollars unless marked as thousands.

An insured in care recovers to $H$ with probability 0.30, remains in care with probability 0.60, and dies with probability 0.10. The annual effective interest rate is 4%. Benefits at year-end are $8,000 in care and $2,000 on death. Next-anniversary reserves are $2,000 in $H$, $5,000 in care, and zero in death. No premium is due in care. Find the current reserve. (Answer: $8,269.23)
From $H$, a contract has one-year probabilities 0.70 to $H$, 0.20 to care, and 0.10 to death. At 5% annual effective interest, year-end benefits are zero in $H$, $5,000 in care, and $1,000 on death. Next-anniversary reserves are $3,000 in $H$, $8,000 in care, and zero in death. A $500 premium is due immediately in $H$. Find $V_H$. (Answer: $4,071.43)
An insured in severe care has probabilities 0.10 of recovery to moderate care, 0.70 of remaining in severe care, and 0.20 of death. The annual effective interest rate is 5%. Year-end benefits are $9,000 in moderate care, $15,000 in severe care, and $4,000 on death. Next-anniversary reserves are $7,000 in moderate care, $12,000 in severe care, and zero in death. There is no premium in care. Find the severe-care reserve. (Answer: $20,285.71)
A two-year contract has states $H,C,D$. In each year, from $H$ the probabilities to $H,C,D$ are 0.75, 0.20, 0.05; from $C$ they are 0.30, 0.60, 0.10. Death is absorbing. The annual effective interest rate is zero. A year-end benefit is $6,000 in $C$, $2,000 on death, and zero in $H$. A $1,000 premium is due at the start of each year in $H$ only. All reserves at the end of year 2 are zero. Calculate the time-0 reserve for an insured in $H$. (Answer: $1,285.00)

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