Discrete Payment Timing for LTC Benefits

Why Payment Timing Matters

Long-term care eligibility is determined by a person's state over time, but a contract may pay benefits only at scheduled dates. A transition into care can occur between two payment dates; a recovery or death can occur before the next one. Therefore, the probability of having entered care is generally not the probability of being eligible when a payment is due.

This tutorial considers hypothetical contracts with monthly or annual payments in arrears. It distinguishes a fixed installment payable only when the insured is eligible on the payment date from a benefit that accrues in proportion to time spent eligible during a period and is paid at that period's end. The contract wording determines which convention applies. The calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A payment is made in arrears when it is paid at the end of the period to which it relates. Under a payment-date eligibility convention, a scheduled installment is paid only if the insured is in an eligible state on that date. Under an accrual convention, an amount is earned in proportion to time spent in eligible states during the period, then paid at the period's end. These conventions can produce different expected payments even with the same transition model.

State Probabilities at Discrete Payment Dates

Assume the insured's status follows a time-homogeneous continuous-time Markov chain with a finite set of states. Let $Q$ be its generator matrix, whose off-diagonal entries are transition intensities and whose rows sum to zero. Let $\boldsymbol{\pi}(0)$ be the row vector of probabilities of the states at issue. The state-probability vector at time $t$ is $\boldsymbol{\pi}(t)=\boldsymbol{\pi}(0)e^{Qt}$.

Let $\mathbf{b}_j$ be a column vector giving the scheduled benefit payable in each state at payment date $t_j$. An ineligible state has benefit zero. With effective annual interest rate $i$, the expected present value (EPV) of the scheduled payments is:

\operatorname{EPV}=\sum_j v^{t_j}\boldsymbol{\pi}(0)e^{Qt_j}\mathbf{b}_j, \qquad v=\frac{1}{1+i}.

For monthly payments, $t_j=j/12$ when time is measured in years. For annual payments, $t_j=j$. The discount exponent uses the actual time from issue to payment, not the number of payment periods interpreted as years.

Key distinction: A fixed installment due at $t_j$ is weighted by the probability of being eligible at $t_j$. It is not generally weighted by the probability of having ever entered an eligible state by $t_j$. If the payment instead reflects eligible time accumulated during a period, use expected time in eligible states over that period and discount the resulting payment at the period's end.

A Two-Transition Model for Care Eligibility

For clear calculations, consider three states: healthy $H$, eligible for care $C$, and dead $D$. Assume the insured starts healthy, transitions from $H$ to $C$ at constant intensity $\lambda$, and transitions from $C$ to $D$ at constant intensity $\mu$. There is no recovery, no direct transition from $H$ to $D$, and no benefit in $H$ or $D$. These are hypothetical assumptions for the examples, not universal LTC contract features.

Write $p_H(t)$ and $p_C(t)$ for the probabilities of occupying states $H$ and $C$ at time $t$. The transition equations and initial conditions are:

p_H'(t)=-\lambda p_H(t),\qquad p_C'(t)=\lambda p_H(t)-\mu p_C(t),\qquad p_H(0)=1,\quad p_C(0)=0.

Solving the first equation gives $p_H(t)=e^{-\lambda t}$. To obtain $p_C(t)$, multiply its differential equation by the integrating factor $e^{\mu t}$:

\frac{d}{dt}\left(e^{\mu t}p_C(t)\right) =\lambda e^{(\mu-\lambda)t}.

When $\lambda\ne\mu$, integrate from 0 to $t$, using $p_C(0)=0$, then multiply by $e^{-\mu t}$. This gives:

p_C(t)=\frac{\lambda}{\mu-\lambda} \left(e^{-\lambda t}-e^{-\mu t}\right).

This formula counts only people in care at time $t$. Those who entered care but subsequently died are not included. When $\lambda=\mu$, the corresponding limit is $p_C(t)=\lambda t e^{-\lambda t}$. In more general multi-state models, use the state-probability vector rather than this special three-state formula.

Worked Example: Monthly Installments Based on Status at Month-End

Method. Value each installment at its scheduled date using the care-state occupancy probability at that date. A hypothetical insured starts in $H$ and follows the model above, with constant annual intensities $\lambda=12\ln 2$ and $\mu=12\ln 4$. A benefit of $900 is payable at the end of each of the first two months only if the insured is in $C$ then. The effective annual interest rate is $i=12\%$.

At $t=k/12$, the exponentials simplify to $e^{-\lambda t}=2^{-k}$ and $e^{-\mu t}=4^{-k}$. Also, $\lambda/(\mu-\lambda)=1$. Therefore:

p_C(1/12)=2^{-1}-4^{-1}=0.5-0.25=0.25, \qquad p_C(2/12)=2^{-2}-4^{-2}=0.25-0.0625=0.1875.

The monthly discount factors are $v^{1/12}=1.12^{-1/12}=0.9906004$ and $v^{2/12}=1.12^{-2/12}=0.9812894$. Thus the expected present value is:

\begin{aligned} \operatorname{EPV} &=900(0.25)(0.9906004)+900(0.1875)(0.9812894)\\ &=222.8851+165.5926\\ &=\$388.48. \end{aligned}

The final amount is rounded to the nearest cent. A check is that the first payment's expected amount before discount is $900(0.25)=\$225$, not $900: payment is conditional on being in care at month-end. Common mistake avoided: treating everyone who entered care during the month as eligible at month-end ignores the possibility of death after care entry.

Benefits Accrued During a Period, Paid at Its End

Some contracts define the amount due by reference to eligible time during the period, rather than requiring the insured to be eligible at the payment date. If the benefit accrues at rate $r_s$ per year in state $s$, then the payment at the end of period $(t_{j-1},t_j]$ is the accumulated state-specific rate integrated over that period. Its expected present value is:

v^{t_j}\int_{t_{j-1}}^{t_j}\boldsymbol{\pi}(0)e^{Qs}\mathbf{r}\,ds,

where $\mathbf{r}$ is the column vector of accrual rates by state. The discount factor is constant at the period-end payment date; the accrued amount itself depends on occupancy throughout the period. If the rate is zero in every ineligible state, only eligible-state time contributes. This formula assumes the contract prorates benefits by eligible time and pays the accrued amount at period end. It does not describe a fixed installment payable only if eligible at that date.

Worked Example: Monthly Benefit Accrued for Care Time

Method. Integrate the probability of occupying the eligible state across the month, multiply by the annualized accrual rate, and discount the resulting payment at month-end. Use the same hypothetical model, starting in $H$, with $\lambda=12\ln 2$ and $\mu=12\ln 4$ per year. A benefit accrues at $1,200 per month of care, prorated exactly by time in $C$, and the accrued amount for the first month is paid at month-end. The effective annual interest rate is $12\%$.

The monthly accrual rate is equivalent to $12(1200)=\$14{,}400$ per year in $C$. Under these intensities, $p_C(s)=e^{-\lambda s}-e^{-\mu s}$. The expected time in care during the first month, measured in years, is:

\begin{aligned} \int_0^{1/12}p_C(s)\,ds &=\frac{1-e^{-\lambda/12}}{\lambda} -\frac{1-e^{-\mu/12}}{\mu}\\ &=\frac{0.5}{12\ln 2}-\frac{0.75}{12\ln 4}\\ &=0.0150283\text{ years}. \end{aligned}

The expected accrued payment before discount is $14{,}400(0.0150283)=\$216.41$. The payment is made at month-end, so:

\operatorname{EPV} =14{,}400(0.0150283)(1.12^{-1/12}) =\$214.37.

The final amount is rounded to the nearest cent. It is reasonable that the expected payment is less than $1,200: the insured begins healthy, so expected time in care is only about $0.0150283(12)=0.18034$ months. Common mistake avoided: paying a full monthly installment whenever care is reached would not match the specified prorated-accrual provision.

Annual Payments and a Payment-Date Check

The same reasoning applies to annual installments. For a fixed annual payment, use the care-state probability at each anniversary; do not substitute expected care time during the year unless the contract defines the benefit as accruing over that year. The following example also shows how survival in the eligible state affects later installments.

Worked Example: Two Annual Payments in Arrears

Method. Calculate occupancy at each anniversary, then discount each expected payment from that anniversary. A hypothetical insured starts in $H$ and has constant annual transition intensities $\lambda=\ln 2$ from $H$ to $C$ and $\mu=\ln 4$ from $C$ to $D$. A $6,000 installment is paid at each of years 1 and 2 only if the insured is in $C$ on that date. The effective annual interest rate is $8\%$.

Here $\lambda/(\mu-\lambda)=1$, so the care-state probabilities are:

p_C(1)=e^{-\ln 2}-e^{-\ln 4}=0.5-0.25=0.25, \qquad p_C(2)=e^{-2\ln 2}-e^{-2\ln 4}=0.25-0.0625=0.1875.

The expected payments at the two dates are $6{,}000(0.25)=\$1{,}500$ and $6{,}000(0.1875)=\$1{,}125$. Discounting each from its own anniversary gives:

\begin{aligned} \operatorname{EPV} &=\frac{1{,}500}{1.08}+\frac{1{,}125}{1.08^2}\\ &=1{,}388.89+964.51\\ &=\$2{,}353.40. \end{aligned}

The final amount is rounded to the nearest cent. The year-2 probability is lower because some people who entered care have died by that date. Common mistake avoided: discounting both installments by $1.08$ would incorrectly treat the year-2 payment as if it were paid after only one year.

Common Timing Errors

A reliable valuation starts by translating the contract language into a payment rule. In particular, identify whether eligibility is tested at a payment date, whether benefits accrue during a period, and exactly when cash is paid. The expected present value then follows from the rule and the state probabilities.

List the payment dates.
Express each date in years from issue, such as $j/12$ for month $j$.

Translate the benefit condition.
For a fixed installment, identify the states that qualify on the payment date. For an accrued benefit, identify the state-specific accrual rates and the period over which they apply.

Find the relevant state probability or expected occupancy.
Use occupancy at the payment date for fixed installments, or integrate occupancy over the accrual period for prorated benefits.

Discount from the actual payment date.
Apply $v^{t_j}$ to a payment made at $t_j$, including a payment accumulated over an earlier period.

A frequent incorrect calculation replaces the probability of being eligible at a scheduled date with the probability of ever having entered care by then. In the two-transition model, the latter is $1-e^{-\lambda t}$; the former is $p_C(t)$. They differ whenever exit from care, such as death, is possible. Another error is to take a continuous-time present value of a care benefit when the contract instead pays a discrete installment. Continuous accrual and discrete payment are different cash-flow patterns, even when their expected amounts are related.

Key takeaway: Continuous-time transitions determine who occupies each state, but the contract determines how that occupancy becomes a payment. Match the state-probability calculation to the benefit rule, and discount at the date cash is actually paid.

Check Your Understanding

For each hypothetical contract, use the stated payment convention and interest rate. Amounts in the answers are rounded to the nearest cent.

An insured starts healthy in the three-state model with $\lambda=\ln 2$ and $\mu=\ln 4$ per year. A $8,000 fixed installment is payable at year 1 only if the insured is in care then. If $i=10\%$, find its EPV. (Answer: $1,818.18)
An insured starts healthy with $\lambda=12\ln 2$ and $\mu=12\ln 4$ per year. A $1,200 fixed installment is payable at the end of month 1 only if the insured is in care then. If $i=12\%$, find its EPV. (Answer: $297.18)
With $\lambda=\ln 2$, $\mu=\ln 4$, and $i=8\%$, a hypothetical insured receives $4,000 at each of years 1 and 2 if in care on the respective payment dates. Find the EPV. (Answer: $1,568.93)
An insured starts healthy with $\lambda=12\ln 2$, $\mu=12\ln 4$ per year. A benefit accrues at $2,400 per month of care, prorated by eligible time in the first month and paid at month-end. If $i=12\%$, find the EPV using the exact time-accrual convention. (Answer: $428.75)
At year 1, the care-state probabilities in a hypothetical multi-state model are 0.12 for care level 1 and 0.08 for care level 2. A fixed annual arrears benefit at year 1 is $6,000 in level 1 and $10,000 in level 2, with zero benefit in all other states. If $i=5\%$, find its EPV. (Answer: $1,447.62)

What You'll Learn