Deferred and Recurrent LTC Claims

Introduction

Long-term care coverage may not pay as soon as a person needs care. A claim can be deferred because care starts after the policy takes effect, and an elimination period can require a spell of care to continue for a specified time before benefits begin. If care ends and later resumes, the contract must also specify whether the interruption resets the elimination period.

These timing rules affect both which periods are payable and how heavily each payment is discounted. The key is to map the care spells and qualification rules onto a timeline before valuing benefits. This tutorial considers hypothetical contracts with continuous benefit payments and focuses on deferred onset and recurrent care. Unless stated otherwise, ignore expenses, benefit caps, premium cash flows, and mortality.

Definition: A deferred claim is a claim whose care spell begins after the valuation time or policy effective time. An elimination period is a specified period of qualifying care during which no benefit is paid. A recurrent claim occurs when a person leaves the qualifying care state and later returns to it. Under a fresh-elimination rule, each recurrent spell must satisfy a new elimination period. Under a continuous-eligibility rule, a temporary interruption does not reset qualification, so benefits can resume immediately when qualifying care resumes. The contract wording determines which rule applies.

Mapping Care Spells to Payable Intervals

Measure time from the valuation date. Suppose a continuous benefit is paid at rate $b$ dollars per year, and the force of interest is constant at $\delta$. If benefits are payable from time $a$ to time $c$, where $0\leq a<c$, their present value is

\operatorname{PV}[a,c] =\int_a^c b e^{-\delta t}\,dt =\frac{b}{\delta}\left(e^{-\delta a}-e^{-\delta c}\right), \qquad \delta>0.

The endpoints are measured from the valuation date, not from the start of the most recent care spell. If the care spell starts at time $s$ and the elimination period is $e$, the earliest payable time under a fresh-elimination rule is $s+e$, provided the spell lasts beyond that time. Thus, for a spell ending at $r$, the payable interval is $[s+e,r]$ when $s+e<r$, and there is no payment from that spell otherwise.

For several spells, apply the rule to each spell separately. With a fresh elimination period, each spell has its own qualification clock. Under continuous eligibility, the first spell satisfies the elimination period, and a later spell can resume payment immediately at its onset if the contract treats the interruption as not ending eligibility. No benefit is paid during a period with no care unless the contract explicitly provides otherwise.

Place every spell on the valuation timeline.
Record each onset and end time, including any gaps without care.

Apply the contract’s qualification rule.
For a fresh elimination period, shift each spell’s payable start forward by the elimination period. For continuous eligibility, determine whether a resumed spell inherits qualification.

Value each payable interval.
Use the interval’s absolute times in the discounting formula, then add the present values.

Why the Continuous-Benefit Formula Works

A continuous payment at rate $b$ over a short interval of length $dt$ around time $t$ is approximately $b\,dt$. Discounting that payment to time zero multiplies it by $e^{-\delta t}$. Adding the discounted payments over the entire payable interval gives the integral in the formula. Since an antiderivative of $e^{-\delta t}$ is $-e^{-\delta t}/\delta$, evaluating between $a$ and $c$ gives $b(e^{-\delta a}-e^{-\delta c})/\delta$. The formula assumes a constant payment rate and constant force of interest over the interval; with a varying rate or force, the relevant integral must be used instead.

A related result is useful when the care duration is uncertain. Suppose care begins at a known time $s$, the duration $D$ from care onset has an exponential distribution with intensity $\gamma>0$, and the elimination period is $e$. The benefit is paid only if $D>e$, and then continuously until recovery. The expected present value is

\begin{aligned} E[\operatorname{PV}] &=b e^{-\delta s} E\left[\int_e^D e^{-\delta u}\,du\, \mathbf{1}_{\{D>e\}}\right]\\ &=b e^{-\delta s}\int_e^\infty e^{-\delta u}P(D>u)\,du\\ &=\frac{b e^{-\delta s}e^{-(\delta+\gamma)e}}{\delta+\gamma}. \end{aligned}

The second line follows by integrating the discounted payment at each time $u\geq e$ only when the person is still in care at $u$. For an exponential duration, $P(D>u)=e^{-\gamma u}$, which gives the final line. This result assumes that duration is measured from the care onset, the future duration has the stated exponential distribution, and benefits stop at recovery. It is not the value of a guaranteed benefit for a fixed period.

Worked Example: A Deferred Spell With Uncertain Duration

A hypothetical policy pays continuously at $b=\$12{,}000$ per year. Care begins $0.5$ years after valuation, and the care duration $D$, measured from that onset, is exponential with intensity $\gamma=0.16$ per year. The elimination period is $e=0.25$ years. The force of interest is $\delta=0.04$. Find the expected present value.

Method. Use the exponential-duration formula. The delay to care onset is $s=0.5$, so the discount from valuation to onset is $e^{-0.04(0.5)}=e^{-0.02}$. The combined rate after onset is $\delta+\gamma=0.04+0.16=0.20$.

The expected present value is

\begin{aligned} E[\operatorname{PV}] &=\frac{12{,}000\,e^{-0.04(0.5)}e^{-(0.04+0.16)(0.25)}}{0.04+0.16}\\ &=60{,}000e^{-0.02}e^{-0.05}\\ &=60{,}000e^{-0.07}\\ &=60{,}000(0.93239382)\\ &=\$55{,}943.63. \end{aligned}

The value is rounded to the nearest cent. The calculation includes both the delay before care begins and the elimination period after onset. As a check, the value is below $b/(\delta+\gamma)=\$60{,}000$, because the onset delay and positive elimination period both reduce the expected value. A common mistake is to apply the elimination period from the valuation date; here it is measured from care onset, so the two timing factors are distinct.

Comparing Fresh and Continuous Eligibility

A recurrent spell does not have a universal treatment. In particular, the phrase “elimination period” alone does not settle whether a relapse must satisfy a new period. A fresh-elimination provision treats each new spell as a new qualification period. A continuous-eligibility provision can preserve qualification after the first spell, but it does not create payments during the recovery gap. The following example isolates this distinction.

Worked Example: A Relapse With Two Possible Rules

A hypothetical contract pays continuously at $\$20{,}000$ per year. The force of interest is $\delta=0.05$. Care occurs from time $0$ to time $1.0$, followed by recovery from $1.0$ to $1.2$, and then care resumes from $1.2$ to $2.2$. The elimination period is $0.25$ years. Find the present value under (a) a fresh-elimination rule and (b) continuous eligibility through the interruption.

First spell. The first spell begins at time $0$, so its payable interval is $[0.25,1.0]$. Its present value is

\begin{aligned} \operatorname{PV}_1 &=\frac{20{,}000}{0.05} \left(e^{-0.05(0.25)}-e^{-0.05(1.0)}\right)\\ &=400{,}000(0.9875778005-0.9512294245)\\ &=400{,}000(0.0363483760)\\ &=\$14{,}539.3504. \end{aligned}

(a) Fresh elimination. The second spell begins at $1.2$, so its new elimination period ends at $1.2+0.25=1.45$. The payable interval is $[1.45,2.2]$:

\begin{aligned} \operatorname{PV}_{2,\mathrm{fresh}} &=\frac{20{,}000}{0.05} \left(e^{-0.05(1.45)}-e^{-0.05(2.2)}\right)\\ &=400{,}000(0.9300657465-0.8958341353)\\ &=400{,}000(0.0342316112)\\ &=\$13{,}692.6448. \end{aligned}

Adding the unrounded components gives $14{,}539.3504+13{,}692.6448=\$28{,}231.9952$, or $\$28{,}232.00$ to the nearest cent.

(b) Continuous eligibility. The first spell has already satisfied the elimination period. Under the stated rule, benefits resume as soon as care resumes at time $1.2$, so the second payable interval is $[1.2,2.2]$:

\begin{aligned} \operatorname{PV}_{2,\mathrm{continuous}} &=\frac{20{,}000}{0.05} \left(e^{-0.05(1.2)}-e^{-0.05(2.2)}\right)\\ &=400{,}000(0.9417645336-0.8958341353)\\ &=400{,}000(0.0459303983)\\ &=\$18{,}372.1593. \end{aligned}

The total is $14{,}539.3504+18{,}372.1593=\$32{,}911.5097$, or $\$32{,}911.51$ to the nearest cent. The larger value reflects the additional payable interval from $1.2$ to $1.45$; neither rule pays during the recovery gap from $1.0$ to $1.2$.

Common mistake and correction. Treating continuous eligibility as though it imposed a fresh period would incorrectly delay the second payment start to $1.45$. Conversely, treating the fresh-elimination clause as continuous eligibility would value the second interval from $1.2$ and overstate that component by $\$4{,}679.51$. Always identify the rule before setting the interval endpoints.

Weighting a Possible Recurrent Claim

Sometimes recurrence is possible but not certain. If an initial spell is certain and a second spell occurs with probability $q$, the expected present value is the value of the first spell plus $q$ times the conditional value of the second spell, provided the second spell’s timing and benefit amount are fixed conditional on recurrence. This uses linearity of expectation; it does not require the two spell values to be independent when the first value is fixed.

Worked Example: A Recurrent Spell That Occurs With Probability Less Than One

A hypothetical policy pays $\$10{,}000$ per year continuously, with force of interest $\delta=0.04$. The insured is in care from time $0$ to $0.8$. A recurrence begins at time $1.0$ with probability $0.30$, and, if it occurs, care continues to time $1.8$. Apply a fresh elimination period of $0.2$ years to each spell. Find the expected present value.

First spell. The payable interval is $[0.2,0.8]$, so

\begin{aligned} \operatorname{PV}_1 &=\frac{10{,}000}{0.04}(e^{-0.04(0.2)}-e^{-0.04(0.8)})\\ &=250{,}000(e^{-0.008}-e^{-0.032})\\ &=250{,}000(0.9920319148-0.9685065821)\\ &=\$5{,}881.3332. \end{aligned}

Conditional recurrence value. If the second spell occurs, its fresh elimination period ends at $1.0+0.2=1.2$, making the payable interval $[1.2,1.8]$:

\begin{aligned} \operatorname{PV}_{2\mid\mathrm{recurrence}} &=250{,}000(e^{-0.04(1.2)}-e^{-0.04(1.8)})\\ &=250{,}000(e^{-0.048}-e^{-0.072})\\ &=250{,}000(0.9531337871-0.9305308958)\\ &=\$5{,}650.7228. \end{aligned}

The recurrence contributes $0.30(5{,}650.7228)=\$1{,}695.2168$ in expected present value. Therefore, the total is $5{,}881.3332+1{,}695.2168=\$7{,}576.5500$, or $\$7{,}576.55$ to the nearest cent.

The probability applies to the entire conditional value of the second spell, not to its benefit rate or its duration. The payable interval is still determined first, using the fresh elimination period. The final value is between the first-spell value alone and the value if recurrence were certain, which provides a useful check.

Common Timing Traps

Deferred and recurrent claims are especially sensitive to time origins and contract wording. The following checks help prevent errors:

Separate policy delay from elimination. A delay to care onset discounts the claim from the valuation date; an elimination period is measured from the relevant care onset.
Use absolute endpoints in discounting. A recurrent spell starting at time $1.2$ with a $0.25$-year fresh elimination period begins paying at time $1.45$, not $0.25$.
Do not pay through a care-free gap. Continuous eligibility can preserve qualification, but it does not by itself imply a benefit during recovery.
Check whether a spell qualifies at all. If its duration is no longer than its elimination period, it generates no benefit under a fresh-elimination rule.
Weight uncertain recurrence once. If recurrence occurs with probability $q$, multiply its conditional present value by $q$; do not also reduce the benefit rate by $q$.

The distinction between fresh elimination and continuous eligibility is contract-specific, so a valuation must follow the stated provision rather than assume one interpretation. Correctly identifying the payable intervals is the central step in valuing these benefits, aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payable period or periods before calculating or interpreting the value.

A hypothetical contract pays continuously at $\$6{,}000$ per year. Care lasts from time $0.4$ to $1.4$, the elimination period is $0.2$ years, and $\delta=0.05$. Find the present value. (Answer: $\frac{6{,}000}{0.05}(e^{-0.05(0.6)}-e^{-0.05(1.4)})=\$4{,}566.21$.)
Care occurs over $[0,0.9]$ and $[1.1,1.8]$, with a $0.2$-year elimination period and fresh elimination after relapse. State the payable intervals. (Answer: $[0.2,0.9]$ and $[1.3,1.8]$.)
A hypothetical benefit pays $\$8{,}000$ per year. Care begins at time $0.6$, its duration from onset is exponential with intensity $\gamma=0.18$, the elimination period is $0.4$ years, and $\delta=0.02$. Find the expected present value. (Answer: $\frac{8{,}000e^{-0.02(0.6)}e^{-(0.02+0.18)(0.4)}}{0.02+0.18}=40{,}000e^{-0.092}\approx \$36{,}484.21$.)
A hypothetical contract pays $\$12{,}000$ per year continuously at $\delta=0.03$. The first care spell is $[0,0.5]$; a recurrence occurs with probability $0.25$, beginning at time $0.8$ and ending at $1.3$ if it occurs. A fresh elimination period of $0.1$ years applies to each spell. Give the expected present value. (Answer: $\frac{12{,}000}{0.03}\left[(e^{-0.003}-e^{-0.015})+0.25(e^{-0.027}-e^{-0.039})\right]$.)

What You'll Learn