Maximum Benefit Periods and Claim Termination

Why a Maximum Benefit Period Matters

A disability income contract may promise a benefit for as long as a claim continues, but only up to a stated maximum duration. The claim may end earlier if the claimant recovers or dies; it may also be cut short if the contract expires. These limits affect both the expected amount paid and the timing of payments. The maximum period is measured from the start of the claim, not automatically from the issue date.

This tutorial values hypothetical benefits under a disability model with active, disabled, and dead states. It focuses on a single claim spell: once recovery ends the claim, no later disability spell is included. All examples state their payment timing and assumptions explicitly. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A maximum benefit period $T$ is the longest duration for which a claim can generate benefits, measured from the claim’s start time $s$. If the contract ends at time $H$, the maximum possible payment duration for that claim is $\min(T,H-s)$, provided $s<H$. Actual payments can stop sooner on recovery or death.

Claim-Start Valuation Under Constant Intensities

First consider a claim that has just started. Assume that, while disabled, the claimant has constant recovery intensity $\rho$ and constant mortality intensity $\mu$. Recovery and death are mutually exclusive exits from the disabled state; once either occurs, benefits stop. Assume a constant force of interest $\delta$. For a continuous benefit paid at rate $b$ per year, the probability that the claimant is still disabled $t$ years after claim start is $e^{-(\rho+\mu)t}$.

The relevant payment horizon is $D=\min(T,H-s)$. If the contract has no finite expiry before the maximum period, take $D=T$. For a claim starting at issue, its expected present value is:

\operatorname{EPV}_{\text{claim start}} = b\int_0^D e^{-\delta t}e^{-(\rho+\mu)t}\,dt = b\frac{1-e^{-(\delta+\rho+\mu)D}}{\delta+\rho+\mu}.

This formula requires constant intensities and a constant continuous benefit rate over the payment period. It does not apply unchanged to a benefit paid only at discrete dates, a benefit whose amount changes during the claim, or a model with time-varying intensities. In those cases, value each payment date or integrate using the appropriate time-dependent survival probability.

Find the claim’s available duration.
Use the earlier of the maximum benefit period and the time remaining to contract expiry: $D=\min(T,H-s)$.

Find the disabled-state survival probability.
With constant recovery and disabled-state mortality intensities, the probability of remaining disabled to duration $t$ is $e^{-(\rho+\mu)t}$.

Discount and value the benefits.
For continuous payments, integrate the benefit rate times disabled-state survival and discount over $0\leq t\leq D$. For discrete payments, include only payment dates at which the claimant remains eligible.

Why the Continuous-Rate Formula Holds

At a time $t$ after claim start, a payment over the next short interval of length $dt$ is made only if the claimant is still disabled at time $t$. Under the stated assumptions, the probability of that event is $e^{-(\rho+\mu)t}$. The payment amount over that short interval is approximately $b\,dt$, and its discount factor from claim start is $e^{-\delta t}$. Thus the expected present value contributed by the interval is $b e^{-(\delta+\rho+\mu)t}dt$. Adding these contributions over the permitted duration gives the integral.

Writing $k=\delta+\rho+\mu$, with $k>0$, the integral evaluates as:

b\int_0^D e^{-kt}\,dt = b\left[-\frac{1}{k}e^{-kt}\right]_0^D = b\frac{1-e^{-kD}}{k}.

The recovery and mortality intensities enter as termination hazards: either exit removes the claimant from the benefit-paying state. Discounting is separate and adds $\delta$ to the exponent. A maximum benefit period truncates the integration interval; it is not another termination intensity.

Worked Example: Continuous Benefit With Recovery and Death

A hypothetical claim pays continuously at a rate of $12,000 per year for at most two years. The force of interest is $\delta=0.04$, the recovery intensity is $\rho=0.10$, and disabled-state mortality intensity is $\mu=0.02$, all per year. The contract does not expire during the two-year period. Find the expected present value at claim start.

Method. Use the continuous benefit formula. The payment duration is $D=2$, and the combined exponent rate is $k=0.04+0.10+0.02=0.16$.

Substitute the inputs:

\operatorname{EPV} =12{,}000\frac{1-e^{-0.16(2)}}{0.16} =12{,}000\frac{1-e^{-0.32}}{0.16}.

Using $e^{-0.32}=0.7261490$, the survival-and-discount factor integrated over the period is $(1-0.7261490)/0.16=1.7115685$ years. Therefore, the expected present value is $12{,}000(1.7115685)=\$20{,}538.82$, rounded to the nearest cent.

Check. The value is less than the undiscounted maximum of $12{,}000(2)=\$24{,}000$, as it must be: discounting and possible claim termination both reduce the value.

Common mistake avoided. Discounting only for interest would omit the probability that the claimant recovers or dies. The correct exponent uses $\delta+\rho+\mu$, not $\delta$ alone.

Discrete Payments and Claim Termination

For a benefit paid at discrete times, the payment is included only if the claimant is still eligible at the payment time. Suppose payments of amount $B$ are made at durations $t_1,\ldots,t_m$ after claim start, and each date is within the maximum period and contract term. Under the same constant-intensity assumptions, the expected present value is:

\operatorname{EPV}_{\text{claim start}} = B\sum_{j=1}^{m} e^{-\delta t_j}e^{-(\rho+\mu)t_j} = B\sum_{j=1}^{m}e^{-(\delta+\rho+\mu)t_j}.

This expression assumes that a scheduled payment is made if the claimant is disabled immediately before that payment. It does not prorate payments for a claim that terminates between payment dates. If a contract instead pays a proportionate amount on termination, that additional payment must be valued separately.

Worked Example: Annual Payments for a Limited Number of Years

A hypothetical policy pays $5,000 at the end of each year for at most three years after a claim begins. The benefit is payable only if the claimant is disabled immediately before each payment. Assume $\delta=0.05$, $\rho=0.10$, and $\mu=0.03$, all constant per year, and no earlier contract expiry. Find the value at claim start.

Method. The possible payment times are $1,2,3$ years after claim start. The combined rate is $k=0.05+0.10+0.03=0.18$. At each payment date, multiply the benefit by the probability of remaining disabled and the discount factor.

\begin{aligned} \operatorname{EPV} &=5{,}000\left(e^{-0.18(1)}+e^{-0.18(2)}+e^{-0.18(3)}\right)\\ &=5{,}000\left(e^{-0.18}+e^{-0.36}+e^{-0.54}\right)\\ &=5{,}000(0.8352702+0.6976763+0.5827483)\\ &=5{,}000(2.1156948) =\$10{,}578.47. \end{aligned}

The three discounted eligibility factors are $0.8352702$, $0.6976763$, and $0.5827483$. Their sum is $2.1156948$; multiplying by $5,000 gives $10,578.47, rounded to the nearest cent.

Check. The expected present value is below the maximum undiscounted total of $15,000. The first payment is not certain: recovery or death can occur before the end of the first year.

Common mistake avoided. Multiplying the three payments by discount factors alone would incorrectly assume that every claimant remains eligible through every payment date.

Claims Beginning at Different Times

A claim that begins after issue must be discounted from its own start time back to issue. Its remaining payment duration is also constrained by contract expiry. If a claim begins at time $s$, the latest possible payment time is $H$, so its maximum duration is $D(s)=\min(T,H-s)$. The value at issue of a continuous benefit, conditional on a claim beginning at $s$, is therefore:

e^{-\delta s}\,b\frac{1-e^{-(\delta+\rho+\mu)D(s)}}{\delta+\rho+\mu}.

If the claim can begin at several possible times $s_j$ with mutually exclusive probabilities $p_j$ specified at issue, the expected value is the sum of each conditional value times its probability. These probabilities must be defined consistently: if they are unconditional probabilities of the onset events from issue, do not multiply them by a separate survival-to-onset factor. If onset is instead modeled by an intensity among active lives, the probability density of onset must include the probability of remaining active until onset.

For example, under a first-onset model with constant disability incidence intensity $\alpha$ and active-state mortality intensity $\nu$, starting active at issue, the density of first disability onset at time $s$ is $\alpha e^{-(\alpha+\nu)s}$. This follows because onset at $s$ requires remaining active to $s$, with probability $e^{-(\alpha+\nu)s}$, followed by an onset in an interval of length $ds$, with probability approximately $\alpha ds$. An issue valuation integrates the claim-start value against that density over the eligible onset times. This expression presumes constant active-state intensities and no other exits from the active state.

Worked Example: Two Possible Claim Start Times Before Expiry

A hypothetical contract expires at time $H=3.5$ years. Conditional on a claim occurring, its onset is at year 1 with probability $0.30$, or at year 2 with probability $0.70$; these are mutually exclusive probabilities stated at issue. The benefit is $4,000 per year, paid continuously while disabled, for at most $T=2$ years from onset. Assume $\delta=0.04$, $\rho=0.06$, and $\mu=0.02$, all constant per year. Find the expected present value at issue, conditional on a claim occurring at one of the stated times.

Method. First determine the duration available for each onset, then value at claim start and discount back to issue. The combined rate is $k=0.04+0.06+0.02=0.12$.

For onset at year 1, the contract has $3.5-1=2.5$ years remaining, so the full two-year maximum period applies: $D(1)=\min(2,2.5)=2$. Its value at claim start is:

4{,}000\frac{1-e^{-0.12(2)}}{0.12} =4{,}000\frac{1-e^{-0.24}}{0.12} =\$7{,}112.40.

Discount this value one year to issue: $e^{-0.04}(7{,}112.40)=\$6{,}833.52$. Weighting by its onset probability gives $0.30(6{,}833.52)=\$2{,}050.06$.

For onset at year 2, only $3.5-2=1.5$ years remain before expiry. Thus $D(2)=\min(2,1.5)=1.5$. Its value at claim start is:

4{,}000\frac{1-e^{-0.12(1.5)}}{0.12} =4{,}000\frac{1-e^{-0.18}}{0.12} =\$5{,}490.99.

Discount two years to issue: $e^{-0.08}(5{,}490.99)=\$5{,}068.83$. Weighting by its onset probability gives $0.70(5{,}068.83)=\$3{,}548.18$. The total conditional expected present value is $\$2{,}050.06+\$3{,}548.18=\$5{,}598.23$, rounded to the nearest cent. Intermediate values are displayed to cents; the total is computed using unrounded values.

Check. The later claim has a smaller value at claim start because it has only 1.5 years before expiry, rather than the full two-year maximum. Its value is also discounted for two years, not one.

Common mistake avoided. Giving the year-2 claim the full two-year period would ignore expiry. That incorrect calculation gives approximately $6,645.96 at issue, rather than the correct $5,598.23.

Common Timing and Eligibility Traps

The calculation is most reliable when claim duration, contract duration, and payment timing are kept distinct. The maximum period limits how long a claim can pay; it does not guarantee that the full period will be paid. Recovery and death may terminate the claim earlier, and contract expiry may truncate the maximum period. For a claim starting at $s$, the correct available duration is based on $H-s$, not $H$.

Resetting the maximum period at issue. Measure $T$ from claim onset. A two-year maximum does not mean two years from policy issue if the claim begins later.
Ignoring contract expiry. Use $D(s)=\min(T,H-s)$. A claim beginning near expiry may have substantially less than $T$ years of possible benefits.
Discounting but not allowing for termination. For continuous benefits, recovery and disabled-state mortality affect the probability of payment throughout the period. For discrete benefits, they affect the probability of eligibility at each payment time.
Applying onset probabilities twice. Determine whether given probabilities already represent onset events from issue, or whether onset must be modeled using an incidence intensity and active-state survival.
Changing the payment convention silently. A benefit paid at the end of a year only if the claimant is disabled then is different from a continuous benefit or a benefit prorated on recovery.

These distinctions matter in valuation because two claims with the same benefit amount and maximum period can have different values if they start at different times, face different remaining contract terms, or use different payment conventions. Always identify the valuation date first, then the claim start, benefit horizon, possible termination events, and payment dates.

Check Your Understanding

For each problem, apply the stated payment and termination assumptions. Answers are rounded to the nearest cent where a dollar value is requested.

A continuous benefit of $10,000 per year is payable for at most 1.5 years from claim start. Assume $\delta=0.03$, $\rho=0.05$, and $\mu=0.02$, all constant per year, with no earlier contract expiry. Find the expected present value at claim start. (Answer: $13,929.20)
A claim pays $3,000 at the end of each year for at most two years, provided the claimant is still disabled immediately before payment. Assume $\delta=0.04$, $\rho=0.06$, and $\mu=0.02$, all constant per year. Find the expected present value at claim start. (Answer: $5,020.64)
A continuous benefit of $8,000 per year has a maximum period of two years. A claim begins at year 1.5, and the contract expires at year 3. Assume $\delta=0.05$, $\rho=0.04$, and $\mu=0.01$, all constant per year. Find the value at issue, conditional on this claim start. (Answer: $10,338.18)
A contract expires at year 4. Conditional on a claim, onset occurs at year 1 with probability 0.40 or year 2 with probability 0.60. A continuous benefit of $2,000 per year is payable for at most 1.5 years. Assume $\delta=0.03$, $\rho=0.05$, and $\mu=0.02$, all constant per year. Find the expected present value at issue, conditional on a claim. (Answer: $2,655.57)
A contract expires at year 3. Conditional on a claim, onset occurs at year 0.5 with probability 0.25 or year 2.5 with probability 0.75. A continuous benefit of $5,000 per year is payable for at most one year. Assume $\delta=0.04$, $\rho=0.03$, and $\mu=0.03$, all constant per year. Find the expected present value at issue, conditional on a claim. (Answer: $2,820.83)

What You'll Learn