Choosing Payment Timing and Disability Assumptions

Why Payment Timing and Disability Assumptions Must Match

A disability income value depends not only on how much is payable, but also on when a payment is made and which disability conditions trigger it. A benefit payable at the start of a period may use the insured’s state at that date; a benefit payable at the end may depend on the state then, the time spent disabled during the period, or a contractual test such as satisfying a waiting period. Those are different valuation rules, not interchangeable descriptions of one cash flow.

This tutorial focuses on setting out those rules clearly before calculating an expected present value. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A payment in advance is made at the beginning of a specified period. A payment in arrears is made at the end of that period. The payment date alone does not determine eligibility: the contract must also specify the disability state or disability duration that qualifies for payment.

State the Payment Rule Before Valuing

For a simple two-state model, let $A$ denote active and $D$ denote disabled, with death an absorbing state if mortality is included. Suppose a fixed amount $B$ is payable at time $t$ exactly when the policyholder is in $D$ at that time. If the annual effective interest rate is $i$, then $v=1/(1+i)$. The expected present value of payments at specified dates is obtained by multiplying each payment amount by its probability of occurring and discounting it to time 0.

\operatorname{EPV}=\sum_t v^t E[C_t]

Here $C_t$ is the payment at time $t$. If $C_t=B$ when the insured is disabled and $0$ otherwise, then $E[C_t]=B P(D\text{ at }t)$. This compact expression requires the stated eligibility rule. It does not apply unchanged when the payment depends on time spent disabled, a qualifying period, a partial-disability definition, or another condition.

The distinction between the common state-based conventions can be written as follows for payments at annual dates $0,1,\ldots,n-1$ in advance, or $1,2,\ldots,n$ in arrears:

\operatorname{EPV}_{\text{advance}}=B\sum_{k=0}^{n-1}v^k P(D\text{ at }k), \qquad \operatorname{EPV}_{\text{arrears}}=B\sum_{k=1}^{n}v^k P(D\text{ at }k).

These formulas describe benefits conditional on being disabled at the respective payment dates. They do not assert that every real contract uses that test. In particular, a payment in arrears for a period of disability might instead be based on disability during the period, accrued duration, or a waiting-period rule. That contractual detail must be represented in $C_t$.

Why the Expected-Present-Value Formula Holds

Let each scheduled payment be a random variable $C_t$, and discount it by the deterministic factor $v^t$. The present value is the sum of the discounted payments. Taking expectations and using the linearity of expectation gives the valuation formula, whether or not payments at different dates are dependent:

E\!\left[\sum_t v^t C_t\right] =\sum_t E[v^t C_t] =\sum_t v^t E[C_t].

If payment $B$ is made exactly when the insured is in $D$, define the indicator $I_t$ to be 1 when the insured is in $D$ at time $t$, and 0 otherwise. Then $C_t=BI_t$, so $E[C_t]=B E[I_t]=B P(D\text{ at }t)$. The derivation shows why both the payment date and the event that triggers payment matter: changing either changes the discounted cash flow or its probability.

Worked Example: Annual Payments in Advance and in Arrears

A hypothetical contract pays $10,000 annually for three payment dates if the insured is disabled on each applicable date. Payments in advance are at times 0, 1, and 2; payments in arrears are at times 1, 2, and 3. The insured is active at time 0. At each year-end, an active insured becomes disabled with probability 0.10. A disabled insured recovers with probability 0.25, remains disabled with probability 0.70, and dies with probability 0.05. Use an annual effective interest rate of 5%. Find both expected present values.

Method. First calculate the probabilities of being disabled at each payment date, then discount each expected payment. Death is absorbing, and the transition probabilities apply once per year.

At time 0, $P(D)=0$. At time 1, the insured started active, so $P(D\text{ at }1)=0.10$, and $P(A\text{ at }1)=0.90$.

For time 2, an insured can be disabled after starting active, or remain disabled after starting disabled:

P(D\text{ at }2)=0.90(0.10)+0.10(0.70)=0.09+0.07=0.16.

The active probability at time 2 is $0.90(0.90)+0.10(0.25)=0.835$. Thus:

P(D\text{ at }3)=0.835(0.10)+0.16(0.70)=0.0835+0.112=0.1955.

With $v=1/1.05=0.95238095$, $v^2=0.90702948$, and $v^3=0.86383760$, the advance value is:

10{,}000\left[0(1)+0.10v+0.16v^2\right] =10{,}000(0.09523810+0.14512472) =\$2{,}403.63.

The arrears value is:

10{,}000\left[0.10v+0.16v^2+0.1955v^3\right] =10{,}000(0.09523810+0.14512472+0.16888025) =\$4{,}092.43.

Rounded to the nearest cent, the expected present values are $2,403.63 in advance and $4,092.43 in arrears. The larger arrears value does not mean arrears are inherently more valuable: this contract’s arrears schedule includes a third possible payment at time 3, while the advance schedule includes a date-0 payment that cannot be made because the insured starts active. The date sets differ as well as the payment convention.

Common mistake avoided: multiplying the advance value by $1+i$ to obtain the arrears value. That shortcut would require the same contingent payments shifted exactly one period. Here the dates and disability probabilities included in the two schedules are not the same.

Transition Timing Is Not the Same as Disability Duration

A discrete-time transition model records a state at specified observation dates. If the insured is active at the start of a year and disabled at year-end, that observation establishes a change of state during the year, but does not tell us the exact onset date. For a payment that accrues in proportion to time disabled, a year-end state indicator alone is insufficient.

Worked Example: A Midyear Onset and a Year-End Payment

In a hypothetical one-year contract, an active insured becomes disabled during the year with probability 0.20. Conditional on becoming disabled, the onset time $T$ is uniformly distributed over the year, so $T\sim U(0,1)$. There is no recovery or death before year-end. The benefit accrues continuously at $12,000 per year while disabled and is paid at time 1. Use an annual effective interest rate of 5%. Find the expected present value.

Method. Calculate expected disabled duration, convert that duration to the amount accrued, and discount the year-end payment. Conditional on a claim, the disabled duration is $1-T$, with expected value $1-E[T]=1-0.5=0.5$ years.

The unconditional expected disabled duration is $0.20(0.5)=0.10$ years. Therefore the expected accrued benefit is:

\$12{,}000\text{ per year}\times 0.10\text{ years}=\$1{,}200.

The benefit is paid at time 1, so:

\operatorname{EPV}=\frac{\$1{,}200}{1.05}=\$1{,}142.86.

This is also a check on the duration calculation: the 0.20 probability of a claim is multiplied by half a year of expected disability conditional on a claim, not by a full year. If the contract instead paid a fixed $12,000 year-end installment whenever the insured was disabled at year-end, its expected present value would be $0.20(\$12{,}000)/1.05=\$2{,}285.71$. That is a different benefit rule, not another way to value the duration-accrued benefit.

Common mistake avoided: treating the year-end disability probability as the probability that a full year of benefit has accrued. A transition observed during the year does not imply disability for the entire year.

The Disability Definition Determines the Eligible State

The label “disabled” must correspond to the contract’s definition. For example, inability to perform the insured’s own occupation can identify a broader eligible group than inability to perform any occupation for which the insured is reasonably suited. A valuation model must use transition assumptions that match the definition used to determine payment. If a definition changes during a claim, eligibility may also change over time; that requires a corresponding state or eligibility rule rather than a single unchanging disabled state.

A broader definition will often produce a higher probability of eligibility, but that conclusion depends on the definitions and assumptions. A comparison is valid only when the other model features—such as payment dates, benefits, interest, and recovery assumptions—are held constant.

Worked Example: Comparing Two Disability Definitions

Consider a hypothetical two-year contract with payments of $24,000 at the end of each year if the insured meets the disability definition on that date. The insured starts active. Under either definition, an active insured becomes eligible for benefits over a year with probability $q$; a disabled insured remains disabled to the next year with probability 0.80 and otherwise recovers to active. There is no mortality. The own-occupation definition has $q=0.12$, while the narrower any-occupation definition has $q=0.07$. Use an annual effective interest rate of 4%. Compare the expected present values.

Method. For a given $q$, the probability of being disabled at time 1 is $q$. At time 2, disability can continue from time 1 or begin during the second year after recovery or continued active status. Thus:

P(D\text{ at }2)=0.80q+(1-q)q=1.80q-q^2.

For the own-occupation definition, $P(D\text{ at }1)=0.12$ and $P(D\text{ at }2)=1.80(0.12)-(0.12)^2=0.216-0.0144=0.2016$. With $v=1/1.04$, the expected present value is:

\$24{,}000\left(\frac{0.12}{1.04}+\frac{0.2016}{1.04^2}\right) =\$24{,}000(0.11538462+0.18639053) =\$7{,}242.60.

For the any-occupation definition, $P(D\text{ at }1)=0.07$ and $P(D\text{ at }2)=1.80(0.07)-(0.07)^2=0.126-0.0049=0.1211$. Therefore:

\$24{,}000\left(\frac{0.07}{1.04}+\frac{0.1211}{1.04^2}\right) =\$24{,}000(0.06730769+0.11196302) =\$4{,}302.50.

The own-occupation value is $7,242.60 and the any-occupation value is $4,302.50, rounded to the nearest cent. The difference follows from the stipulated eligibility probabilities; the calculation does not establish that these values apply to any particular product or population.

Common mistake avoided: applying the same transition probability to both definitions without justification. The definition determines who qualifies, so the probability assumptions must be appropriate to that eligibility test.

Practical Checks Before Finalizing a Value

Before evaluating a disability benefit, make the timing and eligibility assumptions visible. The following sequence helps prevent a correct calculation from valuing the wrong contract.

Write the benefit rule.
State the amount, payment dates, and whether eligibility depends on status at a date, duration during a period, or another contractual test.

Specify the disability definition.
Identify which insureds qualify and whether eligibility can change during a claim.

Match transitions to observation times.
Distinguish a transition during a period from a state observed at its beginning or end. Do not infer onset duration from an endpoint state alone.

Discount each expected payment.
Use the actual payment time and the appropriate probability or expected accrued amount for that payment.

A useful exam check is to ask whether moving a payment date would leave its eligibility condition unchanged. If not, the change affects both discounting and the contingent payment itself. Similarly, replacing a duration-accrued payment with a fixed installment based on year-end status changes the benefit rule, even if both payments occur at the same date.

Key takeaway: A disability benefit value is determined jointly by the payment date, the event that makes payment due, and the disability assumptions used to model that event. State all three before calculating an expected present value.

Check Your Understanding

For each problem, state the timing rule you use before calculating. Round dollar answers to the nearest cent.

A benefit of $5,000 is payable at the start of each of two years if the insured is disabled on the payment date. The insured is known to remain disabled at both dates. Compare the present values of payments in advance and payments in arrears at an annual effective interest rate of 4%, using two dates in each schedule. (Answer: Advance: $9,807.69; arrears: $9,430.47.)
An insured starts active. At each year-end, an active insured becomes disabled with probability 0.15; a disabled insured remains disabled with probability 0.60 and otherwise recovers. There is no mortality. A benefit of $8,000 is paid at the ends of years 1 and 2 if disabled then. At an annual effective interest rate of 3%, calculate the expected present value. (Answer: $2,805.17.)
A two-year contract pays $20,000 at each year-end if the insured meets the disability definition. The insured starts active; the annual active-to-disabled probability is 0.10 under a broad definition and 0.05 under a narrow definition. A disabled insured remains disabled with probability 0.75, otherwise recovering; there is no mortality. At 5% annual effective interest, find the expected present value under each definition. (Answer: Broad definition: $4,897.96; narrow definition: $2,494.33.)
With probability 0.25, an insured becomes disabled during a year; conditional on onset, the onset time is uniform over that year. There is no recovery or mortality. A benefit accrues at $6,000 per year while disabled and is paid at year-end. At 6% annual effective interest, find its expected present value. (Answer: $707.55.)
An insured starts active and has an annual probability of 0.20 of becoming disabled from active; a disabled insured remains disabled to the next year with probability 0.50 and otherwise recovers. There is no mortality. A $10,000 benefit is paid only at time 2 if disabled then. At 4% annual effective interest, calculate its expected present value. (Answer: $2,403.85.)

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