Periodic Benefits And Claim Outgo

Why Payment Timing Matters

A death claim is generally associated with a single event, while disability income or long-term care benefits may produce a series of payments. For recurring benefits, annual claim outgo depends not only on the benefit amount but also on which dates are payment dates and what condition makes an installment payable. A benefit payable at every quarter-end while the insured is disabled is different from a single payment if the insured becomes disabled at any time during the year.

In an annual profit calculation, claim outgo is the total amount paid during the policy year under the stated cash-flow convention. When adding installments in this tutorial, the amounts are nominal dollars paid on their scheduled dates; no interest is accumulated or discounted between those dates. Thus, take $i=0$ for these annual outgo calculations. If a profit test assigns a particular annual timing point to claim payments, that timing convention must also be applied consistently; the nominal total alone does not determine investment income.

Definition: A periodic benefit is a benefit payable in recurring installments according to a specified schedule. For an installment due at time $t_j$, define an eligibility indicator $I_j$ to equal 1 if the contractual eligibility condition is satisfied on that payment date, and 0 otherwise. If the installment amount is $b_j$, total claim outgo over the year is $C=\sum_j b_j I_j$. Therefore, its expected amount is $E[C]=\sum_j b_jP(I_j=1)$.

The eligibility rule must be stated before probabilities are calculated. For example, a contract might pay only if the insured is alive and disabled at the end of each quarter. A person who becomes disabled and recovers before a quarter-end would not qualify at that date under this rule. Another contract could pay for a benefit period once disability has begun, even if the person later recovers. Those are different benefit designs and require different indicators.

State Probabilities at Payment Dates

A convenient representation uses a discrete-time state model. The states might be active, disabled, and dead. Let $p_k$ be a row vector containing the probabilities of being in each state immediately after the $k$th transition, and let $P$ be the transition matrix for one payment period. Each row of $P$ gives the probabilities of the next state conditional on the current state. If the initial state distribution is $p_0$, then $p_k=p_0P^k$. In particular, the disabled-state component of $p_k$ is the probability that an installment contingent on being disabled at time $k$ is payable.

This model assumes that, conditional on the current state, the next-period transition probabilities do not depend on earlier states or durations. It also assumes that the stated benefit is paid exactly when the specified eligibility condition holds. If transition probabilities vary by duration or the contract uses another trigger, the state model or eligibility rule must be adjusted.

E[C]=\sum_{j=1}^{m} b_jP(\text{eligible at }t_j)

This result does not require the payment indicators at different dates to be independent. The expected total is the sum of the expected installment amounts by linearity of expectation. The probabilities at later dates must still reflect the transition process, including recovery, death, or other exits that affect eligibility.

Worked Examples

Worked Example: Two Separate Quarterly Installments

A hypothetical policyholder is active at the start of a year. At each quarter-end, an active insured remains active with probability 0.90, becomes disabled with probability 0.08, and dies with probability 0.02. A disabled insured recovers to active with probability 0.20, remains disabled with probability 0.70, and dies with probability 0.10. A benefit of $1,250 is paid at each of the first two quarter-ends if the insured is disabled then. Find expected outgo through the second quarter-end. Assume $i=0$ for aggregation and no payment after death.

Method. Find the probability of disability at each payment date, then multiply each probability by the installment amount and add. This method avoids treating the two installments as one payment triggered by disability at either date.

At the first quarter-end, the insured is disabled only if the transition from active is to disabled. Therefore,

$$ P(D_1)=0.08. $$

At the second quarter-end, there are two mutually exclusive paths to disability: active at the first quarter-end and then disabled, or disabled at the first quarter-end and still disabled. The probabilities of those paths are $0.90(0.08)=0.072$ and $0.08(0.70)=0.056$, respectively. Hence,

$$ P(D_2)=0.072+0.056=0.128. $$

The expected first installment is $1{,}250(0.08)=\$100.00$, and the expected second installment is $1{,}250(0.128)=\$160.00$. Thus, expected outgo is $\$100.00+\$160.00=\boxed{\$260.00}$.

Check. The probability of being disabled at the second date, $0.128$, includes insureds who recovered and are active at that date only if they become disabled again in the same transition; that possibility is already represented through the active-to-disabled transition path. The calculation counts a payment at each date separately.

Worked Example: Four Quarterly Payments With Recovery and Death

A hypothetical disability-income contract starts with an active insured. At each quarter-end, an active insured remains active with probability 0.88, becomes disabled with probability 0.10, and dies with probability 0.02. A disabled insured recovers to active with probability 0.30, remains disabled with probability 0.65, and dies with probability 0.05. The contract pays $400 at each quarter-end if the insured is disabled at that date. Calculate expected claim outgo for the four quarter-ends. Assume $i=0$; death is absorbing and no payment is due after death.

Method. Propagate the active and disabled probabilities from one quarter-end to the next, then sum the four expected installments. The death probability is accounted for by the fact that dead insureds are not in either eligible state.

Let $a_k$ and $d_k$ be the probabilities of being active and disabled at quarter-end $k$. Initially $a_0=1$ and $d_0=0$. At the first quarter-end, $a_1=0.88$ and $d_1=0.10$.

At the second quarter-end, active status can arise from an active insured who stays active or a disabled insured who recovers. Disability can arise from an active insured who becomes disabled or a disabled insured who remains disabled:

a_2=0.88(0.88)+0.10(0.30)=0.7744+0.0300=0.8044, \qquad d_2=0.88(0.10)+0.10(0.65)=0.0880+0.0650=0.1530.

For the third quarter-end,

$$ a_3=0.8044(0.88)+0.1530(0.30)=0.707872+0.045900=0.753772, $$ $$ d_3=0.8044(0.10)+0.1530(0.65)=0.080440+0.099450=0.179890. $$

For the fourth quarter-end,

$$ d_4=0.753772(0.10)+0.179890(0.65) =0.0753772+0.1169285=0.1923057. $$

The expected number of payable installments is $d_1+d_2+d_3+d_4=0.10+0.1530+0.179890+0.1923057=0.6251957$. Therefore expected claim outgo is $400(0.6251957)=\$250.07828$, or $250.08 to the nearest cent.

Check. Each disabled-state probability is at most 1, and the probability of disability increases here because the active-to-disabled inflow exceeds the loss of disabled lives through recovery and death over these payment dates. The final amount is the sum of four expected installments, not the probability of ever becoming disabled multiplied by $400.

Worked Example: Installments Paid at the Start of Each Period

A hypothetical insured is already disabled at the start of a one-year period. At each quarter, a disabled insured remains disabled through the next quarter with probability 0.90, recovers with probability 0.08, and dies with probability 0.02. A $600 installment is paid at the start of each quarter if the insured is disabled at that instant. There are four installments, at the start of quarters 1 through 4. Assume $i=0$. Find expected outgo.

Method. Since payments are in advance, determine eligibility at the start of each quarter. The initial payment is certain; later payments depend on remaining disabled through the preceding quarters. Recovery and death both end eligibility.

At the start of quarter 1, the insured is disabled with probability 1. At the starts of quarters 2, 3, and 4, the probabilities of still being disabled are $0.90$, $0.90^2=0.81$, and $0.90^3=0.729$, respectively. The expected number of payments is therefore $1+0.90+0.81+0.729=3.439$. Expected claim outgo is $600(3.439)=\boxed{\$2{,}063.40}$.

If installments were instead payable at quarter-ends, eligibility would be evaluated after each quarter’s transition, and the four probabilities would be $0.90$, $0.81$, $0.729$, and $0.6561$. The expected amount would then be $600(3.0951)=\$1{,}857.06$. The difference arises solely from payment timing and the eligibility dates specified.

Why the Expected-Outgo Formula Holds

For each scheduled payment date $t_j$, the actual installment is $b_jI_j$: it equals $b_j$ when eligible and zero otherwise. The expected value of that installment is $b_jP(I_j=1)$, because $I_j$ is an indicator. Adding all actual installments gives $C=\sum_j b_jI_j$. Taking expectations and using linearity yields the expected-outgo formula.

Specify the payment dates.
State whether installments are paid at the start or end of each period and how many dates fall within the policy year.

State the eligibility test.
For each date, identify the required state or event, such as being alive and disabled on that date.

Calculate eligibility probabilities.
Use the applicable transition probabilities from the initial state through each payment date.

Sum expected installments.
Multiply each date’s eligibility probability by its benefit amount and add the results.

In a profit vector, claim outgo is an insurer cash outflow, so its sign is negative when profit is recorded from the insurer’s perspective. The expected claim amount calculated here is the positive magnitude of that outflow. Applying the sign convention after calculating the amount helps prevent a negative claim cost from being mistaken for a benefit to the insurer.

Common Mistake: One Trigger Versus Repeated Payments

A frequent error is to calculate the probability that disability occurs at least once and multiply it by one installment, even when the contract pays at multiple dates while disability persists. In the first worked example, the probability of disability at one or both of the first two quarter-ends is $0.08+0.90(0.08)=0.152$. Multiplying by $1,250 gives the wrong result $1{,}250(0.152)=\$190.00$ for a contract that pays at each qualifying date. That calculation counts only one payment per insured.

The contract specifies separate payments at both dates, so the correct calculation is $1{,}250(0.08+0.128)=\$260.00$. The first payment and the second payment can both be made to the same insured. Their indicators may be dependent, but that does not prevent adding their expected amounts. Conversely, if the contract truly paid only once upon becoming disabled at either date, the probability of at least one qualifying event would be appropriate.

Key takeaway: For recurring benefits, model each scheduled installment separately. The expected annual outgo is the sum of installment amounts multiplied by the probabilities that the contract’s eligibility conditions hold on their respective payment dates.

Check Your Understanding

For each problem, identify the payment dates and eligibility rule before calculating. Amounts are nominal annual outgo with $i=0$.

A policy starts active. At each of two quarter-ends, an active insured stays active with probability 0.85, becomes disabled with probability 0.12, and dies with probability 0.03. A disabled insured stays disabled with probability 0.80, recovers with probability 0.10, and dies with probability 0.10. A $500 installment is paid at each quarter-end if disabled. Find expected outgo. (Answer: $159.00)
An insured starts disabled. At each of three payment dates, a disabled insured remains disabled with probability 0.70, recovers with probability 0.20, and dies with probability 0.10. A $900 payment is made at each date if disabled then. Find expected outgo. (Answer: $1,379.70)
A policy starts active. At each of two year-ends, an active insured stays active with probability 0.90, becomes disabled with probability 0.08, and dies with probability 0.02. A disabled insured recovers with probability 0.25, stays disabled with probability 0.65, and dies with probability 0.10. The benefit is $2,000 at each year-end if disabled. Find expected outgo. (Answer: $408.00)
An insured starts disabled. At each quarter the insured remains disabled with probability 0.80, recovers with probability 0.15, and dies with probability 0.05. Four $1,000 installments are paid in advance at the starts of the four quarters if disabled at that instant. Find expected outgo. (Answer: $2,952.00)

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