Defining States From Actuarial Conditions

Why State Definitions Matter

A state model begins with a decision about what the contract needs to distinguish. Is the insured currently eligible for a disability benefit? Has a long-term-care condition been met? Is the policy still in force? These questions concern the insured’s condition at a particular time, not the sequence of events that led there.

The state definitions must reflect the contract being valued. If two conditions lead to different contractual benefits, treating them as one state can hide a material distinction. Conversely, adding unnecessary states makes the model harder to interpret without adding useful information. The first task is therefore to define states that are precise, non-overlapping, and sufficient for the contractual distinctions under study.

Definition: A state is a defined condition of a policy or insured at a particular time. A state space is the set of all states used by the model. The states are mutually exclusive if no policy can satisfy two state definitions at once, and exhaustive if every policy covered by the model satisfies at least one state definition at every time in the model’s scope.

“Active,” “disabled,” and “dead” are labels, not complete definitions by themselves. The contract’s terms must say what counts as active or disabled. For example, the definition of disability might depend on the insured’s ability to perform an occupation, a waiting period, or a required level of care. The state description should state the relevant conditions rather than leave them to interpretation.

Build States as a Partition of Conditions

A reliable way to define states is to begin with the conditions that affect the contract, then assign every possible status to one and only one category. For a simple income-protection contract, a possible classification is: alive and not meeting the contractual disability test; alive and meeting that test; or dead. If the contract also distinguishes a disabled waiting period from a period in which benefits are payable, those may need to be separate states.

The partition requirement can be expressed with an indicator for each state. Let \(I_j(t)\) equal 1 when the policy is in state \(j\) at time \(t\), and 0 otherwise. If the states are mutually exclusive and exhaustive, exactly one of the indicators equals 1:

\sum_{j=1}^{m} I_j(t)=1

This identity is a useful check: a sum of 0 means the state definitions leave a possible condition uncovered; a sum greater than 1 means some condition has been assigned to overlapping states. If the state conditions are mutually exclusive and exhaustive, they also partition the population at a fixed time. For example, the probability of being in any one of the states is the sum of the probabilities of the individual states.

Practical rule: Define the states using the conditions that determine contractual cash flows or policy status. Then check both directions: no policy can meet two definitions at once, and no in-scope policy can fall outside all definitions.

Current Condition and Benefit Eligibility

A common modelling decision is whether to define states by a person’s physical or occupational condition, by the contract’s benefit status, or by both. The appropriate choice depends on whether the distinction changes the contractual payments or other quantities the model must represent.

For instance, an insured may satisfy the contract’s disability test before the benefit waiting period has ended. The insured is disabled, but a benefit is not yet payable. If the model only needs to distinguish disability from non-disability, a single disabled state may suffice. If the model must calculate whether monthly payments are currently due, it may need separate states for disabled-but-waiting and disabled-with-benefits-payable.

Worked Example: Separating Disability From Payment Status

A hypothetical income-protection contract defines disability as inability to perform the insured’s regular occupation. A benefit becomes payable after 60 consecutive days of disability. For this model, benefits are payable while the insured remains disabled after completing the waiting period. Define states and classify the insured at days 0, 40, 60, and 75 if disability begins on day 1 and continues through day 75. Assume the insured is alive throughout.

Method. First identify which distinctions affect the contract. The insured can be alive and not disabled, disabled while the waiting period is incomplete, or disabled with benefits payable. Death must also be covered for an in-force policy model.

State definitions. Let state 1 be alive and not meeting the disability test. Let state 2 be alive, disabled, and still within the 60-day waiting period. Let state 3 be alive, disabled, and past the waiting period, with benefits payable. Let state 4 be dead.

Day 0. Disability has not begun, so the insured is alive and not disabled: state 1.

Day 40. The insured has been disabled for 40 consecutive days, which is less than 60. The insured is in state 2; no benefit is payable yet.

Day 60. The insured has completed 60 consecutive days of disability. Under the stated contract convention, benefits become payable after completion of the waiting period, so the insured is in state 3 at day 60.

Day 75. Disability continues and the waiting period has been completed, so the insured remains in state 3.

Check. At each listed time the insured satisfies exactly one state definition. The important distinction is that disability alone does not imply that a benefit is already payable. A common mistake is to label day 40 as “benefit payable” because the insured is disabled; the correction is to apply the 60-day contractual waiting condition as a separate criterion.

A Short Justification of the Partition Check

The indicator identity follows directly from the two partition requirements. Mutual exclusivity means at most one state indicator can equal 1. Exhaustiveness means at least one must equal 1. Together these imply exactly one indicator is 1, so their sum equals 1. Conversely, if the indicators are defined to be 1 precisely when their corresponding state applies, a sum of 1 at every in-scope time rules out both gaps and overlaps.

\text{at most one}+\text{at least one} \quad\Longrightarrow\quad \sum_{j=1}^{m} I_j(t)=1

This is a check on the state definitions, not a formula for how likely a policy is to occupy each state. It does not describe how a policy moves between states. Those are separate modelling questions.

When Does a Contract Need More States?

A state should preserve the information needed for the purpose of the model. A useful test is to ask whether two apparently different conditions have the same contractual consequences for everything being valued. If the answer is no, combining them may lose information.

Suppose a long-term-care contract pays a daily benefit only when the insured meets a specified care-eligibility test and the elimination period has been satisfied. If the model must determine current benefit payments, “meets the care test but still in the elimination period” and “meets the care test and benefit is payable” are different conditions. By contrast, if a calculation only needs to count how many insureds meet the care test, those conditions might be grouped for that limited purpose.

The state space is therefore defined relative to the model’s purpose and the contract’s terms. Adding a new state is not automatically better; each added state must represent a meaningful distinction. Definitions should also make clear whether death ends the relevant policy coverage and whether statuses such as lapsed or expired are within the model’s scope. If they are in scope, they must be accounted for; if they are excluded, state that limitation.

Worked Example: Choosing States for a Care Benefit

A hypothetical long-term-care contract pays a monthly benefit only while the insured is alive, meets a two-part care-eligibility test, and has completed a 3-month elimination period. The model is intended to calculate whether a benefit is due at each monthly valuation date. Define a suitable state space and classify these records: (A) alive, does not meet the test; (B) alive, meets the test for 2 months; (C) alive, meets the test for 4 months; (D) dead.

Method. Because the model needs to identify when a monthly benefit is due, eligibility and completion of the elimination period must be represented separately. The contract’s rule makes the difference between 2 and 4 months material.

Define state 1 as alive and not meeting the care-eligibility test. Define state 2 as alive, meeting the test, but not yet having completed the 3-month elimination period. Define state 3 as alive, meeting the test, and having completed the elimination period, so the benefit is due under the stated conditions. Define state 4 as dead.

Record A meets state 1’s definition. Record B meets the test but has completed only 2 of the required 3 months, so it belongs to state 2 and no benefit is due. Record C has met the test for 4 months, so the elimination period has been completed and it belongs to state 3; a benefit is due, subject to the stated conditions. Record D belongs to state 4.

Check. Each record falls into exactly one state, and all four listed conditions are covered. The common mistake is to treat every eligible insured as benefit-payable; the correction is to include completion of the elimination period in the definition of the payable state.

Worked Example: Classifying a Portfolio at a Valuation Date

A hypothetical insurer reviews eight policies at a particular valuation date. For this exercise, use four states: alive and not disabled; disabled within the waiting period; disabled with benefits payable; and dead. The records are: two alive and not disabled; three disabled, of whom two are still within the waiting period and one has completed it; one disabled and benefit-payable; and two dead. Classify the policies and verify that the state definitions cover the portfolio without overlap.

Method. Assign each policy to the state matching its status at the valuation date, then total the state counts and compare with the portfolio size.

State 1 has 2 policies. State 2 has 2 policies, because those two are disabled but have not completed the waiting period. State 3 has 1 policy from the group of three that completed the waiting period, plus the separately stated 1 benefit-payable policy, for a total of 2. State 4 has 2 policies.

The state-count total is \(2+2+2+2=8\), equal to the portfolio size. The proportions are therefore \(2/8=0.25\) in each of the four states.

Check. The counts sum to 8, and each record has been assigned once. A common mistake is to place all three policies in the “disabled” group into the waiting-period state; that ignores the one that has completed the waiting period. The correction is to classify each policy by both its disability condition and its payment status.

Common State-Definition Mistakes

Using labels without criteria. “Disabled” is ambiguous unless the relevant contractual disability test is specified.
Confusing a condition with a payment rule. A person can meet a disability or care test while a waiting or elimination period still prevents payment.
Leaving out an in-scope condition. If the model includes policies after death, lapse, or expiry, those conditions need to be accounted for in its definitions.
Allowing overlap. Definitions such as “alive” and “disabled” overlap if both are listed as states without specifying that one means alive and not disabled.
Adding distinctions without a purpose. Separate states are most useful when the contract or the model’s objective requires different treatment.

Key takeaway: A well-defined state space is a partition of the in-scope contract conditions: every policy is in one state and only one state at any given time. Make state definitions precise enough to preserve the contractual distinctions needed for the valuation.

Check Your Understanding

For each problem, define or apply states using the contractual conditions given.

A hypothetical disability contract pays after 45 consecutive days of disability. An insured has been disabled for 30 days and is alive. Under a model with states “not disabled,” “disabled but waiting,” “disabled with benefit payable,” and “dead,” identify the state and whether a benefit is due. (Answer: disabled but waiting; no benefit is due.)
A care contract’s benefit requires both eligibility and completion of a 2-month elimination period. An insured is alive, meets the eligibility test, and has met it for exactly 1 month. Which condition must a state definition include to distinguish this insured from one who is already receiving benefits? (Answer: whether the 2-month elimination period has been completed.)
A state space is defined as “alive” and “alive and disabled” for an income-protection model. Identify the defect and give a corrected three-state definition that includes death. (Answer: the states overlap; use alive and not disabled, alive and disabled, and dead.)
At a valuation date, a portfolio has 5 active policies, 2 disabled in a waiting period, 4 disabled with benefits payable, and 1 dead policy. How many policies are in scope, and what fraction are disabled with benefits payable? (Answer: 12 policies; \(4/12=1/3\).)
A model defines states by whether an insured meets a care test, but its purpose is to calculate current benefits that also depend on a completed elimination period. What is the modelling concern, and what distinction should be considered? (Answer: the states may not retain enough information to determine payment; distinguish eligible-but-within the elimination period from eligible with benefits payable.)

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