From Contract Rules to Allowed State Changes
A state model is useful only if its allowed changes reflect the contract or process being modelled. A policyholder might move from active coverage to disability, recover, or die, but not every pair of states necessarily represents a possible direct change. In particular, a contract may require a person to pass through an intermediate status before entering another state.
The key distinction is between a direct transition and a route. A direct transition changes the state from one specified state to another without an intervening state in the model. A route can contain several direct transitions. Reaching a state by a route does not by itself mean that a direct transition to that state is allowed.
This tutorial concerns the structure of the model, not the probability or speed of a change. An allowed transition need not occur for every person, or even occur in a particular observation. A forbidden transition, by contrast, is excluded by the model as specified. No interest, mortality, or probability distribution assumption is needed to decide which changes the contract permits; those assumptions are relevant to other calculations.
Read the Contract Conditions Literally
Start by stating what each state means and then translate each relevant contract rule into a direct-change decision. Avoid adding a transition merely because two states can occur at different times in a person’s history. Ask instead: Can the model move from the first state to the second in one change, under the stated rules?
A useful way to organize the decisions is to list each state pair and classify it as permitted or forbidden. A permitted pair becomes a direct link in the model’s directed graph. If the only allowed route between two states passes through a third state, include the links along that route, not an extra direct link. A state may also be absorbing if the scenario says there is no permitted move out of it; this conclusion follows from the stated rules, not simply from the state’s name.
Worked Example: Disability Coverage With Recovery
Hypothetical contract and model assumptions. A disability policy is represented by three states: \(A\), active and not disabled; \(D\), disabled and eligible for the stated disability status; and \(X\), deceased. The model allows a healthy active person to become disabled or die, and a disabled person to recover to active status or die. Death is permanent. No separate “recovered” state is used: recovery returns the person to \(A\).
Method. Test each possible change against the stated rules, including whether it is direct. The interest rate, mortality law, and probability distribution are not specified because this is a structural classification, not a valuation.
From \(A\), becoming disabled is explicitly allowed, so \(A \to D\) is permitted. Death from active status is also allowed, so \(A \to X\) is permitted.
From \(D\), recovery returns the person directly to active status, so \(D \to A\) is permitted. A disabled person can also die, so \(D \to X\) is permitted.
The deceased state has no permitted exit. Thus \(X \to A\) and \(X \to D\) are forbidden. The model contains no separate transition \(A \to A\) or \(D \to D\): remaining in the current state is not a change to another state.
Check. The complete list of permitted direct changes is \(A \to D\), \(A \to X\), \(D \to A\), and \(D \to X\). The only route from \(A\) to \(X\) that passes through disability is \(A \to D \to X\), but \(A \to X\) is separately permitted by the contract. These are two distinct ways the model can represent a history.
Common mistake and correction. A modeller might wrongly omit \(A \to X\), reasoning that every death must follow disability because the model includes a disability state. The rules say that an active person may die, so the direct link \(A \to X\) must be included. Conversely, a history that includes disability does not make the direct link \(A \to X\) redundant: one link represents death without an intervening disabled state, while the other route includes it.
When an Intermediate State Is Required
Some contract conditions explicitly impose a sequence. For instance, a hypothetical long-term care benefit might recognize a moderate-impairment status before a severe-care status. If the model records that status and the contract rules require it to occur first, a direct change from independent to severe-care status is forbidden. The route through moderate impairment is permitted.
The requirement is about the states and rules chosen for the model. If a modeller uses fewer, broader states, a change that was represented as two transitions in a detailed model may appear as one transition in the broader model. That does not make the detailed model wrong; the two models record different levels of detail. The modeller must be consistent about whether the intermediate condition is represented.
Worked Example: Long-Term Care With a Required Assessment Stage
Hypothetical contract and model assumptions. A long-term care benefit is represented by \(I\) (independent), \(M\) (assessed moderate impairment), \(C\) (receiving the defined custodial-care status), and \(X\) (deceased). The hypothetical contract requires a person to meet the moderate-impairment assessment before entering \(C\). From \(I\), a person may enter \(M\) or die. From \(M\), a person may recover to \(I\), enter \(C\), or die. From \(C\), a person may return to \(M\) or die. Death is permanent. No other direct changes are allowed.
Method. Apply the transition rules to each origin state. The required assessment means that \(I \to C\) is not a direct permitted change. This example specifies no interest rate, mortality assumption, or distribution because no numerical valuation is being performed.
From \(I\), the permitted direct changes are \(I \to M\) and \(I \to X\). The direct change \(I \to C\) is forbidden because it would skip the required assessment state.
From \(M\), the permitted changes are \(M \to I\), \(M \to C\), and \(M \to X\). From \(C\), the permitted changes are \(C \to M\) and \(C \to X\). From \(X\), there are no permitted exits.
A person can nevertheless move from \(I\) to \(C\) over a route: \(I \to M \to C\). That route has two direct changes, both permitted. It does not create a direct \(I \to C\) transition. If the person instead moves from \(M\) back to \(I\), a later route to \(C\) must again pass through \(M\).
Check. The model respects the assessment rule because every route into \(C\) from \(I\) includes \(M\). It also allows \(C \to M\), as specified, without requiring a return to \(I\).
Common mistake and correction. It is incorrect to say “\(I\) can reach \(C\), therefore \(I \to C\) is permitted.” Reachability through \(M\) is not direct permission. Record \(I \to M\) and \(M \to C\), and mark the direct pair \(I \to C\) forbidden.
A Precise Test for Direct Changes and Routes
For a finite collection of states, let \(E\) be the set of permitted ordered pairs. An ordered pair \((i,j)\) belongs to \(E\) exactly when the model permits a direct change from \(i\) to \(j\). A route from \(i\) to \(j\) is a sequence of states \(i=i_0,i_1,\ldots,i_n=j\) for which each consecutive pair \((i_{r-1},i_r)\) is permitted.
The distinction follows directly from these definitions. A direct transition requires the single pair \((i,j)\) to belong to \(E\). A route with an intermediate state \(k\) requires \((i,k)\) and \((k,j)\) to belong to \(E\). Those two conditions do not imply that \((i,j)\) belongs to \(E\). Therefore, if a contract requires an intermediate state, include the required consecutive pairs and exclude the direct pair that skips it.
Write each state in terms of the conditions that distinguish it from the others. Check that the contract rules can be interpreted using those definitions.
For each state, identify the changes the hypothetical contract permits and any conditions that must be met first.
Mark a direct pair permitted only if the rules allow that one-step change. Mark pairs that skip a required status forbidden.
Confirm that required multi-step routes use permitted links, and that any state with no allowed exits has no outgoing links.
Worked Example: A Required Waiting Status Before Benefits
Hypothetical contract and model assumptions. A short-term disability product is modelled with \(A\) (active and not disabled), \(W\) (disabled during an unpaid waiting period), \(B\) (disabled with the benefit payable), \(R\) (recovered), and \(X\) (deceased). A new disability episode must enter \(W\); the benefit becomes payable only after the waiting period is completed, represented by a change from \(W\) to \(B\). A person may recover from \(W\) or \(B\) to \(R\), and may die from any living state. A person in \(R\) may return to \(A\). No other direct changes are allowed. The example treats waiting-period completion as a state change and specifies no interest, mortality law, or probability distribution.
Method. Translate the waiting-period rule first, then list the other explicitly permitted changes. The central test is whether the model can move directly from \(A\) to \(B\).
From \(A\), disability begins in \(W\), so \(A \to W\) is permitted and \(A \to B\) is forbidden. Death while active is permitted, so \(A \to X\) is also allowed.
From \(W\), the waiting period can be completed, giving \(W \to B\). Recovery before benefit payment gives \(W \to R\), and death gives \(W \to X\). From \(B\), recovery gives \(B \to R\), and death gives \(B \to X\). From \(R\), return to active status gives \(R \to A\), while death gives \(R \to X\). The deceased state has no exits.
The benefit-paying state is reachable from active status by the route \(A \to W \to B\), but the contract does not permit the direct pair \(A \to B\). A person who recovers during the waiting period follows \(A \to W \to R\); that history does not include a benefit-paying state.
Check. Every route from \(A\) into \(B\) passes through \(W\). The waiting-period condition is therefore represented without accidentally paying benefits immediately upon disability.
Common mistake and correction. Treating “becomes disabled and is later paid” as one direct change \(A \to B\) erases the waiting-period status. Keep \(W\) as an intermediate state and use \(A \to W\) followed, if the waiting period is completed, by \(W \to B\).
Structural Possibility Is Not Likelihood
A permitted transition describes what the model allows, not how often it happens. This distinction matters when building a model from contract wording. A rare event can still require a permitted link if the contract allows it. Conversely, a transition should not be added merely because it seems plausible in real life when the model’s stated rules exclude it.
A useful final review is to examine each proposed link in both directions of the reasoning:
- For every included link: identify the rule that permits that direct change.
- For every required intermediate state: check that no direct link skips it.
- For every forbidden link: make sure the prohibition comes from the stated contract or modelling assumptions, not an unstated intuition.
- For every terminal state: verify whether the rules truly prohibit all exits.
These checks prevent two common structural errors: allowing a contractually impossible shortcut and excluding a change simply because another route is also possible. The chosen state definitions and the allowed links must work together. If a rule cannot be represented clearly, reconsider whether the states distinguish the conditions the contract needs to distinguish.
Check Your Understanding
For each hypothetical model, decide which direct changes are permitted and distinguish them from routes through intermediate states.
- A product has states \(A\) (active), \(D\) (disabled), and \(X\) (deceased). Active people may become disabled or die; disabled people may recover to active or die; death is permanent. Is \(A \to X\) permitted, and is \(A \to D \to X\) a direct transition or a route? (Answer: \(A \to X\) is permitted; \(A \to D \to X\) is a route through \(D\), not a direct transition.)
- A care model has \(I\) (independent), \(M\) (moderate impairment), and \(C\) (custodial care). The rules permit \(I \to M\) and \(M \to C\), and require \(M\) before \(C\). Can a person move from \(I\) to \(C\) over time, and is \(I \to C\) directly permitted? (Answer: Yes, by the route \(I \to M \to C\); no, the direct transition \(I \to C\) is forbidden.)
- A waiting-period model allows \(A \to W\), \(W \to B\), and recovery \(W \to R\). Benefits are payable only in \(B\), after passing through \(W\). A modeller adds \(A \to B\) because a person can eventually receive benefits. What is the structural error? (Answer: The direct link \(A \to B\) skips the required waiting state; use the route \(A \to W \to B\).)
- A hypothetical model allows \(P \to Q\), \(Q \to R\), and \(P \to R\). The contract permits direct movement from \(P\) to \(R\), and also allows an optional intermediate status \(Q\). Is the direct link \(P \to R\) redundant or forbidden because the route through \(Q\) exists? (Answer: Neither; \(P \to R\) remains a permitted direct transition, distinct from the route \(P \to Q \to R\).)
- A model defines \(X\) as deceased and states that death is permanent. A proposed graph contains \(X \to A\), where \(A\) is active. What must be changed? (Answer: Remove \(X \to A\); under the stated assumptions, \(X\) has no permitted exit.)