Workshop: Constructing a Model From a Contract Description

From Contract Wording to a State Model

A contract description is written in benefit language: it may refer to an active policyholder, someone receiving disability benefits, a person entering long-term care, or death. A multiple-state model translates those descriptions into a precise set of states and permitted movements between them. The translation matters because an omitted state or an impossible transition can make the intensity matrix inconsistent with the contract.

This tutorial uses the row-current, column-destination convention. The order of the states must be declared and then used consistently in the transition graph, initial distribution, and intensity matrix. All rates in the examples are hypothetical annual intensities. They are not observed data or assumptions about an actual insurer.

Definition: A state is a mutually exclusive condition used to describe the status of a person at a given time. The set of states should cover every status included in the model. An allowed transition is a movement from one state to another. With a fixed ordering \(1,\ldots,n\), the intensity \(\lambda_{ij}(t)\), for \(i\ne j\), is the instantaneous rate of transition from current state \(i\) to destination state \(j\) at time \(t\). The intensity matrix \(Q(t)\) places \(\lambda_{ij}(t)\) in row \(i\), column \(j\); each diagonal entry is the negative sum of the other entries in its row.

The contract description does not always specify every possible transition explicitly. A model therefore needs stated assumptions. For example, a model might allow recovery from disability but assume that a person cannot move directly from active status to death without first being represented by the model’s direct active-to-dead transition. Whether a path is included is a modeling decision that must match the intended scope.

Before filling in rates, settle four questions: what statuses matter, which transitions are allowed, what is the state order, and what status or distribution applies at time zero? Then write the matrix and check it against those decisions.

A Construction and Checking Routine

Extract the statuses.
List the distinct contract conditions that affect the model. Make them mutually exclusive: a person cannot occupy two listed states at once. Include death when it is part of the contract’s model.

List permitted transitions.
For each current state, identify every allowed destination. A missing transition is represented by an off-diagonal zero, not by an unmentioned or guessed rate.

Specify the initial state.
Give either a definite starting state or a distribution across the ordered states. Check that any initial probabilities are nonnegative and sum to one.

Fill and check the intensity matrix.
Put each allowed rate in its current-state row and destination-state column. For each row, set the diagonal to the negative of the sum of its off-diagonal entries, then verify that the row sums to zero.

For a time-homogeneous model, the rates do not vary with calendar time, so the matrix is written \(Q\). If rates vary with calendar time, write \(Q(t)\) and specify how the rates depend on \(t\). In either case, the matrix must satisfy the same row-by-row checks at each time at which it is defined.

The standard validity checks are direct. Every off-diagonal entry must be nonnegative; a negative off-diagonal “rate” would not describe a possible transition intensity. Each diagonal entry must be nonpositive and equal to the negative sum of its row’s off-diagonal entries. Thus every row sums to zero. A state with no permitted exits has an all-zero row; this is an absorbing state in the intensity model.

Worked Contract Translations

Worked Example: Active, Disabled, and Dead

A hypothetical income-protection contract distinguishes active policyholders, disabled policyholders, and deceased policyholders. A disabled person may recover to active status or die. An active person may become disabled or die. Assume constant annual intensities: active-to-disabled \(0.08\), active-to-dead \(0.01\), disabled-to-active \(0.20\), and disabled-to-dead \(0.04\). Death is permanent. Construct the model and check it.

Method and assumptions. Treat each status as one state, assume these are the only permitted transitions, and use the state order active, disabled, dead. The rates are constant over time and are annual intensities.

Transition graph. From active, the permitted destinations are disabled and dead. From disabled, the permitted destinations are active and dead. There are no exits from dead. In particular, the graph contains no direct active-to-active or disabled-to-disabled transition; remaining in a state is not an off-diagonal transition.

Initial specification. Suppose the policyholder is active at time zero. In the stated order, the initial row distribution is \((1,0,0)\). Its entries are nonnegative and sum to \(1\).

Matrix construction. The active row has off-diagonal rates \(0.08\) and \(0.01\), whose sum is \(0.09\); its diagonal is therefore \(-0.09\). The disabled row has off-diagonal rates \(0.20\) and \(0.04\), whose sum is \(0.24\); its diagonal is \(-0.24\). The dead row has no exits, so all its entries are zero.

Q= \begin{pmatrix} -0.09 & 0.08 & 0.01\\ 0.20 & -0.24 & 0.04\\ 0 & 0 & 0 \end{pmatrix}, \qquad \boldsymbol{\pi}(0)=(1,0,0).

Check. The row sums are \(-0.09+0.08+0.01=0\), \(0.20-0.24+0.04=0\), and \(0+0+0=0\). All off-diagonal entries are nonnegative, the diagonals are nonpositive, and the dead row is zero as required.

Common mistake avoided. Setting the active diagonal to \(-0.08\) would account for disability but omit death. That row would sum to \(-0.08+0.08+0.01=0.01\), not zero. The diagonal must include every permitted exit from active status.

Worked Example: Long-Term Care With an Uncertain Starting Status

A hypothetical long-term care model has three states: independent, receiving home care, and dead. A person who is independent may begin home care or die. A person receiving home care may return to independence or die. Assume constant annual intensities of \(0.08\) from independent to home care, \(0.02\) from independent to dead, \(0.12\) from home care to independent, and \(0.10\) from home care to dead. At the model’s starting time, records imply probabilities \(0.85\) of independent status, \(0.15\) of home care, and \(0\) of death. Specify and check the model.

Method and assumptions. Use the state order independent, home care, dead. Assume the stated transitions are the only permitted ones, rates are constant annual intensities, and death is absorbing.

Initial distribution. The row vector in the declared order is \((0.85,0.15,0)\). The entries are nonnegative and \(0.85+0.15+0=1\), so it is a valid initial distribution. It represents uncertainty about the starting status; it does not change the transition rates.

Matrix construction. The independent-state exit rates sum to \(0.08+0.02=0.10\), so the independent diagonal entry is \(-0.10\). The home-care exit rates sum to \(0.12+0.10=0.22\), so that diagonal entry is \(-0.22\). Death has no exits.

Q= \begin{pmatrix} -0.10 & 0.08 & 0.02\\ 0.12 & -0.22 & 0.10\\ 0 & 0 & 0 \end{pmatrix}, \qquad \boldsymbol{\pi}(0)=(0.85,0.15,0).

Check. The row sums are \(-0.10+0.08+0.02=0\), \(0.12-0.22+0.10=0\), and \(0\). The initial entries sum to \(1\), and all off-diagonal rates are nonnegative.

Common mistake avoided. The initial distribution is not an extra row of \(Q\), nor should it be confused with the rates in the matrix. It records where the modeled lives start; the matrix records the assumed transitions from each state.

Worked Example: Separating Rehabilitation From Long-Term Disability

A hypothetical disability contract distinguishes active status, short-term rehabilitation while disabled, long-term disability, and death. From active status, transition to rehabilitation has annual intensity \(0.10\), and transition to death has intensity \(0.01\). From rehabilitation, recovery to active status has intensity \(0.25\), progression to long-term disability has intensity \(0.15\), and death has intensity \(0.02\). From long-term disability, recovery to active status has intensity \(0.05\), and death has intensity \(0.06\). Assume no other transitions and constant annual intensities. Construct the model for a person active at time zero.

Method and assumptions. Use the order active, rehabilitation, long-term disability, dead. The contract’s distinction between rehabilitation and long-term disability is represented by separate states, since the described transition routes and rates differ by status. Death is absorbing.

Transition graph. Active may move to rehabilitation or death. Rehabilitation may move to active, long-term disability, or death. Long-term disability may move to active or death. No direct active-to-long-term-disability transition is specified, so its intensity is zero in this model.

Initial specification. Active at time zero gives the row distribution \((1,0,0,0)\).

Matrix construction. The exit-rate total from active is \(0.10+0.01=0.11\). From rehabilitation it is \(0.25+0.15+0.02=0.42\). From long-term disability it is \(0.05+0.06=0.11\). Negating these totals gives the respective diagonal entries. The zero for active-to-long-term disability records the omitted route.

Q= \begin{pmatrix} -0.11 & 0.10 & 0 & 0.01\\ 0.25 & -0.42 & 0.15 & 0.02\\ 0.05 & 0 & -0.11 & 0.06\\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad \boldsymbol{\pi}(0)=(1,0,0,0).

Check. The row sums are \(-0.11+0.10+0+0.01=0\), \(0.25-0.42+0.15+0.02=0\), \(\;0.05+0-0.11+0.06=0\), and \(0\). All off-diagonal entries are nonnegative, and the initial distribution sums to \(1\).

Common mistake avoided. Combining both disability states into one would discard the distinction between rehabilitation and long-term disability, including their different permitted routes and rates. Separate states preserve the distinctions specified in the hypothetical contract.

Why the Diagonal Is Determined by the Exits

The diagonal is not an additional transition rate. It is fixed by the requirement that the total instantaneous rate of leaving a state equals the sum of the rates to its possible destinations. The row-sum condition expresses that accounting consistently.

To see why, consider a person currently in state \(i\), with finitely many possible destinations \(j\ne i\). Over a short interval of length \(h\), the probability of a transition from \(i\) to \(j\) is \(\lambda_{ij}(t)h+o(h)\), where \(o(h)\) denotes a term that becomes negligible relative to \(h\) as \(h\) tends to zero. The probability of staying in \(i\) is therefore one minus the sum of the probabilities of moving to another state, up to terms negligible relative to \(h\). Thus its first-order change is the negative of the total exit rate.

\sum_{j\ne i}\lambda_{ij}(t) \quad\Longrightarrow\quad q_{ii}(t)=-\sum_{j\ne i}\lambda_{ij}(t), \qquad \sum_j q_{ij}(t)=0.

This also explains why an absorbing state has a zero row: there are no allowed exits to sum, so its diagonal is zero. This justification concerns the intensity matrix’s construction; it does not calculate finite-time transition probabilities.

Modeling Judgments and Common Traps

A state model is an abstraction of contract wording, not merely a list of labels. The state definitions must be detailed enough to represent the distinctions relevant to the model. If the contract description treats two conditions differently—for example, by giving them different permitted future transitions—combining them may lose necessary information. Conversely, adding states without a contractual or modeling reason can make the specification harder to interpret without improving it.

Keep one state order throughout. If the matrix order is active, disabled, dead, then every row and column and the initial distribution must follow that order.
Distinguish “no transition” from “no exit.” A zero off-diagonal entry means that particular route is not allowed in the model. A whole zero row means the current state has no allowed exits.
Do not put a positive exit rate on the diagonal. The diagonal is the negative sum of all off-diagonal exit rates in that row.
State the time basis. An intensity stated per year must be used consistently as an annual rate; do not silently reinterpret it as a monthly rate.
Do not hide starting uncertainty. If the person’s starting status is uncertain, show the distribution in the same state order and check its sum.

If a contract specifies that a rate depends on how long a person has already been in a condition, it may not be enough to label the state only by that condition. A duration-sensitive description may require a refined state representation or an explicit time dependence, as appropriate to the model assumptions. Do not silently use a constant, status-only rate if the contract description says otherwise.

Key takeaway: A reliable contract translation states the statuses, allowed routes, state order, time basis, and initial condition before writing the matrix. Check every off-diagonal rate, calculate each diagonal from the complete list of exits, and verify every row sum is zero.

Check Your Understanding

For each problem, write the states in the requested order before checking the proposed model.

A hypothetical contract has active, disabled, and dead states in that order. Active-to-disabled and active-to-dead intensities are \(0.07\) and \(0.01\); disabled-to-active and disabled-to-dead intensities are \(0.18\) and \(0.03\). Death is absorbing, and the insured is active initially. Give the intensity matrix and initial distribution. (Answer: \(Q=\begin{pmatrix}-0.08&0.07&0.01\\0.18&-0.21&0.03\\0&0&0\end{pmatrix}\), \(\boldsymbol{\pi}(0)=(1,0,0)\).)
With state order active, disabled, dead, is the proposed disabled row \((0.14,-0.20,0.06)\) valid? Explain by checking its destinations and row sum. (Answer: Yes. The off-diagonal entries are nonnegative, the diagonal is \(-(0.14+0.06)=-0.20\), and the row sum is zero.)
A long-term care model uses independent, home care, dead in that order. At time zero, probabilities are \(0.70\), \(0.20\), and \(0.10\), respectively. Is this a valid initial distribution? (Answer: Yes; all entries are nonnegative and \(0.70+0.20+0.10=1\).)
For states active, rehabilitation, long-term disability, dead, the active row is proposed as \((-0.10,0.08,0.01,0.01)\). The contract permits active-to-rehabilitation at \(0.08\), active-to-dead at \(0.01\), and no direct active-to-long-term-disability route. Identify the error and give the corrected row. (Answer: The proposed row incorrectly assigns \(0.01\) to active-to-long-term disability. The corrected row is \((-0.09,0.08,0,0.01)\).)
A model has two living states, independent and care, followed by dead. Its care row is \((0.09,-0.15,0.06)\), and its dead row is \((0,0,0)\). Under the stated order and row-current, column-destination convention, check whether these rows are valid. (Answer: Both rows are valid: \(0.09-0.15+0.06=0\), the care exits are nonnegative, and the dead row has no exits.)

What You'll Learn