Initial State Distributions at Policy Issue

Why the Starting State Matters

A long-term care model describes how a policyholder may move among health and care states over time. The first projection question is not only which transitions are possible, but also where the policyholder is at issue. A model for applicants who are all healthy at issue starts differently from one for a hypothetical group that includes people already receiving home care. Even if both groups use the same subsequent transition assumptions, their projected care-state probabilities need not be the same.

An initial-state distribution records the probabilities assigned to the possible states at the time the policy is issued. These probabilities can represent uncertainty about an individual’s starting state, or the composition of a group of policies. They must be based on the model’s state definitions and the contract’s issue conditions. For example, a state representing a person already in facility care should not receive positive issue probability if the contract is assumed to be issued only to people who are not receiving care.

Throughout the worked examples, the state order is \(A,H,F,D\): active and not receiving care; receiving home care; receiving facility care; and dead. Death is absorbing, so a person in state \(D\) remains there. The examples use hypothetical transition probabilities for one-year periods. A transition matrix is assumed to have nonnegative entries and each row must sum to one. Its row for a given state describes probabilities conditional on starting that period in that state.

From Issue Conditions to a Projection

Suppose the transition assumptions are represented by a one-year matrix \(P\). If the same matrix applies in each year, then the state-probability vector after \(n\) years is \(\boldsymbol{\alpha}P^n\). If transition assumptions depend on age or duration, use the appropriate matrix for each period in chronological order; for example, the two-year projection is \(\boldsymbol{\alpha}P_1P_2\). These formulas require that the matrix entries are valid transition probabilities for the specified period and that the state definitions remain consistent.

In a continuous-time model, the analogous transition matrix from issue to time \(t\) is written \(P(0,t)\). If the intensity matrix is constant, \(P(0,t)=\exp(Qt)\), where \(Q\) is the generator with nonnegative off-diagonal intensities and rows summing to zero. If intensities vary with age or duration, the transition matrix must reflect that variation over the interval. In either setting, the initial distribution enters on the left, as a row vector.

Why the Matrix Product Works

The row-vector formula follows from conditioning on the state at issue. Let \(I_j\) mean that the policyholder is in state \(j\) at issue, and let \(K_k(t)\) mean that the policyholder is in state \(k\) at time \(t\). For each issue state \(j\), \(P_{jk}(0,t)=P(K_k(t)\mid I_j)\). The events \(I_1,\ldots,I_m\) partition the possible issue states, so the law of total probability gives

The right-hand side is exactly component \(k\) of the row-vector product \(\boldsymbol{\alpha}P(0,t)\). Thus, each projected state probability is a weighted average of the conditional probabilities associated with the possible issue states. This also explains why a change in the initial distribution can change the projection even when the transition assumptions stay fixed.

Underwriting Groups and Alternative Starting Conditions

Underwriting can affect the model in two distinct ways. First, different groups may have different initial-state distributions. Second, the groups may have different transition assumptions after issue. These effects should not be conflated. If the groups share the same transition matrix, their projections differ only because they start with different state mixes. If they have group-specific transition matrices as well, each group must be projected using its own assumptions.

Known Issue State and Multiple Periods

Sometimes the issue state is known rather than uncertain. If all policies start in home care, the initial vector is \((0,1,0,0)\), not a population-wide average. Multiplication then selects the home-care row for the first period. For later periods, the resulting distribution must be multiplied by the next period’s transition matrix. This ordering matters: the first matrix represents the first interval after issue, and the second represents the following interval.

Checks and Interpretation

Before using an initial distribution in a valuation or projection, confirm that the issue-state probabilities represent the population actually being modeled. A group of newly issued policies may have a different starting mix from a group observed after a period of selection, and a model for in-force policies should not silently be given the issue distribution of new business. If the issue condition rules out a state, assign it probability zero unless the model explicitly includes a different interpretation.

Several checks help catch errors:

• The components of the initial vector are nonnegative and sum to one.
• The state order in the vector matches the order of rows and columns in every matrix.
• Every transition matrix has rows that sum to one, including the absorbing death row.
• The future-state vector is calculated by multiplying a row vector on the left.
• The projected components are nonnegative and sum to one, apart from any stated rounding.

A particularly common exam error is to use a conditional probability as if it were unconditional. For example, an entry in the active-state row is the probability of a destination state given active status at the start; it is not automatically the probability for a mixed issue group. The correction is to weight each applicable row by that state’s initial probability and add the results. Also check the time unit: a one-year transition matrix cannot be applied as though it described a month or a full five-year interval.