Disability-State Transition Probabilities

From Transition Intensities to Future-State Probabilities

A disability model describes movement among states over time. For example, a person who is active may become disabled, recover to active status, or die; a disabled person may recover or die. The transition intensities describe instantaneous rates of movement, while a transition probability answers a different question: what is the probability of occupying each state after a specified period?

This distinction matters in valuation. A benefit may depend on whether a person is disabled at a future date, not merely on whether disability began at some point during the period. Transition probabilities account for the entire sequence of possible state changes before that date, subject to the model assumptions.

Model and Matrix Setup

Assume a time-homogeneous continuous-time Markov model: transition intensities are constant over the period, and conditional on the current state, the future state development does not depend on the earlier path. Allow transitions from Active to Disabled at intensity \(\alpha\), from Active to Dead at intensity \(\mu_A\), from Disabled to Active at intensity \(\rho\), and from Disabled to Dead at intensity \(\mu_D\). Death is absorbing, so there are no transitions out of Dead.

An intensity is a rate per unit time, not a probability over a whole year. For instance, an intensity of 0.2 per year does not mean that the one-year transition probability is exactly 0.2. Other transitions can occur during the year, and the person may change state more than once.

With states ordered A, D, X, the generator matrix \(Q\) records the transition intensities off the diagonal. Each diagonal entry is the negative sum of the other entries in its row. This makes the row sums of \(Q\) equal to zero.

For a time-homogeneous model, the transition probability matrix is the matrix exponential \(P(t)=\exp(Qt)\). It can be characterized by the Kolmogorov differential equations. With the row convention used here, the forward equation is \(P'(t)=P(t)Q\), and the initial condition is \(P(0)=I\), the identity matrix. These equations describe how the probabilities change as the time horizon increases.

If the initial state is known to be \(i\), the required probabilities are row \(i\) of \(P(t)\). If the initial state is uncertain, let \(\boldsymbol{\pi}(0)\) be a row vector of initial-state probabilities. Then the future distribution is \(\boldsymbol{\pi}(t)=\boldsymbol{\pi}(0)P(t)\). This multiplication averages the transition probabilities over the possible starting states.

A Useful Closed Form in a Symmetric Case

A closed form makes the matrix exponential especially transparent when the disability and recovery intensities are equal and the two mortality intensities are equal. Let \(\alpha=\rho=b\) and \(\mu_A=\mu_D=m\), where \(b\) and \(m\) are constant rates. The transient part of the generator, for A and D, is then \(\begin{pmatrix}-(b+m)&b\\b&-(b+m)\end{pmatrix}\).

The vectors \((1,1)\) and \((1,-1)\) are eigenvectors of this two-state transient matrix. Their respective eigenvalues are \(-m\) and \(-(m+2b)\). Therefore the corresponding components of the matrix exponential are \(e^{-mt}\) and \(e^{-(m+2b)t}\). Combining the components gives the probabilities of being in A or D, before adding the absorbing death state.

The derivation also gives useful checks. The probability of remaining alive from either transient starting state is \(g\), because the first two entries of either transient row sum to \(g\). The death probability is consequently \(1-g\). Each transient row sums to \(g+(1-g)=1\), while the Dead row is \((0,0,1)\).

Working With an Initial Distribution

A starting population may contain people in more than one state. The correct calculation is to multiply the initial distribution by \(P(t)\), not to select a single row unless everyone starts in the same state. For a distribution \((\pi_A,\pi_D,0)\), the first future-state probability is \(\pi_Ap_{AA}(t)+\pi_Dp_{DA}(t)\). This is a weighted sum of conditional probabilities.

Transition Over Multiple Periods

For a time-homogeneous model, transition matrices satisfy the Chapman–Kolmogorov relationship \(P(s+t)=P(s)P(t)\). To see why, condition on the state occupied at time \(s\). The probability of moving from \(i\) to \(j\) over \(s+t\) is the sum, over each possible intermediate state \(k\), of the probability of reaching \(k\) at time \(s\) multiplied by the probability of reaching \(j\) from \(k\) over the next \(t\) units. Matrix multiplication performs exactly this sum.

This identity allows a long-period calculation to be checked by splitting the period into shorter intervals. It does not mean that the state at the intermediate date can be ignored: every possible intermediate state contributes to the product.

Interpreting and Checking the Results

A transition probability \(p_{AD}(t)\) concerns the state at the end of the period. It includes paths that may involve multiple transitions, such as Active-to-Disabled-to-Active-to-Disabled, when those paths are permitted by the model. It is therefore different from a probability defined by a particular transition history, such as the probability of a first disability onset. State occupancy and the history of how that state was reached are not interchangeable.

For a three-state disability model, useful checks include:

• Every entry of \(P(t)\) must be between 0 and 1, inclusive.
• Each row must sum to 1, because the model includes all possible states at time \(t\).
• The Dead row must remain \((0,0,1)\), because death is absorbing.
• At \(t=0\), \(P(0)=I\): everyone is in their starting state with probability 1.
• For a specified initial distribution, \(\boldsymbol{\pi}(0)P(t)\) must also have nonnegative entries summing to 1.

These checks can expose errors in row and column orientation, omitted death probabilities, or incorrect use of intensities as finite-period probabilities. In applications, the resulting state probabilities can be combined with state-contingent amounts and payment timing to evaluate expected cash flows. The probability calculation itself should come first; payment amounts and discounting are separate valuation steps.