December 17, 2016

Actuarial computations involving life insurance generally require computing probabilistic trajectories involving only two possible states: at any moment in time the person is either alive or dead. Actuarial computations involving disability insurance, long-term care insurance, or other long-term insurance based on health require more complex modeling, however, in which the probabilistic trajectories involve more than two states. This Demonstration illustrates a three-state model in which the person is either healthy, ill, or dead. You set seven parameters as follows to create a set of transition rates between the three states. The parameters ... and ... determine the "force of mortality" as a function of age. The parameters ... and ... similarly determine the "force of illness" as a function of age. The parameters ... and ... determine the relationship between age and the likelihood that an ill person will transition back to health. The parameter ... adjusts the "as if" mortality of an ill person, making their mortality rate equal to what it would be at ... more years than their actual age. The Demonstration then numerically solves coupled differential equations to produce three views of the resulting probabilities that a person will be healthy, ill, or dead over ages 0 to 120. The "trajectories" view directly shows the probabilities. The "cumulative" view shows the probability (in blue) that the person will be healthy and the probability (in red) that the person will be ill. The sum of these two probabilities is the probability that the person will have survived. The "Δ trajectories" view shows the change in the probability of each state over time. The "Markov" view shows the values of the Markov transition matrix for a user-determined age of the person.