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.
The Demonstration assumes that transitions between the healthy and ill states, the healthy and dead states, and the ill and dead states are all determined by modified Gompertz models. A Gompertz model is one in which the fraction of people changing their status (the logarithmic derivative of status) is a weighted exponential function of time (), where is the weighting parameter and determines the pace of exponential change. Although widely used in actuarial science and generally providing a plausible parameterized description of human mortality, the Gompertz model cannot be correct at high age values. This theoretical problem exists because when exceeds a critical value (), the function value—the fraction of people changing their status—is greater than one. To prevent this impossibility from occurring as the user changes the parameters, this Demonstration slightly modifies the Gompertz formula. The logarithmic derivative is set equal to an age-based weighted average of the Gompertz function and a constant value. The constant value is set to the value the Gompertz function attains when it is 99% of the critical value. The weighting is the value for a given age of a cumulative logistic distribution centered at 98% of the critical value and with a standard deviation equal to 6.3% of that critical value.
The default values for the Gompertz mortality parameters are the best fit (using Mathematica's built-in function NonlinearModelFit) for data from the 2001 CSO Mortality Tables.
The net premium to break even on a policy that pays in the event of illness and that does not require payment of a premium in the event the person is ill should equilibrate (a) the present value of the probability-weighted state of being ill multiplied by the payment in the event the person is ill; and (b) the present value of premiums multiplied by the probability that the person is healthy. Computation of actuarially fair premiums for "real" illness-based insurance policies is considerably more complicated since such policies generally have deductibles based on the number of days of prior illness and often have limits based on the total number of days of illness.