Mortality and Morbidity Models: Laws, Selection and Multiple Decrements
Mortality laws and selection
This domain tests parametric mortality laws, selection and multi-state or multiple-decrement models. Gompertz law sets μ_x=Bc^x, giving exponentially increasing mortality. Makeham adds a constant: μ_x=A+Bc^x. Survival is t p_x=exp(-integral from 0 to t of μ_{x+s} ds). Selection means recently underwritten lives have lower mortality than the ultimate table: q_{[x]+r}<q_{x+r} during the select period. After the select period, select and ultimate rates converge.
Multiple decrements and states
A multiple-decrement table has several exits, such as death and withdrawal. Dependent probabilities sum to the total exit probability, while associated single-decrement probabilities assume only one cause operates. A multiple-state model allows movement between states, for example healthy, sick and dead, and requires transition intensities. Morbidity models value sickness benefits using probabilities of being in a state and the timing of benefit payments.
Worked example
Under Makeham with A=0.001, B=0.0001 and c=1.1, at age 60, μ_60=0.001+0.0001(1.1^60). Since 1.1^60 is about 304.48, μ_60=0.001+0.030448=0.031448. The one-year survival probability is approximately exp(-0.031448)=0.9690, so q_60 is about 0.0310. If a select table gives q_[60]=0.020, the select period is showing materially lighter mortality than the ultimate rate.
Traps
Treating μ_x as q_x; ignoring the select period; adding dependent decrement probabilities as if they were independent; and using a single-decrement table when causes compete. In multi-state models, a transition intensity is not the same as an annual transition probability.