Disability Income Insurance and Multi-State Models

Disability Income (DI) Model

Disability income insurance replaces a portion of income during periods of disability. The standard three-state model has states: active (healthy), disabled, and dead. Transition intensities include: sigma_x (active to disabled), rho_x (disabled to active/recovered), mu_x^a (active to dead), and mu_x^d (disabled to dead). Benefits are paid as an annuity while in the disabled state. The elimination period (waiting period) is the duration of continuous disability before benefits begin, typically 30, 60, 90, or 180 days.

Premium and Reserve Calculation

The net annual premium P satisfies P * a-bar_x^{00} = b * a-bar_x^{01}, where a-bar_x^{00} is the annuity value while healthy and a-bar_x^{01} is the expected present value of benefits while disabled, both computed from the transition probabilities. Reserves depend on the current state: the active reserve is the prospective value given the insured is healthy, and the disabled reserve (typically larger) applies when the insured is currently disabled. Exam LTAM tests the three-state disability model, including Thiele's equations for each state and reserve calculations at various durations.