Multiple Decrement Models: Competing Risks for LTAM

Multiple Decrement Tables

A multiple decrement model tracks an individual subject to several causes of exit. Cause j has a dependent probability of decrement q_x^(j) and a force of decrement mu_x^(j). The total force is mu_x^(tau) = sum of mu_x^(j) over all causes. The probability of decrement from cause j in the multiple decrement environment differs from the single-decrement (associated) probability q_x'^(j) because other decrements compete. Converting between dependent and independent rates is essential for building multiple decrement tables from separate single-cause studies.

Conversions and Applications

Under the UDD assumption in the multiple decrement table, q_x^(j) = q_x'^(j) * (1 minus 0.5 * sum of other q_x'^(k)). With two decrements and UDD: q_x^(1) = q_x'^(1) * (1 minus 0.5 * q_x'^(2)). In actuarial applications, common decrements include death, withdrawal, disability, and retirement. The absolute rate of decrement (dependent probability) is used for expected cost calculations. Exam LTAM tests conversions, construction of multiple decrement tables, and their application in pricing and reserving.