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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Projecting and valuing cashflows contingent on multiple decrement events

In a multiple decrement table for active employees with two causes of exit, death (d) and withdrawal (w), the force of decrement from cause j at age x is denoted mu_j(x) and the total force is mu(x) = mu_d(x) + mu_w(x). Which expression gives the dependent probability that a life aged x exits by cause w within the next t years?

The dependent probability of exit by withdrawal within t years is the integral over s from 0 to t of the all-cause survival probability to s multiplied by the withdrawal force at age x+s, because the life must survive every cause before leaving by withdrawal.

  1. AThe integral from 0 to t of s p_x^(τ) times mu_w(x+s) dsCorrect
  2. BThe integral from 0 to t of s p_x^(w) times mu_w(x+s) ds, ignoring other causes
  3. Ct p_x^(τ) times mu_w(x+t)
  4. D1 minus the exponential of minus the integral of mu_w(x+s) ds from 0 to t
  5. t p_x^(τ) multiplied by the integral of mu_w(x+s) ds from 0 to t

Explanation

The dependent probability of leaving by cause w in (0,t) requires survival in all causes to time s, then exit by w at s. This gives the integral of s p_x^(τ) mu_w(x+s) ds. Option 3 is the independent (single-decrement) rate, which ignores the other cause; option 2 is only a density at one time point.

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