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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Assurance and annuity functions involving two lives

Two lives (x) and (y) have independent future lifetimes. Let T_xy be the time until the first death and T_x̄ȳ the time until the last death. Which statement is correct?

The first-death time T_xy exceeds t only if both lives survive to t. With independent lives, the probability is the product t_p_x × t_p_y. The sum overcounts, and the last-survivor probability needs the inclusion-exclusion correction.

  1. AP(T_xy > t) = t_p_x + t_p_y
  2. BP(T_xy > t) = t_p_x × t_p_yCorrect
  3. CP(T_x̄ȳ > t) = t_p_x × t_p_y
  4. DP(T_x̄ȳ > t) = t_p_x + t_p_y
  5. P(T_xy > t) = 1 − t_p_x × t_p_y

Explanation

The joint-life status survives only while both lives survive, so with independence t_p_xy = t_p_x × t_p_y. The last-survivor status survives if at least one survives, so its survival probability is t_p_x + t_p_y − t_p_x t_p_y, not the sum or the product.

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