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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models

Which recursive relationship between curtate expectations of life at successive ages is correct?

The correct recursion is e_x = p_x (1 + e_{x+1}). The life earns one complete year only if it survives the year, with probability p_x, and then has expected further curtate lifetime e_{x+1}. Without the factor p_x on the 1, survival credit would be wrongly given.

  1. Ae_x = p_x (1 + e_{x+1})Correct
  2. Be_x = q_x + p_x e_{x+1}
  3. Ce_x = p_x e_{x+1}
  4. De_x = 1 + p_x e_{x+1}
  5. e_x = p_x + e_{x+1}

Explanation

e_x is the sum over t≥1 of t p_x = p_x + p_x × sum over s≥1 of s p_{x+1} = p_x(1 + e_{x+1}). Option D wrongly credits the year of survival even if the life dies within the year. Option C omits the one year counted for surviving to x+1.

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