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

Let K_x be the curtate future lifetime of a life aged x, and T_x the complete future lifetime. Which statement about the relationship between them is correct?

The curtate future lifetime is the integer part (floor) of the complete future lifetime. With deaths uniformly spread within each year, the average fractional part is one half, so the curtate expectation is approximately the complete expectation minus one half.

  1. AK_x = ceiling(T_x), so e_x exceeds e°_x by about one half
  2. BK_x = floor(T_x), so e_x is approximately e°_x − 1/2 under uniform distribution of deathsCorrect
  3. CK_x = T_x always for integer ages
  4. DK_x = floor(T_x), so e_x is approximately e°_x + 1/2
  5. K_x = round(T_x), so e_x = e°_x exactly

Explanation

K_x is the integer part of T_x. Under uniform distribution of deaths within each year, the fractional part averages 1/2, so e°_x ≈ e_x + 1/2, i.e. e_x ≈ e°_x − 1/2. Options using ceiling or adding 1/2 reverse the direction.

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