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

In standard survival-model notation, for a life aged x with future lifetime random variable T_x, which expression correctly gives the force of mortality mu_x in terms of the survival function S_x(t) = P(T_x > t)?

The force of mortality equals minus the derivative of the log of the survival function at t = 0. It is the instantaneous rate of death at age x, f divided by S, so the sign must be negative because S is decreasing.

  1. Amu_x = -d/dt ln S_x(t) evaluated at t = 0Correct
  2. Bmu_x = d/dt S_x(t) evaluated at t = 0
  3. Cmu_x = ln S_x(1)
  4. Dmu_x = 1 - S_x(1)
  5. mu_x = S_x(t) / (1 - S_x(t)) evaluated at t = 0

Explanation

The force of mortality is f_x(t)/S_x(t) at t = 0, which equals minus the derivative of ln S_x(t) at t = 0. Option B omits the division by S_x(0) = 1 only in the sign: the derivative of S is negative, so mu_x would be negative. The others describe probabilities or odds, not an instantaneous rate.

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