IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models
For a continuous lifetime random variable T with density f(t) and survival function S(t), the force of mortality (hazard rate) at time t is defined as:
The force of mortality equals the density divided by the survival function, f(t)/S(t). It measures the instantaneous death rate among those still alive at time t, which is also the negative derivative of the log of S(t).
- Af(t) × S(t)
- Bf(t) / S(t)Correct
- Cf(t) / (1 − S(t))
- DS(t) / f(t)
- −f(t) × ln S(t)
Explanation
The hazard rate is the instantaneous rate of failure conditional on survival to t, so μ(t) = f(t)/S(t) = −d/dt ln S(t). Dividing by 1 − S(t) would condition on death having occurred by t, which is wrong.
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