IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Concepts of survival models
Under a survival model with force of mortality μ(x) = B·c^x (Gompertz law, B > 0, c > 1), which statement about the survival function is correct?
Integrating the Gompertz hazard B·c^t from 0 to x gives B(c^x - 1)/ln c, so the survival function is exp[-B(c^x - 1)/ln c]. The ln c must divide, not multiply, because the integral of c^t is c^t/ln c.
- AS(x) = exp[-B(c^x - 1)/ln c]Correct
- BS(x) = exp[-B·c^x]
- CS(x) = exp[-B(c^x - 1)·ln c]
- DS(x) = (c^x - 1)/(B·ln c)
- S(x) = exp[-B·x·ln c]
Explanation
S(x) = exp(-∫0^x B c^t dt). The integral of c^t is (c^x - 1)/ln c, so S(x) = exp[-B(c^x - 1)/ln c]. Option 3 multiplies by ln c instead of dividing, which is the integration error.
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