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Different shades of risk: mortality trends implied by term insurance prices. (English) Zbl 1467.91140

The force of mortality is modelled as \[ \mu_t(x_0) = \mu(x_0+t,Y_t,\Gamma_t) = e^{b(x_0+t)} Y_t + \Gamma_t + D_t(x_0)\;, \] where \(x_0\) is the age at time zero, the catastrophe component \(\Gamma_t\) is a shot noise process, the baseline process \(Y_t\) is a Cox–Ingersoll–Ross type model and \(D_t\) is some temporary component. The probability to survive for an \(x_0+t\) year old at time \(t\) until time \(T\) is then \[ E\Bigl[\exp\Bigl\{-\int_t^T \mu(x_0+s,Y_s,\Gamma_s) \;d s\Bigr\} \Bigm| \mathcal{G}_t, \tau_{x_0} > t\Bigr]\;, \] where \(\tau_{x_0}\) is the time of death and \(\mathcal{G}_t\) is the information available at time \(t\). The parameters of five versions of the model are estimated with the generalised method of moments from market prices of insurance contracts and the company effects are determined. It seems that the catastrophe effect or mortality trends are not significant. The base risk seems to dominate.

MSC:

91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics
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