A note on stochastic survival probabilities and their calibration

In this note we use doubly stochastic processes (or Cox processes) in order to model the evolution of the stochastic force of mortality of an individual aged x. These processes have been widely used in the credit risk literature in modelling the default arrival, and in this context have proved to be quite flexible and useful. We investigate the applicability of these processes in describing the individual's mortality, and provide a calibration to the Italian case. Results from the calibration are twofold. Firstly, the stochastic intensities seem to better capture the development of medicine and long term care which is under our daily observation. Secondly, when pricing insurance products such as life annuities, we observe a remarkable premium increase, although the expected residual lifetime is essentially unchanged.

Modelling stochastic mortality for dependent lives

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to rep- resent mortality risk. This paper represents a .rst attempt to model the mortality risk of couples of individuals, according to the stochastic inten- sity approach. We extend to couples the Cox processes set up, namely the idea that mortality is driven by a jump process whose intensity is itself a stochastic process, proper of a particular generation within each gen- der. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. We also provide a methodology for fitting the joint survival function by working separately on the (analytical) copula and the (analytical) mar- gins. First, we calibrate and select the best fit copula according to the methodology of Wang and Wells (2000b) for censored data. Then, we provide a sample-based calibration for the intensity, using a time- homogeneous, non mean-reverting, affine process: this gives the marginal survival functions. By coupling the best fit copula with the calibrated mar- gins we obtain a joint survival function which incorporates the stochastic nature of mortality improvements. Several measures of time dependent association can be computed out of it. We apply the methodology to a well known insurance dataset, using a sample generation. The best fit copula turns out to be a Nelsen one, which implies not only positive dependency, but dependency increasing with age.stochastic mortality, bivariate mortality, copula functions, longevity risk.

A nonparametric visual test of mixed hazard models

We consider mixed hazard models and introduce a new visual inspection technique capable of detecting the credibility of our model assumptions. Our technique is based on a transformed data approach, where the density of the transformed data should be close to the uniform distribution when our model assumptions are correct. To estimate the density on the transformed axis we take advantage of a recently defined local linear density estimator based on filtered data. We apply the method to national mortality data and show that it is capable of detecting signs of heterogeneity even in small data sets with substantial variability in observed death rates

Archimedean copulas derived from utility functions

The inverse of the (additive) generator of an Archimedean copula is a strictly decreasing and convex function, while utility functions (applying to risk averse decision makers) are nondecreasing and concave. This provides a basis for deriving an inverse generator of an Archimedean copula from a utility function. If we derive the inverse of the generator from the utility function, there is a link between the magnitude of measures of risk attitude (like the very common Arrow-Pratt coefficient of absolute risk aversion) and the strength of dependence featured by the corresponding Archimedean copula. Some new copula families are derived, and their properties are discussed. A numerical example about modelling dependence of coupled lives is included

Modelling stochastic mortality for dependent lives

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining an increasing reputation as a way to represent mortality risk. This paper is a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. We also provide a methodology for fitting the joint survival function by working separately on the (analytical) marginals and on the (analytical) copula. First, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the marginal survival functions. Then we calibrate and select the best fit copula according to the Wang and Wells [Wang, W., Wells, M.T., 2000b. Model selection and semiparametric inference for bivariate failure-time data. J. Amer. Statis. Assoc. 95, 62–72] methodology for censored data. By coupling the calibrated marginals with the best fit copula, we obtain a joint survival function, which incorporates the stochastic nature of mortality improvements. We apply the methodology to a well known insurance data set, using a sample generation. The best fit copula turns out to be one listed in [Nelsen, R.B., 2006. An Introduction to Copulas, Second ed. In: Springer Series], which implies not only positive dependence, but dependence increasing with age

Modelling stochastic bivariate mortality

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to represent mortality risk. This paper represents a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. On the theoretical side, we extend to couples the Cox processes set up, i.e. the idea that mortality is driven by a jump process whose intensity is itself a stochastic process, proper of a particular generation within each gender. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. On the calibration side, we fit the joint survival function by calibrating separately the (analytical) copula and the (analytical) margins. First, we select the best fit copula according to the methodology of Wang and Wells (2000) for censored data. Then, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the analytical marginal survival functions. Coupling the best fit copula with the calibrated margins we obtain, on a sample generation, a joint survival function which incorporates the stochastic nature of mortality improvements and is far from representing independency.On the contrary, since the best fit copula turns out to be a Nelsen one, dependency is increasing with age and long-term dependence exists