Abstract
We introduce a model for the mortality rates of multiple populations. To build the proposed model we investigate to what extent a common age effect can be found among the mortality experiences of several countries and use a common principal component analysis to estimate a common age effect in an age–period model for multiple populations. The fit of the proposed model is then compared to age–period models fitted to each country individually, and to the fit of the model proposed by Li and Lee (2005).Although we do not consider stochastic mortality projections in this paper, we argue that the proposed common age effect model can be extended to a stochastic mortality model for multiple populations, which allows to generate mortality scenarios simultaneously for all considered populations. This is particularly relevant when mortality derivatives are used to hedge the longevity risk in an annuity portfolio as this often means that the underlying population for the derivatives is not the same as the population in the annuity portfolio.
Highlights
Mi mi = βi Li Ui Ui Li βi = βi Li Li βi = βi Λi βi with Λ = LL The eigenvalues of mi mi are on the diagonal of the matrix Λi, and the first age effect β1i is the eigenvector corresponding to the largest eigenvalue of mi mi
We call the model in (2) common age effect model of order p. Note that in this model the period effects κci are still depending on the specific population
Our aim is to find an orthogonal matrix β and diagonal matrices Λi such that Qi := mi mi = βΛi β ∀ i = 1, . . . , I. This is equivalent to finding an orthogonal matrix β such that β Qi β = Λi is a diagonal matrix for all i = 1, . . . , I
Summary
TT We use data from the Human Mortality Database Average log mortality rate for a life aged x in population i: 1T mi (x) = T mi (x, tk ). Mi mi = βi Li Ui Ui Li βi = βi Li Li βi = βi Λi βi with Λ = LL The eigenvalues of mi mi are on the diagonal of the matrix Λi , and the first age effect β1i is the eigenvector corresponding to the largest eigenvalue of mi mi. We call the model in (2) common age effect model of order p. Note that in this model the period effects κci are still depending on the specific population
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