Mathematical and Statistical Methods for Actuarial Sciences and Finance

Lee-Carter error matrix simulation:

heteroschedasticity impact on actuarial valuations

Valeria D’Amato and Maria Russolillo

Abstract.Recently a number of approaches have been developed for forecasting mortality. In

this paper, we consider the Lee-Carter model and we investigate in particular the hypothesis

about the error structure implicitly assumed in the model specification, i.e., the errors are ho-

moschedastic. The homoschedasticity assumption is quite unrealistic, because of the observed

pattern of the mortality rates showing a different variability at old ages thanyounger ages.

Therefore, the opportunity to analyse the robustness of estimated parameter is emerging. To

this aim, we propose an experimental strategy in order to assess the robustness of the Lee-Carter

model by inducing the errors to satisfy the homoschedasticity hypothesis. Moreover, we apply

it to a matrix of Italian mortality rates. Finally, we highlight the results through an application

to a pension annuity portfolio.

Key words:Lee-Carter model, mortality forecasting, SVD

1 Introduction

The background of the research is based on the bilinear mortality forecasting methods.

These methods are taken intoaccount to describe the improvements in the mortality

trend and to project survival tables. We focus on the Lee-Carter (hereinafter LC)

method for modelling and forecasting mortality, described in Section 2. In particular,

we focus on a sensitivity issue of this model and in order to deal with it, in Section 3,

we illustrate the implementation of an experimental strategy to assess the robustness

of the LC model. In Section 4, we run the experiment and apply it to a matrix of Italian

mortality rates. The results are applied to a pension annuity portfolio in Section 5.

Finally, Section 6 concludes.

2 The Lee-Carter model: a sensitivity issue

The LC method is a powerful approach to mortality projections. The traditional LC

model analytical expression [7] is the following:

ln

(

Mx,t

)

=αx+βxκt+Ex,t, (1)

M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010