Mortality Rates Of Spanish Dependents: A Joint Correction Approach

The quantification of mortality, especially the one corresponding to the dependents insured, is nowadays a core problem when pricing long term insurance. The usual fit in Spain shows a big gap, even more important when adjusting older ages. What we show and present in the following pages is a joint correction of the extramortality rates departing from the well-known Rickayzen and Walsh reciprocal function of one plus an exponential. When comparing it wit the usual additive and multiplicative corrections, the joint one yields better results, fitting better to the real Spanih sutuation and thus should be choosen and applied by the insurance sector when working on the subject. INTRODUCTION A key element in the pricing of long term care insurance is the quantification of mortality, especially the one corresponding to the dependents insured. In the actuarial literature on mortality of the dependents insured, there is an unanimous opinion in regard to the mortality rates of dependents are different and higher than those of the overall mortality expressed in general mortality tables used by insurers for the valuation of normal hazards, and of course, appreciably higher than the mortality of insured autonomous. The following expression is therefore assumed: d a q q q x x x > > (1) A different question is, and resolved through different procedures too, the treatment given to the mortality rates when it comes to measure and to express in a mathematical way the present asseveration. Then, there is no single approach but, in contrast, different corrections can be found of the overall mortality for heads in situation of dependence. The first step in the delivery of adjusted mortality rates consists in having a mortality table of heads in a state of dependence, with gross values, whose definition corresponds to the contractually used in the policy as the one generating benefits in a long term care situation, and that employs equivalent criteria for obtaining the ranking of the long term care situation to those used in the derivation of the incidence rates. The starting raw data come from French survey HID 98-01 . Differences in the mortality of dependents, according to its level, induce not just higher and different overmortality values but also the functional expressions in which these values are modelled, so it’s usually associated additive adjustments to the great dependencies and multiplicative to the less severe ones, as can be seen in (Gatenby 1991). Once raw mortality of dependents data are derived, different formulas are proposed for correcting these mortality rates; the chosen procedure will minimize the squared deviations function. Base table for correction The mortality of dependents is usually derived from overall mortality statistics or mortality of healthy heads. In this case annuity tables are chosen because in this business insurer must also face the survival risk, as (Pitacco 2002) evidences. In the present paper we will fit French statistics, HID 98-01, corrected for Spanish population, using as base statistics the PERMF-00P tables. Adjustment of mortality rates There are different ways to adjust the mortality rate of the insured in a long-term care situation, all of them using as a starting point the following expression due to (Ainslie 2000). ( ) ( ) d x x q q x x δ β α + = × + (2)