7.3 Stochastic Mortality Models
7.3.1 Poisson and Binomial Models
“The assumption of binomial or Poisson type randomness is the basis of (most) grouped mortality analyses” (Alho (2005, p.33)). The GLM methodology we use for modelling mortality was set out in Chapter 4. In Section 4.2.4, a Poisson GLM for modelling Dx,t (where Dx,t is the number of deaths observed among a group of
lives aged x in year t) was proposed (see Equation (4.15)) in whichDx,t is assumed
to follow a Poisson distribution with mean −ln 1−qs x,t
Ec
x,teη (where Ex,tc is the
central exposed to risk for lives aged x in year t, treated as an offset, and η is the linear predictor), that is,
Dx,t ∼Pois −ln 1−qsx,t
Ecx,teη
. (7.50)
Appropriateness of the Poisson distribution for Dx,t is based on a number of
assumptions, including that the lives under observation all have the same probability of dying during the year of observation (that is, the lives are homogeneous) and that occurrences of deaths are mutually independent. The assumption of equal probabilities of death is made more tenable by dividing the lives under observation into groups possessing similar characteristics (such as sex and policy type) and allowing different mortality rates for each group. However, in spite of the efforts of insurers, it is still not always the case that the homogeneity assumption will hold,
Stochastic Sub-Models 131 in which case heterogeneity is said to be present in the data.
If all of the lives under consideration in a mortality investigation are truly ho- mogeneous, then a binomial or Poisson assumption may be seen to be reasonable. If, however, despite stratification in an attempt to achieve homogeneity, there re- mains heterogeneity between the lives within the sub-classes, then the underlying mean mortality rate is not constant for all of the lives under investigation, and this will lead to over-dispersion in both the binomial and the Poisson mortality models. Pollard (1970) demonstrated that this was true for the binomial model and a similar demonstration can be given to show that heterogeneity also leads to over-dispersion in the Poisson model.
It is common practice among actuaries to assume that different Life Insurance contracts are independent. This is done for simplicity. However, there are many cases where the independence assumption clearly does not hold. For example:
• Family members who hold policies in the same portfolio: family members are likely to be exposed to similar risks of death (for example, house fires, car accidents, contagious diseases, lifestyle factors etc). Furthermore, studies, such as that conducted by Jagger and Sutton (1991), have shown that there is an increased risk of mortality among people whose spouses have died in the period immediately following the death. Although spouses are rarely considered in the same model (males and females are usually considered in separate mortality models due to mortality differences between the sexes), it is not unreasonable to believe that there might be some instances where a parent and a child or two siblings of the same sex, living at the same address, both hold policies in the same portfolio.
• Policies written on the same life will be dependent.
• In areas with a high density of insured lives (for example, capital cities), catas- trophes (including epidemics) can lead to an accumulation of mortality claims for the insurer.
Besides these three, there may be many other causes of a lack of independence among lives.
The presence of any of the causes of mortality dependence mentioned is likely to give rise to over-dispersion. This is because all of the abovementioned scenarios imply a positive correlation between the deaths occurring in the respective scenarios. For the variance to decrease due to a mortality dependency relationship, it would require a negative correlation between the mortality rates of the lives involved. It is possible to theorise some situations where this might occur. For example, the accidental death of a factory worker at work may result in improved workplace safety,
132 Stochastic Sub-Models thus reducing the mortality rates of his or her co-workers. If any such situations do exist, then they will tend to reduce the variance, cancelling out some of the over-dispersion that may be present. Situations where under-dispersion occurs may exist, but are considered unlikely.
Consequently, there are several situations where the Poisson assumptions may be violated and which may give rise to over-dispersion. In such situations, it is inappropriate to use a Poisson error GLM to model Dx,t and an alternative model
that allows for over-dispersion should be used. Such models were discussed in Section 4.3.2.
Pollard (1970) and Daw (1974) both conducted analyses of insured life and popu- lation mortality data. Both authors found evidence of over-dispersion in their data. Based on this, there is reason to believe that over-dispersion may be present in the mortality data used in this thesis and it is necessary to test for it. We investigate this in the next section.