286 T.Z. Sithole et al. Insurance: Mathematics and Economics 27 2000 285–312
Renshaw et al. 1996 suggested a modelling structure in the framework of generalised linear models, which incorporates both the age variation in mortality and the underlying trends in the mortality rates. In this paper, we
use this modelling structure to investigate mortality trends for immediate annuitants and life office pensioners. We focus on the projected forces of mortality and mortality improvement factors derived from the appropriate model
for each experience.
Section 2 covers the current practice of the CMI in projecting mortality rates. In Section 3, the modelling structure suggested by Renshaw et al. 1996 is given. The application of the structure in modelling immediate annuitants’
and pensioners’ mortality experiences is discussed in Section 4, while in Section 5 we give a brief outline of a procedure for modelling select mortality and how this can be applied to the annuitants’ select data.
2. Current CMI practice in projecting mortality
The current practice of the CMI is to graduate the force of mortality at age x, µ
x
, by fitting the “Gompertz– Makeham” class of formulae:
µ
x
= GMr, s =
r−1
X
i=1
α
i
x
i
+ exp
s−1
X
j =0
β
j
x
j
, 2.1
with the convention that when r = 0, the polynomial term is absent, and when s = 0, the exponential term is absent Forfar et al., 1988.
Tables resulting from the graduation, referred to as base tables, are then projected by applying time reduction factors, RFx, t , for an ultimate life attaining exact age x at time t, where t is measured in years from an appropriate
origin. The projected mortality rate at time t is q
x,t
= q
x,0
RFx, t , 2.2
where q
x ,0
is the rate of mortality from the new base table for the appropriate experience. Section 4 of CMI Report 10 contains a full description of this method of projecting mortality rates.
The CMI have recently proposed a new mortality improvement model for pensioners and annuitants CMI Report 17, to be used with mortality tables based on the 1991–1994 mortality experiences. The form of the model assumes
that at each age, the limiting rate of mortality is non-zero and that the rate of mortality decreases to its limiting value by exponential decay. There is a further assumption that a given percentage of the total future decrease in mortality
will occur in the first 20 years, with the percentage varying by age.
The mortality improvement model adopted by the CMI is RFx, t = αx + [1 − αx][1 − f x]
t 20
, 2.3
where αx =
c,
x 60, 1 +
1 − cx − 110 50
, 60 ≤ x ≤ 110,
1, x 110,
f x =
p, x 60,
[110 − xp + x − 60q] 50
, 60 ≤ x ≤ 110,
q, x 110,
with c = 0.13, p = 0.55 and q = 0.29.
T.Z. Sithole et al. Insurance: Mathematics and Economics 27 2000 285–312 287
The new model is such that the rate of improvement in mortality is assumed to depend on both age and time for lives aged between 60 and 110 years only. At ages below 60 years, the rate of improvement is assumed to depend
only on time, while no improvement is assumed for lives aged 110 years and above. The same factors apply for all experiences, male and female, for data based on lives and amounts.
3. Modelling with respect to age and time
