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Citation: Butt, Z. and Haberman, S. (2010). A comparative study of parametric mortality
projection models (Actuarial Research Paper No. 196). London, UK: Faculty of Actuarial
Science & Insurance, City University London.
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City Research Online
Faculty of Actuarial
Science and Insurance
Actuarial Research Paper
No. 196
A Comparative Study of Parametric
Mortality Projection Models
Zoltan Butt
Steven Haberman
August 2010
Cass Business School
106 Bunhill Row
London EC1Y 8TZ
Tel +44 (0)20 7040 8470
ISBN 978-1-905752-29-4 www.cass.city.ac.uk
“Any opinions expressed in this paper are my/our own and not necessarily
those of my/our employer or anyone else I/we have discussed them with.
You must not copy this paper or quote it without my/our permission”.
1
A comparative study of parametric mortality projection models
Abstract
The relative merits of different parametric models for making life expectancy and annuity value
predictions at both pensioner and adult ages are investigated. This study builds on current published
research and considers recent model enhancements and the extent to which these enhancements address the
deficiencies that have been identified of some of the models. The England & Wales male mortality
experience is used to conduct detailed comparisons at pensioner ages, having first established a common
basis for comparison across all models. The model comparison is then extended to include the England &
Wales female experience and both the male and female USA mortality experiences over a wider age range,
encompassing also the working ages.
Key words and phrases: Mortality forecasting; binomial response models; age-period effects; age-period-
cohort effects; forecast statistics; model and forecast comparison; back-fitting
1. Introduction
In this paper, we contribute to the debate on the relative merits of various
extrapolation models used as a means of projecting future mortality rates. We focus, in
particular, on comparing the key indices of life expectancy and annuity value predictions,
as computed by the cohort method. In formulating our approach, we establish a common
basis for comparison across models, and this means that noteworthy differences in the
predicted indices of interest may be directly attributable to the choice of model predictor
structure.
The details of the models and methodology are set out systematically in Section 2,
which is supported by technical Appendices, A & B, for completeness. The models
include a group of 4 parametric predictor models based on, and including the Lee &
Carter (1992) bilinear structure, with the optional inclusion of a second pair of age-period
components and the capture of cohort effects; together with a further group of 8 linear
parametric predictors based on Cairns et al. (2009) and including extensions due to Plat
(2009).
A comparative study of the models, using the England & Wales 1961-2007 male
mortality experience, restricted to pensioner ages is reported in Section 3. The age
restriction is imposed in order to accommodate the models due to Cairns et al. (2009),
denoted by M5-M8, and which are designed for use at pensioner ages only. Results based
on the different stages of model building are set out in Section 2 and are presented
pictorially. Diagnostic checks on each model and the accompanying random walk period
index model are conducted by monitoring residual plots. Life expectancy and annuity
model predictions are examined for robustness by systematically truncating the time span
of the data at the two extremities, before repeated modelling.
In Section 4, the age restriction imposed in Section 3 is lifted and further
comparative studies are reported. These are again conducted following the different
stages set out in Section 2, using both the England & Wales and USA mortality
experiences for each gender and involving a wider age span that includes the working
ages as well as pensioner ages.
2
A detailed discussion of the issues arising is presented in Section 5, followed by a
summary in Section 6.
2. Methodology
2.1 Data array
We denote a rectangular mortality data array, partitioned into unit square cells of
size one year by
12 01
, , : age , ,... , period , ,...,
x
txt xt k n
de xxx x ttt t
where
xt
d
- reported number of deaths
xt
e
- matching initial exposures to the risk of death
xt
- 0/1 weights to indicate empty or omitted data cells
When initial exposures are required for analysis and only central exposures are available,
as in this paper, we approximate the initial exposures to the risk of death by adding half
the matching reported numbers of deaths to the central exposures (e.g. Section 2.2, Forfar
et al. 1988).
2.2 Model structures
We target and project the probability of death
xt
q
throughout. A common basis
for comparison across all models is established by using the log-odds function to link
xt
q
to the parametric predictor structure
xt
in all cases, so that, typically, for any model H
:log
1
xt
xt
xt
q
H
q
.
The log-odds function is also chosen because of the historical ties with the early actuarial
work of Perks (1932).
The following predictor structures are compared
:
xt x x t
LC
1
:
xt x x t t x
H
(0)
:
xt x x t x t x
M
(1)(1) (2)(2)
2:
xt x x t x t
LC
