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Comparison of different stochastic
mortality models
1
Professor David Blake
Director
Pensions Institute
Cass Business School
November 2013
Agenda� What needs to be modelled
� Main classes of stochastic mortality models
� Extrapolative models: Two general families
2
� Integrating the two general families of extrapolative models
� Conclusion
What needs to be modelled
3
Measures of mortality
4
Mortality rates at different ages
5
Mortality rates approx. log-linear at high ages
6
Main classes of stochastic mortality
models
7
Main classes of stochastic mortality models
� ‘Process-based’ models
�Model process of dying or improvement� e.g., RMS model
� ‘Causal’ or ‘explanatory’ models
�Model causes of death using exogenous
8
�Model causes of death using exogenous explanatory variables
� e.g. macro-economic variables or socio-economic indicators
� ‘Extrapolative’ projection models
� Purely data-driven
�Will only be reliable if the past trends continue:� medical advances can invalidate extrapolative projections by changing the trend
Process-based models� Cause-of-death models
� Total mortality rate is decomposed amongst a
number of diseases
�Models are fit and projected stochastically for
each underlying cause
Causes then re-aggregated to give a forecast for
9
�Causes then re-aggregated to give a forecast for
total mortality or life expectancy
�Highly subjective:
� large number of competing processes need to be
calibrated from sparse data
� decisions on the likely path of medical progress
need to be made.
Process-based models
� Cause-of-improvement models (e.g.,RMS)
�Mortality improvement attributed to (‘vitagion’)
categories of causes:
�Changing lifestyle trends
� Improvements in general health environment
10
� Improvements in general health environment
�Progress in medical intervention
�Regenerative medicine
�Retardation of ageing
Timeline into the future
11Source: RMS (2010) “Longevity Risk”
Exogenous causal models
� Mortality rates for different causes regressed on
different macro- economic variables such as
GDP growth, inflation and unemployment
�GDP growth directionally correlated with
mortality improvements
12
� Allowing for macro health indicators (especially,
smoking history) can account for most mortality
differences between men and women
� Post code as indicator of social class
�used in annuity pricing
Main extrapolative models� Lee-Carter class of models:
�Non-parametric or factorial models
�No functional form used for age effects
�No smoothness across ages or years
� P-spline class of models:
13
�Smoothness across years and ages
�Not common outside UK
� Cairns-Blake-Dowd (CBD) class of models:
�Parametric or formulaic models
�Specific functional form for each age effect
�Smoothness across ages in same year
Extrapolatative models:
Two general families
14
Two general families of extrapolative model
15
Lee-Carter class of models
16
Lee-Carter class of models
17
CBD class of models
18
Integrating the two general families of
extrapolative models
19
Integrating the two general families of extrapolative models
( ) (1) (2) (3)log , ( ) ( )x t t t t x
m t x x x x xα κ κ κ γ+
−= + + − + − +
( ) ( )( )2(1) (2) (3)α κ κ κ γ+ += + + − + − + − +
� Plat (2009):
� O’Hare and Li (2010):
20
( ) ( )( )2(1) (2) (3)log , ( ) ( ) ( )x t t t t x
m t x x x x x x xα κ κ κ γ+ +
−= + + − + − + − +
(1) (2) (3) (4)
centre young oldlogit ( , ) ( ) ( ) ( )x t t t t t x
q t x x x x x x xα κ κ κ κ γ+ +
−= + + − + − + − +
� Boerger at al. (2011):
A General Procedure for building stochastic mortality models
� Recently, there has been a proliferation of new mortality models
� Some of these models are “black-box algorithms” such as principal components analysis (PCA):� Involving terms that lack “demographic significance”
21
� Involving terms that lack “demographic significance”
� Others have added new functional terms which attempt to “fix” a problem with an existing model�Good, but has the appearance of being “ad hoc”
� Therefore need a “general procedure” which provides structure to the model building process
� This requires an explicit “toolkit” of functions
Model selection criteria� Adequacy:
� There should be a sufficient number of terms to capture all significant
structure in, and provide a good fit to the data
� Parsimony:
� Have the smallest number of terms and free parameters necessary
� trade off with the adequacy of the model
� Demographic significance:� Demographic significance:
� Models should be biologically reasonable
� Terms allow identification with underlying biological and socio-economic
processes occurring in the population
� Completeness:
� Models should span entire age range and not be limited to a subset of
ages by construction
� Models should include allowance for cohort effects and be able to
separate these from age/period terms
22
General model building procedure
23
Final UK model has αx, γt-x and 7
age-period terms
24
Improvements in goodness of fit at different stages of the GP
25
Heat map of residuals at stage 8
26
Conclusion
27
� Big explosion of research in last decade on
stochastic mortality models
� New models have tried to overcome the
weaknesses of existing models
� Too many models?
Conclusion
28
� Too many models?
� Now an attempt to reduce the number of models
by having a ‘general procedure’ for building a
model
� Going forward, the models will be country-
specific, but consistent with a commonly agreed
‘general procedure’
Thank you!
Longevity 10:
29
Longevity 10:
Tenth International Longevity Risk and
Capital Markets Solutions Conference
3-4 September 2014
Santiago, Chile
http://www.cass.city.ac.uk/longevity-10
References
� Lee, R. D., Carter, L. R., 1992. Modeling
and forecasting U.S. mortality. American
Statistical Association 87 (419), 659- 671
� Renshaw, A., Haberman, S., 2006. A
cohort-based extension to the Lee-Carter cohort-based extension to the Lee-Carter
model for mortality reduction factors.
Insurance: Mathematics and Economics 38
(3), 556-570.
30
References
� Cairns, A., Blake, D., Dowd, K., 2006. A
two-factor model for stochastic mortality with
parameter uncertainty: Theory and
calibration. Journal of Risk and Insurance
73 (4), 687-718.73 (4), 687-718.
� Cairns, A., Blake, D., Dowd, K., Coughlan,
G., Epstein, D., Ong, A., Balevich, I., 2009.
A quantitative comparison of stochastic
mortality models using data from England
and Wales and the United States. North
American Actuarial Journal 13 (1), 1-35.
31
References
� Cairns, A., Blake, D., Dowd, K., Coughlan,
G., Epstein, D., Khallaf-Allah, M. , 2011.
Mortality Density Forecasts: An Analysis of
Six Stochastic Mortality Models, Insurance:
Mathematics and Economics 48 (2011) Mathematics and Economics 48 (2011)
355–367
� Plat, R., 2009. On stochastic mortality
modeling. Insurance: Mathematics and
Economics 45 (3), 393-404.
32
References
� O’Hare, C., Li, Y., 2012. Explaining young
mortality. Insurance: Mathematics and
Economics 50 (1), 12–25.
� Boerger, M., Fleischer, D., Kuksin, N., Mar
2013. Modeling the Mortality Trend under 2013. Modeling the Mortality Trend under
Modern Solvency Regimes, U. Ulm
� Hunt, A., and Blake, D. (forthcoming). A
General Procedure for Building Mortality
Models, North American Actuarial Journal
(http://pensions-
institute.org/workingpapers/wp1301.pdf)33