Comparison of different stochastic

mortality models

1

Professor David Blake

Director

Pensions Institute

Cass Business School

d.blake@city.ac.uk

November 2013

Agenda� What needs to be modelled

� Main classes of stochastic mortality models

� Extrapolative models: Two general families

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� Integrating the two general families of extrapolative models

� Conclusion

What needs to be modelled

3

Measures of mortality

4

Mortality rates at different ages

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Mortality rates approx. log-linear at high ages

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Main classes of stochastic mortality

models

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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

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�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

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�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

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� 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

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� 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:

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�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

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Two general families of extrapolative model

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Lee-Carter class of models

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Lee-Carter class of models

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CBD class of models

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Integrating the two general families of

extrapolative models

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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):

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( ) ( )( )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”

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� 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

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General model building procedure

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Final UK model has αx, γt-x and 7

age-period terms

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Improvements in goodness of fit at different stages of the GP

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Heat map of residuals at stage 8

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Conclusion

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� 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

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� 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.

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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.

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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.

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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