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Modeling and Forecasting Mortality Rates Modeling and Forecasting Mortality Rates Modeling and Forecasting Mortality Rates Modeling and Forecasting Mortality Rates

Andrew Smeed and Marie-Claire Koissi | Department of Mathematics, University of Wisconsin - Eau Claire

A comparative Study with the A comparative Study with the A comparative Study with the A comparative Study with the LeeLeeLeeLee----Carter Carter Carter Carter Stochastic ModelStochastic ModelStochastic ModelStochastic Model

Past mortality models have failed to capture fluctuations and drasticdecline in human mortality rate. Thus, the human life expectancy wasunderestimated in general. This justified the need for more accuratemodels.

The stochastic model proposed by Lee and Carter (1992) seems to fitwell data from several countries. It is however still assessed andextensions are still proposed to improve the model’s efficiency.

This project examines the performance of the Lee-Carter model on datafrom selected countries in North America (Canada and USA) andEastern Europe. We use the Singular Value Decomposition (SVD) andthe Maximum Likelihood (MLE) methods.

The LC model seems to capture the change in mortality in Canada andUSA, but presents some insufficiencies for the selected easternEuropean countries.

Beta parameter

USA, Canada

INTRODUCTION

describes average shape of age profile

METHODOLOGY

With the constraints, we can derive the expression for age-dependent parameter ��

SINGULAR VALUEDECOMPOSITION (SVD)

MAIN REFERENCES

ACKNOWLEDGEMENTS

Andrew Smeed received financial support through UWEC Blugold scholarship, which is greatly

acknowledged. The authors also thank the Human Mortality Database for providing the data for this

study.

BACKGROUND

THEMODEL: LEE-CARTER

THEMAXIMUM LIKELIHOOD ESTIMATE (MLE)

RESULTS

DATA

The number of deaths for people age x in year t , ��,� , follows a Poisson distribution with parameter λ�,� =��,��,�

��,�~Poisson���,� �,��

� ��,� ���,� ���,���,�����,���,�!

where ��,� = death rates for age x at year t�,� = exposure to risk for age x at year t

Assuming independent observations, the Full log-likelihood is

���,� ln �,��,�

!� �,��,�

!�ln���,�!��,�

We maximize the full log-likelihood with respect to λ�,� and obtain

estimates �"�, #$�, and %&�

�"� � �1/)��ln ��,��

*+� ln ��,� ! �"� � ,-.-/-0 1 ,2.2/201. . . . . 1,4.4/40

#$� � .-∑ �.-�66

� 7�1: 9, 1: 1�:;��7-�

where U is the matrix with .6 as components

The parameters #�and %� are estimated using the SVD method

%&� � ,-��.-�66

/-0 � ,-:;� 7- /-0

<= >?,@ � A? 1 B? C@1 D?,@

The Lee-Carter model for mortality forecasting provides the central death rates >?,@ of people age x at time t as follows

describes pattern of deviation from age profile when the time index %� varies

describes variation in level of mortality with time t

D?,@ ~ N(0, σ2) (White Noise, error term)

Model Constraints (for uniqueness of solutions)

∑ %�� � 0 and ∑ #�� � 1

0 10 20 30 40 50 60 70 80 90 100-0.05

0

0.05

0.1

0.15

0.2

Ages

Beta

LC Beta Parameters

Bulgaria

Hungary

Lithuania

Russia

Zero line

Beta parameter

Bulgaria, Hungary, Lithuania and Russia,

• Aro and Pennanen. 2011, “A user-friendly approach to stochastic mortality modelling”. European Actuarial Journal, 1:151-167, 2011. ISSN 2190-9733.

• Lee, R. D., 2000. The Lee-Carter Method for Forecasting Mortality, with Various Extensions and Applications. North American Actuarial Journal, (4) 1, 80-91.

• Lee, R. D., Carter, L. R., 1992. Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association. (87) 419, 659-675.

• Koissi, M-C.; Shapiro, A.; Högnäs, G. 2006. Evaluating and Extending the Lee-Carter Model for Mortality Forecasting: Bootstrap Confidence Interval. Insurance: Mathematics and Economics, 38:1-20

Source: Human Mortality Database, www.mortality.org• Countries:

North America: US, Canada, Eastern Europe: Bulgaria, Hungary, Lithuania and Russia

• Ages: 0 to 100+, divided into 24 group (0,1], (1,5], (5,10], (10, 15]….• Years 1933-2010. 78 years total

MODELING

Time index parameter, kappa

USA, Canada

FORECASTINGˆ ˆAR IM A on { } forecasts { , 0}

t t sk k s

+⇒ >

1

, 1

ˆ ˆARIMA(0,1,0) :

ˆ ˆˆˆ log( ) ( )

t t t

x t x x t

k k c

m b k c

ε

α

+

+

= + +

⇒ = + +

Fitted and forecasted Time index parameter,

kappa, USA

Past mortality models underestimated• Decline in old-age mortality• Life expectancy (how long people were going to live on average).

Improvements in longevity during the 20th century• In developed countries, average life expectancy has increased by

1.2 months per year• Globally, life expectancy at birth has increased by 4.5 months per

year

Parties affected by longevity risks• Pension funds: Corporate or Government Sponsored• Annuity Providers: Insurance or Reinsurance companies• Individuals: outlive their resources and be forced to reduce their

living standards (or experience poverty at advanced ages)• Annuity providers and Pension funds may have inadequate

reserves.

Past mortality models

• De Moivre (1729):

• Gompertz (1825):

• Makeham (1860):

• Weibull (1951):

• Kannisto (1992):

• Heligman & Pollard (1980): (8 parameters)

1)( −−= xwxµ

xbx

x BCae ==µ

bx

x aec +=µ

b

x ax=µ

bx

bx

xae

aec

++=

1µ

xFxEBx

x

x HGeDAq

q C

++=−

−+2))/(log()(

1