1 Modeling Coherent Mortality Forecasts using the Framework of Lee-Carter Model Presenter: Jack C. Yue /National Chengchi University, Taiwan Co-author:

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Modeling Coherent Mortality Forecasts using the Framework of Lee-Carter

Model

Presenter: Jack C. Yue /National Chengchi University, Taiwan

Co-author: Sharon S. Yang /National Central University Yi-Ping Chang /Soochow University

Yu-Yun Yeh/Cathay Life Insurance Company

Sept. 26, 2009

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Outline

Introduction The Coherent Mortality Modeling Analysis of the Coherent Mortality

Modeling Mortality Forecasts: Coherent Models vs.

Single Population Conclusions and Discussions

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Introduction

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Motivation Men and women in a country or people in nearby

countries share comparable living conditions and are likely to have similar mortality behaviors.

For example, the correlation coefficients of the life expectancy for the male and female between Canada and the U.S. are 0.9922 and is 0.9926, respectively.

The correlation between male and female for Canada and U.S. are 0.9701 and 0.9666.

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Life expectancy of male and female aged 65 for Canada and the US

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 200512

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14

15

16

17

18

19

20

21

Year

Life

Exp

ecta

ncy

USAFUSAMCanadaFCanadaM

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Coherent Mortality

Wilson (2001) pointed out a global convergence in mortality and mentioned that it is improper to prepare mortality forecasts for individual nation in isolation from one another.

Li and Lee (2005) mentioned that mortality patterns in closely related populations are likely to be similar in some respects and differences are unlikely to increase in the long run.

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The Coherent Mortality Modeling

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Lee-Carter Model Lee and Carter (1992) proposed the following

mortality model for U.S.,

where

Central Death Rate of age x, at time t

Intensity of Mortality at time t (linear!)

Average Mortality of age x

Tendency of Mortality change for age x

xttxxxtm )ln(

tx

x

xtm

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Empirical Evidence for LC Model LC Model provides fairly accuracy forecasts for

the countries such as the U.S. and Japan.

It works well for a single population, one sex or two sexes combined.

However, using the LC model to forecast two-sex mortality of a population has been a problem.

The values of can be very different for the male and female. (Divergence problem!)

x

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Converging Mortality Forecast The gaps of life expectancy between developed

and developing countries have been decreasing since the second half of the 20th century.

Li and Lee (2005) think that a long-term divergence in life expectancy is unlikely. They extended the LC model and proposed using the model for a group of populations with similar socioeconomic conditions.

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Coherent Mortality Model(Augmented Common Factor LC Method) Li and Lee (2005) modified the original LC

model to multiple populations, assuming that they have same and .

They suggest using the “Explanation Ratio” (similar to R2 in regression) to check if combining populations is appropriate, comparing to modeling the single population.

They found that the proposed model worked well if combining 15 low mortality countries as a group.

x t

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Our proposed Study

In this study, our goal is also to explore the coherent mortality model. In specific, our study has different focus:

Estimation method

Mortality of the elderly

Combining sexes or countries

Goodness-of-fit

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Our proposed Study(Estimation Method)

Unlike the ordinary least squares is used in Li and Lee (2005), we suggest using the maximum likelihood estimation (MLE), assuming that the numbers of deaths follow Poisson distributions.

Due to the nature of nonlinear optimization, the recursive procedure of Newton method is used. The initial values of the parameters are obtained from the singular value decomposition (SVD).

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Our proposed Study(Elderly Data) The patterns of mortality improvement are

changing over time and the different age groups have quite different experiences.

For example, the reduction rate of mortality for the younger group is decreasing, but that for the elderly is increasing.

We will focus on the data of ages 65~99

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Our proposed Study(Grouping and Goodness-of-fit) We want to know if male and female in a

country, or different countries in a region fits better as a group.

Li and Lee (2005) did not suggest any tests of goodness-of-fit. We want to know if there are any systematic errors which can help to improve the modeling. We shall look at the residuals of the coherent model.

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Empirical Analysis of the Coherent Model

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Data and Evaluation Methods

We use the data of Canada and U.S., in Human Mortality Database (HMD).

Five-year age groups, years 1950~2005. The log-likelihood, mean absolute percentage

error (MAPE),

and AIC (BIC) are used to evaluate the model fit.

1

ˆ1

100%n

i i

i i

Y YMAPE

n Y

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Parameter estimates of coherent LC and original LC models (US Male 65~99)

1 2 3 4 5 6 7-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

age-group

Alpha

Single PopulationCoherent Country GroupCoherent Gender Group

1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

age-group

Beta


1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005-20

-15

-10

-5

0

5

10

15

Year

Kappa


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Parameter estimates of coherent LC and original LC models (US Female 65~99)

1 2 3 4 5 6 7-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

age-group

Alpha


1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

age-group

Beta


1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005-20

-15

-10

-5

0

5

10

15

Year

Kappa


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Empirical Results of Parameters For the cases of U.S. 65~99 (years 1950-2000),

the estimated results of and from combining countries and those from combining genders look similar, unlike those from the single population.

It seems that the coherent model produce similar estimates, no matter if the group variable is sex or country.

The intercepts are almost identical.

x t

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Log-likelihood for Different Coherent Groups

Country Gender Coherent group All Ages 0~29 30~64 65~99

USA

MaleCountry -10887.0 -2989.2 -3392.1 -3243.2

Gender -1210.03 -2292.5 -3476.7 -6019.0

FemaleCountry -9251.7 -2102.4 -2632.7 -4015.5

Gender -12048.0 -1784.2 -3314.4 -6462.2

Canada

MaleCountry -13027.0 -3526.5 -3157.6 -3825.2

Gender -16122.0 -2940.8 -3096.1 -9123.2

FemaleCountry -9756.1 -2477.6 -2559.6 -4265.1

Gender -16678.0 -2121.8 -3223.8 -10071.8

Note: The numbers in red are preferred.

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Mortality Forecasts: Coherent Models vs. Single Population

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Forecasting Mortality

We use the prediction error to evaluate the coherent model and the single LC model.

The data of years 1950-2000 are used to reach parameter estimates, and the data of years 2001-2005 are treated as “unknown.” Also, we focus on the elderly, ages 65-99.

Unlike using the “R2” or log-likelihood, we can use the predicted MAPE to evaluate the models, without adjusting degree of freedom.

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Forecast MAPE for Higher Ages (65-99): Coherent Groups vs. Single Population Country Gender MAPE

USA

Male

Coherent Country Group 1.18

Coherent Gender Group 2.86

Single Population 1.90

Female




Canada

Male




Female




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Forecasting Mortality (conti.)

The predicted MAPEs are all very small, indicating the LC-type models are a good candidate to model the elderly mortality rates.

On average, combining countries has smaller predicted MAPE than combining gender, similar to the result in estimation.

Combining countries almost dominates the single population, except for the case of U.S. female.

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Forecasting Mortality (conti.)

In addition to the predicted MAPE, we also compare the variances of predicted mortality rates for the coherent model to those for the single population model.

As expected, since the coherent model use more samples (i.e., combining country or gender), the variances and the prediction intervals are smaller.

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95% Confidence Interval of Simulated Mortality (U.S. Male Aged 65). Coherent Country Model

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95% Confidence Interval of Simulated Mortality (U.S. Male Aged 65). Single Mortality Model

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95% Confidence Interval of Simulated Mortality (U.S. Male Aged 65, 75, 85).

Age 85

Age 75

Age 65

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95% Confidence Interval of Simulated Annuity Value (U.S. Male Aged 65) Coherent Country Model

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95% Confidence Interval of Simulated Annuity Value (U.S. Male Aged 65) Single Mortality Model

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Conclusion

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Conclusion

The coherent model is to treat populations with similar socioeconomic conditions as a group.

We found that the data of Canada and U.S. do support the coherent model, and it seems that treating Canada and U.S. as a group is more appropriate (than combining gender).

We provide an alternative approach in this study, in addition to the “R2” measure.

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Discussions How do we choose a group of populations

with similar conditions?

What measures can we use? (trial-and-error?)

Should we check if and are consistent? There are problems in goodness-of-fit.

Does it matter?

Should we select the age groups? Are the parameters fixed?

If not, what can we do?

x t

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Documents

1 Modeling Coherent Mortality Forecasts using the Framework of Lee-Carter Model Presenter: Jack C. Yue /National Chengchi University, Taiwan Co-author: