Modeling%20Coherent%20Mortality%20Forecasts%20using%20the%20Framework%20of%20Lee-Carter%20Model

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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
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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
  

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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!)

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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.

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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 6599

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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 19502005.
• The log-likelihood, mean absolute percentage
  error (MAPE),
• and AIC (BIC) are used to evaluate the model
  fit.

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Parameter estimates of coherent LC and original
LC models (US Male 6599)
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Parameter estimates of coherent LC and original
LC models (US Female 6599)
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Empirical Results of Parameters

• For the cases of U.S. 6599 (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.

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Log-likelihood for Different Coherent Groups
Country Gender Coherent group All Ages 029 3064 6599
USA Male Country -10887.0 -2989.2 -3392.1 -3243.2
USA Male Gender -1210.03 -2292.5 -3476.7 -6019.0
USA Female Country -9251.7 -2102.4 -2632.7 -4015.5
USA Female Gender -12048.0 -1784.2 -3314.4 -6462.2
Canada Male Country -13027.0 -3526.5 -3157.6 -3825.2
Canada Male Gender -16122.0 -2940.8 -3096.1 -9123.2
Canada Female Country -9756.1 -2477.6 -2559.6 -4265.1
Canada Female 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
USA Male Coherent Gender Group 2.86
USA Male Single Population 1.90
USA Female Coherent Country Group 1.03
USA Female Coherent Gender Group 5.42
USA Female Single Population 0.89
Canada Male Coherent Country Group 2.86
Canada Male Coherent Gender Group 2.90
Canada Male Single Population 5.50
Canada Female Coherent Country Group 2.16
Canada Female Coherent Gender Group 1.36
Canada Female Single Population 3.69
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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?

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Thank you!!