forecasting mortality rates using the lee-carter model

FORECASTING PHILIPPINE MORTALITYRATES USING THE LEE-CARTER MODEL

CIOLO MIGUEL C. CALMAMONICA E. REVADULLA

A Special Problem Presented to theFaculty of the Mathematics Division

Institute of Mathematical Sciences and PhysicsCollege of Arts and Sciences

University of the Philippines Los Banos

In Partial Fulfillment of the Requirementsfor the Degree of

B.S. Applied MathematicsActuarial Science Option

April 2014

The Faculty of the Mathematics DivisionInstitute of Mathematical Sciences and Physics

College of Arts and SciencesUniversity of the Philippines Los Banos

hereby accepts the Special Problem entitled

FORECASTING PHILIPPINE MORTALITY RATESUSING THE LEE-CARTER MODEL

by

CIOLO MIGUEL C. CALMAMONICA E. REVADULLA

In Partial Fulfillment of the Requirementsfor the Degree of

B.S. Applied MathematicsActuarial Science Option

Prof. Jonathan B. MamplataSpecial Problem Adviser

Date

Prof. Jonathan B. MamplataHead, Mathematics Division

Date

Dr. Chrysline Margus N. PinolDirector, Institute of Mathematical Sciences and Physics

Date

ii

ABSTRACT

FORECASTING PHILIPPINE MORTALITY RATES USING THELEE-CARTER MODEL

Ciolo Miguel C. CalmaMonica E. RevadullaUP Los Banos, April, 2014

Adviser:Prof. Jonathan B. Mamplata

This study described the application of the Lee-Carter model to age-specific deathrates by gender in the Philippines. These rates are available for the period that goesfrom 1960 to 2009. The mortality index, average age-specific mortality, the deviationin mortality for each age interval and gender, and the shape and sensitivity coefficientsfor sixteen age groups are obtained through the Lee-Carter method. The forecastedmortality rates for all genders from 2014 to 2018 were calculated using Holts LinearExponential Smoothing Method. Finally, the conversion from death rates to deathprobabilities, the projection of life expectancies and the construction of the life tableswas formulated through Chiangs formula for abridged life tables. As calculated, thelife expectancy for individuals ages 70 and above for all genders is 10, and the lifeexpectancy for infants for all years and gender range from 65-70. The death ratefor infants for the male range from 8-9, for female, 11-15, and 9-13 for the wholepopulation.

iii

Acknowledgements

This paper will not be possible without the help of the these special people:

To my Parents and siblings, Myriam Cases Calma and Virgilio Yonzon Calma, Manang Toyette,Manong Dave and Pipi, thank you for always giving me your full support and trust to the times that I am notin your sight. Giving your full trust and hope to me are the most uncountable blessing that you can ever give to meand I will always cherish that.

To Professor Jonathan B. Mamplata, for being meticulous, opinionated, challenging and supportive specialproblem adviser. Thank you for painstakingly scrutinizing our special problem from Day 1 up to the very end. Thankyou also for being a good friend to me. Always have the passion to nurture,and instill your knowledge to the futureActuarial Science majors in UPLB.

To all of my Actuarial Science professors, Prof. Crisanto A. Dorado, Prof. Lester Charles A. Umaliand soon-to-be Prof. Jeric S. Alcala, for being the first stepping stone of my Actuarial Science career. Thankyou also for your patience and the passion in teaching the best subjects ever! Also, thank you for being cool and notboring teachers.

To the Society of Applied Mathematics of UPLB (SAM-UP) and Rocesians United in UP (ROUND-UP),for enhancing my social skills at its best and for being a family away from home. I will always treasure and willremember all the fun times and the life lessons that you have given to me. To my Batchmates Petere Excellentiae,for the fun and crazy times that we had especially during our aplikante days. It may be hard for us to be at thesame place at the same time but youll always be PeTex in my heart. Let us always AIM FOR EXCELLENCE!

To Pogi Productions staff, for giving me the chance to explore more options which is ACTING but later ondid not worked for me. Also, for the chance to be more creative and making a group presentation more colorful andfun but still academic. It is with full hope that this production will soon be a hit to all UPLB constitutents.

To SAM-UPs Executive Committee A.Y. 2014-2015 : Erika, Adrian, Jomi, Roel, Arden, Joanne,Gideon,and Nigel, for allowing me to allot time for my Special Problem during our convention. You guys provedto me that diversity is never a hindrance for unity. Thank you for giving me the chance to share with you yourwillingness to serve our beloved organization.

To Miss Abigail Tayas and Miss Krizza Calingasan, for their never ending hospitality everytime I get tostay to their apartment. Thank you!

To my homies Aldous, Jboy, Kristel,Sir Athan and Jet, for being the coolest housemate I have ever had.I will always miss the crazy nights, the spicy caldereta, poker, gossips, and dramas that we all shared in our veryspecial apartment.To my roomate, Jerson, for being a cool and good listener and a brother to me. I will be foreverbe thankful to that. Always remember the bro code: Bros before hoes!

To my SP partner, the gorgeous Monica E. Revadulla for doing the major works in our paper and also forbeing understanding friend to me.Our crazy, sleepless and stressful days and nights are over. I will always be hereto support your passion to teach as your way of serving the people.

And of course to our Great God Almighty, for always hearing my prayers to give me the wisdom, knowledgeand the courage in every journeys of my life. Amen!

Ciolo Miguel Cases Calma

iv

This paper would not have been possible without the help of the individuals listed below:

To my father, Benny, thank you for driving me all the way to the National StatisticsOffice in Sta. Mesa, Manila to get the official data needed to start the study. Spending fourlong boring hours in a fast food chain while waiting me finish scribbling all the data is greatlyappreciated. To my mother, Cristy, thank you for being my constant encouragement.

To one of the creators of the Lee-Carter Model, Mr. Ronald Lee, thank you forquickly responding to my e-mails and patiently answering all questions about the study.Your intelligent answers to the seemingly simple questions truly kept the study in the righttrack. Thank you for creating such a well-devised model which would surely be useful indemography.

To our adviser, Mr. Jonathan Mamplata, thank you for the patience you showedthroughout the checking of the paper. I learned a lot from your guidance on the flow of thestudy and the correctness of the construction of the paper itself.

To my partner in this study, Ciolo, thank you for giving me a head start on what weare supposed to do: it would have taken me more than a month to think of a topic. Thankyou for staying strong through my nagging and panic-stricken days. Your perseverance inlearning Latex is greatly appreciated. Flying solo, I think it would take me a year to finishthe manuscript just because of the program. Also, thumbs up to your presentation at theMSP Calabarzon Event.

Last, but the very best, to my partner in every aspect of life, Daphne, thank you forbuying me my favorite dinner while I was cramming the first draft of the study: your concernand care helped me go through the night. Also, staying with me in my boiling apartment asI scratched my head throughout Kelangan ko mag-SP nights is greatly appreciated. I loveyou.

Monica Encinas Revadulla

v

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 Introduction 1

1.1 Background of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Objective of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Theoretical Framework 5

2.1 Age-Specific Death Rates . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Lee-Carter Model . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Second Estimation of the parameter k . . . . . . . . . . . . . . . . . 10

2.5 Holts Linear Exponential Smoothing Method (LES) . . . . . . . . . 11

2.6 Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6.1 The Construction of a Life Table . . . . . . . . . . . . . . . . 14

2.6.2 Statistical Inference on Life Tables . . . . . . . . . . . . . . . 16

vi

3 Solving for the Parameters of the Lee-Carter Model 19

3.1 Solving for the parameters of the Lee-Carter model . . . . . . . . . . 19

3.1.1 Estimation of the k parameter . . . . . . . . . . . . . . . . . . 27

4 Holts Linear Exponential Smoothing Method (LES) 30

4.1 Forecasting mortality rates . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Computing new age-specific death rates . . . . . . . . . . . . . . . . . 31

5 Life Tables 33

5.1 Constructing the life table . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Standard error and confidence intervals . . . . . . . . . . . . . . . . . 35

6 Summary and Conclusions 36

7 Recommendation 39

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

vii

List of Tables

3.1 Death rates for male and its corresponding natural logarithm for 1960 20

3.2 Natural logarithm of death rates for male from 1960-1970 . . . . . . . 21

3.3 Parameter ax for male . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 The centralized death rates for the Male population from 1960 - 1970 22

3.5 the diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 The parameter bx for the male population . . . . . . . . . . . . . . . 25

3.7 The ax and bx parameters fore male, female and total population inthe Philippines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 The forecasted parameter k for male, female and total population inPhilippines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 The Mean Squared Error, ,and for male, female and total Population 31

4.3 The new age-specific death rates of male for 2014 up to 2018 . . . . . 32

5.1 Life table for the year 2014 for the male population . . . . . . . . . . 34

5.2 Statistical Inference of forecasted rates for the year 2014 of the malepopulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.1 The age-specific male population from 1960 - 1975 . . . . . . . . . . . 42

7.2 The age-specific male population from 1976 - 1990 . . . . . . . . . . . 42

7.3 The age-specific male deaths from 1960 - 1975 . . . . . . . . . . . . . 43


viii



7.7 The age-specific female deaths from 1960 - 1975 . . . . . . . . . . . . 44




7.11 The age-specific total deaths from 1960 - 1975 . . . . . . . . . . . . . 46




7.15 The age-specific male death rates from 1960 - 1975 . . . . . . . . . . 48

7.16 The age-specific male death rates from 1976 - 1990 . . . . . . . . . . 48

7.17 The age-specific death rates from 1991 - 2005 . . . . . . . . . . . . . 49

7.18 The age-specific death rates from 2006 - 2009 . . . . . . . . . . . . . 49

7.19 The age-specific female death rates from 1960 - 1975 . . . . . . . . . 49




7.23 The age-specific total death rates from 1960 - 1975 . . . . . . . . . . 51




7.27 The age-specific male logarithm of death rates from 1960 - 1975 . . . 53

ix




7.31 Life table for the year 2015 of the male population . . . . . . . . . . . 55




7.35 Life table for the year 2014 of the female population . . . . . . . . . . 57





7.40 Life table for the year 2014 of the total population . . . . . . . . . . . 60









x

7.49 Statistical Inference of forecasted rates for the year 2014 of the femalepopulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64





7.54 Statistical Inference of forecasted rates for the year 2014 of the totalpopulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66





xi

List of Figures

3.1 The first element of the 50 50 orthogonal matrix U . . . . . . . . . 23

3.2 The first elements of the 16 x 16 matrix V T . . . . . . . . . . . . . 23

3.3 ax parameter for male, female and the total population in the Philippines 26

3.4 bx parameter for men, women and the total population in the Philippines 27

3.5 First estimation of the parameter k for male, female and total population 28

3.6 Second estimation of the parameter k for male, female and total pop-ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xii

Chapter 1

Introduction

In this Chapter, an overview of this Special Problem is presented. A brief

introduction to the concepts regarding mortality rates and life expectancy are dis-

cussed.Also, presented in this chapter are the objectives of the study, the statement

of the problem, the scopes and limitations and the review of related literature.

1.1 Background of the Study

Life expectancy was slightly increased during the transition between the 20th

and the 21st century. According to the World Health Organization (WHO), life

expectancy in the Philippines increased from 63 to 66 from 1990 to 2011 [1].

In the fields of Demography and Actuarial Science, there have been many

attempts to find an appropriate model that represents mortality. Traditionally, a

1

parametric curve, like the ones suggested by De Moivre, Gompertz and Weibull, was

used to fit annual death rates [7].

Over the past years, a great number of new approaches were developed in order

to forecast mortality by using stochastic models. The Lee-Carter model became one

of the most well-known models and it is applied in different countries around the

world to forecast age specific death rates [2]. The method proposed by Lee and

Carter (1992) has become the leading statistical model of mortality forecasting in

the demographic literature. It was used as a benchmark for recent Census Bureau

population forecasts of the United States [14].

Lee and Carter developed their approach specifically for U.S. mortality data,

1933-1987. However, the method is now being applied to all-cause and cause-specific

mortality data from many countries and time periods, all well beyond the application

for which it was designed [14].

1.2 Statement of the Problem

The Lee-Carter model is a numerical algorithm used in mortality forecasting

and life expectancy forecasting. This study focused on the application of the Lee-

Carter model in forecasting and estimating the mortality rates in the Philippines.

This study would also construct life tables and compute life expectancies using the

2

forecasted mortality rates.

1.3 Significance of the Study

Mortality data are essential in the measurement of disease and consequently

health in the planning of public health care. Studying trends in mortality over time

helps to understand how the health status of the population is changing and assists

in the evaluation of the health system.

Mortality data also provide a basis for investigating the incidence of disease,

its severity and the quality of life before death [1].

Lee-Carter model and its extensions have been used by actuaries for multiple

purposes like longevity risk and annuity pricing [16].The model has been used by the

United States Social Security Administration, the US Census Bureau, and the United

Nations. It has become the most widely used mortality forecasting technique in the

world today [3].

1.4 Objective of the Study

The primary goal of the study is to forecast the Philippine mortality rates

using the Lee-Carter model. Specifically, this study aims to:

3

1. Estimate the index of level of mortality

2. Solve the parameters of the Lee-Carter Model;

3. Use Holts Linear Exponential Smoothing method to forecast the index of level

of mortality;

4. Compute for the life expectancy using the forecasted probability of death;

5. Construct life tables using the forecasted mortality rates.

4

Chapter 2

Theoretical Framework

In this Chapter, the concepts used in this Special Problem are discussed. The

average age-specific mortality and the deviations are discussed by the Lee-Carter

model, the Singular Value Decomposition and the mortality rates are forecasted using

Holts Linear Method.

2.1 Age-Specific Death Rates

An age specific death rate, mx,t, is the ratio of the number of deaths within a specified

age group in a specific geographic area during a certain period of time to the corre-

sponding population at risk of the same group, in the same geographic area during

the specified time period of study [1].

5

mx,t =Dx,tNx,t

(2.1)

where

mx,t -is the age specific death rate at age group x at year t

Dx,t -is the number of deaths at age group x at year t

Nx,t -is the population at age group x at year t

2.2 The Lee-Carter Model

The Lee-Carter model is a demographic and statistical model used in mortality fore-

casting and life expectancy forecasting [2]. It is a method used for long-run forecasts

of the level and age pattern of mortality based on a combination of statistical time

series methods and a simple approach to dealing with the age distribution of mortal-

ity. Fitting into historical data, the method describes the logarithm of a time series of

age-specific death rates as the sum of an age-specific component that is independent

of time and another component that is the product of a time-varying parameter re-

flecting the general level of mortality, and an age-specific component that represents

how rapidly or slowly mortality at each age varies when the general level of mortality

changes. The resulting estimate of the time varying parameter is then modeled and

forecast as a stochastic time series using standard methods [3].

6

The models basic premise is that there is a linear relationship among the logarithm

of age-specific death rates mx,t and two factors: the initial age interval x and year t.

Information is distributed in age intervals, so the interval that begins with age x will

be called x age interval [7]. The equation describing this is as follows:

mx,t = eax+bxkt+x,t , t = 1, 2, 3, ...., (2.2)

Taking the natural logarithm of both sides:

fx,t = ln(mx,t) = ax + bxkt + x,t, t = 1, 2, 3, ...., (2.3)

where

mx,t -is the age specific death rate for the x interval

kt -is the mortality index in the year t. This would capture 80% - 90%

of the historical mortality trend

ax -is the average age-specific mortality

bx -is the deviation in mortality due to changes in the kt index.

This describes the amount of mortality change at a given age

for one unit of total mortality change.

x,t -is the random error assuming normal distribution N(,)

-is the beginning of the last age interval

7

Definition 2.1. Average age-specific mortality

The average age-specific mortality, ax, is given by:

ax =

nt=1

lnmx,t

n(2.4)

The Lee-Carter model cannot fit by simple regression because there is no observed

variable on the right side. Nonetheless, a least-square solution exists and can be found

using the first element of the singular value decomposition or principal components

[4]. In order to standardize the matrix undergoing singular value decomposition and

to assure that a unique solution for the bx and kt for the system of equations of the

model,without loss of generality, proposed the following constraints:

x=1

bx = 1 andnt=1

kt = 0 (2.5)

This simply means that the total amount of morality change at a given age for one

unit of total mortality change is 1 and the total mortality index is zero.

2.3 Singular Value Decomposition

Definition 2.2. Singular Value Decomposition

Suppose M is an m x n matrix. Then there exists a factorization of the form

M = UV T (2.6)

8

where

U -is an m m orthogonal matrix

-is an m n matrix with non-negative numbers on the diagonal.

V T -is the conjugate transpose of the n n orthogonal matrix V

Writing in matrix form, we have,

U =

u1,1 u1,2 u1,m

u2,1 u2,2 u2,m...

.... . .

...

um,1 um,2 um,m

=

1,1 1,2 1,n

2,1 2,2 2,n...

.... . .

...

m,1 m,2 m,n

V T =

v1,1 v1,2 v1,n

v2,1 v2,2 v2,n...

.... . .

...

vn,1 un,2 vn,n

(2.7)

Such factorization is called the singular value decomposition of M [5]. The di-

agonal entries i,i, i = 1, 2, ...,m of are the singular values of M . These singular

values are are the square roots of the eigenvalues used to obtain matrix U , and listed

in descending order, and the diagonal matrix is uniquely determined by M .

The singular value decomposition Method factorization results in a real or complex

orthogonal matrices U and V T , where U*UT = V *V T = I, where I is the identity

matrix. The parameter bx is solved by the following formula:

bx =v1,xnj=1

v1,j

(2.8)

9

while the first estimation of the parameter k is calculated by:

kt = vS(1)Ut,1 x, t = 1, 2, ....., n (2.9)

where

v =n

j=1 vj,1

1,1 is the first element of the diagonal matrix s from

2.4 Second Estimation of the parameter k

A second stage estimate of k is calculated by finding the value of k which, for a

given population age distribution and the previously computed coefficients of ax and

bx, which would produce the exact observed number of total deaths for the year in

observation [7]. The second estimate of k is computed such that:

Dt =nx=1

eax+bx+ktNx,t (2.10)

where

Dt is the total death in year t

Nx,t is the population of age interval x at year t, x =1,2,...,

10

2.5 Holts Linear Exponential Smoothing Method

(LES)

Time series is defined to be the set of data points whick are analyzed using statistical

techniques to obtain the statistics and characteristics of data [8].

Time series analysis deals with two goals: first is to determine the characteristics

the data exhibits and second is to forecast. For the methods for this to be utilized,

patterns of the observed time series data such as trend and seasonality should be

studied first. Analysis for such time series patterns are done to examine and remove

such properties to make the given data stationary which is a requirement for time

series techniques.

There are various time series techniques known to model and forecast future events

based on the given data. One such model is the linear exponential smoothing method.

Linear Exponential Smoothing Method (LES) method allows the forecasting data

with a trend. It computes an evolving trend equation through the data using a special

weighting function that places the greatest emphasis on the most recent time periods

[10]. This method involves a forecast equation and two smoothing equations which

must be updated every period: one equation for the level and one equation to capture

the trend [9].

11

The level equation is given by

lt = yt + (1 )(lt1 + bt1) (2.11)

where

yt -is the set of actual values

bt -is the estimate of the trend or slope of the series at time t

-is the smoothing parameter for the level, where 0 1

The level equation shows that lt is the weighted average of observations yt and the

within sample one step ahead forecast for time t, given by lt1+ bt1. The computation

includes the weight of the previous level, lt1 and previous trend, bt1.

The trend equation is given by

bt = (lt lt1) + (1 )bt1 (2.12)

where

lt -is the estimate level of the series at time t

-is the smoothing parameter for the trend, where 0 1

The trend equation shows that bt is a weighted average of the estimated trend at

time t on lt - lt1 and bt1, the previous estimate of the trend. The value of b0 is the

difference between the second and the first actual values y2 - y1.

12

The forecast equation, the general formula for computing the next forecast data

value is given by:

yt+h|t = lt + hbt (2.13)

where

h -is the forecast horizon

y -is the forecasted values

Since the Linear Exponential Smoothing Method forecasts data with trend, the

forecast function is not flat, but trending. The h-step ahead forecast is equal to the

last estimated level, lt, added to h times the last estimated trend value, bt. Thus, the

forecasts are a linear function of h. The error correction form of the level and the

trend equations show the adjustments in terms of the h-step forecast errors:

lt = lt1 + bt1 + et (2.14)

bt = bt1 + et (2.15)

where

et = yt - yt|t1

The smoothing parameters and are computed by minimizing the Mean Squared

13

Error (MSE) of the data through the Generalized Reduced Gradient (GRG2) Algo-

rithm. The MSE is given the formula:

MSE =

nt=1

e2t

n(2.16)

2.6 Life Tables

2.6.1 The Construction of a Life Table

A life table is a table of statistics relating to life expectancy and mortality for a given

category of people. It is based on the mortality experience of a population during

a relatively short period of time. Generally, a life table shows, for each age or age

group, the probability of surviving any particular year of age and the remaining life

expectancy for people at different ages or age groups [11].

Life tables can be constructed using projections of future mortality rates, but more

often they are constructed through the use of age-specific mortality rates. Life tables

are usually constructed separately for men and for women because of their substan-

tially different mortality rates. Other characteristics can also be used to distinguished

different risks, such as smoking status, occupation and socio-economic class [12].

14

Construction of a life table depends on the presentation of the data. A complete

life table is used for data with single years of age, and an abridged life table is used

for data with group of ages, which is more common [15].

The abridged life table is constructed through the following formulas and defini-

tions:

qi =nimi

1 + (1 ai)nimi i = 0, 1, ..., 1 (2.17)

where

qi is the probability that an individual will die in the ith interval

ni is the length of the interval

mi is the death rate in the interval

ai is the average of the fractions lived by the individuals that died in the

interval

di = liqi (2.18)

di is the number of artificial cohort dying in ith interval

li is the number of artificial cohort at the start of the interval

Li = ni(li di) + ainidi, i = 0, 1, 2, ....., w 1 (2.19)

Li is the number of years lived in the interval

by the artificial cohort

15

Lw =lwmw

(2.20)

Ti =wk=i

Lk (2.21)

ei =Tili

(2.22)

w is the number of intervals

Ti is total number of years lived by individuals from the artificial cohort

ai attaining the age that starts the interval

ei is the expectation of life at the age that starts the interval

2.6.2 Statistical Inference on Life Tables

Each figure in a life table is an estimate of the corresponding unknown true

value. Thus, statistical inference regarding these unknown values may be made on

the basis of observed quantities.

An essential element required in making statistical inference is the standard

error of the estimate. Specifically, inference will be made about two categories of esti-

mated parameters: the probability of dying in the age group x, and its corresponding

expectation of life [13].

The probability of dying, qi, and the probability of survival, pi on an age

interval are complementary to one another, making the sample variances of their

16

estimates are equal. The sample variance of qi and pi is denoted by s2qi

and s2pi ,

respectively. Thus, we have

s2qi = s2pi

(2.23)

In a life table, the estimate qi is derived from certain mortality information,

which would be manipulated to get the sample variance of qi. The equation for the

variance of qi is

s2qi =1

Diq2i (1 qi) (2.24)

And the 95% confidence interval for qi is

(qi 1.96sqi , qi + 1.96sqi) (2.25)

The square root of s2qi would give the standard error, sqi , of qi.

The variance of the life expectancy is computed by

s2ei =1i=

p2i [(1 i)ni + ei+1]2s2pi (2.26)

The square root of s2ei would give the standard error, sei , of ei

The 95% confidence interval for ei is

17

(ei 1.96sei , ei + 1.96sei) (2.27)

An observed expectation of life is a sample mean of future lifetime. Thus, sta-

tistical tests based on normal distribution may be used in making inference regarding

expectation of life at a particular age, or in comparing expectations of life of two or

more populations.

18

Chapter 3

Solving for the Parameters of the

Lee-Carter Model

In this chapter, the steps needed to obtain the necessary calculations for the

ax, bx and kt parameters for the Male, Female and Total Population of the Philippines

from 1960 to 2009 were presented.

3.1 Solving for the parameters of the Lee-Carter

model

The Lee-Carter model shows that there is an equation for each time and age

interval. As a result, a system of simultaneous equations needs to be solved so as to

estimate the values of ax , bx and kt, which are the solutions to the system. A group

19

of death rates with different r age groups that were analyzed in different moments

creates a system of equations containing 2r + n unknown factors that correspond to

the sum of the r values of ax , r values of bx and n values of kt , and r n equations.

The logarithm of the death rates is calculated.

Year Age Male logMale

1960 0-1 0.0698787 -2.660994404

1960 1-4 0.013353008 -4.316013615

1960 5-9 0.002917171 -5.837140825

1960 10-14 0.001342569 -6.613170559

1960 15-19 0.001200554 -6.72497215

1960 20-24 0.001770714 -6.336372552

1960 25-29 0.002227692 -6.106789315

1960 30-34 0.002844456 -5.86238356

1960 35-39 0.003311874 -5.710241122

1960 40-44 0.004677319 -5.365030242

1960 45-49 0.005573339 -5.189760874

1960 50-54 0.007753851 -4.85956571

1960 55-59 0.008798441 -4.733180745

1960 60-64 0.017332353 -4.055180393

1960 65-69 0.016255512 -4.119323236

1960 >70 0.052303832 -2.950685635

Table 3.1: Death rates for male and its corresponding natural logarithm for 1960

Table 3.1 shows the death rates for male, computed from the age specific

number of deaths and the estimated population of the same age interval for 1960.

The first interval is for infants where observed mortality is significantly higher than

the rest of the age groups and the second interval is for early childhood.

Table 3.2 shows the sorted data to ease further computations for the male

population. This contains the natural logarithm of all age specific death rates from

20

Age logm1960 logm1961 logm1962 logm1963 logm1964 logm1965 logm1966 logm1967 logm1968 logm1969 logm19700-1 -2.660995 -2.788522 -2.614783 -2.629055 -3.021058 -3.04528 -3.053338 -3.055663 -3.059738 -3.061518 -2.8107351-4 -4.316014 -4.67117 -4.749695 -4.714931 -4.705819 -4.605005 -4.624796 -4.807929 -4.7915 -4.919944 -4.9780965-9 -5.837141 -6.085285 -6.020898 -6.078217 -5.935295 -5.96722 -6.091424 -6.082354 -5.991465 -6.165719 -6.287868

10-14 -6.61317 -6.76349 -6.710989 -6.749802 -6.683335 -6.754079 -6.723209 -6.673301 -6.571283 -6.6958 -6.73561915-19 -6.724972 -6.636725 -6.605447 -6.636499 -6.566495 -6.548948 -6.481376 -6.526153 -6.437752 -6.454457 -6.33982820-24 -6.336372 -6.005816 -6.00369 -6.016682 -6.004268 -6.076895 -6.088617 -6.080787 -5.991465 -6.034125 -5.93620225-29 -6.106789 -5.764001 -5.727261 -5.780907 -5.805523 -5.794353 -5.820773 -5.735015 -5.713833 -5.77072 -5.84205730-34 -5.862383 -5.634271 -5.596931 -5.625729 -5.663842 -5.649984 -5.667739 -5.596412 -5.472671 -5.604671 -5.63782135-39 -5.710241 -5.426815 -5.332268 -5.386514 -5.356634 -5.384789 -5.347045 -5.303902 -5.31852 -5.350147 -5.36979240-44 -5.36503 -5.398234 -5.301457 -5.39285 -5.288616 -5.268311 -5.305217 -5.217804 -5.115996 -5.18332 -5.09787745-49 -5.189761 -5.012224 -5.043702 -5.096659 -4.938013 -4.916888 -4.993903 -4.935592 -4.840893 -4.906852 -4.90116550-54 -4.859566 -4.823654 -4.750865 -4.868935 -4.6353 -4.666998 -4.672024 -4.664136 -4.575612 -4.678673 -4.61652155-59 -4.733181 -4.499728 -4.599526 -4.647592 -4.465935 -4.436017 -4.461902 -4.304173 -4.268698 -4.29134 -4.37438160-64 -4.055181 -4.312229 -4.096692 -4.18237 -3.80584 -3.913441 -4.050662 -3.986074 -3.952845 -4.033301 -4.1379465-69 -4.119323 -4.052442 -4.104721 -4.139961 -3.956599 -3.723593 -3.6633 -3.511854 -3.448289 -3.535075 -3.695456>70 -2.950686 -2.315235 -2.332829 -2.40184 -2.478098 -2.398648 -2.449082 -2.36192 -2.389233 -2.519524 -2.913381

Table 3.2: Natural logarithm of death rates for male from 1960-1970

1960 up to 2009. These values are the entries for a 16 50 matrix A, where singular

value decomposition is performed.

At first, there is no unique solution for the system. Thus, it is necessary to specify

that the sum of the mortality changes is 1 and the sum of the mortality index is 0.

ax is only a simple arithmetic average over time for the natural logarithms of the age

specific mortality rates.

The ax parameter in the Lee-Carter model is the average age specific mortality

rates for the historical data, which is sorted per year. Table 3.3 shows the variable a

which would be the ax vector for the male population. Each element of a is computed

by taking the mean of the natural logarithm of death rates from 1960 up to 2009 on

a specified age group.

Once the matrix a values are determined, the system can be rewritten as:

M = M a = bk (3.1)

This centralizes the death rates in the matrix M by subtracting the vector a to all

21

Age ax0-1 -3.488815921-4 -5.48049865-9 -6.62845394

10-14 -7.024372215-19 -6.5819346620-24 -6.0719168625-29 -5.869231530-34 -5.6904475235-39 -5.4378337240-44 -5.196961845-49 -4.8688410450-54 -4.5469764455-59 -4.2542121660-64 -3.8613183265-69 -3.58330512>70 -2.57409308

Table 3.3: Parameter ax for male

columns of the matrix M .

Age logm1960 logm1961 logm1962 logm1963 logm1964 logm1965 logm1966 logm1967 logm1968 logm1969 logm19700-1 0.8278 0.7003 0.8740 0.8598 0.4678 0.4435 0.4355 0.4332 0.4291 0.4273 0.67811-4 1.1645 0.8093 0.7308 0.7656 0.7747 0.8755 0.8557 0.6726 0.6890 0.5606 0.50245-9 0.7913 0.5432 0.6076 0.5502 0.6932 0.6612 0.5370 0.5461 0.6370 0.4627 0.3406

10-14 0.4112 0.2609 0.3134 0.2746 0.3410 0.2703 0.3012 0.3511 0.4531 0.3286 0.288815-19 -0.1430 -0.0548 -0.0235 -0.0546 0.0154 0.0330 0.1006 0.0558 0.1442 0.1275 0.242120-24 -0.2645 0.0661 0.0682 0.0552 0.0676 -0.0050 -0.0167 -0.0089 0.0805 0.0378 0.135725-29 -0.2376 0.1052 0.1420 0.0883 0.0637 0.0749 0.0485 0.1342 0.1554 0.0985 0.027230-34 -0.1719 0.0562 0.0935 0.0647 0.0266 0.0405 0.0227 0.0940 0.2178 0.0858 0.052635-39 -0.2724 0.0110 0.1056 0.0513 0.0812 0.0530 0.0908 0.1339 0.1193 0.0877 0.068040-44 -0.1681 -0.2013 -0.1045 -0.1959 -0.0917 -0.0713 -0.1083 -0.0208 0.0810 0.0136 0.099145-49 -0.3209 -0.1434 -0.1749 -0.2278 -0.0692 -0.0480 -0.1251 -0.0668 0.0279 -0.0380 -0.032350-54 -0.3126 -0.2767 -0.2039 -0.3220 -0.0883 -0.1200 -0.1250 -0.1172 -0.0286 -0.1317 -0.069555-59 -0.4790 -0.2455 -0.3453 -0.3934 -0.2117 -0.1818 -0.2077 -0.0500 -0.0145 -0.0371 -0.120260-64 -0.1939 -0.4509 -0.2354 -0.3211 0.0555 -0.0521 -0.1893 -0.1248 -0.0915 -0.1720 -0.276665-69 -0.5360 -0.4691 -0.5214 -0.5567 -0.3733 -0.1403 -0.0800 0.0715 0.1350 0.0482 -0.1122>70 -0.3766 0.2589 0.2413 0.1723 0.0960 0.1754 0.1250 0.2122 0.1849 0.0546 -0.3393

Table 3.4: The centralized death rates for the Male population from 1960 - 1970

This system provides a unique solution when these constraints are included. The

bk and kt parameters are to be determined by using the Singular Value Decomposition

(SVD). This method is used to obtain the exact fitting of least squares.

22

Through SVD, M can be written as the product of two matrices.

M = [mi,j] i = 1, 2, , (3.2)

In M, the (i, j) element is the sum of the product of row i of B and row j of K

which can be written as

mi,j =rt=1

Bi,jKj,iT (3.3)

Therefore, the decomposition creates r terms that exactly fit the mi,j element of the

M matrix. Lee and Carter suggest to only take account of the first order approxi-

mation of the resulting vectors.

For the Male Population, the results of the Singular Value Decomposition are:

Figure 3.1: The first element of the 50 50 orthogonal matrix U

Figure 3.2: The first elements of the 16 x 16 matrix V T

To perform SVD, the rows of the matrix must be greater than its columns, thus

the transpose of M is the matrix is taken. The results of the SVD in M are stored in

23

1 8.5377830142 1.9639674683 1.0788841864 0.8705818655 0.840851646 0.6495616297 0.5657378198 0.4889255619 0.39498197610 0.27475492411 0.23905404112 0.22888097713 0.18766861814 0.16936588115 0.1373642716 0.114767783

Table 3.5: the diagonal matrix

a unitary 50 50 matrix U , a column vector s, and a 16 16 matrix V T . Figures

3.1,and 3.2 and Figure 3.5 shows a part of the matrix U , matrix V T and matrix s.

To find the univariate time series kt for the data, the first order approximation of the

Singular Value Decomposition of matrix A is taken, that is, the first column of U , the

column vector s and the first row of V T . The first row of matrix V T is used in the

computation of each age interval in bx. The elements of bx are computed by dividing

each corresponding age specific element of V T to the sum of its elements. The same

process is done for the female and total population

Table 3.6 shows the ax and bx estimations for male, female and the total pop-

ulation. Higher values of bx appear in the 0-4 interval, which means that, in such

interval, mortality varies substantially when the general mortality index kt changes.

24

Age bx0-1 0.2525831

1-4 0.3304692

5-9 0.2208131

10-14 0.1212668

15-19 0.0742751

20-24 0.0673185

25-29 0.047476

30-34 0.0390297

35-39 0.0330816

40-44 0.0126288

45-49 -0.0021691

50-54 -0.0235981

55-59 -0.0490273

60-64 -0.0477929

65-69 -0.0694024

>70 -0.0069522

Table 3.6: The parameter bx for the male population

The old ages display lower parameters which means that mortality slightly varies in

that period of time.

25

Age Group Male Male Female Female Total Totalax bx ax bx ax bx

0-1 -3.48882 0.252583 -3.77405 0.148731 -3.61774 0.2017191-4 -5.4805 0.330469 -5.59969 0.199163 -5.53694 0.2675614-9 -6.62845 0.220813 -6.84302 0.136403 -6.72704 0.180191

10-14 -7.02437 0.121267 -7.30471 0.074358 -7.15073 0.09865815-19 -6.58193 0.074275 -7.09231 0.064035 -6.8015 0.06944720-24 -6.07192 0.067319 -6.78958 0.066427 -6.36794 0.06497425-29 -5.86923 0.047476 -6.56184 0.0656 -6.15634 0.05454630-34 -5.69045 0.03903 -6.26195 0.078201 -5.93772 0.05435435-39 -5.43783 0.033082 -5.99056 0.065903 -5.67026 0.04997340-44 -5.19696 0.012629 -5.74492 0.047624 -5.42604 0.02983745-49 -4.86884 -0.00217 -5.43036 0.034369 -5.10827 0.01445550-54 -4.54698 -0.0236 -5.14822 0.013193 -4.80664 -0.0070855-59 -4.25421 -0.04903 -4.86439 -0.00192 -4.51823 -0.0268560-64 -3.86132 -0.04779 -4.39049 0.000848 -4.09875 -0.0234865-69 -3.58331 -0.0694 -4.02074 -0.00898 -3.78575 -0.03643

Table 3.7: The ax and bx parameters fore male, female and total population in thePhilippines

Figure 3.3: ax parameter for male, female and the total population in the Philippines

Figure 3.4 shows the estimations of the shape parameter ax. These estimations

show in a way in which mortality behaves through intervals. Figure 3.4 shows the

significant difference of the mortality changes between the male and the female, mean-

ing at most age intervals of adulthood, male deaths generally occur more than female

deaths.

26

Figure 3.4: bx parameter for men, women and the total population in the Philippines

3.1.1 Estimation of the k parameter

In the following stage, first estimations and re-estimations of the general index

of mortality are calculated for the Philippines.

The elements of the time series kx are computed by the product of the first column

of U , the sum of the elements of the first row of V t, and the column vector s. For

further accuracy, the time series must be re-estimated to fit to the corresponding

yearly total number of deaths and the population, and the computed parameters ax

and bx. The same process is done for the Female and Total Population.

Comparing the general mortality indices of the male, female and total population,

the series of these general indices clearly tend to decrease, although not monotonically,

over time. For the first half of the period, both figures show the notable increase of

27

Figure 3.5: First estimation of the parameter k for male, female and total population

female mortality over men, which decreased significantly on the lower half of the

period.

At this point, the modeling of the parameter k as a time series process can already

be done. Instead, a second stage estimate of k should be calculated by finding the

value of k which, for a given population age distribution and the previously estimated

coefficients ax and bx , produces exactly the observed number of total deaths for the

observed year. This is done since the first projections of k may cause deviation on

the projections. The second estimation for k can be obtained by:

Dt =nx=1

eax+bxktNx,t (3.4)

where

Dt -is the total deaths in year t

28

Nx,t -is the population at age interval x in year t

Expanding the formula would produce a number of equations equivalent to the

number of years in observation.

Figure 3.6: Second estimation of the parameter k for male, female and total popu-lation

29

Chapter 4

Holts Linear Exponential Smoothing

Method (LES)

4.1 Forecasting mortality rates

The second estimation of the parameter k is used for the forecast. Holts Linear

Exponential Smoothing Method was used in the said forecast. Using the forecast,

trend and level equations given by LES, the second estimation of the mortality index

k, mortality rates from 2014 to 2018 are computed.

Year Male Female Total2014 -2.76448294 -1.18343741 -2.358412312015 -2.65899668 -0.71670650 -1.950598662016 -2.55351042 -0.24997559 -1.542785022017 -2.44802416 0.21675532 -1.134971372018 -2.34253790 0.68348622 -0.72715772

Table 4.1: The forecasted parameter k for male, female and total population inPhilippines

30

MSEMale 0.23728213 0.14823359 0.09994884

Female 0.31308186 0.65536894 0.14242946Total 0.1 0.46290818 -1.54278502

Table 4.2: The Mean Squared Error, ,and for male, female and total Population

To get a more accurate forecast, the Mean Squared Error (MSE) for Male, Female

and Total was minimized by the Solver for Microsoft Excel to get the appropriate

values for and which were used in the trend, level and forecast equations of

Holts Linear Exponential Smoothing Method.

4.2 Computing new age-specific death rates

The future age-specific death rates can be computed from the forecasted mortality

index k for 2014 up to 2018. In order to do so, the forecast values of k would be

substituted in the formula:

mx,n+h = mx,nebx(kn+hkn) (4.1)

where

mx,n -is the age specific death rate at the last year, 2009

h -is the difference of the forecasted year in year 2009

31

x 2014 2015 2016 2017 2018 bx mx,20090-1 0.008294 0.008386 0.008480 0.008575 0.008671 0.25258 0.010911-4 0.000819 0.000831 0.000843 0.000856 0.000868 0.33047 0.001175-9 0.000449 0.000453 0.000458 0.000462 0.000467 0.22081 0.00057

10-14 0.000475 0.000478 0.000480 0.000483 0.000486 0.12127 0.0005415-19 0.001011 0.001014 0.001017 0.001021 0.001024 0.07428 0.001120-24 0.001643 0.001648 0.001652 0.001657 0.001662 0.06732 0.0017725-29 0.002123 0.002127 0.002131 0.002136 0.002140 0.04748 0.0022330-34 0.002677 0.002681 0.002686 0.002691 0.002695 0.03903 0.0027935-39 0.003535 0.003540 0.003546 0.003551 0.003556 0.03308 0.0036640-44 0.005036 0.005038 0.005041 0.005044 0.005047 0.01263 0.0051145-49 0.007696 0.007695 0.007694 0.007694 0.007693 -0.00217 0.0076850-54 0.12327 0.012314 0.012301 0.012289 0.012276 -0.0236 0.0120255-59 0.019271 0.019230 0.019188 0.019147 0.019106 -0.04903 0.0182760-64 0.029751 0.029688 0.029626 0.029563 0.029501 -0.04779 0.0282565-69 0.039481 0.039361 0.039240 0.039120 0.039001 -0.0694 0.03662>70 0.089556 0.089528 0.089501 0.089474 0.089446 -0.00695 0.08888

Table 4.3: The new age-specific death rates of male for 2014 up to 2018

32

Chapter 5

Life Tables

5.1 Constructing the life table

The average of the fractions lived by the individuals that died in the interval, ax is

computed by the exact number of age specific deaths and several other statistics and

data concerning mortality. The exact value for ax can only be calculated from the

full death records. Since this is estimated on forecasted data, it is impossible to have

specific number of deaths.

From the forecasted values of k, the age specific death rates and probabilities of

death are computed. The death rates are computed using the calculated b parameter,

the death rates and the parameter k on the last year where data is available, 2009.

The probability of death, qx, is computed using the years between age intervals and

the previously computed death rates.

33

The life table is constructed in the following order: Age, length of the interval

n, fraction of the last age interval of life ai , forecasted death rates mx , probability

of death qx , alive artificial cohort lx , deaths in the artificial cohort di , years lived

in the interval by cohort, Li , total number of years lived by individuals in cohort

at start of age interval Ti and the expectation of life ei. The statistical inference is

computed in a separate table. [12]

Age n ax mx qx lx dx Lx Tx ex0-1 1 0.09 0.008271 0.008209 100000 821 99253 6778051 67.780511041-4 4 0.4 0.000816 0.003258 99179 323 395941 6678798 67.340802515-9 5 0.4 0.000448 0.002236 98856 221 493616 6282857 63.55570933

10-14 5 0.5 0.000475 0.002370 98635 234 492590 5789241 58.6936620815-19 5 0.5 0.001010 0.005037 98401 496 490766 5296651 53.8271777520-24 5 0.5 0.001642 0.008174 97905 800 487526 4805885 49.0870396125-29 5 0.5 0.002122 0.010552 97105 1025 482964 4318359 44.4709845130-34 5 0.5 0.002676 0.013290 96080 1277 477210 3835395 39.9185660835-39 5 0.5 0.003534 0.017515 94804 1660 469867 3358185 35.4225426940-44 5 0.5 0.005035 0.024862 93143 2316 459926 2888318 31.0094569445-49 5 0.5 0.007696 0.037753 90827 3429 445564 2428392 26.7363337650-54 5 0.5 0.012330 0.059807 87398 5227 423924 1982827 22.6872397255-59 5 0.5 0.019282 0.091974 82171 7558 391962 1558903 18.9713772560-64 5 0.5 0.029766 0.138522 74614 10336 347229 1166941 15.6397729465-69 5 0.5 0.039510 0.179793 64278 11557 292498 819711 12.75258518>70 20 0.5 0.089563 1.000000 52721 52721 527213 527213 10

Table 5.1: Life table for the year 2014 for the male population

34

If the infant mortality rate is less than 0.02, ai=0.09. From Table 5.1, the death

rate for the first age interval is 0.008, which is less than 0.02. The values of ai for

young childhood intervals is 0.4 and adult intervals are around 0.5. The artificial

cohort, lx, is set to be 100,000 at the start of the age interval.

5.2 Standard error and confidence intervals

Age SE (qx) Confidence Interval for qx px Var(px) G(x) SE (ex) Confidence Interval for ex0 to 1 0.000285721 0.007671378 0.008791403 0.991769 0.000010 458183937.63 0.515714 66.768487 68.7900851 to 4 0.00018128 0.002914581 0.003625199 0.996730 0.000010 428631533.67 0.473087 66.413836 68.2683385 to 9 0.00015041 0.00194658 0.002536188 0.997759 0.000010 374575542.38 0.425933 62.721930 64.39158610 to 14 0.000154941 0.002069755 0.002677123 0.997627 0.000010 311465106.82 0.379117 57.951970 59.43811115 to 19 0.00022578 0.004598821 0.005483878 0.994959 0.000010 259240302.86 0.335030 53.172075 54.48539220 to 24 0.000287872 0.007615734 0.008744191 0.991820 0.000010 212495483.49 0.293833 48.512898 49.66472325 to 29 0.000327985 0.009914041 0.011199741 0.989443 0.000010 171064683.80 0.255402 43.972461 44.97363530 to 34 0.000369518 0.0125709 0.01401941 0.986705 0.000010 134535897.08 0.219309 39.491035 40.35072735 to 39 0.000426131 0.016685825 0.018356257 0.982479 0.000010 102766521.84 0.185555 35.061413 35.78878940 to 44 0.000510235 0.02386537 0.025865489 0.975135 0.000010 75715413.94 0.154346 30.709753 31.31478945 to 49 0.000632447 0.036513022 0.038992216 0.962247 0.000010 53361259.29 0.125991 26.492377 26.98626150 to 54 0.000802044 0.058220267 0.061364282 0.940208 0.000010 35625286.44 0.100781 22.492789 22.88785155 to 59 0.001007945 0.089952782 0.093903927 0.908072 0.000010 22300298.63 0.078823 18.819826 19.12881060 to 64 0.001264405 0.13597732 0.140933786 0.861544 0.000010 12886791.83 0.059405 15.525658 15.75852465 to 69 0.001514212 0.176703646 0.182639358 0.820328 0.000010 6759105.37 0.040444 12.674837 12.83337670 up 0 1 1 0.000000 0.000000 0.00 0.000000 10.000000 10.000000

Table 5.2: Statistical Inference of forecasted rates for the year 2014 of the malepopulation

35

Chapter 6

Summary and Conclusions

This study computed for the log rates of the corresponding age-specific death rates

of male, female and total population. The computation showed that the first interval

is for infants where observed mortality is significantly higher than the rest of the age

groups and the second interval is for early childhood.

The log rates of the age specific death rates from 1960 up to 2009 were used to

obtain matrix A as a prerequisite in performing the Singular Value Decompostion.

The parameter ax where obtained by taking the average over time for the natural

logarithm of the age specific mortality rates. The computed ax parameters ranged

from -3 up to -7. The significant difference of the mortality changes between the male

and female was also shown in the study. It is observed that at most age intervals of

adulthood, male deaths generally occur more than female deaths.

The Singular Value Decomposition (SVD) was used to determine the parameters

36

of bk and kt. For the parameter bk, higher values appeared in the 0-4 interval, which

means that, in such interval, mortality varies substantially when the general mortality

index kt changes. The old ages (50 and above) showed lower parameters which means

that mortality slightly varies during that period.

Comparing the general mortality indices of the male, female and total population,

the series of these general indeces clearly tend to decrease, although not monotonically,

over time. For the first half of the period, it is observed that there is a notable

increase of female mortality over men and decreased significantly on the lower half of

the period.

New age specific death rates were computed using the Holts Linear Exponential

Smoothing Method (LES). The Mean Squared Error (MSE) was minimized to get

the appropriate values of the parameter and which will be used in the trend,

level and forecast equations of LES. The computed and for male is 0.23728213

and 0.14823359 respectively. The computed and for female is 0.31308186 and

0.65536894 respectively. The computed and for total population is 0.1 and -

0.46290818 respectively.

The computed age specific death rates from 2014 up to 2018 showed a little sig-

nificant changes per interval. Also, parameter bx is negative in the 50 and above

intervals.

The computed age specific death rates were then used in constructing a life table

37

showing the mortality rate and the life expectancy at birth. It is obtained in the

table that the life expectancy of male and female is aproximately 67 and 69 years old,

respectively.

38

Chapter 7

Recommendation

This study considered data from 1960 up to 2009 with sixteen age groups and used

Holts Linear Exponential Smoothing Method in forecasting the mortality index. Fu-

ture researchers interested in extending the Lee - Carter model may consider the

following recommendations:

Shorten the length of Historical data to avoid outliers and dicrepancy of

the data (ex. massacres).

Look for more recent data of death rates for better accuracy and precision

of results.

Convert the abridged life table into Illustrative Life (ILT) table using the

graduation process since insurance companies rely more on ILT.

Use Space-State Model (SSM) or other forecasting method in determining

future mortality index

39

References

[1] Boyle, P. and Parkin, D.M. Statistical Methods for Registries. International Agency

for Research on Cancer, 150 cours Albert Thomas, 69372 Lyon Cedex 08, France

[2] Lee, R. (2003). Reflections on Inverse Projection: Its Origins, Development, Exten-

sions, and Relation to Forecasting. University of California, Berkely

[3] Lee, R. The Lee-Carter Method for Forecasting Mortality, with Various Extensions

and Applications. University of California, Berkely.

[4] Wang, C. and Liu, Y., (2010). Comparisons of Mortality Modelling and Forecasting-

Empirical Evidence from Taiwan. International Research Journal of Finance and Eco-

nomics. Issue 37.

[5] Skoufranis, P. (2010). Singular Value Decomposition. University of Georgia.

[6] Baker, K. (2005). Singular Value Decomposition Tutorial

[7] Andreozzi, L., Blacona, M. and Arnesi, N. (2008). The Lee Carter method for esti-

mating and forecasting mortality: an application for Argentina. National University

of Rosario, Argentina.

[8] Mentzer, S. (2004). Time Series Forecasting Techniques. SAGE Publications, Inc.

40

[9] Kalekar, P. (2004). Time Series Forecasting using Holt-Winters Exponential Smooth-

ing. Kanwal Rekshi School of Information Technology.

[10] NCSS Statistical Software. Exponential Smoothing - Trend. Chapter 466, pp. 466-1 to

466-9.

[11] Rossa, A. (2011). Future Life Tables based on the Lee-Carter methodology and their

application to calculating the pension annuities. Acta Universitatis Lodziensis, Folia

Oeconomica 250.

[12] Chiang, C. Life Table and Mortality Analysis. World Health Organization

[13] Andreev, E. and Shkolnikov, V. (2010). Spreadsheet for calculation of confidence limits

for any life table or healthy-life table quantity. Max-Planck-Institut fur demografishe

Forschung.

[14] Girosi, F. and King, G. (2007). Understanding the Lee-Carter Mortality Forecasting

Method. Harvard University

[15] Cabigon, J. (2009). 2000 Life Table Estimates for the Philippines and Provinces By

Sex. University of the Philippines Diliman, Quezon City.

[16] Richards, SJ and Currie, ID (2009). Longetivity Risk and Annuity Pricing with Lee-

Carter Model.Sessional Meeting Paper. United Kingdom

41

APPENDIX A

Male 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 19750-1 456 539 441 457 671 705 731 754 791 796 566 586 705 739 762 7751-4 1430 2109 2297 2379 2329 2417 2511 2610 2650 2821 2570 2659 2640 2716 2799 28735-9 1509 2173 2241 2313 2385 2453 2544 2645 2689 2874 2810 2852 2889 2932 3022 308110-14 1488 1888 1936 1987 2035 2107 2171 2233 2147 2355 2204 2314 2420 2535 2616 271115-19 1773 1529 1612 1688 1667 1733 1798 1866 1990 2007 1746 1828 1909 1997 2053 211320-24 1509 1165 1218 1277 1352 1402 1456 1512 1645 1633 1584 1613 1638 1666 1718 176825-29 1237 924 961 1001 1103 1141 1182 1226 1290 1319 1431 1456 1481 1506 1552 157130-34 935 771 791 813 908 939 970 1004 1026 1074 1163 1209 1254 1302 1342 137235-39 839 685 695 706 752 775 799 826 975 883 980 1011 1041 1072 1104 111840-44 595 631 638 643 636 650 669 687 777 727 711 757 800 848 873 89645-49 621 567 578 589 539 553 566 580 652 609 612 627 641 656 675 70050-54 417 455 475 493 438 452 467 482 523 508 505 519 535 550 566 57855-59 369 366 371 378 330 344 359 373 428 402 443 450 455 460 475 49860-64 214 275 290 305 237 247 258 269 308 292 361 368 376 384 396 40965-69 168 179 186 196 164 170 176 183 204 198 238 251 264 278 287 290>70 309 179 192 205 226 226 228 230 243 243 394 389 388 289 297 421

Table 7.1: The age-specific male population from 1960 - 1975

Male 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 19900-1 788 810 820 838 832 800 815 830 843 855 866 876 884 891 9301-4 2931 3060 3050 3121 3118 3175 3182 3185 3186 3200 3257 3310 3358 3401 34325-9 3194 3240 3324 3400 3411 3493 3586 3683 3781 3860 3889 3913 3933 3949 414410-14 2713 2880 2914 2981 3049 3122 3186 3248 3312 3385 3467 3559 3656 3754 381715-19 2144 2250 2276 2329 2578 2663 2755 2849 2940 3023 3095 3160 3221 3286 337320-24 1881 1890 1912 1956 2220 2276 2335 2399 2468 2545 2629 2721 2815 2905 297325-29 1663 1665 1685 1724 1926 1985 2037 2085 2133 2185 2241 2300 2364 2433 252430-34 1356 1440 1457 1491 1528 1598 1675 1752 1826 1094 1952 2004 2052 2100 215735-39 1094 1170 1184 1211 1233 1275 1322 1375 1433 1497 1567 1642 1719 1793 184240-44 963 945 956 978 1051 1082 1109 1136 1166 1201 1243 1287 1341 1399 146945-49 700 720 729 745 829 861 899 939 978 1013 1044 1071 1097 1127 116950-54 613 630 637 652 686 705 716 716 722 765 784 798 826 833 95855-59 525 540 546 559 531 548 550 581 591 601 620 662 650 713 74760-64 394 450 455 466 443 446 454 446 452 482 496 507 530 533 57565-69 350 315 319 326 351 358 367 336 341 350 373 384 412 412 421>70 438 405 364 373 448 459 487 532 527 573 590 603 629 653 651

Table 7.2: The age-specific male population from 1976 - 1990

42

APPENDIX B

Male 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 19750-1 31.9 33.2 32.3 33.0 32.7 33.5 34.5 35.5 37.1 37.3 34.1 34.5 38.2 39.4 36.6 37.61-4 19.1 19.7 19.9 21.3 21.1 24.2 24.6 21.3 22.0 20.6 17.7 21.1 25.6 26.7 21.8 19.35-9 4.4 4.9 5.4 5.3 6.3 6.3 5.8 6.0 6.7 6.0 5.2 5.6 7.0 6.5 6.4 5.710-14 2.0 2.2 2.4 2.3 2.5 2.5 2.6 2.8 3.0 2.9 2.6 2.7 3.2 3.0 3.0 2.915-19 2.1 2.0 2.2 2.2 2.3 2.5 2.8 2.7 3.2 3.2 3.1 3.3 3.8 3.3 3.4 3.920-24 2.7 2.9 3.0 3.1 3.3 3.2 3.3 3.5 4.1 3.9 4.2 4.8 5.0 4.2 4.9 4.925-29 2.8 2.9 3.1 3.1 3.3 3.5 3.5 4.0 4.3 4.1 4.2 4.6 4.3 3.8 4.3 4.530-34 2.7 2.8 2.9 2.9 3.2 3.3 3.4 3.7 4.3 4.0 4.1 4.5 4.7 4.1 4.5 4.135-39 2.8 3.0 3.4 3.2 3.5 3.6 3.8 4.1 4.8 4.2 4.6 4.9 5.3 4.7 5.2 5.140-44 2.8 2.9 3.2 2.9 3.2 3.3 3.3 3.7 4.7 4.1 4.3 4.7 5.0 4.8 5.3 5.145-49 3.5 3.8 3.7 3.6 3.9 4.0 3.8 4.2 5.2 4.5 4.6 4.7 5.4 5.3 5.8 5.750-54 3.2 3.7 4.1 3.8 4.3 4.2 4.4 4.5 5.4 4.7 5.0 5.2 5.9 5.6 6.3 5.955-59 3.2 4.1 3.7 3.6 3.8 4.1 4.1 5.0 6.0 5.5 5.6 5.8 6.2 6.0 6.6 6.360-64 3.7 3.7 4.8 4.7 5.3 4.9 4.5 5.0 5.9 5.2 5.8 6.2 7.1 7.4 8.7 8.965-69 2.7 3.1 3.1 3.1 3.1 4.1 4.5 5.5 6.5 5.8 5.9 5.7 6.2 6.5 7.1 7.0>70 16.2 17.7 18.6 18.6 19.0 20.5 19.7 21.7 22.2 19.6 21.4 23.2 26.8 25.6 28.6 25.5

Table 7.3: The age-specific male deaths from 1960 - 1975

Male 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 19750-1 43.3 42.5 41.8 40.9 39.2 37.9 36.1 37.7 33.4 31.6 30.6 29.6 27.4 25.0 23.0 37.61-4 21.0 20.3 19.4 21.7 19.7 21.1 21.0 23.9 21.4 22.7 19.6 22.5 17.5 16.1 12.9 19.35-9 6.4 6.4 6.1 6.0 5.5 5.4 5.5 6.1 5.7 6.0 5.7 6.8 6.1 5.6 4.9 5.710-14 3.1 3.2 3.2 3.2 3.2 3.1 3.1 3.3 3.1 3.2 3.1 3.1 3.2 3.1 3.1 2.915-19 4.1 4.1 4.3 4.5 4.5 4.3 4.6 4.7 4.7 5.2 5.0 5.0 5.0 4.8 4.5 3.920-24 5.2 5.5 5.9 6.1 6.1 6.1 6.4 6.8 7.0 7.6 7.5 7.4 7.3 7.3 7.0 4.925-29 4.8 5.4 5.7 5.9 6.2 6.3 6.5 7.1 7.2 6.2 8.2 8.0 8.2 8.0 7.6 4.530-34 4.6 4.7 5.0 5.3 5.8 5.9 6.2 6.6 7.0 7.4 7.7 7.8 7.7 7.5 7.8 4.135-39 5.5 5.6 5.4 5.6 5.8 5.7 5.7 6.4 6.5 7.4 7.8 8.1 8.1 8.1 7.7 5.140-44 5.8 6.1 6.0 6.1 6.7 6.6 7.1 7.2 7.2 7.5 7.3 7.3 7.7 7.7 8.0 5.145-49 6.2 6.4 6.2 6.6 6.9 7.1 7.6 7.8 8.0 8.6 8.6 8.7 8.8 8.7 8.9 5.750-54 6.6 6.6 7.0 7.1 7.7 7.7 8.3 8.5 8.6 9.1 9.3 9.7 9.7 10.2 10.1 5.955-59 7.0 7.4 7.4 7.5 7.6 7.8 8.5 8.8 8.8 9.7 10.0 10.2 10.1 10.7 10.7 6.360-64 9.8 9.6 9.0 8.3 9.0 8.9 9.8 10.0 10.0 10.4 10.7 10.9 11.3 11.9 12.1 8.965-69 8.0 8.9 8.7 9.6 9.8 9.8 9.4 9.8 9.5 10.8 11.0 11.1 11.4 12.2 11.7 7.0>70 28.0 28.4 27.4 30.2 33.0 32.8 35.6 37.8 38.0 41.3 40.9 41.4 42.7 45.5 45.8 25.5


43

Male 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 20050-1 20.1 21.5 20.2 18.2 17.9 17.8 16.5 16.7 14.8 16.3 15.4 13.9 13.3 13.2 12.81-4 9.9 11.6 9.1 7.7 7.9 8.5 6.8 6.8 5.6 6.4 6.4 6.0 5.6 4.8 5.05-9 4.4 4.8 4.0 3.5 3.7 4.1 3.5 3.7 3.2 3.3 3.3 3.1 3.1 2.9 3.010-14 2.9 3.1 2.9 2.9 2.8 3.0 2.8 3.0 2.8 2.8 3.0 2.8 2.7 2.7 2.815-19 4.1 4.3 4.4 4.1 4.0 4.3 4.2 4.2 4.2 4.3 4.4 4.3 4.3 4.5 4.620-24 7.0 6.7 6.7 6.3 5.9 6.2 6.0 6.1 6.0 6.5 6.8 6.7 6.8 7.1 7.025-29 7.6 7.4 7.4 7.2 7.1 7.5 7.4 7.4 7.2 7.3 7.4 7.5 7.5 7.8 8.330-34 7.7 7.8 7.7 7.6 7.3 7.7 7.8 7.9 7.7 8.1 8.6 8.6 8.7 8.9 8.735-39 7.9 8.3 8.4 8.4 8.3 8.9 8.7 9.3 9.2 9.4 9.8 9.8 10.2 10.5 10.940-44 8.2 8.5 8.8 9.2 8.9 9.6 9.8 10.2 10.2 10.8 11.2 11.6 12.0 12.2 12.645-49 8.7 9.1 9.7 10.1 10.6 11.1 11.4 12.3 12.3 12.7 13.4 14.1 14.4 14.9 15.750-54 10.0 10.7 11.0 11.3 11.3 12.1 12.4 13.5 13.7 14.9 15.9 16.8 17.3 17.8 18.655-59 10.8 11.7 12.2 13.1 13.2 14.1 14.3 15.1 15.2 15.8 16.6 17.6 18.0 19.2 21.260-64 12.0 13.0 13.6 14.8 14.9 16.1 16.5 17.3 17.9 18.8 19.4 20.3 20.4 21.1 21.365-69 11.9 12.7 13.5 14.4 15.3 16.0 16.5 17.2 17.8 18.5 20.0 21.2 21.4 21.9 23.5>70 44.7 47.8 49.1 52.5 53.3 57.1 56.7 59.3 59.4 61.1 64.2 67.9 67.5 67.8 73.9


Male 2006 2007 2008 2009

0-1 12.8 12.8 13.1 12.5

1-4 5.3 4.8 4.9 5.4

5-9 3.2 3.0 2.9 3.1

10-14 2.9 2.7 2.8 2.9

15-19 4.9 5.1 5.1 5.3

20-24 7.2 6.9 7.2 7.6

25-29 8.5 8.4 8.5 8.5

30-34 8.9 8.6 9.0 9.2

35-39 11.1 11.0 11.0 11.0

40-44 12.9 12.8 13.2 13.5

45-49 16.3 16.1 16.5 17.1

50-54 19.1 19.6 20.1 21.1

55-59 21.9 22.4 23.6 24.4

60-64 22.5 22.8 24.1 26.2

65-69 24.3 24.4 25.4 26.5

>70 76.3 76.7 81.3 84.8


Female 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 19750-1 23.6 24.1 24.1 24.3 23.9 24.4 24.8 25.2 22.7 26.5 23.9 25.2 27.6 28.5 26.9 27.61-4 16.6 17.5 17.6 19.1 18.4 21.6 21.3 18.8 18.4 17.4 15.1 18.2 22.3 23.6 18.9 17.05-9 3.3 3.7 4.2 4.7 5.1 4.9 5.2 4.6 4.8 4.5 4.0 4.4 5.5 5.2 5.0 4.610-14 1.4 1.4 1.7 1.5 1.8 1.8 2.0 1.8 2.1 1.9 1.9 1.8 2.3 2.1 2.1 2.115-19 1.6 1.4 1.5 1.2 1.6 1.6 1.7 1.8 2.1 1.9 2.6 2.0 2.4 2.2 2.4 2.620-24 2.2 2.0 2.3 2.0 2.2 2.1 2.0 2.0 2.4 2.3 2.3 2.4 2.7 2.6 2.8 2.825-29 2.3 2.4 2.6 2.3 2.4 2.5 2.5 2.5 2.8 2.5 2.4 2.3 2.6 2.6 2.7 2.830-34 2.4 2.5 2.7 2.3 2.6 2.4 2.6 2.6 3.0 2.6 2.6 2.7 3.1 2.9 2.9 2.835-39 2.7 2.7 3.1 2.7 3.0 2.9 2.9 3.1 3.9 3.1 2.9 3.0 3.5 3.3 3.6 3.440-44 2.4 2.7 2.9 2.6 2.9 2.7 2.7 2.8 3.3 2.9 2.9 2.9 3.2 3.3 3.5 3.345-49 2.7 3.1 3.0 2.8 2.9 3.0 3.0 3.1 3.6 3.1 3.1 3.1 3.5 4.0 3.6 3.550-54 2.5 2.7 3.0 2.8 3.2 3.1 3.1 3.2 3.9 3.1 3.1 3.3 3.9 3.8 4.1 3.755-59 2.9 2.5 2.7 2.7 2.8 3.0 3.0 3.3 4.1 3.6 3.6 3.7 3.9 3.8 4.2 4.160-64 2.9 3.2 3.5 3.5 3.8 3.8 3.1 3.8 4.5 3.9 4.1 4.3 5.2 4.7 5.9 6.365-69 2.5 2.6 2.6 2.6 2.5 3.2 3.4 3.9 4.9 4.3 4.5 4.2 4.8 5.0 5.3 5.3>70 17.3 18.6 19.6 19.0 20.3 22.6 21.4 22.8 24.5 20.5 21.0 23.2 27.5 26.0 28.9 25.2

Table 7.7: The age-specific female deaths from 1960 - 1975

44

Female 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 19750-1 31.5 33.9 31.8 30.9 26.5 26.5 25.6 26.6 23.5 23.0 21.6 21.4 19.9 18.1 16.6 27.61-4 18.5 18.4 16.9 19.0 16.3 17.8 18.3 20.4 17.9 19.8 16.9 20.0 14.9 14.0 11.0 17.05-9 5.1 5.3 4.7 4.6 4.0 4.1 4.3 4.6 4.3 4.9 4.5 5.3 4.9 4.4 3.8 4.610-14 2.2 2.4 2.4 2.4 2.1 2.2 2.2 2.3 2.5 2.5 2.4 2.5 2.6 2.4 2.3 2.115-19 2.8 3.0 2.6 2.8 2.6 2.6 2.6 2.7 2.5 2.9 2.8 2.7 2.7 2.6 2.5 2.620-24 2.9 3.2 3.0 3.0 3.0 2.9 2.9 3.1 3.0 3.2 3.2 3.1 3.1 3.1 3.0 2.825-29 2.9 3.3 3.0 3.1 3.0 3.1 3.2 3.3 3.0 3.6 3.5 3.4 3.4 3.3 3.3 2.830-34 3.1 3.1 2.9 3.0 2.9 3.1 3.1 3.2 3.2 3.5 3.6 3.5 3.5 3.4 3.4 2.835-39 3.5 3.6 3.3 3.3 3.1 3.1 3.0 3.3 3.2 3.7 3.7 3.7 3.8 3.8 3.7 3.440-44 3.5 3.8 3.5 3.5 3.4 3.4 3.6 3.6 3.5 3.6 3.6 3.7 3.6 3.6 3.7 3.345-49 3.7 3.9 3.7 3.6 3.5 3.7 3.7 4.0 3.7 4.2 4.2 4.2 4.2 4.2 4.1 3.550-54 4.1 4.4 4.2 4.3 4.1 4.2 4.4 4.5 4.5 4.9 5.1 5.1 5.2 5.4 5.3 3.755-59 4.3 4.8 4.4 4.4 4.3 4.4 4.5 4.9 4.7 5.2 5.3 5.3 5.3 5.6 5.7 4.160-64 7.1 7.2 6.6 5.6 5.6 5.6 6.1 6.2 6.0 6.4 6.4 6.6 6.6 7.0 7.0 6.365-69 5.8 6.7 6.3 6.8 6.6 6.8 6.4 6.8 6.4 7.2 7.3 7.6 7.8 8.3 7.8 5.3>70 28.0 30.0 28.0 29.9 30.5 31.1 33.2 35.1 35.7 39.4 39.5 39.9 41.7 44.6 44.8 25.2


Female 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 20050-1 14.2 15.3 14.5 12.8 12.7 12.7 11.6 11.5 10.4 11.4 10.7 9.9 9.5 9.3 8.91-4 8.2 9.8 7.7 6.3 6.1 7.0 5.5 5.5 4.5 5.3 5.3 4.9 4.7 3.8 4.15-9 3.2 3.7 3.2 2.7 2.5 3.1 2.6 2.9 2.1 2.4 2.4 2.3 2.2 2.2 2.310-14 2.2 2.5 2.2 2.0 2.0 2.1 1.9 2.1 1.8 1.9 2.0 2.0 2.0 1.9 2.015-19 2.3 2.6 2.3 2.3 2.2 2.3 2.2 2.3 2.1 2.2 2.4 2.3 2.3 2.5 2.520-24 2.7 2.9 2.8 2.7 2.7 2.7 2.6 2.8 2.6 2.8 3.1 2.9 3.0 3.0 3.125-29 3.1 3.0 3.2 3.2 3.1 3.1 3.1 3.2 3.0 3.2 3.4 3.1 3.4 3.5 3.630-34 3.4 3.5 3.5 3.6 3.4 3.4 3.4 3.6 3.4 3.7 3.9 4.0 4.0 4.1 4.035-39 3.7 3.9 3.9 3.8 4.1 4.1 4.2 4.3 4.3 4.5 4.8 4.8 4.8 5.0 5.340-44 3.8 3.9 4.1 4.3 4.3 4.5 4.6 4.8 4.9 5.3 5.5 5.9 5.8 6.0 6.145-49 4.1 4.2 4.6 4.6 4.9 5.1 5.4 5.6 5.7 6.2 6.5 6.9 7.0 7.2 7.850-54 5.1 5.4 5.5 5.5 5.5 5.7 5.9 6.3 6.6 7.3 7.7 8.2 8.4 8.7 9.255-59 5.5 6.1 6.2 6.3 6.5 6.9 6.8 7.3 7.4 7.7 8.2 8.7 8.8 9.4 10.360-64 6.9 7.4 7.8 8.4 8.2 9.0 9.1 9.5 9.7 10.4 10.6 11.2 11.0 11.5 11.965-69 7.8 8.3 8.7 9.1 9.3 10.1 10.4 10.9 11.2 11.7 12.6 13.7 13.6 14.0 14.8>70 44.0 47.0 48.5 51.7 54.2 58.0 58.2 59.7 60.6 63.3 66.7 72.9 71.7 72.9 79.8


Female 2006 2007 2008 2009

0-1 9.0 8.9 9.3 9.2

1-4 4.4 3.9 4.1 4.4

5-9 2.4 2.1 2.2 2.3

10-14 2.1 1.9 2.1 2.0

15-19 2.5 2.6 2.7 2.8

20-24 3.2 3.3 3.3 3.4

25-29 3.8 3.7 3.7 4.0

30-34 4.2 4.1 4.3 4.4

35-39 5.4 5.5 5.5 5.5

40-44 6.4 6.3 6.5 7.0

45-49 8.1 8.0 8.4 8.7

50-54 9.7 10.0 10.2 10.7

55-59 11.0 11.1 11.8 12.6

60-64 12.5 12.5 13.3 14.3

65-69 15.0 15.3 15.6 16.2

>70 82.9 84.5 89.9 93.4


45

Total 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 19750-1 55.5 57.3 56.4 57.3 56.6 58.0 59.3 60.7 59.8 63.7 58.0 59.7 65.7 67.9 63.5 65.21-4 35.7 37.2 37.5 40.4 39.5 45.8 46.0 40.1 40.4 38.0 32.8 39.3 47.9 50.3 40.7 36.45-9 7.7 8.6 9.6 10.0 11.4 11.2 11.0 10.6 11.5 10.6 9.2 10.1 12.5 11.7 11.5 10.310-14 3.4 3.6 4.1 3.9 4.4 4.2 4.6 4.6 5.1 4.8 4.5 4.5 5.5 5.1 5.1 5.015-19 3.7 3.4 3.7 3.5 4.0 4.1 4.4 4.5 5.2 5.1 5.6 5.3 6.2 5.5 5.8 6.420-24 4.8 4.9 5.3 5.1 5.5 5.3 5.3 5.5 6.5 6.2 6.5 7.2 7.7 6.9 7.8 7.825-29 5.0 5.3 5.7 5.4 5.7 5.9 6.0 6.5 7.0 6.6 6.5 6.9 6.9 6.4 7.0 7.330-34 5.1 5.3 5.7 5.2 5.8 5.7 5.9 6.3 7.3 6.5 6.8 7.1 7.8 7.0 7.4 6.935-39 5.5 5.8 6.4 5.9 6.6 6.5 6.7 7.3 8.7 7.3 7.5 8.0 8.8 8.0 8.8 8.640-44 5.2 5.5 6.1 5.5 6.1 6.0 6.1 6.6 8.0 6.9 7.2 7.6 8.2 8.0 8.8 8.445-49 6.2 6.9 6.7 6.4 6.8 7.1 6.8 7.2 8.7 7.6 7.7 7.8 8.8 9.3 9.4 9.250-54 5.8 6.4 7.1 6.6 7.4 7.3 7.4 7.7 9.3 7.8 8.1 8.5 9.8 9.4 10.4 9.655-59 6.1 6.6 6.4 6.3 6.6 7.1 7.2 8.4 10.1 9.1 9.2 9.5 10.1 9.8 10.7 10.460-64 6.6 6.9 8.3 8.1 9.1 8.7 7.6 8.8 10.4 9.0 9.8 10.5 12.4 12.1 14.6 15.265-69 5.2 5.7 5.7 5.7 5.7 7.3 7.9 9.4 11.4 10.1 10.4 9.9 11.0 11.6 12.4 12.3>70 33.5 36.3 38.2 37.5 39.3 43.1 41.1 44.5 46.7 40.0 42.3 46.4 54.3 51.6 57.5 50.7

Table 7.11: The age-specific total deaths from 1960 - 1975

Total 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 19900-1 74.8 76.3 73.6 71.8 65.7 64.4 61.7 64.3 56.9 54.6 52.3 51.1 47.2 43.0 39.61-4 39.5 38.7 36.3 40.7 36.0 38.9 39.3 44.3 39.3 42.5 36.5 42.5 32.4 30.1 23.95-9 11.5 11.7 10.8 10.6 9.4 9.5 9.8 10.7 9.9 10.9 10.2 12.1 11.0 10.0 8.710-14 5.3 5.6 5.6 5.6 5.3 5.2 5.3 5.7 5.6 5.6 5.6 5.7 5.7 5.5 5.515-19 6.9 7.0 6.9 7.4 7.0 6.9 7.2 7.3 7.2 8.1 7.8 7.7 7.8 7.3 7.020-24 8.0 8.7 9.0 9.1 9.0 9.0 9.4 9.8 10.0 10.8 10.7 10.5 10.4 10.3 10.025-29 7.7 8.7 8.7 9.0 9.2 9.4 9.6 10.4 10.2 9.8 11.7 11.4 11.5 11.3 10.930-34 7.7 7.7 7.9 8.2 8.7 9.0 9.3 9.9 10.2 10.9 11.4 11.3 11.2 10.9 11.335-39 9.0 9.2 8.7 8.9 8.9 8.8 8.7 9.7 9.7 11.1 11.4 11.8 11.8 11.8 11.540-44 9.3 9.8 9.5 9.7 10.1 10.0 10.7 10.8 10.7 11.2 10.8 11.0 11.2 11.3 11.745-49 9.8 10.3 9.9 10.1 10.4 10.7 11.3 11.8 11.8 12.8 12.8 12.8 12.9 13.0 12.950-54 10.7 10.9 11.2 11.3 11.8 11.8 12.7 13.0 13.0 13.9 14.5 14.8 14.9 15.6 15.455-59 11.4 12.2 11.8 11.9 11.9 12.3 13.0 13.7 13.6 14.9 15.3 15.5 15.5 16.2 16.460-64 16.9 16.9 15.5 13.9 14.6 14.6 15.8 16.2 16.0 16.8 17.1 17.5 17.9 18.8 19.165-69 13.8 15.6 15.0 16.4 16.4 16.6 15.9 16.6 15.9 18.0 18.3 18.7 19.2 20.4 19.5>70 56.0 58.3 55.4 60.1 63.5 63.9 68.8 73.0 73.7 80.7 80.4 81.3 84.5 90.1 90.7


Total 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 20050-1 34.3 36.8 34.7 31.1 30.6 30.6 28.1 28.2 25.2 27.7 26.1 23.8 22.8 22.6 21.71-4 18.1 21.4 16.9 14.0 14.0 15.5 12.3 12.3 10.1 11.7 11.7 11.0 10.3 8.6 9.25-9 7.5 8.5 7.2 6.1 6.2 7.2 6.1 6.6 5.3 5.7 5.7 5.3 5.3 5.2 5.210-14 5.1 5.5 5.1 5.0 4.8 5.1 4.8 5.0 4.5 4.8 5.0 4.8 4.6 4.7 4.815-19 6.5 6.8 6.7 6.4 6.2 6.6 6.4 6.6 6.4 6.4 6.8 6.6 6.6 6.9 7.120-24 9.7 9.6 9.5 9.0 8.6 8.9 8.5 8.9 8.6 9.4 9.8 9.7 9.8 10.1 10.125-29 10.8 10.4 10.6 10.4 10.1 10.6 10.5 10.6 10.2 10.5 10.7 10.7 10.9 11.2 11.930-34 11.0 11.3 11.2 11.2 10.7 11.1 11.1 11.5 11.1 11.9 12.5 12.6 12.7 13.0 12.735-39 11.7 12.1 12.3 12.2 12.4 13.0 12.9 13.5 13.5 13.9 14.6 14.7 15.0 15.5 16.240-44 12.0 12.4 12.9 13.5 13.2 14.1 14.4 15.0 15.1 16.0 16.7 17.5 17.8 18.3 18.745-49 12.8 13.3 14.3 14.8 15.5 16.1 16.8 17.9 18.0 18.9 19.9 21.0 21.5 22.1 23.550-54 15.1 16.1 16.5 16.9 16.8 17.8 18.3 19.8 20.3 22.2 23.7 25.0 25.8 26.5 27.855-59 16.4 17.8 18.5 19.4 19.7 21.0 21.2 22.4 22.6 23.5 24.7 26.3 26.8 28.6 31.560-64 18.8 20.4 21.4 23.2 23.1 25.1 25.6 26.8 27.6 29.2 30.1 31.5 31.4 32.6 33.265-69 19.7 21.0 22.2 23.5 24.6 26.1 26.9 28.1 29.0 30.2 32.6 34.9 35.0 35.9 38.3>70 88.7 94.8 97.6 104.3 107.5 115.1 114.9 119.1 120.0 124.4 130.9 140.8 139.2 140.7 153.6


46

Total 2006 2007 2008 2009

0-1 21.8 21.7 22.4 21.7

1-4 9.6 8.7 9.1 9.8

5-9 5.6 5.1 5.1 5.4

10-14 5.0 4.7 4.9 4.9

15-19 7.4 7.6 7.8 8.2

20-24 10.4 10.2 10.5 11.0

25-29 12.3 12.1 12.2 12.5

30-34 13.1 12.7 13.2 13.6

35-39 16.5 16.4 16.5 16.5

40-44 19.3 19.1 19.7 20.5

45-49 24.5 24.1 24.9 25.8

50-54 28.8 29.6 30.2 31.8

55-59 32.9 33.5 35.4 37.0

60-64 35.0 35.3 37.5 40.5

65-69 39.3 39.7 41.0 42.8

>70 159.2 161.1 171.2 178.2


47

APPENDIX C

Male 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 19750-1 0.070 0.062 0.073 0.072 0.049 0.048 0.047 0.047 0.047 0.047 0.060 0.059 0.054 0.053 0.048 0.0491-4 0.013 0.009 0.009 0.009 0.009 0.010 0.010 0.008 0.008 0.007 0.007 0.008 0.010 0.010 0.008 0.0075-9 0.003 0.002 0.002 0.002 0.003 0.003 0.002 0.002 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002

10-14 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00115-19 0.001 0.001 0.001 0.001 0.001 0.001 0.002 0.001 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.00220-24 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.003 0.002 0.003 0.003 0.003 0.003 0.003 0.00325-29 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.00330-34 0.003 0.004 0.004 0.004 0.003 0.004 0.003 0.004 0.004 0.004 0.004 0.004 0.004 0.003 0.003 0.00335-39 0.003 0.004 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.004 0.005 0.00540-44 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.00645-49 0.006 0.007 0.006 0.006 0.007 0.007 0.007 0.007 0.008 0.007 0.007 0.007 0.008 0.008 0.009 0.00850-54 0.008 0.008 0.009 0.008 0.010 0.009 0.009 0.009 0.010 0.009 0.010 0.010 0.011 0.010 0.011 0.01055-59 0.009 0.011 0.010 0.010 0.011 0.012 0.012 0.014 0.014 0.014 0.013 0.013 0.014 0.013 0.014 0.01360-64 0.017 0.013 0.017 0.015 0.022 0.020 0.017 0.019 0.019 0.018 0.016 0.017 0.019 0.019 0.022 0.02265-69 0.016 0.017 0.016 0.016 0.019 0.024 0.026 0.030 0.032 0.029 0.025 0.023 0.024 0.024 0.025 0.024>70 0.052 0.099 0.097 0.091 0.084 0.091 0.086 0.094 0.092 0.080 0.054 0.060 0.069 0.089 0.096 0.061

Table 7.15: The age-specific male death rates from 1960 - 1975

Male 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 19900-1 0.055 0.052 0.051 0.049 0.047 0.047 0.044 0.045 0.040 0.037 0.035 0.034 0.031 0.028 0.0251-4 0.007 0.007 0.006 0.007 0.006 0.007 0.007 0.008 0.007 0.007 0.006 0.007 0.005 0.005 0.004

9-May 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.002 0.001 0.002 0.002 0.001 0.00110-14 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00115-19 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.00120-24 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.002 0.00225-29 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.004 0.003 0.003 0.003 0.00330-34 0.003 0.003 0.003 0.004 0.004 0.004 0.004 0.004 0.004 0.007 0.004 0.004 0.004 0.004 0.00435-39 0.005 0.005 0.005 0.005 0.005 0.004 0.004 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.00440-44 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.00545-49 0.009 0.009 0.009 0.009 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.00850-54 0.011 0.010 0.011 0.011 0.011 0.011 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.01155-59 0.013 0.014 0.013 0.013 0.014 0.014 0.015 0.015 0.015 0.016 0.016 0.015 0.016 0.015 0.01460-64 0.025 0.021 0.020 0.018 0.020 0.020 0.022 0.022 0.022 0.021 0.022 0.022 0.021 0.022 0.02165-69 0.023 0.028 0.027 0.029 0.028 0.027 0.026 0.029 0.028 0.031 0.029 0.029 0.028 0.029 0.028>70 0.064 0.070 0.075 0.081 0.074 0.072 0.073 0.071 0.072 0.072 0.069 0.069 0.068 0.070 0.070

Table 7.16: The age-specific male death rates from 1976 - 1990

48

Male 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 20050-1 0.021 0.022 0.020 0.017 0.018 0.018 0.016 0.016 0.015 0.016 0.015 0.014 0.013 0.012 0.0121-4 0.003 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.002 0.002 0.002 0.001 0.001 0.0015-9 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

10-14 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00115-19 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00120-24 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.00225-29 0.003 0.003 0.003 0.003 0.002 0.003 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.00230-34 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.00335-39 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.00440-44 0.005 0.006 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.00545-49 0.007 0.007 0.007 0.007 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.00850-54 0.010 0.011 0.010 0.010 0.011 0.010 0.010 0.011 0.010 0.011 0.011 0.011 0.011 0.012 0.01155-59 0.014 0.014 0.015 0.015 0.016 0.016 0.015 0.016 0.015 0.015 0.015 0.015 0.015 0.019 0.01760-64 0.020 0.020 0.021 0.022 0.023 0.024 0.023 0.024 0.023 0.024 0.023 0.024 0.023 0.025 0.02565-69 0.027 0.030 0.029 0.030 0.034 0.032 0.032 0.032 0.032 0.032 0.033 0.034 0.033 0.038 0.035>70 0.067 0.067 0.069 0.072 0.082 0.085 0.080 0.080 0.076 0.075 0.076 0.076 0.073 0.080 0.084

Table 7.17: The age-specific death rates from 1991 - 2005

Male 2006 2007 2008 2009

0-1 0.012 0.012 0.012 0.011

1-4 0.001 0.001 0.001 0.001

5-9 0.001 0.001 0.001 0.001

10-14 0.001 0.001 0.001 0.001

15-19 0.001 0.001 0.001 0.001

20-24 0.002 0.002 0.002 0.002

25-29 0.002 0.002 0.002 0.002

30-34 0.003 0.003 0.003 0.003

35-39 0.004 0.004 0.004 0.004

40-44 0.005 0.005 0.005 0.005

45-49 0.008 0.007 0.008 0.008

50-54 0.012 0.011 0.012 0.012

55-59 0.017 0.017 0.018 0.018

60-64 0.026 0.026 0.027 0.028

65-69 0.036 0.037 0.036 0.037

>70 0.085 0.082 0.087 0.089

Table 7.18: The age-specific death rates from 2006 - 2009

Female 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 19750-1 0.054 0.047 0.057 0.056 0.036 0.035 0.035 0.034 0.032 0.034 0.045 0.046 0.041 0.040 0.037 0.0361-4 0.012 0.009 0.008 0.008 0.008 0.009 0.009 0.007 0.007 0.006 0.006 0.007 0.009 0.009 0.007 0.0065-9 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

10-14 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00115-19 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00120-24 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.00125-29 0.002 0.002 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.00230-34 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.00235-39 0.003 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.005 0.004 0.003 0.003 0.003 0.003 0.003 0.00340-44 0.004 0.004 0.004 0.004 0.005 0.004 0.004 0.004 0.005 0.004 0.004 0.004 0.004 0.004 0.004 0.00445-49 0.004 0.006 0.005 0.005 0.005 0.005 0.005 0.005 0.006 0.005 0.005 0.004 0.005 0.006 0.005 0.00550-54 0.006 0.006 0.006 0.006 0.007 0.007 0.006 0.006 0.008 0.006 0.005 0.006 0.007 0.006 0.006 0.00555-59 0.008 0.007 0.007 0.007 0.008 0.008 0.008 0.009 0.011 0.008 0.007 0.007 0.008 0.007 0.008 0.00760-64 0.013 0.012 0.013 0.012 0.015 0.015 0.011 0.013 0.016 0.012 0.011 0.011 0.013 0.011 0.014 0.01365-69 0.015 0.015 0.015 0.014 0.014 0.018 0.018 0.020 0.027 0.020 0.018 0.016 0.017 0.017 0.017 0.017>70 0.056 0.107 0.102 0.093 0.080 0.088 0.083 0.087 0.090 0.073 0.052 0.057 0.067 0.087 0.094 0.059

Table 7.19: The age-specific female death rates from 1960 - 1975

49

Female 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 19900-1 0.038 0.042 0.041 0.039 0.033 0.035 0.033 0.034 0.029 0.028 0.026 0.026 0.024 0.021 0.0191-4 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.007 0.006 0.006 0.005 0.006 0.005 0.004 0.0035-9 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

10-14 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00115-19 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00120-24 0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00125-29 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.002 0.001 0.002 0.001 0.001 0.001 0.001 0.00130-34 0.002 0.002 0.002 0.021 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.00135-39 0.003 0.003 0.003 0.003 0.003 0.002 0.002 0.002 0.002 0.003 0.002 0.002 0.002 0.002 0.00240-44 0.004 0.004 0.004 0.004 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.00345-49 0.005 0.005 0.005 0.004 0.004 0.004 0.018 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.00450-54 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.004 0.00555-59 0.008 0.007 0.006 0.006 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.007 0.008 0.00760-64 0.013 0.015 0.012 0.010 0.012 0.012 0.012 0.013 0.012 0.013 0.012 0.012 0.009 0.011 0.01165-69 0.022 0.021 0.017 0.018 0.018 0.018 0.016 0.019 0.018 0.019 0.018 0.018 0.017 0.017 0.016>70 0.064 0.072 0.068 0.071 0.064 0.063 0.066 0.062 0.062 0.067 0.065 0.062 0.064 0.066 0.060

Table 7.20: The age-specific female death rates from 1976 - 1990

Female 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 20050-1 0.016 0.018 0.015 0.013 0.014 0.014 0.012 0.012 0.011 0.012 0.011 0.010 0.010 0.009 0.0091-4 0.002 0.003 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.0015-9 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.000 0.000 0.000 0.000

10-14 0.001 0.001 0.001 0.001 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.00015-19 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00120-24 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00125-29 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00130-34 0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.

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