School of Education, Culture and Communication
Division of Applied Mathematics
Comparison of mortality rate forecasting using the Second Order
Lee–Carter method with different mortality models
by
Hisham Sulemana
MASTER THESIS IN MATHEMATICS/ APPLIED MATHEMATICS
DIVISION OF APPLIED MATHEMATICS
MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN
School of Education, Culture and Communication
Division of Applied Mathematics
Master thesis in mathematics / applied mathematics
Date:
2019-05-27
Project name:
Comparison of mortality rate forecasting using the Second Order Lee–Carter method with dif-ferent mortality models
Author:
Hisham Sulemana
Supervisor(s):
Milica Ranˇci´c and Karl Lundengård
Reviewer: Ying Ni Examiner: Anatoliy Malyarenko Comprising: 30 ECTS credits
Acknowledgements
I want to thank and express my sincere gratitude to my supervisors Milica Ranˇci´c and Karl Lun-dengård for their patience, motivation, guidance and support that they gave to me throughout the thesis. I would like to thank the reviewer, Dr Ying Ni for her support. I want to express my sincere gratitude and thanks to the examiner, Professor Anatoliy Malyarenko for his patience, guidance, motivation and immense knowledge I have gained from him throughout my studies. Finally, I want to thank my family for their encouragement and support throughout my studies.
Abstract
Mortality information is very important for national planning and health of a country. Mortal-ity rate forecasting is a basic contribution for projection of financial improvement of pension plans, well-being and social strategy planning. In the first part of the thesis, we fit the selec-ted mortality rate models, namely the Power-exponential function based model, the Modified Perks model and the Heligman and Pollard (HP4) model to the data obtained from the Human Mortality Database [22] for the male population ages 1–70 of the USA, Japan and Australia. We observe that the Heligman and Pollard (HP4) model performs well and better fit the data as compared to the Power-exponential function based model and the Modified Perks model. The second part is to systematically compare the quality of the mortality rate forecasting using the Second order Lee–Carter method with the selected mortality rate models. The results indicate that Power-exponential function based model and the Heligman and Pollard (HP4) model give a more reliable forecast depending on individual countries.
Keywords: Mortality rate, Power-exponential function based model, Modified Perks model, Heligman and Pollard (HP4) model, Model fitting, Forecasting, Central death rate, Second order Lee–Carter method, Mortality indices, Comparison.
Contents
1 Introduction 9
1.1 Contribution Statement . . . 9
1.2 A Review of Mortality Rate Models . . . 10
1.3 Approaches to Forecasting Mortality Rate . . . 14
2 Selected Mortality Rate Models 15 2.1 The Survival Function . . . 15
2.2 Mortality Data . . . 17
2.3 Model Fitting . . . 17
2.4 Power-Exponential Function Based Model . . . 18
2.5 Modified Perks Model . . . 21
2.6 Heligman and Pollard (HP4) Model . . . 24
2.7 Comparison Between Models . . . 27
3 Selected Approach to Forecasting 32 3.1 Second Order Lee–Carter Model . . . 32
3.1.1 Forecasting Mortality Indices . . . 35
4 Forecasting and Comparison of Models 38 4.1 Mortality Rate Forecasting . . . 38
4.3 Comparison of Forecast Mortality Rate Models . . . 53
5 Conclusion and Future Work 54
5.1 Project Summary . . . 54 5.2 Future Work . . . 55
6 Summary of reflection of objectives in the thesis 56
6.1 Objective 1: Knowledge and understanding . . . 56 6.2 Objective 2: Methodological knowledge . . . 56 6.3 Objective 3: Critically and Systematically Integrate Knowledge . . . 57 6.4 Objective 4: Independently and Creatively Identify and Carry out Advanced
Tasks . . . 57 6.5 Objective 5: Present and Discuss Conclusions and Knowledge . . . 57 6.6 Objective 6: Scientific, Social and Ethical Aspects . . . 57
Bibliography 57
List of Figures
2.1 The logarithm of raw data and mortality rates fitted by the Power-exponential function based model for the male population of the USA in the years 1980, 1995 and 2005. . . 19 2.2 The logarithm of raw data and mortality rates fitted by the Power-exponential
function based model for the male population of Japan in the years 1980, 1995 and 2005. . . 20 2.3 The logarithm of raw data and mortality rates fitted by the Power-exponential
function based model for the male population of Australia in the years 1980, 1995 and 2005. . . 20 2.4 The logarithm of raw data and mortality rates fitted by the Modified Perks
model for the male population of the USA in the year 1980, 1995 and 2005. . . 22 2.5 The logarithm of raw data and mortality rates fitted by the Modified Perks
model for the male population of Japan in the year 1980, 1995 and 2005. . . 22 2.6 The logarithm of raw data and mortality rates fitted by the Modified Perks
model for the male population of Australia in he year 1980, 1995 and 2005. . . 23 2.7 The logarithm of raw data and mortality rates fitted by the Heligman and
Pol-lard (HP4) model for the male population of the USA in the years 1980, 1995 and 2005. . . 25
2.8 The logarithm of raw data and mortality rates fitted by the Heligman and Pol-lard (HP4) model for the male population of Japan in the years 1980, 1995 and 2005. . . 26 2.9 The logarithm of raw data and mortality rates fitted by the Heligman and
Pol-lard (HP4) model for the male population of Australia in the years 1980, 1995 and 2005. . . 26 2.10 Comparison between the models for the male population of the USA in 1980. . 27 2.11 Comparison between the models for the male population of the USA in 1995. . 28 2.12 Comparison between the models for the male population of the USA in 2005. . 28 2.13 Comparison between the models for the male population of Japan in 1980. . . . 29 2.14 Comparison between the models for the male population of Japan in 1995 . . . 29 2.15 Comparison between the models for the male population of Japan in 2005. . . . 30 2.16 Comparison between the models for the male population of Australia in 1980. . 30 2.17 Comparison between the models for the male population of Australia in 1995 . 31 2.18 Comparison between the models for the male population of Australia in 2005. . 31 4.1 Comparison of fitted and forecast mortality rates for the male population of the
USA for the raw data in the year 2000–2010 with 95% predictive confidence intervals. . . 39 4.2 Comparison of fitted and forecast mortality rates for the male population of
the USA for the Power-exponential model in the year 2000–2010 with 95% predictive confidence intervals. . . 40 4.3 Comparison of fitted and forecast mortality rates for the male population of
the USA for Modified Perks model in the year 2000–2010 with 95% predictive confidence intervals. . . 40 4.4 Comparison of fitted and forecast mortality rates for the male population of the
USA for Heligman and Pollard (HP4) model in the year 2000–2010 with 95% predictive confidence intervals. . . 41
4.5 Comparison of fitted and forecast mortality rates for the male population of Japan for the raw data in the year 2000–2010 with 95% predictive confidence intervals. . . 41 4.6 Comparison of fitted and forecast mortality rates for the male population of
Japan for the Power-exponential model in the year 2000–2010 with 95% pre-dictive confidence intervals. . . 42 4.7 Comparison of fitted and forecast mortality rates for the male population of
Japan for Modified Perks model in the year 2000–2010 with 95% predictive confidence intervals. . . 42 4.8 Comparison of fitted and forecast mortality rates for the male population of
Japan for Heligman and Pollard (HP4) model in the year 2000–2010 with 95% predictive confidence intervals. . . 43 4.9 Comparison of fitted and forecast mortality rates for the male population of
Australia for the raw data in the year 2000–2010 with 95% predictive confid-ence intervals. . . 43 4.10 Comparison of fitted and forecast mortality rates for the male population of
Australia for the Power-exponential model in the year 2000–2010 with 95% predictive confidence intervals. . . 44 4.11 Comparison of fitted and forecast mortality rates for the male population of
Australia for Modified Perks model in the year 2000–2010 with 95% predictive confidence intervals. . . 44 4.12 Comparison of fitted and forecast mortality rates for the male population of
Australia for Heligman and Pollard (HP4) model in the year 2000–2010 with 95% predictive confidence intervals. . . 45 4.13 Estimated from 1970–2010 and forecast for 2011–2050 using the first mortality
index (k1) of the USA with 95% predictive intervals for both raw data and the
4.14 Estimated from 1970–2010 and forecast for 2011–2050 using the second mor-tality index (k2) of the USA with 95% predictive confidence intervals for both
raw data and the three models. . . 48 4.15 Estimated from 1970–2010 and forecast for 2011–2050 using the first mortality
index (k1) of Japan with 95% predictive confidence intervals for both raw data
and the three models. . . 49 4.16 Estimated from 1970–2010 and forecast for 2011–2050 using the second
mor-tality index (k2) of Japan with 95% predictive confidence intervals for both raw
data and the three models. . . 50 4.17 Estimated from 1970–2010 and forecast for 2011–2050 using the first mortality
index (k1) of Australia with 95% predictive confidence intervals for both raw
data and the three models. . . 51 4.18 Estimated from 1970–2010 and forecast for 2011–2050 using the second
mor-tality index (k2) of Australia with 95% predictive confidence intervals for both
List of Tables
2.1 Estimate parameters for the Power-exponential function based model for the USA, Japan and Australia in the year 1980, 1995 and 2005 . . . 19 2.2 Estimate parameters for the Modified Perks model for the male population of
the USA, Japan and Australia in the year 1980, 1995 and 2005 . . . 21 2.3 Estimate parameters for the HP4 model for the USA, Japan and Australia in the
year 1980, 1995 and 2005. . . 25 4.1 Standard error estimate (see) of forecast mortality indices. . . 53
Chapter 1
Introduction
In the field of actuarial science, demography, national planning and health, mortality inform-ation is a vital component. Mortality projections are a basic contribution for projections of financial improvement of pension plans, well-being and social strategy planning. Governments and insurance companies mostly depend on good mortality rate forecasting for pension calcu-lation. Mortality rate forecasting is fundamental for long-term planning of a country. Along these lines, proposing a better model for mortality rate forecasting can encourage a country to develop a good quality of life [24]. In this thesis, we will centre on the types of mortality rates that insurance companies are interested in to be able to make a future decision. The main objective of this thesis is to systematically compare the results of mortality rate forecasting by applying the Second Order Lee–Carter method with different mortality rate models.
After an introduction, we will talk briefly regarding our contribution to the subject, review several models for modelling mortality rates and then discuss approaches to forecasting mor-tality rate. In Chapter 2, we will focus on specific mormor-tality rate models, namely, the Power-exponential function based model [25], the Modified Perks model [37] and the Heligman and Pollard (HP4) model [26]. In Chapter 3, we will present the approach that we will use to forecast the mortality rate data and the general description of mathematical and computational tools. In Chapter 4, we will discuss results regarding mortality rate forecasting, analysis of forecast mortality indices and comparison of forecast mortality rate models. We conclude in Chapter 5 by discussing project summary and future work.
1.1
Contribution Statement
We fit the three mortality models, namely the Power-exponential function based model, the Modified Perks model and the Heligman and Pollard (HP4) model to the data obtained from the Human Mortality Database [22] for the male population ages 1–70 of the USA, Japan and Australia in the year 1980, 1995 and 2005. We observe that the Heligman and Pollard (HP4) model fits the data better for the three stages of an individual’s lifespan as compared to the other two models for the USA, Japan and Australia. Based on central mortality rates found by fitting the three models to data for the USA, Japan and Australia in the years 1970–2000, we used the Second Order Lee–Carter model to forecast 10 years into the future. The forecast (prediction)
with 95% confidence intervals is compared to fitted mortality rates for the years 2000 and 2010. We observed a decline in mortality for the three stages of an individual’s lifespan when we compare the forecast in the year 2010 to the fitted in the year 2000 for the USA, Japan and Australia. Again in the case of USA and Australia, when we compare the forecast to the fitted in the year 2010, we observe almost the same pattern for the infant and senescent mortality but slightly different with the peak of the "accident hump" for the Power-exponential function based model and the Heligman and Pollard (HP4) model. In the case of Japan, when we compare forecast to the fitted in the year 2010, we observe almost the same pattern for the three stages of an individual’s lifespan for each of the three models. We estimated the mortality indices from 1970–2010 and then forecast into the future from 2011–2050 with 95% confidence interval and we observe that from 1970–2010 for the first mortality indices for the USA, Japan and Australia, mortality tends to decline year by year for both the raw data and the three models and for the forecast area we observe that both the prediction for the raw data and the three models are indicating that mortality will decline significantly by the year 2050 and the predictions are almost the same. But for the second mortality indices, we observe that they exhibit a less systematic pattern as compared to the first mortality indices for the USA, Japan and Australia for the estimated area and for the forecast area we observed different predictions for both the raw data and the three models as compared to the first mortality indices. Finally, we systematically compare the quality of the forecasts using the standard error estimates of the forecast mortality indices and the results indicate that the performance of each model depends on the individual countries.
1.2
A Review of Mortality Rate Models
Definition 1. In modelling future lifetime, the mortality rate is the basis and essential concept. Let mortality rate at age x be denoted by µ(x) and future lifetime at birth is denoted by T0. We
can now define mortality rate at age x as [9]: µ (x) = lim
dx→0+ 1
dxPr[T0≤ x + dx | T0> x]. Equivalently, we can also define mortality rate as:
µ (x) = lim
dx→0+ 1
dxPr[Tx≤ dx], where Txis future lifetime at age x.
Mortality rates are conventionally arduous to predict due to health (disease, accidents, change in lifestyle), conflicts, and other reasons. We have different types of mortality rates, namely, infant mortality rate, crude mortality rate, maternal mortality rate, and age-specific mortality rate. We will be looking at the age-specific mortality rate for men. There are many models that can model the mortality rate. Here we will review several models that appear in the literature. Later we will select a few and examine their suitability for mortality rate forecasting.
In 1825, Benjamin Gompertz was the first person to propose the law of mortality. He no-ticed that the human mortality rate increase exponentially with age. During that time many other people were interested in constructing life tables for actuarial purposes, but Gompertz’s investigation was able to reduce large life tables to a notably simple form. Gompertz law of mortality has become the most often used model of senescence [4, 8, 33]. Mathematically, Gompertz law of mortality is expressed by:
µ (x) = becx, (1.1)
where b and c are constant in which b represent the mortality at age 0, c is the rate of demo-graphic aging or the change of mortality rate and x is the age of the population. Equation (1.1) shows that mortality rate increases progressively with age [7, 8]. Makeham [30] extended the Gompertz model by adding a constant a, which is known as Gompertz–Makeham law defined as:
µ (x) = a + becx.
The constant a can be explained as the risk of death from all causes which do not depend on age, representing non-senescent deaths, for example, from accidents. Another model known as Double Geometric [30] which is an extension of Gompertz and Gompertz–Makeham model [30] is mathematically expressed as :
µ (x) = a + b1bx2+ c1cx2.
Oppermann [27] propose a 3-parameter formula:
µ (x) = ax−1/2+ b + cx1/3, which only applies for young ages, x≤ 20.
Thiele [34] also propose a seven parameter formula: µ (x) = a1eb1x+ a2e−b2
(x−c)2
2 + a3eb3x, (1.2)
which covers the three stages of human life. In equation (1.2), the first term denotes the infant mortality, the second term is the "accident hump " and the last term is the Gompertz law [30]. Moreover, there are two parametric frailty models proposed by Perks which is a modifica-tion of Gompertz law that assumes that frailty follows a gamma distribumodifica-tion. Frailty is an unobservable univariate statistical variable Z which defines the implications for standard life table methods [37]. We will only discuss the final expression of mortality rate. The first one is Gompertz-gamma, known as Perks (P) and it is expressed as:
µ (x) = a 1 + eb−cx.
be expressed mathematically as:
µ (x) = a
1 + eb−cx + d.
The Perks (P) and Modified Perks (MP) model, both uses frailty distribution to measure het-erogeneity in mortality data [37]. Since Modified Perks (MP) is one of the selected models, see Chapter 2 for more details. There is another parametric frailty model known as Gompertz-inverse Gaussian that combines with Gompertz law and it is expressed as:
µ (x) = e
a−bx
√
1 + e−c+bx.
According to previous studies, the model does not perform better with respect to mortality rates at the higher ages [37].
Weibull [36] also proposed a model with two parameters which formally was used for model-ling only failure rate of machines, but now can also be used for mortality and human population growth modelling [21]. Mathematically it is expressed as:
µ (x) =a b x b a−1 .
Lundengård et al [25] proposed the Power-exponential function based model which has five different parameters. Mathematically, it is expressed as:
µ (x) = c1
xe−c2x+ a1 xe
−a2xa3,
where parameters c1, c2and (a1, a2, a3) explain the qualitative properties of the curve [25]. We
will present more details regarding the model since it is one of the selected models to be used in this thesis.
Another model is known as the logistic model with three parameters. This model is used in many disciplines including mortality forecasting [5, 12, 18]. According to Thatcher [5], when he used logistic model for his data he observed that it predicted better at the higher ages. It is defined as:
µ (x) = ae
bx
1 +acb(ebx− 1).
A similar model to the logistic model known as the log-logistic model which has two paramet-ers. It is expressed as:
µ (x) = abx
a−1
Another model proposed by Harald Hannerz [19] is expressed by: µ (x) =g(x)e G(x) 1 + eG(x), where g(x) =a1 x2+ a2x+ a3e cx, and G(x) = a0− a1 x + a2x2 2 + a3 c e cx,
where a0 is a level parameter, a1, a2 and a3 are profile parameters and c is a constant held in
common by the two mortality schedules. This model was used in fitting the Swedish female population [19].
Finally, four different models proposed by Heligman and Pollard (HP) [26], each model con-tains Gompertz law as a particular case and covers all the three stages of human life.
HP1 formula is: µ (x) = a1(x+a2) a3 + b1e−b2ln x b3 2 + c1cx2. HP2 is expressed by: µ (x) = a1(x+a2) a3 + b1e−b2ln x b3 2 + c1c x 2 1 + c1cx2 . HP3 is defined as: µ (x) = a1(x+a2) a3 + b1e −b2ln x b3 2 + c1c x 2 1 + c3c1cx2 . HP4 is expressed as: µ (x) = a1(x+a2) a3 + b1e −b2ln x b3 2 + c1c xc3 2 1 + c1cx2c3 .
All HP models contain three parts and each part represents a different stage of mortality. The first part indicates a fall in mortality at infant and early childhood ages. The second part rep-resents the mortality of middle ages often known as the "accident hump". The last part is the Gompertz law [32]. We will discuss HP4 in detail in the next chapter since it is one of the selected models.
1.3
Approaches to Forecasting Mortality Rate
Since Gompertz proposed his mortality rate model in 1825, many mortality models have been proposed afterwards. Mortality modelling has become an interesting area for most research-ers. Mortality rate forecasting is used by actuaries for calculation of insurance premiums and pension funds, whiles it is used by demographers for population projections [16]. According to [16] there are three approaches to forecasting mortality rate . These approaches are expectation, explanation, and extrapolation.
Starting with expectation approach to forecasting mortality rate, this approach is based on the personal opinion of experts. Majority of the official from the statistical agencies has offered pri-ority to this approach [15, 16]. Previously, actuaries also used to rely on this approach. Expert opinion with regards to their experience in the field is considered as an advantage but it is fun-damentally subjective and thus has a potential for bias, which is considered a disadvantage [16]. Considering the second approach, explanatory approach of forecasting mortality are mostly established on structural models. The structural models describes the underlying causes of mortality. This approach to forecasting mortality rate is still to be developed in reality because human life is too complicated to model all the factors. Because of response techniques and limiting factors are taken into consideration, this approach is considered as an advantage but it cannot be used for long term forecasting [16].
Finally, considering the extrapolation approach, it is the starting point and most reliable ap-proach used by actuaries and demographers for forecasting mortality rate. This apap-proach can either be used for a short or long range of mortality rate forecasting. The extrapolative approach is based on the assumption that future trends will basically be a continuation of the past. This is usually, a reasonable assumption in mortality rate forecasting because of historical consist-encies. In extrapolative mortality rate forecasting, time series methods are often used [16, 17]. The univariate ARIMA modelling often used extrapolation approach [15, 17]. The extrapol-ation approach can be categorized according to the number of factors modelled; 0, 1, 2 or 3 [10, 11, 17]. Parameterization functions are an example of a 1-factor model that is used in fore-casting rate mortality. Lee–Carter model [31] is a 2-factor model which can be computed using the principal component approach that is singular value decomposition (SVD) method. There are many extensions of the Lee–Carter model, one of them is the Second Order Lee–Carter model [3] which we will use for forecasting our mortality data. Most of the developed nations use the Lee–Carter approach to forecasting mortality rate because of its simplicity and also it is easier to interpret the parameters straight forward. Second Order Lee–Carter model [3] will be discussed in details in Chapter 3. Another type of 2-factor model is known as generalized linear modelling (GLM). The 3-factor model, also known as age-period-cohort (APC) models and the reason for this model is to differentiate between age, period and cohort effects in the data [10, 11, 17].
Chapter 2
Selected Mortality Rate Models
Before we focus on how we obtained the mortality data, model fitting for the selected mortality rate models namely the Power-exponential function based model, the Heligman and Pollard (HP4) model and the Modified Perks (MP) model and comparison between the models. We begin with definitions, theorem, proof and properties of the survival function.
See [9] for the following Definitions, Theorem, proof and properties.
2.1
The Survival Function
Definition 2. The life time distribution function, Fx(t), is the probability of an individual at age
xdoes not survive beyond age x + t. In other words, Fx(t) = Pr[Tx≤ t].
Definition 3. The survival function, Sx(t), is the probability that an individual of age x survives
at least t years. In other words,
Theorem 1. The survival function Sxcan be expressed in terms of the mortality rate as: Sx(t) = S0(x + t) S0(t) = exp ( − Z t 0 µ(x+s) ds ) . (2.1) Proof. Sx(t) = 1 − Fx(t) = Pr(T0≤ x + t | T0> x) = S0(x + t) S0(t) . (2.2) From Definition 1: µ (x) = lim dx→0+ 1 dxPr[Tx≤ dx] = lim dx→0+ 1 dx(1 − Sx(dx)). (2.3) Since Sx(dx) =S0(x + dx) S0(x) , (2.4)
then equation (2.3) becomes:
µ (x) = 1 S0(x)dx→0lim+ S0(x) − S0(x + dx) dx = 1 S0(x) − d dxS0(x) ! . (2.5)
From equation (2.5) we can say that:
µ(x+s)=
−dxdS0(x + s) S0(x + s) = −d
dsln S0(x + s). (2.6)
integrating equation (2.6) over (0,t) we get: − Z t 0 µ(x+s)ds = ln S0(x + t) − ln S0(x) = lnS0(x + t) S0(x) = ln Sx(t). (2.7)
Hence, Sx(t) = exp ( − Z t 0 µ(x+s)ds ) = exp ( − Z x+t x µzdz ) . (2.8)
The survival function for a lifetime distribution has the following properties [9]:
Property 1: Sx(0) = 1, the probability that a person at age x survives zero years is one.
Property 2: lim
t→∞Sx(t) = 0, everybody dies eventually.
Property 3: The survival function must be a non-increasing function of t.
2.2
Mortality Data
The data is obtained from the Human Mortality Database [22]. The typical data set consists of the mortality rates for the male population ages 1–70 for the USA, Japan and Australia respect-ively. The selected countries are basically due to quality and reliable data, different population size, population density and different geographical location. All the countries have experienced major mortality improvements for a very long time. We are using male population instead of the female population because most research has shown that the female mortality rate tends to decline year by years especially the "accident hump" compared to the male population. This means that there is an improvement in the lifestyle of the female population compared to the male population. But the author in [2] used mortality rates data for the male population ages 1–100 for the USA, Sweden and Greece.
2.3
Model Fitting
We are going to fit the three models namely the Power-exponential function based model, the Modified Perks (MP) model and the Heligman and Pollard (HP4) model to the data taken from the Human Mortality Database [22]. We will use the same method of fitting that was used by the authors of [1] using software for numerical computing [29] based on the interior reflective Newton method described in [13, 14]. The same data obtained from the Human Mortality Database [22] will be used for mortality rate forecasting using Second Order Lee– Carter model and the results will be presented in Chapter 4. In Sections 2.4, 2.5 and 2.6, we present data obtained by fitting above mention models for the male population ages 1–70 of the USA, Japan and Australia in the year 1980, 1995 and 2005 to be able to observe the performance of each model so that we can make a valid conclusion as regards to improvement in mortality rates for each country. The author in [2] on the other hand observed the year 1970,
2000 and 2010 for both the USA and Sweden and the year 1981, 2000 and 2010 for Greece and only compared between models. But, we will first fit each model separately to the data obtained from the Human Mortality Database [22] to observe how each model display the three patterns of mortality and also the trend of mortality through time using both the raw data and the models. After that, we will then compare between models in the year 1980, 1995 and 2005 for the USA, Japan and Australia to be able to observe which model better fit to the data. See Section 2.7 for comparison between models.
2.4
Power-Exponential Function Based Model
As mentioned in Chapter 1, the Power-exponential function based model is a newly proposed model by Lundengård et al [25] and it describes adequately the three stages of individual lifespan. The model is establish on the basis of power exponential function f (x) = (xe1−x)β
which has also been used for modelling electrostatic discharges [25].
Definition 4. The Power-exponential function based model for a single "hump" is defined as: µ (x) = c1
xe−c2x+ a1 xe
−a2xa3,
where the parameters c1, c2and (a1, a2, a3) explain the qualitative properties of the curve [25].
Theorem 2. Single "hump" survival function Sxof the Power-exponential function based model
is expressed as: Sx(t) = exp ( c1 Ei(c2x) − Ei(c2(x + t)) + a1 (a2a3)a3+1 γ (a3+ 1, a2a3x) − γ(a3+ 1, a2a3(x + t)) ) . whereEi(x) = − Z ∞ −x e−s
s ds is the exponential integral and γ(a,t) =
Z t
0
xa−1e−x dx is the lower incomplete Gamma function as defined in [28].
For more detailed proof of Theorem 2 refer to [25].
In Table 2.1, we show the estimated parameters for the Power-exponential function based model for three years for the USA, Japan and Australia. Note that since the a3parameter is relatively
Country a1 a2 a3 c1 c2 USA 1980 5.1·10−69 0.0500 74.5604 0.0011 0.1023 USA 1995 4.6·10−57 0.0511 61.3917 0.0008 0.1061 USA 2005 1.7·10−40 0.0458 39.8717 0.0006 0.1092 Japan 1980 7.2·10−68 0.0487 72.6228 0.0012 0.1049 Japan 1995 1.1·10−50 0.0463 51.7589 0.0008 0.1080 Japan 2005 4.9·10−33 0.0413 30.3752 0.0005 0.1139 Australia 1980 1.7·10−50 0.0467 52.2802 0.0012 0.1092 Australia 1995 1.5·10−24 0.0392 21.4327 0.0007 0.1104 Australia 2005 2.6·10−24 0.0374 20.5644 0.0005 0.1116
Table 2.1: Estimate parameters for the Power-exponential function based model for the USA, Japan and Australia in the year 1980, 1995 and 2005
In Figures 2.1–2.3, we can observe that both the raw data and the fitted mortality rates obtained by the Power-exponential function based model are lower in 2005 than they were 10 and 25 years ago, which means that there is a significant decrease in mortality for the USA, Japan and Australia. We can see that there is a decline in "accident hump" for each of the three countries. The life of people has improved and also there are no major events like natural disasters and wars of the past decade, therefore a decline in mortality for the entire life span of an individual. Finally, it is observed that the Power-exponential function based model fits well for all the three stages of human life.
Figure 2.1: The logarithm of raw data and mortality rates fitted by the Power-exponential function based model for the male population of the USA in the years 1980, 1995 and 2005.
Figure 2.2: The logarithm of raw data and mortality rates fitted by the Power-exponential function based model for the male population of Japan in the years 1980, 1995 and 2005.
Figure 2.3: The logarithm of raw data and mortality rates fitted by the Power-exponential function based model for the male population of Australia in the years 1980, 1995 and 2005.
2.5
Modified Perks Model
This model is one of the performing models regarding mortality rate forecasting for the higher ages. It uses gamma frailty distribution to measure heterogeneity in mortality data. Frailty is an unobservable univariate statistical variable Z which defines the implications for standard life table methods [37]. We will discuss only the final expression of the mortality rate.
Definition 5. The Modified Perks Model is defined as : µ (x) = a
1 + eb−cx + d.
Theorem 3. The survival function Sxof Modified Perks model is expressed by:
Sx(t) = exp ( −a c " ln ec(x+t)+ eb ! − ln ecx+ eb !# − dt ) .
For more detailed proof of Theorem 3 refer to [2].
In Table 2.2, we show the estimated parameters for the Modified Perks model for three years for the USA, Japan and Australia. The parameters values for a and b for Australia are signific-antly larger than the other two countries, but note that parameters a and b primarily affects the shape for the curve for low and medium ages and since the model fit poorly for the infant and "accident hump" we can expect a lot of variation in those parameters values.
Country a b c d USA 1980 0.0296 8.2087 0.1230 0.0004 USA 1995 0.0580 8.0724 0.1032 0.0003 USA 2005 0.0322 7.5051 0.1037 0.0002 Japan 1980 0.0672 8.4249 0.1148 0.0005 Japan 1995 0.1778 9.3284 0.1070 0.0003 Japan 2005 0.0630 7.8663 0.0989 0.0002 Australia 1980 24734 20.4580 0.1018 0.0005 Australia 1995 80531 20.9346 0.0824 0.0002 Australia 2005 95529 21.2840 0.0806 0.0001
Table 2.2: Estimate parameters for the Modified Perks model for the male population of the USA, Japan and Australia in the year 1980, 1995 and 2005
Observing Figures 2.4–2.6, we can see that human mortality rates declined quite rapidly from the year 1980 to 2005. This decrease in mortality is attributed to an improvement in medical care and lifestyle of the male population for the USA, Japan and Australia. It is clearly observ-able that mortality rates fitted by the Modified Perks model do better at the higher ages, but does not show the infant or the "hump" during adulthood as shown in both the power-exponential function based model and Heligman and Pollard (HP4) model.
Figure 2.4: The logarithm of raw data and mortality rates fitted by the Modified Perks model for the male population of the USA in the year 1980, 1995 and 2005.
Figure 2.5: The logarithm of raw data and mortality rates fitted by the Modified Perks model for the male population of Japan in the year 1980, 1995 and 2005.
Figure 2.6: The logarithm of raw data and mortality rates fitted by the Modified Perks model for the male population of Australia in he year 1980, 1995 and 2005.
2.6
Heligman and Pollard (HP4) Model
Heligman and Pollard [28] proposed a mortality rate model that shows a pattern of mortality in the entire life span of a person.
Definition 6. Heligman and Pollard (HP4) model is expressed by;
µ (x) = a1(x+a2) a3 + b1e −b2ln x b3 2 + c1c xc3 2 1 + c1cx2c3 , (2.9)
where a1, a2, a3, b1, b2, b3, c1, c2, and c3are the estimated parameters. Equation (2.9) has three
parts, each part represents a different stage of mortality. The first part indicates child mortality pattern, in which parameter a1 stands for infant mortality, a2 is the mortality rate of children
at age one and a3 is the rate of decline in mortality of infant. The second part represents the
accident "hump", in which b1 is the severity of the accident "hump", b2 is the spread and b3
is the location of the accident "hump". The third part is Gompertz exponential function which represents the old age, in which c1 indicates the adult mortality of the base level, c2 is the rate
of increase in adult mortality and c3is the curvature [26, 32, 35].
Theorem 4. The survival function Sx of the Heligman and Pollard (HP4) model is expressed
by: Sx(t) = exp ( − 1 a3 −1 ln a1 !1 a3 γ 1 a3, − ln a1(x + t + a2) a3 − γ 1 a3, − ln a1(x + a2) a3 ! − b1b2b3 2 " (x + t)1−2b2− x1−2b2 1 − 2b2 # −
Z
t 0 c1c(x+s)2 c3 1 + c1c (x+s)c3 2 ds ) , where γ(a,t) = Z t 0xa−1e−x dx is the lower incomplete Gamma function as defined in [28]. For more detailed proof of Theorem 4 refer to [2].
Country a1 a2 a3 b1 b2 b3 c1 c2 c3 USA 1980 0.0071 2.3·10−14 0.4115 0.0002 25.8840 20.09351 0.0002 1.0111 1.4356 USA 1995 0.0047 2.2·10−14 0.3942 0.0002 27.8580 19.1595 0.0001 1.0261 1.2460 USA 2005 0.0044 4.3·10−14 0.4370 0.0001 24.5508 20.7319 0.0001 1.0385 1.1649 Japan 1980 0.0046 2.2·10−14 0.2376 0.0005 18.1869 20.9208 6.0·10−5 1.0737 1.0502 Japan 1995 0.0031 2.3·10−14 0.2655 0.0004 16.7470 21.7830 0.0001 1.0164 1.3707 Japan 2005 0.0019 2.3·10−14 0.2338 0.0003 12.3764 23.1329 3.0·10−5 1.1141 0.9632 Australia 1980 0.0077 2.3·10−14 0.4175 0.0013 21.0284 21.4957 0.0002 1.0083 1.5202 Australia 1995 0.0042 2.3·10−14 0.3516 0.0010 7.4833 25.8950 0.0001 1.0116 1.4532 Australia 2005 0.0037 4.2·10−14 0.3884 0.0006 9.6382 25.8008 6.8·10−5 1.0331 1.2103
Table 2.3: Estimate parameters for the HP4 model for the USA, Japan and Australia in the year 1980, 1995 and 2005.
Based on Figures 2.7–2.9, we can observe that mortality rates declined significantly in the past 25 years as shown by the raw data and mortality rates fitted by the Heligman and Pollard (HP4) model for the USA, Japan and Australia. We can see that there is a decline in "accident hump" for each of the three countries. This means that there are improvements in the lifestyle and medical care of the male population in the past 25 years. Heligman and Pollard (HP4) model fit the data very well for all the three stages of human life.
Figure 2.7: The logarithm of raw data and mortality rates fitted by the Heligman and Pollard (HP4) model for the male population of the USA in the years 1980, 1995 and 2005.
Figure 2.8: The logarithm of raw data and mortality rates fitted by the Heligman and Pollard (HP4) model for the male population of Japan in the years 1980, 1995 and 2005.
Figure 2.9: The logarithm of raw data and mortality rates fitted by the Heligman and Pollard (HP4) model for the male population of Australia in the years 1980, 1995 and 2005.
2.7
Comparison Between Models
In Figures 2.10–2.18, we can observe a typical example of the performance of each model with regard to mortality rates for the male population ages 1–70 of the USA, Japan and Australia. In Figures 2.10–2.12, we can see that the Power-exponential function based model and the Helig-man and Pollard (HP4) model describe the infant mortality and "accident hump" very well compared to the Modified Perks model for the USA, Japan and Australia. We can observe that the Modified Perks model only performs better at higher ages for each of the three countries. Mortality rates fitted by the Power-exponential function based model and the Heligman and Pollard (HP4) model shows a decline in the "accident hump" for the male population of USA, see Figure 2.10 and Figure 2.12.
Based on Figures 2.13–2.15, we can observe that the three models have performed well with regard to the decrease in mortality throughout the three stages of human life. This means that there is an improvement in mortality. The "accident hump" has improved for the male popula-tion of Japan, see Figure 2.13 and Figure 2.15.
From Figures 2.16–2.18, we can observe that the three models performed well with regard to the decrease in mortality rates. There is also a decline in "accident hump" for the male pop-ulation of Australia. The Heligman and Pollard (HP4) model better describe and fit the data well compared to the Power-exponential function based model and the Modified Perks model for the USA, Japan and Australia. We can conclude that life expectancy has improved for each of the three countries.
Figure 2.11: Comparison between the models for the male population of the USA in 1995.
Figure 2.13: Comparison between the models for the male population of Japan in 1980.
Figure 2.15: Comparison between the models for the male population of Japan in 2005.
Figure 2.17: Comparison between the models for the male population of Australia in 1995 .
Chapter 3
Selected Approach to Forecasting
3.1
Second Order Lee–Carter Model
Lee and Carter [31] proposed a statistical model for forecasting mortality rate and it has become one of the best methods used in the field of actuarial science and demography. The method de-veloped by Lee and Carter was used specifically for U.S mortality data for the year 1933–1987 [31]. Now, this method is world known and widely used by many countries for mortality fore-casting. There are extensions of the Lee–Carter model in order to improve the method, one of them is the second order Lee–Carter model. In the Lee–Carter model, only the first term of singular value decomposition (SVD) vectors which account for the highest percentage of the explained variance in the logarithm of central death rates are used for forecasting. The second order Lee–Carter model was first modelled by Booth, Maindonald, and Smith [27] to fit the Australian mortality data by including the second term of the singular value decomposition (SVD) vectors in the approximation.
Similarly, Renshaw and Haberman [3] also used the extended singular value decomposition (SVD) method to modelled and forecast mortality data for England and Wales using univariate ARIMA processes. They realized that using the Lee–Carter model to forecast England and Wales males mortality data especially for the year 1950-1998, it failed to capture the crude death rate for ages between 29 and 30 years. This problem was due to death related cases like HIV infection and AIDS and also an increased in the number of suicide cases. The extension of the Lee–Carter model allows flexibility in forecasting change [3, 16, 18].
Definition 7. The Central death rate mxfor age x is defined as the ratio of the number of deaths
within a period of time to the average number alive during that period of time. Mathematically, it is expressed as:
mx=
dx
Lx,
where dxis the number of deaths at age x within a period of time and Lxis the average number
In Chapter 2, we defined mortality rate or the force of mortality in terms of survival func-tion in equafunc-tion (2.5), similarly we can defined mortality rate in term of lx as the instantaneous
rate of decrease in lx divided by the value of lx. In order words,
µ (x) = −1 lx dlx dx = − d dx(ln lx),
where lx is the number who live up to age x. Since death occurs for different ages within
different time interval, suppose we have µ(x+t), this means that death occurs at every age x + t, where 0 < t < 1. The central death rate mxfor age x can be regarded as an average of all deaths
within that time interval. The central death rate mx for age x is therefore approximately equal
to the mortality rate in the middle of that time interval [6]. In order words, mx≈ µ(x+1
2). (3.1)
Definition 8. The Lee–Carter Model [31] is expressed by : ln(mx,t) = ax+ bxkt+ εx,t,
where
• x is the age group ( x = 1, ..., n) • t is the time in years ( t = 1, ..., T )
• mx,t, is the central death rate for age x at time t
• ax, is the average (over time) of the ln(mx,t)
• bx, describes how the mortality at each age varies when the general level of mortality
changes
• kt, describes the variation in the level of mortality in year t
• εx,t, is the error term, which assume to be normally distributed N(0, σε2).
bx and kt are estimated using the singular value decomposition (SVD) approach with the
con-straints that the estimated kt sum up to zero and estimated bxsum up to one. ˆkt can be forecast
using a random walk with drift model.
This first order Lee–Carter model was used by the author in [2] to forecast mortality rates for the USA, Sweden and Greece while we will use the Second Order Lee–Carter model to forecast the mortality rates for the USA, Japan and Australia with 95% confidence interval. Moreover, we will estimate the mortality indices from 1970–2010 and then forecast for 2011–2050 with 95% confidence interval using both the raw data and the three models to be able to observe what will be the future trend with regard to improvement in mortality rates. Also we will systemat-ically compare the quality of the forecast for the raw data and three models using the standard error estimates of the forecast mortality indices for the USA, Japan and Australia. The results
will be presented in Chapter 4.
Definition 9. The Second Order Lee–Carter model [3] is expressed by: ln(mx,t) = ax+ b (1) x k (1) t + b (2) x k (2) t + εx,t, where
• x is the age group ( x = 1, ..., n) • t is the time in years ( t = 1, 2, ..., T )
• mx,t, is the central death rate for age x at time t
• ax, is the average (over time) of the ln(mx,t)
• b(1)x , b(2)x , describes how the mortality at each age varies when the general level of
mor-tality changes
• k(1)t , kt(2), these are mortality indices that describes the variation in the level of mortality in year t
• εx,t, is the error term, which assume to be normally distributed N(0, σε2).
The parameters ax, b (1) x , b (2) x , k (1) t and k (2)
t can be estimated using the singular value
decompos-ition (SVD) approach which involves the following steps [1, 3]:
Step 1. To ensure a unique solution, constraints are imposed on b(1)x , b (2) x , k (1) t and k (2) t which
gives axas the arithmetic mean of ln(mx,t) over time as follows: n
∑
x=1 b(1)x = 1, n∑
x=1 b(2)x = 1, T∑
t=1 k(1)t = 0, T∑
t=1 k(2)t = 0,ˆ ax= 1 T T
∑
t=1 ln(mx,t),where T is an integer representing the number of mortality age profiles available.
Step 2. We center or normalize the data by subtracting ˆax from ln(mx,t) to create the
mat-rix Zx,t as: Zx,t = ln(mx,t) − ˆax= b (1) x k (1) t + b (2) x k (2) t .
Step 3. We apply singular value decomposition (SVD) on Zx,t such that:
SV D(Zx,t) = U SVT.
We obtain ˆb(1)x , ˆb(2)x as the first and second column of U , ˆkt(1) is the first singular value of S
multiplied by the first column of VT and ˆkt(2)is the second singular value of S multiplied by the second column of VT [1, 3]. In other words,
ˆb(1)x = u1,1 u2,1 .. . ux,1 , ˆb(2)x = u1,2 u2,2 .. . ux,2 , ˆk(1)t = s1· [v1,1 v2,1 · · · vT,1] , ˆk(2)t = s2· [v1,2 v2,2 · · · vT,2] .
3.1.1
Forecasting Mortality Indices
To forecast future mortality rates using the Second Order Lee–Carter model is similar to the first order Lee–Carter model. After estimating ˆb(1)x , ˆb(2)x , ˆkt(1) and ˆk
(2)
t using singular value
decomposition (SVD) method, Lee and Carter assume that ˆb(1)x and ˆb (2)
x remains constant over
time and forecast the future values of adjusted ˆk(1)t and ˆkt(2). Since ˆk(1)t and ˆkt(2)are uncorrelated, we assume that they are independent so we forecast each separately. There are different ways to forecast mortality indices using time series models. However, in practice, random walk with drift is the most appropriate model used for forecasting future mortality indices. Estimating ˆk(1)t and ˆk(2)t using random walk with drift model are as follows [1, 3, 12].
ˆk(2) t = k (2) t−1+ d(2)+ ε (2) t , (3.3)
where d(1), d(2) are the drift terms and εt(1), εt(2) are the error terms. εt(1) and εt(2) are assumed to be normally distributed N(0, σε2). The maximum likelihood estimates
ˆ d(1)= ˆk (1) T − ˆk (1) 1 T− 1 , (3.4) and ˆ d(2)= ˆk (2) T − ˆk (2) 1 T− 1 . (3.5)
Considering only the first mortality index ˆk(1)t , the variance of error between the mortality indices values can be estimated as:
see(1) 2 = 1 T− 2 T−1
∑
t=1 ˆk(1) t+1− ˆk (1) t − ˆd(1) 2 . (3.6)We can compute the variance of error for the first forecast horizon as: var ˆk(1) t =see(1) 2 ∆t, (3.7)
and it standard error estimate as: SD ˆ kt (1) =see(1) √ ∆t, (3.8)
where ∆t is the forecast horizon. At 95% confidence interval with z0.025= 1.96, we can compute
the confidence interval of ˆkt(1)[1] as: ˆ kt (1) ± 1.96 · SDkˆt (1) . (3.9)
We substitute for k(1)t−1 in equation (3.18) by shifting back in time one period and also put the estimate of the drift ˆd(1) in other to forecast two periods ahead as:
ˆk(1)t−1= ˆk(1)t−2+ ˆd(1)+ εt−1(1), (3.10) ˆkt(1)= ˆkt−1(1) + ˆd(1)+ εt(1) =ˆk(1)t−2+ ˆd(1)+ εt−1(1)+ ˆd(1)+ εt(1) = ˆkt−2(1) + 2 ˆd(1)+εt−1(1) + εt(1) . (3.11)
Since we have data of mortality age profiles up to time T , to forecast the parameter ˆk(1)t at the next time period T + ∆t, we iterate equation (3.18) ∆t times and we get:
ˆk(1) T+∆t = ˆk (1) T + ∆t ˆd (1)+ ∆t
∑
l=1 εT(1)+ l − 1 = ˆk(1)T + ∆t ˆd(1)+ q ∆tεt(1). (3.12)Since the mean is 0 and assumed to be independent with the same variance, we ignore the error term, then we get a forecast point estimates which follow a straight line as a function of ∆t and gradient ˆd(1) [1, 12]. ˆkT(1)+∆t = ˆk(1)T + ∆t ˆd(1) = ˆk(1)T + ∆tˆk (1) T − ˆk (1) 1 T− 1 . (3.13)
To forecast the mortality index ˆk(1)t , we extrapolate from a straight line and then we drawn through the first ˆk1and the last ˆkT points and forget all other ˆk(1)t points [12].
Following the above procedures, ˆkT(2)+∆t is given by: ˆk(2) T+∆t = ˆk (2) T + ∆t ˆd (2) = ˆk(2)T + ∆tˆk (2) T − ˆk (2) 1 T− 1 . (3.14)
Therefore using the mortality indices, we can calculate the forecast mortality rates as:
Chapter 4
Forecasting and Comparison of Models
In this Chapter, we will discuss results regarding mortality rate forecasting, analysis of forecast mortality indices and comparison of forecast mortality rate models.
4.1
Mortality Rate Forecasting
In this section, we will compare the forecasts of mortality rate based on the three different models. We will use data obtained from the Human Mortality Database [22] and based on central mortality rates found by fitting the three models to data for the three countries in the years 1970–2000. We used the fitted models to generate new data set using the approximation given by (3.1) and then we apply the Second Order Lee–Carter method described in Chapter 3 to forecast the mortality rates for the male population of the USA, Japan and Australia 10 years into the future to be able to make a better analysis on the performance of each model. The forecast (prediction) with 95% confidence intervals is compared to fitted mortality rates for the years 2000 and 2010 for the three models of the USA, Japan and Australia.
In the case of USA in Figures 4.1–4.4, we can observe that the forecast (predicted mortal-ity rates) compare to the fitted mortalmortal-ity rates in the year 2010 exhibit almost the same pattern for the infant and the senescent mortality but slightly different with the peak of the "accident hump" for the Power-exponential function based model and the Heligman and Pollard (HP4) model c in the year 2010. Comparing the forecast (predicted mortality rates) in the year 2010 to the fitted mortality rates in the year 2000 we can observe a decline in mortality for the infant, "accident hump" and the senescent mortality for the Power-exponential function based model, the Heligman and Pollard (HP4) model and the Modified Perks model.
Regarding Japan in Figures 4.5–4.8, we can observe a decline in mortality rates when com-paring the forecast (predicted mortality rates) in the year 2010 to the fitted mortality rates in the year 2000 for the three models. Both the forecast and the fitted mortality rates in the year 2010 exhibit almost the same pattern for each of the three models throughout the three stages of an individual’s lifespan.
In the case of Australia in Figures 4.9–4.12, we can actually observe a decline in mortality rates when comparing forecast (predicted mortality rates) in the year 2010 to the fitted mortal-ity rates in the year 2000 for each of the three models. Looking at the three models the mortalmortal-ity rates for both the forecast and the fitted in the year 2010 for the infant and senescent mortality displays almost the same pattern, but different with the peak of the "accident hump" for the Power-exponential function based model and the Heligman and Pollard (HP4) model. Finally, the Power-exponential function based model and the Heligman and Pollard (HP4) model dis-play better the mortality rate curve compared to the Modified Perks model for the USA, Japan and Australia.
Figure 4.1: Comparison of fitted and forecast mortality rates for the male population of the USA for the raw data in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.2: Comparison of fitted and forecast mortality rates for the male population of the USA for the Power-exponential model in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.3: Comparison of fitted and forecast mortality rates for the male population of the USA for Modified Perks model in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.4: Comparison of fitted and forecast mortality rates for the male population of the USA for Heligman and Pollard (HP4) model in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.5: Comparison of fitted and forecast mortality rates for the male population of Japan for the raw data in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.6: Comparison of fitted and forecast mortality rates for the male population of Ja-pan for the Power-exponential model in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.7: Comparison of fitted and forecast mortality rates for the male population of Japan for Modified Perks model in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.8: Comparison of fitted and forecast mortality rates for the male population of Japan for Heligman and Pollard (HP4) model in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.9: Comparison of fitted and forecast mortality rates for the male population of Aus-tralia for the raw data in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.10: Comparison of fitted and forecast mortality rates for the male population of Aus-tralia for the Power-exponential model in the year 2000–2010 with 95% predictive confidence intervals.
Figure 4.11: Comparison of fitted and forecast mortality rates for the male population of Aus-tralia for Modified Perks model in the year 2000–2010 with 95% predictive confidence inter-vals.
Figure 4.12: Comparison of fitted and forecast mortality rates for the male population of Aus-tralia for Heligman and Pollard (HP4) model in the year 2000–2010 with 95% predictive con-fidence intervals.
4.2
Analysis of Forecast Mortality Indices
From the data obtained from the Human Mortality Database [22], the mortality indices are estimated from 1970–2010 and then forecast into the future from the year 2011–2050 for the three countries. The Figures shown below are both the estimated and the forecast of the raw data and the three models using the mortality indices of the USA, Japan and Australia with 95% confidence intervals.
From the year 1970–2010 in Figures 4.13, 4.15 and 4.17, the first mortality indices (k1) show a
decline in mortality for both the raw data and three models as we can observe that the mortality tend to decline year by year for the USA, Japan and Australia this is as a result of big improve-ment in medical care and also there are no major events like natural disasters and wars for the past 40 years. Regarding the estimated area for the second mortality indices (k2) in Figures
4.14, 4.16 and 4.18, we can observe a less systematic pattern compared to the first mortality indices (k1) for each of the three countries. Observing the forecast period from 2011–2050 for
the first mortality indices (k1) in Figures 4.13, 4.15 and 4.17, we observe that the prediction for
the raw data and the three models are almost the same at 95% confidence interval and also we can observe that by the year 2050 mortality rates will decline significantly as compared to the forecast period for the second mortality indices (k2) in Figures 4.14, 4.16 and 4.18.
Figure 4.13: Estimated from 1970–2010 and forecast for 2011–2050 using the first mortality index (k1) of the USA with 95% predictive intervals for both raw data and the three models.
Figure 4.14: Estimated from 1970–2010 and forecast for 2011–2050 using the second mortality index (k2) of the USA with 95% predictive confidence intervals for both raw data and the three
Figure 4.15: Estimated from 1970–2010 and forecast for 2011–2050 using the first mortality index (k1) of Japan with 95% predictive confidence intervals for both raw data and the three
Figure 4.16: Estimated from 1970–2010 and forecast for 2011–2050 using the second mortality index (k2) of Japan with 95% predictive confidence intervals for both raw data and the three
Figure 4.17: Estimated from 1970–2010 and forecast for 2011–2050 using the first mortality index (k1) of Australia with 95% predictive confidence intervals for both raw data and the three
Figure 4.18: Estimated from 1970–2010 and forecast for 2011–2050 using the second mortality index (k2) of Australia with 95% predictive confidence intervals for both raw data and the three
4.3
Comparison of Forecast Mortality Rate Models
The most reliable way to compare the quality of the mortality rate forecast for the three models to each other is to compute and compare the standard error estimates of the forecast mortality indices by equation (3.6) in Chapter 3. Table 4.1 shows the results of the standard error estim-ates for the raw data and the three models. A more reliable forecast is the one with the smaller standard error estimate.
Standard error estimate (see) USA Japan Australia
see(1) see(2) see(1) see(2) see(1) see(2)
Raw Data 0.1422 0.0755 0.2211 0.1145 0.1734 0.2191
Power-Exponential Function 0.1492 0.0911 0.2175 0.1188 0.1722 0.1717 Modified Perks 0.1458 0.0743 0.2231 0.1091 0.1869 0.1322 Heligman and Pollard (HP4) 0.1424 0.0903 0.2223 0.1168 0.1691 0.1937
Table 4.1: Standard error estimate (see) of forecast mortality indices.
In Table 4.1, considering only the first standard error estimate see(1)of the forecast mortality in-dices for the USA, Japan and Australia. In the case of USA, we can see that the raw data gives a smaller standard error estimate as compared to the three models but when comparing the three models, the Heligman and Pollard (HP4) model performs better than the Power-exponential function based model and Modified Perks model. In Japan, the Power-exponential function based model tends to give a smaller standard error estimate as compared to the raw data, the Heligman and Pollard (HP4) and the Modified Perks model. We can observe that in Australia, the Heligman and Pollard (HP4) model gives a smaller standard error estimate as compared to the Power-exponential function based model and Modified Perks model.
Regarding the second standard error estimate see(2)of the forecast mortality indices, we can see that the Modified Perks model gives a smaller standard error estimate as compared to Power-exponential function based model and the Heligman and Pollard (HP4) model for the USA, Japan and Australia. This scenario happens because the Modified Perks model does not fit the data well for the infant mortality and the "accident hump" it only fit well at the higher ages. The second standard error estimates mostly capture the forecast of the infant mortality and the "accident hump". In this case, we only compare the Power-exponential function based model, the Heligman and Pollard (HP4) model and the raw data. We can observe that in the USA and Japan the raw data has a small standard error estimate as compared to the Power-exponential function based model and the Heligman and Pollard (HP4) model but when comparing the two models, the Heligman and Pollard (HP4) model gives a more reliable forecast than the Power-exponential function based model. In the case of Australia, the Power-exponential func-tion based model gives a smaller standard error estimate compared to the raw data and the Heligman and Pollard (HP4) model. We observe that the performance of a model depends on individual countries .
Chapter 5
Conclusion and Future Work
5.1
Project Summary
In the field of actuarial science and demography, mortality information is a vital component for national planning and health. Population forecasting is fundamental for long term planning of a country [24].
In Chapter 1 of the thesis, we review several models of mortality rates that appear in liter-ature and also present approaches to forecasting mortality rates.
In Chapter 2, selected mortality rate models, namely the Power-exponential function based model, Modified Perks model and the Heligman and Pollard (HP4) model are discussed. In our discussion, we talk about how we obtained our mortality data, model fitting and comparison between the models for the male population ages 1–70 of USA, Japan and Australia in the year 1980, 1995 and 2005. We observed that the Heligman and Pollard (HP4) model better fits the data compared to the Power-exponential function based model and Modified Perks model. In Chapter 3, we discussed procedures and techniques of Second Order Lee–Carter method of forecasting mortality rates, which we used for forecasting the mortality rate data for USA, Japan and Australia.
In Chapter 4, we used the fitted models to generate new data set using the approximation given by (3.1) and then we apply the Second Order Lee–Carter method to forecast (predict) the mortality rates 10 years into the future. Then the forecast (prediction) with 95% confidence in-tervals is compared to fitted mortality rates for the years 2000 and 2010. The mortality indices are estimated from 1970–2010 and then forecast into the future from 2011–2050 for the male population of the USA, Japan and Australia . We computed and compared the quality of the forecast by estimating the standard errors of the forecast mortality indices for the raw data and three models.
We can conclude that the Power-exponential function based model and the Heligman and Pol-lard (HP4) model display the mortality rate curve for the infant, "accident hump" and the higher ages better as compared to the Modified Perks model. Regarding the mortality indices, we ob-serve that the estimated area for the second mortality indices exhibit less systematic pattern compared to the first mortality indices for the USA, Japan and Australia and for the forecast area we observe that the raw data and the three models predict almost the same at 95% con-fidence interval and also indicating that mortality will decline significantly by the year 2050 compared to second mortality indices for each of the three countries. Comparing the quality of the mortality rates forecast, we observe that the Power-exponential function based model and the Heligman and Pollard (HP4) model gives more reliable forecast depending on individual countries when using Second Order Lee–Carter method of forecasting mortality rate for the male population ages 1–70.
5.2
Future Work
In this thesis, we only compare the quality of the forecast using three mortality rate models. We suggest that more mortality rate models together with the three models used in this thesis should be applied to a larger number of data in different countries to be able to ascertain the best model to use when considering the Second Order Lee–Carter method for forecasting mortality rate.
We also suggest that the Second Order Lee–Carter method should be extended to the third or up to the fifth order in order to improve the fitting result since it accounts for some percent-age of the total variance explained by the approximation.
Since we only use the ages 1–70 and forecasted 10 years into the future, we suggest that a different scenario should be applied either short or long term with different age range. For example, forecasting 5 years or 20 years into the future with ages 1–100 with many countries. Finally, if you are only interested in higher ages, we suggest that the Modified Perks model will be better because it is a simple model with few parameters.
Chapter 6
Summary of reflection of objectives in the
thesis
6.1
Objective 1: Knowledge and understanding
The topic of the thesis clearly shows the knowledge and understanding of applied mathemat-ics. Having previously studied a course in actuarial mathematics it helps me to understand my thesis topic and also give me a fair idea on what to include in the work. The student gained more knowledge and ideas by reading many scientific papers and books related to the topic. Most of these scientific papers and books have been cited in the work. Since the student already studied courses in applied mathematics, he was able to overcome the mathematical aspect of the thesis.
6.2
Objective 2: Methodological knowledge
The student learns more regarding mathematical methodology and theoretical ideas in mortality rate forecasting and also how to implement and analyzed results using MATLAB. The student is not familiar with MATLAB but with the help of my supervisors and the student previous knowledge in Python programming, the student was able to overcome this challenge.
6.3
Objective 3: Critically and Systematically Integrate
Know-ledge
The student first discussion with supervisors gave him a fair idea of the topic. The supervisors suggested many literature references to the student. The student also carried out his own re-search by reading many books and scientific papers related to the topic of which many are cited in the thesis. To write on a particular subject or chapter the student read different sources to gather different knowledge and ideas as to how to go about it independently.
6.4
Objective 4: Independently and Creatively Identify and
Carry out Advanced Tasks
In this thesis, most of the work related to the choices of content was independently presented by the student. Ideas and discussions between the student and the supervisors by having meetings improved the quality of the final results. The student found more than one way to improve the Lee–Carter method in literature, but the student decided to use one of them which is the Second Order Lee–Carter method.
6.5
Objective 5: Present and Discuss Conclusions and
Know-ledge
In this thesis, theoretical knowledge from different sources is used. With regards to most of the Chapters, a reader without in-depth knowledge in actuarial mathematics can easily follow and understand. Results are presented in Table and Figures with explanations which makes it easier for the reader to understand. Finally, the student is now very familiar with LaTeX.
6.6
Objective 6: Scientific, Social and Ethical Aspects
Publicly available data for the mortality rates were used. The student has clearly referenced and credited scientific works in this thesis. When possible the original creator/discoverer of a result has been credited. Finally, majority of the MATLAB code is provided by one of my supervisors Karl Lundengård.
