A comparative approach:
Single-population and
multi-population models in five
OECD countries.
Nektaria Themistokleous
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: Nektaria Themistokleous
Student nr: 11376562
Email: n-ektaria@hotmail.com
Date: July 15, 2017
Supervisor: Prof. Dr. Tim Boonen
Statement of Originality
This document is written by Student Nektaria Themistokleous who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
1 Introduction 7
1.1 Historical Background . . . 7
1.2 Longevity Risk, Defined Benefit and Defined Contribution . . . 8
1.3 Aims and Objectives . . . 10
2 Literature Review 11 2.1 Definitions . . . 11
2.1.1 Defined Benefit pension plan . . . 11
2.1.2 Defined Contribution pension plan . . . 11
2.2 Impact of longevity Risk . . . 12
2.3 Micro-longevity and Macro-longevity risk . . . 12
2.4 Pension Systems examined . . . 13
2.4.1 Justification . . . 14
3 Mortality Models: 15 3.1 Lee-Carter mortality forecasting methodology under a Poisson setting 17 3.1.1 Method . . . 17 3.1.2 Estimation . . . 18 3.1.3 Forecasting . . . 19 3.2 Cairns-Blake-Dowd model . . . 20 3.2.1 Method . . . 20 3.2.2 Estimation . . . 21 3.2.3 Forecasting . . . 21 3.3 Plat model . . . 22 3.3.1 Method . . . 22 3.3.2 Estimation . . . 24 3.3.3 Forecasting . . . 25 3.4 Li-Lee model . . . 25 3.4.1 Method . . . 26 3.4.2 Estimation . . . 26 3.4.3 Forecasting . . . 27
4 Mortality models: comparison and application in the selected coun-tries 28 4.1 Lee-Carter . . . 28
4.2 Cairns-Blake-Dowd model . . . 36
4.3 Plat model . . . 38
4.5 Lee-Carter, CBD and Plat model: A comparison . . . 44
5 Model selection 48 5.1 In-sample forecast performance . . . 48
5.1.1 Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC) . . . 48
5.1.2 Standardized Residual . . . 51
5.1.3 Discussion . . . 54
5.2 Out-of-sample forecast performance . . . 54
5.2.1 Mean Absolute Percent Error (MAPE) . . . 55
5.2.2 Root Mean Squared Forecast Error (RMSFE) . . . 56
5.2.3 Discussion . . . 57
6 Conclusion 59
7 References 61
Acknowledgments
Firstly, I would deeply like to express my greatest gratitude to my supervisor, Dr. Tim Boonen, for his constant support and guidance. His professionalism and moti-vation throughout the course of my thesis was priceless to me, whilst his feedback was fundamental to the completion and correct implementation of my thoughts. It was truly of great significance to myself, especially at hard times. Furthermore, I would like to acknowledge and thank the University of Amsterdam for the facilities and access to relevant material provided to me. In addition, I would like to express my sincere appreciation to my friend Constantinos for his support during the dif-ficult times that I encountered. Last but certainty not least, my love and honest gratitude goes to my family for their reciprocal and endless support that fueled the completion of this thesis.
Abstract
It is widely accepted that the increasing longevity risk, observed dur-ing the years leaddur-ing to the turn of the millennium, has had a wide impact on pension funds and insurance companies decision making, in terms of capital reserve prediction accuracy. In this thesis, a multi-population model (i.e. Li-Lee) is compared against three single-population models (Lee-Carter, CBD and Plat) in a quest to identify the most reliable, less volatile and superior model in terms of goodness of fit, thus effectively contributing in tackling longevity consequences. This is done, through several in-sample and out-of-sample tests to ensure reliability. The re-sults extracted in this thesis are in line with an array of literature (i.e. Oeppen, 2002, Boonen and Li, 2017), fundamentally supporting the ef-fectiveness of the Li-Lee model as the better model in capturing the aforementioned factors.
1
Introduction
1.1
Historical Background
From the beginning of the 21st century it is widely accepted that longevity and mortality are directly linked with financial assets and liabilities (Thomsen and An-dersen, 2007). The most compelling evidence in support of this theory is based on the fact that longevity “is a huge risk that is proving to be highly burdensome” not only to individuals and pension funds, but to private corporations and insurance companies as well (Blake et al., 2013).
There has been an ongoing debate amongst scholars and pension fund policy makers regarding the duration of retirement periods and the strategic approach that governments should follow in dealing with increasing longevity risk and mortality trends. According to Olshansky et al., (2005), life expectancy is steadily increasing. Now a 65-year-old woman has a 50/50 chance of living to the age of 85, whilst a 65-year-old man has a 50/50 chance of living to the age of 82.
Additionally, English and Welsh women are expected to live until the age of 94 while men are expected to live until the age of 90, with 39% of women and 32% of men expected to live to the age of 100 in 2112 (Mortality.org, 2017). In 1880, 1.7 million Americans were aged 65 or more corresponding to 3.39% of the population while in 1950 the number increased to 12.7 million corresponding to 8.34% of the population. Namely, by the turn of the millennium Americans aged 65 or older reached approximately 35 million, which is equivalent to 12.37% of the population (Ssa.gov, 2017). On the other hand, Dowd et al., (2010) argue that life expectancy has not been just increasing steadily but much faster than originally anticipated, a notion backed by (Gulland, 2016).
Putting the above into perspective, even though projections for life expectancy can be made, the magnitude of such increase is highly uncertain. In other words, rising life expectancy is exponential and policymakers are finding it more and more difficult to accurately predict future mortality rates. This uncertainty is widely known amongst actuaries as longevity risk, a variable that this thesis is heavily con-cerned with.
1.2
Longevity Risk, Defined Benefit and Defined
Contribu-tion
According to Tan et al., (2015), the fact that individuals tend to live longer (i.e. mortality improvements) puts pressure on governments, life insurance companies and pension funds, to truly take into account longevity risk. Firstly, mortality risk can minimize the effect of several macroeconomic threats such as inflation, higher taxes or market losses since the negative effects of these three financial risks are minimized if people don’t live longer than expected. Kontis et al., (2017) argue that individuals must commit to saving withdraws for longer periods of time, making them vulnerable to inflation risk or investment losses and higher taxes. In other words, living longer results to higher health care demand and at the same time pen-sion funds are obliged to pay for lengthier old-age penpen-sions in an attempt to prevent old age poverty and maintain a high standard of living. For instance, in 2045 indi-viduals aged 60 or older are projected to proportionally outnumber children for the first time in history (Kontis et al., 2017). As a consequence of this discrepancy, a shortage (i.e. shrinkage) in government income (i.e. taxes) is inevitable. As a result states will find it much more difficult to remunerate pensions to retirees.
On the other hand, longevity can be considered not only as a negative (i.e. to pension funds), but as a positive indicator as well. Namely, living longer is pros-perous when it comes to health care thus the global standard of living increases. However, this added prosperity that is upon future generations is not without its costs. For instance, medical costs are getting more and more expensive and demand for health services is growing rapidly. Due to the fact that individuals tend to live longer, social benefits such as retirement pensions are extending over longer time periods. As a result, in order to maintain a high level of effectiveness, social security systems are regularly altered by policy makers (Creedy, 1998).
Moreover, in the future, the majority of countries will be faced with aging popula-tion complicapopula-tions and rising life expectancy. Although this can happen in different time frames, as well as variant severity, it is definitely an unavoidable upcoming phenomenon (World population aging, 1950 - 2050, 2017). This increasing trend shown in Figure 1 is the root cause of longevity risk that affects pension funds and insurance companies in the modern era.
The above situation generates difficulties in pricing and reserving calculations due to increased life expectancy as well as the decrease in the number of employee taxpayers. Furthermore, longevity risk affects pension funds via several channels.
Figure 1: Data Source: World Bank Group - International Development, Poverty, Sustainability. This Figure demonstrates the increase in life expectancy for the five featured countries, for the years 1960-2014. A similar surge in life expectancy is evident in all countries, with little notable difference in their gradients. In 1960 the life expectancy for all countries is around 70-75 whilst in 2014 it increases to 83-87. The five countries examined (Switzerland, United Kingdom, Japan, Canada, Netherlands) are the ones used throughout the duration of this thesis.
Namely, due to the fact that individuals live longer and the active labor force (i.e. taxpayers) is shrinking in comparison to retirees, it is more difficult to price and reserve calculations, decreasing the dependency ratio (Mircea et al., 2014). Addi-tionally, longevity risk affects pension funds through increasing liabilities as indi-viduals live longer and pension funds are forced to offer people benefits for longer than originally intended (Denuit et al., 2007). If longevity risk is not taken into consideration by pension funds, insurance companies and governments then severe consequences are inevitable. Namely, the lack of funds to cover the old age popu-lation with premiums will be the main cause of problems such as volatile financial stability for pension funds as well as for life insurance companies, eventually leading to insolvency risks (Antolin, 2007).
Longevity risk appears mainly in Defined Benefit (DB) pension plans, in the sec-ond pillar of the pension systems, which is the occupational/worker pension pillar. It’s main objective is to maintain a high standard of living. The contemporary trend in pension plans is indicative of a shift from Defined Benefit (DB) pension plans to Defined Contribution (DC) pension plans (Gerrans and Clark, 2013). The main reason for this shift is the high risks that Defined Benefit pension plans are now seen to impose on corporate sponsors. It must be noted that this mainly concerns the countries to be examined for the purpose of this thesis.
1.3
Aims and Objectives
In this thesis longevity trends of five selected countries are discussed and com-pared whilst the main differences and similarities in terms of life expectancy are identified. After calculating and assessing the projected longevity trends for all five countries, an in-sample and out-of-sample forecasting performance analysis will be conducted, with primary aim to assess a multi-population model against a single-population model. As a multi-single-population model, the Li-Lee model is selected for its accuracy whilst it is widely used by the Actuarial Society and more precisely, by the Royal Dutch Actuarial Association for the Projection Table AG2014 (3) and Projection Table AG2016 (4) (Cox et al., 2013). Namely, the evaluation will lay its foundations on whether the multi-population model is less volatile and has better goodness of fit, in predicting life expectancy, than the single-population model, and critically evaluate the following question:
What are the differences between fitting a multi-population model in terms of goodness of fit and volatility in making life expectancy projec-tions, against a single population model?
For the purpose of this thesis, the longevity trends are examined and projections are made in five OECD (Organisation for Economic Co-operation and Development) countries (i.e. the Netherlands, Switzerland, Canada, Japan and UK). The idea be-hind choosing the above countries is for comparison purposes, in longevity trends of five countries with similar Pension systems, where the main part of their occupa-tional scheme (2nd pillar) is a defined benefit pension system.
The period 1950-2010 has been selected to make the projections of future life ex-pectancies. The reason for not utilizing available data before 1950 is the occurrence of the First and Second World War, which influenced life expectancy and increased mortality rates for several years. The data utilized for this thesis consists of mortal-ity rates, number of deaths and exposures to risk published by the Human Mortalmortal-ity Database (HMD). The data published by the HMD is of high quality and is used across a wide range of research studies (i.e. Giacometti et al., 2012, Russolillo et al., 2011, Zhao et al., 2013) and therefore suitable for this thesis.
2
Literature Review
The surge in longevity risk, due to increased life expectancy and consequently lower mortality rates, has sparked an ongoing debate amongst actuaries and poli-cymakers regarding how this increase should be confronted. In the following para-graphs, you can expect a thorough analysis of the theoretical background and em-pirical research regarding the impact of longevity risk in selected countries. A sum-mary of the pension systems of the five OECD countries (Netherlands, Switzerland, Canada, Japan, UK) is given. The objective is to show that all five countries are separated into three pillars and interpret the main similarities between them.
2.1
Definitions
2.1.1 Defined Benefit pension plan
"Defined benefit pension schemes provide a periodic pension at pensionable age as a flat rate benefit or as a function of an individual’s employment and earnings history" (McGillivray, 2006). In real terms the benefit is the individuals retirement income and the ability to know how much he/she will receive upon retirement based on a computation formula taking into consideration previous earnings and duration. Essentially it is a monthly payment provided by an employer to an employee initiat-ing when the employee retires. The definition of defined benefit is indicative of the fact that the monthly payment an employee receives is defined while the benefits are fixed, linked to macro longevity and growth rate tax base.
2.1.2 Defined Contribution pension plan
Defined contribution plans are based on the ability of the plan member to peri-odically contribute a fix sum of his salary to the pension fund. Then the aggregate contributions are invested in numerous assets, picked based on their market availabil-ity. In other words, the DC plan essentially consists of the employers’ contributions added to the employees’ contributions plus the investment return. The contribu-tions are fixed on a monthly basis unlike the pension benefit, which is not. Defined Contribution pension plans allows the employee to freely choose between investing in riskier assets or more conservative ones. The primary difference between DC and DB is that the former provides individuals with a freedom of choice in either investing in risky or risk free assets, depending on individual risk tolerance.
2.2
Impact of longevity Risk
In this Section an overview of the impact of longevity risk on pension funds will be discussed and evaluated. Insurance companies, employees and employers are all stakeholders affected by the augmentation of longevity risk whilst the com-plexity of this issue is recognized by scholars all over the globe. Namely, Blake et al., (2014) find that longevity risk is a significant issue for insurance companies and pension funds and propose the implementation of "Government-issued Longevity bonds" as a measure to rectify and spread longevity risk consequences across future generations hence, dispersing its impact. Even though actuaries are aware of this increasing trend in liabilities of pension funds, the extent to which a pension fund or an insurance company should account for this increase depends on several factors such as risk tolerance (Hari et al., 2008). Furthermore, since there is no open and real longevity market, to price longevity risk, researchers turn to an alternative the-oretical approach. The method introduced by Plat (2011), offers a comprehensive solution in determining risk market prices through the mortality model developed by himself. An in depth evaluation of Plats’ (2011) approach is presented in Section 3.3.
In addition, the gravity model is used to estimate the mortality rate of several re-lated European populations. For instance, Dowd et al., (2011) use the gravity model to “handle the interdependence between the mortality rates of two related popula-tions”, England and Wales. Namely, they find that by using the gravity model they smoothed some volatilities when forecasting secondary populations (i.e. the smaller population sample - Wales).
For the purpose of this thesis the Lee-Carter, Plat, Cairns-Blake-Dowd and Li-Lee models are introduced to assess the mortality pattern for The Netherlands, Canada, Switzerland, UK and Japan. The aforementioned models have been used by scholars all over the world for predicting and modeling mortality in countries on a global level and are well recognized in the actuarial field.
2.3
Micro-longevity and Macro-longevity risk
Longevity risk can be separated into two risks; macro-longevity risk and micro-longevity risk (Hari, et al., 2008). Macro-micro-longevity risk, does not specify how long will individuals live in the future, while micro-longevity does specify future survival rates. However, even with known future survival probabilities the time of death is still uncertain. In addition, macro-longevity and parameter risk are important elements in large portfolios whereas micro-longevity risk could be described as
“neg-ligible”. In addition, Antolin (2007) argues that macro-longevity risk also affects Defined Benefit pension funds internationally and supports such notion by arguing that increasing life expectancy is a global phenomenon. For example the Spanish life expectancy has risen by 39 years from 1908 to 2003 whist in Italy in the same period it increased by 38 years (Antolin, 2007). Similarly, according to the World Bank (2017), Japanese individuals lived to 83.84 on average in 2015 compared to 67.67 in 1960.
2.4
Pension Systems examined
The pension systems of the countries examined during the course of this thesis share similar characteristics. All five countries have implemented a pension system mainly made up of three pillars, generally serving the same purpose. In addition, all pension systems are based on a defined benefit approach whilst, all of them, are structured on Pay-as-you-go (PAYG) system in the 1st pillar scheme, which es-sentially translates to the sole dependency of retirees on today’s workers for their benefits. In addition, the 2nd pillar is made up from contributions both on the em-ployees and employers side and aims to provide a basic level of earning replacement during retirement. Lastly, the 3rd pillar is predominantly optional and acts as an
additional layer of insurance if individuals desire it.
The 1st pillar of each country is the state/public or basic scheme. For instance,
the Dutch 1st pillar’s main objective is to prevent old-age poverty and is available to
everyone who lives or lived in the Netherlands (Høj Jørgensen, 2017). The same ob-jective is shared amongst the 1stpillar of Switzerland, Canada and the UK. However, Japan’s 1st pillar is primarily the same as the other countries with the exception that it consists of two main tiers; the flat rate, basic scheme and the earnings plan (employees’ pension scheme).
The 2nd pillar amongst countries serves the same fundamental purpose; to
pro-vide a basic level of earning replacement during retirement. The Canadian, Dutch and Swiss 2nd pillar are the same, whereas the Japanese version differs slightly. The
Japanese 2nd pillar is made of the Employees’ Pension Insurance (Kosei Nenkin Ho-ken) whereas all the labor force is faced with the same contribution rate (8.675%) and no discounts apply to workers at the lower ends of the income spectrum (Lee, 2008). On the other hand, the British 2nd pillar is made of state owned or private pensions and the default scheme is known as the State Second Pensions (S2P). It covers employees but not self-employed and workers might decide to opt-out of the
S2P and contribute in a private pension of the third pillar (Pensions at a Glance, 2015).
Voluntary and mostly offered by private insurance companies, the 3rdpillar serves as an additional insurance layer for individuals that want it. However, regarding Japan, policy makers see the implementation of the 3rd pillar as remedy (through higher saving rates) towards the issue of financial sustainability as well as intergener-ational equity (Chia, 2015). It should be noted that, in the Netherlands, the Defined Benefit schemes approximately cover 94% of the employees whereas the other 6% is covered by defined contribution schemes. However, there is evidence of an inverting trend indicating a shift from DB to DC (Gerrans and Clark, 2013).
2.4.1 Justification
Having defined the two types of occupational pension systems, our primary focus is shifted towards the Defined Benefit pension where longevity risk mainly exists. One of the main aims and objectives of this thesis is to estimate future life expectan-cies thus, the Defined Contribution pension plan is not considered. In addition the 2nd pillar of the selected counties (i.e. Netherlands, Japan, Switzerland, Canada, UK) is the main area of attention in this thesis since it is regarded as the work-ers/occupational pension system, or to put it simply, the defined benefit pension plan. Although, the recent financial crisis has extremely impacted the performance of pension funds all over the world, in recent days a clear shift from Defined Benefits to Defined Contribution plans is notable. This is primarily due to the fact that the expenses and long-term obligations associated with running a Defined Benefit plan are expensive and partly attributed to the increase in workforce mobility. The De-fined Benefit scheme protects individuals against longevity risk where in a DeDe-fined Contribution scheme participants take on longevity risk under the assumption that wealth is not annuitized. The challenging circumstances that individuals face to perfectly manage their investments and withdraw from their retirement accounts makes annuitizing wealth particularly difficult.
3
Mortality Models:
Over the course of this thesis four models are used for estimating and forecasting mortality:
1. Lee-Carter (LC) model
2. The Cairns-Blake-Dowd (CBD) model 3. Plat model
4. Li-Lee model
To begin with, the LC model is considered to be the benchmark when it comes to stochastic mortality modeling. The LC model is a single factor model which assumes no smoothness throughout ages hence, coping with improving mortality across various ages is not possible. Nonetheless it is widely used due to its simplistic nature (Lee and Carter, 1992). Similarly, another significant drawback of the LC model is the assumption that mortality improvement across ages is linear and not variable (Lee and Miller, 2001). In order to account for the aforementioned inability of the LC model to cope with improving mortality over the years and the fact that it assumes common mortality improvements across ages, the CBD model is intro-duced. Nevertheless, the data used throughout this thesis are in line and a good fit for the LC model.
The CBD model is a two-factor model, which unlike the LC model, assumes and accounts for smoothness throughout years and ages, however it does not assume smoothness between different age groups. A slightly lesser goodness of fit compared to the LC model is observed for some countries used for the purpose of this thesis (see Section 5). Additionally, the CBD is considered to be a relatively simple model in terms of fitting whilst because of its two period-effect parameters, it’s highly reliable in capturing improvements in mortality across different years (Gbari and Denuit, 2016).
To contextualize, due to the fact that the Lee-Carter model is a single fac-tor model, the CBD model adds an additional dimension to the modeling process. Namely, the CBD has a direct impact on mortality proportional to age by introduc-ing a second stochastic factor. Thus, it takes into account mortality change across different ages intensifying the reliability factor of the model (Cairns et al., 2006). Consequently, to account for the lesser goodness of fit that the CBD outputs, the Plat model is considered.
Namely, the Plat model is introduced primarily because of its capability to com-bine the CBD model with numerous attributes of the LC model producing a suitable model for all age ranges, capturing the cohort effect (Plat, 2009). In comparison to the LC and CBD model, the Plat model should give a better fit for the used data, principally because of it’s ability to account for various cohort effects. These cohort effects have been identified as a vital element in a mortality modeling and play a significant role in producing accurate forecasts (Gov.uk, 2017).
Lastly the Li-Lee model is a multi-population model and it’s primarily used to add robustness and consistency to this study, while being able and suitable for fore-casting joint mortality rates. Furthermore, the Li-Lee model is an extension of the LC model in a way that it allows for more than one population to be considered as a joint mortality model (Li and Lee, 2005). In order to assess the main objec-tive of this thesis, the Li-Lee model is used to compare a single population model against a multi-population model to evaluate their volatility levels and goodness of fit. Nonetheless some correlative relationships between mortality improvements of both populations is expected.
After evaluating the four models, the AIC (Akaike Information Criterion) and the BIC (Bayesian Information Criterion) as well as the standardized residuals methods are introduced, aiming to assess the robustness, goodness of fit and consistency of the models. Both criteria are often overlapping in the literature for their ability to punish models with surplus parameters. In simpler words, the smaller the AIC and BIC criteria the higher the robustness of the model. Namely, to satisfy the assump-tions and theoretical approach of this thesis, the multi-population model is expected to have a smaller criterion in comparison to the remaining three models. Lastly, two out-of-sample forecasting methods are used to add reliability to the model while efficiently testing out of sample performance (MAPE, RMSFE - See Chapter 5).
The available data for the selected countries are:
Country Period Ages for all four models
Netherlands 1950-2010 0-110 years old
Japan 1950-2010 0-110 years old
U.K. 1950-2010 0-110 years old
Canada 1950-2010 0-110 years old
Switzerland 1950-2010 0-110 years old
3.1
Lee-Carter mortality forecasting methodology under a
Poisson setting
In 1992 the Lee-Carter (LC) model was created in order to forecast U.S. mor-tality. The method created by Lee and Carter (1992) has become the dominant statistical model of mortality forecasting in the demographic literature (Wisniowski et al., 2015). Nevertheless, researchers have used the LC model on a global scale - for instance Italy and India - for forecasting mortality, establishing it as global model, irrespective of its US oriented foundations (i.e. Chavhan and Shinde, 2016, Maccheroni and Nocito, 2017). For the purpose of this thesis the model is deployed to estimate the longevity trends of Canada, Netherlands, Japan, UK and Switzer-land. Over the years, the Lee-Carter model has been commonly used to forecast mortality in several countries. This method gained popularity primarily because of its simplistic nature, fitting capabilities, the fact that its parameters have clear interpretation but most importantly, the Lee-Carter method models the logarithmic hazard rates, meaning that hazard rates themselves are guaranteed to be positive.
3.1.1 Method
According to D’Amato et al., (2011), the Lee-carter model is given by the fol-lowing equation:
where mx,t = Dx,t
Ex,t
Parameters αx and βx are age dependent parameters and κt is a stochastic
pro-cess with a time trend. Renshaw and Haberman (2003) support that the Poisson form of the Lee-Carter model (by the term Poisson form, a Poisson heteroskedastic error term is implied) is far more accurate compared to if death normality was as-sumed.
Dx,t and Ex,t show the deaths and exposures respectively. Exposures Ex,t, is the
total time that people of ages [x, x+1] have lived during the period [t, t+1] whereas deaths Dx,t is the amount of deaths between people of ages [x, x+1] in the period
[t, t+1]. For example, D37,1982 represents everyone who died between the 1st of
Jan-uary 1982 and the 1st of January 1983 and at the time of death he/she was aged between 37-38.
κt determines the change in mortality rates over time where the effect of
dif-ferent ages changes with βx. Since the formula of the Lee-Carter model is
over-parameterized, parameters κt and βx are subject to the following constraints in
order to guarantee model identification:
2010 X t=1950 κt= 0, 110 X x=1 βx = 1.
Given the two constraints, the following two relations can be concluded: 1. αx = T1
P
t
logmx,t =: logmx,t for all x,
2. κt =
P
x
(logmx,t - logmx,t) for all t. 1
3.1.2 Estimation
According to Brillinger (1986), a Poisson distribution for the number of deaths is appropriate to approach the Lee-Carter model.
Dx,t ∼ Poisson(Ex,tmx,t(φ)) given that mx,t(φ) = exp(αx + βxκt)
The Maximum Likelihood Estimation is utilized to project the parameters αx, βx
and κt which, according to Renshaw and Haberman (2003) and supported by Alho
(2000), allows for non-additive heteroscedasticity. Using the log-likelihood func-tion, it is assumed that φ serves as the full set of parameters (φ = {(αx, βx, κt, x ∈
{1, ..., 110}, t ∈ {1950, ..., 2010}) which are αx, βx and κt. The log-likelihood
func-tion is shown by:
L(φ; D, E) = P
x,t
Dx,tlog[Ex,tmx,t(φ)] − Ex,tmx,t(φ) − log(Dx,t!),
where D = Dx,t∼ Poisson(mx,tEx,t), given x ∈ {1, ..., 110} and t ∈ {1950, ..., 2010}.
In order to achieve the estimates for ˆαx, ˆβx and ˆκt, maximizing over φ is
es-sential. In addition, parameters ˆαx and ˆβx can be considered as constant across
time. A bivariate vector time series is created by the estimated parameter (κt) and
a multivariate approach is used to model it.
For the purpose of this thesis the degree of fitness of the Lee-Carter model will be demonstrated in a death rate matrix of Canada, Netherlands, Japan, Switzerland and UK for the time period 1950-2010. Consequently, the κt values for the next 100
years (up to and until year 2110) are estimated.
Data for the estimation, death rates per calendar year and death rates per age for each of the five countries assessed in this thesis, are withdrawn from the Human mortality Database and separated into males and females. The number of deaths and exposure to risk are denoted in two 1×1 matrices, in which the first number indicates the age interval whilst the second indicates the time interval.
3.1.3 Forecasting
Forecasting stochastic mortality modeling is of high priority, due to the rapid increase in life expectancy (Booth and Tickle, 2008). According to Lee and Carter (1992), a random walk with drift is considered as a fair fit for the projections of the mortality of the Lee-Carter model, using the ARIMA(0,1,0) process:
with δ representing the drift parameter and ξtbeing a Gaussian white noise
pro-cess having a variance σ2 κ.
Since the data up to t0 is calculated, future κt values are forecasted using the
following expression: κt0+n=n·δ + κt0 + n P j=1 ξj.
The reason behind the choice of the specific ARIMA process is mainly due to its simplistic nature while used by many scholars (Kim and Choi, 2010, Leng and Peng, 2016), making the forecasts comparable with literature.
3.2
Cairns-Blake-Dowd model
In contrast with the Lee-Carter model, the CBD model is a two-parameter model following a random walk with drift, assuming a constant drift rate and a correla-tive relationship amongst parameters (Sweeting, 2011). Essentially, it can be said that the CBD model is a modernized version of the Lee-Carter model contributing in mortality forecasting in older ages. Pension funds, life insurance companies and private annuity suppliers find its contributions highly valuable, especially for pricing longevity bonds (Macceroni and Nocito, 2017).
3.2.1 Method
The CBD model according to Cairns et al., (2006), is given by:
log qx,t 1−qx,t = κ (1) t + κ (2) t (x − ¯x) + x,t, x,t,i i.i.d. ∼ N (0, σ2 x,t,i) where:
• qx,t denotes the 1-year mortality (death) probability for an individual with age
x at time t. In Section 3.1.1 regarding the Lee-Carter model the central death rate mx,t is used whereas the 1-year mortality probability is used for the CBD
model. According to, Li, De Waegenaere et al., (2017) these two parameters are linked through the following expression:
qx,t = 1 - exp(-mx,t)
• 1 − qx,t denotes the survival rate (percentage of people who survived) for an
• κ(1)t and κ(2)t indicate two stochastic processes defining the mortality indexes, • ¯x is the average age for a specific range of ages,
• log qx,t
1−qx,t = logit qx,t, which denotes the logit transformation of qx,t
(Mac-cheroni and Nocito, 2017). In general the logit transformation of δ (assuming that δ is a real number), is given by log 1−δδ .
• x,t indicates the error term.
The CBD model differs in comparison to other mortality models, because of its ability to avoid identifiability issues, hence no parameter constraints are required. Additionally the CBD model is "new-data-invariant", meaning that when an extra year of mortality data is published, the stochastic model is accordingly updated, hence no affect is carried over on previous years indexes (Zhu et al., 2017). Further-more, the new-data-invariant capabilities of the CBD model are fundamental since it would be almost impossible to "track an index if its historical values" are altered on a constant basis (Chan et al., 2014).
3.2.2 Estimation
As analyzed by Li et al., (2015), the estimation of the parameters of CBD model is made by the Maximum Likelihood Estimation which is the most commonly used method for stochastic mortality modeling (similar to the Lee-Carter model). The MLE is given by:
L(φ; D, E) = P
x,t
Dx,tlog[Ex,tmx,t(φ)] − Ex,tmx,t(φ) − log(Dx,t!),
where φ serves as the full set of parameters (φ = {(αx, βx, κt, x ∈ {1, ..., 110}, t
∈ {1950, ..., 2010}).
3.2.3 Forecasting
For forecasting purposes the CBD model (Section 3.2.1) is written as follows:
qx,t = e κ(1) t +κ (2) t (x−¯x) 1+eκ (1) t +κ (2) t (x−¯x) .
Using the least squares method, the stochastic processes are obtained for param-eters κ(1)t and κ(2)t . Historical data are then used to obtain the average age for a specific range of ages (¯x) at the 95% confidence level.
Accordingly, the parameters vector is projected ~kt = κ (1) t , κ
(2)
t to observe the
forecasted values by modeling ~kt as a two-dimensional random walk with drift.
In line with the Lee-Carter model, ~kt+1 = ~kt + µ + C Z(t + 1), where
• µ is a (2x1) vector,
• C is a triangular (2x2) matrix,
• Z(t) is a two-dimensional standard normal random variable (Cairns et al., 2006).
The (2x1) vector µ has a negative value (upper value) and a positive value (lower value). An example of such a vector is given when data throughout the 1950-2010 period is used (60 observations are given) for the Netherlands:
ˆ
µ = 0.000346−0.0489
3.3
Plat model
The Plat mortality model is an extension of the Cairns Blake Dowd model (CBD) and the Lee-Carter model that considers the cohort effect (Plat, 2009). The objective of Plat model is to eliminate the disadvantages of Lee-Carter and CBD model, and to create a model with four stochastic components leading to a correlation between ages. According to Roach and Gampe (2004), age-dependent mortality is a crucial variable in parameter estimation and forecasting. Additionally, there is always an age-dependent component in the human death rate growing exponentially with age. As discussed in Section 2.2, theoretical assumptions are used to price longevity since there is no real market for longevity. Plat, (2011) introduced a risk neutral pricing approach which assumes that risk is not of importance thus not affecting behavioral patterns of investors. On the other hand, expected value is considered to be of great importance.
3.3.1 Method
The Plat model is given by the equation:
log(mx,t) = ax + κ (1) t + κ (2) t (¯x - x) + κ (3) t (¯x - x)+ + γt−x
where (¯x − x)+ = max(¯x - x, 0) and a
x is an age dependent parameter similar
to the Lee-Carter model. Furthermore, parameters (κ(1)t , κ(2)t , κ(3)t , and γt−x) can be
considered as four stochastic parameters.
The above model consists of: • The static age parameter αx,
• three age period parameters (κ(1)t , κ(2)t , κ(3)t ),
• three age-adjusting parameters (βx(1) = 1, βx(2) = ¯x - x, βx(3) = (¯x − x)+),
• The cohort effect γt−x.
Factor κ(1)t shows the degree of change in mortality levels amongst all age groups. Following Cairns et al., (2006b) methodology and approach, the long-run stochastic process concerning the κ(1)t factor must not be mean reverting. The fact that there is no real anticipation of enhanced mortality improvements (in the future) being canceled-out by slower mortality improvements (in further future), is the reason behind the non-reverting mean assumption. On the other hand, mortality changes vary amongst ages and the κ(2)t parameter allows for such phenomenon thus reflect-ing historical literature which states that improvement rates are diversified amongst different age groups. In addition historical evidence shows that some times, mortal-ity rates at ages up to 45 can vary significantly. Examples affecting such dynamics include AIDS, violence or alcoholism; hence the κ(3)t parameter is used to capture such issues (Plat, 2009).
The Plat model is over-parameterized (similar to the Lee-Carter in Section 3.1.1), which essentially denotes that dissimilar parameterizations can result to similar val-ues for log(mx,t). It must be said that the succeeding parameterization can produce
closely identical values for log(mx,t):
˜ γt−x = γt−x + ψ1 + ψ2·(t-x), ˜ κt1 = κt1+ψ1 - d·˜x·ψ2, ˜ κt2 = κ2t+d·ψ2, ˜ αx = αx + (1-d)·ψ2,
Thus, parameters κ(1)t , κ(2)t and κ(3)t are subject to the following constraints in order to guarantee model identification:
X t κ(1)t = 0, X t κ(2)t = 0, X t κ(3)t = 0.
The following three constraints demonstrate that the cohort effect (γt−x)
fluctu-ates around 0: tn−x1 X c=t1−xk γc= 0, tn−x1 X c=t1−xk cγc= 0, tn−x1 X c=t1−xk c2γc= 0. 3.3.2 Estimation
Similar to the Lee-Carter model in Section 3.1.2, the number of deaths are es-timated using a Poisson distribution and the estimation is carried out using the Maximum Likelihood function.
Dx,t ∼ Poisson(mx,tEx,t).
Dx,t is the number of deaths for age x and time t whereas Ex,t denotes exposure
to risk (see Section 3.1.1).
An established fitting methodology for the Lee-Carter model, formed through using the Poisson model, is described by Brouhns et al., (2002). It edges other ap-proaches by considering heteroskedasticity across different ages. This method has gained in popularity thereafter (hence the reason for using Poisson distribution in the Plat model). This is evident by its inclusion in several academic papers such as, Renshaw and Haberman (2006) and Cairns et al., (2007).
According to Plat (2009), parameter φ is fitted with maximum likelihood estima-tion where the log-likelihood funcestima-tion of Dx,t ∼ Poisson(mx,tEx,t) is given as follows:
L(φ; D, E) = P
x,t
Dx,t log[Ex,tmx,t(φ)] - Ex,tmx,t(φ) - log(Dx,t!),
where φ is defined similarly to Section 3.1.2.
In order to estimate parameters κ(1)t , κ(2)t and κ(3)t Singular Value Decomposition (SVD) has been used since it allows for heteroscedasticity.
3.3.3 Forecasting
The forecasting procedure is carried out by implementing an ARIMA(pi, qi, di)
process with drift for parameters κ(1)t , κ(2)t , κ(3)t and γt−x of the Plat model. For
the Plat model the implementation of an ARIMA(0,1,0) with non-zero intercept has been selected for the parameter κ(1)t (as defined in Section 3.1.3 for forecasting parameter κt), whereas an ARIMA(1,0,0) process has been selected for fitting
pa-rameters κ(2)t , κ(3)t and γt−x (Plat, 2009).
3.4
Li-Lee model
The aim of the Li-Lee model is to compare similar populations since in the past few years, mortality patterns for several countries are indicative of strong similarities. The main question of this thesis is concerned with whether using a multi-population model for countries with similar pension systems makes predictions less volatile and more reliable, thus the Li-lee model is the appropriate model to assess and evaluate this research objective.
The Li-lee model is a multi-population mortality model that can be considered as an extension of the well-known Lee-Carter model. According to Li and Lee (2005), the Lee-Carter model is applied to a population group (i.e. different countries) where common factors are extracted and tested to assess if it aids single population modeling.
Oeppen, (2002) was the first to introduce the idea of multi-population mortality models. The Li-Lee model along with several other models was introduced since then. Technological growth in medicine, positive alterations in standard of living,
and state regulation are all factors that directly affect mortality and are likely to escalate across countries that share similar characteristics such as geographical loca-tion, economic situation or a common cultural identity. Therefore, multi-population models have the potential to make mortality forecasting more accurate.
3.4.1 Method
The Li-Lee model is given by the formula:
log(mx,t,i) = αx,i + BxKt + βx,iκt,i + x,t,i
for age x ∈ {xmin, ..., xmax} and year t ∈ {tmin, ..., tmax},
where αx,i, βx,i and κt,i are "country specific parameters", Bx, Kt are
com-mon parameters to all countries i and x,t,i is the age and country error term.
(x,t,i i.i.d.
∼ N (0, σ2
x,t,i)). According to Li and Lee, (2005), parameter κt,i is
consid-ered to be stationary and fluctuates around zero for each country. In order to solve the identifiability problems the following constraints are considered:
X x Bx = 1, X x βx,i = 1 ∀i, X t Kt= 0, X t κt,i = 0 ∀i . 3.4.2 Estimation
In order to estimate the Li-Lee model, the Lee-Carter model must be estimated first. This is done by using death rates from the sample population, which estimates Kt and Bx. κt,i is obtained by dividing the average with the log death rates for
ev-ery country and lastly, βx,i and αx,i are estimated by running the Lee-Carter model,
focusing on the residuals obtained from the log mortality rates minus the estimates of the previous parameters. Consequently,
Parameters βx,i and κt,i can be estimated similar to the Lee-Carter model using
a Singular Value Decomposition (SVD) per population i. Parameter αx,i should be
estimated independently for each population in the group, therefore minimizing the total error (Li and Lee, 2005). Hence, SVD aims to minimize the squared estimation errors (x,t,i), thus the following OLS (ordinary least-squares) regression is obtained:
min (
T
P
t
(log(mx,t,i) - αx,i - KtBx)2
) .
Taking into account that the Li-Lee model is an extension of the Lee-Carter model the constraint
tmax
P
t=tmin
Kt= 0 is used to simplify the squared estimation error:
αx,i = T P t=0 log(mx,t,i) T +1
which represents the average across time of log(mx,t,i).
3.4.3 Forecasting
According to Li and Lee (2005) the forecasting procedure is computed by apply-ing a random walk with drift (RWD). For the common parameter Kt the following
time series process is used:
Kt= d + Kt−1+ tσ,
where d 6= 0 is a scalar representing the drift term, meaning that the forecast value would be reliant to historical evidence and σ is the standard deviation consid-ering changes in Kt (Enchev et al., 2016) .
The same forecasting procedure used for the common parameter Kt, will be
used for the country specific parameter κt,i. To be more accurate, x,t,i (see Section
3.4.1) is used to account for the difference between the rate of change in a specific country’s death rates and the rate of change in the common death rates. However, should these differences remain in the long run, the forecasts will be altered. For the approach to be successful then no difference should exist between specific country’s death rates and the rate of change in the common death rates (i.e. No x,t,i). To put
it simply, parameter κt,i must be stationary and this can only happen by assuming
4
Mortality models: comparison and application in
the selected countries
For the stochastic mortality modeling of the four selected mortality models, nu-merous packages in R have been used. Namely:
Using the package "forecast" in R the following information are extracted: • Projections of mortality rates in a matrix form mx,t,
• Forecasted values for κt using a random walk with drift,
• Forecasted values for γt−x(cohort effect) using an ARIMA model (for the Plat
model).
Furthermore, for producing Period Life Tables, the Package "LifeTables" is used.
4.1
Lee-Carter
Principally, Figure 2 demonstrates the log-death rates for Canada and the Nether-lands (which are selected randomly out of the five countries), separated in males-females and the total population, using data from 1950 to 2010. Similar trends for the log-death rates for the selected countries are indicative as worldwide mortality at all ages is steadily decreasing (Wong-Fupuy and Haberman, 2004). Some ran-dom variations especially between ages 0-12 and for old ages 100+ are observed and this is an expected outcome especially when considering the vulnerability these age groups have in diseases and other forms of death related causes. Moreover, Figure 2 suggests that females have a lower mortality than males.
The above results are in line with the results of D’Amato et al., (2011) who studied the life expectancy of the Italian population and Wiśniowski et al., (2015), who implemented the Lee-Carter model by justifying the advantages of a Bayesian approach for the population forecasting in British Pension Funds.
Furthermore, Figure 3 demonstrates the components of the Lee-Carter model (ax, bx, kt) for Canada and the Netherlands classed for male (blue), females (red),
and total of the population (black). Parameter ax represents the “average trend of
the logarithms of mx”. (Hainaut, D. 2012). In essence, parameter αxspecifies the age
specific mortality rate and it is notable that both countries experience similar trends and results. An increasing trend for both the countries for males and females can
Figure 2: Log of death rates for Canada (top graph) and the Netherlands (bottom graph) categorized to time t and age x and separated into male, female and the total population. The different colors (Red to Purple) indicate the chronological range from 1950 to 2010. This clearly shows the decrease in log death rates across the years.
Figure 3: Parameters αx, βx and κt for Canada (top graph) and the Netherlands
(bottom graph). Parameters αx and κt show similar trends for the two countries,
whereas parameter βxfor the Netherlands exhibit some random variations in regards
be concluded for parameter ax and is related to the positive relationship that death
probabilities share with age. A point to be made here is that for both the selected countries, for males, ax seems higher between ages 21-25 compared to females. This
can be attributable to the fact that males tend to get into dangerous activities be-tween the ages in question like driving a fast car, or engaging in risk-heavy activities.
Parameter bx is indicative of how much each age contributes to the decline in
mortality thus, the random variations in parameter βx are mainly a consequence of
each country’s diverse contribution impact in declining mortality. Since parameter bx is a function of age x, it doesn’t change with time t but is does change with age
x (it is directly correlated). It is observed that the relative speed of change of bx
is different for the two selected countries. A common trend regarding the selected countries is observed in the decreasing mortality rate. Namely the reduction is fairly constant for ages 0-20 and 60-110 where a linear reduction is noted, whereas some fluctuations are shown between ages 20-60 signifying that mortality improvements occur mainly between ages 20-60. This is a consequence of the Lee-Carter model’s inability to perform well in young (younger than 20) and old (older than 60) ages (Lee and Miller, 2001).
Mortality rates beyond 60 are becoming less and less effective to overall mortal-ity enhancements. This is an expected outcome since the elderly have the bigger probability of dying. This results are in line with (Zhao, 2012), who utilized the Lee-Carter model “for analyzing short-base-period data” for China, supporting the notion that for old ages, bx declines linearly. Furthermore, Koissi et al., (2006), are
supportive of the aforementioned view and conclude that in elderly ages mortal-ity improvements are less effective. They reach such verdict by using the Singular Value Decomposition (SVD), the Weighted Least Square (WLS) and the Maximum Likelihood estimate (MLE) to forecast mortality rates for four European countries through the Lee-Carter model.
Parameter κt, is demonstrative of decreasing mortality and both countries share
similar trends. Moreover, kt is decreasing as time rises and shows little signs of
volatility. Namely, as time goes by, mortality rates decrease and ultimately are the main cause of longevity risk. Hainaut (2012), supports that in the French con-text, the curve steepness is decreasing but with several volatile periods between 1946–1963 and 1964–2007. Figure 3 (Pg. 30) is indicative of a similar case regard-ing the Netherlands in which from 1955-1975 (especially males) the steepness of this curve is volatile and shares little similarities with other countries that experience linear reduction.
Figure 4: Forecast of parameter κt for the Canadian (top graph) and Dutch
(bot-tom graph) male population, producing an index for the general level of mortality forecasted up to 2110. In theory, κt changes with time (time-varying parameter),
and this can be confirmed empirically by the linear trend observed, demonstrating improvements for the two selected countries. Shade in the fan charts represents prediction in the 95% confidence interval.
Figure 4 shows the forecasted values of kt for the years up to 2110 using an
ARIMA(0,1,0) model. Lee and Carter (1992) used the ARIMA(0,1,0) model, and because their approach is highly compatible with the background of this thesis, their approach is implemented. The package “forecast” in R is used to forecast the results and the predicted values of kthave been rescaled to zero in the last observed year. A
downward linear trend for both countries is evident based on the time series model. As shown, kt declines linearly from 1950-2010, which is not in line with previous
literature since the pattern of change in life expectancy is not often linear (Oeppen, 2002). However, Lee and Carter (1992) find similar results in their study in years 1900-1989. Forecasting mortality trend and life expectancy is of high importance for pension funds and insurance companies since it can be a major factor in forecasting longevity risk. (Leng and Peng, 2016).
Namely, the Netherlands is considered an exception to the general parallel and upward trend regarding life expectancy at birth for both men and women (Mortal-ity.org, 2017). The Dutch life expectancy has risen to 82.7 years in 2009, a vigorous increase from the 76.5 years average in 1970. Up until the turn of the millennium the annual increase of male life expectancy was lesser in comparison to the 2000’s and onwards where it was fairly larger. On the other hand, the female life expectancy was stagnant between the 80’s and the 2000’s. To put everything into perspective, other Western European countries had healthier, in terms of volatility/linearity, and larger improvements in mortality compared to the Netherlands (Stoeldraijer et al., 2013).
Table 1: Increasing life expectancy, Females
1980 2000 2010 2030 2050 2070 2090 2100 2110 Japan 81.5 84.2 85.8 88.4 90.8 93 95.1 96.1 97 Canada 80.2 83.7 85.1 87.1 88.9 90.5 91.8 92.3 93.9 Netherlands 79.2 82.4 83.8 85.5 87.1 88.7 89 90.5 91.1 Switzerland 80.5 84 85.4 87.5 89.4 91.1 92.4 93.1 93.7 UK 79 82.9 84.1 85.5 87 88.6 90 90.7 91.4
Table 1 demonstrates period life expectancy at birth, for females, for all five countries. This is calculated using the Lee-Carter mortality model, where after forecasting the κt values using the
ARIMA(0,1,0) model period life Tables were produced for the dates specified in Table 1. Up to, and including 2010, the historical data is shown whereas from 2030 - 2110 the predicted/fitted
values are shown. Evidently, Japan has the higher life expectancy on average, whereas the Netherlands and UK have the lowest.
Figure 5: Historical and forecasted (up to 2110) death rates for 65 years old individ-uals for Canada. This Figure is separated into male, female and total population. Notably, at the age of 65 a linear decrease in death rates is evident.
Figure 6: Life expectancy in Canada for the years 1950-2110 where shade in the fan represents predictions in the 95% confidence interval. A rising life expectancy trend is evident over the future decades. When exogenous factors affecting mortality amongst males and females are absent in the first year of life, male mortality is 25%-30% higher compared to female mortality.
6. Table 1 shows life expectancy for females in years 1980, 2000, 2010, 2030, 2050, 2070, 2090, 2100, 2110, based on the Lee-Carter model, demonstrating the increase in life expectancy for selected countries. According to Ipss.go.jp (2017) forecasts, life expectancy for Japan in 2050 for females will be 89.22 whereas Table 1 shows that, based on the population sample assessed in this thesis, the Japanese life expectancy is calculated to be 90.8. The minor difference might be attributed to dissimilar data used, since the National Institute of Population and Social Security Research uses historical data including the 1st and 2nd world wars, whereas this thesis evaluated
and analyzed data from 1950 onwards.
Moreover, Knoema (2017) estimated that for females in the Netherlands, life ex-pectancy in 2050 will be 87.6 years whereas Table 1 indicates a minor difference of 0.5 (i.e. 87.1) years old showing a relatively accurate estimation of the results when compared to previous literature. The small difference can be attributed to different data used by Knoema (2017) compared to this research.
To support the above statements and in line with the expectations set up in this thesis, the life expectancy forecast for Canada shows an upward trend according to Figure 6, which transcribes to an increase in the life expectancy in the upcoming years up to 2110. Additionally, females have a higher life expectancy than males and according to Robson (2017); the biological advantage of women is taken for granted since male mortality is higher than females since the beginning of life. Furthermore, Figure 5 demonstrates that between years 1950-2010 male death rates are higher in comparison with female death rates, however there is some indication that in following years such relationship might be altered. That is mainly due to the labor intensive work that men had to go through in the past decades leading up to the turn of the millennium, however as time goes by, there is strong indication that the gap is shrinking, converging towards equilibrium.
4.2
Cairns-Blake-Dowd model
Figure 7 shows the historical and forecasted values of the parameters κ(1)t and κ(2)t for Canada and the Netherlands based on the Cairns-Blake-Dowd stochastic mortality model. These two countries are randomly selected out of the five evalu-ated in this thesis. Parameter κ(1)t illustrates the mortality improvements. As shown in Figure 7, both Canada and the Netherlands, experience mortality improvements which translates to decreasing death rates. As a result, the downward sloping trend and the negative value are justified.
The historical data trend of the parameter κ(2)t can be interpreted to, for older ages, death rates are improving at a slow-moving pace. According to Currie and Schwandt (2016), amongst elderly people, medical science innovations might not have a significant effect in reducing mortality rates. Namely, the elderly are of-ten faced with multiple health threaof-tening illnesses, therefore threating one leaves them exposed to others. As a result medical knowledge and technology are far be-hind when it comes to reducing mortality rates amongst the elderly. A significant amount of time and investment is necessary before reaching the point where medical innovation can positively affect death rates in higher ages. In the mean time, mor-tality rates will hardly be changed and indicate insignificant improvements amongst the elderly (Currie and Schwandt, 2016).
Cairns et al.,(2016) find in their study of English and Welsh males, that when t in parameter κ(2)t surpasses the age of 113 results to "deteriorating" mortality, which translates to increasing mortality rates rather than decreasing.
Moreover, Figure 7 is illustrative of the forecasted values of the two CBD stochas-tic components κ(1)t and κ(2)t . The two parameters will continue to move on the same trajectory as the historical data trend (1950-2010). In addition, parameter κ(1)t is less uncertain (narrower margin) than parameter κ(2)t which has a wider confidence inter-val. This a consequence of continues improvements in mortality in the near future (up to 2060). However, the extent of this improvement is somehow uncertain, hence the existence of longevity risk. As regards to the wider confidence interval of pa-rameter κ(2)t , the extent of medical innovation is uncertain thus the widened interval.
Figure 7: Forecasted values for parameters κ(1)t and κ(2)t according to the CBD model fitted to the Canadian (top graph) and Dutch (bottom graph) male population for ages 50-90 and for years up to 2060. Shades in Figure represent the 80% and 95% confidence interval. The reason why parameter κ(1)t is reversed when compared to parameter κ(2)t is because of κ(1)t negative value, whereas parameter κ(2)t has a positive value (see Section 3.2.3). This results shows consistency between the historical trends and forecasted values. It is predicted that both parameters will continue to move on the same trajectory.
4.3
Plat model
Parameter αx in Figure 8 is similar to parameter αx of the Lee-Carter (Figure
3). It has an increasing trend for ages 55-90 which is an indication that death rates increase with age. Thus, αx is a static age function showing the effect of mortality
throughout different age groups. Evidently this is a sign of no mortality improve-ments as older ages are approached. Namely, this lack of improvement is to be expected since older people are more vulnerable to diseases. For the remaining four countries examined (Netherlands, Canada, UK, Switzerland) a similar trend for pa-rameter αx for males is observed, signifying no improvement in old age mortality
rates.
Just like in the case of the Lee-Carter (Figure 3) and CBD (Figure 7) model, parameter κ(1)t shows the overall decrease in mortality rates over time. For the Japanese males, a decreasing trend is observed indicating a worldwide improvement in mortality rates. Individuals tend to live longer and this is deduced from parameter κ(1)t of Figure 8. Plat, (2009) found similar trends for parameter κ(1)t while exam-ining parameter uncertainty using the Plat model in the US context. In addition, parameter κ(2)t is reflective of the positive relationship amongst age and decreasing death rates. According to Figure 8 results, Japanese κ(2)t trend demonstrates the decrease in mortality from 1985 onwards.
Furthermore, γt−xconsiders the cohort effect analogous to the model of Cairns et
al. (2007). The trend extracted from this process should be totally random and have no conceivable trend. As a result, a mean with a random trend is indicated in Figure 9 for parameter γt−x. The primary observation to be made is that an unspecified
trend is observed for Japan (Figure 8) regarding the cohort effect, suggesting that the trends related to the cohort effect in the Japanese context are not as significant. Cairns et al. (2009), in the comparative study involving the US and the UK, find that the cohort effect has "less well defined pattern" in the United States resulting in "a greater degree of randomness".
Figure 8: Parameters for Plat model fitted to the Japanese male population for ages 55-90 during 1950-2010 (for κ(1)t and κ(2)t ) and 1860-1950 (for the cohort effect). Parameter αx describes the general pattern of mortality by age, thus interpreted as
an age effect. Parameter κ(1)t describes the general pattern of mortality by period (i.e. period effect) whereas parameter κ(2)t allows for mortality variations regarding different ages, hence its volatile and not linear trend. Lastly, γt−x accounts for the
Figure 9: Cohort indexes forecasts based on the Plat model, for males in Canada (top graph) and the Netherlands (bottom graph), aged 55-90 during the period 1850-2050. Shades in this Figure represents the 80% and 95% confidence intervals. An improbable augmentation in mortality rates is evident for the Plat model fore-casts (see figure 15). This is a consequence of its connection to the estimated cohort effect γt−x which is indicative of a sharp upward slopping trend between period
4.4
Li-Lee model
Figures 10 and 11 show the common time and age parameters for the selected populations. A set of populations (i.e. the five selected countries), for which fore-casts for mortality are to be made, are considered. In order to best describe the mortality alterations of the group as a whole, Bx and Kt are picked. The ordinary
Lee-Carter method is utilized to obtain the common parameters Bx and Kt, where
Kt is calibrated on the life expectancy average and consequently modeled as a
Ran-dom Walk with Drift (RWD) in a quest to predict the recurrent trend in future mortality change.
More specifically, Figure 10, is indicative of a steady decline in mortality, es-pecially after 1975 where a more steep and less volatile decline is notable. This is translated to uniform improvement in mortality worldwide. This supports the research of Enchev et al., (2016), who find similar results regarding six European countries. However, they find that the steep and less volatile decline starts in 1970.
Additionally, Figure 11, shows another common parameter, this time associated with age rather than time. Namely, a declining trend is again distinguishable in-dicating that in smaller ages, a more obvious mortality improvement is notable in comparison with older ages. This is interpreted to the fact that older individuals’ mortality improvements are harder to achieve primarily because the elderly are far more susceptible to illnesses and other life threatening diseases.
Figure 12 shows the observed and forecasted log death rates, fitted according to the Li-Lee multi-population mortality model. Compared to the Lee-Carter, CBD and Plat model (see Figures 13-15), both models (including Li-Lee) offer similar results thus adding extra robustness. The difference between the aforementioned Figures and Figure 12 lies in the fact that, the Li-Lee model has a narrower variance (i.e. narrower confidence intervals).
Despite the fact that all four models predict similar logarithmic death rate trends, the Li-Lee model enjoys less parameter risk, primarily due to larger observation pool included in the sample. Moreover, it considers the fact that mortality rates for all five countries are probably keen to converging in the long run. As a result, the Li-Lee model common factor has performed superbly for all five countries grouped together.
Li and Lee, (2005) in their original paper examined the performance of 15 low-mortality countries and conclude that the common factor, based on the Li-Lee model,
has performed adequately for all 15 countries as a group. In fact, if a research was to be made to forecast a specific country, like for instance the US, they argue that it would act as an improvement if the study was comparative between two or more countries sharing similar characteristics (Li and Lee, 2005).
Figure 10: Estimated common parameter Kt computed based on the Li-Lee
multi-population mortality model for years 1950-2010. It shows the common
mor-tality index of the selected populations (Japan, Netherlands, Canada, UK, Switzerland), for the selected years (1950-2010).
Figure 11: Estimated common parameter Bx computed based on the Li-Lee
multi-population mortality model for ages 60-90 for the five countries selected (Japan, Netherlands, Canada, UK, Switzerland).
Figure 12: Forecasting of death rates based on the Li-Lee model for years 2010-2180 and for ages x =30 (green), x =55 (orange) and x=85 (blue), in a logarithmic scale. In fact the logarithmic scale is used to aid result representation. The thinner lines represent historical rates for years 1950-2010. The 95% confidence interval is representative of forecasted death rates. A declining trend is observed across years indicating improvements in mortality. This common trend (common for the Nether-lands, Canada, Japan, Switzerland and the UK) assures that long-run forecasts are less volatile whilst permitting for short-run variation.
4.5
Lee-Carter, CBD and Plat model: A comparison
In order to further compare the three single-population models used in this the-sis, this Section shows the main differences and similarities regarding mortality rates.
Future mortality rates up to 2110 are demonstrated in Figures 13-15, for the 3 single population models evaluated. The death rates are given by mx,t whereas
mortality rates are given by qx,t. Those two variables are interconnected since qx,t
= 1 - exp(-mx,t) for age x and time t.
According to Mortality.org (2017), mortality rates are decreasing globally, a fact that is supported by the downward slopping and linear trend observed in Figures 13, 14 and 15. However, small differences are observed between the countries and this is mainly due to different contributions (i.e. Statistical significance) that each country makes in decreasing mortality. Nevertheless on average, developed coun-tries, tend to experience decreasing mortality rates that are yielded by technological breakthroughs and medical innovation (Jayachandran et al., 2010). In addition, confidence intervals differ amongst models whereas the gradient (steepness) of the trends is dissimilar.
In accordance with historical evidence, fan charts at age 85 show evidence of a wider confidence interval, for CBD (Figure 14) and Plat (Figure 15) models, com-pared to age 65. In other words, the wider the the confidence interval the more uncertain and volatile future predictions are, resulting to a lower confidence level. Furthermore, for the Lee-Carter model (Figure 13), graphs indicate a narrower mar-gin for age 85 compared to age 65. A reasonable interpretation of this would be that, the dataset used in this thesis is not suited for the model. Its worth mention-ing that, for the Lee-Carter model, the margin width is proportional with the age effect, βx and taking into account that the LC model is not meant for predictions
regarding the elderly, fans are narrower rather than wider for ages 85 and above. On the contrary, the CBD and Plat model indicate superior data fit for old ages. The forecasts extracted from the Plat model (Figure 15) show an implausible increase regarding mortality rates. The central trend is related to the estimated cohort effect γt−x for the Plat model (see Figure 9) which shows a steep upward trend between
1935 and 1955.
As discussed in previous lines, models vary in terms of volatility and mortal-ity rate predictions across the ages, thus it should be noted that researchers must implement a wide variety of models to account for these diversities (Cairns et al.,
2011). In essence, a combination of models should be used when forecasting to take advantage of different attributes available from different models. In support of the data in the above Figures, Cairns et al., (2011) compare 6 mortality models (includ-ing LC, CBD and Plat in the English and Welsh context) for males and find several diversities in their fan charts amongst models. Differences include the inability of the LC model to accurately perform in old ages.
Figure 13: Fan charts indicating mortality rates qx,t for ages x = 65 (bottom), x
= 75 (middle) and x = 85 (top) extracted from the Lee-Carter model fitted to the UK (top-left), Japan (top-right), Netherlands (low-left), and Switzerland (low-right), for male population aged 50 to 90 for the period 1950-2010. The dots represent the historical mortality rates for the time period 1950-2010. The fan shades show prediction intervals at the 50%, 80% and 95% confidence level.
Figure 14: Fan charts indicating mortality rates qx,tfor ages x = 65 (bottom), x = 75
(middle) and x = 85 (top) extracted from the Cairns Blake Dowd model fitted to the UK (top-left), Japan (top-right), Netherlands (low-left), and Switzerland (low-right), for male population aged 50 to 90 for the period 1950-2010. The dots represent the historical mortality rates for the time period 1950-2010. The fan shades show prediction intervals at the 50%, 80% and 95% confidence level.
