Coherent mortality analysis of Curacao

(1)

Coherent Mortality Analysis Of

Cura¸

cao

Arthur Rosenda

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: Arthur Rosenda

Student nr: 5634989

Email: Arrose@gmail.com

Date: July 6, 2014

Supervisor: Prof. dr. ir. Michel H. Vellekoop Second reader: Prof. dr. Rob Kaas

(2)

Introduction

In the publication ”World Population Ageing: 1950-2050 ” the United Nations provides the results of a comprehensive study on the global trends of aging. The UN (2002) report captured some important elements on the trends of global aging:

• That global aging is the result of two factors that affect the make up of the population of countries, namely, a decline in fertility rates and a similar decline in mortality rates. • Aging already heavily impacts developed nations but the rate by which developing nations

will be impacted by this phenomenon will be greater. The share of elderly persons in rela-tion to the total popularela-tion will increase at a faster pace in developing narela-tions compared to industrialized nations, leaving these nations in development with little time to adjust to the impending demographic changes.

• As the ratio young-old shifts in the world populations, societies will be hard pressed to finance social security and health care cost of this increasingly elderly population. As the number of working-age persons decreases the tax base which is required to finance the costs attached to population aging is also diminishes, adversely affecting inflow from taxes. • That population aging is a continuous process that will in the twenty first century manifest

itself at a higher pace than in the twentieth century. It is our collective challenge as human beings to the find the proper balance which will guarantee the financial security and dignity as we age.

Aging, or otherwise known as demographic transition, also has implications for Cura¸cao. The fact that the population is aging affects the entire social system; the cost of financing state and private pension arrangements is set to increase. That people live longer does not per definition entail that the additional years lived are healthy. The elderly-sick will undoubtedly strain the health care budgets.

The impact of aging is measurable by the, often belated, actions of Governments to mitigate its socio-economic effects. Since 1974 the pensionable age has been 60 years in Cura¸cao but due to the budgetary strain of an increasing elderly population the Government increased the pensionable age to 65, in early 2013. The measures already taken are not guaranteed to mitigate future adverse developments in fertility and mortality rates and as the demographic transition continues additional policy measures will be required. In order to make informed decisions, the policymakers of Cura¸cao need reliable long range mortality forecasts. This research is intended both as a reference point for future analysis of mortality rates by policy makers and researchers in the area of demography and actuarial sciences.

(5)

1.1 Research Question

The central question in this research is how to forecast Cura¸cao mortality rates using a Lee-Carter method with relatively few available data points. And, given the relationship between Cura¸cao and The Netherlands, to what extent can coherent mortality forecasts for Cura¸cao, be improved using Dutch data.

1.2 Scope and Aim of Research

This research aims to analyse past developments in mortality rates of Cura¸cao and make forecasts of these rates into the future. Little, if any, research has been done on mortality forecasting for Cura¸cao or for that matter on such small populations. Adequate mortality forecasts are contingent on the availability of historical data and since the data from Cura¸cao is limited, this thesis aims to quantify the longevity risk that Cura¸cao faces on the medium to longterm. In addition, this thesis explores whether the mortality rates of the Netherlands, in a coherent model framework, can help in forecasting mortality rates of Cura¸cao.

1.3 Reseach Method

This thesis is limited to quantitative research. The mortality data are calibrated to mathematical models and these models in turn are used to make forecasts. The data, information on the models, statistical applications, and general information used in this thesis is collected from multiple sources, such as the internet and scientific periodicals.

(6)

Chapter 2

Background and Available Data

2.1 The Pension System of Cura¸

cao

The pension system of Cura¸cao is based on three pillars namely: • State Pension

• Supplemental (Collective) Pension Arrangements • Private Insurance

The state pension provides a base level income to the elderly while the supplemental pension provides an addition this base pension to a maximum of 70% of the last earned pay. The supplemental pension is usually a collective pension arrangement derived from a employer-employee relationship but can also have a private, non-collective, character. Private pensions are non-collective by definition and also serve as a supplement to both the state and collective pension plans. In order to remain tax advantageous, income derived from state and supplemental pensions plans must not exceed 70% of the last earned pay. If this requirement is met, premiums paid to fund the supplemental pension plan are tax deductible.

2.2 State Pension

The state pension ”Algemene Ouderdomsverzekering” (AOV) entitles all residents of Cura¸cao to a basic old age pension. The AOV is comparable to the AOW public insurance of the Netherlands and the Social Security Insurance in the United States and aims to provide the elderly with a minimum income level based on residency. From the age of 15 until the age of 65 residents accrue 2% of old age pension per year. If the residency period is uninterrupted a resident can accrue a maximum of 50 years which grants a full pension. The maximum gross AOV pension amounts to Naf 862 a month. The AOV is a pay-as-you-go system that relies on premiums paid by the employed to finance the pensions of the pensionable. The premium amounts to 15% payable over a base income up to Naf 100,000. The employer pays 9% of this premium and the employee pays the balance of 6%. An additional 1% in premium is payable on income in excess of the aforementioned income base. In case a resident dies before attaining the age of 65, through the ”Algemene Weduwen- en Wezenverzekering”(AWW) the spouse\partner and orphans are provided with a widow(er)\orphan pension. All residents from the age of 15 are insured for an AWW.

2.3 Employer Pension Arrangements

The employer pension plan is based on employment relationship between the employer and the employee. These pension plans tend to be collective plans (re)insured in a pension fund or with

(7)

an insurer. The risks that are insured typically are old age, mortality and disability and are mitigated by an old age pension, widow(er)\orphan pension and disability pension, respectively. The pension plans most prevalent in Cura¸cao are defined contribution plans and defined benefit plans. In a defined contribution plan the premium level paid into the pension plan is guaranteed. In contrast, for a defined benefit plan the benefit level is guaranteed. In essence a defined contribution plan is a premium commitment while the defined benefit plan is ben-efit commitment. A review of the ordinance that regulates government employees’ pensions (Pensioenlandsverordening overheidsdienaren, 2013) reveals that the Government changed the pensionable age for the state pension to age 65 but did not adjust upward the pensionable age for the government employees. The failure to synchronize the pensionable age, at least for government employees, creates a financial dilemma. A government employee that retires at 60 has to wait an additional 5 years to receive the complement to his supplemental pension. While receiving less income the retiree is still liable to pay both the employers’ and the employee’s portion of the premium payable for the state pension until the state pensionable age.

2.4 Private Insurance

Private insurance is supplementary to the state and collective pension arrangements. Like the state and collective pension plans, private insurance is a method to mitigate old age risk, mor-tality risk and disability risk. The difference between private and state and collective pensions is that the tax treatment is disadvantageous and does not allow for tax deductibility of premiums.

2.5 Mortality and Population Data Cura¸

cao

The Cura¸cao mortality and population data were provided by the Central Bureau of Statis-tics of Cura¸cao. In 1998 Cura¸cao automated the process of recording its population data and adopted the administrative system used by Municipalities in the Netherlands. This administra-tive system is commonly known by its acronym GBA, which stands for the Gemeentelijke Basis Administratie. Prior to the implementation of this electronic system, mortality and population data were recorded on paper, thus limiting access to data predating the implementation year of the GBA.

The demography data of Cura¸cao per gender classification (male\female) consists of: • Observed deaths as per the last birthday (Dx,t). The time series used span the years

1998 ≤ t ≤ 2012 and ages 0 ≤ x ≤ 100.

• Population\Exposure data (E_x,t) as per the first of January. The time series used span the years 1992 ≤ t ≤ 2012 and ages 0 ≤ x ≤ 100.

2.6 Mortality and Population Data Netherlands

In the Netherlands, the Central Bureau of Statistiek (CBS) is tasked with the collection and dissemination of statistical data. The institution collects statistical data on a range of social and economic fields of interest including demography. Mortality and population time series ranging back to 1950 are available to the general public through the electronic database StatLine.

The Dutch demography data per gender classification (male\female) consists of:

• Observed deaths as per the last birthday (Dx,t). The time series used span the years 1950 ≤ t ≤ 2012 and ages 0 ≤ x ≤ 99.

• Population\Exposure data (E_x,t) as per the first of January. The time series used span the years 1950 ≤ t ≤ 2012 and ages 0 ≤ x ≤ 99.

(8)

The lack of, electronically available, historical data for Cura¸cao imposes a constraint on the data set suitable for research. While the observation period for the Netherlands dates back to 1950, the data for Cura¸cao is lower bound at 1998. In addition, with a maximum age category of 99+ years, the ages for which the Netherlands records mortality is broader than Cura¸cao (85+ years). For the purpose of comparison between The Netherlands and Cura¸cao, the smaller data set of Cura¸cao limits the observation period and age categorization at 1998 ≤ t ≤ 2012 and 0 ≤ x ≤ 85 for the mortality and population data sets of both countries.

2.7 Development of Mortality Rates

Due to the small population size of Cura¸cao the data set contains numerous zero valued mortality observations and exhibits a lot of volatility. This volatility is best illustrated graphically by surface plots of the log of the mortality rates; with the central rate of mortality rate mxtat age x for time period t approximated by force of mortality:

mxt= Dx,t Ex,t

≈ µxt.

The jaggedness of the log mortality surface graphs for male and female (Figure 2.1) provides an indication of the variability in the data. In the instances where Dx,t = 0 so ln(mx,t) = −∞ the observations are excluded from the plot.

Ages (0−85)

Years (1998−2012)

Logr ates

Mortality Surface Females

(a)

Ages (0−85)

Years (1998−2012)

Logr ates

Mortality Surface Males

(b) Figure 2.1: Mortality Surfaces (Log)

In the 3 dimensional graphs the white-areas denote the observations where mx,t = 0. This phenomena is prevalent in childhood and in adolescent years.

In addition, in figure 2.2, for the bookend years of the observation period the mortality rates were plotted in a scatter plot together with the locally weighted fitted line in order to get a general indication of the development of the mortality rates in time.

Not much inference can be drawn from the scatter plots of the selected years other than that it reaffirms the variability of the mortality rates. However, the smoothed data (solid lines), reveal some general patterns. In general male mortality is higher than female mortality for both years and it is noticeable that there is a sizable incidence of infant mortality. Childhood mortality is minimal or even zero at certain ages (missing data points) but at adolescence and early adulthood mortality risk becomes more prevalent and the mortality rates increase again. At adulthood (x ≥ 40) a log-linear mortality trend sets in for both genders.

(9)

−6 −4 −2

0 20 40 60 80

Age

Death Rate (log)

female male Mortality Rate 1998 (a) −7 −6 −5 −4 −3 −2 0 20 40 60 80 Age

Death Rate (log)

female male

Mortality Rate 2012

(b) Figure 2.2: Mortality Graphs for Selected Years (Log)

For a clearer indication of the graduation of mortality rates the mean of the log mortality rates can be determined and plotted for each gender. The time-average mortality gives a central tendency of the mortality over the period in observation.

−7 −6 −5 −4 −3 −2 0 20 40 60 80 Age

Death Rate (log)

female male

Mean Log Rate 2012−1998

(a) −7 −6 −5 −4 −3 −2 0 20 40 60 80 Age

Death Rate (log)

female male

Mean Log Rate 2012−1998 −Smoothed

(b) Figure 2.3: Mean Death Rates (Log)

Over the entire observation period, on average the mortality rates for males is higher than that of females at most ages. For males the customary accident hump in early adulthood is more pronounced while for females the hump is less distinct. The graph for both genders is discontinuous because for females age x = 8 and males age x = 9 not a single death occurrence is observed over the entire observation period 1998-2012.

While averaging over the time period takes out some of the volatility out of the mortality rates of Cura¸cao, the graphs for Cura¸cao are not as smooth as similar graphs for the Netherlands figure 2.4. The differences in population sizes become evident in figure 2.4; small changes in mortality have more impact on the rates of the Cura¸cao than that of The Netherlands.

On average the death rates of The Netherlands are considerably lower than the mortality rates of Cura¸cao. Between the age of 0 to 40 years the difference in mortality rates is substantial, beyond age 40 the difference in mortality rates diminishes.

(10)

−8 −6 −4 −2 0 20 40 60 80 Age

Death Rate (log)

curacao netherlands Female (a) −8 −6 −4 0 20 40 60 80 Age

Death Rate (log)

curacao netherlands

Male

(b)

Figure 2.4: Mean Death Rates Cura¸cao and The Netherlands

definitive indication into aging of the population of Cura¸cao. To get a better indication whether the population of Cura¸cao is aging we can examine the time-development of the aging indicators.

2.8 Aging Indicators

Based on the 1998-2012 time-period the population totals for Cura¸cao and the Netherlands were calculated. The results are displayed in figure 2.5. The population of The Netherlands has seen a steady increase over this period, while the population of Cura¸cao decreases to a minimum of 126,715. The decrease in the population of Cura¸cao is primarily due to negative net-migration (Immigration - Emigration). According to The World Bank during the five year period between 1999-2003, on balance 6,070 more persons emigrated than immigrated. After 2002 the population totals started to increase again, the average population over the observation period was 138,500.

130000 135000 140000 145000 150000 2000 2004 2008 2012 year totalpop (a) 15800000 16000000 16200000 16400000 16600000 2000 2004 2008 2012 year totalpop (b)

(11)

The demographic makeup of the population according to gender is depicted in the population pyramid graphs in figures for both populations for the years 1998 and 2012, figures 2.6 and 2.7.

Curacao 1998 4.2 2.2 0.2 0 1 2 3 4 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−44 85+

Male Age Female

% % 4.7 4.2 4.3 3.9 3 3 3.7 3.8 3.4 3.1 2.5 2 1.8 1.4 1 0.6 0.3 0.2 4.6 4.2 4.1 3.8 3.1 3.6 4.5 4.7 4.2 3.7 3.1 2.4 2.1 1.7 1.3 1 0.6 0.7 (a) Curacao 2012 4.2 2.2 0.2 0 1 2 3 4 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−44 85+

Male Age Female

% % 3 3.3 3.6 3.8 2.6 2.3 2.2 2.7 3.3 3.6 3.5 3 2.7 2.1 1.5 1.1 0.6 0.4 3 3.1 3.3 3.7 2.8 2.9 3.1 3.6 4.2 4.7 4.6 3.8 3.4 2.8 2 1.5 1 0.9 (b) Figure 2.6: Population Pyramid Cura¸cao

The Netherlands 1998 4.2 2.2 0.2 0 1 2 3 4 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−44 85+

Male Age Female

% % 3.2 3.2 3 3 3.2 4.1 4.3 4.1 3.8 3.7 3.5 2.6 2.2 1.9 1.5 1.1 0.6 0.4 3 3.1 2.9 2.9 3.1 4 4.1 4 3.7 3.6 3.3 2.5 2.3 2.2 2 1.7 1.2 1 (a) The Netherlands 2012 3.2 2.2 1.2 0.2 0 1 2 3 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−44 85+

Male Age Female

% % 2.8 3 3.1 3.1 3.2 3 3 3.2 3.9 3.9 3.7 3.3 3.2 2.6 1.9 1.3 0.9 0.6 2.7 2.8 3 2.9 3.1 3 3 3.2 3.8 3.8 3.6 3.3 3.2 2.6 2 1.7 1.3 1.3 (b) Figure 2.7: Population Pyramid The Netherlands

The pyramids make evident that there is an upward shift in the distribution of the population totals for both countries. For Curacao it is noticeable the population totals decrease in the late teens and early adulthood. This is attributable to the annual brain drain that takes place when college aged students and young professionals emigrate to study and/or work; most emigrate to The Netherlands.

According to (UN, 2002) a decline in fertility rates combined with a decline in mortality rates underlie the process of aging, otherwise defined as demographic transition. In order to measure demographic transition, the UN provides indicators of population, of which a non exhaustive list is included below:

• The aging index; is calculated as the number of persons 60 years old or over per hundred persons under age 15.

• The total dependency ratio; this is the number of persons under age 15 plus persons aged 65 or older per one hundred persons in ages 15 to 64. It is the sum of the youth dependency ratio and the old-age dependency ratio.

(12)

• Life expectancy at a specific age; is the average number of additional years a person of that age could expect to live if current mortality levels observed for ages above that age were to continue for the rest of that person’s life. In particular, life expectancy at birth is the average number of years a newborn would live if current age-specific mortality rates were to continue

For the period 1998 to 2012 the aging criteria mentioned above are calculated in order to determine the level of aging the population of Cura¸cao has undergone; the aging indicators for The Netherlands are used as a reference.

2.8.1 Aging Index and Total Dependency Ratio

In the figure below the Aging Index and the Old-Age Dependencies are plotted for Cura¸cao and The Netherlands. The aging index is a ratio of the persons over the age of 60 divided by the number of of persons below the age of 15. This measure gives an indication of the balance between young people and old people. Preferably this balance is tilted toward a young society with an index value < 50.

40 50 60 70 80 90 2000 2004 2008 2012 Year Inde x Curacao Netherlands Aging Index

(a) Aging Index

14 16 18 20 22 24 2000 2004 2008 2012 Year Ratio Curacao Netherlands

Old−age Dependency Ratio

(b) Old-Age Dependency Ratio Figure 2.8: Aging Indicators

The aging index for Cura¸cao has been increasing steadily since 1998, the rate level off but the general trend is upward. The same is true for The Netherlands, as a developed nation the aging index is higher and trends upward.

While not all elderly people are dependent, the Old-Age Dependency Rate gives an indication of the support the elderly might need. This rate has also been on a steady increase for both Cura¸cao and The Netherlands.

2.8.2 Life Expectancy

In order to estimate the development of life expectancy, the period life expectancy at ages 0, 25, 45 and 65 were calculated for the years 1998 to 2012. In figure 2.9 the life expectancy over this time period is depicted together with a linear trend line.

(13)

The trend for male and females at birth and at pensionable age are upward which is indicative of a decrease in mortality rates over time.

1998 2002 2006 2010 70 72 74 76 78

Life Expectancy New Born

Year Y ears Female Male (a) e0 1998 2002 2006 2010 13 14 15 16 17

Life Expectancy Pensioner

Year Y ears Female Male (b) e65

Figure 2.9: Life Expectancy Cura¸cao

In comparison with the life expectancy of The Netherlands, figures 2.10 and 2.11, the life expectancy for the population of Curacao is substantially lower for both males and females. The life expectancy for the Netherlands also shows an increasing trend over the period 1998–2012 for new born and at pensionable age. Based on the slope of the trend line for Dutch males at age 65 we can conclude that the mortality rate of the elderly has been improving considerably.

1998 2002 2006 2010

77

78

79

80

Life Expectancy Females x = 0

Year Y ears Curacao Netherlands 1998 2002 2006 2010 70 72 74 76 78

Life Expectancy Males x = 0

Year

Y

ears

Curacao Netherlands

(14)

1998 2002 2006 2010

16.0

16.5

17.0

17.5

Life Expectancy Females x = 0

Year Y ears Curacao Netherlands 1998 2002 2006 2010 13.0 14.0 15.0 16.0

Life Expectancy Males x = 0

Year

Y

ears

Curacao Netherlands

(15)

Chapter 3

Literature Review

Since the Lee-Carter model was published in 1992, the model has been widely adopted in academia and business alike. In particular, in the fields of demography and actuarial science the model has been applied to forecast mortality trends for individual countries, to make gender and cause-specific rates of mortality forecasts. However, the applicability of the model is not limited to forecasting of mortality rates of single countries and it has also been used to analyze mortality trends in groups of countries like the OECD countries.

The body of literature on the Lee-Carter model, and subsequent extensions on the model, provides an invaluable insight into the strengths and weaknesses of the model. While not perfect, the model’s simplicity and relative ease of implementation make it a popular model to fit and forecast mortality.

3.1 The Model

By design the Lee-Carter model is capable to extrapolate an (improving) mortality trend in time. If the rate of mortality for age x and time t is defined as mx,t, the model proposed by Lee and Carter (1992) fits a matrix of mortality rates as

ln mx,t= ax+ bxkt+ x,t.

The model equates mortality rates to age and time factors with the factors ax and bx as age specific constants that vary over time and ktas an index of the mortality level that varies across ages. The model parameters are defined as follows:

kt is an indicator of the level of mortality at time t;

bx is a measure of the rate of change of the mortality level at age x; ax describes the time-independent pattern of the mortality curve at age x,

x,t encapsulates the variability not explained by the other terms. The error terms are i.d.d. gaussian with mean µ = 0 and variance σ2.

To find a unique solution Lee and Carter (1992) imposed restrictions on the model parameters bx and kt and applied a criterion to fit the model. The parameters are scaled such that

ω X x=0 bx= 1 x = 0 . . . ω and, T X t=1 kt= 0 x = 0 . . . T.

(16)

Age x is maximized at ω for the model and the time step t reaches its maximum at T . Given these constraints the sum of errors squared, P

xt2x,t = P

xt(ln mx,t − ax + bxkt)2, is minimized using Singular Value Decomposition (SVD) to obtain optimal set of parameters bx and kt. The limits placed on the other parameters lead to a shape factor ax that equates to ax= _T1 PT_t=1ln mx,t.

Lee and Carter fitted their model to mortality data of the combined sexes of the United States over the base period 1933 to 1987. The selection of a base period of 55 years was after Lee and Carter (1992) concluded that a shorter base period (10 20 years) adds variability to fitted models and the forecasts. In a second estimation stage the parameter kt is adjusted such that the fitted number of death occurrences ( bDt) equal the actual number of death occurrences (Dt).

The Lee-Carter model in essence has a single time dependent parameter kt, which can be estimated in order to perform interpolation/extrapolation of log mortality rates. These log mor-tality rates subsequently serve as the basis to derive other actuarial quantities of interest such as life expectancy.

Lee and Carter performed model identification methods to find an appropriate model to forecast the mortality level kt and concluded that this parameter is best modeled by a random walk with drift (RDW). The definition of RDW model for the level of mortality is

kt= kt−1+ d + t, where d is the drift parameter and t are i.d.d.

In addition, Lee and Carter (1992) expressed a preference for the forecasts (ln ˆmx,T) at jump-off year T to equal the average mortality rates (ax) and not the actual rates (ln mx,T); consequently accepting a discontinuity between the actual rates and the forecasted mortality rates in the jump-off year.

In summary the Lee Carter model hinges on the choice of: • base period selection (fitting period),

• method to fit the model parameters,

• second stage estimation method of kt ( jump-off choice) and forecasting. 3.1.1 Base Period Selection

In a study of the mortality decline among the G7 countries Tuljapurkar et al. (2000) were the first to adjust base period and allow the fitting period of the model to commence in the year 1950. The study applied the Lee-Carter model over the period 1950-1994 and concluded that the G7 countries share a common pattern of mortality decline. In their analysis Tuljapurkar et al. (2000) noted that on the long term, the time dependent mortality level factor kt was (highly) linear in its decline. In addition, Tuljapurkar et al. (2000) concluded that the age dependent factor bx, while containing some country specific historical and social characteristics, was also dependent on the selection of the base period. They noted that while the general patterns for bx before and after 1950 were similar, the period prior to 1950 exhibited steeper childhood mortality.

In an evaluation of the Lee-Carter model Lee and Miller (2001) proved the existence of a structural change in the rate of decline of mortality rates between the periods 1900-1950 and 1950-1995. For the sake of prudence, Lee and Miller included this reduction in mortality rates in the estimation process of bx by adjusting the start of the fitting period to the year 1950. This adjustment did not invalidate the use of a fixed bx in the Lee-Carter model but rather

(17)

invalidated the assumption of a fixed bx over excessively long time frames. With the publication of (Lee and Miller, 2001) one of the intellectual fathers of the Lee-Carter model, accepted the base period adjustment of Tuljapurkar et al. (2000) as an improvement of the original model.

3.1.2 Methods to Fit the Model

To fit the model parameters Lee and Carter applied Singular Value Decomposition (SVD) on the mean centered mortality rates

¯

mx,t= ln(mx,t) − ax,

with ax defined as _T1 PTt=1ln mx,t. However, SVD is not suitable to fit the mortality data of Cura¸cao because of the ”zero-cell” problem. This term was coined by Wilmoth (1993) and refers to the problem of having to take the logarithm of zero as part of the process to find the optimal model parameters. Instead of SVD, Wilmoth suggests two additional computational methods to estimate the model parameters, namely the Weighted Least Square (WLS) method and Maximum Likelihood (MLE) method, both of which provide a solution to the zero-cell problem.

To circumvent taking the logarithms of zero the Lee-Carter model can be fitted using Weigthed Least Squares (WLS). WLS expands the sum of squared errors equation to allow for weighting by the number of deaths Dx,t

X

xt

Dx,t(ln mx,t− ax− bxkt)2

In the instances where the central rate of mortality (mx,t) equals zero, the observations are replaced by dummy data. As a result of the weighting and the use of dummy data the optimiza-tion process is not hampered by the zero-cell problem. To find the parameter estimates ax, bx and kt the derivative with respect to the parameters are set to zero and solved for each of the parameters. This leads to a set of equations that require numerical algorithms to be solved.

Besides solving the zero-cell problem, according to (Wilmoth, 1993) WLS has another ad-vantage. Since more weight is given to ages where the incidence of death is high the model tends to fit better at those ages as opposed to the ages where the incidence of death is lower.

Maximum Likelihood Estimation (MLE) provides another avenue to estimate the model parameters. In a MLE framework, the death count, Dx,t is assumed to be Poisson distributed with mean λx,t= mx,tEx,t. This leads to optimization of to the following log-likelihood equation,

maxX

xt

Dx,tln(λx,t) − λx,t,

where λx,t equates to the Lee-Carter model parameters as λx,t = eax+bxktEx,t.

By substituting the Lee-Carter equivalent of λx,tin the log-likelihood equation and maximiz-ing with respect to ax, bxand ktthe optimal model parameters are found. Wilmoth (1993) asserts that the WLS and MLE methods obtain parameters that are comparable but not identical.

The fact that WLS and MLE yields similar results for the fitted parameters does entail that the methods are comparable. While the WLS eliminates the zero-cell problem and has easily definable first order derivates, the bias created by the estimator nullifies these advantages (Girosi and King, 2007). According to Girosi and King, the WLS is not a sound statistcal method because it incorrectly assumes heteroskedastic disturbances with variances proportional to the

(18)

death count 1/Dx,t; inverse proportionality of the variance of the disturbance to Dx,t does not hold. In addition, for the likelihood estimation to be sound the weights can not be a random variable of the system. A second disqualifier of the WLS method is that optimization problem has multiple minima and numerical optimization routines not suitable to deal with this problem will yield a sub-optimum as the optimal solution. (Girosi and King, 2007).

3.1.3 Second Stage Estimation Method and Forecasting

Lee and Carter suggested the re-estimation of the level of mortality index kt such that the actual number of deaths observed equal the fitted number of deaths in a particular year. The re-estimated ktare then forecasted using a random walk model with a drift parameter,

kt= kt−1+ d + t,

while the parameters ax and bx are held constant. As a consequence of this approach, the mortality rates at the jump-off year t = T are fitted rates and do not equal the observed mortality rates. The model lacks a continuous transition between the actual rates and the forecasted rates which creates bias in the jump-off year; with bias defined as the difference between the actual rates and the fitted rates ln(mx,T) − ln( ˆmx,T) at the jump-off year.

Lee and Carter were aware that their approach created a discontinuity between actual and forecasted rates which can be solved by adjusting the model such that ax = ln(mx.T). But the authors were not prepared to trade-off the goodness of fit of the model in order to remove the bias in the jump-off year. Lee-Carter were convinced that adjusting kt to the annual death count (Dt) and equating ax to the yearly average mortality would have a negligible effect on life expectancy.

In an assessment various methods to forecast mortality in time, Bell (1997) compared group of models which included the original Lee-Carter model and a bias adjusted extension of the Lee-Carter model. Bell concluded that the Lee-Carter model with a bias adjustment, such that ax= ln(mx.T), outperforms the original Lee-Carter model. As matter of fact, the bias adjusted model outperformed all models he considered. The other models under consideration by Bell included the principal component model of Bozik and Bell (1987) and the model of Heligman and Pollard (1980).

A evaluation of the Lee-Carter model by Lee and Miller (2001) quantifies the bias created in jump-off year in units of years of the life expectancy at birth (e0). They calculated that the jump-off related error amounted to 0.6 years of life expectancy. Lee and Miller (2001) proves that the decision of Lee and Carter not to start the model forecasts at the last observed rates of the base period was erroneous.

3.2 Evaluating Lee-Carter Model

With the purpose of making a detailed assessment of the Lee-Carter model Lee and Miller (2001) examined the forecast errors of the model against the forecast errors of the Social Security Agency of the United States (SSA).

In an initial assessment of the Lee-carter model, Lee and Miller (2001) evaluated the impact of setting ax equal to the average of the log mortality rates. They concluded that a better option would be to set ax equal to the actual death rates in the jump-off year. By setting ax equal to average of the log of the mortality rates Lee and Carter (1992) introduced, in the jump-off year. This bias amounted to 0.6 years in terms of life expectancy.

Based on this analysis Lee and Miller (2001) concluded that forecast using data to fit the model prior to 1946 tended to underestimate the actual life expectancy for 1998 by 5 years while esimates using data post 1946 were within a bandwith of 2 years of the actual e0 for 1998. In addition the authors noted that for 15% of the time the true e0 was not straddled by the 95%

(19)

confidence interval boundaries and that the Lee-Carter model exhibited an overall downward bias.

An analysis of the average rate of decline in unisex mortality lead Lee and Miller (2001) to confirm the initial finding of Tuljapurkar et al. (2000) that the mortality rates of the United States have undergone a structural change which can be remedied by adjusting the fitting period of the model to the year 1950.

Deviating from the original Lee-Carter model, Lee and Miller (2001) modified the model by: • adjusting the base period to 1950 due to changes in the rate of change in mortality rates. • removing the jump-off bias by adjusting ax to equal the actual mortality rate in the

jump-off year with kt= 0

• k_t is re-estimated to equal the life expectancy at birth (e0) in the jump-off year.

Overall, Lee and Miller (2001) rated the forecasts created by the Lee-Carter model better than the SSA forecasts based on lower average error and mean squared error measures. Lee and Miller consider the Lee-Carter model, as redefined by themselves, as a suitable model for baseline planning.

That the Lee-Carter is a suitable reference point model to forecast mortality is not corrobo-rated in a multi-country comparison of various variants and extension of the Lee-Carter model performed by Booth et al. (2006). Booth et al. evaluated the following five mortality forecasting alternatives:

1. Lee Carter model (1992) 2. Lee-Miller variant (2001)

3. Booth-Macdonald-Smith variant (2006) 4. Hyndman-Ullah model (2007)

5. De Jong-Tickle model (2006)

The evaluation lead Booth et al. (2006) to the conclusion that the variants and extensions on Lee-Carter outperform the basic Lee Carter method in forecasting log mortality rates by 61% but do not make significantly better forecasts of life expectancy. The claim by Lee and Miller (2001) that the Lee-Carter model or for that matter their variant is a reference point for planning is disputed by Booth et al. (2006). They are of the opinion that if forecasts of log mortality rate is the basis for evaluation, both the original Lee-Carter model and the Lee-Miller variant are not optimal.

3.3 Coherent Mortality Forecasting for a Group of Populations

Li and Lee (2005) extends the traditional Lee-Carter model to be able to forecast mortality for a group of populations. According to the authors, the world is become increasingly intertwined through advancement in (communication) technology and transportation, increased global trade and convergence of morbidity. The implication of this increased interconnectivity is that mortal-ity patterns, such as life expectancy, show convergence for populations of similar characteristics. Li and Lee (2005) contend that the ordinary Lee-Carter model has proven adequate to forecast mortality for a single population but that the model is inadequate to forecast the mortality of a group of populations because individual population patterns are counter intuitively perpetuated into the future. As a solution, Li and Lee (2005) propose a coherent mortality forecast method as an extension on the Lee-Carter method.

(20)

3.3.1 Common Factor Model

The Li and Lee (2005) common factor model imposes as a restriction that all populations in the group have an identical parameter Bx and Kt but allows for a population specific parameter ax,i. As such the common factor model is defined as,

ln mx,t,i = ax,i+ BxKt+ x,t,i,

where mx,t,i is the mortality rate at age x, for time period t and for population i.

In order to obtain the common factors Bxand Kt, the Lee and Miller (2001) variant of the Lee and Carter (1992) method has to be applied to the aggregate of the individual populations that comprise the group. The Lee and Miller (2001) variant entails that parameter Ktis re-estimated to fit the average life expectancy of the group. The parameter Bx, like in basic Lee-Carter method, is a measure of the rate of change of the mortality level at age x for the group. As with the Lee-Carter method the parameters Kt and Bx are constrained to sum to zero and unity respectively. The imposition of the constraint on Kt implies that

ax,i= PT

t=1ln mx,t,i

T , assuming that x,t,i= 0.

The authors suggest the application of SVD to solve for the common factors Bx and Kt and measure the quality of the common factor model by an explanatory ratio per population. The explanation ratio for the common factor model

RC_i = 1 − Pω x=0 P t=0(ln mx,t,i− ax,i− BxKt)2 Pω x=0 P t=0(ln mx,t,i− ax,i)2 ,

quantifies the percentage of the variance that is explained by the common factor model for pop-ulation i. The explanation ratio RC_i is not an absolutism in its interpretation; expert judgment of the ratio can qualify a low ratio as acceptable (Li and Lee, 2005). To forecast the group mortality level parameter Kt, Li and Lee (2005) use a random walk with drift.

3.3.2 The Augmented Common Factor Model

Li and Lee (2005) do not advocate using the common explanation ratio as a goodness of fit test, but nevertheless argue that a low ratio can be a reason to augment the common factor model with population specific factors.

If judgment deems the common explanation ratio ( RC_i ) to be too small, the common factor model can be augmented with a specific factors bx,i and kt,i for population i. Augmenting the common model provides us with the augmented common factor model:

ln mx,t,i= ax,i+ BxKt+ bx,ikt,i+ x,t,i

In order to estimate the specific factor SVD is applied to the residuals of the common factor model, ln mx,t,i − ax,i− BxKt. Since the specific parameters minimize the common factor residuals, the interpretation differs from that of the common factor parameters. The specific parameter bx,i measures the change in mortality level between the group and population i for age x (Li and Lee, 2005). They assert that because the age vector bx,i changes signs, a second stage estimation of kt,i to match life expectancy is ineffective. An upward readjustment of kt,i and a positive bx,i increases the log mortality rates while a negative bx,i has the opposite effect on log mortality rates.

Comparable to the explanation ratio for the common factor model the Li-Lee coherent model defines an augmented common factor ratio which provides an indication how well augmented common model works for a particular population i. The ratio is defined as follows:

(21)

RAC_i = 1 − Pω

x=0 P

t=0(ln mx,t,i− ax,i− BxKt− bx,ikt,i)2 Pω

x=0 P

t=0(ln mx,t,i− ax,i)2

.

By definition the R_iAC should be greater than RC_i because the added parameters seek to explain the residuals of the common factor model.

3.3.3 Forecasting

The augmented mortality model as defined by the Li-Lee method, ln mx,t,i= ax,i+ BxKt+ bx,ikt,i+ x,t,i, 0 ≤ t ≥ T

has a common and a specific time dependent scalar that has to be projected into the future in order to obtain forecast of the mortality rates.

According to Li and Lee (2005), the common factor scalar Kt can be modeled using the customary random walk with a drift parameter but the specific factor kt should be bounded and therefore can not be forecasted using a random walk with a drift parameter. Instead, the authors suggest the use of an autoregresive regressive model with a single lag (AR(1)). This entails that the model to forecast kt is defined as,

kt,i = c0,i+ c1,ikt−1,i+ σii,t,

where c0,i and c1,i are coefficients and σi is the standard deviation of the autoregressive model. In order for the scalar kt,i to have a trend that is bounded on the short-term the coefficient c1,i has to be smaller than 1, (Li and Lee, 2005), and larger than −1.

The Lee-Li augmented common model defines forecasts of log mortality rates for population i within a group as:

ln mx,t,i = ln mx,T ,i+ Bx[Kt− KT] + bx,i[kt,i− kT ,i] t > T. 3.3.4 Application of the Coherent Model

(Li and Lee, 2005) used the coherent method to forecast combined sexes mortality and to forecast mortality of a group of countries with low mortality and concluded that the coherent method worked well in both instances. They placed the basis for the success of the model in the existence of a clear coherent trend among the sexes and the populations of the low-mortality populations examined.

The augmented model seeks to exploit commonalities among populations such as geographic location or social and economic conditions or past mortality patterns. However, the definition of a group is purposely left open ended by Li and Lee. Cura¸cao and the Netherlands are not on the same socio-economic level neither are they geographically in the vicinity of each other. However, the countries do share a historical bond and have open border which allows for a free movement of citizen between the two nations. This warrants the question if there exists a shared mortality pattern among the countries.

3.4 Forecasting By Mixing Mortality Data

The coherent mortality forecasting extension of the Lee-Carter method is suitable to forecast the mortality of a small population as part of a larger group. If there are similarities between the smaller population and other populations that comprise the group, these commonalities translate into a representative common factor which can be used to make mortality forecast for the smaller population. The coherent method is not the only method that makes it possible to project the mortality for populations with sparse data or that have data affected by random

(22)

fluctuations. Ahcan et al. (2014) offer a method that tries to replace the mortality data of the small population with the (weighted-average) mortality data of a basket reference populations and the original population. The data of the reference populations and that of the smaller populations are mixed using a credibility approach. The Lee-Carter method is applied on the replicated data set in order to forecast.

3.4.1 Average Mortality Assessment

There are 3 alternative methods to determine the average mortality data. These alternatives attempt to replicate properties of the smaller populations such as:

• the probability distribution, • the central mortality rates or

• the improvements (change) in mortality.

Ahcan et al. (2014) concluded that the method replicating the mortality improvements pro-vided the best results. Therefore this alternative is used to illustrate the general application of the method to mix mortality for small populations.

Let mi_x,t denote the central mortality rate and 4mi_x,t = mi_x,t− mi

x,t+1 be the change in the central mortality for age x, year t and country i. If the small population being replicated is given by i = 0, then Ahcan et al. (2014) defines the average annual variation in mortality for the other populations i = 1, . . . , n as: 4mAve_x,t = n X i=1 wi4mix,t = mAvex,t − mAvex,t+1.

The weights wi determine which of the alternate populations are relevant to determine the average annual variation. The appropriate weights are found by optimization of

min{wi} T X x=0 T X t=0 (4m0_x,t− 4mAve_x,t )2.

The conditions wi ≥ 0 for i = 1, . . . , n and Pn

i=1wi = 1 are placed as constraints on the optimization process.

3.4.2 Mixed Mortality Data

The initial step provided the set of weights (wi) based on the alternative that relies on the change in mortality rates to find the optimal set of weights. In the second step, credibility weighting is used to mix the mortality data for the small population (m0_x,t) and the average mortality data (mAve_x,t ) to create an approximate replication of the mortality data of the small population. The replicated mortality data is defined as:

mrep_x,t = m0_x,tzx,t+ mAvex,t (1 − zx,t)

The credibility weight zx,t is the ratio of the exposure (E) of the small population to the total exposure for all populations, whereby the alternative populations are weighted by the optimal set of weights, as described in the preceding section.

zx,t =

E_x,t0 E_x,t0 +Pn

i=1wiEx,ti .

(23)

Evaluating the 3 methods to replicate mortality data listed in (3.4.1), Ahcan et al. (2014) conclude that the best results are obtained by the method that replicates mortality improve-ments. They recommend this procedure for small populations but cast doubt on the application of this method for larger populations or populations that do not have volatile mortality patterns in its mortality data. According to Ahcan et al. (2014), replication of the data advantageously increases the sample size for small populations but it also smoothes the data which leads to information loss. The disadvantage of smoothing of the erratic original data can outweigh the advantage of increasing the sample size.

For the selection of the basket of countries with which to replicate the data Ahcan et al. (2014) advocate a selection of countries with similar socio-economic factors. But the authors are not clear which social or economic factors are relevant to the selection process. Therefore, as a suggestion, all countries in the same geographically area should be added to the basket of countries without pre-selection.

(24)

Chapter 4

Fitting the Coherent Model to

Cura¸

cao Data

4.1 LifeMetrics Package

The LifeMetrics software package (version 2.0) is used to calibrate the Lee-Carter model to the data of Cura¸cao. Besides model calibration capabilities, LifeMetrics is able to forecast and simulate mortality rates. The selection of stochastic mortality models available in LifeMetrics extends beyond the Lee-Carter model since with LifeMetrics four additional models can be calibrated.

LifeMetrics initializes the parameters of the Lee-Carter model prior to optimization, the initialization of the parameters bx and kt do not pose a problem but the application initializes ax as follows: ax= PT t=0 ln(mx,t)wx,t PT t=0wx,t f or x = 0, . . . , ω.

Because the data set of mortality rates for Cura¸cao contains zeros, the shape parameter vector ax contains initial values of negative infinity in instance where the log of mx,t= 0 is taken.

The optimization routine does not end until the difference of subsequent likelihood totals meets a threshold and with each iteration of the optimization routine the parameter kt gets optimized over the years (matrix rows) and bx over the ages (matrix columns). The improved parameter estimates for bx and kt serve to improve the initial estimate of ax. If the sum of the observed death count is

s1 = T X

t=0

wx,t(Dx,t)

and the sum of the fitted death count is

s2 = T X

t=0

wx,t(Ex,tektbx),

then after each iteration the shape parameter is the logarithm of the ratio of the observed death count s1 to the fitted death count s2,

(25)

The estimation of axbecomes problematic because the mortality data for Cura¸cao contains ages (x = 9 for males and x = 10 for females) where not a single incidence of mortality is observed. As a result s1 = 0 and consequently ax equates to negative infinity.

In order to be able to fit the data of Cura¸cao a solution for the initialization and optimization of parameter axis needed. The following approaches have been applied to resolve the shortcoming of LifeMetrics as it pertains to data containing observed death counts of zero.

• Approximation of zero values • Interpolation of death count

• Subset and substitute dummy value substitute 4.1.1 Approximation of zero values

As part of the fitting process the logarithms of mortality rates mxt = D_E_x,tx,t are taken during initialization and iterative optimization of the parameter ax. For mx,t to equal zero it is a necessary condition that the numerator (Dx,t) equal zero. To avoid a death count of zero, all instances where Dx,t= 0 can be approximated with a small non integer value Dx,t = 10−3. The use of an approximate value for zero entails that the logarithm of zero is not taken anywhere during the fitting process. This approach leads to development of the parameters bx and kt, as depicted in figures 4.1 and 4.2.

0 20 40 60 80 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 bx − Female (Substitution) Age bx 1998 2002 2006 2010 −6 −4 −2 0 2 4 kt − Female (Substitution) Year kt

(26)

0 20 40 60 80 −0.8 −0.6 −0.4 −0.2 0.0 bx − Male (Substitution) Age bx 1998 2002 2006 2010 −4 −2 0 2 4 6 8 kt − Male (Substitution) Year kt

Figure 4.2: Males: Approximate Dx,t= 0 with Dx,t = 10−3

4.1.2 Interpolation of zero values

Instead of substituting fictive data when Dxt = 0, it is also a possibility to use interpolation methods to replace the zero values. In order, to interpolate the death count the function fill.zero of the package Demography was modified and applied. This interpolation method first seeks to interpolate zero valued observations over the rows (ages) of a particular data set but because the data from Cura¸cao has entire rows with zeros the function was modified to also interpolate over the columns (years). The interpolation of over the years follows an initial pass over the ages.

The interpolation method uses subsequent non zero data points (x, t) to interpolate inter-mediate zero values. If a data point for interpolation falls outside the range of non zero points (x, t), the nearest extreme is used. For a set of data points Dx,t, a interpolated data point is defined by, Dx,i=      Dx,t i < t 1 2(Dx,tf + Dx,T) t < i < T ; Dx,T i > T

over the rows and over the columns

Di,t =      Dx,t i < x 1 2(Dx,tf + DX,t) x < i < X; DX,T i > X.

(27)

0 20 40 60 80 −0.10 −0.05 0.00 0.05 0.10 bx − Female (Interpolation) Age bx 1998 2002 2006 2010 −6 −4 −2 0 2 4 6 kt − Female (Interpolation) Year kt

Figure 4.3: Females: Interpolation of Dx,t = 0

0 20 40 60 80 −0.05 0.00 0.05 bx − Male (Interpolation) Age bx 1998 2002 2006 2010 −2 0 2 4 6 8 kt − Male (Interpolation) Year kt

Figure 4.4: Males: Interpolation of Dx,t= 0

4.1.3 Dummy substitution of zero values

As expected, the zero values in the data set predominately occur in the childhood years because the incidence of death is low or non existent. By sub-setting the data to ages x = 25 . . . ω, the childhood and teenage years are excluded thus circumventing the years in which the frequency of death count zero are high, including the ages where no deaths have been observed over the entire period under consideration (zero rows). The starting age of x = 25 has been chosen because it is the age at which participation into the pension arrangement is allowed. Subsetting the data to x = 25 does not completely exclude all zero observations in the data subset but make the use of dummy substitution possible.

Wilmoth (1993) described a Weighted Least Square method to fit the Lee-Carter model which relied on the use of dummy data and weight to circumvent the zero cell problem. This principle can also be applied in this instance by applying the the following substitution rule to the data subset:

fxt= (

ln(mx,t) and wx,t= 1 if mx,t > 0

9999 and wx,t= 0 if mx,t = 0

Based on the subset method the fit of the parameters are depicted in 4.5 and 4.6.

The interpolation and approximation methods are viable methods but manipulate the orig-inal data set more than the age based subsetting method. The selected age of 25 is well beyond

(28)

30 40 50 60 70 80

−0.04

0.00

0.04

0.08

bx − Female (Data Subset)

Age bx 1998 2002 2006 2010 −10 −5 0 5

kt − Female (Data Subset)

Year

kt

Figure 4.5: Females: Subset to x = 25

30 40 50 60 70 80

−0.05

0.00

0.05

0.10

bx − male (Data Subset)

Age bx 1998 2002 2006 2010 −2 0 2 4 6

kt − male (Data Subset)

Year

kt

Figure 4.6: Males: Subset to x = 25

the problematic childhood and teenage years. In addition, the chosen age, coincides threshold age which grants admission to an employers’ pension scheme. Hence forth, a subset of the data set of the Netherlands and Cura¸cao will be used to fit the coherent model.

4.2 Application Common Factor Model

In order to fit the coherent mortality model as defined by Li and Lee the demographic data for Cura¸cao and the Netherlands has to be aggregated over the 1998-2012 period such that coherent death counts (D) and exposures (E) equate to the sum of the populations;

Dcoh= Dcur+ Dnld and

Ecoh= Ecur+ Enld.

On this aggregate data set the Lee-Carter method is applied to obtain the common factors Kt and Bx. Contrary to the original Lee-Carter method the model is not fitted using Singular Value Decomposition but instead Maximum Likelihood Estimation with a Poisson distributed death count. The graphs of the parameters for the common factor model for females and males are depicted in figures 4.7 and 4.8, respectively.

Due to the disparity in population size between Cura¸cao and The Netherlands it is expected that the fitted parameters for the common model are similar to the parameters of the individual Lee-Carter model for The Netherlands. A graphical comparison corroborates this supposition (See figures 4.10 and 4.11). Even though the coherent model allows for a country specific mor-tality rates, the average log mormor-tality implied by the coherent model is compared to the average log mortality of the Netherlands ax,i, where i = nld.

(29)

30 40 50 60 70 80 0.005 0.015 0.025 Bx − Female (Coherent x = 25) Age bx 1998 2002 2006 2010 −5 0 5 Kt − Female (Coherent x = 25) Year kt

Figure 4.7: Common Factor Model

30 40 50 60 70 80 0.010 0.015 0.020 0.025 Bx − Male (Coherent x = 25) Age bx 1998 2002 2006 2010 −10 −5 0 5 10 Kt − Male (Coherent x = 25) Year kt

Figure 4.8: Males: Common Factor Model

30 40 50 60 70 80 −8 −7 −6 −5 −4 −3

Ax − Female (Coherent vs. Netherlands)

Age ax Coherent Netherlands 30 40 50 60 70 80 0.005 0.015 0.025

Bx − Female (Coherent vs. Netherlands)

Age

Bx

Coherent Netherlands

As expected there is not a noticeable difference between common factor parameters Bx, Kt of the coherent model and the customary Lee-Carter model parameters if applied solely on data of The Netherlands.

To get a sense of the how the common factor model performs, the common factor explanation ratio is calculated for each population and gender. The common factor model as defined in paragraph 3.3.1 has been adapted to accommodate the fact that the data set for Cura¸cao contains zeros. In this instance dummy substitution and weighting is applied to circumvent

(30)

1998 2002 2006 2010

−5

0

5

Kt − Female (Coherent vs. Netherlands)

Year

Kt

Figure 4.10: Females: Common Factor Model

30 40 50 60 70 80 −7 −6 −5 −4 −3 −2

Ax − Male (Coherent vs. Netherlands)

Age ax Coherent Netherlands 30 40 50 60 70 80 0.010 0.015 0.020 0.025

Bx − Male (Coherent vs. Netherlands)

Age bx Coherent Netherlands 1998 2002 2006 2010 −10 −5 0 5 10

Kt − Male (Coherent vs. Netherlands)

Year

Kt

Figure 4.11: Males: Common Factor Model

taking logarithm of zero. The common factor explanation ratio is redefined as:

RC_i = 1 − Pω

x=0 PT

t=0wx,t,i(ln mx,t,i− ax,i− BxKt)2 Pω

x=0 PT

(31)

where,

ln(mx,t,i) = (

ln(mx,t,i) and wx,t,i = 1 if mx,t,i> 0

99 and wx,t,i = 0 if mx,t,i= 0

As is evident in table 4.1, the common factor explanation ratio is reasonably high for The Netherlands. For the Dutch females the common factor model could be applied to forecast. For Dutch males, the explanation ratio is more ambiguous and warrants caution.

Table 4.1: Common Factor Model Ratios (%) Fit Period 1998-2012 Gender Cura¸cao The Netherlands

Male 0.29 73.85

Female 3.30 58.6

Fit Period 1950-2012 Gender Cura¸cao The Netherlands

Male na 82.36

Female na 85.74

However, for Cura¸cao, coherent forecasting based on the common factor alone is not recom-mended because the explanation ratios are very low. As a reference the common factor for the Dutch data was calculated over the period 1950-2012. The explanatory value of the common factor model improves as the fitting period is increased; the Lee-Carter method and Lee-Carter model extensions perform better with longer time-series.

4.3 Augmented Common Factor Model

Expanding the common factor model with the (country) specific factor bx,ikt,i produces the augmented common factor model as defined in section 3.3.2. The added factors attempt to describe the residual matrix of the common factor model. If the random variable,

Dx,t,i ∼ P(Ex,t,ieax,i+BxKt+bx,ikt,i).

The common factor BxKtis already known which allows for a redefinition of the random variable Dx,t,i in terms of newly defined exposure,

Dx,t,i ∼ P( ˇEx,t,ieax,i+bx,ikt,i). where ˇEx,t,i is defined as,

ˇ

Ex,t,i = Ex,t,ieBxKt.

In essence, the Lee- Carter method is applied with new exposures in order to obtain the country specific factors. The specific factor for Cura¸cao and the Netherlands have been fitted and the explanation ratios for the augmented common factor model have been tablulated in table 4.2.

As expected, adding the specific factors improves the explanatory value of the models over every fitting period. However, is a model with additional parameters such as the augmented coherent model more beneficial than a simpler model with less parameters like the Lee-Carter model.

(32)

Table 4.2: Augmented Factor Model Ratios Fit Period 1998-2012 Gender Cura¸cao The Netherlands

Male 8.2 76.4

Female 14.54 62.5

Male na 86.62

Female na 87.15

4.4 Explanation Factor Ordinary Lee-Carter Model

The common factor model is equivalent to the Lee-Carter model in terms of parameters. The low explanation ratios for the common factor model indicate that this model is not suitable to model the mortality rates of Cura¸cao. The addition of the specific factors improves the explanatory performance of the model but at the same time detracts from the overall simplicity of the model. In order to evaluate the merit of the augmented model, the explanation ratios of this model are compared to the ordinary Lee-Carter model. The latter is an often used reference model.

Table 4.3: Lee-Carter Model Ratios Fit Period 1998-2012 Gender Cura¸cao The Netherlands

Male 8.09 73.9

Female 15.92 58.59

Male na 82.42

Female na 85.78

If the explanation ratios for the augmented model in table 4.2 are compared to the explana-tion ratios in table 4.3 it can be concluded that augmented common factor, in most instances, is a better model with higher ratios with the exception of the ratio for females over the fit period 1998-2012. However, adding complexity to the model by expanding the number of pa-rameters does not significantly improve the results of the augmented common factor model when compared to the ordinary Lee-Carter model.

4.5 Replicating Mortality Rate Improvements

The chosen method of Ahcan et al. requires that the mortality rate of the profile country (Cura¸cao) is replicated by a mix of mortality rates of a group of populations and that of the profile country. In a two step process, first the average mortality of a pool of countries is de-termined. This step is followed by a credibility weighting approach to replicate the mortality of Cura¸cao by a combination of mortality rates of the pool of countries and the rates of Cura¸cao. The pool of countries consists of a subset of countries with data series spanning years 1998-2008 available on the Human Mortality Database (HMD).

As described in section 3.4.2, the average annual variation in the mortality rates is defined as:

(33)

4mAve_x,t = n X

i=1

wi4mix,t,

where i equates to the countries Netherlands (nld), Belarus (blr), Belguim(bel), France (fr), Portugal (prt), Israel (isr), Estonia (est), Luxembourg (lux), Slovakia (svk), Bulgaria (bul), Solvenia.

The country weights wiare obtained by minimizing the distance between the annual improve-ment of the mortality of the profile country Cura¸cao and the average of the pool of countries:

min{wi} T X x=0 T X t=0 (4m0_x,t− 4mAve x,t )2.

The credibility weights (zx,t) are consequently defined as a ratio of exposures (E) of the Cura¸cao and the pool of countries:

zx,t= Ecur x,t E_x,tcur+Pprt i=nldwiEx,ti .

The average mortality mortality rates in conjunction with the credibility weights yield the replicated mortality rates based on the following formula:

mrep_x,t = mcur_x,tzx,t+ mAvex,t (1 − zx,t)

The above described method was applied for Cura¸cao over the years 1998-2008 and in table 4.4 the optimal weights that yield the average mortality (mAve_x,t ) are tabulated. For comparison the method was also applied by mixing the plain mortality rates. This variation is identical in its application but for the fact that the central mortality rates (mx,t) and not the annual variation thereon (4mx,t) are used to derive average mortality.

Table 4.4: Multi Country Optimal weights wi and average credibility weights z

Method (wm) Method (w4m)

Pool Male Female Male Female

nld 0.2382 0.1435 0 0.0096 blr 0.2609 0.2276 0 0.5052 bel 0 0.1790 0 0 f r 0 0 0 0 prt 0 0.2323 0 0 isr 0 0 0 0.4852 est 0 0.1682 0.3417 0 lux 0 0 0.6583 0 svk 0 0 0 0 bul 0.1433 0 0 0 svn 0 0.0494 0 0 ¯ z 0.016 0.14 0.141 0.018 (1 − ¯z) 0.984 0.986 0.859 0.982

The method that applies weights wmto obtain mAve_x,t yields a more diverse basket of countries than the method that uses mortality improvements to obtain the weights. The w4m method

(34)

does place more weight on the data of Cura¸cao for both genders. No sensible conclusion can be inferred from the selection of the countries with which to weight the average mortality

By graphing the actual rates and the rates replicated with the wm and w4m methods we can visually inspect the effect of replication. The year 2003 was graphed as the median year in the calibration period 1998-2008.

30 40 50 60 70 80

0.00

0.05

0.10

0.15

Female Mortality Data 2003

Age Centr al Mor tality Rate Orginal Rep m Rep Delta m 30 40 50 60 70 80 0.00 0.05 0.10 0.15 0.20

Male Mortality Data 2003

Age Centr al Mor tality Rate Orginal Rep m Rep Delta m

Figure 4.12: Comparison Central Mortality Rates

From figure 4.12 it is noticeable that replication has a smoothing effect on the data. Much of the variability of the original data is removed. Graphing the replicated does not provide an measure of the goodness of fit the replicated data. In order to check the goodness of fit Ahcan et al. suggest the comparison of the Mean Squared Error (MSE) of the mortality projections based on the original data (M SE[0]_{) and the projections based on replicated data (M SE}[R0]_). As mentioned earlier, data of the years 1998-2008 are used for calibration and data of the years 2009-2012 are used to backtest. For the replicated data the mortality projections over 2009-2012 were determined by applying the Lee-Carter method.

Table 4.5: MSE Ratios

Method (wm) Method (w4m)

Male Female Male Female

M SE[0] 0.0024 0.0026 0.0024 0.0026

M SE[R0] 0.00103 0.00058 0.00090 0.00056 M SE[0]

M SE[R0] 24.14% 22.26% 23.22% 21.42%

Both methods of replication provide satisfactory MSE ratios below 100%. Because of the lower MSE ratios it can be concluded that the w4m is the better method to replicate the mortality data of Cura¸cao than the wm method.

By calculating the life expectancy of the projections based on the original data and both replication methods over the backtest period 2009-2012, we can discern the applicability of these methods. In graph 4.13, the life expectancy at age x = 25 for both genders are depicted. The projected life expectancy trends upward similar to the observed trend in life expectancy.

The projections entail best estimates (mean values) based on 5000 simulations. The repli-cating methods provide a mixed picture with the w4m forecasting the life expectancy better for females and the wm method performing better for males. The fact that best estimate life

(35)

expectancy for males based on the w4m method exceeds does not preclude the fact that the observed life expectancy is straddled by its 95%-99% confidence intervals.

Comparing the life expectancy at a single point in time should neither lead to the conclusion that the wm method outperforms the w4m, the MSE ratio purports otherwise and compasses all time periods and observations. The w4mmethod should lead to less deviation from the observed life expectancy if the life expectancy at all ages is considered.

2000 2020 2040 2060 52 53 54 55 56 57 58 59

Life Expectancy x = 25 Females

Age Lif e Expectancy ex_(mxt_rep) ex_(delta_mxt_rep) ex_(obs_1998_2012) 2000 2020 2040 2060 45 50 55

Life Expectancy x = 25 Males

Age Lif e Expectancy ex_(mxt_rep) ex_(delta_mxt_rep) ex_(obs_1998_2012)

Figure 4.13: Comparison Central Mortality Rates

Table contains 4.6 the average period life expectancy over the fit period 1998-2008 and the forecast years 2009-2062, of which the years 2009-2012 represent the backtest period. The life expectancy based on the replication methods are inline with life expectancy observed during the fit period. In the last year of the backtest period the maximum deviation from the observed life expectancy is 0.7 years based on the w4m method. Based on forecasts of either replication methods, a person age x = 25 can expect to exceed the age of 80 years.

Table 4.6: Average and Forecasted Life Expectancy x = 25

Observed Method (wm) Method (w4m)

Male Female Male Female Male Female

Mean Observed 1998-2008 47.9 53.2 47.4 53.2 48.2 53.1

2012 (Forecast) 49.7 54.3 (49.4) (54.6) (50.4) (54.3)

2062 (Forecast) na na (56.3) (58.9) (57.6) (58.3)

The mixing mortality is more effective if the pool of countries from which the mortality data can be replicated from is sufficiently large and diverse. Therefore, expanding the number of countries available as a source to replicate the mortality data of Cura¸cao might improve the fit of the replicated data further. However the choice of countries needs to include more countries with small populations, countries in development and countries in the geographic vicinity of Cura¸cao. The more similarities the pool of replicating countries have with the profile country in terms geography and socio-economic factors, the better results this method yields.

Coherent mortality analysis of Curacao