The gravity model : using a multi-population CBD mortality model to forecast Dutch mortality

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J.M. den Hollander

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: J.M. den Hollander Student nr: 10681434

Email: martjandenhollander@gmail.com Date: January 5, 2015

Supervisor: Dr. T.J. Boonen Second reader: F. van Berkum

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This thesis introduces a term to the Cairns-Blake-Dowd (CBD) mortality model, to couple Dutch mortality to European mortality. This idea was explored before in an APC (Age-Period-Cohort) model. The resulting model was named the gravity model, which is also the name that will be used for the model in this thesis. The rationale behind the gravity model, is that culturally, economically and technologically similar and connected populations, such as the Dutch and the European population, should not exhibit widely different mortality statistics. As no longevity boost is present in other European countries, and the Dutch longevity boost followed a period of relative stagna-tion in mortality improvement, assuming a coupling between Dutch and European data could lead to a better model fit.

The gravity model results in Financial Components of Liabilities (FCLs) that are similar to those of the CBD model. In Solvency and Capital Requirement (SCR) cal-culations, the gravity model produced some deviations from the CBD model. The SCR calculations seem to be hindered by a lack of tail dependence in the gravity model.

The gravity model compares favorably with respect to AG and CBS mortality tables for the Dutch population, when they are estimated on similar data sets. More recent AG and CBS mortality tables produce a lower mortality than the gravity model, especially for Dutch males. This is due to the fact that the most recent mortality observations (2010 – 2013) show a stronger mortality improvement than was predicted by the gravity model.

Several tests, including normality tests, back tests and robustness tests, have been performed. In the back tests and robustness tests, the gravity model shows better results than the CBD model. However, lack of tail dependence between European and Dutch mortality poses a problem. Also, it is unclear whether the model is parsimonious or if at least one coupling term should be removed from the model.

Keywords Mortality model, CBD model, Gravity model, Multi-population mortality model, European mortality, Dutch mortality, Longevity risk, Mortality Risk, FCL calculation, SCR calculation

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Among other reasons, the introduction of the Solvency II framework has fueled actuaries interests in mortality projection models. Recently, this has led to several proposals for mortality trend modelling[4][3][17]. No model has currently been universally recognized as being the best in business. Model performance depends on the data set and the cri-terium for goodness used [5][8][9].

For Dutch insurers, using an internal model for mortality trends has proved to be prob-lematic. One reason for this, is that mortality trend in the Netherlands changed drasti-cally around 2002[2]. This is often attributed to improved medical facilities, especially for cardiovascular disease. The trend break presents two problems:

1. The amount of data, on which a model can be fitted, is limited. Pre-2002 data may not be considered relevant anymore for future mortality projections. However, if a model is fit on post-2002 data exclusively, results may turn out to be biologically implausible.

2. Any mortality projection model, when fitted to data between 1970 and 2002, fails to predict the 2002 trend break. This leads to problems when back testing a model. As can be seen in figure1.1, however, no profound trend break is visible in 2002 for most European countries[13]. Rather, life expectancies in European countries seem to share a common trend, and up to converge towards each other. Short term divergences are possible, but on the long run a narrowing down of life expectancies is visible. Given the ongoing economic and political integration within Europe, and the proliferation of (medical) science and technology, this pattern is reasonable. Full convergence, however, is not to be expected in the near future, due to factors like cultural differences, local eating habits, local environmental conditions and uneven wealth distribution.

The differences in life expectancy between national populations in Europe is typically smaller than differences that can be expected between subgroups of any national pop-ulation. (For example: high education vs low education, smoking vs non-smoking, etc.) Because of these limited differences, it may be possible to model mortality for national populations, based on the common European trend. By modelling the Dutch mortality as a deviation from the common European trend, the aforementioned problems due to the 2002 trend break can be mitigated.

1.2 Content of this thesis

In this thesis, the 2006 CBD model[4] will be extended to accommodate convergence of mortality in a relatively small population towards the mortality in a closely connected, but larger, population. This model, named the gravity model[10], will be used to model

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2 J.M. den Hollander — The gravity model

Figure 1.1: Life expectancy at birth, male, for several European countries.

the Dutch population mortality in context of the European mortality.

The goal of the thesis is to establish if it is possible to produce a biologically plausible and Solvency II-compliant Dutch mortality forecast from a gravity model, when applied on a European and a Dutch data set. In order to explore this question, the following questions are addressed:

• Historically, how do Dutch mortality observations compare to European observa-tions?

• Can a stochastic mortality model be fitted to European mortality observations? • Can the chosen stochastic model be expanded to model Dutch mortality as a

deviation from European mortality?

• How do results from this model compare to results from a single population CBD model, or from those of a CBD model in which the European and Dutch mortality progressions merely correlate?

• Can time dynamics be added to the model, in order to obtain a biologically plau-sible and Solvency II compliant prediction?

• When fitted on mortality data of other small European countries, how do results compare to the Dutch results?

• How do forecasts from this model compare to predictions from Dutch actuarial (AG) and statistics (CBS) agencies?

• How do results from this model compare to the Solvency II standard model or other stochastic models, in terms of the SCR/FCL ratio?

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2.1 Mortality in actuarial practice

In modern actuarial practice, mortality probabilities are considered to be both time-and age-dependent, time-and modelled stochastically[20]. The most natural coordinate for modelling human mortality is the force of mortality µ(x; t). µ(x; t)dt is defined as the probability of death for an x-year old between times t and t + dt, conditional on survival until time t. So µ(x; t) can be expressed in terms of the survival function s(x; t) and the mortality probability density function f (x; t) as follows:

µ(x; t) = f (x; t)

s(x; t). (2.1)

Defining b as the moment of birth, survival function s(x; t) can be defined as the probability that a person with age x at time t will die later than at time t:

s(x; t) = Z ∞

t

f (u − b; u)du. (2.2)

Or, in terms of the mortality cumulative density function F (x; t): F (x; t) = 1 − s(x; t) =

Z t 0

f (u − b; u)du. (2.3)

µ(x; t) can then be expressed as: µ(x; t) = f (x; t) s(x; t) = F0(x; t) s(x; t) = − s0(x; t) s(x; t) = − d dt[log s(x; t)]. (2.4) This leads to the following expression for s(x; t):

s(x; t) = exp − Z t 0 µ(u − b; u)du . (2.5)

Data is typically available in terms of exposures to risk E(x; t) and observed death counts D(x; t). The observed death count is simply the number of individuals that die within cell (x; t). That is, the number of individuals that have an age between x and x + 1 and die between times t and t + 1. The exposure in a cell is the total amount of time spent alive by all individuals in that cell. For example, if an individual turns 70 at April 1st 2014 and dies October 1st 2014, then this person adds 0.25 to E(69; 2014), 0.50 to E(70; 2014), 1 to D(70; 2014) and 0 to all other E(x; t) and D(x; t).

µ(x; t) is assumed to be constant within a cell (x; t):

µ(x; t) = µx,t. (2.6)

Now, suppose a person, marked with i, is exactly x years old at time t. Let Ti be the

amount of time that the person i is alive in the cell (x, t). According to equation2.5, the 3

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probability of living another ti years, where ti is smaller than 1 year, and then dying,

is equal to µx,t· exp(−µx,t· ti). The probability for surviving for ti= 1 year is equal to

exp(−µx,t· ti). Plugging in ti = 1 leads to the one-year mortality rates, typically used

by insurers:

q(x; t) = 1 − exp(−µx,t). (2.7)

To estimate µx,t, MLE techniques can be used. Let ˆµx,t be the maximum likelihood

estimator for µx,t. The likelihood function for a population is equal to:

L(µx,t) = Y i µDi(x;t) x,t · exp (µx,t· Ti(x; t)) = µ P iDi(x;t) x,t · exp µx,t· X i Ti(x; t) ! (2.8) = µD(x;t)_x,t · exp (µx,t· E(x; t)) .

And the log-likelihood is given by:

l = log(L) = D(x; t) · log(µx,t) − (µx,t· E(x; t)). (2.9)

Equating the derivative of the log-likelihood function to 0 gives:

0 = ∂ ∂µx,t [l] = ∂ ∂µx,t [log(L)] = ∂ ∂µx,t [D(x; t) · log(µx,t) + µx,t· E(x; t))] (2.10) = D(x; t) µx,t − E(x; t). So: ˆ µx,t = D(x; t) E(x; t) = m(x; t). (2.11)

Here m(x; t) is the crude death rate in cell (x; t). Replacing µx,t by its MLE, the

one-year mortality rate can be expressed in terms of the crude death rates:

q(x; t) ≈ 1 − exp(−m(x; t)). (2.12)

2.2 The CBD model for human mortality

In 2006, Cairns, Blake and Dowd introduced a two-factor model for mortality[4]. This model exploits the (near) linearity of the logit of the one-year mortality probabilities q(x; t) as a function of age, at the pension ages, which is shown in figure2.1. The CBD model will serve as a basis for the model outlined in this thesis.

The functional form of the CBD model is as follows:

κt= (X0X)−1X0Yt, (2.13) where κt= " κ(1)_t κ(2)_t # ,

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and Yt=    logit q(60; t) .. . logit q(90; t)   .

Unless indicated otherwise, in this thesis ¯x = 75, as to indicate the middle of the age interval to which the CBD model is applied.

After fitting the CBD model parameters κt, mortality predictions are made by

projecting the time-dependent factors κ(i)_t . In their paper, Cairns et al chose a two-dimensional random walk with drift (RWD) for the time dynamics of the κ(i)_t :

κ(1)_t = µ(1)+ κ(1)_t−1+ C(11)Z_t(1)

κ(2)_t = µ(1)+ κ(2)_t−1+ C(12)Z_t(1)+ C(22)Z_t(2) (2.14) Z_t(i)∼ N (0, σ2_i).

Note that the choice for this RWD model has a large impact on the limiting behavior of the model. Depending on the estimates for the drift factor µ, logit q(x; t) will diverge to either plus or minus infinity, possibly depending on age x. This leads to the q(x; t) tending to either 0 or 1. This limiting behavior need not be a drawback for the CBD model, because the actuarial practice is typically only concerned with life expectancies in the next one hundred years.

Several improved versions of the CBD model have been proposed[3][17]. These mod-els typically expand the age range of the model by assuming piecewise linearity. Some others take into account the observed convexity of the logit q(x; t) as a function of age x, that depending on the data set may be statistically relevant. For simplicity, the original CBD model will be used in this thesis.

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6 J.M. den Hollander — The gravity model 60 65 70 75 80 85 90 −5 −4 −3 −2 Age logit q(x,2009) Dutch males European females

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3.1 Defining European mortality

All mortality data used in this thesis is publicly available from the Human Mortality Database (HMDB). For each country, both exposures[13] and death counts[12] were downloaded, with a cell resolution of 1 year in both time and age.

The HMDB provides no mortality data on Europe as a whole. Therefore, Euro-pean data was retrieved by adding exposure and deaths cell-wise over several EuroEuro-pean countries c: EEU R(x; t) = X c Ec(x; t) (3.1) DEU R(x; t) = X c Dc(x; t). (3.2)

There is no objective standard on what countries to include into Europe, from a mortality standpoint. Typical definitions for Europe include the EU-12, the EU-27, or all countries that are located on the European continent. In this thesis, the criteria for inclusion of a country into the European data set are as follows:

• The country should be located on the European continent. Iceland is considered to be part of the European continent.

• The country should not have been subject to any recent wars, non-democratic government changes or non-democratic governing, as these events can strongly impact death counts over an extended period of time. Based on this criterion, former Soviet countries are excluded from the data set. West-Germany is included, whereas East-Germany is not. Spain, Portugal and Greece, that changed to a democratic government in the 1970s, are included.

• Data should be available through the HMDB for the period 1960 up to and in-cluding 2009, with a 1x1 (age and year) cell resolution and split out to men and women.

Based on these criteria, the European data set will consist of data from the following countries: Austria, Belgium, West-Germany1, Denmark, Spain, Finland, France, Eng-land and Wales, ScotEng-land, IreEng-land, IceEng-land, Italy, Luxembourg, the NetherEng-lands, Norway, Portugal and Sweden. For males, the total exposure, summed over ages 60 and to and including 90, amounts to slightly over 38 million years in 2009; for females total expo-sure is slightly over 47 million in 2009. Dutch expoexpo-sure is approximately 1.6 million in

1

After the German reunification, West-Germany is understood to consist of the regions of Germany that previously formed West-Germany. This data set is readily available through the HMDB.

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Figure 3.1: Total exposure in years, males aged 60-90, per country.

2009 for males and 1.9 million for females.

The division of the exposure over all countries for males is displayed in figure3.1. For females the graph takes a similar shape. European exposure is dominated by exposure in West-Germany, France, England and Wales, Spain and Italy. The Netherlands makes up for less than 5 percent of the exposure.

One could argue if the Netherlands should be a part of the European data set, as it introduces an automatic correlation between Dutch and European mortality. Given the small contribution of the Netherlands to the European data set, this is a minor issue. In this thesis, the Netherlands have been included.

3.2 Comparing European and Dutch mortality

There are several ways of comparing mortality in different countries. One way of com-paring life expectancies, is to look at the flat table life expectancy at birth. This metric estimates the life expectancy of a new-born, if mortality probabilities were fixed at time of birth: e0(t) = 1 2+ ω X x=0 x Y i=0 (1 − q(i; t)). (3.3)

In figure3.2, it can be seen that in 1960 Dutch life expectancy was 4.5 years (males) and 3.5 years (females) higher than the European average. European life expectancy steadily improved in a near-linear way, whereas Dutch life expectancy stagnated at times. For males, a period of stagnation has occurred between 1960 and approximately 2000, after which the Dutch life expectancy converges to the European average. For females, a strong stagnation takes place between 1980 and 2000. In the beginning of the 1990s, European average life expectancy surpassed the Dutch life expectancy. Since 2000, Dutch life expectancy is strengthening and converging to the European average.

It is also possible to look at the time series of the observed one-year mortality proba-bilities q(x; t) for a certain age. In figure3.3, a comparison in made between the q(70; t)

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1960 1970 1980 1990 2000 2010 66 68 70 72 74 76 78 Year e0 (in y ears) European males Dutch males (a) Males 1960 1970 1980 1990 2000 2010 72 74 76 78 80 82 Year e0 (in y ears) European females Dutch females (b) Females

Figure 3.2: Flat table life expectancy e0(t) for Dutch and European males and females,

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10 J.M. den Hollander — The gravity model 1960 1970 1980 1990 2000 2010 −3.8 −3.6 −3.4 −3.2 −3.0 Year logit(q(70,t)) European males Dutch males

(a) European and Dutch males

1960 1970 1980 1990 2000 2010 −4.4 −4.2 −4.0 −3.8 −3.6 −3.4 Year logit(q(70,t)) European females Dutch females

(b) European and Dutch females

Figure 3.3: Progression of the logit of the q(x; t) at age 70, between 1960 and 2009.

for European and Dutch males and females. Like in the life expectancy graphs above, Dutch mortality probabilities were lower than European probabilities in 1960, and are approximately equal in current times. The same patterns of stagnation are visible in the mortality probabilities as there are in the life expectancies.

Similar figures can be made for other ages. For age 80, figure3.4gives approximately the same picture, except the 1960-1990 mortality bump for Dutch males is missing and replaced by a plateau. Current mortality probabilities for 80-year olds are slightly higher in the Netherlands than in the average of Europe, especially for males.

At age 90, as can be seen in figure 3.5, hardly any mortality improvement can be seen for Dutch males, at least up until 2005. In 2009, mortality probabilities for 90-year old males are higher in the Netherlands than in the average of Europe.

3.3 Fitting the CBD-model

Another way of comparing Dutch and European mortality near and past the pension age, is by estimating the κ(1)_t and κ(2)_t parameters in the CBD model:

logit q(x; t) = log q(x; t) 1 − q(x; t) = κ (1) t + κ (2) t (x − ¯x) (3.4)

Here parameter κ(1)_t can be interpreted as a general measure for mortality in the age bracket from age 60 up to and including age 90. Parameter κ(2)_t is the additional amount of logit mortality probability per year of age gained. A higher value for κ(2)_t indicates a mortality probability that rises stronger with age.

Using OLS techniques, the estimates for the κ(1)_t and κ(2)_t parameters for males are summarized in figure 3.6. Like the life expectancy and the mortality probability, the Dutch κ(1)_t parameters show stagnation compared to the European parameters in the 20th century, before catching on in the first decade of the 21st century. The Dutch κ(1)_t parameters have no long-time divergence from the European values in the used time frame. Note, however, that the Dutch κ(2)_t parameters have been diverging from the

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1960 1970 1980 1990 2000 2010 −2.6 −2.4 −2.2 −2.0 Year logit(q(80,t)) European males Dutch males

1960 1970 1980 1990 2000 2010 −3.2 −3.0 −2.8 −2.6 −2.4 Year logit(q(80,t)) European females Dutch females

Figure 3.4: Progression of the logit of the q(x; t) at age 80, between 1960 and 2009.

1960 1970 1980 1990 2000 2010 −1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1.0 Year logit(q(90,t)) European males Dutch males

1960 1970 1980 1990 2000 2010 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.2 Year logit(q(90,t)) European females Dutch females

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12 J.M. den Hollander — The gravity model 1960 1970 1980 1990 2000 2010 −3.2 −3.0 −2.8 −2.6 −2.4 Year European males Dutch males

(a) Estimated κ(1)t values

1960 1970 1980 1990 2000 2010 −3.8 −3.6 −3.4 −3.2 −3.0 Year European females Dutch females (b) Estimated κ(2)t values

Figure 3.6: Estimated values for κ(1)_t and κ(2)_t between 1960 and 2009, for Dutch and European males.

European parameters for approximately twenty years since the late 1980s. However, the parameter estimates never drift very far apart.

The estimates for the κ(1)_t and κ(2)_t parameters for females are summarized in figure

3.7. Again, the κ(1)_t parameters give information similar to the life expectancy and the mortality probability. The Dutch values for the κ(1)_t parameters however do not show consistent convergence towards the European values. The κ(2)_t parameters for females are relatively stable over time and of similar value.

European κ(1)_t parameters for both males and females tend to go down in time near-linearly. The κ(2)_t parameters show a weak, near-linear trend.

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1960 1970 1980 1990 2000 2010 0.090 0.095 0.100 0.105 0.110 0.115 Year European males Dutch males

(a) Estimated κ(1)_t values

1960 1970 1980 1990 2000 2010 0.110 0.115 0.120 0.125 Year European females Dutch females (b) Estimated κ(2)_t values

Figure 3.7: Estimated values for κ(1)_t and κ(2)_t between 1960 and 2009, for Dutch and European females.

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Chapter 4

The model

4.1 Gravity model for multiple populations

In their 2011 paper, Dowd et al propose a gravity model[10]. In this model, the mortality in two populations was considered simultaneously. The first population (males, aged 60-84, from England and Wales) was far larger than the second population (insured males, aged 60-84, from England and Wales). The populations were closely related, so a long-term divergence in mortality behaviour was considered unlikely. For the second population, κ(i)_t were modeled with mean reversion to the κ(i)_t of the first population. Though the principle can be extended to other mortality models, Dowd et al used an APC-model (age, period and cohort) to fit mortality data:

log(mx,t) = βx+ n−1α κt+ n−1α γc. (4.1)

Dowd et al use the following time dynamics in the κ(i)_t for the larger population 1 and smaller population 2:

κ(pop−1)_t = κ(pop−1)_t−1 + µ(pop−1)+ C(1,1)Z_t(pop−1)

κ(pop−2)_t = κ(pop−1)_t−1 + φ(κ(pop−1)_t−1 − κ(pop−2)_t−1 ) + µ(pop−2) (4.2)

+ C(2,1)Z_t(pop−1)+ C(2,2)Z_t(pop−2).

They prescribe a similar type of time dynamics in their cohort variables γc.

The CBD-model has two κ(i)_t , describing the general mortality improvements in the age group 60 90, and the age differentials in this age group. In order to allow both Dutch κ(i)_t parameters towards their European counterparts, the aforementioned time dynamics will be applied to both Dutch κ(i)_t parameters1. This leads to:

logit qpop−1(x; t) = log

qpop−1(x; t)

1 − qpop−1(x; t)

= κ(1)_t + κ(2)_t (x − ¯x) logit qpop−2(x; t) = log

qpop−2(x; t) 1 − qpop−2(x; t) = κ(3)_t + κ(4)_t (x − ¯x) (4.3) and

1_{In the mathematical formulary, these parameters will be referred to as κ}(3)

t and κ

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t . Occasionally,

for linguistic ease, these same parameters will be referred to as the Dutch κ(1)_t and κ(2)_t .

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+ C(1,2)Z_t + C(2,2)Z_t κ(3)_t = κ(3)_t−1+ µ(3)+ φ(1)(κ(1)_t−1− κ(3)_t−1) (4.4) + C(1,3)Z_t(1)+ C(2,3)Z_t(2)+ C(3,3)Z_t(3) κ(4)_t = κ(4)_t−1+ µ(4)+ φ(2)(κ(2)_t−1− κ(4)_t−1) + C(1,4)Z_t(1)+ C(2,4)Z_t(2)+ C(3,4)Z_t(3)+ C(4,4)Z_t(4). Or, equivalently: κt= Φ0κ(t−1)+ µ + C0Zt, (4.5) with Φ =     1 0 φ(1) ₀ 0 1 0 φ(2) 0 0 1 − φ(1) ₀ 0 0 0 1 − φ(2)     (4.6) and C =     C(1,1) C(1,2) C(1,3) C(1,4) 0 C(2,2) C(2,3) C(2,4) 0 0 C(3,3) C(3,4) 0 0 0 C(4,4)     . (4.7)

In the described model, there are six parameters that cause the coupling between the European (population 1) and Dutch (population 2) mortalities:

• Parameters φ(1) _{and φ}(2) _{cause the Dutch mortality to converge to the European}

mortality. These parameters can reflect the ongoing political and social unifica-tion of Europe. High values for these variables indicate that there exists a strong mechanism for Dutch mortality to converge to European mortality. Values near zero indicate a lack of such a mechanism. If parameters φ(1) and φ(2) are zero, Dutch and European mortality can strongly diverge on the long run.

• C(1,3)_{, C}(1,4)_{, C}(2,3) _{and C}(2,4) _{determine the correlation in the innovations for the}

European and Dutch mortality. Drivers for these correlations are random processes that effect European mortality and Dutch mortality in a similar way. Examples include technological and health care advances, weather influences and epidemics. It is interesting to look at the expectation values for the κ(i)_t that drive the model. For the European population the expectation value can be calculated easily by recursion. For i = 1 and i = 2, with T indicating the time of the last observation and C(j,i) = 0 for j > i, the expectation values are:

E h κ(i)_t i = E  κ(i)_t−1+ µ(i)+ 4 X j=1 C(j,i)Z_t(i)  

= E[κ(i)_t−1] + µ(i)

= . . . (4.8)

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For the Dutch population, with i = 3 and i = 4, the recursion relations can be used as well, albeit slightly more laborious:

E h κ(i)_t i = E  κ(i)_t−1+ φ(i−2)

κ(i−2)_t−1 − κ(i)_t−1+ µ(i)+

4 X j=1 C(j,i)Z_t(i)  

=1 − φ(i−2)Ehκ(i)_t−1i+ µ(i)+ φ(i−2)Ehκ(i−2)_t−1 i = 1 − φ(i−2) E h 1 − φ(i−2)

κ(i)_t−2+ µ(i)+ φ(i−2)E

h κ(i−2)_t−2 ii + µ(i)+ φ(i−2) κ(i−2)_T + (t − T − 1)µ(i−2)

=1 − φ(i−2)2Ehκ(i)_t−2i+1 − φ(i−2)φ(i−2)Ehκ(i−2)_t−2 i + 1 + 1 − φ(i−2) µ(i)+ φ(i−2) κ(i−2)_T + (t − T − 1)µ(i−2) (4.9) =1 − φ(i−2)2Ehκ_t−2(i) i+1 +1 − φ(i−2)µ(i)

+ φ(i−2)κ(i−2)_T 1 + 1 − φ(i−2)

+ φ(i−2)µ(i−2)(t − T − 1) + (t − T − 2)1 − φ(i−2)

= . . .

=1 − φ(i−2)t−Tκ(i)_T +µ(i)+ φ(i−2)κ(i−2)_T

t−T −1 X k=0 1 − φ(i−2)k + φ(i−2)µ(i−2) t−T −1 X m=0 (t − T − 1 − m)1 − φ(i−2)m.

Solving the summations and rearranging gives the following expression for the expecta-tion value, for φ(i−2) 6= 0:

E[κ(i)_t ]

=1 − φ(i−2)(t−T )κ(i)_T

+µ(i)+ φ(i−2)κ(i−2)_T 1 − 1 − φ

(i−2)t−T φ(i−2) + µ(i−2)φ (i−2)_{(t − T ) − 1 + 1 − φ}(i−2)t−T φ(i−2) (4.10) = 1 − φ(i−2) (t−T ) κ(i)_T + µ(i−2)(t − T ) +

µ(i)+ φ(i−2)κ(i−2)_T − µ(i−2)1 − 1 − φ

(i−2)t−T

φ(i−2) .

It can be seen, that for t T , the expectation value for the Dutch κ(i)_(t) converge to the European κ(i)_(t), unless φ(i−2) = 0. Therefore, European and Dutch mortality paths are expected not to diverge in the long run.

4.2 Non-convergent models: the case Φ = 1

Some limit cases of the model are of special interest, and will be estimated along with the full model for comparison purposes. These cases are discussed below.

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κ_t = κ_t−1+ µ(1)+ C(1,1)Z_t

κ(2)_t = κ(2)_t−1+ µ(2)+ C(1,2)Z_t(1)+ C(2,2)Z_t(2)

κ(3)_t = κ(3)_t−1+ µ(3)+ C(3,3)Z_t(3) (4.11)

κ(4)_t = κ(4)_t−1+ µ(4)+ C(3,4)Z_t(3)+ C(4,4)Z_t(4)

This model allows for diverging and non-correlated European and Dutch mortality paths. The expectation values for the κ(i)_t are as follows, with T indicating the time of the last observation and C(j,i)_{= 0 for j > i:}

E[κ(i)_t ] = E  κ(i)_t−1+ µ(i)+ 4 X j=1 C(j,i)Z_t(i)  

= E[κ(i)_t ] + µ(i) (4.12)

= κ(i)_T + (t − T )µ(i).

Unless µ(1) = µ(3) and µ(2) = µ(4), the European and Dutch mortality will diverge in the future. Note that this divergence need not be harmful for actuarial practices.

4.2.2 The correlated population model

Assuming φ(1) = φ(2) = 0, the model will reduce to two CBD models with correlated innovations, describing European and Dutch mortality respectively:

κ(1)_t = κ(1)_t−1+ µ(1)+ C(1,1)Z_t(1)

κ(2)_t = κ(2)_t−1+ µ(2)+ C(1,2)Z_t(1)+ C(2,2)Z_t(2)

κ(3)_t = κ(3)_t−1+ µ(3)+ C(1,3)Z_t(1)+ C(2,3)Z_t(2)+ C(3,3)Z_t(3) (4.13)

κ(4)_t = κ(4)_t−1+ µ(4)+ C(1,4)Z_t(1)+ C(2,4)Z_t(2)+ C(3,4)Z_t(3)+ C(4,4)Z_t(4).

The expectation values for the κ(i)_t are equal to those calculated in the equation

4.12. Therefore, this model allows for correlated European and Dutch mortality paths, though these paths will still diverge on the long term.

4.2.3 Examples

To give a feel for the how the gravity model and the single and correlated population models compare, a single simulation was run. In this simulation, the three models were fit to the available data in the time range between 1970 and 2009. Details of the fitting process and the estimated model parameters are found in chapter 5. Afterwards, 400 random numbers were drawn from a standard normal distribution2 for the Z_t(i) with t between 2010 and 2109. Based on the κ(i)₂₀₀₉, the estimated model parameters and the Z_t(i), κ(i)_t is simulated by each of the three models. The results are shown in the figures below.

2

It could be argued that the t-distribution should be used, as the model parameters are estimated. For mathematical ease, the normal distribution is used, which is justified when using a sufficiently large data set.

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(a) Estimated and projected κ(1)_t values

(b) Estimated and projected κ(2)_t values

Figure 4.1: Estimated (based on observations) and projected values for κ(1)_t and κ(2)_t between 1960 and 2109, for Dutch and European males.

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In this chapter, the gravity model and the two single population models described in the previous sections, will be fitted. In all cases, the estimation of the parameters will follow the procedure outlined appendix B of the 2011 paper by Dowd et al[10].

To fit the model, only data from the time period 1970-2009 is used. Looking at the data presented in chapter 3, the 1960-1969 data seems to deviate from the later trend, especially for males. In chapter 9, tests will be performed in order to determine the robustness of the model with respect to the chosen time frame.

5.1 Fitting the single population and correlated

popula-tion models

The single population model is the easiest model to fit. First of all, the drift parameters are estimated as follows. Referring to the 1970 observation as t0and to the last available

observation (2009) as T , the following expression gives the estimate for the µ(i):

ˆ µ(i)= 1 T − t0− 1 T X t=t0+1 κ(i)_t − κ(i)_t−1= 1 39 κ(i)_T − κ(i)_t 0 . (5.1)

In the single population and correlated population models, this expression holds for all i

Secondly, given the estimated factors for µ(i), matrix C can be estimated. First, the residuals are calculated:

ˆ

D_t(i)= κ(i)_t − κ(i)_t−1− ˆµ(i). (5.2) Based on the residuals, the variance-covariance matrix is constructed as follows:

ˆ V(i,j)= 1 T − t0− 1 T X t=t0+1 ˆ D(i)_t · ˆD(j)_t (5.3) For the single population model, no correlation is modeled for innovations between the European and Dutch mortality. Therefore, this model has the restriction ˆV(i,j)= 0 if i represents the European population (i is 1 or 2) and j represents the Dutch popu-lation (j is 3 or 4) or vice versa.

Then, matrix ˆC is the Cholesky decomposition of matrix ˆV : ˆ

V = ˆC · ˆC0 (5.4)

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The estimated µ(i) parameters for the single population model and the correlated population model are as follows.

Males Females ˆ µ(1) -0.0186 -0.0205 ˆ µ(2) 0.000380 0.000171 ˆ µ(3) -0.0145 -0.0156 ˆ µ(4) 0.000581 -0.00000880

Matrix ˆC for the single population model is as follows for males and females:

ˆ Cmale =     0.0182 0.000646 0 0 0 0.000507 0 0 0 0 0.0274 0.000723 0 0 0 0.000970     ˆ Cfemale=     0.0175 0.000784 0 0 0 0.000407 0 0 0 0 0.0214 0.000985 0 0 0 0.00125    

Matrix ˆC for the correlated population model is as follows for males and females:

ˆ Cmale=     0.0182 0.000646 0.0189 0.000710 0 0.000507 −0.00929 0.000274 0 0 0.0174 0.000509 0 0 0 0.000791     ˆ Cfemale=     0.0175 0.000784 0.00953 0.000823 0 0.000407 −0.00287 −0.000113 0 0 0.0190 0.000682 0 0 0 0.00118    

5.2 Fitting the gravity model

In the gravity model, the Dutch mortality is made to converge towards the European mortality via the parameters φ(1) and φ(2). Therefore, for the Dutch mortality, ˆµ(i) and

ˆ

D(i)_t must be estimated given a certain φ(1) and φ(2).

Following Dowd et al, ˆµ(i) for i = 3 or i = 4, conditional on the φ(i) is estimated as follows: µ(3)|φ(1) = 1 T − t0− 1 T X t=t0 κ(3)_t −1 − φ(1) κ(3)_t−1− φ(1)κ(1)_t−1 µ(4)|φ(2) ₌ 1 T − t0− 1 T X t=t0 κ(4)_t −1 − φ(2)κ(4)_t−1− φ(2)_κ(2) t−1 . (5.5) ˆ

D(i)_t for i = 3 or i = 4, conditional on the φ(i) is then estimated as follows:

ˆ D_t(3)|φ(1) _{= κ}(3) t − 1 − φ(1)κ(3)_t−1− φ(1)_κ(1) t−1− µ(3)|φ(1) ˆ D_t(4)|φ(2) = κ(4)_t −1 − φ(2) κ(4)_t−1− φ(2)κ(2)_t−1− µ(4)|φ(2). (5.6)

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T − t0− 1_t=t

0+1

Again, ˆC|φ(1), φ(2) will be the Cholesky decomposition of ˆV(i,j)|φ(1)_{, φ}(2)_.

Now, it is necessary to estimate the values for φ(1) and φ(2). Again, following Dowd et al, a maximum likelihood estimation is performed. Defining ˆW(i,j)|φ(1)_{, φ}(2) _{as the}

inverse matrix of ˆV(i,j)|φ(1)_{, φ}(2)_{, the log-likelihood function is as follows:}

lφ(1), φ(2)= −T − t0− 1 2 ln h det ˆV(i,j)|φ(1)_{, φ}(2)i _(5.8) −1 2 4 X i=1 4 X j=1 T X t=t0+1 ˆ_D(i) t |φ(i−2) · ˆD_t(j)|φ(j−2)· ˆW(i,j)|φ(1), φ(2) + const.

Subsequently, the likelihood function is maximized numerically by calculating its value for numerous φ(1), φ(2)∈ [0, 1]. The following maximum likelihood estimates were found for φ(1) and φ(2):

Males Females ˆ

φ(1) 0.038 0.005 ˆ

φ(2) 0 0.0859

Parameter φ(1) is the most interesting of the two parameters, as it quantifies the coupling between general mortality levels of the European and Dutch population κ(1)_t and κ(3)_t . For men, ˆφ(1)has a value of 0.038. According to the formula for the expectation values as determined in equations4.9and4.10, Dutch observations and trend contribute to the projections a factor 1 − φ(1)(t−T )

while European observations contribute a fac-tor other 1 − 1 − φ(1)(t−T ). Based on that formula, current European observations will become the dominant factor in predicting Dutch male mortality approximately twenty years from time T . The expectation value for κ(1)_t for Dutch and European males, based on the estimated parameters, evolves as shown in figure 5.1:

For females, ˆφ(1) = 0.005 is found, which does not lead to a convergence in the near future, as can be seen in figure 5.2.

Parameter ˆφ(2) quantifies the coupling between the age dependency of mortality within the Dutch and European populations. For males, the maximum likelihood is reached if zero coupling is assumed. As the parameters κ(2)_t and κ(4)_t for males are seen to diverge between approximately 1990 and 2009, this is not surprising. As can be seen in figure 5.3, this leads to diverging expectation values for κ(2)_t and κ(4)_t for males. For females however, ˆφ(2) indicates a fairly strong coupling, as can be seen in figure 5.4.

When using a restricted model for females, having φ(2) _{= 0 is estimated, ˆ}_φ(1)_{is found}

to be 0.013, which is considerably higher than in the non-restricted model, though still far lower than for males. Whether or not a restricted model should be used, is discussed further in chapter 9.

Conditional on φ(1) and φ(2), the estimated µ(i) parameters for the are largely com-parable to the values estimated in the single population models.

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Figure 5.1: Estimated values and expected values for κ(1)_t between 1960 and 2060, for Dutch and European males.

Figure 5.2: Estimated values and expected values for κ(1)_t between 1960 and 2060, for Dutch and European females.

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Figure 5.3: Estimated values and expected values for κ(2)_t between 1960 and 2060, for Dutch and European males.

Figure 5.4: Estimated values and expected values for κ(2)_t between 1960 and 2060, for Dutch and European females.

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24 J.M. den Hollander — The gravity model Males Females ˆ µ(1) -0.0186 -0.0205 ˆ µ(2) 0.000380 0.000171 ˆ µ(3) -0.0147 -0.0158 ˆ µ(4) 0.000581 -0.0000446

Matrix ˆC for the gravity model is as follows for males and females:

ˆ Cmale=     0.0182 0.000646 0.0180 0.000710 0 0.000507 −0.00773 0.000274 0 0 0.0177 0.000542 0 0 0 0.000769     and ˆ Cfemale =     0.0175 0.000784 0.00952 0.000838 0 0.000407 −0.00282 −0.0000674 0 0 0.0190 0.000609 0 0 0 0.00116     .

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6.1 Comparison against CBS and AG forecasts

Biennially, the Dutch Statistics agency (CBS) and the Dutch Actuarial Association (AG) publish their respective mortality prognoses for the Dutch populations. These prognoses are heavily relied upon by the Dutch government, that bases policies on them. Dutch insurance companies and pension funds often use one of these two prognosis tables to base FCL calculations on. The CBS and AG models are quite different in nature from each other, and from the CBD and gravity model.

6.1.1 The AG models

The AG-2012 model is a deterministic model, that was specifically designed to fit the strong mortality improvement for the Dutch population that manifested at the begin-ning of the new millennium[1]. On the short term, the model fit the post-2000 mortality improvement trend. On the long term, the model approaches a goal table, which is reached in 2062. This goal table is based on the long term (1987/1988 - 2010/2011) observed mortality improvement.

The AG-2014 model is a stochastic mortality model[15], based on the Li-Lee model[16]. Like the gravity model, the AG-2014 model is based on both European and Dutch mor-tality observations, although a different set of countries is chosen to represent Europe. In the Li-Lee model, the force of mortality µ(x; t) is modeled as follows:

log µ(x; t) = log µEU R(x; t) + log µN L(x; t)

log µEU R(x; t) = Ax+ BxKt (6.1)

log µN L(x; t) = ax+ bxkt.

For the Kt and kt the following dynamics are assumed:

Kt+1= Kt+ θ + t+1

kt+1= c · kt+ δt+1. (6.2)

Here t+1 and δt+1 are normally distributed. The dynamics are fitted on mortality

observations made between 1970 and 2013. The countries included in the West-European whose mortality statistics are included, apart from The Netherlands, are Austria, Bel-gium, Denmark, England and Wales, France, Finland, Germany (West-Germany before 1989), Ireland, Iceland, Norway, Sweden and Switzerland.

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6.1.2 The CBS models

The CBS-2010 model is not fully disclosed in their 2010 publication[11]. However, from the publication it can be deduced, that the CBS model is based on estimating mortal-ity probabilities for various causes of death, such as lung cancer, heart and coronary disease, etc. Cohort-dependent lifestyle choices like smoking and overeating, leading to obesity, are considered when calculating morbidity probabilities for these possible causes of death. Furthermore, migration trends are taken into accounts.

The CBS-2012 model is described in a 2012 CBS publication[19]. In the CBS-2012 model, death resulting from lung cancer and other smoking-related mortality are eled separately from non-smoking-related death. The smoking-related deaths are mod-eled by an APC-model (which was briefly described in section 4.1). The non-smoking-related deaths are modeled by an Li-Lee model, using the same dynamics in the Ktand

kt as the AG-2014 model. Denmark, Finland, France, Germany, Italy, Norway, Spain,

Sweden, Switzerland and England and Wales are the countries on which the European mortality is based. The model is fitted on data from the time period 1970-2011.

The CBS-2012 and AG-2014 model are both strongly based on the Li-Lee model, and used roughly the same data sets. Therefore it can be expected that the CBS-2012 and AG-2014 model produce roughly the same results. In the CBS-2012 model, some non-linearity can be expected due to the cohort effects in smoking deaths.

A note on the CBS age convention

In the age definition of actuarial mortality prognosis tables, q(x; t) refers to the proba-bility of someone, being alive and of exactly age x years on January 1st of year t, dying at some time in year t. Had this person survived during year t, his average age during year t would have been x + 0.5. As CBS prognosis tables and observations have not been tailored for actuarial practice, a different age convention is used. In recent publications on CBS Statline, the group (x; t) includes all people that reach, or would have reached, age x at some point during year t. This means, that the average age of this group dur-ing year t was exactly x years, which is half a year lower than is the case in actuarial practice. Therefore, when interpreting CBS observations and the CBS-2014 prognosis table, half a year is subtracted from the age as reported by CBS. The CBS-2012 table follows an age convention than coincides with the actuarial practice.

6.1.3 Comparison

To directly compare the results from the AG and CBS model to the CBD-model and gravity model results, it would be preferable to use the κ(i)_t factors that follow from the CBS and AG prognosis tables. In order to do this, first the logit of the q(x; t) are calculated, after which OLS is used to estimate the linear age-dependency in the logit q(x; t).

As noted before, the κ(i)_t factors typically capture the behavior of the observed mor-tality probabilities in the pension age segment quite well. However, the AG and CBS models need not preserve this property. Non-linear and piecewise linear behavior can be observed in the results of these models. Especially the AG prognoses for females tend to demonstrate convexity in the later years. Piecewise linear behavior occurred primarily in the less recent AG-2012 and CBS-2012 model, as can been seen in figures6.1. Strong piecewise linear behavior was also found to be present in the short term predictions of the AG-2012 model. Therefore, instead of using the κ(i)_t factors, comparisons will be made using the logit of q(x; t) for ages 65 and 85.

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60 65 70 75 80 85 90 −6 −5 −4 −3 −2 Age logit q(x;t) AG−2012 AG−2014 CBS−2010 CBS−2012 (a) Males 60 65 70 75 80 85 90 −6 −5 −4 −3 −2 Age logit q(x;t) AG−2012 AG−2014 CBS−2010 CBS−2012 (b) Females

Figure 6.1: Values for logit q(x; t) in 2060, as forecasted by the AG-2012, AG-2014, CBS-2010 and CBS-2012 models, for Dutch males and females.

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28 J.M. den Hollander — The gravity model 2010 2020 2030 2040 2050 2060 −5.6 −5.4 −5.2 −5.0 −4.8 −4.6 −4.4 Year logit q(x;t) AG−2012 AG−2014 CBS−2010 CBS−2012 Gravity CBD

Figure 6.2: Values for the logit of q(65; t) between 2010 and 2062, as forecasted by six models, for Dutch males.

Figures 6.2 and 6.3 contain the forecasts for the logit of q(x; t) for Dutch males produced by the different models. At age 65, the projected logit of the q(x; t) is simi-lar to the recent AG-2014 model projections and fairly simisimi-lar to projections from the CBD and CBS-2012 model. The CBS-2010 predicts a far lower mortality improvement, whereas the AG-2012 model forecasts a greater improvement. At age 85, the gravity model produces a time progression that is similar to that of the CBS-2010 model. The more recent AG-2014 and CBS-2012 models forecast a greater mortality improvement, whereas the CBD model predicts a lower improvement. The AG-2012 model forecasts a mortality improvement that is initially quite strong, and stagnates around 2050.

The progressions for female mortality are shown in figures 6.4 and 6.5. As was the case with the 65 year old males, the gravity model produces a projection that is similar to that of the AG-2014 for 65 year old females. The CBS-2012 model produces a similar projection in the later years, after an initial stagnation. The less recent AG-2012 and CBS-2010 models forecast a lower mortality improvement. For 85 year old females, the CBD and gravity models are in good agreement with the AG-2014 and CBS-2012 models. The CBS-2012 model forecasts a slightly higher mortality for 85 year old females near the year 2030. This seems to be a cohort effect similar to the stagnant mortality improvement for the 65 year olds. The AG-2012 model follows the trend of

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2010 2020 2030 2040 2050 2060 −3.0 −2.8 −2.6 −2.4 −2.2 −2.0 Year logit q(x;t) AG−2012 AG−2014 CBS−2010 CBS−2012 Gravity CBD

Figure 6.3: Values for the logit of q(85; t) between 2010 and 2062, as forecasted by six models, for Dutch males.

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30 J.M. den Hollander — The gravity model 2010 2020 2030 2040 2050 2060 −5.8 −5.6 −5.4 −5.2 −5.0 −4.8 Year logit q(x;t) AG−2012 AG−2014 CBS−2010 CBS−2012 Gravity CBD

Figure 6.4: Values for the logit of q(65; t) between 2010 and 2062, as forecasted by six models, for Dutch females.

the gravity model, but forecasts a stagnation starting from approximately 2040. The CBS-2010 model predicts a lower mortality improvement than the other models.

Summarizing, for males at higher ages, the gravity model underestimates the mor-tality improvement compared to the most recent AG and CBS models. At lower ages, the forecasts are in good agreement. For females, the gravity model and the AG-2014 model are in good agreement. At longer horizons, the gravity model is in decent agree-ment with the CBS-2012 model. At shorter horizons, the CBS-2012 model shows cohort effects that cannot be produced by the gravity model.

6.2 Comparison against recent mortality observations

The gravity model is fitted on data from between 1970 and 2009. More recent (2010 up until 2013) mortality observations have since become available for the Netherlands, via CBS Statline[18]. It is possible to compare predictions based on the gravity model to the realized mortality rates. Given the short interval on which new observations are available, the gravity models forecasts will not deviate strongly from the CBD model for the Dutch population.

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2010 2020 2030 2040 2050 2060 −3.4 −3.2 −3.0 −2.8 −2.6 −2.4 Year logit q(x;t) AG−2012 AG−2014 CBS−2010 CBS−2012 Gravity CBD

Figure 6.5: Values for the logit of q(85; t) between 2010 and 2062, as forecasted by six models, for Dutch females.

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Year Actual κ(1) CBD model Gravity model 2010 -3.210 -3.199 -3.201

2011 -3.262 -3.213 -3.217 2012 -3.256 -3.228 -3.233 2013 -3.281 -3.242 -3.250

Year Actual κ(2) CBD model Gravity model 2010 0.1120 0.1129 0.1129

2011 0.1142 0.1135 0.1135 2012 0.1142 0.1141 0.1141 2013 0.1136 0.1147 0.1147

Table 6.1: Forecasted κ(i) values by the CBD and gravity models, and estimations from actual recent observations, for Dutch males.

Year Actual κ(1) CBD model Gravity model 2010 -3.679 -3.695 -3.695

2011 -3.691 -3.710 -3.711 2012 -3.687 -3.726 -3.728 2013 -3.705 -3.741 -3.744

Year Actual κ(2) CBD model Gravity model 2010 0.1150 0.1169 0.1171

2011 0.1138 0.1168 0.1173 2012 0.1137 0.1168 0.1175 2013 0.1135 0.1168 0.1177

Table 6.2: Forecasted κ(i) values by the CBD and gravity models, and estimations from actual recent observations, for Dutch females.

Tables 6.1and 6.2contain the values for κ(1)_t and κ(2)_t for males and females, calcu-lated from the mortality observations. Furthermore, the table contains the expectation values for the parameters based on both the CBD and gravity models.

For Dutch males, the observed mortality improvement was stronger than forecasted by the gravity model. The estimated recent values for κ(2)_t are currently in line with the gravity model forecast. For Dutch females, the gravity model so far has overestimated the mortality improvement. This overestimation was slightly stronger than the under-estimation in the under-estimation for Dutch males. The recent values for κ(2)_t are quite far off from the predicted value.

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7.1 Table closing: the Kannisto method

The CBD model, and subsequently the gravity model, are specified only for the pen-sion age range, where the logit of the one-year mortality probabilities has a near-linear behavior. Outside this age range, the linearity can be seen to break down. This can be observed in the figure7.1.

At the lower ages, the age-dependence of the mortality rates follows a rather com-plex pattern, consisting of a relativity high mortality in the first years after birth, a low childhood mortality and an elevated mortality in adolescence and early adulthood. From mid-adulthood up until pension years, mortality rates approximately follow a piecewise linear increase. Then, from the age of 90 years, the mortality rate increase in age tends to slow down. Low-age mortality is of minor interest to FCL and SCR calculations. Therefore, they will not be looked into any further in this chapter. High-age mortality, however, must be handled when calculating FCL or SCR.

Typically, the high-age segment of mortality prognoses are not based directly on mortality observations, as few observations are available. An often used method to estimate high-age mortality rates, was pioneered by Kannisto[14]. The Kannisto method relies on logistic regression in order to extrapolate high-age mortality rates from known rates. Kannisto fits the force of mortality µ(x; t) at high age to the logistic function:

µ(x; t) = λ1exp(λ2x) 1 + λ1exp(λ2x)

(7.1) Parameters λ1 and λ2 are estimated by performing OLS on the logit of the µ(x; t):

logit (µ(x; t)) = log

µ(x; t) 1 − µ(x; t)

= log (λ1exp(λ2x)) = log(λ1) + λ2x (7.2)

In accordance with the AG-2014 table closing, the fit is based on crude death rates of the age range x ∈ {80, 81, . . . , 90}.

In figure 7.2, an example of the Kannisto method is shown for European males, based on 2009 data. As an estimation for the µ(x; t), the crude death rates m(x; t) have been used.

7.2 FCL calculations

Using the Kannisto method for closing the table, FCL calculations can be performed on paying annuities depending on lives in the pensioner age range. These calculations can

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34 J.M. den Hollander — The gravity model 0 20 40 60 80 100 −8 −6 −4 −2 0 Age logit q(x,2009)

(a) Males, ages 0-105

0 20 40 60 80 100 −8 −6 −4 −2 0 Age logit q(y ,2009) (b) Females, ages 0-105 60 70 80 90 100 −4 −3 −2 −1 0 Age logit q(x,2009) (c) Males, ages 60-105 60 70 80 90 100 −5 −4 −3 −2 −1 Age logit q(y ,2009) (d) Females, ages 60-105

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Figure 7.2: Observations of logit q(x; t) and Kannisto fit, European males, in 2009.

be used as a means of comparing the gravity model to the CBD model or other models. Furthermore, the calculations are required to properly assess the SCR calculations in the next paragraph.

A general formula for the FCL of a simple life insurance policy, without expenses or premium payments, can be written as follows:

F CL =

∞

X

i=0

B(i) (1 + r(i)))−ip(i). (7.3) Here, B(i) corresponds to the benefit that is to be paid at time i and r(i) corre-sponds to the zero-coupon interest rate with maturity i. The factor p(i) correcorre-sponds to the probability that the benefit at time i must be paid. Time i can, in principle, be any time unit, but the unit of years is used here.

For the following calculations, the FCL of a paying annuity is calculated. The annu-ity is paying in arrears, such that B(0) = 0. B(i) = 1 is chosen for all benefit payments after t = 0, such that the FCL is calculated per unit. For the interest rates, r(i) = 0.03 is chosen for all maturities, for a flat 3% per annum interest rate. A flat interest rate is, in these calculations, preferred over an interest rate term structure, so that mortality effects in the FCL are focused on.

p(i) is chosen to be the probability that a person of age x at time t will still be alive at time (t + i). For calculating this probability, gravity and CBD model best estimate projections will be used. The Kannisto method, fitted on the age range from 80 years up to and including 90 years, will be used to close the projected tables. Estimations for the µ(x; t), will be based on the projected q(x; t), using that

ˆ µ(x; t) = m(x; t) ≈ − log 1 1 − q(x; t) . (7.4)

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Figure 7.3: Projected FCL per 1 euro nett annuity, for Dutch males aged 60, 70 and 80 years

No experience mortality factors have been used. Furthermore, no expenses have been taken into consideration.

The projected FCLs, with valuation dates in the years 2010 up to 2050, of a 1 euro paying annuity for Dutch males and females of age 60, 70 and 80, are shown in figures7.3

and7.4. For Dutch males, the gravity model predicts slightly higher FCLs than the CBD model for all ages, especially in later years. This was expected, as the κ(1)_t parameter estimates are uniformly lower for the gravity model than for the CBD model, and the κ(2)_t estimates are equal. For Dutch females, the gravity model and CBD model produce FCLs that are approximately equal. At higher ages, the FCLs based on the CBD model are slightly higher than the FCLs based on the gravity model. This is due to the fact that the mortality improvement of the gravity model benefits the younger age groups as a result of higher κ(2)_t estimates.

7.3 SCR calculations

Under Solvency II, the SCR is the capital required by an insurer, in order to be able to fulfill its obligations in one year time, with a 99.5% probability. As this quantity is extremely difficult to calculate, typically the SCR is broken down into several SCRs for risk types, and further into sub-risk SCRs. For each sub-risk, an SCR is calculated, based on the 99.5% quantile of the one year loss distribution[7]. The total SCR is then calculated, based on assumed risk and sub-risk correlations. For pension funds and pen-sion insurers, longevity risk is one of the most important sub-risks. In this paragraph, SCR calculations for longevity risk will be made, using the standard model, the gravity model and the CBD model.

The standard model for the longevity risk SCR is based on recalculating the FCL of the portfolio, using shocked mortality rates. The shocked rates are 20% lower than the best estimate mortality rates. The SCR is then calculated as the difference between

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Figure 7.4: Projected FCL per 1 euro nett annuity, for Dutch females aged 60, 70 and 80 years

the recalculated FCL and the FCL based on Best Estimate mortality rates.

The gravity model, as well as the CBD model, can be used to perform SCR cal-culations. As these models contain several κ(1)_t parameters, and the impact of these parameters can only be evaluated in the context of the underlying liabilities, it is not immediately clear how a 99.5% scenario can be calculated. For both models, the follow-ing procedure is followed:

1. The model parameters are estimated, based on the historical data.

2. A sufficiently large number of drawings for the κ(i)_t parameters is made. In this thesis, 10,000 drawings are used. The drawings follow equation 4.4, where the uncertainty is introduced via the standard normally distributed Z_t(i).

3. To limit calculation time, the drawings with the 90% highest values for the κ(3)_t parameters are not used. These drawings are unlikely to give rise to a 99.5% scenario, as the κ(3)_t are expected to have the highest impact on the FCL. For the CBD model, this step is omitted as calculation times are shorter for that model. 4. For the remaining drawings, re-estimate the model, adding the drawing as an extra

year of data.

5. Deduce the mortality prognosis table based on the re-estimated model, and cal-culate the FCL.

6. The scenario that is at the 95% quantile of the FCL distribution of the remain-ing scenarios, constitutes the SCR scenario. Different liabilities will give rise to different FCL distributions, and therefore to different SCR scenarios.

For the SCR calculation, the value of an annuity, paying one euro in arrears starting on December 31st 2010, is calculated using both the best estimate and shocked mortality

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Figure 7.5: Nett annuity SCR/FCL ratio for Dutch males in 2009.

rates. The FCL of the annuity is calculated per January 1st 2010, and a flat 3% interest rate is used. The calculation is performed on lives of all ages in range between 60 and 89 years. The results are summarized in figures7.5and7.6. SCRs are expressed in terms of the SCR to FCL ratio. For the standard model calculations, mortality prognoses by the gravity model are used for the FCL calculations and as basis for the shocked mortality rates. Using CBD model prognoses instead produced very similar results.

For Dutch males, the gravity model yields SCR/FCL ratios between 3.5% (at 60 years) and 10.1% (at 89 years). This is significantly higher than the CBD model ratios (2.5% to 8.0%). Using the standard model results in far higher SCR/FCL ratios, espe-cially at higher ages. For Dutch females, standard model and CBD model SCR/FCL ratios are very similar to the ratios found for Dutch males. However, contrary to the Dutch males, the gravity model produces slightly lower SCR/FCL ratios than the CBD model.

A reason for the different behavior for the SCR/FCL ratio between Dutch males and Dutch females can be found in the model estimations for the φ(1) parameters. In the 1000 scenarios from which the SCR is calculated, the φ(1) parameters are re-estimated. In the case of Dutch females, these parameters strongly tend to be larger than the orig-inal φ(1) parameter, leading to more convergence to the European mortality prognoses than in the FCL calculation. For Dutch males, the re-estimated φ(1) parameters tend to be smaller than the original estimation, leading to a weaker convergence.

In an SCR calculation, following the procedure outlined above, both Dutch and European mortality rates are shocked in the first year. However, these shocks need not be of equal magnitude. The Dutch mortality rates are the determining factor in the FCL distribution, and thereby for selecting the SCR scenario. It is very likely that the simultaneous shock to the European mortality rates is far less severe. Stronger convergence to European mortality rates will then be a moderating factor in an SCR calculation. The correlation between European and Dutch mortality in tail risk scenarios

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Figure 7.6: Nett annuity SCR/FCL ratio for Dutch females in 2009.

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Chapter 8

Model estimation for other

European countries

The gravity model is a model that forces long term convergence of a small populations mortality to the mortality of a larger, related population. In Europe, there are various countries that have a similar exposure to mortality as the Netherlands. It is interesting to have a look at the mortality data for those countries, and to the gravity and CBD model fits to this data. In this chapter, the mortality of Austria, Belgium, Denmark and Portugal will be considered.

As was previously done with the European and Dutch data, the exposures and death counts per year and one-year age bracket are used to estimate one-year mortality prob-abilities for each country and sex. After this, OLS regression is performed to calculate the CBD-model parameters κ(1)_t and κ(2)_t .

The results for males of this are summarized in figure 8.1. For κ(1)_t , the progressions of Belgium, Portugal and Austria have followed the European progression quite strongly for the last decades. Denmark follows a mortality progression akin to the Dutch progres-sion. For Denmark and the Netherlands, decades of stagnation of mortality improvement are followed by adherence to the European progression. The Danish mortality figures, however, are still lagging behind compared to the Dutch and European mortality.

For κ(2)_t , all countries, as well as the European average, show a weak upward trend over the last decades.

The κ(1)_t and κ(2)_t factors for females are displayed in figure 8.2. Similar to the re-sults for males, the κ(1)_t factors of Belgium, Austria and Portugal follow the European factors closely. The Danish progression is somewhat similar to the Dutch progression, albeit that Danish mortality improvement stalled approximately a decade before Dutch mortality improvement did. Also, Danish mortality is currently lagging behind the Eu-ropean average, whereas Dutch mortality is currently comparable to EuEu-ropean mortality. For κ(2)_t no clear common trend is visible. European κ(2)_t have a weak upward trend, but trend for individual countries can deviate strongly. Most prominently, Danish κ(2)_t show a steady decrease in the last four decades of the 20th century, after which it rapidly goes up.

The estimated CBD single population model and gravity model parameters, for these four populations, are summarized in tables8.1and 8.2. Several interesting points arise from these values. First of all, with the exception of Denmark, the values for ˆφ(1) are typically much higher for other countries than for the Netherlands. This reflects

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(a) Estimated κ(1)t values

(b) Estimated κ(2)_t values

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(a) Estimated κ(1)_t values

(b) Estimated κ(2)_t values

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ˆ

µ , females 0.000169 0.0000913 -0.0000134 0.000344 Table 8.1: Single population parameter estimates. Austria Belgium Denmark Portugal

ˆ µ(3), males 0.00106 0.0127 -0.00778 0.00493 ˆ µ(4), males 0.000637 0.000483 -0.000784 0.000788 ˆ µ(3)_{, females} _-0.0118 _-0.00828 _-0.00572 _0.0238 ˆ µ(4), females 0.00182 0.000736 -0.000391 0.000843 ˆ φ(1)_{, males} _0.34 _0.45 _0.08 _0.34 ˆ φ(2), males 0.25 0.08 0.30 0.23 ˆ φ(1), females 0.35 0.24 0.04 0.42 ˆ φ(2), females 0.12 0.51 0.03 0.11

Table 8.2: Gravity population parameter estimates.

the fact that the κ(1)_t progressions for these countries typically follow the European κ(1)_t progression more closely than the Dutch or the Danish κ(1)_t .

Secondly, the role of the ˆµ(3) parameters in the gravity model is different than in the single-population CBD model. For countries with high values for ˆφ(1), the gravity term can have a strong effect on the ˆµ(3) parameter, which is estimated conditional on

ˆ

φ(1). The ˆµ(3) parameter can no longer be viewed as describing the drift in the mortality progressions. This makes it more difficult to interpret the gravity model estimations as compared to the single population CBD model.

Thirdly, the values for ˆφ(2) vary greatly among the populations. For Dutch males, its value is zero and would be negative if the model would have allowed for that. For Belgian females, ˆφ(2) = 0.51 is found, which causes the European κ(2)_t value to have a greater influence on the Belgian κ(2)_t+1 than the Belgian κ(2)_t . This, combined with the fact that no strong common trends are observed in the behavior of the κ(2)_t , raises the question if the κ(2)_t factors should be estimated through the gravity model. Possibly, simply using a estimation with a fixed φ(2) = 0 would be better for at least some pop-ulations. This topic is explored further in section 9.4.

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Chapter 9

Model testing

9.1 Normality testing

One test that can be performed to assess the validity of the model, is whether the model residues are distributed conform the specified distribution. In the case of both the gravity model and the single population CBD-model, the error terms are multi-variate normally distributed. To test whether the residues are actually normally dis-tributed, a multivariate version of the Shapiro-Wilks test, developed by Villasenor-Alva and Gonzalez-Estrada[21], is used. This normality test is implemented in the R module mvShapiroTest, which is freely available.

The zero hypothesis of the test is, that tested stochastic variables are multivariate normally distributed. The alternative hypothesis is, that this is not the case. The zero hypothesis is rejecting in favor of the alternative hypothesis if the reported p-value is lower than a chosen significance level. The significance level indicates the probability of unjustly rejecting the zero hypothesis of normality.

Table 9.1 lists the p-values for the multivariate Shapiro-Wilk tests on the Dutch, Belgian, Austrian, Danish and Portuguese populations in the gravity model. At a signif-icance level of 0.05, the zero hypothesis of normality is not rejected for any population .As a reference, the p-values for the multivariate Shapiro-Wilk tests on the European and national populations in the single-population CBD-model are listed in table 9.2. Again, the hypothesis of normality is not rejected for any population at a significance level of 0.05.

9.2 Tail risk behaviour

In the SCR calculation procedure, outlined in chapter 7, a drawing from the tail of the joint κ(i)_t distribution is used as a pseudo-observation for the first year for which no observation is available, after which the model is re-estimated. The ordering of the scenarios was subsequently done by calculating the FCLs, given the observations and the added pseudo-observation.

The Netherlands Austria Belgium Denmark Portugal

Male 0.7215 0.5072 0.8111 0.3774 0.5738

Female 0.6388 0.5212 0.4936 0.1190 0.4777

Table 9.1: The p-values for the multivariate normality tests, when using the gravity model.

The gravity model : using a multi-population CBD mortality model to forecast Dutch mortality

J.M. den Hollander

Contents

1.1

Motivation

1.2

Content of this thesis

2.1

Mortality in actuarial practice

2.2

The CBD model for human mortality

3.1

Defining European mortality

3.2

Comparing European and Dutch mortality

3.3

Fitting the CBD-model

Chapter 4

The model

4.1

Gravity model for multiple populations

4.2

Non-convergent models: the case Φ = 1

5.1

Fitting the single population and correlated

popula-tion models

5.2

Fitting the gravity model

6.1

Comparison against CBS and AG forecasts

6.2

Comparison against recent mortality observations

7.1

Table closing: the Kannisto method

7.2

FCL calculations

7.3

SCR calculations

Chapter 8

Model estimation for other

European countries

Chapter 9

Model testing

9.1

Normality testing

9.2

Tail risk behaviour