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The Gravity Model — Using a Multi-Population CBD Mortality Model to Forecast Dutch Mortality

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The Gravity Model — Using a Multi-Population CBD Mortality Model to Forecast Dutch Mortality The gravity model | Using a multi-population CBD mortality model to forecast Dutch mortality 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: [email protected] Date: January 5, 2015 Supervisor: Dr. T.J. Boonen Second reader: F. van Berkum The gravity model | J.M. den Hollander iii Abstract In the first decade of the 21st century, the longevity of the Dutch, especially males, experienced a strong boost. This boost, which is largely absent in other popula- tions in Europe, caused Dutch insurers problems when developing a Solvency II internal model for life risk. The longevity boost makes it difficult to fit a stochastic mortality model to Dutch data, as pre-2002 and post-2002 data prescribe different trends. 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 Contents 1 Introduction1 1.1 Motivation..................................1 1.2 Content of this thesis.............................1 2 Theory3 2.1 Mortality in actuarial practice........................3 2.2 The CBD model for human mortality...................4 3 Data 7 3.1 Defining European mortality........................7 3.2 Comparing European and Dutch mortality.................8 3.3 Fitting the CBD-model........................... 10 4 The model 14 4.1 Gravity model for multiple populations................... 14 4.2 Non-convergent models: the case Φ = 1................... 16 4.2.1 The single population model.................... 17 4.2.2 The correlated population model.................. 17 4.2.3 Examples............................... 17 5 Fitting the models 19 5.1 Fitting the single population and correlated population models..... 19 5.2 Fitting the gravity model.......................... 20 6 Comparing model results to other sources 25 6.1 Comparison against CBS and AG forecasts................ 25 6.1.1 The AG models............................ 25 6.1.2 The CBS models........................... 26 6.1.3 Comparison.............................. 26 6.2 Comparison against recent mortality observations............. 30 7 FCL and SCR calculations 33 7.1 Table closing: the Kannisto method..................... 33 7.2 FCL calculations............................... 33 7.3 SCR calculations............................... 36 8 Model estimation for other European countries 40 9 Model testing 44 9.1 Normality testing............................... 44 9.2 Tail risk behaviour.............................. 44 9.3 Model self-consistency............................ 45 9.4 Model parsimoniousness: φ(2) restriction.................. 48 9.5 Robustness tests............................... 48 iv The gravity model | J.M. den Hollander v 9.6 Back testing.................................. 49 9.7 Biological reasonability............................ 52 10 Conclusions 57 vi J.M. den Hollander | The gravity model Chapter 1 Introduction 1.1 Motivation 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 figure 1.1, 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 1 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? Chapter 2 Theory 2.1 Mortality in actuarial practice In modern actuarial practice, mortality probabilities are considered to be both time- and age-dependent, 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
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