GUIBERT Quentin

Countries with common features in terms of social, economic and health systems generally have mortality trends which evolve in a similar manner. Drawing on this, many multi-population models are built on a coherence assumption which inhibits the divergence of mortality rates between two populations, or more, on the long run. However, this assumption may prove to be too strong in a general context, especially when it is imposed to a large collection of countries. We also note that the coherence hypothesis significantly reduces the spectrum of achievable mortality dispersion forecasts for a collection of populations when comparing to the historical observations. This may distort the longevity risk assessment of an insurer. In this paper, we propose a new model to forecast multiple populations assuming that the long-run coherent principle is verified by subgroups of countries that we call the "locally coherence" property. Thus, our specification is built on a trade-off between the Lee-Carter's diversification and Li-Lee's concentration features and allows to fit the model to a large number of populations simultaneously. A penalized vector autoregressive (VAR) model, based on the elastic-net regularization, is considered for modeling the dynamics of common trends between subgroups. Furthermore, we apply our methodology on 32 European populations mortality data and discuss the behavior of our model in terms of simulated mortality dispersion. Within the Solvency II directive, we quantify the impact on the longevity risk solvency capital requirement of an insurer for a simplified pension product. Finally, we extend our model by allowing populations to switch from one coherence group to another. We then analyze its incidence on longevity hedges basis risk assessment. JEL Classification: C18, C32, C53, J11.

Forecasting mortality rates is a problem which involves the analysis of high-dimensional time series, especially in multi-populations modeling. Most of usual mortality models propose to decompose the mortality rates into several latent factors to reduce this complexity. These approaches, in particular those using cohort factors, have a good fit, but they are less reliable for forecasting purposes. One of the major challenges is to determine the spatial-temporal dependence structure between mortality rates given a relatively moderate sample size. This paper proposes a large vector autoregressive (VAR) model fitted on the diferences in the log-mortality rates, ensuring the existence of long-run relationships between mortality rate improvements. Our contribution is threefold. First, sparsity, when fitting the model, is ensured by using high-dimensional variable selection techniques without imposing arbitrary constraints on the dependence structure. The main interest is that the structure of the model is directly driven by the data, in contrast to the main mortality forecasting models. Hence, this approach is more versatile and would provide good forecasting performance for any considered population. Additionally, our estimation allows a one-step procedure, as we do not need to estimate hyperparameters. The variance-covariance matrix of residuals is then estimated through a parametric form. Secondly, our approach can be used to detect nonintuitive age dependence in the data, beyond the cohort effect which is captured by our model. Third, our approach is natural to model the several populations in long run perspectives. Finally, in an out-of-sample forecasting study for mortality rates, we obtain rather good performances and more relevant forecasts compared to classical mortality models using the French, US and UK data. We also show that our results enlighten the so-called cohort effect for these populations.

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Long Term Care insurance contracts are complex insurance products covering an individual for which pricing and reserving issues are traditionally addressed by the introduction of multi-state models. This type of model allows one to describe the transitions of each insured through different states that correspond to events determining, under the terms of the contract, the respective commitments of the parties. The description of insurance contracts through multi-state models is the subject of several studies in the actuarial literature (cf. [21, 31] or [14]). To implement this approach to pricing or reserving, actuaries need to establish suitable statistical bases.

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This paper focuses on the construction of incidence laws for total loss of autonomy over the period 2010-2012 using data from national hospitalization databases (PMSI 2008-2013). Our results are decomposed according to two types of dependence: cognitive dependence (or dementia) and physical dependence. Women have a slightly higher risk of entering dementia, whereas the risk for physical dependence is higher for men. The incidence of all-cause dependence is comparable between men and women. The results suggest a slowing down of the incidence beyond the age of 90 and a convergence of men and women at older ages. The implications of these results for extrapolation to older ages are discussed.

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Studying Long Term Care (LTC) insurance requires modeling the lifetime of individuals in presence of both terminal and non-terminal events which are concurrent. Although a nonhomogeneous semi-Markov multi-state model is probably the best candidate for this purpose, most of the current researches assume, maybe abusively, that the Markov assumption is satisfied when fitting the model. In this context, using the Aalen-Johansen estimators for transition probabilities can induce bias, which can be important when the Markov assumption is strongly unstated. Based on some recent studies developing non-Markov estimators in the illness-death model that we can easily extend to a more general acyclic multi-state model, we exhibit three non-parametric estimators of transition probabilities of paying cash-flows, which are of interest when pricing or reserving LTC guarantees in discrete time. As our method directly estimates these quantities instead of transition intensities, it is possible to derive asymptotic results for these probabilities under non-dependent random right-censorship, obtained by re-setting the system with two competing risk blocks. Inclusion of left-truncation is also considered. We conduct simulations to compare the performance of our transition probabilities estimators without the Markov assumption. Finally, we propose a numerical application with LTC insurance data, which is traditionally analyzed with a semi-Markov model.

This paper focuses on the construction of incidence laws for total loss of autonomy over the period 2010-2012 using data from national hospitalization databases (PMSI 2008-2013). Our results are decomposed according to two types of dependence: cognitive dependence (or dementia) and physical dependence. Women have a slightly higher risk of entering dementia, whereas the risk for physical dependence is higher for men. The incidence of all-cause dependence is comparable between men and women. The results suggest a slowing down of the incidence beyond the age of 90 and a convergence of men and women at older ages. The implications of these results for extrapolation to older ages are discussed.

Studying Long Term Care (LTC) insurance requires modeling the lifetime of individuals in presence of both terminal and non-terminal events which are concurrent. Although a nonhomogeneous semi-Markov multi-state model is probably the best candidate for this purpose, most of the current researches assume, maybe abusively, that the Markov assumption is satisfied when fitting the model. In this context, using the Aalen-Johansen estimators for transition probabilities can induce bias, which can be important when the Markov assumption is strongly unstated. Based on some recent studies developing non-Markov estimators in the illness-death model that we can easily extend to a more general acyclic multi-state model, we exhibit three non-parametric estimators of transition probabilities of paying cash-flows, which are of interest when pricing or reserving LTC guarantees in discrete time. As our method directly estimates these quantities instead of transition intensities, it is possible to derive asymptotic results for these probabilities under non-dependent random right-censorship, obtained by re-setting the system with two competing risk blocks. Inclusion of left-truncation is also considered. We conduct simulations to compare the performance of our transition probabilities estimators without the Markov assumption. Finally, we propose a numerical application with LTC insurance data, which is traditionally analyzed with a semi-Markov model.

Forecasting mortality rates is a problem which involves the analysis of high-dimensional time series, especially in multi-populations modeling. Most of usual mortality models propose to decompose the mortality rates into several latent factors to reduce this complexity. These approaches, in particular those using cohort factors, have a good fit, but they are less reliable for forecasting purposes. One of the major challenges is to determine the spatial-temporal dependence structure between mortality rates given a relatively moderate sample size. This paper proposes a large vector autoregressive (VAR) model fitted on the diferences in the log-mortality rates, ensuring the existence of long-run relationships between mortality rate improvements. Our contribution is threefold. First, sparsity, when fitting the model, is ensured by using high-dimensional variable selection techniques without imposing arbitrary constraints on the dependence structure. The main interest is that the structure of the model is directly driven by the data, in contrast to the main mortality forecasting models. Hence, this approach is more versatile and would provide good forecasting performance for any considered population. Additionally, our estimation allows a one-step procedure, as we do not need to estimate hyperparameters. The variance-covariance matrix of residuals is then estimated through a parametric form. Secondly, our approach can be used to detect nonintuitive age dependence in the data, beyond the cohort effect which is captured by our model. Third, our approach is natural to model the several populations in long run perspectives. Finally, in an out-of-sample forecasting study for mortality rates, we obtain rather good performances and more relevant forecasts compared to classical mortality models using the French, US and UK data. We also show that our results enlighten the so-called cohort effect for these populations.

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Data quality is an overarching concern when it comes to building a mortality model or prospective mortality tables. This is even more significant when these procedures are based on a small population, as data may show major random fluctuations due to a lack of information for particular ages. Such situations arise frequently with the entry into force of Solvency II as insurers shall consider their own data sets, limited in size, to build best estimate tables. Since parametric methods are too rough to capture a realistic mortality pattern in two dimensions, the mortality profile is quite often adjusted using exogenous information, such as a table based on a national population. In light of this, the aim of this paper is to discuss the problem of choosing appropriate estimators for two-dimensional mortality rates or death rates in the presence of independent censoring. Indeed, practitioners currently use the Hoem estimator or the Kaplan-Meier estimator split by generation without questioning their relevance and reliability. We propose in this paper a comparative analysis of these estimators and try to give some criteria to choose one approach over another, and give some figures based on a real insurance portfolio and simulated data. Finally, we provided some non-parametric estimators for a direct estimation of death rates both with the cohort and the period approaches.

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The implementation of Solvency II leads actuaries to question the good adequacy between models and data. Therefore, this thesis aims to study several statistical approaches, often unknown to practitioners, allowing the use of multi-state methods to model and manage individual risks in insurance. Chapter 1 presents the general context of this thesis and positions its main contributions. We discuss the basic concepts related to the use of multi-state models in insurance and describe the classical inference techniques adapted to the data encountered, whether Markovian or non-Markovian. Finally, we present how these models can be used for credit risk management. Chapter 2 focuses on the use of non-parametric inference methods for the construction of incidence laws in LTC insurance. Since several input causes are likely to be involved and of interest to actuaries, we focus on a method used for estimating continuous-time Markovian multi-state models. We then compare these estimators to those classically used by survival analysis practitioners. This second approach can have significant biases because it does not allow to correctly understand the possible interaction between the causes. In particular, it includes an independence hypothesis that cannot be tested in the context of competing risks models. Our approach then consists in measuring the error made by practitioners when constructing incidence laws. A numerical application is then considered on the basis of data from an insurer With the implementation of the Solvency II framework, actuaries should examine the good adequacy between models and data.

We use a brand new data-set built from French supervisory reports to investigate the drivers of the participation rates served on euro-denominated life-insurance contracts over the period 1999-2013. Our analysis confirms practitioners’ insight on the alignment with the 10-years French government bond, yet we show that on aggregate, insurers serve less than this target. Our data indicate that financial margins are more strictly targeted than participation. We find evidence that lapses are fairly uncorrelated with participation, suggesting other levers to pilot surrenders. If higher asset returns can imply better yield for policyholders, riskier portfolios do not translate into better participation.

This paper presents a model for the determination and forecast of the number of defaults and credit changes by estimating a reduced-form ordered regression model with a large data set from a credit insurance portfolio. Similarly to banks with their classical credit risk management techniques, credit insurers measure the credit quality of buyers with rating transition matrices depending on the economical environment. Our approach consists in modeling stochastic transition matrices for homogeneous groups of firms depending on macroeconomic risk factors. One of the main features of this business is the close monitoring of covered firms and the insurer’s ability to cancel or reduce guarantees when the risk changes. As our primary goal is a risk management analysis, we try to account for this leeway and study how this helps mitigate risks in case of shocks. This specification is particularly useful as an input for the Own Risk Solvency Assessment (ORSA) since it illustrates the kind of management actions that can be implemented by an insurer when the credit environment is stressed.

The implementation of Solvency II leads actuaries to question the good adequacy between models and data. Therefore, this thesis aims to study several statistical approaches, often unknown to practitioners, allowing the use of multi-state methods to model and manage individual risks in insurance. Chapter 1 presents the general context of this thesis and positions its main contributions. We discuss the basic concepts related to the use of multi-state models in insurance and describe the classical inference techniques adapted to the data encountered, whether Markovian or non-Markovian. Finally, we present how these models can be used for credit risk management. Chapter 2 focuses on the use of non-parametric inference methods for the construction of incidence laws in LTC insurance. Since several input causes are likely to be involved and of interest to actuaries, we focus on a method used for estimating continuous-time Markovian multi-state models. We then compare these estimators to those classically used by survival analysis practitioners. This second approach can have significant biases because it does not allow to correctly understand the possible interaction between the causes. In particular, it includes an independence hypothesis that cannot be tested in the context of competing risks models. Our approach then consists in measuring the error made by practitioners when constructing incidence laws. A numerical application is then considered on the basis of data from a long term care insurer.

The Solvency 2 prudential framework, which will come into force at the beginning of 2016, imposes an instantaneous and highly analytical view of the risks carried in order to determine the minimum capital requirement ("pillar 1") that an insurer must hold. While this approach has the advantage of providing a normative framework as well as a certain exhaustiveness of analysis, in practice it leads to the design of models that are ill-suited to the multi-year projection of the balance sheet and solvency because they are not sufficiently robust. Additional analyses ("pillar 2"), which aim to measure the impact of the strategic plan and management actions on the insurer's solvency, are also required. They are part of a more global approach allowing to provide adequate information on the deformation of the distribution of key balance sheet elements (net asset value, capital requirement, coverage ratio, etc.) over time for the steering of the organization. The challenge is then, on the basis of a global vision of the risks (pricing, provisioning, commercial, hedging and financial risks) to propose prospective models flexible enough to allow these projections as well as an analysis of the impact of the company's management actions. The objective of this book is to describe such approaches (model building and hypothesis justification) and to propose realistic illustrations. Its ambition is to provide a set of aggregated modeling tools for the insurer's balance sheet that are better able to account for dynamic aspects and thus provide a tool for operational management of the business.

This paper proposes to illustrate the implementation of non-parametric estimation methods, introduced in the framework of Markovian multi-state models with censoring, to construct laws of experience applicable in the presence of several concurrent events. This situation arises in practice in long term care insurance when it is necessary to distinguish the incidence laws by pathology. Therefore, rather than applying techniques usually used by practitioners and consisting in observing marginally each cause of entry into dependence, the approach described allows to estimate globally all the entry laws by cause and to correctly apprehend the interdependence between each of them. This work provides a comparison of the results obtained by these two approaches at the level of the provision to be made in order to justify a marginal approach, which is simpler to implement in practice.

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