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This paper considers variable annuity (VA) contracts embedded with guaranteed minimum accumulation benefit (GMAB) riders when policyholder’s proceeds are taxed upon early surrender or maturity. These contracts promise the return of the premium paid by the policyholder, or a higher rolled-up value, at the end of the investment period. A partial differential equation valuation framework which exploits the numerical method of lines is used to determine fair fees that render the policyholder and insurer breakeven. Two taxation regimes are considered: one where capital gains are allowed to offset losses and a second where gains do not offset losses. Most insurance providers highlight the tax-deferred features of VA contracts. We show that the regime under which the insured is taxed significantly impacts prices. If losses are allowed to offset gains then this enhances the market, increasing the policyholder’s willingness to participate in the market compared to the case when losses are not allowed to offset gains. With fair fees from the policyholder’s perspective, we show that the net profit is generally positive for insurance companies offering the contract as a naked option without any hedge. We also show how investment policy, as reflected in the Sharpe ratio, impacts and interacts with policyholder persistency., SCOPUS: ar.j, info:eu-repo/semantics/published

We explore a stochastic model that enables capturing external influences in two specific ways. The model allows for the expression of uncertainty in the parametrisation of the stochastic dynamics and incorporates patterns to account for different behaviours across various times or regimes. To establish our framework, we initially construct a model with random parameters, where the switching between regimes can be dictated either by random variables or deterministically. Such a model is highly interpretable. We further ensure mathematical consistency by demonstrating that the framework can be elegantly expressed through local volatility models taking the form of standard jump diffusions. Additionally, we consider a Markov-modulated approach for the switching between regimes characterised by random parameters. For all considered models, we derive characteristic functions, providing a versatile tool with wide-ranging applications. In a numerical experiment, we apply the framework to the financial problem of option pricing. The impact of parameter uncertainty is analysed in a two-regime model, where the asset process switches between periods of high and low volatility imbued with high and low uncertainty, respectively., SCOPUS: ar.j, info:eu-repo/semantics/published

The study of mortality is an ever-active field of research, and new methods or combinations of methods are constantly being developed. In the actuarial domain, the study of phenomena disrupting mortality and leading to excess mortality, as in the case of COVID-19, is of great interest. Therefore, it is relevant to investigate the extent to which an epidemiological model can be integrated into an actuarial approach in the context of mortality. The aim of this project is to establish a method for the study of excess mortality due to an epidemic and to quantify these effects in the context of the insurance world to anticipate certain possible financial instabilities. We consider a case study caused by SARS-CoV-2 in Belgium during the year 2020. We propose an approach that develops an epidemiological model simulating excess mortality, and we incorporate this model into a classical approach to pricing life insurance products., SCOPUS: ar.j, info:eu-repo/semantics/published

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In this article, we focus on death-linked contingent claims (GMDBs) paying a random financial return at a random time of death in the general case where financial returns follow a regime switching model with two-sided phase-type jumps. We approximate the distribution of the remaining lifetime by either a series of Erlang distributions or a Laguerre series expansion, whose capability to fit the tail of the observed mortality data turns out to be much better than the commonly used series of exponential distributions. More precisely, we develop a Laurent series expansion of the discounted Laplace transform of the subordinated process at an Erlang distributed time, which leads to explicit formulae for European-type GMDB as well as related risk measures such as the Value-at-Risk (VaR) and the Conditional-Tail-Expectation (CTE). We further concentrate upon path-dependent GMDBs with lookback features like dynamic fund protection or dynamic withdrawal benefits, by relying on a Sylvester equation approach. The advantage of our approaches is that our results are of semi-closed form, avoiding numerical Fourier inversion or Monte-Carlo simulation, leading to fast evaluation. This is necessary in risk-management, in particularly for nested simulation in the framework of Solvency II. Several numerical experiments are included. Our results have implications beyond life-insurance and GMDBs, namely in all situations where randomization or Erlangization replaces known quantities, like, for example, model parameters, by random variables. In Finance, it is for example well-known that a random maturity time leads to much more convenient valuation formulas that well approximate its non-random counterpart., SCOPUS: ar.j, info:eu-repo/semantics/published

Exposure simulations are fundamental to many xVA calculations and are a nested expectation problem where repeated portfolio valuations create a significant computational expense. Sensitivity calculations which require shocked and unshocked valuations in bump-and-revalue schemes exacerbate the computational load. A known reduction of the portfolio valuation cost is understood to be found in polynomial approximations, which we apply in this article to interest rate sensitivities of expected exposures. We consider a method based on the approximation of the shocked and unshocked valuation functions, as well as a novel approach in which the difference between these functions is approximated. Convergence results are shown, and we study the choice of interpolation nodes. Numerical experiments with interest rate derivatives are conducted to demonstrate the high accuracy and remarkable computational cost reduction. We further illustrate how the method can be extended to more general xVA models using the example of CVA with wrong-way risk., SCOPUS: ar.j, info:eu-repo/semantics/published

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This paper proposes a new boosting machine based on forward stagewise additive modeling with cost-complexity pruned trees. In the Tweedie case, it deals directly with observed responses, not gradients of the loss function. Trees included in the score progressively reduce to the root-node one, in an adaptive way. The proposed Adaptive Boosting Tree (ABT) machine thus automatically stops at that time, avoiding to resort to the time-consuming cross validation approach. Case studies performed on motor third-party liability insurance claim data demonstrate the performances of the proposed ABT machine for ratemaking, in comparison with regular gradient boosting trees., SCOPUS: ar.j, info:eu-repo/semantics/published

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We investigate the effect of part-time and full-time work on health using a Markov framework and generalized linear models to smooth the resulting crude rates. The Chapman-Kolmogorov equations are used for a general solution. We apply this model to assess a partial early retirement incentive in the Netherlands, known as ‘the generation pact’. The smoothed rates imply that working part time does not necessarily mean a better health for the elderly. In fact, men are healthier when working full time, while women fall sick more often when working full time but recover more often as well. However, when comparing the future rates of a person currently aged 50 working full time and using the generation pact, both the recovery and the morbidity rates drop when starting the generation pact. A stylized assessment of the costs associated yields that costs drop by half when using the generation pact., SCOPUS: ar.j, info:eu-repo/semantics/published

We study the impact of housing wealth and individual preferences on demand for annuities and long-term care insurance (LTCI). We build a multi-state lifecycle model that includes longevity risk and health shocks. The preference is represented by a recursive utility function that separates risk aversion and elasticity of intertemporal substitution (EIS). When health shocks are considered, a higher level of risk aversion lowers the annuity demand, while a lower level of the EIS has the opposite effect. The impact diminishes with a weaker bequest motive, more liquid wealth, or access to LTCI, all of which increase the demand for annuities. The presence of home equity can enhance annuity demand, but the enhancement is marginal when LTCI is available. The presence of home equity has a crowding-out effect on LTCI demand, and the effect is strengthened by a lack of bequest motives or a lower degree of risk aversion. The cash poor but asset rich may demand more LTCI coverage than their renter counterparts to preserve bequests. When both life annuities and LTCI are available, we find that the product demand is robust to changes in risk aversion and the EIS, providing insights into product designs that bundle annuities and LTCI., SCOPUS: ar.j, info:eu-repo/semantics/published

The collateral choice option allows a collateral-posting party the opportunity to change the type of security in which the collateral is deposited. Due to nonzero collateral basis spreads, this optionality significantly impacts asset valuation. Because of the complexity of valuing the option, many practitioners resort to deterministic assumptions on the collateral rates. In this paper, we focus on a valuation model of the collateral choice option based on stochastic dynamics. Intrinsic differences in the resulting collateral choice option valuation and its implications for collateral management are presented. We obtain sensitivities of the collateral choice option price under both the deterministic and the stochastic model, and we show that the stochastic model attributes risks to all involved collateral currencies. Besides an inability to capture volatility effects, the deterministic model exhibits a digital structure in which only the cheapest-to-deliver currency influences the valuation at a given time. We further consider hedging an asset with the collateral choice option by a portfolio of domestic and foreign zero-coupon bonds that do not carry the collateral choice option. We propose static hedging strategies based on the crossing times of the deterministic model and based on variance-minimization under the stochastic model. We show how the weights of this model can be explicitly determined with the semi-analytical common factor approach and we show in numerical experiments that this strategy offers good hedging performance under minimized variance., SCOPUS: ar.j, info:eu-repo/semantics/published

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We investigate the importance of alternative motives for saving and spending patterns after retirement in the Netherlands and Australia. Using an online experimental survey, we elicit the impact on advised spending patterns and underlying saving motives of alternative retirement drawdown designs, ranging from complete flexibility in Australia to full annuitization in the Netherlands, and of major life events such as becoming frail or losing a spouse. We find that important motives for spending and saving after retirement in both countries are the desire to remain financially independent and to ensure that it is possible to enjoy life now as well as later. However, consistent with differences in real world pension settings, life span risk is more important and liquidity less important in Australia than in the Netherlands. With the exception of inter vivos transfers to a surviving spouse, the bequest motive is not important in either country. Our results suggest that an important reason for the widespread behaviour of retirees to hold on to their wealth might be the desire to hold precautionary savings for health and long-term care expenditures. We also find that individuals revise their saving motives in anticipation of major life events but are less responsive to variation in ‘experimental’ retirement drawdown arrangements, remaining aligned to the prevailing institutional arrangements in their country., SCOPUS: ar.j, info:eu-repo/semantics/published

Thanks to its outstanding performances, boosting has rapidly gained wide acceptance among actuaries. To speed up calculations, boosting is often applied to gradients of the loss function, not to responses (hence the name gradient boosting). When the model is trained by minimizing Poisson deviance, this amounts to apply the least-squares principle to raw residuals. This exposes gradient boosting to the same problems that lead to replace least-squares with Poisson Generalized Linear Models (GLM) to analyze low counts (typically, the number of reported claims at policy level in personal lines). This paper shows that boosting can be conducted directly on the response under Tweedie loss function and log-link, by adapting the weights at each step. Numerical illustrations demonstrate similar or better performances compared to gradient boosting when trees are used as weak learners, with a higher level of transparency since responses are used instead of gradients., SCOPUS: ar.j, info:eu-repo/semantics/published

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In this paper, we study a generalized version of the expected time in the red and the expected area in red introduced in Loisel (2005). We derive some expressions for this risk measure both in light tailed and heavy tailed cases for general risk processes. Then, some further expressions are obtained for Lévy processes together with an upper bound in the light-tailed case and an additional result for large initial capitals in the subexponential case., info:eu-repo/semantics/published

Telematicsdevices installed in insured vehicles provide actuaries with new risk factors, such as the time of the day, average speeds, and other driving habits. This paper extends the multivariate mixed model describing the joint dynamics of telematics data and claim frequencies proposed by Denuit et al. (2019a) by allowing for signals with various formats, not necessarily integer-valued, and by replacing the estimation procedure with the Expected Conditional Maximization algorithm. A numerical study performed on a database related to Pay-How-You-Drive, or PHYD motor insurance illustrates the relevance of the proposed approach for practice., SCOPUS: ar.j, info:eu-repo/semantics/published

In this note, we derive upper bounds on Kendall’s tau and Spearman’s rho for multivariate zero-inflated continuous variables often encountered in insurance. A lower bound for Spearman’s rho is also established in the bivariate case. These bounds are easy to compute and can be estimated from a data set of zero-inflated random vectors as illustrated in this note with a motor insurance portfolio., SCOPUS: ar.j, info:eu-repo/semantics/published

Spearman’s rho is one of the most popular dependence measures used in practice to describe the association between two random variables. However, in case of at least one random variable being discrete, Spearman’s correlations are often bounded and restricted to a sub-interval of [- 1 ,1]. Hence, small positive values of Spearman’s rho may actually support a strong positive dependence when getting close to its highest attainable value. Similarly, slight negative values of Spearman’s rho can actually mean a strong negative dependence. In this paper, we derive the best-possible upper and lower bounds for Spearman’s rho when at least one random variable is discrete. We illustrate the obtained lower and upper bounds in some situations of practical relevance., SCOPUS: ar.j, info:eu-repo/semantics/published

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The global reforms to public pension schemes over the last thirty years have progressively reduced individuals’ post-retirement social security income. In order to compensate for this, individuals join pension funds and individual plans to increase their wealth at retirement. These types of fully funded plans generally give individuals the opportunity to withdraw the capital accumulated into their scheme or to convert it into an annuity. In this paper, we analyse individuals’ post-retirement choices to allocate the wealth at retirement between consumption, risk-free investments and a life annuity. We develop a discrete time optimisation model, in a deterministic framework, with a constant relative risk aversion (CRRA) utility function. We study the effect of a bequest motive and the annuity rate used by the insurer on the optimal choice. Several numerical applications are presented to illustrate the optimal annuitisation decision results and the optimal consumption paths., SCOPUS: ar.j, info:eu-repo/semantics/published

For an insurance company, one of the main objectives in risk management is to assess the risk associated with an insurance portfolio. In insurance mathematics, ruin theory is a traditional tool for performing such an evaluation. In the context of this theory, the time evolution of the reserve of an insurance company is modelled by a long-term risk process. In order to assess the riskiness of such a process, an appropriate risk measure is needed. However, there is no single number that captures all the characteristics of the risk associated with the process. Hence, there is a need to develop multiple risk measures, and to select an appropriate risk measure depending on the context in mind. Another issue related to many risk management situations is the need to properly measure the dependency between two random variables. Standard dependence measures include Spearman rank correlation coefficient and Kendall rank correlation coefficient. In this context, the case where at least one of the random variables is discrete is qualitatively different from the continuous case. This situation occurs, for example, when one wishes to assess the dependence between the annual number of claims for policyholders who experienced at least one claim and the average claim severity. In this case, the results of dependency measures need to be interpreted differently since small values of these measures may actually support strong dependence. This thesis consists of two parts. The first part is devoted to ruin theory. Although ruin theory is the subject of abundant literature, questions remain unanswered. We aim to give answers to questions related to computation, properties and applications of ruin-based risk measures within both continuous and discrete-time models. In the second part, we derive bounds on Spearman's rho in case of at least one random variable is discrete. A motivation for this is given in terms of applications of practical relevance., Doctorat en Sciences, info:eu-repo/semantics/nonPublished

Many OECD countries have addressed the issue of increased longevity by mainly increasing the retirement age. However, this kind of reforms may lead to substantial transfers from those with shorter lifespans to those that will live longer than the average, as they do not necessarily take into account the socio-economic differences in mortality. The contribution of our paper is therefore twofold. Firstly, we illustrate how both a Defined Benefit and a Notional Defined Contribution pay-as-you-go scheme can put the lower social economic classes at a disadvantage, when compared to the actuarially fair pensions. In contrast to that, higher classes experience a gain. This is due to the fact that mortality rates per socio-economic class are not considered by either scheme. Consequently, we propose a model that determines the parameters for each scheme and class which would render the pensions fairer even when no socio-economic mortality differences are considered., SCOPUS: ar.j, info:eu-repo/semantics/published

Recent pension reforms in Europe have implemented a link between retirement age and life expectancy. The accurate forecast of life tables and life expectancy is hence paramount for governmental policy and financial institutions. We developed a multi-population mortality model which includes a cause-specific environment using Archimedean copulae to model dependence between various groups of causes of death. For this, Dutch data on cause-of-death mortality and cause-specific mortality data from 14 comparable European countries were used. We find that the inclusion of a common factor to a cause-specific mortality context increases the robustness of the forecast and we underline that cause-specific mortality forecasts foresee a more pessimistic mortality future than general mortality models. Overall, we find that this non-trivial extension is robust to the copula specification for commonly chosen dependence parameters., SCOPUS: ar.j, info:eu-repo/semantics/published

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Modern data science tools are effective to produce predictions that strongly correlate with responses. Model comparison can therefore be based on the strength of dependence between responses and their predictions. Positive expectation dependence turns out to be attractive in that respect. The present paper proposes an effective testing procedure for this dependence concept and applies it to compare two models. A simulation study is performed to evaluate the performances of the proposed testing procedure. Empirical illustrations using insurance loss data demonstrate the relevance of the approach for model selection in supervised learning. The most positively expectation dependent predictor can then be autocalibrated to obtain its balance-corrected version that appears to be optimal with respect to Bregman, or forecast dominance., SCOPUS: ar.j, info:eu-repo/semantics/published

This paper proposes a multistate model with a Semi-Markov dependence structure describing the different stages in the settlement process of individual claims in general insurance. Every trajectory, from reporting to closure is combined with a modeling of individual link ratios to obtain the ultimate cost of each claim. Analytical expressions are derived for the moments of ultimate amounts whereas quantile risk measures can be obtained by simulation. In the 1-year view, the proposed matrix calculations avoid the simulation-within-simulation issue and offer a tractable evaluation method. A case study illustrates the relevance of the proposed approach., SCOPUS: ar.j, info:eu-repo/semantics/published

Actuarial ratemaking is usually performed at product and guarantee level, meaning that each product and guarantee is considered in isolation. Moreover, independence between policyholders is generally assumed. In this paper, we propose a multivariate Poisson mixture, with random effects correlated using a hierarchical structure, to accommodate for the dependence that may exist between unobserved risk factors across Home and Motor insurance and between policyholders from the same household. The hierarchical structure accounts for the fact that Home insurance covers the whole household, whereas Motor insurance policies are subscribed by specific policyholders within the household. The model allows to periodically correct the a priori expected claim frequencies using the reported number of claims in any of the considered products. Applications show that the impact of the number of claims reported in Motor insurance on the number of claims expected in Home insurance is larger than the other way around. Moreover, an out-of-sample analysis validates an improved predictive power. Also, the model allows to identify more rapidly the riskiest households., SCOPUS: ar.j, info:eu-repo/semantics/published

Boosting techniques and neural networks are particularly effective machine learning methods for insurance pricing. Often in practice, the sum of fitted values can depart from the observed totals to a large extent. The possible lack of balance when models are trained by minimizing deviance outside the familiar GLM with canonical link setting has been documented in Wüthrich (2019, 2020, 2021). The present paper aims to further study this phenomenon when learning proceeds by minimizing Tweedie deviance. It is shown that minimizing deviance involves a trade-off between the integral of weighted differences of lower partial moments and the bias measured on a specific scale. Hence, there is no guarantee that the sum of fitted values stays close to observed totals if the latter bias term is dominated by the former one entering deviance. Autocalibration is then proposed as a remedy. This new method to correct for bias adds an extra local GLM step to the analysis with the output of the first step as only predictor. Theoretically, it is shown that it implements the autocalibration concept in pure premium calculation and ensures that balance also holds on a local scale, not only at portfolio level as with existing bias-correction techniques., SCOPUS: ar.j, info:eu-repo/semantics/published

In this paper, we propose and study a risk model with two types of claims in which the insurer may invest into a prevention plan which decreases the intensity of large claims without impacting the small claims. We identify a necessary and sufficient condition for insurers to use prevention if there is no surplus. If, in addition, the severity of large claims dominates that of small claims by the harmonic mean residual life (HMRL) order, insurers invest more in prevention in the presence of a surplus. Finally, we characterize the asymptotic optimal prevention strategy when the initial surplus tends to infinity in the two main cases where both claim types are light-tailed and where one of them is light-tailed and the other one is heavy-tailed., SCOPUS: ar.j, info:eu-repo/semantics/published

We fielded an online survey in the Netherlands and Australia to explore the influence of an implied endorsement nudge, conveyed by a government-regulated drawdown from pension wealth, on spending patterns in retirement. The implied endorsement nudge was effective. It influenced the preferred retirement spending patterns of around 30 percent of survey participants, particularly those with fewer financial resources and low pension capability. Australian participants were more likely to follow the nudge where it was framed as implicit government advice, while the Dutch were more likely to respond to a suggestion that the nudge was a recommendation from peers. Our results provide support for a regulated drawdown rule as part of a strategy to guide spending patterns in retirement., SCOPUS: ar.j, info:eu-repo/semantics/published

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