Astin Bull Research Articles

Mortality prediction using survival energy models with functional data analysis

The survival energy model (SEM), as originally introduced by Shimizu et al. (ASTIN Bull 51(1):191–219, 2020), is designed to characterize human bioenergetics by employing diffusion processes or inverse Gaussian processes. While parametric models have been employed to articulate the SEM, they exhibit inherent sensitivity in their parameters and hyperparameters, which in turn introduces issues of instability in estimation and prediction. In this paper, we demonstrate that the utilization of functional data analysis techniques for nonparametric estimation and prediction of critical functions within the SEM leads to a substantial enhancement in prediction performance.

High-cardinality categorical covariates in network regressions

High-cardinality (nominal) categorical covariates are challenging in regression modeling, because they lead to high-dimensional models. For example, in generalized linear models (GLMs), categorical covariates can be implemented by dummy coding which results in high-dimensional regression parameters for high-cardinality categorical covariates. It is difficult to find the correct structure of interactions in high-cardinality covariates, and such high-dimensional models are prone to over-fitting. Various regularization strategies can be applied to prevent over-fitting. In neural network regressions, a popular way of dealing with categorical covariates is entity embedding, and, typically, over-fitting is taken care of by exploiting early stopping strategies. In case of high-cardinality categorical covariates, this often leads to a very early stopping, resulting in a poor predictive model. Building on Avanzi et al. (ASTIN Bull, 2024), we introduce new versions of random effects entity embedding of categorical covariates. In particular, having a hierarchical structure in the categorical covariates, we propose a recurrent neural network architecture and a Transformer architecture, respectively, for random-effects entity embedding that give us very accurate regression models.

On the impact of outliers in loss reserving

The sensitivity of loss reserving techniques to outliers in the data or deviations from model assumptions is a well known challenge. It has been shown that the popular chain-ladder reserving approach is at significant risk to such aberrant observations in that reserve estimates can be significantly shifted in the presence of even one outlier. As a consequence the chain-ladder reserving technique is non-robust. In this paper we investigate the sensitivity of reserves and mean squared errors of prediction under Mack’s Model (ASTIN Bull 23(2):213–225, 1993). This is done through the derivation of impact functions which are calculated by taking the first derivative of the relevant statistic of interest with respect to an observation. We also provide and discuss the impact functions for quantiles when total reserves are assumed to be lognormally distributed. Additionally, comparisons are made between the impact functions for individual accident year reserves under Mack’s Model and the Bornhuetter–Ferguson methodology. It is shown that the impact of incremental claims on these statistics of interest varies widely throughout a loss triangle and is heavily dependent on other cells in the triangle. Results are illustrated using data from a Belgian non-life insurer.

Estimation error and bootstrapping in the chain-ladder model of Mack

In 2006 there was quite some discussion on how to estimate the conditional estimation error in the chain-ladder (CL) model of Mack. Buchwalder, Buhlmann, Merz and Wuthrich (BBMW) (ASTIN Bull 36(2):521–542, 2006) proposed another estimator than the one derived by Mack (ASTIN Bull 23(2):213–225, 1993). These two estimators are also found in a broader context by new authors in recent papers. In the present paper we examine the theoretical properties of the two estimators and come to the conclusion that the BBMW estimator has some major deficiencies compared with the Mack estimator. It takes much less information of the observed triangle into account, the averaging is done over inappropriate sets and it does not properly fit to the Mack CL-model.

Investing in your own and peers’ risks: the simple analytics of P2P insurance

This paper studies a peer-to-peer (P2P) insurance scheme where participants share the first layer of their respective losses while the higher layer is transferred to a (re-)insurer. The conditional mean risk sharing rule proposed by Denuit and Dhaene (Insur Math Econ 51:265–270, 2012) appears to be a very convenient way to distribute retained losses among participants, as shown by Denuit (ASTIN Bull 49:591–617, 2019). The amount of contributions paid by participants is determined by splitting it into the price of the stop-loss protection limiting the community’s total payout and an appropriate provision for the coverage of the lower layer which is mutualized inside the P2P community. As an application, the paper considers the case of a P2P insurance scheme when losses are modeled by independent compound Poisson sums with integer-valued severities (resulting from discretization). Some extensions are also discussed.

The distortion principle for insurance pricing: properties, identification and robustness

Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia.

A Fuzzy Least-Squares Estimation of a Hybrid Log-Poisson Regression and Its Goodness of Fit for Optimal Loss Reserves in Insurance

In loss reserving, the log-Poisson regression model is a well-known stochastic model underlying the chain-ladder method which is the most used method for reserving purposes. Mack (ASTIN Bull 21(01):93–109, 1991) proved that the log-Poisson model provides the same estimates as the chain-ladder method. So in this article, our objective is to improve the log-Poisson regression model in loss reserving framework. Thereby, we prove the reliability of hybrid models in loss reserving, especially when the data contain fuzziness, for example when the claims are related to body injures (Straub and Swiss in Non-life insurance mathematics, Springer, Berlin, 1988). Thus, we estimate a hybrid generalized linear model (GLM) (log-Poisson) using the fuzzy least-squares procedures (Celmis in Fuzzy Sets Syst 22(3):245–269, 1987a; Math Model 9(9):669–690, 1987b; D’Urso and Gastaldi in Comput Stat Data Anal 34(4): 427–440, 2000; in: Advances in classification and data analysis, Springer, 2001). We develop a new goodness of fit index to compare this new model and the classical log-Poisson regression (Mack 1991). Both the classical log-Poisson model and the hybrid one are performed on a loss reserving data. According to the goodness of fit index and the mean square error prediction, we prove that the new model provide better results than the classical log-Poisson model. This comparison can be extend to any other GLM in loss reserving.

Optimality of Hybrid Continuous and Periodic Barrier Strategies in the Dual Model

Avanzi et al. (ASTIN Bull 46(3): 709–746, 2016) recently studied an optimal dividend problem where dividends are paid both periodically and continuously with different transaction costs. In the Brownian model with Poissonian periodic dividend payment opportunities, they showed that the optimal strategy is either of the pure-continuous, pure-periodic, or hybrid-barrier type. In this paper, we generalize the results of their previous study to the dual (spectrally positive Levy) model. The optimal strategy is again of the hybrid-barrier type and can be concisely expressed using the scale function. These results are confirmed through a sequence of numerical experiments.

Core of the Reinsurance Market with Dependent Risks

Baton and Lemaire (Astin Bull 12:57–71, 1981) proved the nonemptiness of the core of a reinsurance market in which the risks of companies are independent. However, cases involving dependent risks have received increasing concerns in modern actuarial science. In this paper, we investigate the nonemptiness of the core of a reinsurance market where the risks of different companies may be dependent. When the exponential utility function is employed, we find an important property on risk premium and show that the core of the market is always nonempty.

A non-linear mixed model approach for excess of loss benchmark rating

This paper proposes market conform individual benchmark rates for the excess of loss reinsurance of long tail insurance portfolios, that offer market references for the premium rates taking individual contractual conditions into account. The premium rates are expressed in terms of the percentage of the expected premium income of the covered insurance portfolio. We incorporate the specific reinsurance contractual conditions like stabilisation, interest sharing and deposit clauses as well as payment patterns and ’incurred but not (enough) reported’ information. The parameters of the benchmark model are estimated within the framework of non-linear mixed models. This approach allows to correct for different cedent specific conditions in the model, and so we refine the results from Verlaak et al. (Astin Bull 39, 2009) where only one benchmark was proposed for the whole market. The method is applied to the Belgian Motor Third Party Liability XL rates observed from 2001 till 2004.

A credibility approach of the Makeham mortality law

Interest from life insurers to assess their portfolios’ mortality risk has considerably increased. The new regulation and norms, Solvency II, shed light on the need of life tables that best reflect the experience of insured portfolios in order to quantify reliably the underlying mortality risk. In this context and following the work of Buhlmann and Gisler (A course in credibility theory and its applications. Springer, New York, 2005) and Hardy and Panjer (ASTIN Bull 28(2):269–283, 1998), we propose a credibility approach which consists on reviewing, as new observations arrive, the assumption on the mortality curve. Unlike the methodology considered in Hardy and Panjer (1998) that consists on updating the aggregate deaths we have chosen to add an age structure on these deaths. Formally, we use a Makeham graduation model. Such an adjustment allows to add a structure in the mortality pattern which is useful when portfolios are of limited size so as to ensure a good representation over the entire age bands considered. We investigate the divergences in the mortality forecasts generated by the classical credibility approaches of mortality including Hardy and Panjer (1998) and the Poisson-Gamma model on portfolios originating from various French insurance companies.

The impact of covariates on a bonus–malus system: an application of Taylor’s model

Obviously, the design of a bonus–malus system has to take into consideration all rating variables used by the auto insurance carrier. For instance, a single male urban driver is likely to be penalized twice, a priori through explicit surcharges linked to his risk class, and a posteriori through premium increases triggered by the transition rules of the bonus–malus system, creating the possibility of excessive premiums through double-counting for the same reason. Taylor (ASTIN Bull 27:319–327, 1997) developed a Bayesian model to evaluate the impact of covariate rating variables on bonus–malus premium levels, but could not access actual data to implement his model. We present the first real-life application of Taylor’s research, by using a unique database originating from Taiwan and the bonus–malus system in force in this island. Our data combines car and insurance information from the leading insurer in the island with annual mileage readings from a network of repair shops operated by the largest car manufacturer—over a quarter million policy-years. Park et al. (The use of annual mileage as a rating variable. Working paper, 2014), using the same data, used negative binomial regression to prove that mileage is by far the best predictor of accidents. The application of Taylor’s model concludes that the impact of mileage on bonus–malus premium levels is small. Therefore, double-counting should not be considered as a major concern in practice.

On the empirical multilinear copula process for count data

Continuation refers to the operation by which the cumulative distribution function of a discontinuous random vector is made continuous through multilinear interpolation. The copula that results from the application of this technique to the classical empirical copula is either called the multilinear or the checkerboard copula. As shown by Genest and Nešlehová (Astin Bull. 37 (2007) 475–515) and Nešlehová (J. Multivariate Anal. 98 (2007) 544–567), this copula plays a central role in characterizing dependence concepts in discrete random vectors. In this paper, the authors establish the asymptotic behavior of the empirical process associated with the multilinear copula based on $d$-variate count data. This empirical process does not generally converge in law on the space $\mathcal{C}([0,1]^{d})$ of continuous functions on $[0,1]^{d}$, equipped with the uniform norm. However, the authors show that the process converges in $\mathcal{C}(K)$ for any compact $K\subset\mathcal{O}$, where $\mathcal{O}$ is a dense open subset of $[0,1]^{d}$, whose complement is the Cartesian product of the ranges of the marginal distribution functions. This result is sufficient to deduce the weak limit of many functionals of the process, including classical statistics for monotone trend. It also leads to a powerful and consistent test of independence which is applicable even to sparse contingency tables whose dimension is sample size dependent.

Liquidity-adjusted risk measures

Liquidity risk is an important type of risk, especially during times of crises. As observed by Acerbi and Scandolo (Quant Financ 8(7):681–691, 2008), it requires adjustments to classical portfolio valuation and risk measurement. Main drivers are two dimensions of liquidity risk, namely price impact of trades and limited access to financing. The key contribution of the current paper is the construction of a new, cash-invariant liquidity-adjusted risk measure that can naturally be interpreted as a capital requirement. We clarify the difference between our construction and the one of Acerbi and Scandolo (Quant Financ 8(7):681–691, 2008) in the framework of capital requirements using the notion of eligible assets, as introduced by Artzner et al. (Astin Bull 39(1):101–116, 2009). Numerical case studies illustrate how price impact and limited access to financing influence the liquidity-adjusted risk measurements.

A subordinated Markov model for stochastic mortality

In this paper we propose a subordinated Markov model for modeling stochastic mortality. The aging process of a life is assumed to follow a finite-state Markov process with a single absorbing state and the stochasticity of mortality is governed by a subordinating gamma process. We focus on the theoretical development of the model and have shown that the model exhibits many desirable properties of a mortality model and meets many model selection criteria laid out in Cairns et al. (ASTIN Bull 36:79–120, 2006; Scand Actuar J 2:79–113, 2008). The model is flexible and fits either historical mortality data or projected mortality data well. We also explore applications of the proposed model to the valuation of mortality-linked securities. A general valuation framework for valuing mortality-linked products is presented for this model. With a proposed risk loading mechanism, we can make an easy transition from the physical measure to a risk-neutral measure and hence is able to calibrate the model to market information. The phase-type structure of the model allows us to apply the matrix-analytic methods that have been extensively used in ruin theory in actuarial science and queuing theory in operations research (see Asmussen in Applied probability and queues. Wiley, New York, 1987; Asmussen and Albrecher in Ruin probabilities, 2nd edn. World Scientific Publishing, Singapore, 2010; Neuts in Matrix-geometrix solutions in stochastic models. Johns Hopkins University Press, Baltimore, 1981). As a result, many quantities of interest such as the distribution of future survival rates, prediction intervals, the term structure of mortality as well as the value of caps and floors on the survival index can be obtained analytically.

GMWB for life an analysis of lifelong withdrawal guarantees

In this paper, we analyze the latest guarantee feature in the variable annuities market: guaranteed minimum withdrawal benefits for life (GMWB for life) which are also called guaranteed lifelong withdrawal benefits (GLWB). This option gives the client the right to deduct a certain amount annually from the policy’s account value until death—even if a unit-linked account value drops to zero. We show how such products can be analyzed within a general framework presented in Bauer et al. (ASTIN Bull. 38(2):621–651, 2008). We price the embedded guarantee for different product designs and parameters under both, deterministic and optimal client behavior.

Stochastic mortality: experience-based modeling and application issues consistent with Solvency 2

This paper is motivated by the need to have appropriate tools for the valuation of mortality/longevity risks when carrying out an internal assessment of the insurance business, such as a portfolio valuation or a solvency investigation based on internal models. The case of life annuities is in particular addressed, due to the importance of mortality/longevity on the cost of living benefits for the elderly. Our aim is to describe a stochastic mortality model calibrated on the best-estimate life table available to the insurer and to the mortality experienced in the insurer’s portfolio. As we address life annuities, uncertainty of future mortality improvements, i.e. the aggregate mortality/longevity risk, needs in particular to be accounted for. Following Olivieri and Pitacco (ASTIN Bull 39(2):541–563, 2009), we extend some classical results about the modeling of the number of deaths joint to the modeling of parameter uncertainty. We explore the features of two alternative approaches: a Beta-Binomial and a Poisson-Gamma setting; we then focus on the latter only. Thanks to a Bayesian inferential procedure, the parameters, initially assigned referring to the best-estimate life table, are updated according to the mortality experienced in the portfolio. Extending a traditional tool-kit, the framework we define is in particular suitable for practical work. Due to calibration, clearly the scope of the model is within internal assessments. To provide an example of application of the model, we perform a capital assessment. Solvency rules which could be adopted as an alternative to a standard, regulatory requirement are tested. The Solvency 2 requirement for longevity risk is referred to for the design of an internal rule which could get validation for replacing the standard one.

Ruin Analysis of a Threshold Strategy in a Discrete-Time Sparre Andersen Model

In this paper, we extend the methodology of Alfa and Drekic (ASTIN Bull 37:293–317, 2007) to analyze a discrete-time, delayed Sparre Andersen insurance risk model featuring a single threshold level and randomized dividend payments. Using matrix analytic techniques, we construct a set of computational procedures enabling one to calculate probability distributions associated with fundamental ruin-related quantities of interest, namely the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the general model, including the ordinary and stationary Sparre Andersen variants, are examined in several numerical examples.

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161–168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599–616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.

Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness

Mainly due to new capital adequacy standards for banking and insurance, an increased interest exists in the aggregation properties of risk measures like Value-at-Risk (VaR). We show how VaR can change from sub to superadditivity depending on the properties of the underlying model. Mainly, the switch from a finite to an infinite mean model gives a completely different asymptotic behaviour. Our main result proves a conjecture made in Barbe et al. [Barbe, P., Fougères, A.L., Genest, C., 2006. On the tail behavior of sums of dependent risks. ASTIN Bull. 36(2), 361–374].