ارزیابی حق بیمه بر اساس شدت برای محصولات بیمه بیکاری

We present a flexible premium determination method for insurance products, in particular, for unemployment insurance products. The price is determined with the real-world pricing formula and under the assumption that the employment–unemployment progress of an insured person follows an FF-doubly stochastic Markov chain. The stochastic intensity processes are estimated for the German labor market, using Cox’s proportional hazards model with time-dependent covariates on a sample of integrated labor market biographies. The estimation procedure is based on a counting process framework with stochastic compensators, which we show to be naturally connected to the class of FF-doubly stochastic Markov chains. Based on the statistical analysis, the prices are computed using Monte Carlo simulations.

The debt crisis in the Euro zone with Cyprus, Greece, Ireland, Portugal and Spain having applied for financial assistance of the European Stability Mechanism (ESM) or the European Financial Stability Facility (EFSF), respectively, constitutes an immense challenge for Europe and the rest of the world. One of the core structural problems of the affected countries is a troubled labor market with unemployment rates around 25%. Beside the financial drawbacks for the unemployed people, these high levels of unemployment rates burden the public unemployment insurance systems as well as the private insurance sector, which has started to introduce special products against unemployment.The demand on modern, well elaborated and tested mathematical models for premium determination and risk mitigation for these kind of insurance products, but also other insurance products in general, is high and an ongoing field in actuarial research; see e.g. Bacinello et al. (2009), Biagini and Schreiber (2013), Biagini and Widenmann, 2012 and Biagini and Widenmann, 2013 or Møller (1998). In times, where all sectors of an economy are closely connected, one main issue in this context is to consider the insurance market as part of a hybrid market, consisting among others of stocks, equities, commodities, fixed income and insurance products, all influenced by micro- and macro-economic factors. Hence, the correlations and dependencies of models for (unemployment) insurance products and other sources of randomness in hybrid markets need to be investigated thoroughly. In this context, the present paper aims to introduce a flexible premium determination framework for unemployment insurance products, particularly for so-called payment protection insurances (PPIs): given some underlying payment obligation of the insured person, e.g. a loan, the insurance company will take over the installments during an unemployment period. In this way, the financial challenges for the insured persons and the credit default risk for the creditor are both reduced at the same time. Our pricing model is generally based on a two-state switching process with state space {1,2}{1,2} (View the MathML source1=̂ employed, View the MathML source2=̂ unemployed), generated by two stochastic intensity processes. Generally speaking, the intensity of a transition from employment to unemployment at time tt characterizes the conditional instantaneous probability at time tt for an employee to become unemployed, given the currently available information. The intensity of a transition from unemployment to employment is interpreted analogously. In regard to the aforementioned dependencies of the model in hybrid markets, we assume the intensity processes to be driven by individual-related as well as macro- and micro-economic covariate processes; see Eq. (1) in Section 2. An adequate class for the two-state switching process, which allows for stochastic intensity processes, is the class of FF-doubly stochastic Markov chains, introduced by Jakubowski and Niewęgłowski (2010). It extends the notion of classical Markov chains. As a general pricing rule, we adopt the real-world pricing formula; see Platen and Heath (2007). A first model, using FF-doubly stochastic Markov chains and the benchmark approach for pricing PPIs, is proposed by Biagini and Widenmann (2012). However, the intensities there are assumed to be random, but not varying over time. In the present paper we extend this approach in order to address the more realistic case of stochastic intensity processes. However, in this case it is generally not possible to obtain an analytical expression for the insurance premium which will be here computed by using Monte Carlo simulations. In order to calibrate the price for the unemployment insurance products to real data, we estimate the intensity processes using Cox’s proportional hazards model; see Cox, 1972 and Cox, 1975 and Andersen et al. (1993). Our data set is provided by the “Institut für Arbeitsmarkt- und Berufsforschung” (IAB) and contains a sample of integrated labor market biographies, including the duration of employment and unemployment periods between 1975 and 2008 of more than 1.5 million German individuals as well as several useful socio-demographic covariates, such as age, nationality, educational level, regional details, etc. In order to reflect additional dependencies of the intensity processes of macro-economic factors, we also incorporate further covariates such as time series for MSCI-world returns and German unemployment rates. An advantage of using Cox’s proportional hazards model is the availability of adequate implementations; see for example the R-packages corresponding to de Wreede et al. (2010), Jackson (2011) or Aalen et al. (2004). A Bayesian approach is proposed by Kneib and Hennerfeind (2008) and implemented in the software BayesX. Here, a major difficulty is to adequately operationalize the data set with regard to the software packages. Technically, the implemented estimators are based on the theory on multivariate counting processes and their compensators, where the counting process is assumed to count subsequent jumps of the same kind of an underlying multi-state switching process. Given the martingale property of the compensated counting process, estimators for the compensators can be derived. In the present paper we extend the existing theory for (classical) Markov chains1 and show that the class of FF-doubly stochastic Markov chains is the natural candidate for the underlying multi-state switching processes corresponding to the multivariate counting processes with stochastic compensators of the form given by Cox’s proportional hazards model. This relation can easily be extended to general multiplicative hazard models as given in Andersen et al. (1993). In order to test the obtained estimation results, we apply conventional goodness-of-fit methods. The results generally show adequate performance of the estimated model parameters. Moreover, we introduce a further, non-standard method for testing the applicability of the obtained parameters with respect to prediction, by comparing actual and simulated jump times for selected paths from the data set. The results here show good predictive power, which implicates robustness of the Monte Carlo simulations to compute the premiums. A sensitivity analysis of the insurance premiums also confirms these findings. Our approach, therefore, represents a flexible premium determination tool for unemployment insurance products since it takes into account various risk factors. Moreover, it can easily be adapted to model and estimate stochastic intensities and dependence structures in many other different applications of financial and actuarial practice as well as from other fields. The rest of the paper is structured as follows. In Section 2 we introduce the form of the considered unemployment insurance products and derive the pricing formula. The connection between the multivariate counting processes of Cox’s proportional hazards model and the class of FF-doubly stochastic Markov chains is established in Section 3. Additionally, the data set is described and estimation results for the intensity processes are presented. The Monte Carlo simulation procedure is explained in detail in Section 4 followed by a sensitivity analysis of the insurance premiums.

Here we establish an innovative and flexible premium determination framework for unemployment insurance products which incorporates both static and stochastic covariate processes in an effective way. This allows us to adjust the insurance premiums according to individual characteristics of the insured person as well as to micro- and macro-economic risk factors. Within the framework of Cox’s proportional hazards model the intensities are estimated on a data set provided by the IAB. We show that the class of FF-doubly stochastic Markov chains provides appropriate multi-state switching processes, underlying the counting process framework of the estimation procedure. Several goodness-of-fit methods indicate the adequateness of the model assumptions and the estimated results. In particular, with a new goodness-of-fit method we show the good predictive power of the results for the jump times of the underlying multi-state switching process. Further improvements of the model could be to consider other (time-dependent) covariates and a direct parametrization of the likelihood function. We then model dependencies directly in the structure of the stochastic intensity processes and calculate the fair insurance premiums with Monte Carlo arguments. This evaluation method, based on the real-world pricing formula and on stochastic intensities, can easily be adapted to other financial and insurance products traded on the market. Furthermore, our approach for estimating stochastic intensities and the dependence structure of jump times can be used for a wide spectrum of applications. Therefore, we are confident to have provided a flexible and widely employable framework which can be applied beyond the scope of this paper. Role of the funding source. The opinions expressed in this article are those of the authors and do not necessarily reflect the views of Swiss Life Insurance Solutions AG. Moreover, presented simulation concepts are not necessarily used by Swiss Life Insurance Solutions AG or any affiliates. Swiss Life Insurance Solutions AG provides the general research framework of investigating unemployment insurance products but is neither incorporated in the collection, analysis and interpretation of data, nor in the writing of the working paper. Moreover, submission for publication of the working paper is gratefully accepted.