شبیه سازی عددی قطعی موثر مدل های مدیریت دارایی و بدهی تصادفی در بیمه عمر

New regulations, stronger competitions and more volatile capital markets have increased the demand for stochastic asset-liability management (ALM) models for insurance companies in recent years. The numerical simulation of such models is usually performed by Monte Carlo methods which suffer from a slow and erratic convergence, though. As alternatives to Monte Carlo simulation, we propose and investigate in this article the use of deterministic integration schemes, such as quasi-Monte Carlo and sparse grid quadrature methods. Numerical experiments with different ALM models for portfolios of participating life insurance products demonstrate that these deterministic methods often converge faster, are less erratic and produce more accurate results than Monte Carlo simulation even for small sample sizes and complex models if the methods are combined with adaptivity and dimension reduction techniques. In addition, we show by an analysis of variance (ANOVA) that ALM problems are often of very low effective dimension which provides a theoretical explanation for the success of the deterministic quadrature methods.

Much effort has been spent on the development of stochastic asset-liability management (ALM) models for life insurance companies in the last years, see, e.g., Bacinello (2001), Bacinello (2003), Ballotta et al. (2006), Briys and Varenne (1997), De Felice and Moriconi (2005), Gerstner et al. (2008), Grosen and Jorgensen (2000), Miltersen and Persson (2003) and Moller and Steffensen (2007) and the references therein. Such models are becoming more and more important due to new accountancy standards, greater globalisation, stronger competition, more volatile capital markets and long periods of low interest rates. They are employed to simulate the medium and long-term development of all assets and liabilities. This way, the exposure of the insurance company to financial, mortality and surrender risks can be analysed. The results are used to support management decisions regarding, e.g., the asset allocation, the bonus declaration or the development of more profitable and competitive insurance products. The models are also applied to obtain market-based, fair value accountancy standards as required by Solvency II and the International Financial Reporting Standard, see, e.g., Jorgensen (2004) and the references therein. Due to the wide range of path-dependencies, guarantees and option-like features of insurance products, closed-form representations of statistical target figures, like expected values or variances, which in turn yield embedded values or risk-return profiles of the company, are in general not available. Therefore, insurance companies have to resort to numerical methods for the simulation of ALM models. In practice, usually Monte Carlo methods are used which are based on the averaging of a large number of simulated scenarios. These methods are robust and easy to implement but suffer from an erratic convergence and relatively low convergence rates. In order to improve an initial approximation by one more digit precision, Monte Carlo methods require, on average, the simulation of a hundred times as many scenarios as have been used for the initial approximation. Since the simulation of each scenario requires to run over all relevant points in time and all policies in the portfolio of the company, often very long computing times are needed to obtain approximations of satisfactory accuracy. As a consequence, a frequent and comprehensive risk management, extensive sensitivity investigations or the optimisation of product parameters and management rules are often not possible. In this article, we focus on approaches to speed up the simulation of ALM models. To this end, we rewrite the ALM simulation problem as a multivariate integration problem and apply quasi-Monte Carlo (see, e.g., Glasserman (2003) and Niederreiter (1992)) and sparse grid methods (see, e.g., Bungartz and Griebel (2004), Gerstner and Griebel (1998), Gerstner and Griebel (2003), Griebel (2006) and Smolyak (1963)) in combination with adaptivity (Gerstner and Griebel, 2003) and dimension reduction techniques (Acworth et al., 1998 and Moskowitz and Calfisch, 1996) for its numerical computation. Quasi-Monte Carlo and sparse grid methods are alternatives to Monte Carlo simulation, which are also based on a (weighted) average of different scenarios, but which use deterministic sample points instead of random ones. We refer to these methods as deterministic simulation approaches. They can attain faster rates of convergence than Monte Carlo, can exploit the smoothness of the integrand and have deterministic upper bounds on their error. In this way, they have the potential to significantly reduce the number of required scenarios and computing times. For many problems from mathematical finance, the efficiency of these deterministic quadrature methods has already been studied and numerical experiments have showed that they are indeed often faster and more accurate than Monte Carlo simulation. Examples are the pricing of bonds (Albrecher et al., 2004 and Ninomiya and Tezuka, 1996), options (Acworth et al., 1998 and Gerstner and Holtz, 2008) and mortgage backed securities (Caflisch et al., 1997, Gerstner and Griebel, 1998, Gerstner and Griebel, 2003 and Paskov and Traub, 1995). To our knowledge, it is so far not known if deterministic methods can also be successfully applied to ALM simulations. Such simulations are usually much more complex and time-consuming than the examples mentioned above since many different model equations for the capital market, for the management of the insurance company and for the policyholder behaviour are involved. While there are many publications on modeling aspects, only very few focus on the numerical issues which arise in the simulation of ALM models in life insurance. Monte Carlo methods with antithetic variates, Faure low-discrepancy sequences1 and different time discretisation methods were studied in Corsaro et al. (2006). Finite difference methods were considered in Bauer et al. (2007), Jensen et al. (2001) and Tanskanen and Lukkarinen (2003). In order to assess the efficiency of deterministic simulation methods we here use the general ALM model framework developed in Gerstner et al. (2008) as a benchmark. This model includes as special cases many other models (e.g., Bacinello (2001), Bacinello (2003) and Grosen and Jorgensen (2000)) which have been proposed in the literature for the ALM of participating life insurance products. We show in numerical experiments based on different parameter setups how the accuracy of Monte Carlo, quasi-Monte Carlo and sparse grid integration depends on mathematical properties such as the variance and the smoothness of the corresponding integration problem. Our numerical results demonstrate that quasi-Monte Carlo methods based on Sobol sequences and dimension-adaptive sparse grids based on Gauss–Hermite quadrature formulas are often faster and more accurate than Monte Carlo simulation even for complex ALM models with many time steps. It is known that the performance of these deterministic numerical methods is closely related to the effective dimension of the underlying integration problem, see, e.g., Caflisch et al. (1997), Griebel (2006) and Wang and Fang (2003). To study this property with respect to ALM models, we determine the effective dimension in the truncation and in the superposition sense by means of the analysis of variance (ANOVA) decomposition (Caflisch et al., 1997) of the integrand. The results indicate that ALM problems are often of very low effective dimension, which provides a theoretical explanation for the efficiency of the deterministic quadrature methods. In this context we also show that path generating methods for the capital market scenarios have a significant impact on the effective dimension and on the performance of the numerical methods. Thereby, we compare the random walk, the Brownian bridge (Moskowitz and Calfisch, 1996) and two principal component constructions (Glasserman, 2003). The remainder of this article is as follows: First, in Section 2, we describe an abstract representation of stochastic ALM models and introduce our benchmark model. Section 3 then deals with the numerical simulation of ALM models by Monte Carlo and deterministic integration methods. In Section 4, we then present numerical results which illustrate the efficiency of the different numerical approaches as well as the numerical effects which arise due to different parameter setups and path constructions. The article finally closes in Section 5 with concluding remarks.

In this article, we showed that deterministic integration schemes, such as quasi-Monte Carlo and sparse grid methods can considerably accelerate the numerical simulation of stochastic ALM models in life insurance. As a benchmark model for the different numerical approaches, we used a general model framework for the asset-liability management of portfolios of participating life insurance products. The model incorporates fairly general product characteristics, a surrender option, a reserve-dependent bonus declaration, a dynamic capital allocation and a two-factor stochastic capital market model. Numerical experiments with several different parameter setups demonstrate that Sobol quasi-Monte Carlo and dimension-adaptive sparse grid methods often significantly outperform Monte Carlo simulation. The quasi-Monte Carlo method in combination with the Brownian bridge path construction converges nearly independently of the dimension, converges faster and less erratically than Monte Carlo and produces more precise results even for high dimensions and complex models. We furthermore showed that sparse grid integration is not suited as a black box method for the numerical simulation of ALM models in life insurance as its performance is comparably sensitive to different model parameters and components which affect the smoothness of the integrand. However, for ALM models which are sufficiently smooth or which can be transformed such that they are sufficiently smooth, the sparse grid method constitutes an extremely efficient simulation approach and is superior to Monte Carlo and quasi-Monte Carlo methods by several orders of magnitude. To explain the efficiency of the deterministic methods we computed the effective dimension of various ALM problems with and without dimension reduction techniques. We observed that low effective dimension is a common feature of our ALM model problems in the truncation as well as in the superposition sense. We believe that this property is maintained also in more complex ALM models and can be exploited to speed up the numerical simulation of such models in practise. It would be interesting to extend our study in this direction and to consider ALM models which include, e.g., multiple model points with different insurance products, continuing new business rather than a single cohort, additional asset classes, multiple bond terms or more complex management rules which involve optimisation routines (e.g. a duration matching) or depend on ALM results at future points in time (stochastic-in-stochastic simulations). In this article, we focused on a reduction of the overall run times of ALM simulations by the use of deterministic integration methods. Variance reduction techniques, different time discretisation schemes and model approximations will be the topic of future research.