Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance

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.     

Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment

Over the last years, the valuation of life insurance contracts using concepts from financial mathematics has become a popular research area for actuaries as well as financial economists. In particular, several methods have been proposed of how to model and price participating policies, which are characterized by an annual interest rate guarantee and some bonus distribution rules. However, despite the long terms of life insurance products, most valuation models allowing for sophisticated bonus distribution rules and the inclusion of frequently offered options assume a simple Black–Scholes setup and, more specifically, deterministic or even constant interest rates.We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [Bauer, D., Kiesel, R., Kling, A., Ruß, J., 2006. Risk-neutral valuation of participating life insurance contracts. Insurance: Math. Econom. 39, 171–183]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.     

Analyzing legal regulations in the Norwegian life insurance business using a multistage asset–liability management model

The legal regulations for the life insurance business in Norway have recently been, and still are, under revision. The government's intention is to secure the interests of the customers in life insurance companies. However, there has been debate as to whether the regulations really are in the customers' best interest. We apply an asset–liability management (ALM) model to analyze the implications of the regulations. The model is multistage, stochastic and integrates assets and liabilities. We employ a four stage model to analyze the legal regulations, and conclude that the current legal framework is not in the insurance holders' best interests.     

Risk comparison of different bonus distribution approaches in participating life insurance

The fair pricing of explicit and implicit options in life insurance products has received broad attention in the academic literature over the past years. Participating life insurance (PLI) contracts have been the focus especially. These policies are typically characterized by a term life insurance, a minimum interest rate guarantee, and bonus participation rules with regard to the insurer’s asset returns or reserve situation. Researchers replicate these bonus policies quite differently. We categorize and formally present the most common PLI bonus distribution mechanisms. These bonus models closely mirror the Danish, German, British, and Italian regulatory framework. Subsequently, we perform a comparative analysis of the different bonus models with regard to risk valuation. We calibrate contract parameters so that the compared contracts have a net present value of zero and the same safety level as the initial position, using risk-neutral valuation. Subsequently, we analyze the effect of changes in the asset volatility and in the initial reserve amount (per contract) on the value of the default put option (DPO), while keeping all other parameters constant. Our results show that DPO values obtained with the PLI bonus distribution model of Bacinello (2001), which replicates the Italian regulatory framework, are most sensitive to changes in volatility and initial reserves.     

An Asset Liability Management Model for Casualty Insurers: Complexity Reduction vs. Parameterized Decision Rules

In this paper we study possibilities for complexity reductions in large scale stochastic programming problems with specific reference to the asset liability management (ALM) problem for casualty insurers. We describe a dynamic, stochastic portfolio selection model, within which the casualty insurer maximizes a concave objective function, indicating that the company perceives itself as risk averse. In this context we examine the sensitivity of the solution to the quality and accuracy with which economic uncertainties are represented in the model. We demonstrate a solution method that combines two solution approaches: A truly stochastic, dynamic solution method that requires scenario aggregation, and a solution method based on ex ante decision rules, that allow for a greater number of scenarios. This dynamic/fix mix decision policy, which facilitates a huge number of outcomes, is then compared to a fully dynamic decision policy, requiring fewer outcomes. We present results from solving the model. Basically we find that the insurance company is likely to prefer accurate representation of uncertainties. In order to accomplish this, it will accept to calculate its current portfolio using parameterized decision rules.     

The uncertain mortality intensity framework: Pricing and hedging unit-linked life insurance contracts

We study the valuation and hedging of unit-linked life insurance contracts in a setting where mortality intensity is governed by a stochastic process. We focus on model risk arising from different specifications for the mortality intensity. To do so we assume that the mortality intensity is almost surely bounded under the statistical measure. Further, we restrict the equivalent martingale measures and apply the same bounds to the mortality intensity under these measures. For this setting we derive upper and lower price bounds for unit-linked life insurance contracts using stochastic control techniques. We also show that the induced hedging strategies indeed produce a dynamic superhedge and subhedge under the statistical measure in the limit when the number of contracts increases. This justifies the bounds for the mortality intensity under the pricing measures. We provide numerical examples investigating fixed-term, endowment insurance contracts and their combinations including various guarantee features. The pricing partial differential equation for the upper and lower price bounds is solved by finite difference methods. For our contracts and choice of parameters the pricing and hedging is fairly robust with respect to misspecification of the mortality intensity. The model risk resulting from the uncertain mortality intensity is of minor importance.     

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

Consider an insurer who invests in the financial market where correlations among risky asset returns are randomly changing over time. The insurer who faces the risk of paying stochastic insurance claims needs to manage her asset and liability by taking into account of the correlation risk. This paper investigates the impact of correlation risk to the optimal asset–liability management (ALM) of an insurer. We employ the Wishart process to model the stochastic covariance matrix of risky asset returns. The insurer aims to minimize the variance of the terminal wealth given an expected terminal wealth subject to the risk of paying out random liabilities of compound Poisson process. This ALM problem then becomes a linear–quadratic stochastic optimal control problem with stochastic volatilities, stochastic correlations and jumps. The recognition of an affine form in the solution process enables us to derive the explicit closed-form solution to the optimal ALM portfolio policy, obtain the efficient frontier, and identify the condition that the solution is well behaved.     

综合人寿保险精算模型

保险是金融的重要组成部分，国际保险业发展迅速，我国保险业务较晚，资料匮乏，迫切需要引进国外先进的保险经验和保险技术，并结合我国的实际情况加以运用。本文建立了一个综合的人寿保险精算模型，其中包括生存年金，终身寿险和还本部分。通过适当的调整参数进行组合，可以获得不同的保险产品。     

基于混合跳扩散模型的最优投资消费和寿险购买策略研究

本文研究在混合跳扩散模型下投资者分别投资于寿险、零息债券和股票时,关于最优投资消费和寿险购买的随机策略问题。通过构造满足混合跳扩散模型的金融市场、保险市场和可容许策略,在CRRA(constant relative risk aversion)效用下,利用动态规划的方法求解了对应的HJB方程,获得了值函数和最优策略的显式表达式。为了探索模型的有效性,本文给出了相对风险厌恶系数的数值分析以及相关参数对最优策略的影响。     

Evaluation of insurance products with guarantee in incomplete markets

Life insurance products are usually equipped with minimum guarantee and bonus provision options. The pricing of such claims is of vital importance for the insurance industry. Risk management, strategic asset allocation, and product design depend on the correct evaluation of the written options. Also regulators are interested in such issues since they have to be aware of the possible scenarios that the overall industry will face. Pricing techniques based on the Black & Scholes paradigm are often used, however, the hypotheses underneath this model are rarely met.To overcome Black & Scholes limitations, we develop a stochastic programming model to determine the fair price of the minimum guarantee and bonus provision options. We show that such a model covers the most relevant sources of incompleteness accounted in the financial and insurance literature. We provide extensive empirical analyses to highlight the effect of incompleteness on the fair value of the option, and show how the whole framework can be used as a valuable normative tool for insurance companies and regulators.     

The effect of the nature of the liabilities on the solvency and maturity payouts of a UK life office fund: a stochastic evaluation

Several studies have shown that a simulation model of participating life insurance business must reflect explicitly the major decisions which are left to the discretion of management. In the UK these decisions include asset allocation and bonus distribution. It is also common in the UK to use premium bases which do not explicitly account for the expectation that a terminal bonus will be paid, while using a uniform system of reversionary bonus for business of all terms. Here, we show that this results in short-term business being at much greater risk of insolvency than long-term business, under a variety of investment strategies. Our main purpose is to study the effect of small changes to the parameters of the investment model used on the outcomes described above. For this purpose we compare results using two versions of the Wilkie model (Wilkie, A.D., 1986. A stochastic investment model for actuarial use. Transactions of the Faculty of Actuaries 39, 341–403, Wilkie, A.D., 1995. More on a stochastic asset model for actuarial use. British Actuarial Journal 1, 1–168.) We show that small changes in the variances of the economic series modelled have a marked impact on solvency, while changes in maturity values are mostly the result of a small change in the rate of real dividend growth.     

Valuation of life insurance products under stochastic interest rates

In this paper, we introduce a consistent pricing method for life insurance products whose benefits are contingent on the level of interest rates. Since these products involve mortality as well as financial risks, we present an approach that introduces stochastic models for insurance products through stochastic interest rate models. Similar to Black et al. [Black, Fisher, Derman, Emanuel, Toy, William, 1990. A one-factor model of interest rates and its application to treasury bond options. Financ. Anal. J. 46 (January-February), 33-39], we assume that the premiums and volatilities of standard insurance products are given exogenously. We then project insurance prices to extract underlying martingale probability structures. Numerical examples on variable annuities are provided to illustrate the implementation of this method.     

基于个体公平原则的寿险产品定价模型

在无套利框架的基础上,讨论基于个体公平原则下的寿险产品定价问题,即运用倒向随机微分方程理论,将投保人和保险人置于同一系统中进行考虑:首先,根据双方的随机投资决策目标分别建立无套利寿险定价模型和动态资产份额定价模型,得出两个特殊线性倒向随机微分方程的显式解;然后,建立基于个体公平原则的寿险定价模型,从投保人和保险人双方的角度对寿险产品进行公平定价,得出了从供需双方考虑的投资回报定价公式;最后,利用所建立的模型进行案例分析,计算出基于个体公平原则的保费及保险公司的投资策略.该寿险产品定价模型不仅考虑了保险人的意愿,还同时考虑了投保人的实际情况,因此,按此定价理念开发出的保险产品,不仅可以提高产品研发的成功率,而且使得研发出的新产品更能在竞争激烈的保险市场中站稳脚步.     

Agent-based simulation for computational experimentation: Developing an artificial labor market

This paper discusses the creation of an artificial labor market (ALM) as an agent-based simulation model. We trace the development of the ALM by adapting the traditional simulation life cycle into two main parts: the model phase and the simulation phase. In the modeling phase of the life cycle, we focus upon agent representation and specification within the virtual world. In the simulation phase, we discuss the use of scenario planning as the experimentation vehicle. Throughout, we use military recruit market as an example to illustrate the methodology. The benefits of the ALM are (1) it provides a virtual world for continuous computational experimentation, (2) it supports market segmentation by allowing “drilldowns” to finer and finer levels of granularity, and (3) when connected via a common OLAP interface to a “real world” counterpart, it facilitates a tightly integrated, persistent, “sense and respond” decision support functionality.     

On the management of life insurance company risk by strategic choice of product mix,investment strategy and surplus appropriation schemes

The aim of this paper is to analyze the impact of management’s strategic choice of asset and liability composition in life insurance on shortfall risk and the shareholders’ fair risk charge. In contrast to previous work, we focus on the effectiveness of management decisions regarding the product mix and the riskiness of the asset side under different surplus appropriation schemes. We propose a model setting that comprises temporary life annuities and endowment insurance contracts. Our numerical results show that the effectiveness of management decisions in regard to risk reduction strongly depends on the surplus appropriation scheme offered to the customer and their impact on guaranteed benefit payments, which thus presents an important control variable for the insurer.     

应对长寿风险的分红年金:随机精算建模与应用

我国的商业养老保险作为养老金体系的重要组成部分,在实践中的发展比较缓慢,原因之一是保险公司缺乏长寿风险管理的经验。本文将探索我国商业养老保险使用分红年金管理长寿风险的可行性。研究该分红年金在给付规则和分红来源方面的特征,并基于实际数据,构建动态随机死亡率模型和随机收益率模型,采用蒙特卡洛随机模拟方法,比较分红年金和传统年金在待遇分布、资产和损失分布、破产概率等方面的特征,得出分红年金能够在精算公平原则下有效应对长寿风险,并且在待遇给付、偿付能力和盈利能力方面具有明显优势的结论。     

Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.     

The impact of illiquidity on the asset management of insurance companies

This paper investigates optimal asset management strategies for property and casualty insurance companies in illiquid markets. Using a cash-flow based liquidation model of an insurance company, we consider the effects of permanent and temporary price impact as well as commonality in price impact. Focusing on the interaction of a single large investor with the financial market makes the main results generally applicable for any institutional investor with stochastic future liabilities and restrictions on short-sales and financial leverage. Our analysis reveals a clear diversification benefit in illiquid markets apart from the one introduced by Markowitz [Markowitz, H., 1952. Portfolio selection. J. Financ. 7, 77-91]. In the presence of commonality, cash-flow matching is shown to be the optimal strategy for a large investor.     

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.     

Mean‐variance hedging with basis risk

Basis risk arises in a number of financial and insurance risk management problems when the hedging assets do not perfectly match the underlying asset in a hedging program. Notable examples in insurance include the hedging for longevity risks, weather index–based insurance products, variable annuities, etc. In the presence of basis risk, a perfect hedging is impossible, and in this paper, we adopt a mean‐variance criterion to strike a balance between the expected hedging error and its variability. Under a time‐dependent diffusion model setup, explicit optimal solutions are derived for the hedging target being either a European option or a forward contract. The solutions are obtained by a delicate application of the linear quadratic control theory, the method of backward stochastic differential equation, and Malliavin calculus. A numerical example is presented to illustrate our theoretical results and their interesting implications.