ارزیابی ریسک پذیری قراردادهای بیمه زندگی در شرکت های عضو

The valuation of life insurance contracts using concepts from financial mathematics has recently attracted considerable interest in academia as well as among practitioners. In this paper, we will investigate the valuation of participating contracts, which are characterized by embedded interest rate guarantees and some bonus distribution rules. We will model these under the specific regulatory framework in Germany; however, our analysis can be applied to any insurance market with cliquet-style guarantees. We will present a framework, in which different kinds of guarantees or options can be analyzed separately. Also, the practical implementation of such models is discussed. We use two different numerical approaches to derive fair parameter settings of such contracts and price the embedded options. The sensitivity of the contract value with respect to multiple parameters is studied. In particular, we find that life insurers offer interest rate guarantees below their risk-neutral value. Furthermore, the financial strength of an insurance company considerably affects the value of a contract.

Participating life insurance policies often contain an interest rate guarantee. In many products, this guarantee is given on a point-to-point basis, i.e. the guarantee is only relevant at maturity of the contract and not increased by bonus distribution during the term of the contract. In other products (which are predominant e.g. in the German market), there is a so-called cliquet-style guarantee. This means that the policy holders have an account to which each year a certain rate of return has to be credited. Usually, the life insurance companies provide the guaranteed rate of interest plus some surplus on the policy holders’ account every year. Considering the big market share of such products in many countries, the analysis of life insurance contracts with a cliquet-style guarantee is very important. However, a comparatively large portion of the academic literature focuses on guarantees in unit-linked, equity-linked or variable life insurance contracts (e.g. Brennan and Schwartz (1976) or Aase and Persson (1994)). The analysis of participating policies with a clique style guarantee requires a realistic model of bonus payments. For instance the approach presented in Grosen and Jørgensen (2000) explicitly models a bonus account, which permits smoothing the reserve-dependent bonus payments. Smoothing the returns is often referred to as the “average interest principle”. Aside from the guarantee and a distribution mechanism for excessive returns, Grosen and Jørgensen’s model includes the option for the policy holder to surrender and “walk away”. In this case, the policy holder obtains his account value whereas the reserves remain with the company. Since the account value is path dependent, they are not able to present closed form solutions for the risk-neutral value of the liabilities. Monte Carlo methods are used for the valuation and the analysis. Similarly, in Miltersen and Persson (2003) a cliquet-style guarantee, a bonus account and a distribution mechanism are considered. Here the return exceeding the guaranteed level is distributed between the policy holders’ account, the company’s account and an account for terminal bonus. If in some year the return on assets is below the guaranteed rate, the bonus account can be used to fulfil the guarantee. In particular, the bonus account can become negative, but the insurer has to consolidate a negative balance at the end of the insurance period. However, a positive balance is completely credited to the policy holders. In Hansen and Miltersen (2002), a hybrid of the models by Miltersen and Persson (2003) and Grosen and Jørgensen (2000) is presented. They use the same model for the distribution mechanism as in Grosen and Jørgensen (2000), but the account structure from Miltersen and Persson (2003). Besides a variety of numerical results, they focus on the analysis of the “pooling effect”, i.e. they analyze the consequence of pooling the undistributed surplus over two inhomogeneous customers. In Bacinello (2001) and Bacinello (2003), also cliquet-style guarantees are considered. In Bacinello (2001), she prices participating insurance contracts with a guaranteed interest rate in a Black–Scholes market model. Here, as in Miltersen and Persson (2003), the bonus is modeled as a fixed fraction of the excessive return. She finds closed form solutions for the prices of various policies. In Bacinello (2003), she additionally allows for the surrender of the policy and presents numerical results in a Cox–Ross–Rubinstein framework. Grosen et al. (2001) introduce a different numerical approach to their valuation problem from Grosen and Jørgensen (2000) using the Black–Scholes Partial Differential Equation and arbitrage arguments. They show that the value function follows a known differential equation which can be solved by a finite difference method. This approach is extended and generalized in Tanskanen and Lukkarinen (2004). They use a discretization method in order to solve the partial differential equation. Their model permits multiple distribution mechanisms, including those considered in Miltersen and Persson (2003) and Grosen and Jørgensen (2000). However, these models can not be used to analyze some important features of contracts in insurance markets, where accounting rules allow for building and dissolving valuation reserves which can be used to stabilize the return on book values and, thus, the surplus distribution. In this case, insurers should consider the reserve quota when deciding how much surplus is distributed. Thus, the reserve situation is of great influence on the value of an insurance contract. The present paper fills this gap: Surplus at time tt can be determined and credited depending on the development of the assets (book or market value) and any management decision rule based on information available at time tt. Furthermore, minimum surplus distribution laws that exist in many countries may be considered. In particular, our model can represent all relevant features of the German market, including legal and supervisory issues as well as predominant management decision rules. On the other hand, our framework is general enough to include most of the above models and, therefore, products of other insurance markets as special cases. We use a distribution mechanism that is typical for the German market which has been introduced in Kling et al. (2004). As opposed to their work, where the authors investigate, how the different parameters, such as the initial reserve quota, legal requirements, etc., affect the shortfall probability of a contract and how these factors interact, we are interested in the risk-neutral value of the corresponding contracts. The rest of this paper is organized as follows. In Section 2 we introduce our model and the distribution mechanisms, i.e. the rules according to which earnings are distributed among policy holders, shareholders, and the insurance company. The goal of this paper is to find a fair price for an insurance policy using methods from financial mathematics. However, certain conditions must be fulfilled in order to obtain a “meaningful” price. In Section 3, we will discuss under which circumstances a risk-neutral valuation is appropriate. Since the considered insurance contracts are complex and path-dependent derivatives, it is not possible to find closed form solutions for their price. In Section 4, we present a Monte Carlo Algorithm, which allows for the separate valuation of the embedded options, and an extension of the discretization approach presented in Tanskanen and Lukkarinen (2004), that allows us to consider a surrender or walk-away option. Our results are presented in Section 5. Besides the values of the contracts and the embedded options, we examine the influence of several parameters and give economic interpretations. Section 6 closes with a summary of the main results and an outlook for future research.

We presented a model for evaluating and analyzing participating insurance contracts and adapted it to the German regulatory framework. Besides considering obligatory payments (MUST-case), we also included a distribution mechanism which is typical for German insurers (IS-case). We applied the model to valuate and analyze contracts. Furthermore, we discussed under which conditions and prerequisites a risk-neutral approach is meaningful. We presented a cash flow model, which takes into account the special circumstances of the valuation of German insurance contracts and also provides the possibility to separately valuate and analyze embedded options and other components of the contract. Since these types of contracts are complex and path-dependent contingent claims, we relied on numerical methods for the evaluations. Besides an efficient Monte Carlo algorithm, which enables us to consider the contract components separately, we presented a discretization algorithm based on the Black–Scholes PDE, which allows us to include a surrender option. We examined the impact of various parameters on the value of a contract. We found that this value is significantly influenced by the insurer’s financial situation and the provided minimum interest and participation guarantees, whereas the surrender option is of negligible value in most realistic cases. In particular, a contract with a financially strong company in general is more valuable. Under current market conditions, the value of the contract exceeds the initial investment–and therefore the price–of a contract. This is alarming and partially explains current problems of the German life insurance industry. Furthermore, we found the ability to build up and dissolve hidden reserves is crucial for offering these types of contracts. Our model supports the venture of several insurance companies in the German market to provide different target rates of interest for different contract generations with different guarantee levels. However, other factors than just the interaction of target and guarantee rates have to be taken into account in order to not disadvantage either “old” or “new” customers. We particularly found that the interactions of the parameters describing the regulatory framework, the financial market, the insurance company’s situation, and the insurance contract are rather complex. An isolated analysis of the impact of one (set of) parameters does not seem appropriate. Since our results are rather sensitive to changes in the risk free rate of interest rr and since the considered time horizon is rather long, including stochastic interest rates in the model should be a next step. Also, an analysis of the corresponding hedging strategies would be appropriate and of practical relevance.