0تجزیه و تحلیل ریسک و ارزیابی قراردادهای بیمه عمر: ترکیبی از رویکردهای آماری و مالی

In this paper, we analyze traditional (i.e. not unit-linked) participating life insurance contracts with a guaranteed interest rate and surplus participation. We consider three different surplus distribution models and an asset allocation that consists of money market, bonds with different maturities, and stocks. In this setting, we combine actuarial and financial approaches by selecting a risk minimizing asset allocation (under the real world measure View the MathML sourceP) and distributing terminal surplus such that the contract value (under the pricing measure View the MathML sourceQ) is fair. We prove that this strategy is always possible unless the insurance contracts introduce arbitrage opportunities in the market. We then analyze differences between the different surplus distribution models and investigate the impact of the selected risk measure on the risk minimizing portfolio.

Interest rate guarantees are a very common product feature within traditional participating life insurance contracts in many markets. There are two major types of interest rate guarantees: The simplest interest rate guarantee is a so-called point-to-point guarantee, i.e. a guarantee that is only relevant at maturity of the contract. The other type is called cliquet-style (or year-by-year) guarantee. This means that the policy holders have an account to which every year at least a certain guaranteed rate of return has to be credited. Cliquet-style guarantees of course may force insurers to provide relatively high guaranteed rates of interest to accounts to which a big portion of the past years’ surplus has already been credited. Adverse capital market scenarios of recent years appeared to have caused significant problems for insurers offering this type of guarantee. Therefore, the analysis of traditional life insurance contracts with cliquet-style guarantees has become a subject of increasing concern for the academic world as well as for practitioners. There are so-called financial and actuarial approaches to handling financial guarantees within life insurance contracts. The financial approach is concerned with risk-neutral valuation and fair pricing and has been researched by various authors such as Bryis and de Varenne (1997), Grosen and Jørgensen (2000), Grosen and Jørgensen (2002) or Bauer et al. (2006). Note that the concept of risk-neutral valuation is based on the assumption of a perfect (or super-) hedging strategy, which insurance companies normally do not or cannot follow (cf. e.g. Bauer et al. (2006)). If the insurer does not or cannot invest in a portfolio that replicates the liabilities, the company remains at risk and should therefore additionally perform some risk analyses. The actuarial approach focuses on quantifying this risk with suitable risk-measures under an objective ‘real-world’ probability-measure, cf. e.g. Kling et al. (2007a) or Kling et al. (2007b). Such approaches also play an important rolee.g. in financial strength ratings or under the new Solvency II approach. Amongst others, Gatzert and Kling (2007) investigate parameter combinations that yield fair contracts and analyze the risk imposed by fair contracts for various insurance contract models, starting with a simple generic point-to-point guarantee and afterwards analyzing more sophisticated Danish- and UK-style contracts. Kling (2007) focuses on traditional German insurance contracts where the interdependence of various parameters concerning the risk exposure of fair contracts is studied. Gatzert (2008) extends the work from Gatzert and Kling (2007) where an approach to ‘risk pricing’ is introduced using the ‘fair value of default’ to determine contracts with the same risk exposure. However, this risk measure neglects real-world scenarios and is only concerned with the (risk-neutral) value of the introduced default put option. Whilst Gatzert (2008) analyzes some real-world risk generated by the considered contracts, the risk exposure is not incorporated in the pricing procedure. Barbarin and Devolder (2005) introduce a methodology that allows for combining the financial and actuarial approach. They consider a contract similar to Bryis and de Varenne’s (1997) with a point-to-point guarantee and terminal surplus participation. To integrate both approaches, they use a two-step method of pricing life insurance contracts: First, they determine a guaranteed interest rate such that certain solvency requirements are satisfied, using value at risk and expected shortfall risk measures. Second, to obtain fair contracts, they use risk-neutral valuation and adjust the participation in terminal surplus accordingly. In the present work we extend Barbarin and Devolder’s (2005) methodology which then allows the pricing of life insurance contracts in a more general liability framework including in particular typical product features of the German insurance market, and an asset allocation that consists of money market, bonds with different maturities and stocks. We identify parameter combinations that minimize the real world risk without changing the fair value of the contract. We prove that the proposed methodology works unless the insurance contract design introduces arbitrage opportunities. The remainder of this paper is organized as follows. After an introduction of the considered financial market, the insurer’s asset allocation, and different liability models in Section 2, Section 3 presents our methodology of combining the actuarial and financial approach and the theoretical result that the strategy we propose is always possible unless the insurance contracts introduce arbitrage opportunities in the market. In Section 4, we show various numerical results for the introduced liability models, focusing on both, the risk a specific contract design and asset allocation imposes on the insurance company and the valuation of the contract from the client’s perspective. We further investigate how the results depend on the risk measure used. Section 5 concludes.

In this paper, we have analyzed three different types of participating life insurance contracts. The theoretical result from Section 3 shows, that–unless contract design introduces arbitrage to the market–it is always possible to combine actuarial and financial approaches such that a management of the insurer’s risk and a desired contract pricing can be achieved simultaneously. In our numerical analyses, we found that optimal, i.e. risk-minimizing, asset allocations as well as the amount of risk depend heavily on the selected liability modes (i.e. surplus distribution mechanism). Also, the results depend very strongly on the chosen risk measure. Our results indicate that under many circumstances, using the shortfall probability as the sole risk measure can lead to wrong incentives. This should be of interest to practitioners as well as regulators when implementing value at risk-based regulation. Of course, our model and analyses can and should be refined in future research. It would be particularly worthwhile including management rules that allow for path dependent asset allocation strategies. Also, the question what an optimal bond portfolio under a given liability model would look like would be of great interest. Finally, the model could be made more realistic by including surrender and mortality and considering more that just one insurance contract on the insurer’s balance sheet.