معیار ریسک و ارزیابی عادلانه تضمین سرمایه گذاری در بیمه عمر

Investment guarantees are amongst the most important topics in the pricing and management of life insurance. Traditionally, two ways of analyzing the risk are possible: on the one hand, the financial approach based on risk-neutral measure and leading to option pricing and continuous hedging strategy and on the other hand, a more actuarial approach based on ruin probability and distribution of surplus. The purpose of this paper is to try to integrate these two approaches in the management of life insurance contracts with profits. First, we analyze in terms of value at risk and conditional value at risk the effect of putting an investment guarantee. This will be done in an ALM framework, based on different investment strategies of the insurer in terms of risk and matching between assets and liabilities. The liability side will be represented by a guaranteed technical rate; the asset side will be a mix of stocks, cash and bonds in a Gaussian environment with different matching strategies. Consequences of an investment choice in terms of ruin probability and level of solvency will be illustrated. In a second step, fair valuation principles are used in order to compute the market value of the contract and fix the participation rate of the contract

Investment guarantees embedded in classical life insurance contracts are surely nowadays one of the most important challenges for the insurance industry. In order to cope with this risk, two big paradigms seem to be available for the actuary. 1.1. The risk-neutral approach Under classical assumptions on the market (completeness, no arbitrage, …) the price of the guarantee is associated to the price of an option and the machinery of option pricing can be used in order to valuate the product. This philosophy has been used with success as well for equity linked contracts with maturity guarantee (see, for instance, Brennan and Schwartz, 1976, Delbaen, 1986, Aase and Persson, 1994 and Nielsen and Sandmann, 1995) as for life insurances with profit (see, for instance, Bacinello, 2001, Bacinello, 2003 and Grosen and Jorgensen, 2000). In this context, computations take place under a risk-neutral probability measure. We can consider these models as directly inspired by modern finance theory. 1.2. The risk-management approach Modern tools of simulation permit more and more to generate a lot of future scenarios and to build the distribution of the final surplus of the insurer taken into account stochastic financial models. Parameters of the contracts are then fixed in relation with a certain level of solvency. Value at risk concept or more generally risk measures are the central tools (Artzner et al., 1997, Artzner et al., 1999 and Wirch and Hardy, 1999). Of course, computations are done in the real historical world. These models directly linked with dynamic financial analysis (DFA, developed as well in life as in non life insurance-see, for instance, Kaufmann et al., 2001) are in fact close to the classical risk theory and the actuarial well-known concept of probability of ruin. These two methodologies have already been compared in terms of reserving (Boyle and Hardy, 1997). The purpose of this paper is to propose a way to combine these two methodologies in order to fix the technical parameters of a classical life insurance product with profit. In this context we suppose that the insurer guarantees a certain fixed rate on the paid premiums. On top of that, participation is given at maturity if the real investment performances are good. This bonus is expressed as a rate applied to the eventual final surplus of the insurer. So, the decision problems for the insurer in this kind of product consists of choosing two numbers: the guaranteed rate and the participation rate. In order to proceed, we propose to decompose the problems in two steps: First step: In our mind, the choice of a guaranteed rate is directly linked with solvency concerns. Perfect hedging for long periods as in life or pensions contracts is a utopia. Derivative instruments for very long periods as in life or in pension insurance are not common on the markets and self-hedging is too much expensive. A risk-neutral price could give the illusion of an absence of risk like in the Black and Scholes world but in fact risk is still there at maturity if you do not hedge perfectly as requested by the underlying pricing process. All this motivates to use a risk-management approach in order to fix the guaranteed rate for long periods. Second step: Once this technical rate is chosen and in order to fix the second parameter – the participation rate – we can try to build a contract equilibrated between the policyholder and the insurer. Risk-neutral fairness is then used to compute an equilibrated value for the participation rate. By this way, combination of the two paradigms permits to take into account at the same time as well solvency concerns as fair valuation principles. The paper is organized as follows. Section 2 presents the main assumptions on the financial market and the investment strategies. In this context we develop a model with cash, bonds and stocks (cash–bond–stock model, CBS model hereafter) representing better than other simple models (for instance, binomial or geometric Brownian) the fundamental investments used by the insurers as underlying values for life insurance products with profit. Different kinds of bond strategies are especially developed in order to model the influence of matching policies. Section 3 is then devoted to the risk-management analysis in order to select the rate that can be guaranteed. Explicit solutions in the CBS model are given using a “value at risk” approach; not surprisingly the rate is a function of the required level of solvency. Formulas are also given if we choose as criteria the expected shortfall instead of the value at risk. In Section 4, fair valuation principles are then applied in order to determine the equilibrium value of the participation rate corresponding to the “risk-management” value of the guarantee. Forward risk-neutral measure is used to price the contract. Finally, Section 5 presents some numerical illustrations.

6. Conclusions and extensions In this paper, we studied the interrelationship between the risk-management and the pricing of a classical single premium life insurance contract with profit. The pricing is considered in terms of a guaranteed rate on the premium and a participation rate on the financial surplus. We argued it is a utopia to think we can perfectly hedge the liabilities of long-term insurance contracts. There are two main reasons to this. First of all, derivative instruments with terms of 10, 15 or 20 years are not traded on the markets. Secondly, self-hedging would be too expensive. So, at least for these very long-term liabilities, financial risk is unavoidable. By using only the risk-neutral valuation principle to determine the guaranteed rate and the participation rate, we cannot properly, take into account this financial risk. Moreover, using this pricing principle alone, we would have to pick up arbitrarily, a pair of guaranteed and participation rates among the set of pairs consistent with the no arbitrage assumption. Instead, we argued these parameters should be determined by taking into account explicitly the risk the insurer accepts to bear. Moreover, this acceptable level of risk should be set consistently with the risk-management policy of the insurance company, in terms of a risk measure such as the value at risk or the conditional value at risk. In order to answer these points, we suggested to divide our problem in two parts. In the first one, we consider the insurance contract without profit. Since the financial risk comes only from the guaranteed rate, we proposed to fix the guaranteed rate such that the value at risk or conditional value at risk of this modified contract, does not exceed a certain level chosen by the risk-management. In the second part, we fix the participation rate according to the risk-neutral valuation principle and according to the guaranteed rate found in step one. In this way, we have a contract that is simultaneously fairly priced and that exhibits a risk consistent with the risk-management policy. Finally, since the financial risk depends obviously on the financial portfolio, it is important to model realistically this portfolio. Accordingly, we proposed a more sophisticated model that includes investment in cash, stocks and bonds and that allows us to study the effect of the matching policy and the effect of the strategic allocation on the technical parameters. Furthermore, this model has the advantage to offer closed form solution. This model can be extended in a variety of ways and still keeps its explicit solutions. First of all, our portfolio is assumed to be invested in a single zero-coupon bond. We could also have assumed our bond portfolio is invested in several zero-coupon bonds. Secondly, we could also have assumed a more general gaussian model for our financial market. For example, the instantaneous risk-free rate could follow a multi-factor gaussian model. In both cases, we still would have obtained closed form solutions, at the cost of more fastidious calculations. Thirdly, we could even extend our model to a more general non gaussian framework. For example, it certainly would be interesting to model our stock index as a discontinuous Lévy process since empirically, this family of models is much more justified than the log-normal one. Moreover, since these kinds of processes exhibit fat tails in contrast with gaussian model, they should have an important impact on the risk measure, and therefore, on the guaranteed rate. Finally, we could also extend our model to a periodic premium contract. Obviously, in these cases, we cannot anymore expect to obtain closed form solutions and we should admit to rely on numerical techniques.