تجزیه و تحلیل حساسیت از نمونه قرارداد بیمه عمر با توجه به پایه فنی

In [Christiansen, M.C., 2007. A sensitivity analysis concept for life insurance with respect to a valuation basis of infinite dimension. Insurance: Math. Econom. doi:10.1016/j.insmatheco.2007.07.005] a sensitivity analysis concept was introduced for the prospective reserve of individual life insurance contracts as functional of the technical basis parameters such as interest rate, mortality probability, disability probability, et cetera. On the basis of that concept, the present paper gives in addition the sensitivities of the premium level. Applying these approaches, an extensive sensitivity analysis is carried out: A study of the basic life insurance contract types ‘pure endowment insurance’, ‘temporary life insurance’, ‘annuity insurance’ and ‘disability insurance’ identifies their diverse characteristics, in particular their weakest points concerning fluctuations of the technical basis. An investigation of combinations of these insurance contract types shows what synergy effects can be expected by creating insurance packages.

The past has shown that actuarial assumptions such as interest rate or mortality can vary significantly within a contract period. Especially in recent years financial markets have shown a high volatility, and life expectancies in many developed countries increased with an unforeseen rate. Hence, an actuary is well advised to pay attention to the influence such changes can have on premiums or reserves. In Christiansen (2007) a detailed overview is given of the literature addressing the dependency of prospective reserve or premium level on actuarial assumptions: First, there are Lidstone (1905) and his successors Norberg (1985), Hoem (1988), Ramlau-Hansen (1988), and Linnemann (1993), who mainly presented qualitative statements. Second, there are scenario-based approaches, as for example Olivieri (2001) or Khalaf-Allah et al. (2006), but that idea works only for a small number of parameters. A third way is to study sensitivities by means of derivatives, which turned out to be a very efficient concept. References using such an approach are Dienst (1995), Bowers et al. (1997), Kalashnikov and Norberg (2003), and Helwich (2003). All of those studies have in common that they only allowed for a finite number of parameters. In Christiansen (2007) a generalized gradient vector concept was introduced that enables to study sensitivities with respect to an infinite number of parameters. This meets, for example, the more realistic idea of actuarial assumptions (e.g. the mortality) being functions on a real line rather than on a discrete time grid. Christiansen (2007) gave a general formula for the sensitivity of the prospective reserve of an individual insurance contract with respect to changes of the technical basis. The present paper gives in Section 2.2 an analogous formula for the sensitivity of the premium level of a contract. Applying them, Section 3 carries out a sensitivity analysis for various examples of typical life insurance contracts. It turns out that the different contract types have very diverse characteristics. With the calculated sensitivities being linear in the payment streams, the sensitivities of combinations of different insurance contracts are obtained by just adding up their sensitivities, which enables to easily create cancellation effects in order to reduce sensitivities. This simplicity makes the approach presented herein a valuable tool for the designing of insurance packages.