خطر بیمه غیر عمر یک ساله

A major part of the literature on non-life insurance reserve risk has been devoted to the ultimo risk, the risk in the full run-off of the liabilities. This is in contrast to the short time horizon in internal risk models at insurance companies, and the one-year risk perspective taken in the Solvency II project of the European Community. This paper aims at clarifying the one-year risk concept and describing simulation approaches, in particular for the one-year reserve risk. We also discuss the one-year premium risk and its relation to the premium reserve. Finally, we initiate a discussion on the role of risk margins and discounting for the reserve and premium risk, with focus on the Cost-of-Capital method. We show that risk margins do not affect the reserve risk and show how reserve duration can be used for easy calculation of risk margins. 1

In most risk models, non-life insurance risk is divided into reserve risk and premium risk. Reserve risk concerns the liabilities for insurance policies covering historical years, sometimes referred to as the risk in the claims reserve (the provision for outstanding claims). Premium risk relates to future risks, some of which are already liabilities, covered by the premium reserve (the provision for unearned premium and unexpired risks); others relate to policies expected to be written during the risk period, covered by the corresponding expected premium income. (For technical reasons, catastrophe risk is often singled out as a third part of non-life insurance risk, but that lies outside the scope of this paper.) The above-mentioned risks are also involved in the calculation of risk margins for the reserves; we will consider the Cost-of-Capital (CoC) approach, which is mandatory in the Solvency II Draft Framework Directive, EU Commission (2007). In the Discussion paper IASB (2007) on the forthcoming IFRS 4 phase II accounting standard, CoC is one of the listed possible approaches to determine risk margins. In the Solvency II framework, the time horizon is one year, described by the EU Commission (2007) as follows: “all potential losses, including adverse revaluation of assets and liabilities over the next 12 months are to be assessed.” In the actuarial literature, on the other hand, reserve risk has traditionally been discussed in terms of the risk that the estimated reserves will not be able to cover the claims payment during the full run-off of today’s liabilities, which may be a period of several decades; we call this the ultimo risk. If R0R0 is the reserve estimate at the beginning of the year and C∞C∞ are the payments over the entire run-off period, this risk is measured by studying the probability distribution of R0−C∞R0−C∞. This is the approach of the so-called stochastic claims reserving which has been developed in the actuarial literature over the last two decades, by Mack (1993), England and Verrall (2002) and many others. Considering this background, it may not be surprising that it is noted in a study from the mutual insurers organization AISAM-ACME (2007), that “Only a few members were aware of the inconsistency between their assessment on the ultimate costs and the Solvency II framework which uses a one-year horizon”. Furthermore, the study shows that for long-tailed business, the ultimo (or as it is called there: full run-off) approach gives risk estimates that are 2 to 3 times higher than those for the one-year result. We conclude that it is both necessary and important to clarify the difference between the one-year and ultimo perspective. In Section 6 of Dhaene et al. (2008), an approximate rule is given indicating that a one-year certainty level of 99.5% corresponds to a 40-year full run-off level of 81.8%. If we suppose that both risks are normally distributed, this can be seen to correspond to a volatility 2.8 times larger for the full run-off case, which is in line with the AISAM-ACME study. (On the other hand, Dhaene et al., at the end of Section 5, discuss an unclear point in the methodological description of the AISAM-ACME study that might explain some of the differences.) On the methodological side, a few papers on the one-year reserve risk have recently appeared in the literature, see Wütrich et al. (2009) or Merz and Wütrich (2008). In the special case of a pure Chain Ladder estimate, they give analytic formulae for the mean squared error of prediction of the one-year result under an extension of the classic Mack (1993) model for the ultimo result. Wütrich and Bühlmann (in press) model the one-year risk when claims reserves are discounted. Our first aim is to help in clarifying the methodological issues for the one-year approach to reserve risk, and in particular to describe a general simulation approach to the problem. A special case of this approach is the bootstrap methods in the context of Dynamic Financial Analysis which are implemented in some commercial software; cf. Björkwall et al. (in press). In Section 3 we turn to the one-year perspective on premium risk, followed by a discussion in Section 4 on the role of the risk margin for premium and reserve risk. To the best of our knowledge, these issues have not been discussed in the literature before.

In this paper, we have tried to clarify the one-year perspective on reserve risk. We have demonstrated that for the Cost-of-Capital (CoC) method, there is no circular reference in letting the risk margin enter the solvency capital requirement (SCR) calculations. In practice, one has to use a simplified method, for which we have seen that we can omit the risk margin completely from the SCR calculations. We have also given the simplified CoC method an interpretation in terms of reserve duration, which might turn out to be useful. As concerns the premium reserve, we have initiated a discussion of its role in the one-year premium risk and indicated very briefly how a simulation approach might include the premium reserve risk. In this area there is need for further research and numerical tests, before the ideas can be implemented.