خطر بیمه غیر عمر چند سال در مدل رزرواسیون افزایش کاهش

The aim of this paper is to expand on recent contributions in the field of risk modelling for non-life insurance companies by modelling insurance risk in a multi-year context. Academic literature on non-life insurance risk to date has only considered an ultimo perspective (using traditional methods) and, more recently, a one-year perspective (for solvency purposes). However, strategic management in an insurance company requires a multi-year time horizon for economic decision making, providing the motivation for this paper. This is the first paper to derive analytically closed formulae for multi-year non-life insurance risk in the additive loss reserving model as defined by variation of multi-year claims development results. Embedding future accident years leads to an integrated approach to quantifying multi-year risk arising from the settlement of outstanding claims (reserve risk) and future claims yet to occur (premium risk). An application study will serve to illustrate the usefulness of the new multi-year horizon.

Typically, non-life insurance risk is mainly divided into reserve risk and premium risk (see Ohlsson and Lauzeningks (2009)). Reserve risk refers to outstanding payments on claims that have already occurred in prior accident years, whereas premium risk refers to claims yet to occur in future accident years. So far, both premium and reserve risk have been modelled taking the ultimo perspective. This means that the uncertainty in future claim payments is quantified up to final settlement—both for claims already occurred in the past (reserve risk) and claims yet to arise in the future (premium risk). In practice, the separation between reserve risk and premium risk is very strict, and completely different modelling approaches are even applied in some cases (see DAV-Arbeitsgruppe Interne Modelle (2008)). The new regulatory framework, Solvency II, refers to a period of one year, so uncertainty should be quantified on future claim payments in a one-year time horizon (for examples, see Merz and Wüthrich (2008) and Ohlsson and Lauzeningks (2009)). Beyond that, a multi-year perspective is vital for practical decision making such that m∈Nm∈N future calendar years should be taken into account. Management needs answers to questions such as: “How many years of high aggregate losses or adverse claim developments can we withstand at a certain confidence level without requiring external capital”? or “How much risk capital do we need to survive the next five years without external capital supply”? In this contribution, we will present a multi-year approach that can help answer these questions. In addition, Solvency II requires that insurance companies handle risks in their ORSA (own risk and solvency assessment) for a multi-year period (see CEIOPS (2008)); this period usually amounts to three to five years in non-life insurance. Uncertainty in future claim payments should therefore be quantified for both premium risk and reserve risk, resulting in a multi-year insurance risk consisting of multi-year premium and reserve risk over the next mm calendar years (see Diers (2011)). The additive loss reserving method (see Merz and Wüthrich (2010)) represents a classical deterministic reserving method in non-life insurance, which yields best-estimates for future claim payments from outstanding claims. The underlying stochastic model is quite simple, allowing quantification of non-life insurance risk in terms of a multi-year claims development result. Böhm and Glaab (2006) and Merz and Wüthrich (2008) developed an analytical approach towards calculating prediction uncertainty of the one-year claims development result for the chain-ladder method. Similarly, Böhm and Glaab (2006), Mack (2009a) and Merz and Wüthrich (2010) developed an analytical approach for calculating prediction uncertainty of the one-year claims development result for the additive model. Apart from Böhm and Glaab (2006), both cases only dealt with reserve risk while neglecting premium risk. In addition, the results only covered ultimo and one-year risk. To our knowledge, there are as yet no closed formulae for calculating the prediction uncertainty in multi-year claims development results in the additive model. The same applies to the chain-ladder model, although deriving closed analytic formulae would not be possible in this case without resorting to approximations. The aim of this paper is to present the first analytically closed formulae derived for the multi-year non-life insurance risk in the one-year and ultimo perspectives based on additive loss reserving method. The resulting formulae for multi-year prediction errors yield a very simple representation, making them very easy to understand and implement in real life. Apart from that, we have derived an exact calculation of the prediction error of one-year claims development results in arbitrary future calendar years. This paper makes a twofold contribution—first, we present a consistent and integrated approach to calculating premium and reserve risk for arbitrary time horizons (multi-year, one-year and ultimo) in an analytically closed approach, and second, we obtain analytical results that may be used for reference in direct comparison for simulation models. The analytical results may be used in support of multi-year risk models in strategic management decisions and the ORSA process. The rest of the paper is structured as follows: Section 2 introduces the concept of multi-year claims development results for prior and future accident years, which is the basis for quantifying non-life reserve risk, premium risk and insurance risk in a multi-year view. Section 3 presents multi-year non-life insurance risk in the additive loss reserving model, yielding one-year, multi-year, and ultimo results. In addition, Section 3 describes the exact formula for the prediction error of future one-year claims development results in the additive loss reserving model. Section 4 applies the analytical results of the previous section to a case study. Section 5 concludes the paper.

This is the first study to provide a consistent and integrated approach to quantifying prediction uncertainty of multi-year claims development results based on the additive loss reserving method. We have derived analytically closed formulae for prediction errors referring to an arbitrary risk horizon. Accordingly, we can model non-life insurance risk (that is, premium risk and reserve risk) in a closed analytical form for the first time. These results may provide valuable assistance in multi-year business planning, strategic business management and the ORSA process as required in Solvency II. However, limitations also apply to our model. Practitioners often use modifications on original loss reserving methods, such as allowance for tail extrapolation and parameter smoothing, leading to different standard errors compared to the original methods. In addition, the settlement of large claims should be treated separately from attritional claims in loss reserving, as they usually involve different development patterns. Regarding practical application we would therefore recommend using the formula presented here in case of homogeneous and stable portfolios only; these are usually attritional claims. Large claims should be modelled separately. These limitations give rise to additional approaches for further research, the most important and interesting aspect of which would, in our opinion, be the development of closed formulae for multi-year prediction errors in other loss reserving models.