Further Results on Recursive Evaluation of Compound Distributions

Sundt, Bjørn; Jewell, William S.

doi:10.1017/s0515036100006802

Cited by 163 publications

(80 citation statements)

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“…Observe that the initial value (5) simplifies to g 0 = q 0 for f 0 = 0. Sundt and Jewell (1981) showed that (3) holds precisely for the binomial, negative binomial and Poisson distribution (and, as a matter of course, for the degenerate distribution q 0 = 1).…”

Section: Panjer Recursionmentioning

confidence: 97%

Panjer recursion versus FFT for compound distributions

Embrechts

Frei

2008

Math Meth Oper Res

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Section: Panjer Recursionmentioning

confidence: 97%

Panjer recursion versus FFT for compound distributions

Embrechts

Frei

2008

Math Meth Oper Res

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“…The famous Panjer recursion [20,29] is contained in the following theorem: Theorem 4.1 (Extended Panjer recursion). Assume that the probability distribution {q n } n2N 0 of N belongs to the Panjer(a, b, k) class and a P[X 1 = 0] 6 = 1.…”

Section: A Generalization Of the Panjer Recursionmentioning

confidence: 99%

“…All distributions belonging to a Panjer(a, b, k) class were identified by Sundt and Jewell [29] for the case k = 0, Willmot [33] for the case k = 1, and finally Hess, Liewald and Schmidt [11] for general k 2 N 0 . More general relations than (1.2) and corresponding recursion schemes have been considered in articles by Sundt [28], Hesselager [12], and Wang and Sobrero [31].…”

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confidence: 99%

A generalization of Panjer’s recursion and numerically stable risk aggregation

Gerhold

Schmock

Warnung³

2009

Finance Stoch

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Abstract. Portfolio credit risk models as well as models for operational risk can often be treated analogously to the collective risk model coming from insurance. Applying the classical Panjer recursion in the collective risk model can lead to numerical instabilities, for instance if the claim number distribution is extended negative binomial or extended logarithmic. We present a generalization of Panjer's recursion that leads to numerically stable algorithms. The algorithm can be applied to the collective risk model, where the claim number follows, for example, a Poisson distribution mixed over a generalized tempered stable distribution with exponent in (0, 1). De Pril's recursion can be generalized in the same vein. We also present an analogue of our method for the collective model with a severity distribution having mixed support.

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“…This very popular set of claim number distributions was introduced in 1981 by Jewell and Sundt [21]. It is based on the simple recursion…”

Section: On Excess-of-loss Reinsurancementioning

confidence: 99%

On excess-of-loss reinsurance

Albrecher

Teugels

2009

Theor. Probability and Math. Statist.

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Abstract. We discuss a unified framework to analyze the distribution of the number of claims and the aggregate claim sizes in an excess-of-loss reinsurance contract based upon the use of point processes and work out several examples explicitly. We first deal with a single excess-of-loss situation with an extra upper bound on the coverage of individual claims. Subsequently the results are extended to a reinsurance chain with k partners. IntroductionTraditionally, an excess-of-loss reinsurance form covers the overshoot over a certain retention level M for all claims whether or not they are considered to be large. Among the insurance branches where it is used, we mention in particular general liability, and to a lesser extent motor liability ([25]) and windstorm reinsurance ([18]). Because of its transparency, an excess-of-loss treaty has been one of the main research objects in the reinsurance literature right from the beginning (see for instance [3,4]). It is clear that excess-of-loss reinsurance limits the liability of the first line insurer but that he himself will cover all claims below the retention M . In this form, the excess-of-loss reinsurance treaty cover has a number of desirable theoretical properties as explained by Bühlmann in [5] and by Asmussen et al. in [2]. For a combination with quota-share reinsurance, see [6].We will look at a more general form where each claim will be considered between two boundaries: we will call the lower retention u while the upper will be taken to be u + v, indicating by v the range for which the treaty is used. As illustrated below, the notation allows us to deal with any partner in an excess-loss reinsurance chain. If u and v depend on the claim orderings, then this reinsurance form is commonly called drop-down-excessof-loss reinsurance. For a study of such a reinsurance form, we refer to Ladoucette et al. [12].In the following we need some basic notation, first for the original portfolio.• The epochs of the claims are denoted by T 0 = 0, T 1 , T 2 , . . . . Apart from the fact that the epochs form a nondecreasing sequence, we in general do not assume anything specific about their interdependence. The random variables defined by W 0 = 0 and {W i+1 := T i+1 − T i ; i = 0, 1, . . .} are called the waiting times in between successive claims. In some particular cases it might be useful to assume that the sequence {W i ; i ≥ 1} consists of independent random variables all with a common distribution V as a random variable T , i.e. P(T ≤ x) = V (x). In that2000 Mathematics Subject Classification. Primary 62P05, 62H20.

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Further Results on Recursive Evaluation of Compound Distributions

Cited by 163 publications

References 1 publication

Panjer recursion versus FFT for compound distributions

Panjer recursion versus FFT for compound distributions

A generalization of Panjer’s recursion and numerically stable risk aggregation

On excess-of-loss reinsurance

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