Panjer recursion: Difference between revisions

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The '''Panjer recursion''' is an [[algorithm]] to compute the [[probability distribution]] approximation of a compound [[random variable]]

<math>S = \sum_{i=1}^N X_i\,</math>.

where both <math>N\,</math> and <math>X_i\,</math> are [[random variable]]s and of special types. In more general cases the distribution of ''S'' is a [[compound distribution]]. The recursion for the special cases considered was introduced in a paper <ref>{{cite journal|last=Panjer|first=Harry H.|year=1981|title=Recursive evaluation of a family of compound distributions.| journal=ASTIN Bulletin|volume=12|issue=1|pages=22–26|publisher=[[International Actuarial Association]]|url=http://www.casact.org/library/astin/vol12no1/22.pdf|formatdoi=PDF10.1017/S0515036100006796|s2cid=15372040 }}</ref> by [[Harry Panjer]] ([[Distinguished Emeritus professorProfessor]], [[University of Waterloo]]<ref>[http://www.actuaries.org/COUNCIL/Documents/CV_Panjer.pdf CV], actuaries.org; [https://math.uwaterloo.ca/statistics-and-actuarial-science/about/people/harry-panjer Staff page], math.uwaterloo.ca</ref>). It is heavily used in [[actuarial science]] (see also [[systemic risk]]).

== Preliminaries ==

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The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of ''S''. In the following <math>W_N(x)\,</math> denotes the [[probability generating function]] of ''N'': for this see the table in [[(a,b,0) class of distributions]].

In the case of claim number is known, please note the ''De Pril'' algorithm.<ref>Vose Software Risk Wiki: http://www.vosesoftware.com/riskwiki/Aggregatemodeling-DePrilsrecursivemethod.php</ref> This algorithm is suitable to compute the sum distribution of <math>n</math> discrete [[random variables]].<ref>{{Cite journal | doi = 10.1080/03461238.1988.10413837| title = Improved approximations for the aggregate claims distribution of a life insurance portfolio| journal = Scandinavian Actuarial Journal| volume = 1988| issue = 1–3| pages = 6161–68| year = 1988| last1 = De Pril | first1 = N. }}</ref>

== Recursion ==

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The starting value is <math>g_0 = W_N(f_0)\,</math> with the special cases

:<math>g_0=p_0\cdot \exp(f_0 b) \quad \text{ if } \quad a = 0,\,</math>

and

:<math>g_0=\frac{p_0}{(1-f_0a)^{1+b/a}} \quad \text{ for } \quad a \ne 0,\,</math>

and proceed with

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|issue = 4

|pages = 53–72

|doi = 10.21314/JOP.2009.068

|url = http://www.risk.net/journal-of-operational-risk/technical-paper/2160851/a-modified-panjer-algorithm-operational-risk-capital-calculations

|s2cid = 4992848

}}</ref>

|citeseerx = 10.1.1.413.5632}}</ref>

== References ==

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==External links==

*[http://www.vosesoftware.com/ModelRiskHelpriskwiki/index.htm#Aggregate_distributions/Aggregate_modeling_Aggregatemodeling-_Panjer_s_recursive_methodPanjersrecursivemethod.htmphp Panjer recursion and the distributions it can be used with]

[[Category:Actuarial science]]