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 of [[Harry Panjer]]<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 Professor]], [[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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: <math>f_k = P[X_i = hk].\,</math>

In actuarial practice, <math>X_i\,</math> is obtained by discretisation of the claim density function (upper, lower...).

=== Claim number distribution ===

The number of claims ''N'' is a [[random variable]], which is said to have a "claim number distribution", and which can take values 0, 1, 2, .... etc.. For the "Panjer recursion", the [[probability distribution]] of ''N'' has to be a member of the '''Panjer class''', otherwise known as the [[(a,b,0) class of distributions]]. This class consists of all counting random variables which fulfill the following relation:

:<math>P[N=k] = p_k= \left(a + \frac{b}{k} \right) \cdot p_{k-1},~~k \ge 1.\, </math>

for some ''<math>a''</math> and ''<math>b''</math> which fulfill <math>a+b \ge 0\,</math>. The initial value <math>p_0\,</math> is determined such that <math>\sum_{k=0}^\infty p_k = 1.\,</math>

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 = 61–68| year = 1988| last1 = De Pril | first1 = N. }}</ref>

== Recursion ==

The algorithm now gives a recursion to compute the <math>g_k =P[S = hk] \,</math>.

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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[[Image:Expba07.jpg]]

As observed, an issue may arise at the initialization of the recursion. Guégan and Hassani (2009) have proposed a solution to deal with that issue

.<ref>{{cite journal

|last1 = Guégan |first1 = D.

|last2 = Hassani |first2 = B.K.

|title = A modified Panjer algorithm for operational risk capital calculations

|year = 2009

|journal = Journal of Operational Risk

|volume = 4

|issue = 4

|pages = 53–72

|doi = 10.21314/JOP.2009.068

|s2cid = 4992848

|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]]

[[Category:Compound probability distributions]]

[[Category:Theory of probability distributions]]

[[de:Panjer-Algorithmus]]

[[pl:Wzór Panjera]]