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|doi=10.1017/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]]).

where both <math>N</math> and <math>X_i</math> are [[stochastic]].

It 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|format=PDF}}</ref>. It is heavily used in [[actuarial science]].

== Preliminaries ==

We are interested in the compound random variable <math>S = \sum_{i=1}^N X_i\,</math> where <math>N\,</math> and <math>X_i\,</math> fulfill the following preconditions.

=== Claim number distribution ===

<math>N</math> is the "claim number distribution", i.e. <math>N \in \mathbb{N}_0</math>. <math>N</math> is assumed to be independent of the <math>X_i</math>.

=== Claim numbersize distribution ===

Furthermore, <math>N</math> has to be a member of the [[Panjer class]]. The Panjer class consists of all counting random variables which fulfill the following relation:

We assume the <math>X_i\,</math> to be [[i.i.d.]] and independent of <math>N\,</math>. Furthermore the <math>X_i\,</math> have to be distributed on a lattice <math>h \mathbb{N}_0\,</math> with latticewidth <math>h>0\,</math>.

<math>p_k= (a + \frac{b}{k}) \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 value <math>p_0</math> is determined such that <math>\sum_{k=0}^\infty p_k = 1.</math>

: <math>f_k = P[X_i = hk].\,</math>

Sundt proved in the paper <ref>{{cite journal|author=B. Sundt and W. S. Jewell|title=Further results on recursive evaluation of compound distributions|journal=ASTIN Bulletin|volume=12|issue=1|year=1981|pages=27–39|publisher=[[International Actuarial Association]]|url=http://www.casact.org/library/astin/vol12no1/27.pdf|format=PDF}} </ref> that only the [[binomial distribution]], the [[Poisson distribution]] and the [[negative binomial distribution]] belong to the Panjer class. They have the parameters and values as described in the following table. <math>W_N(x)</math> denotes the [[probability generating function]].

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

{| class="prettytable"

! width="60" ! class="hintergrundfarbe6" |Distribution

! width="60" ! class="hintergrundfarbe6" |<math> P[N=k] </math>

! width="60" ! class="hintergrundfarbe6" |<math> a </math>

! width="68" ! class="hintergrundfarbe6" |<math> b </math>

! width="45" ! class="hintergrundfarbe6" |<math> p_0 </math>

! width="35" ! class="hintergrundfarbe6" |<math> W_N(x) </math>

! width="35" ! class="hintergrundfarbe6" |<math> E[N] </math>

! width="35" ! class="hintergrundfarbe6" |<math> Var(N) </math>

|-

|[[Binomial distribution|Binomial]]

|<math>\binom{n}{k} p^k (1-p)^{n-k} </math>

|<math> \frac{p}{p-1} </math>

|<math> \frac{p(n+1)}{1-p} </math>

|<math> (1-p)^n </math>

|<math> (px+(1-p))^{n} </math>

|<math> np </math>

|<math> np(1-p) </math>

|-

|[[Poisson distribution|Poisson]]

|<math> e^{-\lambda}\frac{ \lambda^k}{k!} </math>

|<math> 0 </math>

|<math> \lambda </math>

|<math> e^{- \lambda} </math>

|<math> e^{\lambda(s-1)} </math>

|<math> \lambda </math>

|<math> \lambda </math>

|-

|[[Negative binomial distribution|negative binomial]]

|<math> \frac{\Gamma(r+k)}{k!\,\Gamma(r)}\,p^r\,(1-p)^k </math>

|<math> 1-p </math>

|<math> (1-p)(r-1) </math>

|<math> p^r </math>

|<math> \left( \frac{p}{1 - z(1-p)}\right) ^r </math>

|<math> \frac{r(1-p)}{p} </math>

|<math> \frac{r(1-p)}{p^2} </math>

|-

|}

=== Claim sizenumber distribution ===

We assume the <math>X_i</math> to be [[i.i.d.]] and independent of <math>N</math>. Furthermore the <math>X_i</math> have to be distributed on a lattice <math>h \mathbb{N}_0</math> with latticewidth <math>h>0</math>.

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>

: <math>f_k = P[X_i = hk].</math>

== 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

:<math>g_k=\frac{1}{1-f_0a}\sum_{j=1}^k \left( a+\frac{b\cdot j}{k} \right) \cdot f_j \cdot g_{k-j}.\,</math>

== Example ==

The following example shows the approximated density of <math>\scriptstyle S \,=\, \sum_{i=1}^N X_i</math> where <math>\scriptstyle N\, \sim\, \text{NegBin}(3.5,0.3)\,</math> and <math>\scriptstyle X \,\sim \,\text{Frechet}(1.7,1)</math> with lattice width ''h'' = 0.04. (See [[Fréchet distribution]].)

[[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 ==

<references/>

==External links==

*[http://www.vosesoftware.com/riskwiki/Aggregatemodeling-Panjersrecursivemethod.php Panjer recursion and the distributions it can be used with]

[[Category:Actuarial science]]

[[Category:ProbabilityCompound theoryprobability distributions]]

[[Category:Theory of probability distributions]]