The Panjer recursion is an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

to compute the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

approximation of a compound

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

S = \sum_^N X_i\,

where both

N\,

and

X_i\,

are

s and of special types. In more general cases the distribution of ''S'' is a

compound distribution In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some ...

. The recursion for the special cases considered was introduced in a paper by

Harry Panjer The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable S = \sum_^N X_i\, where both N\, and X_i\, are random variables and of special types. In more general cases the distribution of ...

(

Distinguished Emeritus Professor The ruling made by the judge or panel of judges must be based on the evidence at hand and the standard binding precedents covering the subject-matter (they must be ''followed''). Definition In law, to distinguish a case means a court decides t ...

University of Waterloo The University of Waterloo (UWaterloo, UW, or Waterloo) is a public research university with a main campus in Waterloo, Ontario, Canada. The main campus is on of land adjacent to "Uptown" Waterloo and Waterloo Park. The university also operates ...

). It is heavily used in actuarial science (see also

systemic risk In finance, systemic risk is the risk of collapse of an entire financial system or entire market, as opposed to the risk associated with any one individual entity, group or component of a system, that can be contained therein without harming th ...

Preliminaries

We are interested in the compound random variable

S = \sum_^N X_i\,

where

N\,

and

X_i\,

fulfill the following preconditions.

Claim size distribution

We assume the

X_i\,

to be i.i.d. and independent of

N\,

. Furthermore the

X_i\,

have to be distributed on a lattice

h \mathbb_0\,

with latticewidth

h>0\,

. :

f_k = P_i = hk \,

In actuarial practice,

X_i\,

is obtained by discretisation of the claim density function (upper, lower...).

Claim number distribution

The number of claims ''N'' is a

, which is said to have a "claim number distribution", and which can take values 0, 1, 2, .... etc.. For the "Panjer recursion", the

of ''N'' has to be a member of the Panjer class, otherwise known as the

(a,b,0) class of distributions In probability theory, a member of the (''a'', ''b'', 0) class of distributions is any distribution of a discrete random variable ''N'' whose values are nonnegative integers whose probability mass function satisfies the recurrence formula : \f ...

. This class consists of all counting random variables which fulfill the following relation: :

= p_k= \left(a + \frac \right) \cdot p_,~~k \ge 1.\,

for some

a

and

b

which fulfill

a+b \ge 0\,

. The initial value

p_0\,

is determined such that

\sum_^\infty p_k = 1.\,

The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of ''S''. In the following

W_N(x)\,

denotes the

probability generating function In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often ...

of ''N'': for this see the table in

. In the case of claim number is known, please note the ''De Pril'' algorithm. This algorithm is suitable to compute the sum distribution of

n

discrete random variables.

Recursion

The algorithm now gives a recursion to compute the

g_k =P = hk \,

. The starting value is

g_0 = W_N(f_0)\,

with the special cases :

g_0=p_0\cdot \exp(f_0 b) \quad \text \quad a = 0,\,

and :

g_0=\frac \quad \text \quad a \ne 0,\,

and proceed with :

g_k=\frac\sum_^k \left( a+\frac \right) \cdot f_j \cdot g_.\,

Example

The following example shows the approximated density of

\scriptstyle S \,=\, \sum_^N X_i

where

\scriptstyle N\, \sim\, \text(3.5,0.3)\,

and

\scriptstyle X \,\sim \,\text(1.7,1)

with lattice width ''h'' = 0.04. (See

Fréchet distribution The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\Pr(X \le x)=e^ \text x>0. where ''α'' > 0 is a ...

.) 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 .{{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

References

External links

Panjer recursion and the distributions it can be used with
Actuarial science Compound probability distributions Theory of probability distributions