actuar (version 0.1-3)

panjer: Panjer's recurison formula

Description

Panjer recursion formula to compute the total amount of claims probability function of a portfolio.

Usage

panjer(fx, freq.dist = c("poisson", "negative binomial", "binomial",
"geometric", "logarithmic"),par, p0, TOL = 1e-08, echo= FALSE)

Arguments

fx

a vector of the (discretized) claim amount distribution; first element *must* be $f_X(0)$

freq.dist

name of the counting distribution

par

named list of the parameter(s) of the counting distribution as they are defined in "rdist".

p0

arbitrary amount of probability at zero given to the frequency distribution. It creates zero-modified or zero-truncated distributions

TOL

stop recursion when cumulative probability function is less than TOL away from 1

echo

print the cumulative distribution of the total amount of claims as it is computed.

Value

• A vector representing the probability density function of the total amount of claims.

Details

The formula of the (a, b, 1) class is: $$f_S(x)=\frac{[p_1-(a+b)p_0]f_X(x)+\sum_{y=1}^{\min(x, m)}(a+\frac{by}{x})f_X(y)f_S(x-y)}{1-af_X(0)}$$ For the (a, b, 0) class, the result reduces to $$f_S(x)=\frac{\sum_{y=1}^{\min(x,m)}(a+\frac{by}{x})f_X(y)f_S(x-y)}{1-af_X(0)}$$ The counting variable is a member of the (a, b, 0) family of discrete distributions if p0 is not specified and a member of the (a,b,1) family if p0 is specified. The logarithmic distribution is a limiting case of the negative binomial distribution where the size parameter is equal to 0.

References

Klugman, S.A and Panjer,H.H and Willmot, G.E (2004), Loss Models: from data to decision, Second Edition, Wiley, Sections 4.6.5 - 4.6.7, 6.6 and Appendix B

Examples

### (a,b,0) class with the binomial distribution
(a <- panjer(fx=rep(0.5,2), freq.dist="bin",par=list(size=3, prob=0.5),echo=TRUE))
sum(a)  

### Example 6.18 of "Loss Models" (Second Edition).
### (a,b,1) class with the Extended Truncated Negative Binomial distribution.  
(a <-
panjer(fx=c(0.3, 0.5, 0.2), "negative bin", par=list(size=0.2, prob=0.25), p0=0, TOL=1E-5, echo=TRUE))
plot(a)