ruin {actuar}

Probability of Ruin

Description

Calulation of infinite time probability of ruin in the models of Cramér-Lundberg and Sparre Andersen, that is with exponential or phase-type (including mixtures of exponentials, Erlang and mixture of Erlang) claims interarrival time.

Usage

ruin(claims = c("exponential", "Erlang", "phase-type"), par.claims,
     wait = c("exponential", "Erlang", "phase-type"), par.wait,
     premium.rate = 1, tol = sqrt(.Machine$double.eps),
     maxit = 200L, echo = FALSE)

## S3 method for class 'ruin'
plot(x, from = NULL, to = NULL, add = FALSE,
     xlab = "u", ylab = expression(psi(u)),
     main = "Probability of Ruin", xlim = NULL, ...)

Arguments

claims

character; the type of claim severity distribution.

wait

character; the type of claim interarrival (wait) time distribution.

par.claims, par.wait

named list containing the parameters of the distribution; see Details.

premium.rate

numeric vector of length 1; the premium rate.

tol, maxit, echo

respectively the tolerance level of the stopping criteria, the maximum number of iterations and whether or not to echo the procedure when the transition rates matrix is determined iteratively. Ignored if wait = "exponential".

x

an object of class "ruin".

from, to

the range over which the function will be plotted.

add

logical; if TRUE add to already existing plot.

xlim

numeric of length 2; if specified, it serves as default for c(from, to).

xlab, ylab

label of the x and y axes, respectively.

main

main title.

...

further graphical parameters accepted by curve.

Details

The names of the parameters in par.claims and par.wait must the same as in dexp, dgamma or dphtype, as appropriate. A model will be a mixture of exponential or Erlang distributions (but not phase-type) when the parameters are vectors of length > 1 and the parameter list contains a vector weights of the coefficients of the mixture.

Parameters are recycled when needed. Their names can be abbreviated.

Combinations of exponentials as defined in Dufresne and Gerber (1988) are not supported.

Ruin probabilities are evaluated using pphtype except when both distributions are exponential, in which case an explicit formula is used.

When wait != "exponential" (Sparre Andersen model), the transition rate matrix \boldsymbol{Q} of the distribution of the probability of ruin is determined iteratively using a fixed point-like algorithm. The stopping criteria used is

\max \left\{ \sum_{j = 1}^n |\boldsymbol{Q}_{ij} - \boldsymbol{Q}_{ij}^\prime| \right\} < \code{tol},

where \boldsymbol{Q} and \boldsymbol{Q}^\prime are two successive values of the matrix.

Value

A function of class "ruin" inheriting from the "function" class to compute the probability of ruin given initial surplus levels. The function has arguments:

u

numeric vector of initial surplus levels;

survival

logical; if FALSE (default), probabilities are \psi(u), otherwise, \phi(u) = 1 - \psi(u);

lower.tail

an alias for !survival.

Author(s)

Vincent Goulet [e-mail], and Christophe Dutang

References

Asmussen, S. and Rolski, T. (1991), Computational methods in risk theory: A matrix algorithmic approach, Insurance: Mathematics and Economics 10, 259–274.

Dufresne, F. and Gerber, H. U. (1988), Three methods to calculate the probability of ruin, Astin Bulletin 19, 71–90.

Gerber, H. U. (1979), An Introduction to Mathematical Risk Theory, Huebner Foundation.

Examples

## Case with an explicit formula: exponential claims and exponential
## interarrival times.
psi <- ruin(claims = "e", par.claims = list(rate = 5),
            wait   = "e", par.wait   = list(rate = 3))
psi
psi(0:10)
plot(psi, from = 0, to = 10)

## Mixture of two exponentials for claims, exponential interarrival
## times (Gerber 1979)
psi <- ruin(claims = "e", par.claims = list(rate = c(3, 7), w = 0.5),
            wait   = "e", par.wait   = list(rate = 3), pre = 1)
u <- 0:10
psi(u)
(24 * exp(-u) + exp(-6 * u))/35	# same

## Phase-type claims, exponential interarrival times (Asmussen and
## Rolski 1991)
p <- c(0.5614, 0.4386)
r <- matrix(c(-8.64, 0.101, 1.997, -1.095), 2, 2)
lambda <- 1/(1.1 * mphtype(1, p, r))
psi <- ruin(claims = "p", par.claims = list(prob = p, rates = r),
            wait   = "e", par.wait   = list(rate = lambda))
psi
plot(psi, xlim = c(0, 50))

## Phase-type claims, mixture of two exponentials for interarrival times
## (Asmussen and Rolski 1991)
a <- (0.4/5 + 0.6) * lambda
ruin(claims = "p", par.claims = list(prob = p, rates = r),
     wait   = "e", par.wait   = list(rate = c(5 * a, a), weights =
                                     c(0.4, 0.6)),
     maxit = 225L)