ruin {actuar} R Documentation

## 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 Q of the distribution of the probability of ruin is determined iteratively using a fixed point-like algorithm. The stopping criteria used is

max(rowSum(|Q - Q'|)) < tol,

where Q and Q' 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 vincent.goulet@act.ulaval.ca, 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)
```

[Package actuar version 3.1-4 Index]