actuar (version 0.9-4)

adjCoef: Adjustment coefficient

Description

Compute the adjustment coefficient in ruin theory, or return a function to compute the adjustment coefficient for various reinsurance retentions.

Usage

adjCoef(mgf.claim, mgf.wait = mgfexp(x), premium.rate, upper.bound,
        h, reinsurance = c("none", "proportional", "excess-of-loss"),
        from, to, n = 101)
## S3 method for class 'adjCoef':
plot(x, xlab = "x", ylab = "R(x)",
     main = "Adjustment Coefficient", sub = comment(x),
     type = "l", add = FALSE, ...)

Arguments

mgf.claim

an expression written as a function of x or of x and y, or alternatively the name of a function, giving the moment generating function (mgf) of the claim severity distribution.

mgf.wait

an expression written as a function of x, or alternatively the name of a function, giving the mgf of the claims interarrival time distribution. Defaults to an exponential distribution with parameter 1.

premium.rate

if reinsurance = "none", a numeric value of the premium rate; otherwise, an expression written as a function of y, or alternatively the name of a function, giving the premium rate function.

upper.bound

numeric; an upper bound for the coefficient, usually the upper bound of the support of the claim severity mgf.

h

an expression written as a function of x or of x and y, or alternatively the name of a function, giving function $h$ in the Lundberg equation (see below); ignored if mgf.claim is provided.

reinsurance

the type of reinsurance for the portfolio; can be abbreviated.

from, to

the range over which the adjustment coefficient will be calculated.

n

integer; the number of values at which to evaluate the adjustment coefficient.

x

an object of class "adjCoef".

xlab, ylab

label of the x and y axes, respectively.

main

main title.

sub

subtitle, defaulting to the type of reinsurance.

type

1-character string giving the type of plot desired; see plot for details.

add

logical; if TRUE add to already existing plot.

...

further graphical parameters accepted by plot or lines.

Value

• If reinsurance = "none", a numeric vector of lenght one. Otherwise, a function of class "adjCoef" inheriting from the "function" class.

Details

In the typical case reinsurance = "none", the coefficient of determination is the smallest (strictly) positive root of the Lundberg equation $$h(x) = E[e^{x B - x c W}] = 1$$ on $[0, m)$, where $m =$ upper.bound, $B$ is the claim severity random variable, $W$ is the claim interarrival (or wait) time random variable and $c =$ premium.rate. The premium rate must satisfy the positive safety loading constraint $E[B - c W] < 0$.

With reinsurance = "proportional", the equation becomes $$h(x, y) = E[e^{x y B - x c(y) W}] = 1,$$ where $y$ is the retention rate and $c(y)$ is the premium rate function.

With reinsurance = "excess-of-loss", the equation becomes $$h(x, y) = E[e^{x \min(B, y) - x c(y) W}] = 1,$$ where $y$ is the retention limit and $c(y)$ is the premium rate function.

One can use argument h as an alternative way to provide function $h(x)$ or $h(x, y)$. This is necessary in cases where random variables $B$ and $W$ are not independent.

The root of $h(x) = 1$ is found my minimizing $(h(x) - 1)^2$.

References

Bowers, N. J. J., Gerber, H. U., Hickman, J., Jones, D. and Nesbitt, C. (1986), Actuarial Mathematics, Itasca IL: Society of Actuaries.

Centeno, M. d. L. (2002), Measuring the effects of reinsurance by the adjustment coefficient in the Sparre-Anderson model, Insurance: Mathematics and Economics 30, 37--49.

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

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second Edition, Wiley.

Examples

## Basic example: no reinsurance, exponential claim severity and wait
## times, premium rate computed with expected value principle and
## safety loading of 20\%.
adjCoef(mgfexp, premium = 1.2, upper = 1)

## Same thing, giving function h.
h <- function(x) 1/((1 - x) * (1 + 1.2 * x))
adjCoef(h = h, upper = 1)

## Example 8.7, Klugman et al. (2004)
mgfx <- function(x) 0.6 * exp(x) + 0.4 * exp(2 * x)
adjCoef(mgfx(x), mgfexp(x, 4), prem = 7, upper = 0.3182)

## Proportional reinsurance, same assumptions as above, reinsurer's
## safety loading of 30\%.
mgfx <- function(x, y) mgfexp(x * y)
p <- function(x) 1.3 * x - 0.1
h <- function(x, a) 1/((1 - a * x) * (1 + x * p(a)))
R1 <- adjCoef(mgfx, premium = p, upper = 1, reins = "proportional",
              from = 0, to = 1, n = 11)
R2 <- adjCoef(h = h, upper = 1, reins = "p",
             from = 0, to = 1, n = 101)
R1(seq(0, 1, length = 10))	# evaluation for various retention rates 
R2(seq(0, 1, length = 10))	# same
plot(R1)		        # graphical representation
plot(R2, col = "green", add = TRUE) # smoother function

## Excess-of-loss reinsurance
p <- function(x) 1.3 * levgamma(x, 2, 2) - 0.1
mgfx <- function(x, l)
    mgfgamma(x, 2, 2) * pgamma(l, 2, 2 - x) +
    exp(x * l) * pgamma(l, 2, 2, lower = FALSE)
h <- function(x, l) mgfx(x, l) * mgfexp(-x * p(l))
R1 <- adjCoef(mgfx, upper = 1, premium = p, reins = "excess-of-loss",
             from = 0, to = 10, n = 11)
R2 <- adjCoef(h = h, upper = 1, reins = "e",
             from = 0, to = 10, n = 101)
plot(R1)
plot(R2, col = "green", add = TRUE)