| adjCoef {actuar} | R Documentation |
Compute the adjustment coefficient in ruin theory, or return a function to compute the adjustment coefficient for various reinsurance retentions.
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, ...)
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. |
In the typical case reinsurance = "none", the coefficient of
determination is the smallest (strictly) positive root of the Lundberg
equation
h(x) = E[exp(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[exp(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[exp(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 by minimizing (h(x) - 1)^2.
If reinsurance = "none", a numeric vector of lenght one.
Otherwise, a function of class "adjCoef" inheriting from the
"function" class.
Christophe Dutang, Vincent Goulet vincent.goulet@act.ulaval.ca
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.
## 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)