| aggregateDist {actuar} | R Documentation |
Aggregate Claim Amount Distribution
Description
Compute the aggregate claim amount cumulative distribution function of a portfolio over a period using one of five methods.
Usage
aggregateDist(method = c("recursive", "convolution", "normal",
"npower", "simulation"),
model.freq = NULL, model.sev = NULL, p0 = NULL,
x.scale = 1, convolve = 0, moments, nb.simul, ...,
tol = 1e-06, maxit = 500, echo = FALSE)
## S3 method for class 'aggregateDist'
print(x, ...)
## S3 method for class 'aggregateDist'
plot(x, xlim, ylab = expression(F[S](x)),
main = "Aggregate Claim Amount Distribution",
sub = comment(x), ...)
## S3 method for class 'aggregateDist'
summary(object, ...)
## S3 method for class 'aggregateDist'
mean(x, ...)
## S3 method for class 'aggregateDist'
diff(x, ...)
Arguments
method |
method to be used |
model.freq |
for |
model.sev |
for |
p0 |
arbitrary probability at zero for the frequency
distribution. Creates a zero-modified or zero-truncated
distribution if not |
x.scale |
value of an amount of 1 in the severity model (monetary
unit). Used only with |
convolve |
number of times to convolve the resulting distribution
with itself. Used only with |
moments |
vector of the true moments of the aggregate claim
amount distribution; required only by the |
nb.simul |
number of simulations for the |
... |
parameters of the frequency distribution for the
|
tol |
the resulting cumulative distribution in the
|
maxit |
maximum number of recursions in the |
echo |
logical; echo the recursions to screen in the
|
x, object |
an object of class |
xlim |
numeric of length 2; the |
ylab |
label of the y axis. |
main |
main title. |
sub |
subtitle, defaulting to the calculation method. |
Details
aggregateDist returns a function to compute the cumulative
distribution function (cdf) of the aggregate claim amount distribution
in any point.
The "recursive" method computes the cdf using the Panjer
algorithm; the "convolution" method using convolutions; the
"normal" method using a normal approximation; the
"npower" method using the Normal Power 2 approximation; the
"simulation" method using simulations. More details follow.
Value
A function of class "aggregateDist", inheriting from the
"function" class when using normal and Normal Power
approximations and additionally inheriting from the "ecdf" and
"stepfun" classes when other methods are used.
There are methods available to summarize (summary), represent
(print), plot (plot), compute quantiles
(quantile) and compute the mean (mean) of
"aggregateDist" objects.
For the diff method: a numeric vector of probabilities
corresponding to the probability mass function evaluated
at the knots of the distribution.
Recursive method
The frequency distribution must be a member of the (a, b, 0) or
(a, b, 1) families of discrete distributions.
To use a distribution from the (a, b, 0) family,
model.freq must be one of
"binomial",
"geometric",
"negative binomial" or
"poisson",
and p0 must be NULL.
To use a zero-truncated distribution from the (a, b, 1) family,
model.freq may be one of the strings above together with
p0 = 0. As a shortcut, model.freq may also be one of
"zero-truncated binomial",
"zero-truncated geometric",
"zero-truncated negative binomial",
"zero-truncated poisson" or
"logarithmic",
and p0 is then ignored (with a warning if non NULL).
(Note: since the logarithmic distribution is always zero-truncated.
model.freq = "logarithmic" may be used with either p0 =
NULL or p0 = 0.)
To use a zero-modified distribution from the (a, b, 1) family,
model.freq may be one of standard frequency distributions
mentioned above with p0 set to some probability that the
distribution takes the value 0. It is equivalent, but more
explicit, to set model.freq to one of
"zero-modified binomial",
"zero-modified geometric",
"zero-modified negative binomial",
"zero-modified poisson" or
"zero-modified logarithmic".
The parameters of the frequency distribution must be specified using
names identical to the arguments of the appropriate function
dbinom, dgeom, dnbinom,
dpois or dlogarithmic. In the latter case,
do take note that the parametrization of dlogarithmic is
different from Appendix B of Klugman et al. (2012).
If the length of p0 is greater than one, only the first element
is used, with a warning.
model.sev is a vector of the (discretized) claim amount
distribution X; the first element must be f_X(0) =
\Pr[X = 0].
The recursion will fail to start if the expected number of claims is
too large. One may divide the appropriate parameter of the frequency
distribution by 2^n and convolve the resulting distribution
n = convolve times.
Failure to obtain a cumulative distribution function less than
tol away from 1 within maxit iterations is often due
to too coarse a discretization of the severity distribution.
Convolution method
The cumulative distribution function (cdf) F_S(x) of the
aggregate claim amount of a portfolio in the collective risk model is
F_S(x) = \sum_{n = 0}^{\infty} F_X^{*n}(x) p_n,
for x = 0, 1, \dots; p_n = \Pr[N = n] is
the frequency probability mass function and
F_X^{*n}(x) is the cdf of the nth convolution of
the (discrete) claim amount random variable.
model.freq is vector p_n of the number of claims
probabilities; the first element must be \Pr[N =
0].
model.sev is vector f_X(x) of the (discretized)
claim amount distribution; the first element must be
f_X(0).
Normal and Normal Power 2 methods
The Normal approximation of a cumulative distribution function (cdf)
F(x) with mean \mu and standard deviation
\sigma is
F(x) \approx \Phi\left( \frac{x - \mu}{\sigma} \right).
The Normal Power 2 approximation of a cumulative distribution function (cdf)
F(x) with mean \mu, standard deviation \sigma
and skewness \gamma is
F(x) \approx \Phi \left(%
-\frac{3}{\gamma} + \sqrt{\frac{9}{\gamma^2} + 1 %
+ \frac{6}{\gamma} \frac{x - \mu}{\sigma}} \right).
This formula is valid only for the right-hand tail of the distribution and skewness should not exceed unity.
Simulation method
This methods returns the empirical distribution function of a sample
of size nb.simul of the aggregate claim amount distribution
specified by model.freq and
model.sev. rcomphierarc is used for the simulation of
claim amounts, hence both the frequency and severity models can be
mixtures of distributions.
Author(s)
Vincent Goulet vincent.goulet@act.ulaval.ca and Louis-Philippe Pouliot
References
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994), Practical Risk Theory for Actuaries, Chapman & Hall.
See Also
discretize to discretize a severity distribution;
mean.aggregateDist to compute the mean of the
distribution;
quantile.aggregateDist to compute the quantiles or the
Value-at-Risk;
CTE.aggregateDist to compute the Conditional Tail
Expectation (or Tail Value-at-Risk);
rcomphierarc.
Examples
## Convolution method (example 9.5 of Klugman et al. (2012))
fx <- c(0, 0.15, 0.2, 0.25, 0.125, 0.075,
0.05, 0.05, 0.05, 0.025, 0.025)
pn <- c(0.05, 0.1, 0.15, 0.2, 0.25, 0.15, 0.06, 0.03, 0.01)
Fs <- aggregateDist("convolution", model.freq = pn,
model.sev = fx, x.scale = 25)
summary(Fs)
c(Fs(0), diff(Fs(25 * 0:21))) # probability mass function
plot(Fs)
## Recursive method (example 9.10 of Klugman et al. (2012))
fx <- c(0, crossprod(c(2, 1)/3,
matrix(c(0.6, 0.7, 0.4, 0, 0, 0.3), 2, 3)))
Fs <- aggregateDist("recursive", model.freq = "poisson",
model.sev = fx, lambda = 3)
plot(Fs)
Fs(knots(Fs)) # cdf evaluated at its knots
diff(Fs) # probability mass function
## Recursive method (high frequency)
fx <- c(0, 0.15, 0.2, 0.25, 0.125, 0.075,
0.05, 0.05, 0.05, 0.025, 0.025)
## Not run: Fs <- aggregateDist("recursive", model.freq = "poisson",
model.sev = fx, lambda = 1000)
## End(Not run)
Fs <- aggregateDist("recursive", model.freq = "poisson",
model.sev = fx, lambda = 250, convolve = 2, maxit = 1500)
plot(Fs)
## Recursive method (zero-modified distribution; example 9.11 of
## Klugman et al. (2012))
Fn <- aggregateDist("recursive", model.freq = "binomial",
model.sev = c(0.3, 0.5, 0.2), x.scale = 50,
p0 = 0.4, size = 3, prob = 0.3)
diff(Fn)
## Equivalent but more explicit call
aggregateDist("recursive", model.freq = "zero-modified binomial",
model.sev = c(0.3, 0.5, 0.2), x.scale = 50,
p0 = 0.4, size = 3, prob = 0.3)
## Recursive method (zero-truncated distribution). Using 'fx' above
## would mean that both Pr[N = 0] = 0 and Pr[X = 0] = 0, therefore
## Pr[S = 0] = 0 and recursions would not start.
fx <- discretize(pexp(x, 1), from = 0, to = 100, method = "upper")
fx[1L] # non zero
aggregateDist("recursive", model.freq = "zero-truncated poisson",
model.sev = fx, lambda = 3, x.scale = 25, echo=TRUE)
## Normal Power approximation
Fs <- aggregateDist("npower", moments = c(200, 200, 0.5))
Fs(210)
## Simulation method
model.freq <- expression(data = rpois(3))
model.sev <- expression(data = rgamma(100, 2))
Fs <- aggregateDist("simulation", nb.simul = 1000,
model.freq, model.sev)
mean(Fs)
plot(Fs)
## Evaluation of ruin probabilities using Beekman's formula with
## Exponential(1) claim severity, Poisson(1) frequency and premium rate
## c = 1.2.
fx <- discretize(pexp(x, 1), from = 0, to = 100, method = "lower")
phi0 <- 0.2/1.2
Fs <- aggregateDist(method = "recursive", model.freq = "geometric",
model.sev = fx, prob = phi0)
1 - Fs(400) # approximate ruin probability
u <- 0:100
plot(u, 1 - Fs(u), type = "l", main = "Ruin probability")