aggregateDist: 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 aggregateDist
print(x, ...)
# S3 method for aggregateDist
plot(x, xlim, ylab = expression(F[S](x)),
     main = "Aggregate Claim Amount Distribution",
     sub = comment(x), ...)
# S3 method for aggregateDist
summary(object, ...)
# S3 method for aggregateDist
mean(x, ...)
# S3 method for aggregateDist
diff(x, ...)

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.

Arguments

method: method to be used
model.freq: for "recursive" method: a character string giving the name of a distribution in the $(a, b, 0)$ or $(a, b, 1)$ families of distributions. For "convolution" method: a vector of claim number probabilities. For "simulation" method: a frequency simulation model (see rcomphierarc for details) or NULL. Ignored with normal and npower methods.
model.sev: for "recursive" and "convolution" methods: a vector of claim amount probabilities. For "simulation" method: a severity simulation model (see rcomphierarc for details) or NULL. Ignored with normal and npower methods.
p0: arbitrary probability at zero for the frequency distribution. Creates a zero-modified or zero-truncated distribution if not NULL. Used only with "recursive" method.
x.scale: value of an amount of 1 in the severity model (monetary unit). Used only with "recursive" and "convolution" methods.
convolve: number of times to convolve the resulting distribution with itself. Used only with "recursive" method.
moments: vector of the true moments of the aggregate claim amount distribution; required only by the "normal" or "npower" methods.
nb.simul: number of simulations for the "simulation" method.
...: parameters of the frequency distribution for the "recursive" method; further arguments to be passed to or from other methods otherwise.
tol: the resulting cumulative distribution in the "recursive" method will get less than tol away from 1.
maxit: maximum number of recursions in the "recursive" method.
echo: logical; echo the recursions to screen in the "recursive" method.
x, object: an object of class "aggregateDist".
xlim: numeric of length 2; the $x$ limits of the plot.
ylab: label of the y axis.
main: main title.
sub: subtitle, defaulting to the calculation method.

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 $n$th 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

Vincent Goulet vincent.goulet@act.ulaval.ca and Louis-Philippe Pouliot

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.

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.

Examples

Run this code

## 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)
if (FALSE) Fs <- aggregateDist("recursive", model.freq = "poisson",
                    model.sev = fx, lambda = 1000)
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")

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