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

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 is a member of the (a, b, 0) family of discrete distributions if p0 is NULL and a member of the (a, b, 1) family if p0 is specified.

model.freq must be one of "binomial", "geometric", "negative binomial", "poisson" or "logarithmic" (these can abbreviated). The parameters of the frequency distribution must be specified using names identical to the arguments of functions dbinom, dgeom, dnbinom, dpois and dnbinom, respectively. (The logarithmic distribution is a limiting case of the negative binomial distribution with size parameter equal to 0.)

model.sev is a vector of the (discretized) claim amount distribution X; the first element must be fx(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 a too coarse discretization of the severity distribution.

Convolution method

The cumulative distribution function (cdf) Fs(x) of the aggregate claim amount of a portfolio in the collective risk model is

Fs(x) = sum(n; Fx^*n(x) * pn)

for x = 0, 1, …; pn = Pr[N = n] is the frequency probability mass function and Fx^*n(x) is the cdf of the nth convolution of the (discrete) claim amount random variable.

model.freq is vector pn of the number of claims probabilities; the first element must be Pr[N = 0].

model.sev is vector fx(x) of the (discretized) claim amount distribution; the first element must be fx(0).

Normal and Normal Power 2 methods

The Normal approximation of a cumulative distribution function (cdf) F(x) with mean m and standard deviation s is

F(x) ~= pnorm((x - m)/s).

The Normal Power 2 approximation of a cumulative distribution function (cdf) F(x) with mean m, standard deviation s and skewness g is

F(x) ~= pnorm(-3/g + sqrt(9/g^2 + 1 + (6/g) * (x - m)/s)).

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. simul 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. (2008), Loss Models, From Data to Decisions, Third Edition, Wiley.

Daykin, C.D., Pentik<c3><83><c2><a4>inen, 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); simul.

Examples

## Convolution method (example 9.5 of Klugman et al. (2008))
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
Fs <- aggregateDist("recursive", model.freq = "poisson",
                    model.sev = fx, lambda = 3, x.scale = 25)
plot(Fs)
Fs(knots(Fs))		      # cdf evaluated at its knots
diff(Fs)                      # probability mass function

## Recursive method (high frequency)
## 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)

## 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")

Results