Risk and Ruin

Introduction

This is the second in a four-part series on the actuar package for the R programming language.

Risk Theory

Risk theory refers to a body of techniques to model and measure the risk
associated with a portfolio of insurance contracts. A first approach consists
in modeling the distribution of total claims over a fixed period of time using
the classical collective model of risk theory. A second input of interest to
the actuary is the evolution of the surplus of the insurance company over
many periods of time.

Ruin Theory

In ruin theory, the main quantity of interest is the
probability that the surplus becomes negative, in which case technical ruin of
the insurance company occurs.¹

The Collective Risk Model

Let random variable S represent the aggregate claim amount (or total amount
of claims) of a portfolio of independent risks over a fixed period of time, random variable N represent the number of claims (or frequency) in the portfolio
over that period, and random variable Cj represent the amount of claim j (or
severity). Then, we have the random sum

S = C₁ + … + C_N

where we assume that C1, C2, … are mutually independent and identically
distributed random variables each independent of N.

Risk Functions

discretize

Compute a discrete probability mass function from a continuous cumulative distribution function (cdf) with various methods discretise is an alias for discretize.

discretize(cdf, from, to, step = 1,
method = c("upper", "lower", "rounding", "unbiased"),
lev, by = step, xlim = NULL)

• cdf: an expression written as a function of x, or alternatively the name of a function, giving the cdf to discretize.
• from, to: the range over which the function will be discretized.
• step: numeric; the discretization step (or span, or lag).
• method: discretization method to use (elaborated on below).
• lev: an expression written as a function of x, or alternatively the name of a function, to compute the limited expected value of the distribution corresponding to cdf. Used only with the "unbiased" method.
• by: an alias for step.
• xlim: numeric of length 2; if specified, it serves as default for c(from, to).

Methods

Let F(x) denote the cdf of the distribution to discretize on some interval
(a, b) and f(x) denote the probability mass at x in the discretized distribution.

Upper discretization, or forward difference of F(x):

for x = a, a + h, … , b h. The discretized cdf is always above the true
cdf.

Lower discretization, or backward difference of F(x):

The discretized cdf is always under the true cdf.

Rounding of the random variable, or the midpoint method:

The true cdf passes exactly midway through the steps of the discretized
cdf.

Unbiased, or local matching of the first moment method:

The discretized and the true distributions have the same total probability
and expected value on (a, b).

A graphical comparison of the four methods can be found here.

aggregatedist

Compute the aggregate claim amount cumulative distribution function of a portfolio over a period using one of five methods.

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 "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 simul 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 simul for details) or NULL. Ignored with normal and npower methods.
• p0: arbitrary probability at zero for the frequency distribution. Creates a zeromodified 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.

Ruin Functions

adjcoef

ruin

Approximating to the probability of Ruin