1.Introduction
This vignette will cover the sliced distributions available in the
NetSimR package. It will overview the theory behind these functions,
propose posible uses and illustrate examples. Currently, Sliced
distributions available are LogNormal and Gamma with a Pareto
distribution for the tail.
2.Backround theory
2.1 Sliced distribution backround
At various insurance modelling exercises, it may be the case that a
sliced distribution provides a significantly better fit than a
non-sliced distribution. Tail losses may be very influential on the
premium, as such modelling the tail more accurately would significantly
improve the accuacy of the premium. This becomes more important when
modelling upper layers that will be mainly based on the severity tail.
The analyst may opt to model attritional losses using a regression and
have a single dimensional Pareto covering all cases. Theory suggests
that Pareto distributions provide an appropriate fit for the tail of
distributions and the mean excess function can be used to set the slice
point. The analyst may consider using a truncated or censored
distribution fit for the attritional data, in order to improve the
fit.
2.2 Sliced distribution definition
Let claim severity \(x\) have a
probability density function \(f(x)\)
and a cumulative density funtion \(F(x)\). Let \(s\) be the slice point. We define \(f(x)\) as: \[f(x)= \left[\begin{array}
{rrr}
g(x) & x \le s \\
(1-G(s))*h(x) & x > s
\end{array}\right]
\] \(g(x)\) is the attritional
attritional probability density function, fitted to the full claims. As
suggested this could be fitted to full claims as a censored distribution
or to the claims below \(s\) as a
truncated distribution for a better fit. \(g(x)\) in NetSimR package will either be
Gamma or LogNormal. \(h(x)\) is the
large probability density function. \(h(x)\) in NetSimR package will be a single
parameter Pareto distribution. The scale paramter \(x_{m}\) will be the slice point \(s\). The shape parameter will be estimated
from the claims larger than the slice point. \(H(x)\) and \(G(x)\) are the cumulative density functions
of \(h(x)\) and \(g(x)\) respectively.
We define \(F(x)\) as: \[F(x)= \left[\begin{array}
{rrr}
G(x) & x \le s \\
G(s)+(1-G(s))*H(x) & x > s
\end{array}\right]
\] The mean \(E(x)\) is defined
to be: \[E(x)=Capped Mean (g(x) @ s) +
(1-G(s))*(Mean(h(x))-s)\] Let claims be capped at amount \(c\). The capped claim severity is defined
as \(y=Min(x,c)\).The capped mean \(E(y)\) is defined to be: \[E(y)= \left[\begin{array}
{rrr}
Capped Mean (g(y)@c) & c \le s \\
Capped Mean (g(x) @ s)+(1-G(s))*(Capped Mean (h(y)@c)-s) & c > s
\end{array}\right]
\]
3. Functions usage
NetSimR package provides functions for the mean, capped mean,
exposure curve and increased limit factor curves. These were discussed
in the “Capped mean, Exposure and Increased Limit Factor curves”
vignette and will not be repeated here. The package also provides the
probability density function (pdf), the cumulative probability density
function (cdf) and the inverse cumulative probability function for the
sliced distributions.
The probability density function can be used for optimising
parameters with the maximum likelihood method, when fitting such a
distribution, in cases where fitting the distributions separately is not
appropriate.
The cumulative density function provides probability of claims being
less or more than a particular level. The distribution can be used
alongst methods such as the Panjer recursion, in order to combine it
with the frequency distribution and provide the aggrecate claims
distribution.
The inverse cumulative probability distribution can be used to run
simulations over the defined distribution. Reinsurance strucutures can
then be applied to use the simulation exercise for reinsurance
pricing.
4. Examples
4.1 Mean of a sliced distribution
Here we calculate the expected severity from a sliced lognormal
pareto distributions, sliced at 10,000 currency units, with a Pareto
shape parameter \(shape = 1.2\) and
lognormal parameters \(mu = 6\) and
\(sigma = 1.6\).
library("NetSimR")
round(SlicedLNormParetoMean(6,1.6,10000,1.2),0)
## [1] 2299
Multiplying the above with claim frequency would provide the expected
claims cost. A capped mean of a sliced distribution follows the capped
mean vignette examples.
4.2 Sliced distribution simulations
We will only demonstrate simulating from the inverse distribution,
the probability density and cumulative probability functions’ use is
simpler. Below we simulate from a sliced distribution, applying an each
and every loss excess of loss reinsurance structure.
#set seed
set.seed(1)
#set parameters
numberOfSimulations=20000
freq=15
mu=6
sigma=1.6
s=1000
alpha=1.2
Deductible=10000
limit=100000
XoLCededClaims <- numeric(length = numberOfSimulations)
#loop simulates frequency then severity and applies layer deductible and limit
for (i in 1 :numberOfSimulations){
x=qSlicedLNormPareto(runif(rpois(1,freq)),mu,sigma,s,alpha)
x=ifelse(x<Deductible,0,ifelse(x-Deductible>limit,limit,x-Deductible))
XoLCededClaims[i]=round(sum(x),0)
}
#Visualising Simulation results
head(XoLCededClaims)
## [1] 0 9463 0 0 0 11173
hist(XoLCededClaims, breaks = 100, xlim = c(0,100000))
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0 0 0 5300 0 201873