SlicedDistributions

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: \[E(y)= \left[\begin{array} {rrr} Capped Mean (g(y)@c) & c \le s \\ Capped Mean (g(x) @ s)+(1-G(s))*(Capped Mean (h(y)@c)-s) & c > s \end{array}\right] \]

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))

summary(XoLCededClaims)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       0       0       0    5300       0  201873