ACTU-457 (Mast 724) Risk Theory

aa <- c(0.1,0.2,0.5,0.8,0.9)
qgamma(aa,1,1)+qgamma(aa,1,1)

[1] 0.2107210 0.4462871 1.3862944 3.2188758
[5] 4.6051702

qgamma(aa,2,1)

[1] 0.5318116 0.8243883 1.6783470 2.9943083
[5] 3.8897202

xx <- c(0, 100, 200, 500, 1000)
pdf <- c(0.3, 0.4, 0.15, 0.1, 0.05)
cdf <- c(0.3, 0.7, 0.85, 0.95, 1)
EX <- sum(xx*pdf)
aa <- c(0.22,0.3,0.39,0.8501, 0.95, 0.9999)
(TVaR.39 <- ((0.15*200+0.1*500+0.05*1000)+
        100*(0.7-0.39))/(1-0.39))

aa <- c(0.9, 0.99, 0.999, 0.9999)
ss <- log(1000^2/(1.5*exp(2*(log(1000))))+1)
(VaR.ln <- exp(log(1000)-(0.5*ss^2)+ss*qnorm(aa)))
EY <- exp(log(1000)-0.5*ss^2+1/2*ss^2)
qgamma(aa,1.5,1.5/1000)
(TVaR.ln <- EY*(1-pnorm(qnorm(aa)-ss))/(1-aa))
(TVaR.G <- 1000*(1-pgamma(qgamma(aa,1.5,1.5/1000),2.5,1.5/1000))/(1-aa))

xx <- c(0, 100, 500, 1000, 5000, 10000)
px <- c(0.5, 0.3, 0.1, 0.05, 0.043, 0.007)
sum(xx*px)
sum(xx^2*px)
sum(xx^2*px)-(sum(xx*px))^2
(5000-3000)*0.043+(10000-3000)*0.007
sum(xx*px)-(10000-7000)*0.007

# Simulation
# Finite sum of independent rvs
##
#S = X1 + X2 + X3
##
#Distribution of X1 : log normal
#mu<-log(1000)-0.5
#sig<-1
# Distribution of X1 : gamma
al1<-2.5
be1<-2.5/1000
# Distribution of X2 : gamma
al2<-0.5
be2<-0.5/1000
# Distribution of X3 : log normal
mu<-log(1000)-0.5
sig<-1
##
#Expectations
#
EX1<-al1/be1
EX2<-al2/be2
EX3<-exp(mu+(sig^2)/2)
ES<-EX1+EX2+EX3
##
#Variance
#
VarX1<-EX1/be1
VarX2<-EX2/be2
VarX3<-exp(2*mu+2*(sig^2))-(EX3^2)
VarS<-VarX1+VarX2+VarX3
##
#Simulation
##
#simulation of nsim samples of U (a rv with Unif(0,1))
set.seed(19661122)
nsim<-1000000 # nb of simulations
nrisk<-3 # nb of risks (X1, X2, X3)
matU<-matrix(runif(nsim*nrisk),nsim,nrisk,byrow=TRUE)
# simulation of nsim samples of X1, X2, X3
vX1<-qgamma(matU[,1],al1,be1)
vX2<-qgamma(matU[,2],al2,be2)
vX3<-qlnorm(matU[,3],mu,sig)
vS<-vX1+vX2+vX3
c(EX1,mean(vX1))

[1] 1000.0000  998.6502

c(EX2,mean(vX2))

[1] 1000.0000  999.0069

c(EX3,mean(vX3))

[1] 1000.0000  999.7506

c(ES,mean(vS))

[1] 3000.000 2997.408

c(VarX1,var(vX1))

[1] 400000.0 398986.8

c(VarX2,var(vX2))

[1] 2000000 1985585

c(VarX3,var(vX3))

[1] 1718282 1699641

c(VarS,var(vS))

[1] 4118282 4077440

##
#Approximation of FS
# Build a function which computes
# the approximated values of FS from the sample vS
FSn<-ecdf(vS)
vx<-(0:100)*100
approxFS<-FSn(vx) # computes approximated values of FS(x) at components of vx
plot(vx,approxFS,type="l",main="Approximated values of cdf of S",xlab="x",ylab="")

###Approximation of VaRS and TVaRS
vkap<-(1:99)/100
approxVaRS<-quantile(vS,probs=vkap,type=1,names=FALSE) # Approximated values of VaR of S
plot(vkap,approxVaRS,type="l",main="Approximated values of VaR of S",xlab="kappa",ylab="")

##
funTVaR<-function(kap) mean(vS[vS>quantile(vS,probs=kap,type=1,names=FALSE)])
vkap<-(1:99)/100
approxTVaRS<-sapply(vkap,funTVaR)
matplot(vkap,cbind(approxVaRS,approxTVaRS),type="l",main="Approximated values of VaR and
TVaR of S",xlab="kappa",ylab="")