[SOLVED] R

MAT 4376F/5313F

R Value-at-Risk, Expected Shortfall
Value-at-Risk
Comparison of parametric and nonparametric method for Exponential distribution n=1000;
lambda=2;
p=seq(1,n,by=1)/(n+1);
true.VAR=-log(p)/lambda;
plot(p,true.VAR,type=l);
X=rexp(n,lambda);
hatlambda=1/mean(X);
parametric.VAR=-log(p)/hatlambda;
points(p,parametric.VAR,type=l,col=blue)
nonparametric.VAR=sort(X,decreasing=TRUE);
points(p,nonparametric.VAR,type=p,col=red);

p0=0.05;
-log(p0)/lambda; -log(p0)/hatlambda;
index=floor(n*p0); nonparametric.VAR[index+1];

Comparison of parametric and nonparametric method for Pareto distribution n=1000;
alpha=2;
p=seq(1,n,by=1)/(n+1);
true.VAR=p^(-1/alpha);
plot(p,true.VAR,type=l);
U=runif(n); X=(1-U)^(-1/alpha);
hatalpha=1/mean(log(X));
parametric.VAR=p^(-1/hatalpha);
points(p,parametric.VAR,type=l,col=blue)
nonparametric.VAR=sort(X,decreasing=TRUE);
points(p,nonparametric.VAR,type=p,col=red);

p0=0.05;
p0^(-1/alpha); p0^(-1/hatalpha);
index=floor(n*p0); nonparametric.VAR[index+1];

VaR for danish insurance data n=length(danish);
p=seq(1,n,by=1)/(n+1);
nonparametric.VAR=sort(danish,decreasing=TRUE);
hatalpha=1/mean(log(danish));
Pareto.method=p^(-1/hatalpha);
Normal.method=mean(danish)+sd(danish)*qnorm(1-p);
plot(p,Pareto.method,type=l);
points(p,Normal.method,type=l,col=blue);
points(p,nonparametric.VAR,type=p,col=red)

n=length(danish);
nonparametric.VAR=sort(danish,decreasing=TRUE);

p0=0.05;
index=floor(n*p0);
nonparametric.VAR[index+1]; # nonparametric method
hatalpha=1/mean(log(danish));
p0^(-1/hatalpha); # Pareto method
mean(danish)+sd(danish)*qnorm(1-p0); # normal method

p0=0.01;
index=floor(n*p0);
nonparametric.VAR[index+1]; # nonparametric method
hatalpha=1/mean(log(danish));
p0^(-1/hatalpha); # Pareto method
mean(danish)+sd(danish)*qnorm(1-p0); # normal method

Expected Shortfall
Comparison of parametric and nonparametric method for Exponential distribution n=1000;
lambda=2;
p=seq(1,n,by=1)/(n+1);
true.ES=(-log(p)+1)/lambda;
plot(p,true.ES,type=l);
X=rexp(n,lambda);
hatlambda=1/mean(X);
parametric.ES=(-log(p)+1)/hatlambda;
points(p,parametric.ES,type=l,col=blue);
nonparametric.VAR=sort(X,decreasing=TRUE);
nonparametric.ES=NULL;
for(i in 1:n)
{
value=sum(X[X>=nonparametric.VAR[i]])/n;
nonparametric.ES=c(nonparametric.ES,value);
}
nonparametric.ES=(1/p)*nonparametric.ES;
points(p,nonparametric.ES,type=p,col=red);

plot(p[1:30],true.ES[1:30],type=l);
points(p[1:30],parametric.ES[1:30],type=l,col=blue);
points(p[1:30],nonparametric.ES[1:30],type=p,col=red);