# Efficient tail-loss probability and conditional excess estimation for t-copula model Using stratified importance sampling (SIS) or naive simulation (NV) the tail-loss probabilities and conditional excess values for several threshold values are estimated for a stock portfolio. The logreturns of the stocks are assumed to follow a t-copula model with generalized hyperbolic or t marginals. ```r SISTCopula(n=10^5,npilot=c(10^4,2*10^4),portfobj,threshold=c(0.95,0.9), stratasize=c(22,22),CEopt=FALSE,beta=0.75,mintype=-1) NVTCopula(n=10^5, portfobj, threshold=c(0.95,0.9)) new.portfobj(nu,R,typemg="GH",parmg,c=rep(1,dim(R)[1]),w=c/sum(c)) ``` ## Arguments - `n`: total sample size - `npilot`: size of one or several pilot runs, the sum of them should be smaller than `n`/2 - `portfobj`: object of portfolio parameters - `threshold`: one or several threshold values (they should be ordered) - `stratasize`: a vector of length two holding the number of strata - `CEopt`: TRUE ... minimize the overall error of Conditional Exess estimates, otherwise of tail-loss estimates - `beta`: weight of maximal threshold value used for calculating the intermediate threshold used for selecting the IS density, only used when `length(threshold)>1` - `mintype`: only used when `length(threshold)>1`; 0 ... minimize mean square errors, -1 ... minimize relative MSE, -2 ... minimize the maximal error, -3 minimize the maximal relative error; a positive integer `j` indicates that the variance of the estimate for the j-th threshold is minimized. - `nu`: degrees of freedom of the t-copula - `R`: correlation matrix of the t-copula - `typemg`: type of the marginal distribution, `"GH"` generalized hyperbolic distribution, `"t"` t-distribution - `parmg`: matrix holding in its rows the parameters of the marginal distribution; for the generalized hyperbolic distribution each row holds the parameters lambda, alpha, beta, delta and mu; for the t-distribution each row holds the parameters mu, sigma and nu (degrees of freedom). - `c`: scale factor vector of the portfolio - `w`: portfolio weights ## Returns For the case that the variable `threshold` contains only one value a matrix containing the results for the tail-loss probability in the first row and that of the conditional excess in the second row is returned. In the case that several threshold values are considered, a list consisting of the result matrices for tail-loss probabilities and for conditional excess and the vector of the threshold values is returned. ## Author(s) Ismail Basoglu, Wolfgang Hormann ## Examples ```r R<- matrix( c(1, 0.554, 0.632, 0.419, 0.400, 0.554,1, 0.495, 0.540, 0.479, 0.632,0.495, 1, 0.426, 0.445, 0.419,0.540, 0.426, 1, 0.443, 0.400,0.479, 0.445, 0.443, 1),ncol=5) pmg<- matrix(NA,ncol=5,nrow=5) colnames(pmg) <- c("lambda","alpha","beta","delta","mu") pmg[1,] <- c(-0.602828, 8.52771, -0.533197, 0.014492, -0.000091) pmg[2,] <- c(-1.331923, 2.72759, -2.573416, 0.019891, 0.001388) pmg[3,] <- c(-1.602705, 3.26482, 1.456542, 0.035139, -0.001662) pmg[4,] <- c(-1.131092, 15.13351, -1.722396, 0.014771, 0.001304) pmg[5,] <- c(-0.955118, 31.14005, 0.896576, 0.015362, -0.000238) portfo <- new.portfobj(nu=8.195,R=R,typemg="GH",parmg=pmg,c=rep(1,5),w=rep(0.2,5)) res1<- SISTCopula(n=10^4,npilot=c(10^3,3*10^3),portfobj=portfo,threshold=c(0.97,0.96,0.95,0.94), stratasize=c(22,22),CEopt=FALSE,beta=0.75,mintype=0) res1 SISTCopula(n=10^4,npilot=c(10^3,3*10^3),portfobj=portfo,threshold=0.94, stratasize=c(22,22),CEopt=FALSE) NVTCopula(n=10^4,portfobj=portfo,threshold=c(0.97,0.96,0.95,0.94)) NVTCopula(n=10^4,portfobj=portfo,threshold=0.94) ######## # example with t-marginals R<- matrix( c(1, 0.551, 0.636, 0.421, 0.398, 0.551,1, 0.496, 0.540, 0.477, 0.636,0.496, 1, 0.428, 0.447, 0.421,0.540, 0.428, 1, 0.444, 0.398,0.477, 0.447, 0.444, 1),ncol=5) pmg<- matrix(NA,ncol=3,nrow=5) colnames(pmg) <- c("mu","sigma","nu") pmg[1,] <- c(-0.000258, 0.013769, 1.78) pmg[2,] <- c(0.000794, 0.012166, 2.64) pmg[3,] <- c(-0.000837, 0.019616, 3.25) pmg[4,] <- c(0.001041, 0.009882, 2.67) pmg[5,] <- c(-0.000104, 0.010812, 3.10) portfo <- new.portfobj(nu=7.525,R=R,typemg="t",parmg=pmg,c=rep(1,5),w=rep(0.2,5)) res1<- SISTCopula(n=10^4,npilot=c(10^3,3*10^3),portfobj=portfo,threshold=c(0.97,0.96,0.95,0.94), stratasize=c(22,22),CEopt=FALSE,beta=0.75,mintype=0) res1 SISTCopula(n=10^4,npilot=c(10^3,3000),portfobj=portfo,threshold=0.94,stratasize=c(22,22)) NVTCopula(n=10^4,portfobj=portfo,threshold=c(0.97,0.96,0.95,0.94)) NVTCopula(n=10^4,portfobj=portfo,threshold=0.94) ```

SISTCopula function