PortfolioOptim = function( minriskcriterion = "mES" , MinMaxComp = F, percriskcontribcriterion = "mES" ,
R = NULL, mu = NULL , sigma = NULL, M3=NULL,M4=NULL, alpha = 0.05, alphariskbudget = 0.05,
lower = NULL , upper = NULL, Riskupper = NULL, Returnlower = NULL, RBlower = NULL , RBupper = NULL, precision = 1e-3 ,
controlDE = list( VTR = 0 , NP=200, trace = FALSE ) , heuristic=TRUE ){
# Description:
# This function produces the portfolio that minimimizes
# either the portfolio risk (when MinMaxComp = F) or the portfolio concentration (when MinMaxComp = T)
# subject to
# lower <= weights <= upper
# risk <= Riskupper
# expected return >= Returnlower
# RBlower <= percentage risk <= RBupper
# Input:
# Either the multivariate return series is given or estimates of the mean, covariance, coskewness or cokurtosis
require(zoo); require(PerformanceAnalytics); require(DEoptim)
if( !is.null(R) ){
R = clean.boudt2( R , alpha = alphariskbudget )[[1]];
T = nrow(R);N=ncol(R)
mu = matrix( as.vector(apply(R,2,'mean')),ncol=1);
sigma = cov(R);
M3 = PerformanceAnalytics:::M3.MM(R)
M4 = PerformanceAnalytics:::M4.MM(R)
}else{ N = length(mu) }
if( is.null(lower) ){ lower = rep(0,N) } ; if( is.null(upper) ){ upper = rep(1,N) }
if( is.null(RBlower) ){ RBlower = rep(-Inf,N) } ; if( is.null(RBupper) ){ RBupper = rep(Inf,N) }
if( is.null(Riskupper) ){ Riskupper = Inf } ; if( is.null(Returnlower) ){ Returnlower = -Inf }
switch( percriskcontribcriterion ,
StdDev = { percriskcontrib = function(w){ return( Portsd(w,mu=mu,sigma=sigma) ) }},
GVaR = { percriskcontrib = function(w){ return( PortgausVaR(w,alpha=alphariskbudget,mu=mu,sigma=sigma) ) }},
GES = { percriskcontrib = function(w){ return( PortgausES(w,mu=mu,alpha=alphariskbudget,sigma=sigma) ) }},
mVaR = { percriskcontrib = function(w){ return( PortMVaR(w,mu=mu,alpha=alphariskbudget,sigma=sigma,M3=M3,M4=M4) ) }},
mES = { percriskcontrib = function(w){ return( operPortMES(w,mu=mu,alpha=alphariskbudget,sigma=sigma,M3=M3,M4=M4) ) }}
) #end function that finds out which percentage risk contribution criterion to use
switch( minriskcriterion ,
StdDev = { prisk = function(w){ return( stddevfun(w,mu=mu,sigma=sigma) ) }},
GVaR = { prisk = function(w){ return( gausVaRfun(w,alpha=alpha,mu=mu,sigma=sigma) ) }},
GES = { prisk = function(w){ return( gausESfun(w,mu=mu,alpha=alpha,sigma=sigma) ) }},
mVaR = { prisk = function(w){ return( MVaRfun(w,mu=mu,alpha=alpha,sigma=sigma,M3=M3,M4=M4) )}},
mES = { prisk = function(w){ return( operMESfun(w,mu=mu,alpha=alpha,sigma=sigma,M3=M3,M4=M4)) }}
) #end function that finds out which risk function to minimize
abspenalty = 1e6; relpenalty = 100*N;
if( heuristic ){
objective = function( w ){
w = w/sum(w)
cont = percriskcontrib( w ); percrisk = cont[[3]]; crisk = cont[[2]] ;
if(MinMaxComp){ out = max( crisk ) }else{ out = prisk(w) }
# add weight constraints
out_con = out + out*abspenalty*sum( 1*( w < lower )+1*( w > upper ) )
# add risk budget constraint
con1 = sum( (percrisk-RBupper)*( percrisk > RBupper ),na.rm=TRUE ) ; if( is.na(con1) ){ con1 = 0 } # because of Inf*0
con2 = sum( (RBlower-percrisk)*( percrisk < RBlower ),na.rm=TRUE ); if( is.na(con2) ){ con2 = 0 }
out_con = out_con + out*relpenalty*(con1+con2)
# add minimum risk constraint
con = ( prisk(w) - Riskupper)*( prisk(w) > Riskupper ); if( is.na(con) ){ con = 0 }
out_con = out_con + out*relpenalty*con
# add minimum return constraint
# portfolio return and risk are in the same unit, but portfolio return is typically some orders of magnitude smaller
# say: as a very conservative choice: 100
preturn = sum( w*mu ) ;
con = ( Returnlower - preturn)*( preturn < Returnlower ); if( is.na(con) ){ con = 0 }
out_con = out_con + out*100*relpenalty*con
return(out_con)
}
minw = DEoptim( fn = objective , lower = lower , upper = upper , control = controlDE)
fvalues = minw$member$bestval
diff = as.vector( quantile(fvalues,0.10) - min(fvalues) )
print(c("diff",diff))
best = min( fvalues ) ; print(best)
bestsol = minw ;
while( diff>1e-6 ){
pop = as.matrix(minw$member$pop)
pop[1,] = minw$optim$bestmem;
minw = DEoptim( fn = objective , lower = lower , upper = upper ,
control = list( itermax = 150, NP=as.numeric(nrow(pop)) , initialpop=pop,trace=F ) )
fvalues = minw$member$bestval
diff = best - min(fvalues) ;
if( diff > 0 ){ best = min( fvalues ) ; bestsol = minw ; print(best) }
}
minw = bestsol$optim$bestmem/sum(bestsol$optim$bestmem) ; #full investment constraint
}else{
objective = function( w ){
if(sum(w)==0){w=w+0.001}
w = w/sum(w)
cont = percriskcontrib( w ); percrisk = cont[[3]]; crisk = cont[[2]] ;
if(MinMaxComp){ out = max( crisk ) }else{ out = prisk(w) }
# add risk budget constraint
con1 = sum( (percrisk-RBupper)*( percrisk > RBupper ),na.rm=TRUE ) ; if( is.na(con1) ){ con1 = 0 } # because of Inf*0
con2 = sum( (RBlower-percrisk)*( percrisk < RBlower ),na.rm=TRUE ); if( is.na(con2) ){ con2 = 0 }
out_con = out + out*relpenalty*(con1+con2)
# add minimum risk constraint
con = ( prisk(w) - Riskupper)*( prisk(w) > Riskupper ); if( is.na(con) ){ con = 0 }
out_con = out_con + out*relpenalty*con
return(out_con)
}
if(Returnlower==-Inf){
inittheta = rep(1,N)/N
out = optim( par=inittheta, f = objective, lower = lower, upper = upper )
}else{
Amat = rbind(diag(x =1,nrow=N,ncol=N), diag(x =-1,nrow=N,ncol=N), rep(1,N), rep(-1,N),as.vector(mu))
inittheta = rep(0.001,N);
inittheta[mu==max(mu)] = 1; inittheta = 1-sum(inittheta[mu!=max(mu)] );
out = constrOptim( theta=inittheta, f = objective, grad=NULL,ui=Amat,
ci = c(rep(0,N),rep(-1,N),0.99999,-1.0001,Returnlower) )
}
minw = out$par/sum(out$par)
}
cont = percriskcontrib( minw ); percrisk = cont[[3]]; crisk = cont[[2]] ;
# check
print( "out = list( minw , sum( minw*mu ) , prisk(minw) , percriskcontrib(minw)" )
out = list( minw , sum( minw*mu ) , prisk(minw) , percrisk , crisk )
print( out )
return(out)
}
findportfolio.dynamic = function(R, from, to, names.input = NA, names.assets,
p = 0.95 , priskbudget = 0.95, mincriterion = "mES" , percriskcontribcriterion = "mES" ,
strategy , controlDE = list( VTR = 0 , NP=200 ) )
{ # @author Kris Boudt and Brian G. Peterson
# Description:
#
# Performs a loop over the reallocation periods with estimation samples given by from:to
# It calls the function RBconportfolio to obtain the optimal weights of the strategy.
#
# @todo
#
# R matrix/zoo holding historical returns on risky assets
#
# names vector holding the names of the .csv files to be read
#
# from, to define the estimation sample
#
# criteria the criterion to be optimized
#
# columns.crit the columns of R in which the criteria are located
#
# percriskcontribcriterion risk measure used for the risk budget constraints
#
# strategy = c( "EqualRisk" , "EqualWeight" , "MinRisk" , "MinRiskConc" ,
# "MinRisk_PositionLimit" , "MinRisk_RiskLimit" , "MinRisk_ReturnTarget",
# "MinRiskConc_PositionLimit" , "MinRiskConc_RiskLimit" , "MinRiskConc_ReturnTarget")
# Return:
# List with first element optimal weights per reallocation period and associated percentage CVaR contributions.
# Create a matrix that will hold for each method and each vector the best weights
cPeriods = length(from);
out = matrix( rep(0, cPeriods*(cAssets)) , ncol= (cAssets) );
RCout = matrix( rep(0, cPeriods*(cAssets)) , ncol= (cAssets) );
# first cPeriods rows correspond to cCriteria[1] and so on
# downside risk
alpha = 1 - p;
alphariskbudget = 1-priskbudget;
# Estimation of the return mean vector, covariance, coskewness and cokurtosis matrix
if(strategy=="EqualRisk"){
lower = rep(0,cAssets); upper=rep(1,cAssets)
RBlower = rep(1/cAssets,cAssets) ; RBupper = rep(1/cAssets,cAssets) ;
}
if(strategy=="EqualWeight"){
lower = rep(1/cAssets,cAssets); upper=rep(1/cAssets,cAssets)
RBlower = rep(-Inf,cAssets) ; RBupper = rep(Inf,cAssets) ;
}
if(strategy=="MinRisk" | strategy=="MinRiskConc" | strategy=="MinRisk_ReturnTarget" | strategy=="MinRiskConc_ReturnTarget"){
lower = rep(0,cAssets); upper=rep(1,cAssets)
RBlower = rep(-Inf,cAssets) ; RBupper = rep(Inf,cAssets) ;
}
MinMaxComp = F; mutarget = -Inf;
if( strategy=="MinRiskConc" | strategy=="MinRiskConc_PositionLimit" | strategy=="MinRiskConc_RiskLimit" | strategy=="MinRiskConc_ReturnTarget" ){
MinMaxComp = T;
}
if(strategy=="MinRisk_PositionLimit" | strategy=="MinRiskConc_PositionLimit"){
lower = rep(0,cAssets); upper=rep(0.4,cAssets)
RBlower = rep(-Inf,cAssets) ; RBupper = rep(Inf,cAssets) ;
}
if(strategy=="MinRisk_RiskLimit" | strategy=="MinRiskConc_RiskLimit"){
lower = rep(0,cAssets); upper=rep(1,cAssets)
RBlower = rep(-Inf,cAssets) ; RBupper = rep(0.40,cAssets) ;
}
for( per in c(1:cPeriods) ){
print("-----------New estimation period ends on observation------------------")
print( paste(to[per],"out of total number of obs equal to", max(to) ));
print("----------------------------------------------------------------")
# Estimate GARCH model with data from inception
inception.R = window(R, start = as.Date(from[1]) , end = as.Date(to[per]) );
# Estimate comoments of innovations with rolling estimation windows
in.sample.R = window(R, start = as.Date(from[per]) , end = as.Date(to[per]) );
in.sample.R = checkData(in.sample.R, method="matrix");
# Estimation of mean return
M = c();
library(TTR)
Tmean = 47 # monthly returns: 4 year exponentially weighted moving average
for( i in 1:cAssets ){
M = cbind( M , as.vector( EMA(x=inception.R[,i],n=Tmean) ) ) #2/(n+1)
}
M = zoo( M , order.by=time(inception.R) )
# Center returns (shift by one observations since M[t,] is rolling mean t-Tmean+1,...,t; otherwise lookahead bias)
inception.R.cent = inception.R;
ZZ = matrix( rep(as.vector( apply( inception.R[1:Tmean, ] , 2 , 'mean' )),Tmean),byrow=T,nrow=Tmean);
inception.R.cent[1:Tmean,] = inception.R[1:Tmean, ] - ZZ;
if( nrow(inception.R)>(Tmean+1) ){
A = M[Tmean:(nrow(inception.R)-1),];
A = zoo( A , order.by = time(inception.R[(Tmean+1):nrow(inception.R), ])) ; #shift dates; otherwise zoo poses problem
inception.R.cent[(Tmean+1):nrow(inception.R), ] = inception.R[(Tmean+1):nrow(inception.R), ] - A}
# Garch estimation
S = c();
for( i in 1:cAssets ){
gout = garchFit(formula ~ garch(1,1), data = inception.R.cent[,i],include.mean = F, cond.dist="QMLE", trace = FALSE )
if( as.vector(gout@fit$coef["alpha1"]) < 0.01 ){
sigmat = rep( sd( as.vector(inception.R.cent[,i])), length(inception.R.cent[,i]) );
}else{
sigmat = gout@sigma.t
}
S = cbind( S , sigmat)
}
S = zoo( S , order.by=time(inception.R.cent) )
# Estimate correlation, coskewness and cokurtosis matrix locally using cleaned innovation series in three year estimation window
selectU = window(inception.R.cent, start = as.Date(from[per]) , end = as.Date(to[per]) )
selectU = selectU/window(S, start = as.Date(from[per]) , end = as.Date(to[per]) );
selectU = clean.boudt2(selectU , alpha = 0.05 )[[1]];
Rcor = cor(selectU)
D = diag( as.vector(tail(S,n=1) ),ncol=cAssets )
sigma = D%*%Rcor%*%D
# we only need mean and conditional covariance matrix of last observation
mu = matrix(tail(M,n=1),ncol=1 ) ;
D = diag( as.vector(as.vector(tail(S,n=1) ) ),ncol=cAssets )
sigma = D%*%Rcor%*%D
in.sample.T = nrow(selectU);
# set volatility of all U to last observation, such that cov(rescaled U)=sigma
selectU = selectU*matrix( rep(as.vector(tail(S,n=1)),in.sample.T ) , ncol = cAssets , byrow = T )
M3 = PerformanceAnalytics:::M3.MM(selectU)
M4 = PerformanceAnalytics:::M4.MM(selectU)
mESfun = function(series){ return( operMES(series,alpha=alpha,2) ) }
if(strategy=="MinRisk_ReturnTarget" | strategy=="MinRiskConc_ReturnTarget"){
mutarget = mean( mu );
print( c("Minimum return requirement is" , mutarget) )
}
if(strategy=="EqualWeight"){
sol1 = rep(1/cAssets,cAssets);
switch( percriskcontribcriterion ,
StdDev = { percriskcontrib = function(w){ cont = Portsd(w,mu=mu,sigma=sigma)[[3]] ; return( cont ) }},
GVaR = {percriskcontrib = function(w){ cont = PortgausVaR(w,alpha=alphariskbudget,mu=mu,sigma=sigma)[[3]] ; return( cont ) }},
GES = {percriskcontrib = function(w){ cont = PortgausES(w,mu=mu,alpha=alphariskbudget,sigma=sigma)[[3]] ; return( cont ) }},
mVaR = {percriskcontrib = function(w){ cont = PortMVaR(w,mu=mu,alpha=alphariskbudget,sigma=sigma,M3=M3,M4=M4)[[3]] ; return( cont ) }},
mES = {percriskcontrib = function(w){ cont = operPortMES(w,mu=mu,alpha=alphariskbudget,sigma=sigma,M3=M3,M4=M4)[[3]] ; return( cont ) }
}
)
sol2 = percriskcontrib( sol1 )
solution = list( sol1 , sol2 ) ;
}else{
solution = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = MinMaxComp, percriskcontribcriterion = "mES" ,
mu = mu , sigma = sigma, M3=M3 , M4=M4 , alpha = alpha , alphariskbudget = alphariskbudget ,
lower = lower , upper = upper , Riskupper = Inf , Returnlower= mutarget , RBlower = RBlower, RBupper = RBupper ,
controlDE = controlDE )
solution = list( solution[[1]] , solution[[4]] );
}
out[ per, ] = as.vector( solution[[1]] )
RCout[per, ] = as.vector( solution[[2]] )
}#end loop over the rebalancing periods; indexed by per=1,...,cPeriods
# Output save
rownames(out) = rownames(RCout) = names.input; colnames(out) = colnames(RCout) = names.assets;
EWweights = c( rep(1/cAssets,cAssets) )
EWweights = matrix ( rep(EWweights,cPeriods) , ncol=(cAssets) , byrow = TRUE )
rownames(EWweights) = names.input; colnames(EWweights) = names.assets;
return( list( out, RCout) )
}
clean.boudt2 =
function (R, alpha = 0.01, trim = 0.001)
{
# @author Kris Boudt and Brian Peterson
# Cleaning method as described in
# Boudt, Peterson and Croux. 2009. Estimation and decomposition of downside risk for portfolios with non-normal returns. Journal of Risk.
stopifnot("package:robustbase" %in% search() || require("robustbase",
quietly = TRUE))
R = checkData(R, method = "zoo")
T = dim(R)[1]
date = c(1:T)
N = dim(R)[2]
MCD = covMcd(as.matrix(R), alpha = 1 - alpha)
# mu = as.matrix(MCD$raw.center)
mu = MCD$raw.center
sigma = MCD$raw.cov
invSigma = solve(sigma)
vd2t = c()
cleaneddata = R
outlierdate = c()
for (t in c(1:T)) {
d2t = as.matrix(R[t, ] - mu) %*% invSigma %*% t(as.matrix(R[t,] - mu))
vd2t = c(vd2t, d2t)
}
out = sort(vd2t, index.return = TRUE)
sortvd2t = out$x
sortt = out$ix
empirical.threshold = sortvd2t[floor((1 - alpha) * T)]
T.alpha = floor(T * (1 - alpha)) + 1
cleanedt = sortt[c(T.alpha:T)]
for (t in cleanedt) {
if (vd2t[t] > qchisq(1 - trim, N)) {
cleaneddata[t, ] = sqrt(max(empirical.threshold,
qchisq(1 - trim, N))/vd2t[t]) * R[t, ]
outlierdate = c(outlierdate, date[t])
}
}
return(list(cleaneddata, outlierdate))
}
Ipower = function(power,h)
{
fullprod = 1;
if( (power%%2)==0 ) #even number: number mod is zero
{
pstar = power/2;
for(j in c(1:pstar))
{
fullprod = fullprod*(2*j)
}
I = fullprod*dnorm(h);
for(i in c(1:pstar) )
{
prod = 1;
for(j in c(1:i) )
{
prod = prod*(2*j)
}
I = I + (fullprod/prod)*(h^(2*i))*dnorm(h)
}
}
else{
pstar = (power-1)/2
for(j in c(0:pstar) )
{
fullprod = fullprod*( (2*j)+1 )
}
I = -fullprod*pnorm(h);
for(i in c(0:pstar) )
{
prod = 1;
for(j in c(0:i) )
{
prod = prod*( (2*j) + 1 )
}
I = I + (fullprod/prod)*(h^( (2*i) + 1))*dnorm(h)
}
}
return(I)
}
# Definition of statistics needed to compute Gaussian and modified VaR and ES for the return series of portfolios
# and to compute the contributions to portfolio downside risk, made by the different positions in the portfolio.
#----------------------------------------------------------------------------------------------------------------
m2 = function(w,sigma)
{
return(t(w)%*%sigma%*%w)
}
derm2 = function(w,sigma)
{
return(2*sigma%*%w)
}
m3 = function(w,M3)
{
return(t(w)%*%M3%*%(w%x%w))
}
derm3 = function(w,M3)
{
return(3*M3%*%(w%x%w))
}
m4 = function(w,M4)
{
return(t(w)%*%M4%*%(w%x%w%x%w))
}
derm4 = function(w,M4)
{
return(4*M4%*%(w%x%w%x%w))
}
StdDevfun = function(w,sigma){ return( sqrt( t(w)%*%sigma%*%w )) }
GVaRfun = function(w,alpha,mu,sigma){ return (- (t(w)%*%mu) - qnorm(alpha)*sqrt( t(w)%*%sigma%*%w ) ) }
mVaRfun = function(w,alpha,mu,sigma,M3,M4){
pm4 = t(w)%*%M4%*%(w%x%w%x%w) ; pm3 = t(w)%*%M3%*%(w%x%w) ; pm2 = t(w)%*%sigma%*%w ;
skew = pm3 / pm2^(3/2);
exkurt = pm4 / pm2^(2) - 3; z = qnorm(alpha);
h = z + (1/6)*(z^2 -1)*skew
h = h + (1/24)*(z^3 - 3*z)*exkurt - (1/36)*(2*z^3 - 5*z)*skew^2;
return (- (t(w)%*%mu) - h*sqrt( pm2 ) ) }
resmVaRfun = function(w,alpha,mu,sigma,ressigma,M3,M4){
pm4 = t(w)%*%M4%*%(w%x%w%x%w) ; pm3 = t(w)%*%M3%*%(w%x%w) ; pm2 = t(w)%*%sigma%*%w ; respm2 = t(w)%*%resSigma%*%w ;
skew = pm3 / respm2^(3/2);
exkurt = pm4 / respm2^(2) - 3; z = qnorm(alpha);
h = z + (1/6)*(z^2 -1)*skew
h = h + (1/24)*(z^3 - 3*z)*exkurt - (1/36)*(2*z^3 - 5*z)*skew^2;
return (- (t(w)%*%mu) - h*sqrt( pm2 ) ) }
GESfun = function(w,alpha,mu,sigma,M3,M4){
return (- (t(w)%*%mu) + dnorm(qnorm(alpha))*sqrt(t(w)%*%sigma%*%w)/alpha ) }
operMESfun = function(w,alpha,mu,sigma,M3,M4){
pm4 = t(w)%*%M4%*%(w%x%w%x%w) ; pm3 = t(w)%*%M3%*%(w%x%w) ; pm2 = t(w)%*%sigma%*%w ;
skew = pm3 / pm2^(3/2);
exkurt = pm4 / pm2^(2) - 3; z = qnorm(alpha);
h = z + (1/6)*(z^2 -1)*skew
h = h + (1/24)*(z^3 - 3*z)*exkurt - (1/36)*(2*z^3 - 5*z)*skew^2;
E = dnorm(h)
E = E + (1/24)*( Ipower(4,h) - 6*Ipower(2,h) + 3*dnorm(h) )*exkurt
E = E + (1/6)*( Ipower(3,h) - 3*Ipower(1,h) )*skew;
E = E + (1/72)*( Ipower(6,h) -15*Ipower(4,h)+ 45*Ipower(2,h) - 15*dnorm(h) )*(skew^2)
E = E/alpha
return (- (t(w)%*%mu) - sqrt(pm2)*min(-E,h) ) }
precision = 4;
Portmean = function(w,mu,precision=4)
{
return( list( round( t(w)%*%mu , precision) , round ( as.vector(w)*as.vector(mu) , precision ) , round( as.vector(w)*as.vector(mu)/t(w)%*%mu) , precision) )
}
Portsd = function(w,sigma,precision=4)
{
pm2 = m2(w,sigma)
dpm2 = derm2(w,sigma)
dersd = (0.5*as.vector(dpm2))/sqrt(pm2);
contrib = dersd*as.vector(w)
return(list( round( sqrt(pm2) , precision ) , round( contrib , precision ) , round ( contrib/sqrt(pm2) , precision) ))
}
PortgausVaR = function(alpha,w,mu,sigma,precision=4){
location = t(w)%*%mu
pm2 = m2(w,sigma)
dpm2 = derm2(w,sigma)
VaR = - location - qnorm(alpha)*sqrt(pm2)
derVaR = - as.vector(mu)- qnorm(alpha)*(0.5*as.vector(dpm2))/sqrt(pm2);
contrib = derVaR*as.vector(w)
return(list( round( VaR , precision ) , round ( contrib , precision ) , round( contrib/VaR , precision) ))
}
PortgausES = function(alpha,w,mu,sigma,precision=4){
location = t(w)%*%mu
pm2 = m2(w,sigma)
dpm2 = derm2(w,sigma)
ES = - location + dnorm(qnorm(alpha))*sqrt(pm2)/alpha
derES = - as.vector(mu) + (1/alpha)*dnorm(qnorm(alpha))*(0.5*as.vector(dpm2))/sqrt(pm2);
contrib = as.vector(w)*derES;
return(list( round( ES , precision ) , round( contrib , precision) , round( contrib/ES , precision) ))
}
PortSkew = function(w,sigma,M3)
{
pm2 = m2(w,sigma)
pm3 = m3(w,M3)
skew = pm3 / pm2^(3/2);
return( skew )
}
PortKurt = function(w,sigma,M4)
{
pm2 = m2(w,sigma)
pm4 = m4(w,M4)
kurt = pm4 / pm2^(2) ;
return( kurt )
}
PortMVaR = function(alpha,w,mu,sigma,M3,M4,precision=4)
{
z = qnorm(alpha)
location = t(w)%*%mu
pm2 = m2(w,sigma)
dpm2 = as.vector( derm2(w,sigma) )
pm3 = m3(w,M3)
dpm3 = as.vector( derm3(w,M3) )
pm4 = m4(w,M4)
dpm4 = as.vector( derm4(w,M4) )
skew = pm3 / pm2^(3/2);
exkurt = pm4 / pm2^(2) - 3;
derskew = ( 2*(pm2^(3/2))*dpm3 - 3*pm3*sqrt(pm2)*dpm2 )/(2*pm2^3)
derexkurt = ( (pm2)*dpm4 - 2*pm4*dpm2 )/(pm2^3)
h = z + (1/6)*(z^2 -1)*skew
h = h + (1/24)*(z^3 - 3*z)*exkurt - (1/36)*(2*z^3 - 5*z)*skew^2;
MVaR = - location - h*sqrt(pm2)
derGausVaR = - as.vector(mu)- qnorm(alpha)*(0.5*as.vector(dpm2))/sqrt(pm2);
derMVaR = derGausVaR + (0.5*dpm2/sqrt(pm2))*( -(1/6)*(z^2 -1)*skew - (1/24)*(z^3 - 3*z)*exkurt + (1/36)*(2*z^3 - 5*z)*skew^2 )
derMVaR = derMVaR + sqrt(pm2)*( -(1/6)*(z^2 -1)*derskew - (1/24)*(z^3 - 3*z)*derexkurt + (1/36)*(2*z^3 - 5*z)*2*skew*derskew )
contrib = as.vector(w)*as.vector(derMVaR)
return(list( round( MVaR , precision) , round( contrib , precision ), round (contrib/MVaR , precision ) ) )
}
derIpower = function(power,h)
{
fullprod = 1;
if( (power%%2)==0 ) #even number: number mod is zero
{
pstar = power/2;
for(j in c(1:pstar))
{
fullprod = fullprod*(2*j)
}
I = -fullprod*h*dnorm(h);
for(i in c(1:pstar) )
{
prod = 1;
for(j in c(1:i) )
{
prod = prod*(2*j)
}
I = I + (fullprod/prod)*(h^(2*i-1))*(2*i-h^2)*dnorm(h)
}
}else{
pstar = (power-1)/2
for(j in c(0:pstar) )
{
fullprod = fullprod*( (2*j)+1 )
}
I = -fullprod*dnorm(h);
for(i in c(0:pstar) )
{
prod = 1;
for(j in c(0:i) )
{
prod = prod*( (2*j) + 1 )
}
I = I + (fullprod/prod)*(h^(2*i)*(2*i+1-h^2) )*dnorm(h)
}
}
return(I)
}
PortMES = function(alpha,w,mu,sigma,M3,M4,precision=4)
{
z = qnorm(alpha)
location = t(w)%*%mu
pm2 = m2(w,sigma)
dpm2 = as.vector( derm2(w,sigma) )
pm3 = m3(w,M3)
dpm3 = as.vector( derm3(w,M3) )
pm4 = m4(w,M4)
dpm4 = as.vector( derm4(w,M4) )
skew = pm3 / pm2^(3/2);
exkurt = pm4 / pm2^(2) - 3;
derskew = ( 2*(pm2^(3/2))*dpm3 - 3*pm3*sqrt(pm2)*dpm2 )/(2*pm2^3)
derexkurt = ( (pm2)*dpm4 - 2*pm4*dpm2 )/(pm2^3)
h = z + (1/6)*(z^2 -1)*skew
h = h + (1/24)*(z^3 - 3*z)*exkurt - (1/36)*(2*z^3 - 5*z)*skew^2;
derh = (1/6)*(z^2 -1)*derskew + (1/24)*(z^3 - 3*z)*derexkurt - (1/18)*(2*z^3 - 5*z)*skew*derskew
E = dnorm(h)
E = E + (1/24)*( Ipower(4,h) - 6*Ipower(2,h) + 3*dnorm(h) )*exkurt
E = E + (1/6)*( Ipower(3,h) - 3*Ipower(1,h) )*skew;
E = E + (1/72)*( Ipower(6,h) -15*Ipower(4,h)+ 45*Ipower(2,h) - 15*dnorm(h) )*(skew^2)
E = E/alpha
MES = - location + sqrt(pm2)*E
derMES = -mu + 0.5*(dpm2/sqrt(pm2))*E
derE = (1/24)*( Ipower(4,h) - 6*Ipower(2,h) + 3*dnorm(h) )*derexkurt
derE = derE + (1/6)*( Ipower(3,h) - 3*Ipower(1,h) )*derskew
derE = derE + (1/36)*( Ipower(6,h) -15*Ipower(4,h)+ 45*Ipower(2,h) - 15*dnorm(h) )*skew*derskew
X = -h*dnorm(h) + (1/24)*( derIpower(4,h) - 6*derIpower(2,h) -3*h*dnorm(h) )*exkurt
X = X + (1/6)*( derIpower(3,h) - 3*derIpower(1,h) )*skew
X = X + (1/72)*( derIpower(6,h) - 15*derIpower(4,h) + 45*derIpower(2,h) + 15*h*dnorm(h) )*skew^2
derE = derE+derh*X # X is a scalar
derE = derE/alpha
derMES = derMES + sqrt(pm2)*derE
contrib = as.vector(w)*as.vector(derMES)
return(list( round( MES , precision ) , round( contrib , precision ), round( contrib/MES, precision )) )
}
operPortMES = function(alpha,w,mu,sigma,M3,M4,precision=4)
{
z = qnorm(alpha)
location = t(w)%*%mu
pm2 = m2(w,sigma)
dpm2 = as.vector( derm2(w,sigma) )
pm3 = m3(w,M3)
dpm3 = as.vector( derm3(w,M3) )
pm4 = m4(w,M4)
dpm4 = as.vector( derm4(w,M4) )
skew = pm3 / pm2^(3/2);
exkurt = pm4 / pm2^(2) - 3;
derskew = ( 2*(pm2^(3/2))*dpm3 - 3*pm3*sqrt(pm2)*dpm2 )/(2*pm2^3)
derexkurt = ( (pm2)*dpm4 - 2*pm4*dpm2 )/(pm2^3)
h = z + (1/6)*(z^2 -1)*skew
h = h + (1/24)*(z^3 - 3*z)*exkurt - (1/36)*(2*z^3 - 5*z)*skew^2;
I1 = Ipower(1,h); I2 = Ipower(2,h); I3 = Ipower(3,h); I4 = Ipower(4,h); I6 = Ipower(6,h);
derh = (1/6)*(z^2 -1)*derskew + (1/24)*(z^3 - 3*z)*derexkurt - (1/18)*(2*z^3 - 5*z)*skew*derskew
E = dnorm(h)
E = E + (1/24)*( I4 - 6*I2 + 3*dnorm(h) )*exkurt
E = E + (1/6)*( I3 - 3*I1 )*skew;
E = E + (1/72)*( I6 -15*I4+ 45*I2 - 15*dnorm(h) )*(skew^2)
E = E/alpha
MES = - location - sqrt(pm2)*min(-E,h)
if(-E<=h){
derMES = -mu + 0.5*(dpm2/sqrt(pm2))*E
derE = (1/24)*( I4 - 6*I2 + 3*dnorm(h) )*derexkurt
derE = derE + (1/6)*( I3 - 3*I1 )*derskew
derE = derE + (1/36)*( I6 -15*I4 + 45*I2 - 15*dnorm(h) )*skew*derskew
X = -h*dnorm(h) + (1/24)*( derIpower(4,h) - 6*derIpower(2,h) -3*h*dnorm(h) )*exkurt
X = X + (1/6)*( derIpower(3,h) - 3*derIpower(1,h) )*skew
X = X + (1/72)*( derIpower(6,h) - 15*derIpower(4,h) + 45*derIpower(2,h) + 15*h*dnorm(h) )*skew^2
derE = derE+derh*X # X is a scalar
derE = derE/alpha
derMES = derMES + sqrt(pm2)*derE }else{
derMES = -mu - 0.5*(dpm2/sqrt(pm2))*h - sqrt(pm2)*derh ; }
contrib = as.vector(w)*as.vector(derMES)
return(list( round( MES, precision) , round( contrib , precision ) , round(contrib/MES,precision) ) )
}
centeredmoment = function(series,power)
{
location = mean(series);
out = sum( (series-location)^power )/length(series);
return(out);
}
operMES = function(series,alpha,r)
{
z = qnorm(alpha)
location = mean(series);
m2 = centeredmoment(series,2)
m3 = centeredmoment(series,3)
m4 = centeredmoment(series,4)
skew = m3 / m2^(3/2);
exkurt = m4 / m2^(2) - 3;
h = z + (1/6)*(z^2 -1)*skew
if(r==2){ h = h + (1/24)*(z^3 - 3*z)*exkurt - (1/36)*(2*z^3 - 5*z)*skew^2};
MES = dnorm(h)
MES = MES + (1/24)*( Ipower(4,h) - 6*Ipower(2,h) + 3*dnorm(h) )*exkurt
MES = MES + (1/6)*( Ipower(3,h) - 3*Ipower(1,h) )*skew;
MES = MES + (1/72)*( Ipower(6,h) -15*Ipower(4,h)+ 45*Ipower(2,h) - 15*dnorm(h) )*(skew^2)
MES = - location - (sqrt(m2))*min( -MES/alpha , h )
return(MES)
}
TwoVarPlot <- function(xvar, y1var, y2var, labels, noincs = 5,marks=c(1,2), legpos, leglabs, title)
{
# https://stat.ethz.ch/pipermail/r-help/2000-September/008182.html
# plots an x y1 y2 using left and right axes for the different y's
# rescales y2 to fit in the same space as y1
# noincs - no of divisions in the axis labels
# marks - type of marker for each y
# legpos - legend position
# leglabs - legend labels
# rescale to fit on same axis
scaledy2var <- (y2var - min(y2var)) / (max(y2var) - min(y2var))
scaledy2var <- (scaledy2var * (max(y1var) - min(y1var))) + min(y1var)
# plot it up and add the points
plot(xvar, y1var, xlab=labels[1], ylab="", axes=F, pch=marks[1],main=title,type="l")
lines(xvar, scaledy2var, lty=3 )
# make up some labels and positions
y1labs <- round(seq(min(y1var), max(y1var), length=noincs),2)
# convert these to the y2 axis scaling
y2labs <- (y1labs - min(y1var)) / (max(y1var) - min(y1var))
y2labs <- (y2labs * (max(y2var) - min(y2var))) + min(y2var)
y2labs <- round(y2labs, 2)
axis(1)
axis(2, at=y1labs, labels=y1labs)
axis(4, at=y1labs, labels=y2labs)
mtext(labels[3], side=4, line=2)
mtext(labels[2], side=2, line=2)
box()
legend( legend=leglabs, lty = c(1,3), bty="o", x=legpos)
}
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