sandbox/riskbudgetpaper(superseded)/R_Allocation/EfficientFrontier.R
In braverock/PortfolioAnalytics: Portfolio Analysis, Including Numerical Methods for Optimization of Portfolios
setwd("c:/Documents and Settings/Administrator/Desktop/risk budget programs")
# Equal risk portfolio
cAssets = 4;
p = priskbudget = 0.95;
mincriterion = "mES" ; percriskcontribcriterion = "mES";
# Load programs
source("R_Allocation/Risk_budget_functions.R");
library(zoo); library(fGarch); library("PerformanceAnalytics"); library("PortfolioAnalytics")
clean = TRUE
# Load the data
firstyear = 1976 ; firstquarter = 1; lastyear = 2010; lastquarter = 2;
data = read.table( file= paste("data/","/data.txt",sep="") ,header=T)
date = as.Date(data[,1],format="%Y-%m-%d")
monthlyR = indexes = zoo( data[,2:(1+cAssets)] , order.by = date )
if(clean){
indexes = monthlyR = clean.boudt2(monthlyR,alpha=0.05)[[1]]
}
mu = apply(indexes,2,'mean')
sigma = cov(indexes)
M3 = PerformanceAnalytics:::M3.MM(indexes)
M4 = PerformanceAnalytics:::M4.MM(indexes)
# Summary stats individual assets
head(indexes[,1:2],2); tail(indexes[,1:2],2)
apply(indexes[,1:2],2,'mean')*12
apply(indexes[,1:2],2,'sd')
ES(indexes[,1],method="modified")
ES(indexes[,2],method="modified")
#################################################################################
# Make Exhibit 3 Risk budget paper: Efficient frontier plot
#################################################################################
mESfun = function( series ){ return( operMES( series , alpha = 0.05 , r = 2 ) ) }
assetmu = apply( monthlyR , 2 , 'mean' )
assetCVaR = apply( monthlyR , 2 , 'mESfun' )
minmu = min(assetmu); maxmu = max(assetmu); print(minmu*12); print(maxmu*12);
# Do the optimization: given a return target minimize the largest percentage CVaR in the portfolio
#---------------------------------------------------------------------------------------------------
# unconstrained solution is Minimum CVaR Concentration portfolio (having the equal risk contribution property):
# sol = MinMaxCompCVaRconportfolio(R=monthlyR, Riskupper = Inf ,Returnlower= -Inf )
minmu = min( apply(monthlyR,2,'mean' ));
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = T, percriskcontribcriterion = "mES" ,
Riskupper = Inf , mu = mu, sigma = sigma, M3=M3, M4=M4)
W = as.vector( sol[[1]] ) ; vmu = as.vector( sol[[2]] )
vrisk = as.vector( sol[[3]] ) ; mPercrisk = as.vector( sol[[4]] )
vmaxpercrisk = max( sol[[4]] ) ; vmaxriskcontrib = max( sol[[5]] )
for( mutarget in seq(minmu+0.00001,maxmu,0.00001) ){
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = T, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
# For the highest return targets, a very high penalty parameter is (sometimes) needed
for( mutarget in c(seq( max(vmu) , maxmu, 0.000001),maxmu) ){
print( c("mutarget equal to",mutarget) )
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = T, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget, penalty = 1e9,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
if(clean){
write.csv( W , file = "EffFrontierMinCVaRConc_weights_clean.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaRConc_percrisk_clean.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaRConc_stats_clean.csv" )
}else{
write.csv( W , file = "EffFrontierMinCVaRConc_weights.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaRConc_percrisk.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaRConc_stats.csv" )
}
# Do the optimization: given a return target minimize the portfolio CVaR
#--------------------------------------------------------------------------
# unconstrained solution is Minimum CVaR portfolio (having the property that percentage CVaR corresponds to porfolio weights):
minmu = min( apply(monthlyR,2,'mean' ));
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = F, percriskcontribcriterion = "mES" ,
Riskupper = Inf , mu = mu, sigma = sigma, M3=M3, M4=M4)
W = as.vector( sol[[1]] ) ; vmu = as.vector( sol[[2]] )
vrisk = as.vector( sol[[3]] ) ; mPercrisk = as.vector( sol[[4]] )
vmaxpercrisk = max( sol[[4]] ) ; vmaxriskcontrib = max( sol[[5]] )
for( mutarget in seq(minmu+0.00001,maxmu,0.00001) ){
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = F, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
# For the highest return targets, a very high penalty parameter is (sometimes) needed
for( mutarget in c(seq( max(vmu) , maxmu, 0.000001),maxmu) ){
print( c("mutarget equal to",mutarget) )
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = F, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget, penalty = 1e9,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
if(clean){
write.csv( W , file = "EffFrontierMinCVaR_weights_clean.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaR_percrisk_clean.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaR_stats_clean.csv" )
}else{
write.csv( W , file = "EffFrontierMinCVaR_weights.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaR_percrisk.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaR_stats.csv" )
}
# Do the optimization: given a return target minimize the portfolio standard deviation (use quadprog)
#------------------------------------------------------------------------------------------------------
library(quadprog)
N = 4 ; maxweight = 1; minweight = 0; sumweights = 1;
Amat = rbind( rep(1,N) , diag(x =1,nrow=N,ncol=N) , diag(x =-1,nrow=N,ncol=N) , as.numeric(mu) );
bvec = c( sumweights , rep(minweight,N), rep(-maxweight,N) )
dvec = matrix( rep(0,N) , ncol=1 )
mutarget = -10000;
# min(-d^T b + 1/2 b^T D b) with the constraints A^T b >= b_0.
optw = solve.QP( Dmat = sigma , dvec = dvec , Amat = t(Amat) ,
bvec = matrix( c(bvec,mutarget) , ncol = 1) , meq =1 )$solution
sol = ES(weights=optw, portfolio_method="component", mu = mu, sigma = sigma, m3=M3, m4=M4,invert=FALSE)
W = optw ; vmu = sum(optw*mu)
vrisk = sol$MES ; mPercrisk = as.vector( sol$pct_contrib_MES )
vmaxpercrisk = max( sol$pct_contrib_MES ) ; vmaxriskcontrib = max( sol$contribution )
# unconstrained solution is Minimum CVaR portfolio (having the property that percentage CVaR corresponds to porfolio weights):
minmu = min( apply(monthlyR,2,'mean' ));
for( mutarget in seq(minmu+0.00001,maxmu,0.00001) ){
optw = solve.QP( Dmat = sigma , dvec = dvec , Amat = t(Amat) ,
bvec = matrix( c(bvec,mutarget) , ncol = 1) , meq =1 )$solution
sol = ES(weights=optw, portfolio_method="component", mu = mu, sigma = sigma, m3=M3, m4=M4,invert=FALSE)
W = rbind( W, optw )
mPercrisk = rbind( mPercrisk , as.vector( sol$pct_contrib_MES ) )
vmu = c( vmu , sum(optw*mu) )
vrisk = c( vrisk , sol$MES )
vmaxpercrisk = c( vmaxpercrisk , max( sol$pct_contrib_MES ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol$contribution ) )
}
if(clean){
write.csv( W , file = "EffFrontierMinVar_weights_clean.csv" )
write.csv( mPercrisk , file = "EffFrontierMinVar_percrisk_clean.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinVar_stats_clean.csv" )
}else{
write.csv( W , file = "EffFrontierMinVar_weights.csv" )
write.csv( mPercrisk , file = "EffFrontierMinVar_percrisk.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinVar_stats.csv" )
}
# Plotting
#----------------------------------------------------------------------
# Layout 3
source("R_interpretation/chart.StackedBar.R");
mESfun = function( series ){ return( operMES( series , alpha = 0.05 , r = 2 ) ) }
assetmu = apply( monthlyR , 2 , 'mean' )
assetCVaR = apply( monthlyR , 2 , 'mESfun' )
minmu = min(assetmu); maxmu = max(assetmu); print(minmu*12); print(maxmu*12);
####################################
# Min CVaR Concentration
####################################
labelnames = c( "US bond" , "S&P 500" , "EAFE" , "GSCI" )
clean = TRUE
if(clean){
# Portfolio weights
W_MCC = read.csv( file = "EffFrontierMinCVaRConc_weights_clean.csv" )[,2:(1+cAssets)]
W_minCVaR = read.csv( file = "EffFrontierMinCVaR_weights_clean.csv" )[,2:(1+cAssets)]
W_minVar = read.csv( file = "EffFrontierMinVar_weights_clean.csv" )[,2:(1+cAssets)]
# Percentage CVaR contributions
PercCVaR_MCC = read.csv( file = "EffFrontierMinCVaRConc_percrisk_clean.csv" )[,2:(1+cAssets)]
PercCVaR_minCVaR = read.csv( file = "EffFrontierMinCVaR_percrisk_clean.csv" )[,2:(1+cAssets)]
PercCVaR_minVar = read.csv( file = "EffFrontierMinVar_percrisk_clean.csv" )[,2:(1+cAssets)]
# Summary stats
EffFrontier_stats_MCC = read.csv(file = "EffFrontierMinCVaRConc_stats_clean.csv")[,2:5]
EffFrontier_stats_minVar = read.csv(file = "EffFrontierMinVar_stats_clean.csv")[,2:5]
EffFrontier_stats_minCVaR = read.csv(file = "EffFrontierMinCVar_stats_clean.csv")[,2:5]
}else{
# Portfolio weights
W_MCC = read.csv(file = "EffFrontierMinCVaRConc_weights.csv")[,2:(1+cAssets)]
W_minCVaR = read.csv(file = "EffFrontierMinCVaR_weights.csv")[,2:(1+cAssets)]
W_minVar = read.csv(file = "EffFrontierMinVar_weights.csv")[,2:(1+cAssets)]
# Percentage CVaR contributions
mPercrisk_MCC = read.csv( file = "EffFrontierMinCVaRConc_percrisk.csv" )[,2:(1+cAssets)]
mPercrisk_minCVaR = read.csv( file = "EffFrontierMinCVaR_percrisk.csv" )[,2:(1+cAssets)]
mPercrisk_minVar = read.csv(file = "EffFrontierMinVar_percrisk.csv")[,2:(1+cAssets)]
# Summary stats
EffFrontier_stats_MCC = read.csv(file = "EffFrontierMinCVaRConc_stats.csv")[,2:5]
EffFrontier_stats_minVar = read.csv(file = "EffFrontierMinVar_stats.csv")[,2:5]
EffFrontier_stats_minCVaR = read.csv(file = "EffFrontierMinCVar_stats.csv")[,2:5]
}
vmu_MCC = EffFrontier_stats_MCC[,1] ; vmu_minCVaR = EffFrontier_stats_minCVaR[,1] ;
vrisk_MCC = EffFrontier_stats_MCC[,2] ; vrisk_minCVaR = EffFrontier_stats_minCVaR[,2] ;
vmaxpercrisk_MCC = EffFrontier_stats_MCC[,3] ; vmaxpercrisk_minCVaR = EffFrontier_stats_minCVaR[,3] ;
vriskconc_MCC = EffFrontier_stats_MCC[,4] ; vriskconc_minCVaR = EffFrontier_stats_minCVaR[,4] ;
vmu_minVar = EffFrontier_stats_minVar[,1] ;
vrisk_minVar = EffFrontier_stats_minVar[,2] ;
vriskconc_minVar = EffFrontier_stats_minVar[,4] ;
# The MCC and Min CVaR were obtained with DEoptim. The solutions are not always optimal. Correct for this:
order_MCC = sort(vmu_MCC,index.return=T)$ix ; order_minCVaR = sort(vmu_minCVaR,index.return=T)$ix
vmu_MCC = vmu_MCC[order_MCC] ; vmu_minCVaR = vmu_minCVaR[order_minCVaR]
vrisk_MCC = vrisk_MCC[order_MCC] ; vrisk_minCVaR = vrisk_minCVaR[order_minCVaR]
vmaxpercrisk_MCC = vmaxpercrisk_MCC[order_MCC] ; vmaxpercrisk_minCVaR = vmaxpercrisk_MCC[order_minCVaR]
vriskconc_MCC = vriskconc_MCC[order_MCC] ; vriskconc_minCVaR = vriskconc_minCVaR[order_minCVaR]
W_MCC = W_MCC[order_MCC,] ; W_minCVaR = W_minCVaR[order_minCVaR,]
PercCVaR_MCC = PercCVaR_MCC[order_MCC,] ; PercCVaR_minCVaR = PercCVaR_minCVaR[order_minCVaR,]
# eliminate duplicates in mu
# Determines which elements of a vector or data frame are duplicates of elements with smaller subscripts,
# and returns a logical vector indicating which elements (rows) are duplicates.
a_MCC = duplicated(vmu_MCC) ; a_minCVaR = duplicated(vmu_minCVaR)
vmu_MCC = vmu_MCC[!a_MCC] ; vmu_minCVaR = vmu_minCVaR[!a_minCVaR]
vrisk_MCC = vrisk_MCC[!a_MCC] ; vrisk_minCVaR = vrisk_minCVaR[!a_minCVaR]
vmaxpercrisk_MCC = vmaxpercrisk_MCC[!a_MCC] ; vmaxpercrisk_minCVaR = vmaxpercrisk_minCVaR[!a_minCVaR]
vriskconc_MCC = vriskconc_MCC[!a_MCC] ; vriskconc_minCVaR = vriskconc_minCVaR[!a_minCVaR]
W_MCC = W_MCC[!a_MCC,] ; W_minCVaR = W_minCVaR[!a_minCVaR,]
PercCVaR_MCC = PercCVaR_MCC[!a_MCC,] ; PercCVaR_minCVaR = PercCVaR_minCVaR[!a_minCVaR,]
# eliminate (efficiency: lower mu, higher CVaR alloc
sel_MCC = sel_minCVaR = c(); nmu_MCC = length(vmu_MCC); nmu_minCVaR = length(vmu_minCVaR);
for( i in 1:(nmu_MCC-1) ){
if( any(vriskconc_MCC[(i+1):nmu_MCC]<=vriskconc_MCC[i]) ){ }else{sel_MCC = c(sel_MCC, i) }
}
for( i in 1:(nmu_minCVaR-1) ){
if( any(vrisk_minCVaR[(i+1):nmu_minCVaR]<=vrisk_minCVaR[i]) ){ }else{sel_minCVaR = c(sel_minCVaR, i) }
}
vmu_MCC = vmu_MCC[sel_MCC] ; vmu_minCVaR = vmu_minCVaR[sel_minCVaR]
vrisk_MCC = vrisk_MCC[sel_MCC] ; vrisk_minCVaR = vrisk_minCVaR[sel_minCVaR]
vmaxpercrisk_MCC = vmaxpercrisk_MCC[sel_MCC] ; vmaxpercrisk_minCVaR = vmaxpercrisk_minCVaR[sel_minCVaR]
vriskconc_MCC = vriskconc_MCC[sel_MCC] ; vriskconc_minCVaR = vriskconc_minCVaR[sel_minCVaR]
W_MCC = W_MCC[sel_MCC,] ; W_minCVaR = W_minCVaR[sel_minCVaR,]
PercCVaR_MCC = PercCVaR_MCC[sel_MCC,] ; PercCVaR_minCVaR = PercCVaR_minCVaR[sel_minCVaR,]
wEW <- rep(1/4,4)
muEW = mean(assetmu*12)
outES = ES(weights=wEW, portfolio_method="component", mu = mu, sigma = sigma, m3=M3, m4=M4,invert=FALSE)
riskEW = outES$MES
riskconcEW = max(outES$contribution)
postscript('frontier_fourassets.eps')
layout( matrix( c(1,2,3,1,2,3), ncol = 2 ) , height= c(5,5,1), width=c(1,1) )
ylim = c( min(assetmu) , max(assetmu) )
par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
plot( vrisk_MCC , vmu_MCC*12 , type="l", lty = 1,
main = "" ,
ylab="Annualized mean return" , xlab="Monthly 95% Portfolio CVaR" ,
lwd=2, xlim = c(0.02,0.13) , ylim = (ylim*12+c(0,0.01)) , col="black" )
lines( vrisk_minCVaR , vmu_minCVaR*12, lty = 1, lwd=2, col="darkgray" )
lines( vrisk_minVar , vmu_minVar*12, lty = 3, lwd=2, col="black" )
c = 1; text(y=assetmu[c]*12,x= assetCVaR[c] , label=labelnames[c] , cex = 0.7, offset = 0.2, pos = 1); points(y=assetmu[c]*12,x= assetCVaR[c] )
for( c in 2:cAssets ){
text(y=assetmu[c]*12,x= assetCVaR[c] , label=labelnames[c] , cex = 0.7, offset = 0.2, pos = 3)
points(y=assetmu[c]*12,x= assetCVaR[c] )
};
# Plot also EW portfolio
text(y=muEW,x=riskEW, label="EW" , cex = 0.7, offset = 0.2, pos = 3)
points(y=muEW,x= riskEW )
par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
plot( vriskconc_MCC/vrisk_MCC , vmu_MCC*12 , type="l", lty = 1,
main = "" ,
ylab="Annualized mean return" , xlab="Largest Perc. CVaR contribution" ,
lwd=2, xlim = c(0.2,1.1), ylim = c(ylim*12+c(0,0.01)) , col="black" )
lines( vriskconc_minCVaR/vrisk_minCVaR , vmu_minCVaR*12, lty = 1, lwd=2, col="darkgray" )
lines( vriskconc_minVar/vrisk_minVar , vmu_minVar*12, lty = 3, lwd=2, col="black" )
text(y=muEW,x=riskconcEW/riskEW, label="EW" , cex = 0.7, offset = 0.2, pos = 1)
points(y=muEW,x= riskconcEW/riskEW )
for( c in 1:cAssets ){
text(y=assetmu[c]*12,x= 1 , label=labelnames[c] , cex = 0.7, offset = 0.2, pos = 4)
points(y=assetmu[c]*12,x= 1 )
};
par( mar = c(0,1,1,0) )
plot.new()
legend("center",legend=c("Mean/StdDev" , "Mean/CVaR","Mean/CVaR concentration" ),
lty=c("dashed","solid","solid"), cex=0.8,ncol=3,lwd=2,col=c("black","darkgray","black"))
dev.off()
# cbind( vmu_minVar*12 , vriskconc_minVar/vrisk_minVar ) : 0.0816 is turning point
# cbind( vmu_minCVaR*12 , vriskconc_minCVaR/vrisk_minCVaR ) : 0.0816 is turning point
# Make the weight/CVaR allocation plots
#---------------------------------------
Wsel_MCC = PercCVaRsel_MCC = Wsel_minCVaR = PercCVaRsel_minCVaR = Wsel_minVar = PercCVaRsel_minVar = c();
vmusel_MCC = vrisksel_MCC = vriskconcsel_MCC = vmusel_minCVaR = vrisksel_minCVaR = vriskconcsel_minCVaR = vmusel_minVar = vrisksel_minVar = vriskconcsel_minVar = c();
lm = minmu = min( c(vmu_MCC,vmu_minCVaR) ) ; maxmu = max( c(vmu_MCC,vmu_minCVaR) );
ylim = c( min(assetmu) - 0.0003 , max(assetmu) + 0.0003 )
xlim = c( 0 , max(assetCVaR) + 0.01 )
binning = T # for the weight plots, binned data such that no visual misinterpretation is possible
step = quantile( diff(vmu_MCC) , 0.1 )
if( binning ){
for( rm in seq( minmu+step , maxmu , step ) ){
selection = c(vmu_MCC >= lm & vmu_MCC < rm) ;
if( any(selection) ){
selection = c(1:length(selection))[selection]
selone = sort(vriskconc_MCC[ selection ],index.return=T)$ix[1]
selone = selection[selone]
vmusel_MCC = c( vmusel_MCC , mean(vmu_MCC[selone ] ));
Wsel_MCC = rbind( Wsel_MCC , apply( W_MCC[selone,] ,2,'mean' ))
PercCVaRsel_MCC = rbind( PercCVaRsel_MCC , apply( PercCVaR_MCC[selone,] ,2,'mean' ))
vrisksel_MCC = c( vrisksel_MCC , mean(vrisk_MCC[ selone ]) );
vriskconcsel_MCC = c( vriskconcsel_MCC , mean(vriskconc_MCC[ selone ]) )
}else{
vmusel_MCC = c( vmusel_MCC , NA );
Wsel_MCC = rbind( Wsel_MCC , rep(NA,cAssets) )
PercCVaRsel_MCC = rbind( PercCVaRsel_MCC , rep(NA,cAssets) )
vrisksel_MCC = c( vrisksel_MCC , NA );
vriskconcsel_MCC = c( vriskconcsel_MCC , NA )
}
selection = c(vmu_minCVaR >= lm & vmu_minCVaR < rm) ;
if( any(selection) ){
selection = c(1:length(selection))[selection]
selone = sort(vrisk_minCVaR[ selection ],index.return=T)$ix[1]
selone = selection[selone]
vmusel_minCVaR = c( vmusel_minCVaR , mean(vmu_minCVaR[selone ] ));
Wsel_minCVaR = rbind( Wsel_minCVaR , apply( W_minCVaR[selone,] ,2,'mean' ))
PercCVaRsel_minCVaR = rbind( PercCVaRsel_minCVaR , apply( PercCVaR_minCVaR[selone,] ,2,'mean' ))
vrisksel_minCVaR = c( vrisksel_minCVaR , mean(vrisk_minCVaR[ selone ]) );
vriskconcsel_minCVaR = c( vriskconcsel_minCVaR , mean(vriskconc_minCVaR[ selone ]) )
}else{
vmusel_minCVaR = c( vmusel_minCVaR , NA );
Wsel_minCVaR = rbind( Wsel_minCVaR , rep(NA,cAssets) )
PercCVaRsel_minCVaR = rbind( PercCVaRsel_minCVaR , rep(NA,cAssets) )
vrisksel_minCVaR = c( vrisksel_minCVaR , NA );
vriskconcsel_minCVaR = c( vriskconcsel_minCVaR , NA )
}
selection = c(vmu_minVar >= lm & vmu_minVar < rm) ;
if( any(selection) ){
selection = c(1:length(selection))[selection]
selone = sort(vrisk_minVar[ selection ],index.return=T)$ix[1]
selone = selection[selone]
vmusel_minVar = c( vmusel_minVar , mean(vmu_minVar[selone ] ));
Wsel_minVar = rbind( Wsel_minVar , apply( W_minVar[selone,] ,2,'mean' ))
PercCVaRsel_minVar = rbind( PercCVaRsel_minVar , apply( PercCVaR_minVar[selone,] ,2,'mean' ))
vrisksel_minVar = c( vrisksel_minVar , mean(vrisk_minVar[ selone ]) );
vriskconcsel_minVar = c( vriskconcsel_minVar , mean(vriskconc_minVar[ selone ]) )
}else{
vmusel_minVar = c( vmusel_minVar , NA );
Wsel_minVar = rbind( Wsel_minVar , rep(NA,cAssets) )
PercCVaRsel_minVar = rbind( PercCVaRsel_minVar , rep(NA,cAssets) )
vrisksel_minVar = c( vrisksel_minVar , NA );
vriskconcsel_minVar = c( vriskconcsel_minVar , NA )
}
lm = rm;
}
}else{
vmusel_MCC = vmu_MCC ; vmusel_minCVaR = vmu_minCVaR
Wsel_MCC = W_MCC ; Wsel_minCVaR = W_minCVaR ;
vrisksel_MCC = vrisk_MCC ; vrisksel_minCVaR = vrisk_minCVaR
vriskconcsel_MCC = vriskconc_MCC ; vriskconcsel_minCVar = vriskconc_minCVaR
PercCVaRsel_MCC = PercCVaR_MCC ; PercCVaRsel_minCVar = PercCVaR_minCVar ;
}
library(zoo)
cAssets = ncol(monthlyR)
colorset = gray( seq(0,(cAssets-1),1)/cAssets ) ;
colnames( Wsel_MCC ) = colnames( Wsel_minCVaR ) = labelnames
rownames( Wsel_MCC ) = rownames( PercCVaRsel_MCC ) = round( interpNA(vmusel_MCC)*12,4);
rownames( Wsel_minCVaR ) = rownames( PercCVaRsel_minCVaR ) = round( interpNA(vmusel_minCVaR)*12,4) ;
rownames( Wsel_minVar ) = rownames( PercCVaRsel_minVar ) = round( interpNA(vmusel_minVar)*12,4) ;
colnames( Wsel_minVar ) = labelnames
w.names = c( "US bond" , "S&P 500", "EAFE" , "GSCI" )
namelabels = c("Mean/StdDev" , "Mean/CVaR","Mean/CVaR concentration" )
l = 2
mar1 =c(2,l,2,1.1)
mar2 =c(0,l,2,1)
mar3 = c(3,l+1,3,0.1)
mar4 = c(2,l+1,2,0.1)
source("R_interpretation/chart.StackedBar.R");
# Stacked weights plot:
postscript('stackedweightsriskcont_efficientfrontier_withNA.eps')
layout( matrix( c(1,2,3,4,5,6,7,4), ncol = 2 ) , height= c(1.5,1.5,1.5,0.7), width=1)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(Wsel_minVar ,col=colorset,space=0, main = namelabels[1], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(Wsel_minCVaR,col=colorset,space=0, main = namelabels[2], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(Wsel_MCC ,col=colorset,space=0, main = namelabels[3], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T, legend.loc = NULL,ylim=c(0,1),border = F )
par(mar=mar1 , cex.main=1)
plot.new()
legend("center",legend=w.names,fill=colorset,ncol=4)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(PercCVaRsel_minVar,col=colorset,space=0, main = namelabels[1], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(PercCVaRsel_minCVaR,col=colorset,space=0, main = namelabels[2], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(PercCVaRsel_MCC,col=colorset,space=0, main = namelabels[3], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
dev.off()
postscript('stackedweightsriskcont_efficientfrontier.eps')
layout( matrix( c(1,2,3,4,5,6,7,4), ncol = 2 ) , height= c(1.5,1.5,1.5,0.7), width=1)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(interpNA(Wsel_minVar) ,col=colorset,space=0, main = namelabels[1], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(Wsel_minCVaR),col=colorset,space=0, main = namelabels[2], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(Wsel_MCC) ,col=colorset,space=0, main = namelabels[3], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T, legend.loc = NULL,ylim=c(0,1),border = F )
par(mar=mar1 , cex.main=1)
plot.new()
legend("center",legend=w.names,fill=colorset,ncol=4)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(interpNA(PercCVaRsel_minVar),col=colorset,space=0, main = namelabels[1], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(PercCVaRsel_minCVaR),col=colorset,space=0, main = namelabels[2], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(PercCVaRsel_MCC),col=colorset,space=0, main = namelabels[3], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
dev.off()
#----------------------------------------------------------------------------------------------------
# DISCUSSION RESULTS
# On Raw Data: Discontinuity in the Mean-CVaR frontier of Mean CVaR concentration efficient portfolios:
# Two types of portfolios:
# Lower return targets: portfolio where the component risk of the bond dominates
# Higher return targets: portfolio where the component risk of the US equity dominates
# Shift when:
# > vmu[13:14]
# [1] 0.006796657 0.006806883
# > vriskconc[13:14]
# [1] 0.0082 0.0254
# > vrisk[13:14]
# [1] 0.01544720 0.06036618
# On Cleaned Data: Three segments
# > assetmu
# Bond SP500 EAFE SPGSCI
# 0.07545960 0.10249338 0.08677129 0.05413622
# > assetCVaR
# Bond SP500 EAFE SPGSCI
# 0.02456233 0.09974267 0.10868066 0.12775266
# Unconstrained: Equal Risk Contribution Portfolio:
# > W[1,]
# 0.6431897 0.1337981 0.1044994 0.1185127
#> mPercrisk[1,]
# 0.2516 0.2517 0.2504 0.2463
# > vmu[1]*12
# [1] 0.07773165
#> vrisk[1]
#[1] 0.03467514
#> vriskconc[1]
#[1] 0.0087
# Segment 1: increasing allocation to bonds and decreasing allocation to commodities
# Portfolio risk concentration increases, portfolio risk decreases
# > W[25:27,]
# 0.7052788 0.1584357 0.1321813 0.0041042550
# 0.7085301 0.1597522 0.1313759 0.0003417814
# 0.7025028 0.1628656 0.1344935 0.0001380022
#> vmu[26]*12
#[1] 0.0812571
#> vrisk[26]
#[1] 0.03352778
#> vriskconc[26]
#[1] 0.0112
# Segment 2: increasing allocation to both US equity and EAFE
# Portfolio risk concentration and risk increase together
# > W[138:140,]
# 0.0078323664 0.5106416 0.4814090 1.169711e-04
# 0.0005986683 0.5136429 0.4856355 1.228907e-04
# 0.0002258733 0.5201106 0.4796231 4.051641e-05
# > mPercrisk[138:140,]
# 0e+00 0.4999 0.5000 0
#> vmu[139]*12
#[1] 0.09483605
# > vrisk[139]
# [1] 0.09808784
# > vriskconc[139]
# [1] 0.049
# Segment 3: increasing alllocation to US equity and decreasing to EAFE
# Portfolio risk concentration and risk increase together
# > tail(W,1)
# 5.45402e-05 0.9998762 4.340072e-05 2.585630e-05
braverock/PortfolioAnalytics documentation built on April 18, 2024, 4:09 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
setwd("c:/Documents and Settings/Administrator/Desktop/risk budget programs")
# Equal risk portfolio
cAssets = 4;
p = priskbudget = 0.95;
mincriterion = "mES" ; percriskcontribcriterion = "mES";
# Load programs
source("R_Allocation/Risk_budget_functions.R");
library(zoo); library(fGarch); library("PerformanceAnalytics"); library("PortfolioAnalytics")
clean = TRUE
# Load the data
firstyear = 1976 ; firstquarter = 1; lastyear = 2010; lastquarter = 2;
data = read.table( file= paste("data/","/data.txt",sep="") ,header=T)
date = as.Date(data[,1],format="%Y-%m-%d")
monthlyR = indexes = zoo( data[,2:(1+cAssets)] , order.by = date )
if(clean){
indexes = monthlyR = clean.boudt2(monthlyR,alpha=0.05)[[1]]
}
mu = apply(indexes,2,'mean')
sigma = cov(indexes)
M3 = PerformanceAnalytics:::M3.MM(indexes)
M4 = PerformanceAnalytics:::M4.MM(indexes)
# Summary stats individual assets
head(indexes[,1:2],2); tail(indexes[,1:2],2)
apply(indexes[,1:2],2,'mean')*12
apply(indexes[,1:2],2,'sd')
ES(indexes[,1],method="modified")
ES(indexes[,2],method="modified")
#################################################################################
# Make Exhibit 3 Risk budget paper: Efficient frontier plot
#################################################################################
mESfun = function( series ){ return( operMES( series , alpha = 0.05 , r = 2 ) ) }
assetmu = apply( monthlyR , 2 , 'mean' )
assetCVaR = apply( monthlyR , 2 , 'mESfun' )
minmu = min(assetmu); maxmu = max(assetmu); print(minmu*12); print(maxmu*12);
# Do the optimization: given a return target minimize the largest percentage CVaR in the portfolio
#---------------------------------------------------------------------------------------------------
# unconstrained solution is Minimum CVaR Concentration portfolio (having the equal risk contribution property):
# sol = MinMaxCompCVaRconportfolio(R=monthlyR, Riskupper = Inf ,Returnlower= -Inf )
minmu = min( apply(monthlyR,2,'mean' ));
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = T, percriskcontribcriterion = "mES" ,
Riskupper = Inf , mu = mu, sigma = sigma, M3=M3, M4=M4)
W = as.vector( sol[[1]] ) ; vmu = as.vector( sol[[2]] )
vrisk = as.vector( sol[[3]] ) ; mPercrisk = as.vector( sol[[4]] )
vmaxpercrisk = max( sol[[4]] ) ; vmaxriskcontrib = max( sol[[5]] )
for( mutarget in seq(minmu+0.00001,maxmu,0.00001) ){
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = T, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
# For the highest return targets, a very high penalty parameter is (sometimes) needed
for( mutarget in c(seq( max(vmu) , maxmu, 0.000001),maxmu) ){
print( c("mutarget equal to",mutarget) )
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = T, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget, penalty = 1e9,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
if(clean){
write.csv( W , file = "EffFrontierMinCVaRConc_weights_clean.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaRConc_percrisk_clean.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaRConc_stats_clean.csv" )
}else{
write.csv( W , file = "EffFrontierMinCVaRConc_weights.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaRConc_percrisk.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaRConc_stats.csv" )
}
# Do the optimization: given a return target minimize the portfolio CVaR
#--------------------------------------------------------------------------
# unconstrained solution is Minimum CVaR portfolio (having the property that percentage CVaR corresponds to porfolio weights):
minmu = min( apply(monthlyR,2,'mean' ));
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = F, percriskcontribcriterion = "mES" ,
Riskupper = Inf , mu = mu, sigma = sigma, M3=M3, M4=M4)
W = as.vector( sol[[1]] ) ; vmu = as.vector( sol[[2]] )
vrisk = as.vector( sol[[3]] ) ; mPercrisk = as.vector( sol[[4]] )
vmaxpercrisk = max( sol[[4]] ) ; vmaxriskcontrib = max( sol[[5]] )
for( mutarget in seq(minmu+0.00001,maxmu,0.00001) ){
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = F, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
# For the highest return targets, a very high penalty parameter is (sometimes) needed
for( mutarget in c(seq( max(vmu) , maxmu, 0.000001),maxmu) ){
print( c("mutarget equal to",mutarget) )
sol = PortfolioOptim( minriskcriterion = "mES" , MinMaxComp = F, percriskcontribcriterion = "mES" ,
Riskupper = Inf ,Returnlower= mutarget, penalty = 1e9,
mu = mu, sigma = sigma, M3=M3, M4=M4)
W = rbind( W, as.vector( sol[[1]] ) )
mPercrisk = rbind( mPercrisk , as.vector( sol[[4]] ) )
vmu = c( vmu , as.vector( sol[[2]] ))
vrisk = c( vrisk , as.vector( sol[[3]] ) )
vmaxpercrisk = c( vmaxpercrisk , max( sol[[4]] ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol[[5]] ) )
}
if(clean){
write.csv( W , file = "EffFrontierMinCVaR_weights_clean.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaR_percrisk_clean.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaR_stats_clean.csv" )
}else{
write.csv( W , file = "EffFrontierMinCVaR_weights.csv" )
write.csv( mPercrisk , file = "EffFrontierMinCVaR_percrisk.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinCVaR_stats.csv" )
}
# Do the optimization: given a return target minimize the portfolio standard deviation (use quadprog)
#------------------------------------------------------------------------------------------------------
library(quadprog)
N = 4 ; maxweight = 1; minweight = 0; sumweights = 1;
Amat = rbind( rep(1,N) , diag(x =1,nrow=N,ncol=N) , diag(x =-1,nrow=N,ncol=N) , as.numeric(mu) );
bvec = c( sumweights , rep(minweight,N), rep(-maxweight,N) )
dvec = matrix( rep(0,N) , ncol=1 )
mutarget = -10000;
# min(-d^T b + 1/2 b^T D b) with the constraints A^T b >= b_0.
optw = solve.QP( Dmat = sigma , dvec = dvec , Amat = t(Amat) ,
bvec = matrix( c(bvec,mutarget) , ncol = 1) , meq =1 )$solution
sol = ES(weights=optw, portfolio_method="component", mu = mu, sigma = sigma, m3=M3, m4=M4,invert=FALSE)
W = optw ; vmu = sum(optw*mu)
vrisk = sol$MES ; mPercrisk = as.vector( sol$pct_contrib_MES )
vmaxpercrisk = max( sol$pct_contrib_MES ) ; vmaxriskcontrib = max( sol$contribution )
# unconstrained solution is Minimum CVaR portfolio (having the property that percentage CVaR corresponds to porfolio weights):
minmu = min( apply(monthlyR,2,'mean' ));
for( mutarget in seq(minmu+0.00001,maxmu,0.00001) ){
optw = solve.QP( Dmat = sigma , dvec = dvec , Amat = t(Amat) ,
bvec = matrix( c(bvec,mutarget) , ncol = 1) , meq =1 )$solution
sol = ES(weights=optw, portfolio_method="component", mu = mu, sigma = sigma, m3=M3, m4=M4,invert=FALSE)
W = rbind( W, optw )
mPercrisk = rbind( mPercrisk , as.vector( sol$pct_contrib_MES ) )
vmu = c( vmu , sum(optw*mu) )
vrisk = c( vrisk , sol$MES )
vmaxpercrisk = c( vmaxpercrisk , max( sol$pct_contrib_MES ) )
vmaxriskcontrib = c( vmaxriskcontrib , max( sol$contribution ) )
}
if(clean){
write.csv( W , file = "EffFrontierMinVar_weights_clean.csv" )
write.csv( mPercrisk , file = "EffFrontierMinVar_percrisk_clean.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinVar_stats_clean.csv" )
}else{
write.csv( W , file = "EffFrontierMinVar_weights.csv" )
write.csv( mPercrisk , file = "EffFrontierMinVar_percrisk.csv" )
EffFrontier_stats = cbind( vmu , vrisk , vmaxpercrisk , vmaxriskcontrib)
write.csv( EffFrontier_stats , file = "EffFrontierMinVar_stats.csv" )
}
# Plotting
#----------------------------------------------------------------------
# Layout 3
source("R_interpretation/chart.StackedBar.R");
mESfun = function( series ){ return( operMES( series , alpha = 0.05 , r = 2 ) ) }
assetmu = apply( monthlyR , 2 , 'mean' )
assetCVaR = apply( monthlyR , 2 , 'mESfun' )
minmu = min(assetmu); maxmu = max(assetmu); print(minmu*12); print(maxmu*12);
####################################
# Min CVaR Concentration
####################################
labelnames = c( "US bond" , "S&P 500" , "EAFE" , "GSCI" )
clean = TRUE
if(clean){
# Portfolio weights
W_MCC = read.csv( file = "EffFrontierMinCVaRConc_weights_clean.csv" )[,2:(1+cAssets)]
W_minCVaR = read.csv( file = "EffFrontierMinCVaR_weights_clean.csv" )[,2:(1+cAssets)]
W_minVar = read.csv( file = "EffFrontierMinVar_weights_clean.csv" )[,2:(1+cAssets)]
# Percentage CVaR contributions
PercCVaR_MCC = read.csv( file = "EffFrontierMinCVaRConc_percrisk_clean.csv" )[,2:(1+cAssets)]
PercCVaR_minCVaR = read.csv( file = "EffFrontierMinCVaR_percrisk_clean.csv" )[,2:(1+cAssets)]
PercCVaR_minVar = read.csv( file = "EffFrontierMinVar_percrisk_clean.csv" )[,2:(1+cAssets)]
# Summary stats
EffFrontier_stats_MCC = read.csv(file = "EffFrontierMinCVaRConc_stats_clean.csv")[,2:5]
EffFrontier_stats_minVar = read.csv(file = "EffFrontierMinVar_stats_clean.csv")[,2:5]
EffFrontier_stats_minCVaR = read.csv(file = "EffFrontierMinCVar_stats_clean.csv")[,2:5]
}else{
# Portfolio weights
W_MCC = read.csv(file = "EffFrontierMinCVaRConc_weights.csv")[,2:(1+cAssets)]
W_minCVaR = read.csv(file = "EffFrontierMinCVaR_weights.csv")[,2:(1+cAssets)]
W_minVar = read.csv(file = "EffFrontierMinVar_weights.csv")[,2:(1+cAssets)]
# Percentage CVaR contributions
mPercrisk_MCC = read.csv( file = "EffFrontierMinCVaRConc_percrisk.csv" )[,2:(1+cAssets)]
mPercrisk_minCVaR = read.csv( file = "EffFrontierMinCVaR_percrisk.csv" )[,2:(1+cAssets)]
mPercrisk_minVar = read.csv(file = "EffFrontierMinVar_percrisk.csv")[,2:(1+cAssets)]
# Summary stats
EffFrontier_stats_MCC = read.csv(file = "EffFrontierMinCVaRConc_stats.csv")[,2:5]
EffFrontier_stats_minVar = read.csv(file = "EffFrontierMinVar_stats.csv")[,2:5]
EffFrontier_stats_minCVaR = read.csv(file = "EffFrontierMinCVar_stats.csv")[,2:5]
}
vmu_MCC = EffFrontier_stats_MCC[,1] ; vmu_minCVaR = EffFrontier_stats_minCVaR[,1] ;
vrisk_MCC = EffFrontier_stats_MCC[,2] ; vrisk_minCVaR = EffFrontier_stats_minCVaR[,2] ;
vmaxpercrisk_MCC = EffFrontier_stats_MCC[,3] ; vmaxpercrisk_minCVaR = EffFrontier_stats_minCVaR[,3] ;
vriskconc_MCC = EffFrontier_stats_MCC[,4] ; vriskconc_minCVaR = EffFrontier_stats_minCVaR[,4] ;
vmu_minVar = EffFrontier_stats_minVar[,1] ;
vrisk_minVar = EffFrontier_stats_minVar[,2] ;
vriskconc_minVar = EffFrontier_stats_minVar[,4] ;
# The MCC and Min CVaR were obtained with DEoptim. The solutions are not always optimal. Correct for this:
order_MCC = sort(vmu_MCC,index.return=T)$ix ; order_minCVaR = sort(vmu_minCVaR,index.return=T)$ix
vmu_MCC = vmu_MCC[order_MCC] ; vmu_minCVaR = vmu_minCVaR[order_minCVaR]
vrisk_MCC = vrisk_MCC[order_MCC] ; vrisk_minCVaR = vrisk_minCVaR[order_minCVaR]
vmaxpercrisk_MCC = vmaxpercrisk_MCC[order_MCC] ; vmaxpercrisk_minCVaR = vmaxpercrisk_MCC[order_minCVaR]
vriskconc_MCC = vriskconc_MCC[order_MCC] ; vriskconc_minCVaR = vriskconc_minCVaR[order_minCVaR]
W_MCC = W_MCC[order_MCC,] ; W_minCVaR = W_minCVaR[order_minCVaR,]
PercCVaR_MCC = PercCVaR_MCC[order_MCC,] ; PercCVaR_minCVaR = PercCVaR_minCVaR[order_minCVaR,]
# eliminate duplicates in mu
# Determines which elements of a vector or data frame are duplicates of elements with smaller subscripts,
# and returns a logical vector indicating which elements (rows) are duplicates.
a_MCC = duplicated(vmu_MCC) ; a_minCVaR = duplicated(vmu_minCVaR)
vmu_MCC = vmu_MCC[!a_MCC] ; vmu_minCVaR = vmu_minCVaR[!a_minCVaR]
vrisk_MCC = vrisk_MCC[!a_MCC] ; vrisk_minCVaR = vrisk_minCVaR[!a_minCVaR]
vmaxpercrisk_MCC = vmaxpercrisk_MCC[!a_MCC] ; vmaxpercrisk_minCVaR = vmaxpercrisk_minCVaR[!a_minCVaR]
vriskconc_MCC = vriskconc_MCC[!a_MCC] ; vriskconc_minCVaR = vriskconc_minCVaR[!a_minCVaR]
W_MCC = W_MCC[!a_MCC,] ; W_minCVaR = W_minCVaR[!a_minCVaR,]
PercCVaR_MCC = PercCVaR_MCC[!a_MCC,] ; PercCVaR_minCVaR = PercCVaR_minCVaR[!a_minCVaR,]
# eliminate (efficiency: lower mu, higher CVaR alloc
sel_MCC = sel_minCVaR = c(); nmu_MCC = length(vmu_MCC); nmu_minCVaR = length(vmu_minCVaR);
for( i in 1:(nmu_MCC-1) ){
if( any(vriskconc_MCC[(i+1):nmu_MCC]<=vriskconc_MCC[i]) ){ }else{sel_MCC = c(sel_MCC, i) }
}
for( i in 1:(nmu_minCVaR-1) ){
if( any(vrisk_minCVaR[(i+1):nmu_minCVaR]<=vrisk_minCVaR[i]) ){ }else{sel_minCVaR = c(sel_minCVaR, i) }
}
vmu_MCC = vmu_MCC[sel_MCC] ; vmu_minCVaR = vmu_minCVaR[sel_minCVaR]
vrisk_MCC = vrisk_MCC[sel_MCC] ; vrisk_minCVaR = vrisk_minCVaR[sel_minCVaR]
vmaxpercrisk_MCC = vmaxpercrisk_MCC[sel_MCC] ; vmaxpercrisk_minCVaR = vmaxpercrisk_minCVaR[sel_minCVaR]
vriskconc_MCC = vriskconc_MCC[sel_MCC] ; vriskconc_minCVaR = vriskconc_minCVaR[sel_minCVaR]
W_MCC = W_MCC[sel_MCC,] ; W_minCVaR = W_minCVaR[sel_minCVaR,]
PercCVaR_MCC = PercCVaR_MCC[sel_MCC,] ; PercCVaR_minCVaR = PercCVaR_minCVaR[sel_minCVaR,]
wEW <- rep(1/4,4)
muEW = mean(assetmu*12)
outES = ES(weights=wEW, portfolio_method="component", mu = mu, sigma = sigma, m3=M3, m4=M4,invert=FALSE)
riskEW = outES$MES
riskconcEW = max(outES$contribution)
postscript('frontier_fourassets.eps')
layout( matrix( c(1,2,3,1,2,3), ncol = 2 ) , height= c(5,5,1), width=c(1,1) )
ylim = c( min(assetmu) , max(assetmu) )
par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
plot( vrisk_MCC , vmu_MCC*12 , type="l", lty = 1,
main = "" ,
ylab="Annualized mean return" , xlab="Monthly 95% Portfolio CVaR" ,
lwd=2, xlim = c(0.02,0.13) , ylim = (ylim*12+c(0,0.01)) , col="black" )
lines( vrisk_minCVaR , vmu_minCVaR*12, lty = 1, lwd=2, col="darkgray" )
lines( vrisk_minVar , vmu_minVar*12, lty = 3, lwd=2, col="black" )
c = 1; text(y=assetmu[c]*12,x= assetCVaR[c] , label=labelnames[c] , cex = 0.7, offset = 0.2, pos = 1); points(y=assetmu[c]*12,x= assetCVaR[c] )
for( c in 2:cAssets ){
text(y=assetmu[c]*12,x= assetCVaR[c] , label=labelnames[c] , cex = 0.7, offset = 0.2, pos = 3)
points(y=assetmu[c]*12,x= assetCVaR[c] )
};
# Plot also EW portfolio
text(y=muEW,x=riskEW, label="EW" , cex = 0.7, offset = 0.2, pos = 3)
points(y=muEW,x= riskEW )
par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
plot( vriskconc_MCC/vrisk_MCC , vmu_MCC*12 , type="l", lty = 1,
main = "" ,
ylab="Annualized mean return" , xlab="Largest Perc. CVaR contribution" ,
lwd=2, xlim = c(0.2,1.1), ylim = c(ylim*12+c(0,0.01)) , col="black" )
lines( vriskconc_minCVaR/vrisk_minCVaR , vmu_minCVaR*12, lty = 1, lwd=2, col="darkgray" )
lines( vriskconc_minVar/vrisk_minVar , vmu_minVar*12, lty = 3, lwd=2, col="black" )
text(y=muEW,x=riskconcEW/riskEW, label="EW" , cex = 0.7, offset = 0.2, pos = 1)
points(y=muEW,x= riskconcEW/riskEW )
for( c in 1:cAssets ){
text(y=assetmu[c]*12,x= 1 , label=labelnames[c] , cex = 0.7, offset = 0.2, pos = 4)
points(y=assetmu[c]*12,x= 1 )
};
par( mar = c(0,1,1,0) )
plot.new()
legend("center",legend=c("Mean/StdDev" , "Mean/CVaR","Mean/CVaR concentration" ),
lty=c("dashed","solid","solid"), cex=0.8,ncol=3,lwd=2,col=c("black","darkgray","black"))
dev.off()
# cbind( vmu_minVar*12 , vriskconc_minVar/vrisk_minVar ) : 0.0816 is turning point
# cbind( vmu_minCVaR*12 , vriskconc_minCVaR/vrisk_minCVaR ) : 0.0816 is turning point
# Make the weight/CVaR allocation plots
#---------------------------------------
Wsel_MCC = PercCVaRsel_MCC = Wsel_minCVaR = PercCVaRsel_minCVaR = Wsel_minVar = PercCVaRsel_minVar = c();
vmusel_MCC = vrisksel_MCC = vriskconcsel_MCC = vmusel_minCVaR = vrisksel_minCVaR = vriskconcsel_minCVaR = vmusel_minVar = vrisksel_minVar = vriskconcsel_minVar = c();
lm = minmu = min( c(vmu_MCC,vmu_minCVaR) ) ; maxmu = max( c(vmu_MCC,vmu_minCVaR) );
ylim = c( min(assetmu) - 0.0003 , max(assetmu) + 0.0003 )
xlim = c( 0 , max(assetCVaR) + 0.01 )
binning = T # for the weight plots, binned data such that no visual misinterpretation is possible
step = quantile( diff(vmu_MCC) , 0.1 )
if( binning ){
for( rm in seq( minmu+step , maxmu , step ) ){
selection = c(vmu_MCC >= lm & vmu_MCC < rm) ;
if( any(selection) ){
selection = c(1:length(selection))[selection]
selone = sort(vriskconc_MCC[ selection ],index.return=T)$ix[1]
selone = selection[selone]
vmusel_MCC = c( vmusel_MCC , mean(vmu_MCC[selone ] ));
Wsel_MCC = rbind( Wsel_MCC , apply( W_MCC[selone,] ,2,'mean' ))
PercCVaRsel_MCC = rbind( PercCVaRsel_MCC , apply( PercCVaR_MCC[selone,] ,2,'mean' ))
vrisksel_MCC = c( vrisksel_MCC , mean(vrisk_MCC[ selone ]) );
vriskconcsel_MCC = c( vriskconcsel_MCC , mean(vriskconc_MCC[ selone ]) )
}else{
vmusel_MCC = c( vmusel_MCC , NA );
Wsel_MCC = rbind( Wsel_MCC , rep(NA,cAssets) )
PercCVaRsel_MCC = rbind( PercCVaRsel_MCC , rep(NA,cAssets) )
vrisksel_MCC = c( vrisksel_MCC , NA );
vriskconcsel_MCC = c( vriskconcsel_MCC , NA )
}
selection = c(vmu_minCVaR >= lm & vmu_minCVaR < rm) ;
if( any(selection) ){
selection = c(1:length(selection))[selection]
selone = sort(vrisk_minCVaR[ selection ],index.return=T)$ix[1]
selone = selection[selone]
vmusel_minCVaR = c( vmusel_minCVaR , mean(vmu_minCVaR[selone ] ));
Wsel_minCVaR = rbind( Wsel_minCVaR , apply( W_minCVaR[selone,] ,2,'mean' ))
PercCVaRsel_minCVaR = rbind( PercCVaRsel_minCVaR , apply( PercCVaR_minCVaR[selone,] ,2,'mean' ))
vrisksel_minCVaR = c( vrisksel_minCVaR , mean(vrisk_minCVaR[ selone ]) );
vriskconcsel_minCVaR = c( vriskconcsel_minCVaR , mean(vriskconc_minCVaR[ selone ]) )
}else{
vmusel_minCVaR = c( vmusel_minCVaR , NA );
Wsel_minCVaR = rbind( Wsel_minCVaR , rep(NA,cAssets) )
PercCVaRsel_minCVaR = rbind( PercCVaRsel_minCVaR , rep(NA,cAssets) )
vrisksel_minCVaR = c( vrisksel_minCVaR , NA );
vriskconcsel_minCVaR = c( vriskconcsel_minCVaR , NA )
}
selection = c(vmu_minVar >= lm & vmu_minVar < rm) ;
if( any(selection) ){
selection = c(1:length(selection))[selection]
selone = sort(vrisk_minVar[ selection ],index.return=T)$ix[1]
selone = selection[selone]
vmusel_minVar = c( vmusel_minVar , mean(vmu_minVar[selone ] ));
Wsel_minVar = rbind( Wsel_minVar , apply( W_minVar[selone,] ,2,'mean' ))
PercCVaRsel_minVar = rbind( PercCVaRsel_minVar , apply( PercCVaR_minVar[selone,] ,2,'mean' ))
vrisksel_minVar = c( vrisksel_minVar , mean(vrisk_minVar[ selone ]) );
vriskconcsel_minVar = c( vriskconcsel_minVar , mean(vriskconc_minVar[ selone ]) )
}else{
vmusel_minVar = c( vmusel_minVar , NA );
Wsel_minVar = rbind( Wsel_minVar , rep(NA,cAssets) )
PercCVaRsel_minVar = rbind( PercCVaRsel_minVar , rep(NA,cAssets) )
vrisksel_minVar = c( vrisksel_minVar , NA );
vriskconcsel_minVar = c( vriskconcsel_minVar , NA )
}
lm = rm;
}
}else{
vmusel_MCC = vmu_MCC ; vmusel_minCVaR = vmu_minCVaR
Wsel_MCC = W_MCC ; Wsel_minCVaR = W_minCVaR ;
vrisksel_MCC = vrisk_MCC ; vrisksel_minCVaR = vrisk_minCVaR
vriskconcsel_MCC = vriskconc_MCC ; vriskconcsel_minCVar = vriskconc_minCVaR
PercCVaRsel_MCC = PercCVaR_MCC ; PercCVaRsel_minCVar = PercCVaR_minCVar ;
}
library(zoo)
cAssets = ncol(monthlyR)
colorset = gray( seq(0,(cAssets-1),1)/cAssets ) ;
colnames( Wsel_MCC ) = colnames( Wsel_minCVaR ) = labelnames
rownames( Wsel_MCC ) = rownames( PercCVaRsel_MCC ) = round( interpNA(vmusel_MCC)*12,4);
rownames( Wsel_minCVaR ) = rownames( PercCVaRsel_minCVaR ) = round( interpNA(vmusel_minCVaR)*12,4) ;
rownames( Wsel_minVar ) = rownames( PercCVaRsel_minVar ) = round( interpNA(vmusel_minVar)*12,4) ;
colnames( Wsel_minVar ) = labelnames
w.names = c( "US bond" , "S&P 500", "EAFE" , "GSCI" )
namelabels = c("Mean/StdDev" , "Mean/CVaR","Mean/CVaR concentration" )
l = 2
mar1 =c(2,l,2,1.1)
mar2 =c(0,l,2,1)
mar3 = c(3,l+1,3,0.1)
mar4 = c(2,l+1,2,0.1)
source("R_interpretation/chart.StackedBar.R");
# Stacked weights plot:
postscript('stackedweightsriskcont_efficientfrontier_withNA.eps')
layout( matrix( c(1,2,3,4,5,6,7,4), ncol = 2 ) , height= c(1.5,1.5,1.5,0.7), width=1)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(Wsel_minVar ,col=colorset,space=0, main = namelabels[1], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(Wsel_minCVaR,col=colorset,space=0, main = namelabels[2], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(Wsel_MCC ,col=colorset,space=0, main = namelabels[3], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T, legend.loc = NULL,ylim=c(0,1),border = F )
par(mar=mar1 , cex.main=1)
plot.new()
legend("center",legend=w.names,fill=colorset,ncol=4)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(PercCVaRsel_minVar,col=colorset,space=0, main = namelabels[1], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(PercCVaRsel_minCVaR,col=colorset,space=0, main = namelabels[2], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(PercCVaRsel_MCC,col=colorset,space=0, main = namelabels[3], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
dev.off()
postscript('stackedweightsriskcont_efficientfrontier.eps')
layout( matrix( c(1,2,3,4,5,6,7,4), ncol = 2 ) , height= c(1.5,1.5,1.5,0.7), width=1)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(interpNA(Wsel_minVar) ,col=colorset,space=0, main = namelabels[1], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(Wsel_minCVaR),col=colorset,space=0, main = namelabels[2], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(Wsel_MCC) ,col=colorset,space=0, main = namelabels[3], ylab="Weight allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T, legend.loc = NULL,ylim=c(0,1),border = F )
par(mar=mar1 , cex.main=1)
plot.new()
legend("center",legend=w.names,fill=colorset,ncol=4)
par(mar=mar3 , cex.main=1)
chart.StackedBar2(interpNA(PercCVaRsel_minVar),col=colorset,space=0, main = namelabels[1], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(PercCVaRsel_minCVaR),col=colorset,space=0, main = namelabels[2], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
chart.StackedBar2(interpNA(PercCVaRsel_MCC),col=colorset,space=0, main = namelabels[3], ylab="CVaR allocation", las=1, l=3.9, r=0, cex.axis=1, cex.lab=1, cex.main=1, axisnames=T,legend.loc = NULL,ylim=c(0,1),border = F )
dev.off()
#----------------------------------------------------------------------------------------------------
# DISCUSSION RESULTS
# On Raw Data: Discontinuity in the Mean-CVaR frontier of Mean CVaR concentration efficient portfolios:
# Two types of portfolios:
# Lower return targets: portfolio where the component risk of the bond dominates
# Higher return targets: portfolio where the component risk of the US equity dominates
# Shift when:
# > vmu[13:14]
# [1] 0.006796657 0.006806883
# > vriskconc[13:14]
# [1] 0.0082 0.0254
# > vrisk[13:14]
# [1] 0.01544720 0.06036618
# On Cleaned Data: Three segments
# > assetmu
# Bond SP500 EAFE SPGSCI
# 0.07545960 0.10249338 0.08677129 0.05413622
# > assetCVaR
# Bond SP500 EAFE SPGSCI
# 0.02456233 0.09974267 0.10868066 0.12775266
# Unconstrained: Equal Risk Contribution Portfolio:
# > W[1,]
# 0.6431897 0.1337981 0.1044994 0.1185127
#> mPercrisk[1,]
# 0.2516 0.2517 0.2504 0.2463
# > vmu[1]*12
# [1] 0.07773165
#> vrisk[1]
#[1] 0.03467514
#> vriskconc[1]
#[1] 0.0087
# Segment 1: increasing allocation to bonds and decreasing allocation to commodities
# Portfolio risk concentration increases, portfolio risk decreases
# > W[25:27,]
# 0.7052788 0.1584357 0.1321813 0.0041042550
# 0.7085301 0.1597522 0.1313759 0.0003417814
# 0.7025028 0.1628656 0.1344935 0.0001380022
#> vmu[26]*12
#[1] 0.0812571
#> vrisk[26]
#[1] 0.03352778
#> vriskconc[26]
#[1] 0.0112
# Segment 2: increasing allocation to both US equity and EAFE
# Portfolio risk concentration and risk increase together
# > W[138:140,]
# 0.0078323664 0.5106416 0.4814090 1.169711e-04
# 0.0005986683 0.5136429 0.4856355 1.228907e-04
# 0.0002258733 0.5201106 0.4796231 4.051641e-05
# > mPercrisk[138:140,]
# 0e+00 0.4999 0.5000 0
#> vmu[139]*12
#[1] 0.09483605
# > vrisk[139]
# [1] 0.09808784
# > vriskconc[139]
# [1] 0.049
# Segment 3: increasing alllocation to US equity and decreasing to EAFE
# Portfolio risk concentration and risk increase together
# > tail(W,1)
# 5.45402e-05 0.9998762 4.340072e-05 2.585630e-05
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.