CoCalc -- EfficientFrontier

GitHub Repository: braverock/portfolioanalytics
Path: blob/master/sandbox/paper_analysis/insample/EfficientFrontier_MakePlots.R
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# Paper: Portfolio Optimization with Conditional Value-at-Risk Budgets
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# Boudt, Carl, Peterson (2010)
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# This R script serves to make the exhibits regarding the four asset efficient frontier
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setwd("c:/Documents and Settings/Administrator/Desktop/risk budget programs/insample")
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cAssets = 4; p = priskbudget = 0.95;
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mincriterion = "mES" ; percriskcontribcriterion = "mES";
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# Load programs
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source("Risk_budget_functions.R"); 
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library(zoo); library(fGarch); library("PerformanceAnalytics"); library("PortfolioAnalytics")
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clean = TRUE; CC = T
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#   Load the data
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firstyear = 1976 ; firstquarter = 1; lastyear = 2010; lastquarter = 2; 
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data = read.table( file= paste(getwd(),"/data.txt",sep="") ,header=T)
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date = as.Date(data[,1],format="%Y-%m-%d")
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monthlyR = zoo( data[,2:(1+cAssets)] , order.by = date )
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set.seed(1234)
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if(clean){ monthlyR = clean.boudt2(monthlyR,alpha=0.05)[[1]] }
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mu = apply(monthlyR,2,'mean')
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sigma = cov(monthlyR)
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if(!CC){
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  M3 = PerformanceAnalytics:::M3.MM(monthlyR-matrix( rep(as.numeric(mu),nrow(monthlyR)) , nrow=nrow(monthlyR) , byrow=TRUE)  ); 
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  M4 = PerformanceAnalytics:::M4.MM(monthlyR-matrix( rep(as.numeric(mu),nrow(monthlyR)) , nrow=nrow(monthlyR) , byrow=TRUE)  )
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}else{
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  source( "coskewkurtosis.R"  ) 
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  M3 = coskewCC(monthlyR-matrix( rep(as.numeric(mu),nrow(monthlyR)) , nrow=nrow(monthlyR) , byrow=TRUE)  ); 
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  M4 = cokurtCC(monthlyR-matrix( rep(as.numeric(mu),nrow(monthlyR)) , nrow=nrow(monthlyR) , byrow=TRUE)  );
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}
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N = ncol(monthlyR)
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# Summary stats individual assets
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apply(monthlyR,2,'mean')*12
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apply(monthlyR,2,'sd')*sqrt(12)
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apply(monthlyR,2,'skewness')
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apply(monthlyR,2,'kurtosis')
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ES(monthlyR[,1],method="modified"); ES(monthlyR[,2],method="modified")
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ES(monthlyR[,3],method="modified"); ES(monthlyR[,4],method="modified")
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#################################################################################
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# Make Exhibit 3 Risk budget paper: Efficient frontier plot
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#################################################################################
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# Layout 3 
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source("chart.StackedBar.R"); 
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mESfun = function( series ){ return( operMES(  series , alpha = 0.05 , r = 2 ) ) }
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assetmu = apply( monthlyR , 2 , 'mean' )
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assetCVaR = apply( monthlyR , 2 , 'mESfun' )
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mESfun2 = function( w ){ return( operPortMES(w,mu=mu,alpha=0.05,sigma=sigma,M3=M3,M4=M4)[[1]]   ) }
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assetCVaR = rep(0,4);
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for( i in 1:4 ){
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  w = rep(0,4); w[i]=1;
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  assetCVaR[i] = mESfun2( as.matrix(w) )
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}
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#> assetCVaR
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#[1] 0.01235899 0.12465759 0.27755656 0.20558527
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minmu = min(assetmu); maxmu = max(assetmu); print(minmu*12); print(maxmu*12);
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# Load the data
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labelnames = c( "US bond" , "S&P 500" , "NAREIT" , "GSCI" )
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if(clean){
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 # Portfolio weights
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 W_MCC                 = read.csv( file = "EffFrontierMinCVaRConc_weights_clean_CC.csv" )[,2:(1+cAssets)]
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 W_minCVaR             = read.csv( file = "EffFrontierMinCVaR_weights_clean_CC.csv" )[,2:(1+cAssets)]
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 W_minVar              = read.csv( file = "EffFrontierMinVar_weights_clean.csv" )[,2:(1+cAssets)]
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 # Percentage CVaR contributions
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 PercCVaR_MCC         = read.csv(  file = "EffFrontierMinCVaRConc_percrisk_clean_CC.csv" )[,2:(1+cAssets)]
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 PercCVaR_minCVaR     = read.csv(  file = "EffFrontierMinCVaR_percrisk_clean_CC.csv" )[,2:(1+cAssets)]
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 PercCVaR_minVar      = read.csv(  file = "EffFrontierMinVar_percrisk_clean.csv" )[,2:(1+cAssets)]
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 # Summary stats
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 EffFrontier_stats_MCC = read.csv(file = "EffFrontierMinCVaRConc_stats_clean_CC.csv")[,2:5]
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 EffFrontier_stats_minCVaR = read.csv(file = "EffFrontierMinCVar_stats_clean_CC.csv")[,2:5]
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 EffFrontier_stats_minVar = read.csv(file = "EffFrontierMinVar_stats_clean.csv")[,2:5]
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}else{
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 # Portfolio weights
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 W_MCC                 = read.csv(file = "EffFrontierMinCVaRConc_weights_CC.csv")[,2:(1+cAssets)]
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 W_minCVaR             = read.csv(file = "EffFrontierMinCVaR_weights_CC.csv")[,2:(1+cAssets)]
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 W_minVar              = read.csv(file = "EffFrontierMinVar_weights.csv")[,2:(1+cAssets)]
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 # Percentage CVaR contributions
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 PercCVaR_MCC          = read.csv( file = "EffFrontierMinCVaRConc_percrisk_CC.csv" )[,2:(1+cAssets)]
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 PercCVaR_minCVaR      = read.csv( file = "EffFrontierMinCVaR_percrisk_CC.csv" )[,2:(1+cAssets)]
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 PercCVaR_minVar        = read.csv(file = "EffFrontierMinVar_percrisk.csv")[,2:(1+cAssets)]
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 # Summary stats
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 EffFrontier_stats_MCC       = read.csv(file = "EffFrontierMinCVaRConc_stats_CC.csv")[,2:5]
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 EffFrontier_stats_minCVaR   = read.csv(file = "EffFrontierMinCVar_stats_CC.csv")[,2:5]
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 EffFrontier_stats_minVar = read.csv(file = "EffFrontierMinVar_stats.csv")[,2:5]
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}
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vmu_MCC             = EffFrontier_stats_MCC[,1] ;  vmu_minCVaR             = EffFrontier_stats_minCVaR[,1] ; vmu_minVar     = EffFrontier_stats_minVar[,1] ;
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vrisk_MCC           = EffFrontier_stats_MCC[,2] ;  vrisk_minCVaR           = EffFrontier_stats_minCVaR[,2] ; vrisk_minVar   = EffFrontier_stats_minVar[,2] ;
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vmaxpercrisk_MCC    = EffFrontier_stats_MCC[,3] ;  vmaxpercrisk_minCVaR    = EffFrontier_stats_minCVaR[,3] ;
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vriskconc_MCC       = EffFrontier_stats_MCC[,4] ;  vriskconc_minCVaR       = EffFrontier_stats_minCVaR[,4] ; vriskconc_minVar  = EffFrontier_stats_minVar[,4] ;
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vstdev_MCC = c(); 
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for( r in 1:nrow(W_MCC) ){  wr = matrix( as.numeric(W_MCC[r,]) , ncol = 1 ) ; 
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                            vstdev_MCC = c( vstdev_MCC , sqrt( t(wr)%*%sigma%*%wr ) )  }
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vstdev_minVar = c(); 
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for( r in 1:nrow(W_minVar) ){  wr = matrix( as.numeric(W_minVar[r,]) , ncol = 1 ) ; 
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                            vstdev_minVar = c( vstdev_minVar , sqrt( t(wr)%*%sigma%*%wr ) )  }
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vstdev_minCVaR = c(); 
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for( r in 1:nrow(W_minCVaR) ){  wr = matrix( as.numeric(W_minCVaR[r,]) , ncol = 1 ) ; 
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                            vstdev_minCVaR = c( vstdev_minCVaR , sqrt( t(wr)%*%sigma%*%wr ) )  }
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# The MCC and Min CVaR were obtained with DEoptim. The solutions are not always optimal. Correct for this: 
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order_MCC = sort(vmu_MCC,index.return=T)$ix      ; order_minCVaR = sort(vmu_minCVaR,index.return=T)$ix 
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vmu_MCC = vmu_MCC[order_MCC]                     ; vmu_minCVaR = vmu_minCVaR[order_minCVaR] 
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vrisk_MCC = vrisk_MCC[order_MCC]                 ; vrisk_minCVaR = vrisk_minCVaR[order_minCVaR]     
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vstdev_MCC = vstdev_MCC[order_MCC]                 ; vstdev_minCVaR = vstdev_minCVaR[order_minCVaR]       
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vmaxpercrisk_MCC = vmaxpercrisk_MCC[order_MCC]   ; vmaxpercrisk_minCVaR = vmaxpercrisk_MCC[order_minCVaR] 
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vriskconc_MCC = vriskconc_MCC[order_MCC]         ; vriskconc_minCVaR = vriskconc_minCVaR[order_minCVaR] 
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W_MCC  = W_MCC[order_MCC,]                       ; W_minCVaR  = W_minCVaR[order_minCVaR,]     
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PercCVaR_MCC = PercCVaR_MCC[order_MCC,]          ; PercCVaR_minCVaR = PercCVaR_minCVaR[order_minCVaR,]   
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# eliminate duplicates in mu
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# Determines which elements of a vector or data frame are duplicates of elements with smaller subscripts, 
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# and returns a logical vector indicating which elements (rows) are duplicates.
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a_MCC = duplicated(vmu_MCC)                         ; a_minCVaR = duplicated(vmu_minCVaR)  
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vmu_MCC = vmu_MCC[!a_MCC]                           ; vmu_minCVaR = vmu_minCVaR[!a_minCVaR] 
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vrisk_MCC = vrisk_MCC[!a_MCC]                       ; vrisk_minCVaR = vrisk_minCVaR[!a_minCVaR]
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vstdev_MCC = vstdev_MCC[!a_MCC]                       ; vstdev_minCVaR = vstdev_minCVaR[!a_minCVaR]
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vmaxpercrisk_MCC = vmaxpercrisk_MCC[!a_MCC]         ; vmaxpercrisk_minCVaR = vmaxpercrisk_minCVaR[!a_minCVaR] 
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vriskconc_MCC = vriskconc_MCC[!a_MCC]               ; vriskconc_minCVaR = vriskconc_minCVaR[!a_minCVaR]  
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W_MCC = W_MCC[!a_MCC,]                              ; W_minCVaR = W_minCVaR[!a_minCVaR,]    
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PercCVaR_MCC = PercCVaR_MCC[!a_MCC,]                ; PercCVaR_minCVaR = PercCVaR_minCVaR[!a_minCVaR,] 
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# eliminate (efficiency: lower mu, higher CVaR alloc
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sel_MCC = sel_minCVaR = c(); nmu_MCC = length(vmu_MCC); nmu_minCVaR = length(vmu_minCVaR); 
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for( i in 1:(nmu_MCC-1) ){
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    if( any(vriskconc_MCC[(i+1):nmu_MCC]<=vriskconc_MCC[i]) ){ }else{sel_MCC = c(sel_MCC, i) }
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}
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for( i in 1:(nmu_minCVaR-1) ){
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    if( any(vrisk_minCVaR[(i+1):nmu_minCVaR]<=vrisk_minCVaR[i]) ){ }else{sel_minCVaR = c(sel_minCVaR, i) }
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}
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vmu_MCC = vmu_MCC[sel_MCC]                        ; vmu_minCVaR = vmu_minCVaR[sel_minCVaR]  
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vrisk_MCC = vrisk_MCC[sel_MCC]                    ; vrisk_minCVaR = vrisk_minCVaR[sel_minCVaR] 
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vstdev_MCC = vstdev_MCC[sel_MCC]                    ; vstdev_minCVaR = vstdev_minCVaR[sel_minCVaR] 
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vmaxpercrisk_MCC = vmaxpercrisk_MCC[sel_MCC]      ; vmaxpercrisk_minCVaR = vmaxpercrisk_minCVaR[sel_minCVaR]
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vriskconc_MCC = vriskconc_MCC[sel_MCC]            ; vriskconc_minCVaR = vriskconc_minCVaR[sel_minCVaR] 
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W_MCC = W_MCC[sel_MCC,]                           ; W_minCVaR = W_minCVaR[sel_minCVaR,]
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PercCVaR_MCC = PercCVaR_MCC[sel_MCC,]             ; PercCVaR_minCVaR = PercCVaR_minCVaR[sel_minCVaR,] 
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wEW <- rep(1/4,4)
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muEW       = mean(assetmu*12)
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outES      = ES(weights=wEW, portfolio_method="component", mu = mu, sigma = sigma, m3=M3, m4=M4,invert=FALSE) 
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riskEW     = outES$MES; riskconcEW = max(outES$contribution)
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if(clean){
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   postscript('frontier_fourassets_clean.eps')
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}else{ postscript('frontier_fourassets.eps')  } 
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 layout( matrix(  c(1,2,3,1,2,3),  ncol = 2 ) , height= c(5,5,1), width=c(1,1) )
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 ylim = c( min(assetmu) , max(assetmu) )
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 par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
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 plot( vrisk_MCC , vmu_MCC*12 , type="l", lty = 1, main = "" , 
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          ylab="Annualized mean return" , xlab="Monthly 95% Portfolio CVaR" , 
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          lwd=2, xlim = c(0.01,0.13) , ylim = (ylim*12+c(0,0.01)) , col="black" )
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 lines( vrisk_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=2, col="darkgray" ) 
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 lines( vrisk_minVar , vmu_minVar*12,  lty = 3, lwd=2, col="black" )
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 c = 1; text(y=assetmu[c]*12,x= assetCVaR[c]-0.005  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3); points(y=assetmu[c]*12,x= assetCVaR[c] )  
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 for( c in 2:cAssets ){ 
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   text(y=assetmu[c]*12,x= assetCVaR[c]  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3) 
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   points(y=assetmu[c]*12,x= assetCVaR[c] ) 
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 };
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 # Plot also EW portfolio
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 text(y=muEW,x=riskEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 3)
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 points(y=muEW,x= riskEW , pch=22 ) 
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 text(y=vmu_MCC[1]*12,x=vrisk_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_MCC[1]*12,x=vrisk_MCC[1],  pch=22 ) 
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 text(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1]-0.0035, label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1],  pch=22 ) 
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 text(y=vmu_minVar[1]*12-0.001,x=vrisk_minVar[1]+0.0035, label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_minVar[1]*12,x=vrisk_minVar[1],  pch=22 ) 
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 par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
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 plot( vriskconc_MCC/vrisk_MCC , vmu_MCC*12 , type="l", lty = 1, main = "" , 
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          ylab="Annualized  mean return" , xlab="Largest Perc. CVaR contribution" , 
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          lwd=2, xlim = c(0.2,1.1), ylim = c(ylim*12+c(0,0.01)) , col="black" )
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 lines( vriskconc_minCVaR/vrisk_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=2, col="darkgray" ) 
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 lines( vriskconc_minVar/vrisk_minVar   , vmu_minVar*12 ,  lty = 3, lwd=2, col="black" )
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 text(y=muEW,x=riskconcEW/riskEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=muEW,x= riskconcEW/riskEW , pch=22 ) 
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 text(y=vmu_MCC[1]*12,x=vriskconc_MCC[1]/vrisk_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_MCC[1]*12,x=vriskconc_MCC[1]/vrisk_MCC[1],  pch=22 ) 
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 text(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1]/vrisk_minCVaR[1], label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1]/vrisk_minCVaR[1],  pch=22 ) 
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 text(y=vmu_minVar[1]*12,x=vriskconc_minVar[1]/vrisk_minVar[1], label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_minVar[1]*12,x=vriskconc_minVar[1]/vrisk_minVar[1],  pch=22 ) 
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 for( c in 1:cAssets ){ 
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   text(y=assetmu[c]*12,x= 1  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 4) 
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   points(y=assetmu[c]*12,x= 1 ) 
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 };
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 par( mar = c(0,1,1,0) )
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 plot.new()
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  legend("center",legend=c("Mean-CVaR concentration", "Mean-CVaR","Mean-StdDev"  ),
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   lty=c("solid","solid","dashed"), cex=0.8,ncol=3,lwd=c(4,4,2),col=c("black","darkgray","black"))
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dev.off()
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if(clean){
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   postscript('frontier_fourassets_bis_clean.eps')
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}else{ postscript('frontier_fourassets_bis.eps')  } 
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 layout( matrix(  c(1,2,3,1,2,3),  ncol = 2 ) , height= c(5,5,1), width=c(1,1) )
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 ylim = c( min(assetmu) , max(assetmu) )
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 par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
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 plot( vrisk_MCC , vmu_MCC*12 , type="l", lty = 1, main = "" , 
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          ylab="Annualized mean return" , xlab="Monthly 95% Portfolio CVaR" , 
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          lwd=2, xlim = c(0,0.13) , ylim = (ylim*12+c(0,0.02)) , col="black" )
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 lines( vrisk_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=2, col="darkgray" ) 
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 lines( vrisk_minVar , vmu_minVar*12,  lty = 3, lwd=2, col="black" )
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 c = 1; text(y=assetmu[c]*12,x= assetCVaR[c]-0.005  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3); points(y=assetmu[c]*12,x= assetCVaR[c] )  
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 for( c in 2:cAssets ){ 
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   text(y=assetmu[c]*12,x= assetCVaR[c]  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3) 
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   points(y=assetmu[c]*12,x= assetCVaR[c] ) 
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 };
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 # Plot also EW portfolio
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 text(y=muEW,x=riskEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 3)
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 points(y=muEW,x= riskEW , pch=22 ) 
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 text(y=vmu_MCC[1]*12,x=vrisk_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 4)
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 points(y=vmu_MCC[1]*12,x=vrisk_MCC[1],  pch=22 ) 
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 text(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1]-0.0035, label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1],  pch=22 ) 
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 text(y=vmu_minVar[1]*12-0.0035,x=vrisk_minVar[1], label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_minVar[1]*12,x=vrisk_minVar[1],  pch=22 ) 
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 par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
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 plot( vriskconc_MCC, vmu_MCC*12 , type="l", lty = 1, main = "" , 
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          ylab="Annualized  mean return" , xlab="Largest CVaR contribution" , 
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          lwd=2, xlim = c(0,0.13), ylim = c(ylim*12+c(0,0.02)) , col="black" )
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 lines( vriskconc_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=2, col="darkgray" ) 
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 lines( vriskconc_minVar   , vmu_minVar*12 ,  lty = 3, lwd=2, col="black" )
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 text(y=muEW,x=riskconcEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=muEW,x= riskconcEW , pch=22 ) 
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 text(y=vmu_MCC[1]*12,x=vriskconc_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 2)
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 points(y=vmu_MCC[1]*12,x=vriskconc_MCC[1],  pch=22 ) 
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 text(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1], label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 4)
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 points(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1],  pch=22 ) 
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 text(y=vmu_minVar[1]*12-0.003,x=vriskconc_minVar[1], label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 1)
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 points(y=vmu_minVar[1]*12,x=vriskconc_minVar[1],  pch=22 ) 
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 #for( c in 1:cAssets ){ 
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 #  text(y=assetmu[c]*12,x= 1  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 4) 
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 #  points(y=assetmu[c]*12,x= 1 ) 
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 #};
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 par( mar = c(0,1,1,0) )
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 plot.new()
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  legend("center",legend=c("Mean-CVaR concentration", "Mean-CVaR","Mean-StdDev"  ),
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   lty=c("solid","solid","dashed"), cex=0.8,ncol=3,lwd=c(4,4,2),col=c("black","darkgray","black"))
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dev.off()
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if(clean){
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   postscript('frontier_fourassets_bis_clean.eps')
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}else{ postscript('frontier_fourassets_bis.eps')  } 
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 layout( matrix(  c(1,2,3,1,2,3),  ncol = 2 ) , height= c(5,5,1), width=c(1,1) )
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 ylim = c( min(assetmu) , max(assetmu) )
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 par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
284
 plot( vrisk_MCC , vmu_MCC*12 , type="l", lty = 1, main = "" , 
285
          ylab="Annualized mean return" , xlab="Monthly 95% Portfolio CVaR" , 
286
          lwd=2, xlim = c(0,0.13) , ylim = (ylim*12+c(0,0.02)) , col="black" )
287
 lines( vrisk_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=2, col="darkgray" ) 
288
 lines( vrisk_minVar , vmu_minVar*12,  lty = 3, lwd=2, col="black" )
289

290
 c = 1; text(y=assetmu[c]*12,x= assetCVaR[c]-0.005  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3); 
291
 points(y=assetmu[c]*12,x= assetCVaR[c] )  
292
 for( c in 2:cAssets ){ 
293
   text(y=assetmu[c]*12,x= assetCVaR[c]  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3) 
294
   points(y=assetmu[c]*12,x= assetCVaR[c] ) 
295
 };
296
 # Plot also EW portfolio
297
 text(y=muEW,x=riskEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 3)
298
 points(y=muEW,x= riskEW , pch=22 ) 
299
 text(y=vmu_MCC[1]*12,x=vrisk_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 4)
300
 points(y=vmu_MCC[1]*12,x=vrisk_MCC[1],  pch=22 ) 
301
 text(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1]-0.0035, label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 1)
302
 points(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1],  pch=22 ) 
303
 text(y=vmu_minVar[1]*12-0.0035,x=vrisk_minVar[1], label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 1)
304
 points(y=vmu_minVar[1]*12,x=vrisk_minVar[1],  pch=22 ) 
305

306
 par( mar = c(4.5,5,2,2), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
307

308
 plot( vriskconc_MCC, vmu_MCC*12 , type="l", lty = 1, main = "" , 
309
          ylab="Annualized  mean return" , xlab="Largest CVaR contribution" , 
310
          lwd=2, xlim = c(0,0.13), ylim = c(ylim*12+c(0,0.02)) , col="black" )
311
 lines( vriskconc_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=2, col="darkgray" ) 
312
 lines( vriskconc_minVar   , vmu_minVar*12 ,  lty = 3, lwd=2, col="black" )
313

314
 text(y=muEW,x=riskconcEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 1)
315
 points(y=muEW,x= riskconcEW , pch=22 ) 
316
 text(y=vmu_MCC[1]*12,x=vriskconc_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 2)
317
 points(y=vmu_MCC[1]*12,x=vriskconc_MCC[1],  pch=22 ) 
318
 text(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1], label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 4)
319
 points(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1],  pch=22 ) 
320
 text(y=vmu_minVar[1]*12-0.003,x=vriskconc_minVar[1], label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 1)
321
 points(y=vmu_minVar[1]*12,x=vriskconc_minVar[1],  pch=22 ) 
322

323
 #for( c in 1:cAssets ){ 
324
 #  text(y=assetmu[c]*12,x= 1  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 4) 
325
 #  points(y=assetmu[c]*12,x= 1 ) 
326
 #};
327
 
328
 par( mar = c(0,1,1,0) )
329
 plot.new()
330
  legend("center",legend=c("Mean-CVaR concentration", "Mean-CVaR","Mean-StdDev"  ),
331
   lty=c("solid","solid","dashed"), cex=0.8,ncol=3,lwd=c(4,4,2),col=c("black","darkgray","black"))
332

333
dev.off()
334

335
assetStdDev = sqrt(diag(sigma))
336
StdDevEW = sqrt(matrix( rep(1/4,4) , ncol = 4 )%*%sigma%*%matrix( rep(1/4,4) , ncol = 1 ))
337
if(clean){
338
   postscript('frontier_fourassets_tris_clean.eps')
339
}else{ postscript('frontier_fourassets_tris.eps')  } 
340

341
 layout( matrix(  c(1,2,3,4,1,2,3,4),  ncol = 2 , nrow=4 ) , height= c(5,5,5,1), width=c(1,1) )
342

343
 layout( matrix(  c(1,2,3,4),  ncol = 2 , nrow=2 ) , height= c(5,5), width=c(1,1) )
344

345
 ylim = c( min(assetmu) , max(assetmu) )
346
 par( mar = c(4.5,5,2,0.1), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
347
 plot( vstdev_MCC*sqrt(12) , vmu_MCC*12 , type="l", lty = 1, main = "" , 
348
          ylab="Annualized mean return" , xlab="Annualized Portfolio StdDev" , 
349
          lwd=2, xlim = c(-0.01,0.2) , ylim = (ylim*12+c(0,0.01)) , col="black" )
350
 lines( vstdev_minCVaR*sqrt(12) , vmu_minCVaR*12,  lty = 1, lwd=1, col="darkgray" ) 
351
 lines( vstdev_minVar*sqrt(12) , vmu_minVar*12,  lty = 3, lwd=2, col="black" )
352

353
 c = 1; text(y=assetmu[c]*12+0.0005,x= assetStdDev[c]*sqrt(12)  , label=labelnames[c]  ,
354
       cex = 0.7, offset = 0.2, pos = 4); 
355
 points(y=assetmu[c]*12,x= assetStdDev[c]*sqrt(12) )  
356
 for( c in 2:cAssets ){ 
357
   text(y=assetmu[c]*12,x= assetStdDev[c]*sqrt(12)  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3) 
358
   points(y=assetmu[c]*12,x= assetStdDev[c]*sqrt(12) ) 
359
 };
360
 # Plot also EW portfolio
361
 text(y=muEW,x=StdDevEW*sqrt(12), label="EW"  , cex = 0.7, offset = 0.2, pos = 3)
362
 points(y=muEW,x= StdDevEW*sqrt(12) , pch=22 ) 
363
 text(y=vmu_MCC[1]*12,x=vstdev_MCC[1]*sqrt(12), label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 4)
364
 points(y=vmu_MCC[1]*12,x=vstdev_MCC[1]*sqrt(12),  pch=22 ) 
365
 text(y=vmu_minCVaR[1]*12-0.0035,x=vstdev_minCVaR[1]*sqrt(12)-0.0035, label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 4)
366
 points(y=vmu_minCVaR[1]*12,x=vstdev_minCVaR[1]*sqrt(12),  pch=22 ) 
367
 text(y=vmu_minVar[1]*12+0.0035,x=vstdev_minVar[1]*sqrt(12), label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 2)
368
 points(y=vmu_minVar[1]*12,x=vstdev_minVar[1]*sqrt(12),  pch=22 ) 
369

370
 # Values of 1, 2, 3 and 4, respectively indicate positions below, to the left of, above and to the right 
371

372

373

374

375
 par( mar = c(0,1,1,0.1) )
376
 plot.new()
377
  legend("center",legend=c("Mean-CVaR concentration", "Mean-CVaR","Mean-StdDev"  ),
378
   lty=c("solid","solid","dashed"), cex=0.8,ncol=1,lwd=c(4,2,2),col=c("black","darkgray","black"))
379

380

381
 ylim = c( min(assetmu) , max(assetmu) )
382
 par( mar = c(4.5,5,2,0.1), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
383
 plot( vrisk_MCC , vmu_MCC*12 , type="l", lty = 1, main = "" , 
384
          ylab="Annualized mean return" , xlab="Monthly 95% Portfolio CVaR" , 
385
          lwd=2, xlim = c(-0.01,0.13) , ylim = (ylim*12+c(0,0.01)) , col="black" )
386
 lines( vrisk_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=1, col="darkgray" ) 
387
 lines( vrisk_minVar , vmu_minVar*12,  lty = 3, lwd=2, col="black" )
388

389
 c = 1; text(y=assetmu[c]*12,x= assetCVaR[c]-0.005  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3); points(y=assetmu[c]*12,x= assetCVaR[c] )  
390
 for( c in 2:cAssets ){ 
391
   text(y=assetmu[c]*12,x= assetCVaR[c]  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 3) 
392
   points(y=assetmu[c]*12,x= assetCVaR[c] ) 
393
 };
394
 # Plot also EW portfolio
395
 text(y=muEW,x=riskEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 3)
396
 points(y=muEW,x= riskEW , pch=22 ) 
397
 text(y=vmu_MCC[1]*12,x=vrisk_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 4)
398
 points(y=vmu_MCC[1]*12,x=vrisk_MCC[1],  pch=22 ) 
399
 text(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1]-0.0035, label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 1)
400
 points(y=vmu_minCVaR[1]*12,x=vrisk_minCVaR[1],  pch=22 ) 
401
 text(y=vmu_minVar[1]*12,x=vrisk_minVar[1], label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 4)
402
 points(y=vmu_minVar[1]*12,x=vrisk_minVar[1],  pch=22 ) 
403

404

405

406
 par( mar = c(4.5,5,2,0.1), las=1 ,cex=0.9 , cex.axis=0.9, cex.main=0.95)
407

408
 plot( vriskconc_MCC, vmu_MCC*12 , type="l", lty = 1, main = "" , 
409
          ylab="Annualized  mean return" , xlab="Largest contribution to monthly 95% CVaR" , 
410
          lwd=2, xlim = c(-0.01,0.13), ylim = c(ylim*12+c(0,0.01)) , col="black" )
411
 lines( vriskconc_minCVaR , vmu_minCVaR*12,  lty = 1, lwd=1, col="darkgray" ) 
412
 lines( vriskconc_minVar   , vmu_minVar*12 ,  lty = 3, lwd=2, col="black" )
413

414
 text(y=muEW,x=riskconcEW, label="EW"  , cex = 0.7, offset = 0.2, pos = 1)
415
 points(y=muEW,x= riskconcEW , pch=22 ) 
416
 text(y=vmu_MCC[1]*12,x=vriskconc_MCC[1], label="MCC=ERC"  , cex = 0.7, offset = 0.2, pos = 2)
417
 points(y=vmu_MCC[1]*12,x=vriskconc_MCC[1],  pch=22 ) 
418
 text(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1], label="Min CVaR"  , cex = 0.7, offset = 0.2, pos = 4)
419
 points(y=vmu_minCVaR[1]*12,x=vriskconc_minCVaR[1],  pch=22 ) 
420
 text(y=vmu_minVar[1]*12,x=vriskconc_minVar[1], label="Min StdDev"  , cex = 0.7, offset = 0.2, pos = 2)
421
 points(y=vmu_minVar[1]*12,x=vriskconc_minVar[1],  pch=22 ) 
422

423

424

425
 #for( c in 1:cAssets ){ 
426
 #  text(y=assetmu[c]*12,x= 1  , label=labelnames[c]  , cex = 0.7, offset = 0.2, pos = 4) 
427
 #  points(y=assetmu[c]*12,x= 1 ) 
428
 #};
429
 
430

431

432
dev.off()
433

434

435
# Make the weight/CVaR allocation plots
436
#---------------------------------------
437

438
Wsel_MCC = PercCVaRsel_MCC  = Wsel_minCVaR = PercCVaRsel_minCVaR  = Wsel_minVar = PercCVaRsel_minVar  = c(); 
439
vmusel_MCC = vrisksel_MCC = vriskconcsel_MCC = vmusel_minCVaR = vrisksel_minCVaR = vriskconcsel_minCVaR =  vmusel_minVar = vrisksel_minVar = vriskconcsel_minVar =   c();
440
lm = minmu = min( c(vmu_MCC,vmu_minCVaR) ) ; maxmu = max( c(vmu_MCC,vmu_minCVaR) );
441
ylim = c( min(assetmu) - 0.0003 , max(assetmu) + 0.0003 )
442
xlim = c( 0 , max(assetCVaR) + 0.01 )
443

444
binning = T # for the weight plots, binned data such that no visual misinterpretation is possible
445
step = quantile( diff(vmu_MCC) , 0.1 ) 
446
if( binning ){
447
   for( rm in seq( minmu+step , maxmu , step )  ){
448
      selection    =  c(vmu_MCC >= lm & vmu_MCC < rm) ;
449
      if( any(selection) ){
450
         selection    = c(1:length(selection))[selection]
451
         selone       = sort(vriskconc_MCC[ selection ],index.return=T)$ix[1]
452
         selone       = selection[selone]
453
         vmusel_MCC       = c( vmusel_MCC       ,  mean(vmu_MCC[selone  ] ));
454
         Wsel_MCC         = rbind( Wsel_MCC     ,  apply( W_MCC[selone,] ,2,'mean' ))
455
         PercCVaRsel_MCC  = rbind( PercCVaRsel_MCC  ,  apply( PercCVaR_MCC[selone,] ,2,'mean' )) 
456
         vrisksel_MCC     = c( vrisksel_MCC     ,  mean(vrisk_MCC[ selone ]) );
457
         vriskconcsel_MCC = c( vriskconcsel_MCC ,  mean(vriskconc_MCC[ selone ])  )
458
      }else{
459
         vmusel_MCC       = c( vmusel_MCC       ,  NA );
460
         Wsel_MCC         = rbind( Wsel_MCC     ,  rep(NA,cAssets)  )
461
         PercCVaRsel_MCC  = rbind( PercCVaRsel_MCC  ,  rep(NA,cAssets) ) 
462
         vrisksel_MCC     = c( vrisksel_MCC     ,  NA );
463
         vriskconcsel_MCC = c( vriskconcsel_MCC ,  NA  )
464
      }
465
      selection    =  c(vmu_minCVaR >= lm & vmu_minCVaR < rm) ;
466
      if( any(selection) ){
467
         selection    = c(1:length(selection))[selection]; selone       = sort(vrisk_minCVaR[ selection ],index.return=T)$ix[1]
468
         selone       = selection[selone]
469
         vmusel_minCVaR       = c( vmusel_minCVaR       ,  mean(vmu_minCVaR[selone  ] ));
470
         Wsel_minCVaR         = rbind( Wsel_minCVaR     ,  apply( W_minCVaR[selone,] ,2,'mean' ))
471
         PercCVaRsel_minCVaR  = rbind( PercCVaRsel_minCVaR  ,  apply( PercCVaR_minCVaR[selone,] ,2,'mean' )) 
472
         vrisksel_minCVaR     = c( vrisksel_minCVaR     ,  mean(vrisk_minCVaR[ selone ]) );
473
         vriskconcsel_minCVaR = c( vriskconcsel_minCVaR ,  mean(vriskconc_minCVaR[ selone ])  )
474
      }else{
475
         vmusel_minCVaR       = c( vmusel_minCVaR       ,  NA );
476
         Wsel_minCVaR         = rbind( Wsel_minCVaR     ,  rep(NA,cAssets) )
477
         PercCVaRsel_minCVaR  = rbind( PercCVaRsel_minCVaR  ,  rep(NA,cAssets) ) 
478
         vrisksel_minCVaR     = c( vrisksel_minCVaR     ,  NA );
479
         vriskconcsel_minCVaR = c( vriskconcsel_minCVaR ,  NA )
480
      }
481
      selection    =  c(vmu_minVar >= lm & vmu_minVar < rm) ;
482
      if( any(selection) ){
483
         selection    = c(1:length(selection))[selection]; selone       = sort(vrisk_minVar[ selection ],index.return=T)$ix[1]
484
         selone       = selection[selone]
485
         vmusel_minVar       = c( vmusel_minVar       ,  mean(vmu_minVar[selone  ] ));
486
         Wsel_minVar         = rbind( Wsel_minVar     ,  apply( W_minVar[selone,] ,2,'mean' ))
487
         PercCVaRsel_minVar  = rbind( PercCVaRsel_minVar  ,  apply( PercCVaR_minVar[selone,] ,2,'mean' )) 
488
         vrisksel_minVar     = c( vrisksel_minVar     ,  mean(vrisk_minVar[ selone ]) );
489
         vriskconcsel_minVar = c( vriskconcsel_minVar ,  mean(vriskconc_minVar[ selone ])  )
490
      }else{
491
         vmusel_minVar       = c( vmusel_minVar       ,  NA );
492
         Wsel_minVar         = rbind( Wsel_minVar     ,  rep(NA,cAssets) )
493
         PercCVaRsel_minVar  = rbind( PercCVaRsel_minVar  ,  rep(NA,cAssets) ) 
494
         vrisksel_minVar     = c( vrisksel_minVar     ,  NA );
495
         vriskconcsel_minVar = c( vriskconcsel_minVar ,  NA  )
496
      }
497
      lm = rm; 
498
   }
499
}else{
500
   vmusel_MCC = vmu_MCC              ;   vmusel_minCVaR = vmu_minCVaR 
501
   Wsel_MCC   = W_MCC                ;   Wsel_minCVaR   = W_minCVaR                ;
502
   vrisksel_MCC = vrisk_MCC          ;   vrisksel_minCVaR = vrisk_minCVaR  
503
   vriskconcsel_MCC = vriskconc_MCC  ;   vriskconcsel_minCVar = vriskconc_minCVaR
504
   PercCVaRsel_MCC =  PercCVaR_MCC   ;   PercCVaRsel_minCVar =  PercCVaR_minCVar       ; 
505
}
506

507
library(zoo)
508
cAssets = ncol(monthlyR)
509
 colorset = sort(gray( 1-(seq(1,cAssets,1)/(cAssets))^1.5 )) ; 
510
colnames( Wsel_MCC ) = colnames( Wsel_minCVaR ) = labelnames
511
rownames( Wsel_MCC ) = rownames( PercCVaRsel_MCC ) = round( interpNA(vmusel_MCC)*12,4); 
512
rownames( Wsel_minCVaR ) = rownames( PercCVaRsel_minCVaR ) = round( interpNA(vmusel_minCVaR)*12,4) ;
513

514
rownames( Wsel_minVar ) = rownames( PercCVaRsel_minVar ) = round( interpNA(vmusel_minVar)*12,4) ; 
515
colnames( Wsel_minVar ) = labelnames
516

517

518
w.names = c( "US bond" , "S&P 500", "NAREIT"  , "GSCI" )
519
namelabels = c("Mean-StdDev" , "Mean-CVaR","Mean-CVaR concentration"  )
520
l = 2 ; mar1 =c(2,l,2,1.1) ; mar3 = c(3,l+1,3,0.1)
521

522
source("chart.StackedBar.R"); 
523

524
# Stacked weights plot: 
525
if(clean){
526
   postscript('stackedweightsriskcont_efficientfrontier_withNA_clean.eps')
527
}else{ postscript('stackedweightsriskcont_efficientfrontier_withNA.eps')  } 
528

529
   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)
530

531
   par(mar=mar3 , cex.main=1)
532
   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 )
533
   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 )
534
   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 )
535

536
   par(mar=mar1 , cex.main=1)
537
   plot.new()
538
   legend("center",legend=w.names,fill=colorset,ncol=4)
539

540
   par(mar=mar3 , cex.main=1)
541
   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 )
542
   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 )
543
   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 )
544

545
dev.off()
546

547

548
if(clean){
549
   postscript('stackedweightsriskcont_efficientfrontier_clean.eps')
550
}else{ postscript('stackedweightsriskcont_efficientfrontier.eps')  } 
551

552
   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)
553

554
   par(mar=mar3 , cex.main=1)
555
   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 )
556
   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 )
557
   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 )
558

559
   par(mar=mar1 , cex.main=1)
560
   plot.new()
561
   legend("center",legend=w.names,col=colorset,ncol=4,lwd=6)
562

563
   par(mar=mar3 , cex.main=1)
564
   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 )
565
   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 )
566
   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 )
567

568
dev.off()
569

570

571

572
#----------------------------------------------------------------------------------------------------
573
# DISCUSSION RESULTS
574

575

576
# unconstrained:
577
12*vmu_minVar[1] ; sqrt(12)*vstdev_minVar[1] ; vrisk_minVar[1] ; W_minVar[1,] ; PercCVaR_minVar[1,]; 
578
12*vmu_minCVaR[1]; sqrt(12)*vstdev_minCVaR[1] ;  vrisk_minCVaR[1]; W_minCVaR[1,]; PercCVaR_minCVaR[1,]; 
579
12*vmu_MCC[1]    ; sqrt(12)*vstdev_MCC[1] ;  vrisk_MCC[1]    ; W_MCC[1,]    ; PercCVaR_MCC[1,]; 
580

581
> # unconstrained:
582
> 12*vmu_minVar[1] ; sqrt(12)*vstdev_minVar[1] ; vrisk_minVar[1] ; W_minVar[1,] ; PercCVaR_minVar[1,]; 
583
[1] 0.07705461
584
[1] 0.04967331
585
[1] 0.0277955
586
         V1         V2         V3         V4
587
1 0.8404353 0.03862133 0.03643427 0.08450912
588
         V1         V2         V3        V4
589
1 0.5620443 0.08734358 0.08376784 0.2668442
590
> 12*vmu_minCVaR[1]; sqrt(12)*vstdev_minCVaR[1] ;  vrisk_minCVaR[1]; W_minCVaR[1,]; PercCVaR_minCVaR[1,]; 
591
[1] 0.075321
592
[1] 0.05230409
593
[1] 0.0228343
594
         V1 V2 V3         V4
595
2 0.9771176  0  0 0.02288243
596
         V1 V2 V3         V4
597
2 0.9771177  0  0 0.02288231
598
> 12*vmu_MCC[1]    ; sqrt(12)*vstdev_MCC[1] ;  vrisk_MCC[1]    ; W_MCC[1,]    ; PercCVaR_MCC[1,]; 
599
[1] 0.08714249
600
[1] 0.06588349
601
[1] 0.038659
602
         V1        V2        V3        V4
603
1 0.4733987 0.2070037 0.1843733 0.1352243
604
       V1        V2        V3        V4
605
1 0.25064 0.2505296 0.2497479 0.2490826
606

607

608

609
cbind( vmu_minVar*12 , vriskconc_minVar ) :
610

611
cbind( vmu_minCVaR*12 , vriskconc_minCVaR ) : 0.0816 is turning point
612
# [124,] 0.09017165        0.01210119 is turning point
613

614
cbind( vmu_MCC*12 , vrisk_MCC, vriskconc_MCC ) 
615

616
# Mean/CVaR concentration efficient portfolios
617

618
# On Cleaned Data: Three segments
619

620
> cbind( vmu_MCC*12 , vrisk_MCC, vriskconc_MCC )
621
  [1,] 0.08714249 0.03865900   0.009689492
622
  [2,] 0.08726420 0.03849615   0.009802639
623
#24
624
 [23,] 0.08997886 0.03585947   0.011824072
625
 [24,] 0.09010885 0.03585032   0.011871467
626
 [25,] 0.09024551 0.03589501   0.011974648
627
 [26,] 0.09036701 0.03594272   0.012064923
628
 [27,] 0.09050353 0.03610158   0.012242378
629
 [28,] 0.09062808 0.03621858   0.012405637
630
 [29,] 0.09075394 0.03635143   0.012544703
631
 [30,] 0.09087482 0.03674130   0.012857458
632
# 174
633
[170,] 0.10996933 0.06748652   0.039200481
634
[171,] 0.11008934 0.06789785   0.042263237
635
[172,] 0.11020953 0.06842979   0.045351289
636
[173,] 0.11033011 0.06909579   0.048542053
637
[174,] 0.11045013 0.06985814   0.051639874
638
[175,] 0.11057105 0.07075940   0.054836023
639
[176,] 0.11069142 0.07176853   0.057990740
640

641
[193,] 0.11273404 0.10433383   0.104615672
642
[194,] 0.11273413 0.10433618   0.104617643
643

644
> W_MCC
645
              V1         V2        V3            V4
646
1   4.733987e-01 0.20700375 0.1843733  1.352243e-01
647
3   4.759377e-01 0.20759359 0.1855799  1.308888e-01
648
#24
649
23  5.561089e-01 0.22228044 0.2020208  1.958989e-02
650
24  5.614145e-01 0.22306274 0.2027785  1.274428e-02
651
25  5.651643e-01 0.22387831 0.2036273  7.330121e-03
652
26  5.664972e-01 0.22511059 0.2048818  3.510355e-03
653
27  5.674492e-01 0.22627029 0.2060026  2.778562e-04
654
28  5.623908e-01 0.22852300 0.2080519  1.034299e-03
655
29  5.595434e-01 0.23008873 0.2098842  4.836782e-04
656

657
# 174
658
170 1.444844e-02 0.51421725 0.4710510  2.833091e-04
659
171 1.074035e-02 0.51609949 0.4727863  3.738407e-04
660
172 7.379454e-03 0.51796198 0.4745030  1.555284e-04
661
173 3.671705e-03 0.51988524 0.4762691  1.739490e-04
662
174 2.930070e-04 0.52170814 0.4779540  4.485562e-05
663
175 2.228161e-05 0.50313662 0.4968072  3.392269e-05
664
176 9.137182e-07 0.48289134 0.5170837  2.405561e-05
665

666
199 1.461185e-06 0.01379234 0.9862059  2.992504e-07
667
200 8.716672e-07 0.01377375 0.9862242  1.214785e-06
668

669
# Unconstrained: Equal Risk Contribution Portfolio:
670
# > W_MCC[1,]
671
# 0.4733987 0.2070037 0.1843733 0.1352243
672
#> PercCVaR_MCC[1,]
673
#  0.25064 0.2505296 0.2497479 0.2490826
674
# > vmu_MCC[1]*12
675
# [1] 0.08714249
676
#> vrisk_MCC[1]
677
#[1] 0.038659
678
#> vriskconc_MCC[1]
679
#[1]  0.009689492
680
# Segment 1: increasing allocation to bonds and decreasing allocation to commodities
681
# Portfolio risk concentration increases, portfolio risk decreases
682
# > W_MCC[25:27,]
683
# 0.7052788 0.1584357 0.1321813 0.0041042550
684
# 0.7085301 0.1597522 0.1313759 0.0003417814
685
# 0.7025028 0.1628656 0.1344935 0.0001380022
686
#> vmu_MCC[26]*12
687
#[1] 0.0812571
688
#> vrisk_MCC[26]
689
#[1] 0.03352778
690
#> vriskconc_MCC[26]
691
#[1] 0.0112
692
# Segment 2: increasing allocation to both US equity and EAFE
693
# Portfolio risk concentration and risk increase together
694
# > W_MCC[138:140,]
695
# 0.0078323664 0.5106416 0.4814090 1.169711e-04
696
# 0.0005986683 0.5136429 0.4856355 1.228907e-04
697
# 0.0002258733 0.5201106 0.4796231 4.051641e-05
698
# > PercCVaR_MCC[138:140,]
699
#  0e+00 0.4999 0.5000  0
700
#> vmu_MCC[139]*12
701
#[1] 0.09483605
702
# > vrisk_MCC[139]
703
# [1] 0.09808784
704
# > vriskconc_MCC[139]
705
# [1] 0.049
706
# Segment 3: increasing alllocation to US equity and decreasing to EAFE
707
# Portfolio risk concentration and risk increase together
708
# > tail(W_MCC,1)
709
# 5.45402e-05 0.9998762 4.340072e-05 2.585630e-05
710

711

712

713
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