CoCalc -- Effect_Feasible

GitHub Repository: braverock/portfolioanalytics
Path: blob/master/sandbox/paper_analysis/illustration/Effect_Feasible_Space.R
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setwd("C:\\Documents and Settings\\Administrator\\Desktop\\risk budget programs\\illustration")
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# Bivariate normal example of the effect of constraints on percentage CVaR on the feasible space
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showweightlines = T;
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srt = 0 ; 
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CVaR = function( w , mu , sigma , alpha )
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{
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   out = -t(w)%*%mu + sqrt( t(w)%*%sigma%*%w )*dnorm(qnorm(alpha))/alpha
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   return( out );
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}
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CompCVaR = function( w , mu , sigma , alpha )
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{
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   out = -mu + as.vector(sigma%*%w)*(1/sqrt( t(w)%*%sigma%*%w))*dnorm(qnorm(alpha))/alpha
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   return( w*out );
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} 
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N = 2; rho1 = -0.5 ; rho2 = 0; rho3 = 0.5; alpha = 0.05
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mu1 = as.matrix( rep(0,N) ) ; mu2 = as.matrix( c(1, 0 ) ) ; 
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sigma1 = sigma2 = matrix ( rep(rho1,N^2) , ncol = N )
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sigma3 = sigma4 = matrix ( rep(rho2,N^2) , ncol = N )
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sigma5 = sigma6 = matrix ( rep(rho3,N^2) , ncol = N )
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diag(sigma1) = diag(sigma2) = diag(sigma3) = diag(sigma4) = diag(sigma5) = diag(sigma6) =1;
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sigma2[2,2] = sigma4[2,2] = sigma6[2,2] = 4; 
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sigma2[1,2]=sigma2[2,1]=2*sigma2[1,2];
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sigma4[1,2]=sigma4[2,1]=2*sigma4[1,2];
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sigma6[1,2]=sigma6[2,1]=2*sigma6[1,2];
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w = as.matrix( rep(0.5,2) ) 
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percCompCVaR = function( w = w , mu = mu , sigma = sigma , alpha = alpha ){
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   out1 = CompCVaR( w = w , mu = mu , sigma = sigma , alpha = alpha )
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   out2 = CVaR( w = w , mu = mu , sigma = sigma , alpha = alpha )
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   return(out1/as.numeric(out2))
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}
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vw1 = seq(0,1,0.01)
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constraint = rep(0,length(vw1));
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i=0;
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vpccvar1 = vpccvar2 = vpccvar3 = vpccvar4 = vpccvar5 = vpccvar6 = vpccvar7 = vpccvar8 = vpccvar9 = c();
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for( w1 in vw1 )
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{
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   w2 = 1-w1; i = i+1;
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   w = as.matrix( c(w1,w2) )
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   vpccvar1 = rbind( vpccvar1 , percCompCVaR( w = w , mu = mu1 , sigma = sigma1 , alpha = alpha )[1])
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   vpccvar2 = rbind( vpccvar2 , percCompCVaR( w = w , mu = mu2 , sigma = sigma1 , alpha = alpha )[1])
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   vpccvar3 = rbind( vpccvar3 , percCompCVaR( w = w , mu = mu1 , sigma = sigma2 , alpha = alpha )[1])
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   vpccvar4 = rbind( vpccvar4 , percCompCVaR( w = w , mu = mu1 , sigma = sigma3 , alpha = alpha )[1])
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   vpccvar5 = rbind( vpccvar5 , percCompCVaR( w = w , mu = mu2 , sigma = sigma3 , alpha = alpha )[1])
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   vpccvar6 = rbind( vpccvar6 , percCompCVaR( w = w , mu = mu1 , sigma = sigma4 , alpha = alpha )[1])
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   vpccvar7 = rbind( vpccvar7 , percCompCVaR( w = w , mu = mu1 , sigma = sigma5 , alpha = alpha )[1])
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   vpccvar8 = rbind( vpccvar8 , percCompCVaR( w = w , mu = mu2 , sigma = sigma5 , alpha = alpha )[1])
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   vpccvar9 = rbind( vpccvar9 , percCompCVaR( w = w , mu = mu1 , sigma = sigma6 , alpha = alpha )[1])
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}
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# Plot 9 x 9
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if(showweightlines){ postscript('sensitivity.eps')  }else{ postscript('sensitivity_noweightlines.eps')}
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par( mfrow = c(3,3) , las = 1 , mar=c(2.2,2.5,2,1.5) , cex=1 , cex.main=1)
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# !!!! Setup 1: mu1, sigma1
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plot( vw1 , vpccvar1 , main = expression( mu[1]*"="*mu[2]*"=0 and "*sigma[1]*"="*sigma[2]*"=1") ,
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         type="l",ylab="" )
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text( x = 0.1 , y = 0.95*max(vpccvar1) , labels=expression( rho*"="*-0.5) )
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# %C[2]CVaR <= 60% implies %C[1]CVaR >= 40%
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if(showweightlines){ abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )}
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# Indicate ERC portfolio
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) ); return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma1 , alpha = alpha )[1] ) }
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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print(ERC); print( 1/2 ) ; 
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text( x = ERC , y = 0.5 , labels="ERC" , srt = srt )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 3: mu1, sigma2
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plot( vw1 , vpccvar3 , main = expression( mu[1]*"="*mu[2]*"=0 and "*sigma[1]*"=1, "*sigma[2]*"=2   "), type="l"  )
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text( x = 0.1 , y = 0.95*max(vpccvar3) , labels=expression( rho*"="*-0.5) )
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if(showweightlines){  abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma2 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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print(ERC); print( 2/(1+2) ) ; 
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 2: mu2, sigma1
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plot( vw1 , vpccvar2 , main = expression( mu[1]*"=1, "*mu[2]*"=0 and "*sigma[1]*"="*sigma[2]*"=1") , type="l"  )
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text( x = 0.1 , y = 0.95*max(vpccvar2) , labels=expression( rho*"="*-0.5) )
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if(showweightlines){ abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 ) }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu2 , sigma = sigma1 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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#---------------------------------------------------------------------
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# !!!! Setup 4: mu1, sigma3
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plot( vw1 , vpccvar4 , main = "" , type="l",ylab="" )
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text( x = 0.1 , y = 0.95*max(vpccvar4)  , labels=expression( rho*"="*0) )
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# %C[2]CVaR <= 60% implies %C[1]CVaR >= 40%
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if(showweightlines){  abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )  }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma3 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 6: mu1, sigma4
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plot( vw1 , vpccvar6 , main = "", type="l"  )
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text( x = 0.1 , y = 0.95*max(vpccvar6)  , labels=expression( rho*"="*0) )
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if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )  }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma4 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 5: mu2, sigma3
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plot( vw1 , vpccvar5 , main = "" , type="l"  )
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text( x = 0.1 , y = 0.95*max(vpccvar5)  , labels=expression( rho*"="*0) )
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if(showweightlines){  abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu2 , sigma = sigma3 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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#----
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# !!!! Setup 7: mu1, sigma5
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plot( vw1 , vpccvar1 , main = expression( mu[1]*"="*mu[2]*"=0 and "*sigma[1]*"="*sigma[2]*"=1") ,
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         type="l",ylab="" )
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text( x = 0.1 , y = 0.95*max(vpccvar1) , labels=expression( rho*"="*0.5) )
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# %C[2]CVaR <= 60% implies %C[1]CVaR >= 40%
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if(showweightlines){ abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )}
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# Indicate ERC portfolio
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) ); return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma5 , alpha = alpha )[1] ) }
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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print(ERC); print( 1/2 ) ; 
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text( x = ERC , y = 0.5 , labels="ERC" , srt = srt )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 9: mu1, sigma6
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plot( vw1 , vpccvar9 , main = "" , type="l"  )
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text( x = 0.1 , y = 0.95*max(vpccvar9)  , labels=expression( rho*"="*0.5) )
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if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma6 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 8: mu2, sigma5
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plot( vw1 , vpccvar8 , main = "", type="l" ) 
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text( x = 0.1 , y = 0.95*max(vpccvar8)  , labels=expression( rho*"="*0.5) )
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if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu2 , sigma = sigma5 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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#----------------------------------------------------------------------------------------
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dev.off()
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####################### Focus on effect of mu and sigma
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postscript('sensitivity_rho50.eps') 
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par( mfrow = c(2,2) , las = 1 , mar=c(4.2,4.5,3,1.5)  , cex=1.2 , cex.main=1.2)
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#---------------------------------------------------------------------
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showweightlines = TRUE;
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# !!!! Setup 7: mu1, sigma5
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plot( vw1 , vpccvar7 , main = expression( mu[1]*"="*mu[2]*"=0, "*sigma[1]*"="*sigma[2]*"=1 and "*rho*"=0.5") , type="l",
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   ylab="Perc CVaR asset 1", xlab="Weight asset 1" )
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#text( x = 0.1 , y = 0.95*max(vpccvar7) , labels=expression( rho*"="*0.5) )
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# %C[2]CVaR <= 60% implies %C[1]CVaR >= 40%
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if(showweightlines){ abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )}
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# Indicate ERC portfolio
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) ); return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma5 , alpha = alpha )[1] ) }
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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print(ERC); print( 1/2 ) ; 
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#text( x = ERC , y = 0.5 , labels="ERC" , srt = srt )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 9: mu1, sigma6
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plot( vw1 , vpccvar9 , main = expression( mu[1]*"="*mu[2]*"=0, "* sigma[1]*"=1, "*sigma[2]*"=2 and "*rho*"=0.5") , type="l" ,
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    ylab="Perc CVaR asset 1", xlab="Weight asset 1" )
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#text( x = 0.1 , y = 0.95*max(vpccvar9)  , labels=expression( rho*"="*0.5) )
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if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma6 , alpha = alpha )[1] )
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}
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#h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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#text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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#par( mfrow = c(2,2) , las = 1 , mar=c(4,4.5,2,1.5)  , cex=1.2 , cex.main=1.2)
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# !!!! Setup 8: mu2, sigma5
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plot( vw1 , vpccvar8 , main = expression( mu[1]*"=1, "*mu[2]*"=0,  "*sigma[1]*"="*sigma[2]*"=1 and "*rho*"=0.5"), type="l" , 
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 ylab="Perc CVaR asset 1 ", xlab = "Weight asset 1") 
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#text( x = 0.1 , y = 0.95*max(vpccvar8)  , labels=expression( rho*"="*0.5) )
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if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu2 , sigma = sigma5 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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#text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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# !!!! Setup 8: mu1, sigma3
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plot( vw1 , vpccvar4 , main = expression(mu[1]*"="*mu[2]*"=0, "*sigma[1]*"="*sigma[2]*"=1 and "*rho*"=0"), type="l" , 
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 ylab="Perc CVaR asset 1 ", xlab = "Weight asset 1") 
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#text( x = 0.1 , y = 0.95*max(vpccvar8)  , labels=expression( rho*"="*0.5) )
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if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma3 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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#text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) }
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dev.off()
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# Plot 1 x 3
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postscript('sensitivity_rho50.eps') 
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showweightlines = TRUE
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par( mfrow = c(1,3) , las = 1 , mar=c(2.2,2.5,2,1.5) , cex=1 , cex.main=1)
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# !!!! Setup 7: mu1, sigma5
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plot( vw1 , vpccvar7 , main = expression( mu[1]*"="*mu[2]*"=0, "*rho*"="*0*", "*sigma[1]*"="*sigma[2]*"=1") , type="l" )
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text( x = 0.1 , y = 0.95*max(vpccvar7)  , labels=expression( rho*"="*0.5) )
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# %C[2]CVaR <= 60% implies %C[1]CVaR >= 40%
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if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
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# uniroot: find value for which it is 0.4 and 0.6
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objective = function( w1 ){
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   w = as.matrix( c(w1,1-w1) );
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   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma5 , alpha = alpha )[1] )
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}
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h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
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text( x = ERC , y = 0.5 , labels="ERC" )
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# uniroot: find value for which it is 0.4 and 0.6
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h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
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h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
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if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) ;
363
                     abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 ) }
364

365
# !!!! Setup 9: mu1, sigma6
366
plot( vw1 , vpccvar9 , main = main = expression( mu[1]*"=1, "*mu[2]*"=0, "*rho*"="*0*", "*sigma[1]*"="*sigma[2]*"=1") , type="l"  )
367
text( x = 0.1 , y = 0.95*max(vpccvar9)  , labels=expression( rho*"="*0.5) )
368
if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
369
# uniroot: find value for which it is 0.4 and 0.6
370
objective = function( w1 ){
371
   w = as.matrix( c(w1,1-w1) );
372
   return( h - percCompCVaR( w = w , mu = mu1 , sigma = sigma6 , alpha = alpha )[1] )
373
}
374
h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
375
text( x = ERC , y = 0.5 , labels="ERC" )
376
# uniroot: find value for which it is 0.4 and 0.6
377
h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
378
h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
379
if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) ;
380
                     abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 ) }
381

382

383
# !!!! Setup 8: mu2, sigma5
384
plot( vw1 , vpccvar8 , main = "", type="l" ) 
385
text( x = 0.1 , y = 0.95*max(vpccvar8)  , labels=expression( rho*"="*0.5) )
386
if(showweightlines){   abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )   }
387
# uniroot: find value for which it is 0.4 and 0.6
388
objective = function( w1 ){
389
   w = as.matrix( c(w1,1-w1) );
390
   return( h - percCompCVaR( w = w , mu = mu2 , sigma = sigma5 , alpha = alpha )[1] )
391
}
392
h=0.5; ERC = uniroot( objective , lower = 0 , upper = 1)$root
393
text( x = ERC , y = 0.5 , labels="ERC" )
394
# uniroot: find value for which it is 0.4 and 0.6
395
h=0.4 ; v1 = uniroot( objective , lower = 0 , upper = 1)$root
396
h=0.6; v2 = uniroot( objective , lower = 0 , upper = 1)$root
397
if(showweightlines){ abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) ;
398
                     abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 ) }
399

400
# !!!! Setup 4: mu1, sigma3
401
plot( vw1 , vpccvar4 , main = expression( mu[1]*"="*mu[2]*"=0, "*rho*"="*0*", "*sigma[1]*"="*sigma[2]*"=1") , 
402
      type="l",ylab="Perc CVaR asset 1 \n in function of its weight" )
403
# %C[2]CVaR <= 60% implies %C[1]CVaR >= 40%
404
# uniroot: find value for which it is 0.4 and 0.6
405
objective = function( w1 ){
406
   w = as.matrix( c(w1,1-w1) );
407
   return( 0.5 - percCompCVaR( w = w , mu = mu1 , sigma = sigma3 , alpha = alpha )[1] )
408
}
409
v1 = uniroot( objective , lower = 0 , upper = 1)$root
410
objective = function( w1 ){
411
   w = as.matrix( c(w1,1-w1) );
412
   return( 0.5 - percCompCVaR( w = w , mu = mu1 , sigma = sigma3 , alpha = alpha )[1] )
413
}
414
v2 = uniroot( objective , lower = 0 , upper = 1)$root
415
if(showweightlines){  abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 ) ;
416
                    abline( h = 0.4 , lty = 3) ; abline( h = 0.6 , lty = 3 )  }
417
abline( h=0.5 , lty = 3 )
418

419
# !!!! Setup 5: mu2, sigma3
420
plot( vw1 , vpccvar5 , main = expression( mu[1]*"=1, "*mu[2]*"=0, "*rho*"="*0*", "*sigma[1]*"="*sigma[2]*"=1") , 
421
     type="l" , ylab = "Perc CVaR asset 1 \n in function of its weight" )
422
# uniroot: find value for which it is 0.4 and 0.6
423
objective = function( w1 ){
424
   w = as.matrix( c(w1,1-w1) );
425
   return( 0.5 - percCompCVaR( w = w , mu = mu2 , sigma = sigma3 , alpha = alpha )[1] )
426
}
427
v1 = uniroot( objective , lower = 0 , upper = 1)$root
428
objective = function( w1 ){
429
   w = as.matrix( c(w1,1-w1) );
430
   return( 0.5 - percCompCVaR( w = w , mu = mu2 , sigma = sigma3 , alpha = alpha )[1] )
431
}
432
v2 = uniroot( objective , lower = 0 , upper = 1)$root
433
if(showweightlines){   abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 )   }
434
abline( h=0.5 , lty = 3 )
435

436
# !!!! Setup 6: mu1, sigma4
437
plot( vw1 , vpccvar6 , main = expression( mu[1]*"="*mu[2]*"=0, "*rho*"=0, "* sigma[1]*"=1, "*sigma[2]*"=2   "), 
438
    type="l" , ylab = "Perc CVaR asset 1 \n in function of its weight" )
439
# uniroot: find value for which it is 0.4 and 0.6
440
objective = function( w1 ){
441
   w = as.matrix( c(w1,1-w1) );
442
   return( 0.5 - percCompCVaR( w = w , mu = mu1 , sigma = sigma4 , alpha = alpha )[1] )
443
}
444
v1 = uniroot( objective , lower = 0 , upper = 1)$root
445
objective = function( w1 ){
446
   w = as.matrix( c(w1,1-w1) );
447
   return( 0.5 - percCompCVaR( w = w , mu = mu1 , sigma = sigma4 , alpha = alpha )[1] )
448
}
449
v2 = uniroot( objective , lower = 0 , upper = 1)$root
450
if(showweightlines){   abline( v = v1 , lty = 3) ; abline( v = v2 , lty = 3 )   }
451
abline( h=0.5 , lty = 3 )
452

453
dev.off()
454

455
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