Asset Pricing [Part 2f : Sketching The Traditional Framework]

To arbitrarily round off this series of posts, along which a sketch of the traditional asset pricing framework has been attempted,  a dashboard of simple plots shall be made to summarise some of the implications of extending the 2-asset base case with an additional risky asset. Starting with the base case of 2 risky assets and one riskless asset, we shall once again draw the minimum variance frontier as well as the capital allocation line to locate the tangency portfolio. A pie chart will conveniently summarise the tangency portfolio weights associated with these 2 assets. The base case with two risky assets will then be extended by including additional risky assets, one at a time , holding the variance of each asset’s return constant while varying the mean return of that asset. Assets will be added sequentially to the base case, so that at any time only 3 risky assets are in the universe along with one riskless asset. Barcharts will illustrate the returns and risks on tangency portfolios for each of the extended scenarios. An area chart will summarise how asset weights vary in tangency portfolios across scenarios.

###################################################################################################
###################################################################################################
#Supplement 2
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###################################################################################################</pre>
idx <- seq(from=-0.5,to=0.2,length.out=10)
weights.mat <-ret.mat <- var.mat <- NULL
col <- adjustcolor(col=c('dark green','green','light green'),alpha.f=0.5)
front.col <- rainbow(length(idx))

#base case
 two.asset.data <- list()
 two.asset.data$mean.ret <- matrix(c(0.800,0.0500),nrow=1,ncol=2,dimnames=list(c('mean return'),c('asset 1','asset 2')))
 two.asset.data$cov.matrix <- matrix(c(0.0256,0,0,0.0144),nrow=2,ncol=2,byrow=T,dimnames=list(c('asset 1','asset 2'),c('asset 1','asset 2')))
 two.asset.data$risk.free <- 0.02

base<-Frontiers(two.asset.data)

windows()
 layout(matrix(c(1,1,1,2,1,1,1,3,1,1,1,4,5,5,5,5),4,4,byrow=T),height=c(0.20,0.20,0.2,0.3))
 par(mai=c(0.7,0.7,0.3,0.2))
 Frontier.Draw(two.asset.data,base,'red','new',lty=1,'Base Case\nextended with new asset')
 CAL.Draw(two.asset.data,base,'red',lty=1,tit=0)
 legend('bottom',fill=front.col,legend=round(idx,3),ncol=5,bg='white',bty='n',cex=0.9,title='New asset (with mean defined relative to base portfolio returns)')
 legend('topleft',legend=c('Tangency portfolios','Assets'),pch=c(17,16),bg='white',bty='o',cex=0.9,ncol=1)
 abline(h=-0.065)
 abline(h=-0.12)

for(i in 1:length(idx))
 {
 z.data <- list()
 z.data$mean.ret <- matrix(c(0.800,0.0500,0.8+idx[i]),nrow=1,ncol=3,dimnames=list(c('mean return'),c('asset 1','asset 2','asset 3')))
 z.data$cov.matrix <- matrix(c(0.0256,0,0,0,0.0144,0,0,0,0.02),nrow=3,ncol=3,byrow=T,dimnames=list(c('asset 1','asset 2','asset 3'),c('asset 1','asset 2','asset 3')))
 z.data$risk.free <- 0.02

 front<-Frontiers(z.data)
 Frontier.Draw(z.data,front,col=front.col[i],'add',2,legend=1)
 CAL.Draw(z.data,front,col=front.col[i],lty=1,tit=0)

weights.mat <- cbind(weights.mat,front$tang.weights)
 ret.mat <- cbind(ret.mat,front$tang.ret)
 var.mat <- cbind(var.mat,front$tang.risk)
 }

rownames(weights.mat) <- c('Asset 1','Asset 2','Asset 3')
 colnames(ret.mat) <- round(idx,2)
 colnames(var.mat) <-round(idx,2)
 weights.mat <- t(weights.mat)
 base.weights <- base$tang.weights

ret.mat <- matrix(ret.mat)
 var.mat <- matrix(var.mat)

par(mai=c(0.3,0.3,0.3,0.3))
 pie(cex.main=0.75,main='Base case\ntangency weights',base.weights,col=col[1:2],labels=c(paste('Asset 1\n',base.weights[1]*100,'%'),paste('Asset 2\n',base.weights[2]*100,'%')),cex=0.75,radius=0.85)

par(mai=c(0.5,0.3,0.3,0.3))
 barplot(axes=T,beside=T,horiz=F,ret.mat,cex.axis=0.75,cex.lab=0.75,cex.main=0.75,main='Tangency\nPortfolio Returns',col=front.col,xaxlab=c(1:10))

 par(mai=c(0.7,0.3,0.2,0.3))
 barplot(axes=T,beside=T,horiz=F,var.mat,cex.axis=0.75,cex.lab=0.75,cex.main=0.75,main='Tangency\nPortfolio Risk',col=front.col,xlab='Assets')

par(mai=c(0.7,0.7,0,0.3))
 stackpoly(xlim=c(1,length(idx)+1.5),axis4=F,cex.main=0.85,cex.lab=0.75,cex.axis=0.75,weights.mat,xaxlab=round(idx,2),border='white',stack=F,xlab='Deviations from base',ylab='Weights',lwd=1,col=col)
 legend('right',fill=col,legend=colnames(weights.mat),ncol=1,bg='white',bty='n',cex=0.85,title='Tangency\nPortfolio Weights')

The resulting dashboard of plots follows.

gg

The vertical alignment of the new assets implies that we are varying the mean returns while keeping their risk constant. Each of the minimum variance frontiers and capital allocation lines are drawn for 3 risky assets and one riskless asset. The mean return of each new asset is chosen as a deviation from the mean return of the second risky asset (Asset 2) of the base case scenario. The inputs associated with the assets in the base case scenario remain unchanged throughout. By varying only the mean returns of the additional risky assets and holding constant all other factors across the 10 scenarios, we can examine how optimal asset weights evolve according to return deviations. In the context of the base case scenario, the optimal risky portfolio would be 93.6% invested in asset 1 and 6.4% invested in asset 2. This is intuitive given that asset 1 has a mean return of 0.8 and a risk of 0.0256 while asset 2 enjoys a mean return of 0.05 and a lower risk of 0.0144. Evidently the lower risk of asset 2 does not sufficiently compensate for the small return it generates relative to asset 1 (0.05 vs 0.80). Visually this is corroborated by the proximity of the optimal risky portfolio relative to asset 1 on the MVF spanning the 2 risky assets in the frontier plot. Every time we add a new risky asset to the scenario, we ensure that it has a lower negative deviation or a higher positive deviation from asset 2’s mean return ( essentially from -0.5 to +0.2 relative to the second asset’s mean return). The risk of each and every one of those assets lies between those of the base assets (as evinced from the frontier plot). The barplots confirm the intuition that by adding risky assets with increasingly higher mean returns to the base case scenario, the return on the tangency portfolio increases. Similarly, as we reduce the negative deviations/enhance the positive deviations of the third asset’s mean return relative to the return of the base asset, the optimal weight shifts from base assets to asset 3.

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