Question

1) Rf=2%, Rm= 10%

According to the CAPM model

Expected Return X= 10

Expected Return Y= 6

Portfolios:

A portfolio of stock X and rf with a 50% equal weighted investment yielded a STANDARD DEVIATION of 3.

A portfolio of stock Y and rf with a 25% investment in stock Y yielded a STANDARD DEVIATION of 1.

A portfolio of the market and rf with a 50% investment in the market yielded a STANDARD DEVIATION of 1.

An equally weighted portfolio of stock X and Y yielded a variance of 1.

Find the variance of the portfolio that invests EQUALLY WEIGHTED in X, Y, and Rf.

Find the BETA of the portfolio that invests EQUALLY WEIGHTED in X, Y, and Rf.

Answer 1

Given that Return on X=10%,Rf=2% and Rm=10%. Using CAPM Model, R=Rf+Beta(Rm-Rf), On Substituting, 10%=2%+Beta*(10%-2%). We get Beta of X=1.

Given that Return on Y=6%,Rf=2% and Rm=10%. Using CAPM Model, R=Rf+Beta(Rm-Rf), On Substituting, 6%=2%+Beta*(10%-2%). We get Beta of Y=0.5

Given in a portfolio of X and Rf with equal weights, Standard deviation of portfolio is 3. we know that with Riskfree asset in portfolio, Standard deviation of portfolio = weight of the asset*Standard deviation of the asset. So, 3= 0.5*sd(X). which gives sd(X)= 6

Given in a portfolio of Y and Rf with 25% in Y, Standard deviation of portfolio is 1. we know that with Riskfree asset in portfolio, Standard deviation of portfolio = weight of the asset*Standard deviation of the asset. So, 1= 0.25*sd(Y). which gives sd(Y)= 4

Given in a portfolio of Market and Rf with equal weights, Standard deviation of portfolio is 1. we know that with Riskfree asset in portfolio, Standard deviation of portfolio = weight of the asset*Standard deviation of the asset. So, 1= 0.5*sd(M). which gives sd(M)= 2

Given equally weighted portfolio of X and Y gives variance of 1. So, 1=W(X)^2*sd(X)^2+W(Y)^2*sd(Y)^2+(2*W(X)*sd(X)*W(Y)*sd(Y)*Corelation(X,Y)). where W(X) and W(Y) are weights of X and Y. In sunstituting, we get 1=0.5^2*6^2+0.5^2*4^2+(2*0.5*6*0.5*4*Corelation(X,Y)). So, 1=13+12*Correlation(X,Y). So, Correlation(X,Y)= -1

Variance of portfolio that invests equally in X,Y and Rf is calculated as W(X)^2*sd(X)^2+W(Y)^2*sd(Y)^2+W(Rf)^2*sd(Rf)^2+(2*W(X)*sd(X)*W(Y)*sd(Y)*Corelation(X,Y)+2*W(Y)*sd(Y)*W(Rf)*sd(Rf)*Corelation(Y,Rf)+2*W(X)*sd(X)*W(Rf)*sd(Rf)*Corelation(X,Rf)). we know that standard deviation of Risk free asset is 0. So, sd(Rf)=0. and correlation with riskfree asset is also 0. So, Corelation(Y,Rf) and Corelation(X,Rf) are also 0. So, on substituting, we get variance of portfolio = 0.33^2*6^2+0.33^2*4^2+0+(2*0.33*6*0.33*4*-1)= 0.44

So, Variance of portfolio that invests equally in X,Y and Rf is 0.44

Beta of a portfolio can be calculated as weighted average of Beta of the assets in the portfolio.

So, Beta of portfolio that invests equally in X,Y and Rf is 0.33*1+0.33*0.5+0.33*0= 0.5

So, Beta of portfolio that invests equally in X,Y and Rf is 0.5