Question

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

Expected return on X= 10%. Using CAPM model, 10%= 2%+ Beta(X)*(10%-2%). So, Beta(X) is 1.

Expected return on Y= 6%. Using CAPM model, 6%= 2%+Beta(Y)*(10%-2%). So, Beta(Y)= 0.5

We know that standard deviation of a riskfree asset is 0 and correlation between a risky asset and a riskfree asset is also 0. So, standard deviation (sd) of a portfolio of a risky asset and a riskfree asset is equal to weight of risky asset*standard deviation of risky asset.

So, For sd of portfolio of X and riskfree asset is 3=0.5*sd(X). So, sd(X)= 6.

For sd of portfolio of Y and riskfree asset is 1= 0.25*sd(Y). So, sd(Y)= 4.

For sd of portfolio of market and riskfree asset is 1=0.5*sd(market). So, sd(market)=2.

Given variance of equally weighted portfolios X and Y is 1. So, 1= 0.5^2*6^2+0.5^2*4^2+2*0.5*0.5*correlation(X,Y)*6*4. On solving, we get Correlation(X,Y)= -1.

Variance of portfolio which equally invests in X,Y and riskfree asset can be calculated by the formula: 0.33^2*6^2+0.33^2*4^2+0.33^2*0^2+(2*0.33*0.33*(-1)*6*4)+0+0= 0.44

To calculate Beta of the equally weighted portfolio in X,Y and riskfree asset, lets first calculate the expected return. It will be (1/3)*10%+(1/3)*6%+(1/3)*2%= 6%. So, using CAPM model, 6%= 2%+ Beta*(10%-2%). So, Beta of the required portfolio is 0.5