What is the stock's beta?

Could Someone please solve the following exam question in relation to the Beta & Correlation of stocks?

• Your stock broker has provided the following historical information on the stocks of two companies: Icarus Ltd and Taurus Co. The variance of Icarus’ returns is 7.8%, while the variance of Taurus’ returns is 3.5%. Expected return on Icarus is 12% and expected return on Taurus is 9%. The correlation coefficient between the two stocks’ returns is 0.36. The correlation between Icarus and the market is 0.60, while the correlation between Taurus and the market is 0.70. In addition, you know that the variance of the market index is 1.2%. (i) Calculate the betas of Icarus and Taurus; (10 marks) (ii) Compute the variance of the portfolio consisting of 70% of Icarus and 30% of Taurus; (10 marks) (iii) Calculate the expected return of the above portfolio; (10 marks) (iv) Calculate the beta of the investment evenly split between Icarus stocks, Taurus stocks, market portfolio, and a risk-free asset. (15 marks)

• Answer:

  Assuming standard CAPM model: (i) β[x|m] = Cov[x/m]/D[m] = ρ[x/m]•σ[x]/σ[m] ρ[x/y] = Cov[x/y]/(σ[x]•σ[y]) D=σ² σ=√D ρ[i/m] = 0.60 ρ[t/m] = 0.70 D[i] = 7.8% D[t] = 3.5% D[m] = 1.2% σ[i] = √7.8% ≈ 2.79% σ[t] = √3.5% ≈ 1.87% σ[m] = √1.2% ≈ 1.10% β[i|m] = 0.60•√7.8%/√1.2% ≈ 1.5297 β[t|m] = 0.70•√3.5%/√1.2% ≈ 1.1955 (ii) D[X+Y] = D[X]+D[Y]+2•Cov[X|Y] = D[X]+D[Y]+2•ρ[X|Y]•√D[X]•√D[Y] D[A•X]=A²•D[X] Cov[A•X|B•Y]=A•B•Cov[X|Y] D[A•X+B•Y] = A²•D[X] + B²•D[Y] + 2•A•B•Cov[X|Y] D[A•X+B•Y] = A²•D[X] + B²•D[Y] + 2•A•B•ρ[X|Y]•σ[X]•σ[Y] D[p] = 0.7²•7.8+0.3²•3.5+ 2•0.7•0.3•(√7.8)•(√3.5) ≈ 4.927 (iii) E[A•X+B•Y] = A•E[X]+B•E[Y] r[p] = 0.7•12+0.3•9 = 11.1 (iv) β[f|m] = 0 β[m|m] = 1 β[i|m] ≈ 1.5297 β[t|m] ≈ 1.1955 β[∑]=∑(w•β) β[p|m] = β[f/4+m/4+i/4+t/4] ≈ 0/4 + 1/4 + 1.5297/4 + 1.1955/4 ≈ 0.9313 But as you already know - simple CAPM/beta is dead, used only if you don't have better alternatives or too tired/lazy or don't have enough time to make decision.