Question

There are two assets, 1 and 2, with Betas: β₁=0.8, β₂=1.2 and Alphas: ɑ₁=0.001, ɑ₂=0.002. The variance of the respective idiosyncratic components are: σ²_ε1=0.004, σ²_ε2=0.001. The variance of the market excess return is: σ2_m=0.016. You form a portfolio P with shares of 1 and 2 equal to w₁=0.8 and w₂=0.2 respectively.

1. What is the Alpha of the portfolio (ɑ_P)?

2. What is the Beta of the portfolio (β_P)?

3. What is the variance of the portoflio idiosyncratic component (σ²_εP)?

4. What is the variance of the excess return on the portfolio?

Answer 1

	Asset 1	Asset 2
beta	0.8	1.2
alpha	0.001	0.002
variance of idiosyncratic risk	0.004	0.001
SD of idiosyncratic risk	0.0632	0.0316
portfolio weight	0.8	0.2
variance of the market excess return	0.016
portfolio alpha	0.0012	=0.8*0.001+0.2*0.002
portfolio beta	0.88	=0.8*0.8+0.2*1.2
Variance of portfolio idiosyncratic risk	0.0026	=(0.8^2*0.004+ 0.2^2*0.001)
SD portfolio idiosyncratic risk	0.0510	=sqrt(variance of portfolio idiosyncratic risk)
Variance of portfolio excess return	0.0186	=0.016+0.0026

portfolio alpha= weighted sum of the individual asset alphas
portfolio beta= weighted sum of the individual asset beta
Variance of portfolio idiosyncratic risk= weighted sum of the individual asset idiosyncratic risk. This is derivation of the portfolio variance, where the correlation between the assets is zero. This is because, the idiosyncratic risk is not correlated with any other asset or market risk
Variance of portfolio excess return= Variance of market excess return + Variance of portfolio excess return. This is because, the idiosyncratic risk is not correlated with any other asset or market risk. This again a special case where the partial derivative of idiosyncratic risk and variance of excess return is equal to one.