# Portfolio Variance for two asset portfolio

- For two-asset portfolio
- Var(w
_{A}k_{A}+ w_{B}k_{B}) = w_{A}^{2}σ_{A}^{2}+ w_{B}^{2}σ_{B}^{2}+ 2 w_{A}w_{B}σ_{A}σ_{B}ρ_{AB}

- Var(w
- Where ρ is correlation coefficient between A and B
- w
_{A},w_{B}are weights of the asset A and B- If ρ =1
- Var(w
_{A}k_{A }+ w_{B}k_{B}) = (w_{A}σ_{A}+ w_{B}σ_{B})^{2}

- Var(w
- If ρ <1
- Var(w
_{A}k_{A}+ w_{B}k_{B}) < (w_{A}σ_{A}+ w_{B}σ_{B})^{2}

- Var(w

- If ρ =1
- So there is a risk reduction from holding a portfolio of assets if assets do not move in perfect unison

Example

E(RA) = 10%, σ_{A} = 20%, E(R_{B}) = 10%, σ_{B} = 20%

Assume the weights to be 50 % for A & B

Calculate portfolio returns when:

Case 1: ρ_{AB} = 1,

Case 2: ρ_{AB} = 0,

Case 3: ρ_{AB} = -1

Solution

Expected return =10%*0.5+10%*0.5 = 10% (in all three cases)

Variance

Case 1: (0.52)*(0.22) + (0.52)*(0.22) + 2*0.5*0.5*0.2*0.2*1 = 0.04

Case 2: (0.52)*(0.22) + (0.52)*(0.22) + 2*0.5*0.5*0.2*0.2*0 = 0.02

Case 3: (0.52)*(0a.22) + (0.52)*(0.22) + 2*0.5*0.5*0.2*0.2*-1 = 0.00

Variance

Case 1: (0.52)*(0.22) + (0.52)*(0.22) + 2*0.5*0.5*0.2*0.2*1 = 0.04

Case 2: (0.52)*(0.22) + (0.52)*(0.22) + 2*0.5*0.5*0.2*0.2*0 = 0.02

Case 3: (0.52)*(0a.22) + (0.52)*(0.22) + 2*0.5*0.5*0.2*0.2*-1 = 0.00