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Portfolio Variance for two asset portfolio

  • For two-asset portfolio
    • Var(wAkA+ wBkB) = wA2 σA2 + wB2 σB2 + 2 wA wB σA σB ρAB
  • Where ρ is correlation coefficient between A and B
  • wA ,wB  are weights of the asset A and B
    • If ρ =1
      • Var(wAkA + wBkB) = (wAσA + wBσB)2
    • If ρ <1
      • Var(wAkA+ wBkB) < (wAσA+ wBσB)2
  • So there is a risk reduction from holding a portfolio of assets if assets do not move in perfect unison

Example

E(RA) = 10%, σA = 20%, E(RB) = 10%, σB = 20%
 
Assume the weights to be 50 % for A & B
 
Calculate portfolio returns when:
Case 1: ρAB = 1,
Case 2: ρAB = 0,
Case 3: ρAB = -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





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