Financial Market Theory Thursday, October 5, 2017 Professor Edwin T Burton
Diagram with 2 Assets Asset 2 (μ2, σ2) Asset 1 (μ1, σ1) Mean Where are the portfolios that can be formed from these two assets? Asset 1 (μ1, σ1) Standard Deviation = √(Variance) October 5, 2017
All Portfolios That Can Be Constructed: 𝑃= α ∗ 𝑎𝑠𝑠𝑒𝑡 1 𝑝𝑙𝑢𝑠 1 − α ∗(𝑎𝑠𝑠𝑒𝑡 2) where 0 ≤ α ≤ 1 𝑀𝑒𝑎𝑛 𝑜𝑓 𝑃= α times Mean of Asset 1 plus (1 – α) times Mean of Asset 2 October 5, 2017
Variance of a Portfolio with two assets P2 = (P - P)2 n = {α(X1- 1) + (1-α)(X2 - 2)}2 n October 5, 2017
After some mild heavy lifting: σ 𝑃 2 = α 2 σ 1 2 + 1−α 2 σ 2 2 +2α 1−α Cov(1,2) Note that Cov(1,2) ≡ σ1,2 So: σ 𝑃 2 = α 2 σ 1 2 + 1−α 2 σ 2 2 +2α 1−α σ 1,2 October 5, 2017
Make use of correlation coefficient: σ 𝑃 2 = α 2 σ 1 2 + 1−α 2 σ 2 2 +2α 1−α σ 1,2 ρ 1,2 ≡ σ 1,2 σ 1 σ 2 σ 𝑃 2 = α 2 σ 1 2 + 1−α 2 σ 2 2 +2α 1−α ρ 1,2 σ 1 σ 2 October 5, 2017
σ 𝑃 2 = α 2 σ 1 2 + 1−α 2 σ 2 2 +2α 1−α ρ 1,2 σ 1 σ 2 What happens when ρ1,2 = 0 ? σ 𝑃 2 = α σ 1 + 1−α σ 2 2 Now, take square roots of both sides: σ 𝑃 = α σ 1 + 1−α σ 2 If α= ½, then σ 𝑃 = 1 2 σ 1 + 1 2 σ 2 October 5, 2017
So, the half/half case is: σ 𝑃 = 1 2 σ 1 + 1 2 σ 2 So, the half/half case is: Mean μ2 Asset 2 (μ2, σ2) Asset 1 (μ1, σ1) μ1 σ1 ½ σ1 + ½ σ2 σ2 October 5, 2017
If ρ < 1 σ 𝑃 2 = α 2 σ 1 2 + 1−α 2 σ 2 2 +2α 1−α σ 1,2 σ 𝑃 2 = α 2 σ 1 2 + 1−α 2 σ 2 2 +2α 1−α σ 1,2 The right hand side will be smaller than before This implies a smaller variance of P And a smaller standard deviation October 5, 2017
So, if σ < 1 σ1 ½ σ1 + ½ σ2 σ2 Asset 2 (μ2, σ2) Asset 1 (μ1, σ1) P will lie to the left of the Line joining the Assets Mean μ2 Asset 2 (μ2, σ2) μ1 Asset 1 (μ1, σ1) σ1 ½ σ1 + ½ σ2 σ2 October 5, 2017
Portfolio Choice σ σ Mean Green Curve is Markowitz’s “Efficient Portfolio” set More risk Less risk σ σ October 5, 2017
October 5, 2017