Answer to Question #121366 in Financial Math for Nisha

Question

Let ( , ) ρ = w1 w2
be a portfolio of two securities. Find the value of w1
and w2
in the following
situations:
i) 1 ρ12 = − and ρ is risk-free.
ii) σ1 = σ2
and variance P is minimum.
iii) Variance on P is minimum and 2 ,5.0 ρ12 = − σ1 = and 3 σ2 = .

Answer 1

i. ρ12 = − and ρ is risk-free.

Rp = w₁r₁ + w₂r₂

Variance of the portfolio return

σ²p = w₁σ²₁ + w₂σ²₂ + 2w₁w₂σ₁₂

i) ρ12 = − and ρ is risk-free.

Value of w₁, w₂

w₁r₁ = Rp - w₂r₂

w₁ = Rp /r₁ - w₂r₂/r₁

w₂ = Rp /r₂ – w₁r₁/r₂

ii) σ1 = σ2 and variance P is minimum.

σ = √(w₁σ²₁ + w₂σ²₂ + 2w₁w₂σ₁₂)

σ² = √[(w₁σ²₁ + w₂σ²₂ + 2w₁w₂σ₁₂)]²

σ² = w₁σ²₁ + w₂σ²₂ + 2w₁w₂σ₁₂

w₁σ²₁ + 2w₁w₂σ₁₂ = σ² - w₂σ²₂  

w₁(σ²₁ + 2w₂σ₁₂) = σ² - w₂σ²₂  

w₁= (σ²₁- w₂σ²₂)/ (σ²₁ + 2w₂σ₁₂)

w₂= (σ²₂- w₁σ²₂)/ (σ²₂ + 2w₁σ₁₂)

The answer is the same for different three situations because the portfolio with the two securities hence the minimum variance frontier (MVF) for the three different scenarios portrays a perfect negative correlation (ρ12 = −).