doubleuson
New member
- Joined
- Oct 10, 2018
- Messages
- 2
Howdy everyone,
I've been having a slight issue with one particular minimum variance portfolio question:
The task is to derive an expression for the share of wealth allocated to a risky asset (1) in a portfolio with the minimum possible variance:
This is what is known:
Two risky assets: 1 and 2
Returns: r1 and r2
Variances: σ12 and σ22
Covariance: σ12
------
So far, I have figured out the following:
The weighted share of asset 1 is {1}:
w2 = 1 - w1 {1}
Portfolio return is:
rp = w1r1 + w2r2 {2}
Hence, substituting {1} into {2}:
rp = w1r1 + (1 - w1)r2 {3}
Portfolio variance is:
σp2 = w12σ12 + w22σ22 + 2w1w2σ1σ2ρ12 {4}
Knowing that:
Corr12 = ρ12 = σ12/σ1σ2 {5}
Sub {5} into {4}
σp2 = w12σ12 + w22σ22 + 2w1w2σ1σ2*σ12/σ1σ2
σp2 = w12σ12 + w22σ22 + 2w1w2σ12 {6}
Then, I guess for the portfolio to have the minimum possible variance one needs to optimize {6} subject to the constraint of {1}, giving a Lagrangean function:
y = w12σ12 + w22σ22 + 2w1w2σ12 + λ(w1 +w2 - 1) {7}
Such that:
∂y/∂w1 = 0
∂y/∂w2 = 0
∂y/∂λ = 0
And from here I am stuck. How am I supposed to use the above calculations, especially {7}, to get an expression for w1 ensuring it has minimum portfolio variance?
Thanks in advance.
I've been having a slight issue with one particular minimum variance portfolio question:
The task is to derive an expression for the share of wealth allocated to a risky asset (1) in a portfolio with the minimum possible variance:
This is what is known:
Two risky assets: 1 and 2
Returns: r1 and r2
Variances: σ12 and σ22
Covariance: σ12
------
So far, I have figured out the following:
The weighted share of asset 1 is {1}:
w2 = 1 - w1 {1}
Portfolio return is:
rp = w1r1 + w2r2 {2}
Hence, substituting {1} into {2}:
rp = w1r1 + (1 - w1)r2 {3}
Portfolio variance is:
σp2 = w12σ12 + w22σ22 + 2w1w2σ1σ2ρ12 {4}
Knowing that:
Corr12 = ρ12 = σ12/σ1σ2 {5}
Sub {5} into {4}
σp2 = w12σ12 + w22σ22 + 2w1w2σ1σ2*σ12/σ1σ2
σp2 = w12σ12 + w22σ22 + 2w1w2σ12 {6}
Then, I guess for the portfolio to have the minimum possible variance one needs to optimize {6} subject to the constraint of {1}, giving a Lagrangean function:
y = w12σ12 + w22σ22 + 2w1w2σ12 + λ(w1 +w2 - 1) {7}
Such that:
∂y/∂w1 = 0
∂y/∂w2 = 0
∂y/∂λ = 0
And from here I am stuck. How am I supposed to use the above calculations, especially {7}, to get an expression for w1 ensuring it has minimum portfolio variance?
Thanks in advance.
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