Minimum Variance Portfolio Assistance: derive an expression for the share of wealth allocated to a risky asset

[doubleuson]

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:
r_p = w₁r₁ + w₂r₂ {2}
Hence, substituting {1} into {2}:
r_p = w₁r₁ + (1 - w₁)r₂ {3}

Portfolio variance is:
σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂ {4}

Knowing that:
Corr₁₂ = ρ₁₂ = σ₁₂/σ₁σ₂ {5}
Sub {5} into {4}
σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂*σ₁₂/σ₁σ₂
σ_p² = 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.

 

[JeffM]

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:
r_p = w₁r₁ + w₂r₂ {2}
Hence, substituting {1} into {2}:
r_p = w₁r₁ + (1 - w₁)r₂ {3}

Portfolio variance is:
σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂ {4}

Knowing that:
Corr₁₂ = ρ₁₂ = σ₁₂/σ₁σ₂ {5}
Sub {5} into {4}
σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂*σ₁₂/σ₁σ₂
σ_p² = 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.

It has been decades since I worked with this stuff, but I think, once you simplify the horrible notation economists like to use, that this is a simple minimization problem. However, given my lack of practice with this stuff, let's view this as a joint project. Now assuming

[MATH]r_1,\ r_2,\ \sigma_1^2,\ \sigma_2^2, \text { and } \sigma_{1,2}[/MATH]
are all given, we can simplify our notation by replacing them with a, b, c, d, and e respectively.

[MATH]f = \sqrt{c} \sqrt{d} e.[/MATH]
Because [MATH]w_1 +w_2 = 1[/MATH]
we can further simplify our notation by replacing those two variables with w and ( 1 - w) respectively.

And, in a final simplification, we can say [MATH]\sigma_p^2 = v.[/MATH]
Now equation 6 becomes

[MATH]v = w^2c + (1 - w)^2d + 2w(1 - w)f \implies[/MATH]
[MATH]v = w^2c + (1 - 2w + w^2)d + 2(w - w^2)f \implies[/MATH]
[MATH]v = w^2c + w^2d - 2w^2f + 2w^2(f - d) + d.[/MATH]
[MATH]\therefore \dfrac{dv}{dw} = 2w(c + d - 2f) + 2(f - d).[/MATH]
[MATH]\therefore \dfrac{dv}{dw} = 0 \implies w = \dfrac{2(d - f)}{2(c + d - 2f)} = \dfrac{d - f}{c + d - 2f}.[/MATH]
[MATH]\therefore w = \dfrac{\sigma_2^2 - \sigma_1\sigma_2\sigma_{1,2}}{\sigma_1^2 + \sigma_2^2 - 2\sigma_1 \sigma_2 \sigma_{1,2}}[/MATH]
Please check the algebra. But I think that is all there is to it. If you have questions, please feel comfortable in challenging what I have done.