Minimum Variance Porfolio

Overview


The minimum variance portfolio is the portfolio that achieves the minimum variance, or smallest risk among possible portfolios. It ignores the expected returns of the assets. If any of the assets has zero variance, then the minimum variance portfolio will have zero variance.

Derivation


For this derivation, we set the portfolio weights as a column vector {% w %}, where {% w[i] %} is the ith asset in the portfolio. Then we take {% \textbf{i} %} is a column vector of 1's. (see qian)

Then we wish to minimize the variance of the portfolio, which is calculated as
{% Minimize \; \frac{1}{2} \textbf{w}^T \Sigma \textbf{w} %}
{% subject \; to \; \textbf{w}^T \textbf{i} = 1 %}
(that is, the sum of the weights should equal 1)

Because this is a constrained optimization, it can be solved using the Lagrange Multipliers technique. First we specify the Lagrangian.
{% Lagrangian \; L = \frac{1}{2} \textbf{x}^T \sigma \textbf{w} - \lambda (\textbf{w}^T \textbf{i} - 1) %}
Taking derivatives yields

{% \Sigma \textbf{w} = \lambda \textbf{i} = 0 %}
{% \textbf{w} = \lambda \Sigma^{-1} \textbf{i} %}
{% \lambda = 1 / (\textbf{i}^T \Sigma ^{-1} \textbf{i} ) %}
{% \textbf{w}_{min} = \frac{ \Sigma ^{-1} \textbf{i} }{ \textbf{i}^T \Sigma ^{-1} \textbf{i}} %}

Implementation


The following code using the linear-algebra.mjs library to compute the minimum variance portofolio solution. 


let la = await import('/lib/linear-algebra/v1.0.0/linear-algebra.mjs');

let sigma = [[0.2,0.05],[0.05, 0.12]];
let i = [[1],[1]]
let sigmaInverse = la.inverse(sigma);
let denom = la.multiply(la.transpose(i), la.multiply(sigmaInverse, i));
let numerator = la.multiply(sigmaInverse, i);
let wMin = la.multiply(1/denom[0][0], numerator);
Try it!

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