The Covariance Matrix and Quadratic Forms

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The Covariance Matrix

Let's take a moment to learn a compact way to represent the portfolio variance using matrices and vectors.

Remember that the portfolio variance we calculated for our two-stock portfolio was:

\sigma_P^2 = x_A^2\sigma_A^2 + x_B^2\sigma_B^2 + 2x_Ax_B\mathrm{Cov}(r_A,r_B).

But

\sigma_A^2 = \mathrm{Cov}(r_A,r_A),

\sigma_P^2 = x_A^2\mathrm{Cov}(r_A,r_A) + x_B^2\mathrm{Cov}(r_B,r_B) + 2x_Ax_B\mathrm{Cov}(r_A,r_B).

This expression now has a nice parallel structure. If we create a matrix called the covariance matrix,

\mathbf{P} = \begin{bmatrix} \mathrm{Cov}(r_A,r_A) &amp; \mathrm{Cov}(r_A,r_B)\ \mathrm{Cov}(r_B,r_A) &amp; \mathrm{Cov}(r_B,r_B) \end{bmatrix},

and a vector of weights:

\mathbf{x} = \begin{bmatrix} x_A \ x_B \end{bmatrix},

and do the following matrix multiplication:

\begin{bmatrix} x_A &amp; x_B \end{bmatrix} \begin{bmatrix} \mathrm{Cov}(r_A,r_A) &amp; \mathrm{Cov}(r_A,r_B)\\ \mathrm{Cov}(r_B,r_A) &amp; \mathrm{Cov}(r_B,r_B) \end{bmatrix} \begin{bmatrix} x_A \\ x_B \end{bmatrix}

= \begin{bmatrix} x_A &amp; x_B \end{bmatrix} \begin{bmatrix} x_A\mathrm{Cov}(r_A,r_A) + x_B \mathrm{Cov}(r_A,r_B)\\ x_A\mathrm{Cov}(r_B,r_A) + x_B\mathrm{Cov}(r_B,r_B) \end{bmatrix}

=x_Ax_A\mathrm{Cov}(r_A,r_A) + x_Ax_B\mathrm{Cov}(r_A,r_B) + x_Ax_B\mathrm{Cov}(r_A,r_B) + x_Bx_B\mathrm{Cov}(r_B,r_B)

= x_A^2\sigma_A^2 + x_B^2\sigma_B^2 + 2x_Ax_B\mathrm{Cov}(r_A,r_B),

we see that we recover the expression for $\sigma_P^2$ .

A polynomial where the sums of the exponents of the variables in each term equals 2 is called a quadratic form. An example:

4x^2 - 2xy + 3y^2

The portfolio variance is an example of a quadratic form (remember, $x_A$ and $x_B$ are the variables here):

\sigma_P^2 = x_A^2\sigma_A^2 + x_B^2\sigma_B^2 + 2x_Ax_B\mathrm{Cov}(r_A,r_B)

A quadratic form can always been written as $\mathbf{x}^\mathrm{T}\mathbf{P}\mathbf{x}$ , where P is a symmetric matrix.