# Portfolio theory: confusion about variance-covariance matrix

I am taking an introductory course to finance in my Master's, and wanted to go further in the topic of portfolio theory (I am an engineering bachelor graduate, but as I just hinted, I am new to advanced maths for statistics). My question might look dumb to some of you, but I think my main problem is simply understanding the notation of the variance-covariance matrix. I watched a video saying that when calculating the optimal vector of weights to obtain the minimum-variance portfolio, in the case of many assets, we have: $$\sigma_{portfolio}^2 = w^T\Sigma w$$ $$w^T\textbf{1}=1$$ , where $$w$$ is a column vector of weights, and $$\textbf{1}$$ is (I suppose?) a column vector filled with 1s with the same dimension as $$w$$.

He then goes on to write the Lagrangian as follows: $$\operatorname{L}=w^T\Sigma w + \lambda(w^T -1)$$ And, by deriving the Lagrangian with respect to $$w$$ and setting the derivative to 0, and after a couple of other steps, he ends up with: $$w=\frac{\Sigma^{-1}\textbf{1}}{\textbf{1}^T\Sigma^{-1}\textbf{1}}$$ (If you'd like to see the intermediary steps, please ask).

The reasoning seems pretty clear to me, however I have no idea how to interpret the notation of the variance-covariance matrix in this final equation: does $$\Sigma^{-1}\textbf{1}$$ mean the inverse variance-covariance matrix of vector $$\textbf{1}$$ (which wouldn't make much sense according to the calculations I did for that case), or is it implied that $$\Sigma^{-1}$$ is the inverse variance-covariance matrix of the expected returns of the respective stocks? And, therefore, $$\textbf{1}^T\Sigma^{-1}\textbf{1}$$ would be the elements of the inverse var-covar matrix of expected returns, but as scalars (which would make sense because we can only divide scalars together and not matrices?).

Am I right in this assumption, or am I still missing the point? Again, maybe this question will look awkward to some of you, but I'm new to the topic, and after some research on the web I couldn't find the var-covar notations well explained anywhere.

Thank you!

• $\Sigma^{-1}\textbf{1}$ really is just multiplication of the matrix $\Sigma^{-1}$ with the column vector $\textbf{1}$, so standard multiplication rules apply. When multiplying with a column vector of ones, you get the row sums. Mar 19, 2021 at 14:42

I think what is going on here is the matrix-vector product $$\Sigma^{-1} \mathbf{1}$$ and quadratic form $$\mathbf{1}^T\Sigma^{-1} \mathbf{1}$$ are being used as a convenient way of specifying summation. That is, they bridge matrix-vector notation and summation notation.

Assuming $$\Sigma^{-1} \in \mathbb{R}^{d \times d}$$, then $$\Sigma^{-1} \mathbf{1}$$ is the product of the matrix $$\Sigma^{-1}$$ with the column vector $$\mathbf{1} \in \mathbb{R}^d$$. Hence the matrix-vector $$\Sigma^{-1} \mathbf{1}$$ product is just a column vector in $$\mathbb{R}^d$$ where each element is the sum of the rows of $$\Sigma^{-1}$$.

Without any further context, $$\Sigma^{-1} \mathbf{1}$$ means take your inverse variance-covariance matrix $$\Sigma^{-1}$$ and take the sum of each row, to give a column vector.

Further, the quadratic form $$\mathbf{1}^T \Sigma^{-1} \mathbf{1}$$ is a product of the row vector $$\mathbf{1}^T \in \mathbb{R}^{1 \times d}$$, the matrix $$\Sigma^{-1}$$, and the column vector $$\mathbf{1} \in \mathbb{R}^{d \times 1}$$. Computing this will yield the sum of all elements in $$\Sigma^{-1}$$.

Similar to above, $$\mathbf{1}^T \Sigma^{-1} \mathbf{1}$$ means take your inverse variance covariance matrix $$\Sigma^{-1}$$ and sum all the elements, yielding a scalar.

I guess then what the expression for $$w$$ means is to take the column vector $$\Sigma^{-1} \mathbf{1}$$ and divide each element by the scalar $$(\mathbf{1}^T \Sigma^{-1} \mathbf{1})$$ (or more precisely multiply each element in the column vector by the scalar $$(\mathbf{1}^T \Sigma^{-1} \mathbf{1})^{-1}$$).

You might wonder why do this when we can just use summation notation - you would be surprised just how useful this is when writing vectorised mathematical pseudocode.

• I see now, things are becoming clearer. However one last thing, you're talking about $\Sigma^{-1}$, but $\Sigma^{-1}$ of what vector quantity exactly? I understand that the diagonal of $\Sigma$ contains the variances and the rest of cells the covariances, but of what vector quantity here exactly? How I see things is that $\Sigma$ alone doesn't calculate the (co)variances of anything unless it's implied somehow Mar 19, 2021 at 15:08
• My response only addresses the problematic interpretations of the matrix-vector product and quadratic form in the numerator and denominator respectively of $w$ from the perspective of linear algebra. If you want context specific interpretations concerning $\Sigma$, which I cannot guarantee, but others on the forum may be able to assist with , please supply more info on what is contained in $\Sigma$. Mar 19, 2021 at 15:27
• Alright thanks for the explanations! All clear now Mar 19, 2021 at 16:53