How can I see a covariance matrix, as a standard regression y-Bx

Question

I have 2 assets, asset A and asset B. These assets have a vector of returns, 30 observations each. I calculate the estimated return as mean of the asset vector A and for B as a mean of asset vector B. Now I have rA and rB which I will call now vector $r$ . I build the covariance matrix as

var A B     COV A B

COV A B     var A B

I also have a risk free asset called Rf with $0.5$ return (for example) and I calculate for both (obtaining vector Rf):

rA - Rf
rB - Rf

Now I calculate the weights as a multiplication between covariance matrix and vector Rf and obtain 2 numbers which standardized, the sum is equal to 1 . Let's say $w_1=1.20$ and $w_2=-0.20$ (in financial terms I buy $120\%$ of asset A and shortsell $20\%$ of asset B) then you multiply this weights obtained by the vector of returns $r$ obtaining $1\times 1$ number which gives the return of the portfolio, same, you multiply the weights with variance covariance matrix and you get the variance of the portfolio.

Now the goal is to minimize the variance covariance matrix in order to have the optimal weights which minimize the risk.

Researchers write this problem as $$w^*=\arg\min (w'\Sigma w)$$ given that $\sum w_i=1$ where $w^*$ is the optimal weights vector and $\Sigma$ the var cov matrix.

Now seen this in a regression framework, it is possible to optimize this formulation adding a penalty function $R_o$ (whose intensity is controlled by $\lambda$) such as: $$w^*=\arg\min ((w'\Sigma w) +\lambda\Sigma R_o(w_i))$$

How can I see the covariance matrix as a regression? Which one is the explanatory variable, which is the dependent variable?

Answer 1

MV Optimization can easily be expressed as a regression (OLS) problem.

Firstly unconstrained: If you regress a $N\times1$ (where $N$ is the sample size) vector of ones on the asset returns, the relative regression coefficients are the same as the MVO weights; in fact, they are the solution to the maximization of quadratic utility i.e. $E[r]-\text{risk_aversion}\times E[r^2]$.

So by changing the constant dependent variable, we can arrive at weights that are exactly the same as MVO ones. If you solve MVO weights for a given risk aversion, then compute the portfolio mean and the portfolio variance, then the dependent variable in the regression will be $$\left[(\text{mean}^2+\text{var})/\text{mean}\right]\times\text{ONE}$$ where again $\text{ONE}$ is an $N\times1$ vector.

Introducing constraints is convenient with any quadratic programming engine you have access to.