ok so I have 2 assets, asset A and asset B, this 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 bild now the covariance matrix as
var A B COV A B COV A B var A B
i have as well 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 multipilcation between covariance matrix and vector Rf and obtain 2 numbers which standardized, the sum is equal to 1 . Let 's say w1 is 1.20 and w2 is -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 1x1 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*=argmin (w'Σw) given that Σwi=1
where w* is the otimal weights vector and Σ the var cov matrix.
now seen this in a regression framework, it is possible to optimize this formulation adding a penalty function Ro (which intensity is controlled by lambda) such as : w*=argmin ((w'Σw) +lambdaΣ Ro(wi))
now my question is how can I see the covariance matrix here above mentioned as a regression. I just can't see which is the explanatory variable, and the dependent variable.. i am a little confused.
Thanks in advance for the support.