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.

  • $\begingroup$ Why do you want to see the covariance matrix as a regression? Why do you think it should be? Forget about regression. You can add a penalty based on risk to the objective function if you want to .If anything the relationship is that regression can also be formulated as an optimization problem with a quadratic objective function, to which a penalty term can also be applied if desired. $\endgroup$ – Mark L. Stone Sep 5 '16 at 17:32
  • $\begingroup$ ok, I am showing the difference between L1norm and L2 norm and why I am applying lasso (when covariance matrix is singular ect ect) instead of OLS.. but how do i show the difference between lasso and ols - applied to a covariance matrix? I do not know how to translate the relationship given my data into a quadratic objective function.. I have been researching in this topic only since 2 weeks so I am no expert and do not know how to move from a to b.. $\endgroup$ – domenico Sep 5 '16 at 18:15
  • $\begingroup$ Neither OLS nor Lasso is involved here. Adding a penalty term does not turn this into a regression problem. $\endgroup$ – shadowtalker Sep 5 '16 at 19:51
  • $\begingroup$ ok then the initial statement in the paper of arxiv.org/pdf/0708.0046.pdf is confusing me " We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem." but I do not get their formulation. that is why I asked it here.. $\endgroup$ – domenico Sep 5 '16 at 20:03

MV Optimization can easily be expressed as a regression (OLS) problem. Firstly unconstrained: If you regress a (NX1 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 qudratic utility i.e. E[r]-risk_aversion*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, them compute the portfolio mean and the portfolio variance, then the dependent variable in the regression will be [(mean^2+var)/mean]*ONE, where again ONE is an NX1 vector. Introducing constraints is convenient with any quadratic programming engine you have access to.

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