So, I am reading the blog post about Mean Absolute Deviation portfolio optimization that
claims that the traditional portfolio optimization problem can be expressed as seen below:
I am not sure however that it is 100% correct for example you have (r[i,t]-u[i])*(r[j,t]-u[j]) = w[t]^2 ???
Also I dont like all the summation notation. If the below indeed holds isnt there a way we can
show this with matrix algebra like Variance= transpose(W).COV.W
The standard QP formulation:
First we plug in the definition of the covariances q(i,j), using:
Here the r’s are the returns at each t for each instrument i. The μ's are the means. This results in:
The a(i,t)’s are the mean adjusted returns.
Now it is easy to see how this can be approximated by an LP:
So the resulting L1 norm model looks like: