Mean-Variance model
The Mean-Variance model by Markowitz (Markowitz, H.M. (March 1952)) is practically the ice-cream/umbrella business in higher dimensions. For the mathematical formulation, we need some definitions.
They are explained as follows:
-
By asset
, we mean a random variable with finite variance.
-
By portfolio, we mean the combination of assets:
, where 
, and 
. The combination can be affine or convex. In the affine case, there is no extra restriction on the weights. In the convex case, all the weights are non-negative.
-
By optimization, we mean a process of choosing the best
coefficients (weights) so that our portfolio meets our needs (that is, it has a minimal risk on the given expected return or has the highest expected return on a given level of risk, and so on).
Let 
be the random return variables with a finite variance, 
be their covariance matrix, 
and 
.
We will focus on the following optimization problems:
It is clear that 
is the variance of the portfolio and 
is the expected...
