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...