Maximum likelihood estimates for means and covariances

Consider segment with observations of the discounted prices: , where . Proceeding from practical reasons, we shall assume that the above prices don’t take into account the dividend yields and the risk-free rate. Based on the prices one forms the range of logarithmic returns of length by means of the formula

Sample Expected Growth Rate

The MLE for expected growth rate is the annualized average of .
Sample estimate for expected excess growth rate is
where
denotes risk-free rate;
denotes vector of dividend yields;
stands for vector of ones.

Sample Covariance matrix

The Covariance matrix estimate , calculated in SmartFolio, is an annualized sample covariance matrixof the range .

Sample Mu vector

To estimate one should resort to the following expression: Note.If price process satisfies the analytical model assumptions, then sample Mu vector can be calculated in alternative way from a sample expected growth rate and a sample covariance matrix :

Corresponding sample estimate for excess Mu has the following form: