Forecasts
The inputs to the mean-variance optimization are Forecast Return, Forecast Risk, and the Correlations between the assets. AllocationADVISOR calculates these inputs based on historical values according to the following formulas:

Historical Forecast Return

The historical Forecast Return is the annualized arithmetic mean return. (This number is different from the annualized return in StyleADVISOR, which uses the geometric mean return.)

Number of periods per year

where:

E[R_i] = Forecast return for asset i
r_t = tth arithmetic return
T = number of arithmetic returns

Historical Forecast Risk (Standard Deviation)

The historical Forecast Risk is the annualized standard deviation of the historical Forecast Return.

*

where:

= Forecast risk for asset i
r_t = tth arithmetic return
T = number of arithmetic returns
average arithmetic return

Historical Correlation

Correlation measures how closely related the variances of two series are.

where:

correlation of asset i with asset j
covariance of asset i with asset j
standard deviation of asset i
standard deviation of asset j
tth arithmetic return of asset i
average arithmetic return of asset i
T = number of arithmetic returns