Let denote discounted prices of -th portfolio component at respective times , where .
Symbols and denote respectively simple return and logarithmic return in -th portfolio component over the period to .

Consider a portfolio with constant weights . It means that at the end of each period such portfolio is rebalanced to state .

Portfolio Dynamics

Symbols and denote respectively simple return and logarithmic return in discounted portfolio wealth over the period to . Then

where

denotes the risk-free rate;
denotes vector of dividend yields.
Accordingly, .
Thus, the discounted portfolio wealth at time is equal to , where denotes initial wealth.

Historical Portfolio Excess Growth Rate

Historical portfolio excess growth rate is equal to .

Note.When using SmartFolio it might appear that for the same portfolio value of the historical portfolio excess growth rate differ substantially from the expected excess growth rate, calculated
under the analytical model assumptions. There are three reasons that explain such deviation:

Parameters and used in analytical portfolio don’t correspond to the historical data. It happens when portfolio components have different lengths of historical data; analyzed time period in parameters
estimation settings doesn’t coincide with historical one; sample estimates are modified by some of more advanced estimation methods.

Distribution of log returns for some assets in portfolio significantly deviates from normality. This is often the case when hedge funds or derivatives are included in the portfolio.

Rebalancing period is sufficiently long to violate the approximation of an analytical portfolio, which is rebalanced continually, with a historical simulation, where rebalancing takes place at the end of every period.

Historical Portfolio Volatility

Historical portfolio volatility is defined as

Historical Portfolio Excess Mu

Historical portfolio excess Mu is equal to .

Contribution to Portfolio Risk

For calculation of vector of portfolio components contributions to risk SmartFolio utilizes the following approximate formula:
where , . It is easy to see that for sufficiently small quantity is close to .