Asset prices are usually non-stationary, that is, their statistics, such as mean and variance (mathematical moments) change over time. This could also mean observing some trends or seasonality in the price series (see Chapter 3, Time Series Modeling). By transforming the prices into returns, we attempt to make the time series stationary, which is the desired property in statistical modeling.

There are two types of returns:

• Simple returns: They aggregate over assets; the simple return of a portfolio is the weighted sum of the returns of the individual assets in the portfolio. Simple returns are defined as:

• Log returns: They aggregate over time; it is easier to understand with the help of an example—the log return for a given month is the sum of the log returns of the days within that month. Log returns are defined as:

 P_t is the price of an asset in time t. In the preceding case, we do not consider dividends, which obviously impact the returns and require a small modification of the formulas.

The difference between simple and log returns for daily/intraday data will be very small, however, the general rule is that log returns are smaller in value than simple returns.

In this recipe, we show how to calculate both types of returns using Apple's stock prices.