Measure Financial Returns


There are different ways to measure financial returns and each way has its own characteristics.

Arithmetic returns

The direct way to measure the return from period \(t-1\) to \(t\) is,

\begin{equation} r_t = \frac{p_t - p_{t-1}}{p_{t-1}} = \frac{p_t}{p_{t-1}} - 1. \end{equation}

Where, \(p_t\) is the price of the asset at time \(t\).

Log returns

The log return is defined as follows,

\begin{equation} r_t = \log \left( \frac{p_t}{p_{t-1}} \right) = \log (p_t) - \log (p_{t-1}). \end{equation}

The log return has nice properties, e.g., it is additative,

\begin{equation} r_t + r_{t+1} = \log (p_t) - \log (p_{t-1}) + \log (p_{t+1}) - \log (p_t) = \log (p_{t+1}) - \log (p_{t-1}). \end{equation}

Geometric return

Money-weighted return