There are several ways one can go about when a trading strategy is to be developed. One approach would be to use the price time-series directly and work with numbers that correspond to some monetary value.

For example, a researcher could be working with time-series expressing the price of a given stock, like the time-series we used in the previous article. Similarly, if working with fixed income instruments, e.g. bonds, one could be using a time-series expressing the price of the bond as a percentage of a given reference value, in this case the par value of the bond. Working with this type of time-series can be more intuitive as people are used to thinking in terms of prices. However, price time-series have some drawbacks. Prices are usually only positive, which makes it harder to use models and approaches which require or produce negative numbers. In addition, price time-series are usually non-stationary, that is their statistical properties are less stable over time.

An alternative approach is to use time-series which correspond not to actual values but changes in the monetary value of the asset. These time-series can and do assume negative values and also, their statistical properties are usually more stable than the ones of price time-series. The most frequently used forms used are relative returns defined as

\begin{equation}

r_{\text{relative}}\left(t\right) = \frac{p\left(t\right) - p\left(t-1\right)}{p\left(t-1\right)}

\end{equation}

and log-returns defined as

$$\begin{equation}

r\left(t\right) = \log\left( \frac{p\left(t\right)}{p\left(t-1\right)} \right)

\end{equation}$$

where $p\left(t\right)$ is the price of the asset at time $t$. For example, if $p\left(t\right) = 101$ and $p\left(t-1\right) = 100$ then $r_{\text{relative}}\left(t\right) = \frac{101 - 100}{100} = 1\%$.

There are several reasons why log-returns are being used in the industry and some of them are related to long-standing assumptions about the behaviour of asset returns and are out of our scope. However, what we need to point out are two quite interesting properties. Log-returns are additive and this facilitates treatment of our time-series, relative returns are not. We can see the additivity of log-returns in the following equation.

\begin{equation}

r\left(t_1\right) + r\left(t_2\right) = \log\left( \frac{p\left(t_1\right)}{p\left(t_0\right)} \right) + \log\left( \frac{p\left(t_2\right)}{p\left(t_1\right)} \right) = \log\left( \frac{p\left(t_2\right)}{p\left(t_0\right)} \right)

\end{equation}

which is simply the log-return from $t_0$ to $t_2$. Secondly, log-returns are approximately equal to the relative returns for values of $\frac{p\left(t\right)}{p\left(t-1\right)}$ sufficiently close to $1$. By taking the 1st order Taylor expansion of $\log\left( \frac{p\left(t\right)}{p\left(t-1\right)} \right)$ around $1$, we get

\begin{equation}

\log\left( \frac{p\left(t\right)}{p\left(t-1\right)} \right) \simeq \log\left(1\right) + \frac{p\left(t\right)}{p\left(t-1\right)} - 1 = r_{\text{relative}}\left(t\right)

\end{equation}

Both of these are trivially calculated using Pandas: