A Note on Stocks vs. Returns

Motivation

Before we move on to how to identify integration in our time series, let’s stop for a moment to talk about a common transformation that we see in financial data called returns. Returns are easy to mistake for first differences, and, in fact, they might do away with our integration problems, but they are not the same. Let’s see why.

Stocks vs. returns

Financial markets are a very rich source of time series data. A simple Google search will show how easy it is to have access to stock and commodity price data at a very granular periodicity: daily, hourly, every five minutes, and even every minute. It is only normal that time series analysis is often applied to stock prices. Look at the image below. It shows Brent crude oil’s daily price from January 4, 2021, to mid-June 2023. It might feel familiar: it looks like a random walk, as most stock and commodity prices do.

Rather than working with stock prices, though, practitioners usually prefer stock returns. By return, we understand the profit (or loss) that an investor has made over a period of time by holding onto a given asset. We’ll represent it as a growth rate.

There are good statistical reasons to do it (returns tend to behave better), but the main one is interpretability. A daily variation of 1 USD in the price of a stock needs contextualization: It’s not the same if that stock went from 3 to 4 USD or from 399 to 400 USD. Even then, it wouldn’t be comparable with other stocks with substantially different prices or even with itself across different periods. A 1% increase, however, is much more straightforward to understand while providing a comparable scale across stocks with different prices.

There are two main types of returns: Arithmetic and logarithmic. The differences between them are subtle, but they are different, nonetheless.

Arithmetic returns

We denote the price of an asset at time $t$ as $P_t$. The arithmetic return $$ ...