Linear or logarithmic? – The choice of scale determines the story the graph tells

"Could Inderes write an article about the differences between linear and logarithmic representation, perhaps in the education section?" This was Juurikki's wish on the forum a couple of days ago. So, here's a concise article on the topic for Juurikki and everyone else.

Juurikki and others have surely encountered situations where a stock, on a linear price chart, appears to be rocketing through the roof. Such a development can't be a sign of anything other than a bubble, can it? Before looking at examples, it's good to understand what actually causes the difference.

What do the different scales actually tell us?

On a linear scale, equal price changes appear as equal movements. For example, if a share rises from EUR 10 to EUR 20, the movement is the same as a rise from EUR 100 to EUR 110. In this case, the first increase represents a doubling of the share price, while the latter movement is only a 10% increase.

On a logarithmic scale, equal percentage changes represent equal movements. A ten percent change looks the same regardless of whether the share price is EUR 10 or EUR 100. In short: a logarithmic scale "normalizes" growth – while a linear scale emphasizes absolute euro amounts, a logarithmic scale speaks in percentages. And percentages are the language investors should think in.

What does this look like in practice?

Currently, investors are especially talking about NVIDIA, the AI boom mammoth that has grown into the world's largest company in a relatively short time. Let's look at a linear price chart from the last 16 years.

During the first six years of the review period, the share barely budged in either direction. You'd need a magnifying glass to spot any movements.

However, since the lows seen at the end of 2022, the share has entered an extraterrestrial orbit, seemingly unaffected by gravity for some time. After all, the share has risen by around 1600%. Normally, this would be a cause for raised eyebrows. Have the company's fundamentals followed suit, or has the share price detached from reality, with the valuation swelling to bubble levels? However, the company's EPS has risen from USD 0.174 to around USD 5 over the same period.

What story does the logarithmic scale tell instead?

Now we can see that the share also fluctuated sharply in the last decade. In fact, the share has halved three times between 2010 and 2016, and declines have always been followed by multiplication! The development of the current decade also looks considerably more stable. One even has to look for a moment to find the dip caused by last year's tariff dip on the graph, even though a third of the share was blown away at that time.

In reality, growth may have been quite steady in terms of annual returns throughout the period. However, a linear scale cannot show this because it scales euros—not percentages. The early history of the share disappears from view, or at least becomes distorted.

Why can a linear graph be misleading in the long term?

The same phenomenon is repeated closer to home on our stock exchange. The linear graph of the OMXH GI, i.e., Nasdaq Helsinki's total return index, shows two striking upward movements – the rise from the COVID lows and the rally of the past year.

However, the logarithmic scale reveals that the Euro crisis caused the index to collapse by a third – practically as severely as COVID. The growth of recent years no longer looks as explosive. The linear scale thus exaggerates later movements and flattens early history.

Which one to use and when?

A linear scale is suitable for short-term analysis or when the euro-denominated change is the focus of interest. A logarithmic scale, on the other hand, is a natural choice for longer timeframes and situations where the development of multiple shares is compared on the same graph.

The difference is easy to grasp through everyday life: if a person saves the same amount monthly into a non-interest-bearing bank account, the balance grows linearly. An investor's returns, however, accumulate through compounding interest – percentage-wise. It is therefore only natural to view price charts with a corresponding scale.

So, before interpreting a price chart, first check which scale it's drawn on. It can change the whole story.

Image sources: Investing.com