Date : 29/08/2012

The Generalist

Logarithmic scale

Why using logarithmic scale to display share prices?

By displaying the graph of a share price or any stock index on the Internet, you have probably noticed that there is a whole bunch of various display options.

These options include the type of graph scale. In general we distinguish two types: the linear scale, which partitions the axes of x and y into equal intervals, and the scale called "logarithmic". This scale, a little more exotic, does not partition the y-axis (the vertical axis of the graph) into equal intervals; on the contrary, they seem to be increasingly tightened.

So, why using this logarithmic scale? If you're familiar with the reading of share prices, this question will seem certainly obvious to you... if not, the rest is for you!

To compare the properties of linear and logarithmic scales, we can study one of the most famous stock index of the world: the Dow Jones. This index of the New York Stock Exchange (full name: Dow Jones Industrial Average), is also the oldest in the world because it was created in 1884. Of course, at that time, companies like Microsoft were not yet part of the index! General Electric is the only company to be included in the Dow since its beginning. It should be noted that nowadays, the S&P 500 Index is preferred to measure the performance of the U.S. economy. Indeed, the Dow Jones has several weaknesses, including the fact that it is based on only 30 companies. However, this has no consequences for this article.

Here is the graph of the Dow Jones, in linear scale, from 1929 to today:

Dow Jones, échelle linéaire

Click to view the image in full size (Source: Yahoo! Finance)

This plot "simply" shows the evolution of the index value in 80 years, from almost 0 to its origins to more than 14,000 points in early October 2007.

By looking at this graph, the first thing you notice is that the value of the index seems to remain roughly the same for fifty years... Half a century! Then there are 80 years of exponential increase of its value, punctuated by a few "small" subsidences, corresponding to the 1987 crash, for example. We can notice more clearly the chaotic explosion of the dot-com bubble soon after 2000, and especially the effects of the famous and current subprime crisis from late 2007. Changes undergone by the index from that date seem totally incredible, compared to the "small" crash of 1987.

Actually, there is something wrong with this chart... Ok, the values displayed by the graph are true, but its representation is none the less misleading: indeed, the linear scale of the y-axis only represents the variation of the index value, not proportionnally. In other words, in this graph we can visually evaluate a difference in index points, and not its percentage change. This is the important point of this article: THE critical information that you want to know in order to evaluate the performance of an index or a share in a given period is precisely its percentage evolution! Indeed, just consider a variation of $1 in one day on any share. If the share price is early in the day $1, it will be worth $2 to the end of trading day, which is simply an evolution of 100%... a very interesting investment! On the contrary, if the share price is worth $1000 in the morning, it will be worth $1,001 in the evening, or a variation of 0.1%. So almost no change.

Now that we know that the percentage change in a stock price is the most important information, how to graphically display it? The answer lies in one word: just display the logarithmic plot of the share price.

The logarithm (noted classically "log") is a mathematical function with a very interesting property: it has the power to transform the multiplication into addition. This property is expressed in the following equation, with a and b positive real numbers:

log( a×b ) = log( a ) + log( b )

What connection is there between this property and the graphical representation of our stock price? Actually, a price change of a given percentage can be seen as a multiplication: a given stock price p undergoing a 4% increase will be worth 1.04 × p, and if it undergoes a decrease of 3%, it will only be worth 0.97 × p.

If we call m the multiplicative factor corresponding to the variation of a stock price p, its new value p' will be:

p' = m×p

Let's see now the logarithm of this new value m'! According to the fundamental formula above, we can write:

log( p' ) = log( m×p ) = log( m ) + log( p )

Thus, we realize that to get the logarithm of the value of a stock price following a variation of a given percentage, we just have to add the logarithm of the change to the logarithm of the initial stock price. That's the reason why it's possible to graphically represent the percentage variation of a stock price. In logarithmic scale, the same variation of a stock price graph represents the same percentage change.

Without further ado, here is the comparison of the Dow Jones in linear scale and logarithmic scale:

Dow Jones, échelle linéaire

Linear scale

Dow Jones, échelle logarithmique

Logarithmic scale

The difference between the two curves is more than striking: the logarithmic curve is much more "rough", which allows us to appreciate the fine details of changes over the years, starting with the 1929 crisis, which seemed curiously absent from line graph. We also realize that ultimately, there was a real economic growth between 1945 and 1975! Not for nothing this period is called the postwar boom... Furthermore, this logarithmic graph allows us to better relativize the intensity of the effects of the subprime crisis: they were certainly very important, but proportionally less than the Great Depression that followed 1929.

As a conclusion, the logarithmic scale appears to be better compared to the linear scale: both scales make it possible to visualize the differences of the value of a stock price, but only the logarithmic scale reflects the changes in proportion.

You are now informed for your future courses consultations! ;-)

Reference :

- To browse the course of the Dow Jones on Yahoo! Finance, click on this link.