**Linear Graphs VS Logarithmic Graphs**

January 13, 2018

Very often I see incomprehensible linear graphs with very long historical series published on social networks like the image that I have chosen provocatively as an example for this post.

Excel graphs or even many trading or analysis platforms are linear in scale, meaning that the Y-axis of Cartesian planes have a uniform and linear scale.

HISTORY

Let's take advantage of this to study a little bit of history: Why are they called Cartesian graphs? Because they were invented by Cartesio, you may say perhaps, not quite so, but I can even quote a paragraph from wikipedia on the history of the cartesian plan:

The use of geometric coordinates was first introduced by Nicola d' Oresme, a 14th century mathematician who was based in Paris. The Cartesian adjective refers to the French mathematician and philosopher René Descartes (Known in Italian as Renato Cartesio, In Latin as Renatus Cartesius)which among other things, taking over the studies of Nicola d' Oresme, he worked on the fusion of algebra with Euclidean geometry. These studies were influential in the development of analytical geometry, calculus and cartography.

The idea of this reference system was developed in 1637 in two writings by Cartesio and independently by Pierre de Fermat, even though Fermat did not publish his discovery. In the second part of his Discourse on the method, Cartesio introduces the new idea of specifying the position of a point or an object on a surface using two lines that intersect at a point as measuring instruments, an idea gotten from Geometry.

LINEAR CHARTS

Returning to the usefulness of linear graphs, this is most beneficial when the historical series do not have a long length and therefore the capitalization effect of the returns is negligible, otherwise, the representation becomes not very useful and even misleading.

LINE

But let's start with the order, let's start from the logic of a simple straight line that goes up with a certain relationship between the X-axis and the Y-axis.

Let's try to draw a straight line on the graph that rises by one for each unit of time (X-axis); the mathematical formula is very simple Y=X+100 (assuming it starts from 100).

Practically a straight line is been represented no matter long it may be and it always remains in a Cartesian plane.

So far so good, we have understood that in a Cartesian plane the X-axis is time while the Y-axis represents the value or price of a historical series (I do not want those who work with finances every day, it is fine to start from the basics for those who are not accustomed to these logics).

Now let's assume that instead the historical series rises by 1% every month; will the resulting series always be a straight line?

Apparently yes, if the time represented is short it will look like the line is still straight, but in reality, it is not so because of the capitalization effect.

Let's use an exaggerated example so the logic can be better understood: suppose we have an investment that rises by 10% every month (stay away from those who propose similar returns, even further away from those who sell them to you for sure...), obviously, it's a didactic example: starting from 100, at the end of the first month we will have 110, but at the end of the second month if the line was straight we should have 120, while in reality, we have 121, because we have to make 10% of 110 or +11 compared to the previous month, the third month we will have 133.10 and so on, so the growth doesn't occur in a linear way but increases more thanks to the capitalization effect.

The graph above shows a monthly return of 1% (this is also too good to be true in traditional finance, the only one that will give these returns with continuity will have to take 150 years...) exactly like the previous one, but if you expand the historical series you will notice that it is not a line but a concave with an upward trend, that tends to grow with an exponential trend.

If we exaggerate with the time, the trend becomes even more steep, canceling any change in the distant past and exaggerating the effects of the recent past.

Logarithmic graphs were invented to solve this problem, based on the logarithmic logic introduced for the first time by Nepero in 1600....

The logarithmic graph has the Y-axis which does not increase linearly but increases exponentially, i.e. instead of having a scale of 1,2,3,4.... you have a scale where the exponent changes such as 101,102,103 etc...

As you can see, using the logarithmic graph makes it possible to make a linear historical series that increases with a constant growth rate (e.g. 1%)

Why is a logarithmic graph so much superior to a linear graph?

The reasons are different, the first (1) is representative, i.e. a long-term linear graph makes no sense, because it is not possible to appreciate past oscillations that become irrelevant because the 10% loss of 10 is equivalent to a decrease of 1 unit, while 10% of 1000 is equivalent to a decrease of 100, the percentage is equal but in the graph the difference is considerable;

The second (2) is that loss of recent value seems much worse than loss of past value;

In the graph we assumed using a historical series that alternately increases per month by 15% and the following month drops 9%; despite being a relatively short graph, it seems that the gains and losses are different over time, but in reality the historical series is equal in returns both at the beginning and at the end, as shown on the logarithmic scale graph:

The third reason (3) is that if we compare two historical series that have the same drawdown, but dissimilar values, they do not seem to have been compared correctly because the oscillations appear very different:

The question arises spontaneously:"which of the two historical series is more volatile? I'll probably reply the blue historical series, even if you have understood the logic it is quite possible that you have not fallen for this trivial trap...

If we look at the graph in a logarithmic scale we will discover that the historical series have exactly the same trend, except that one has started at 100 and the other at 50.

So I hope that these small tips on logarithmic graphs will ensure that in the future when you see meaningless linear graphs even on social networks (especially for those who use technical analysis, there is no point wasting time on linear graphs) and I hope that every time you see a linear graph on social networks, you will comment on the post by using the graph to advise the "unfortunate" writer who published it to study the graph.

I will leave you with this teaser, I want to introduce the topic I will be discussing next week:

But if the graph was a historical series with a return of +50% a month and -50% the following month, what will the graph look like?

That is to say: when volatility is very high, returns are not linear at all and losses cost more than profits...

Share with your friends if you think the post is helpful, thank you

DB