*Bubbles*

#### Going parabolic

From my undergraduate days in university studying for a B.Sc. in Math and Physics, I’ve always been interested in how you can tame masses of data with charts and by inspection understand what the chart is telling you. But, for the uninitiated, charts can be deceptively misleading. A superb book that makes the point is Alberto Cairo’s *How Charts Lie* published in 2019.

The problem revolves around how easy it is to glance at a chart and think you understand what it is telling you. What’s our take away from the following ten year chart of the S&P 500?

The blue line hugs the plot and forms a curve. That is, the plot of the hi/low/close is curved. What is that telling us? It suggests prices are accelerating. To my eyes (without thinking about it) it suggests the S&P 500 might be in a bubble and about to drop like a stone.

Chart 1 S&P 500 ten year arithmetic y scale – tall and narrow

I constructed the chart to create this impression. I made it tall and narrow and plotted the y-scale (prices) arithmetically rather than log. That is, the distance from 1000 to 1100 is the same as between 3000 and 3100. That is, the right hand scale is arithmetic and not based on percentage changes.

What is causing the curve? **It’s simply because we are using an arithmetic scale.**

Let’s look at Chart 2. It’s the same data over the same time period. I have re-jigged the chart to shorten it and stretch it out sideways. It’s still curved but, the curved line is not pointing skyward as in chart 1. It still shows a curve but the message is much less dramatic. The chart is still misleading but less so. The curve makes it look like the rate of change in stock prices is accelerating. But, it isn’t. Still, it doesn’t holler out bubble.

The lesson here is that by displaying the chart tall and narrow we can give one impression. By displaying it short and wide, we can create another.

Chart 2 S&P 500 ten year arithmetic y scale – short and wide

Now let’s look at the Chart 3. This is the way the chart should be constructed. **The chart is no longer curved overall. It has its normal ups and downs i.e. volatility.** The secret is that the y-axis scale is changed. It shows equal spaces between equal percentage differences in price. This chart style is called semi-log. That is, it is log on the y scale.

Chart 3 S&P 500 ten year log y scale

The chart shows the rapid rise in prices in the last 12 months. But, we can see that rise in perspective. At least the chart is displayed properly.

One great thing about semi-log charts for investments is that **the steepness of the plot represents the percentage return on investment**. It’s useful, for example, for comparing the return from stocks and bonds. The one fault I find with Chart 3 is that it doesn’t make allowance for inflation, i.e. not adjusted for inflation. That tends to exaggerate steepness of the plot, the rate of return. If the nominal return over a period is 9% and there is 2% inflation, the plot is steeper than it would be in real terms.

For a properly plotted chart of inflation adjusted stock price see an example here from Professor Siegel.

Even Nobel Prize winners get this wrong. Professor Shiller in his book *Irrational Exuberance*, mistakenly uses a chart constructed in the same way as Chart 1 above, to make his case that prices in 1999 were in a bubble. See my post here. His conclusion was right. His evidence was wrong. **With charts we must always be wary of facile conclusions.**

**The term parabolic**

With all of this clear in our heads, we can turn to the erroneous expression ‘prices going parabolic’.

Many of us have played with a magnifying glass outside in the sun. You can focus the sun’s rays and burn a leaf or piece of wood. A lens does that. You can also do it with a concave mirror. The light rays come in parallel and **the parabolic shape of the mirror** focuses the light as in the figure below.

One of the defining features of a parabolic curve is that **as you go to the right on the x axis the curve of the plot flattens out**. That is, adjacent to position B on the chart the curve has a greater bend. Adjacent to position E on the chart, the curve has less bend.

Plot of parabolic curve

When using a plot of stock prices on a semi-log chart **one shape to look out for is a greater bend (not a lesser bend) in the curve as you move out along the x axis**. See link below. As well parabolas are plotted on an arithmetic chart not semi-log. Also, they are U shaped although that is a minor quibble.

**Conclusion**

Like statistics, charts are useful. They can easily lie. It takes skill and knowledge to properly construct charts and properly read them.

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Charts plotted with a log y-axis are useful to help identify bubbles.** **They must show an acceleration** of not just of prices but of the rate of change in prices (increasing curve) on a semi-log** **chart** (log y-axis). This is noteworthy. As I’ve been at pains to point out, **normal stock action plotted on a semi-log chart makes a straight line**, if you smooth out the volatility. When it curves upward, it is some evidence of the madness of crowds setting in. As it gets steeper and steeper it is more likely a bubble. As well, linear digression bands on such charts **help**. They are not definitive, just aids.

To read further specifically about the use of charts to help identify bubbles, See here

To dig into the whole subject of bubbles and what to do about them, see here.

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Check out the **Tags Index** on the right side of the **Home page** that goes from ‘**accounting goodwill**’ to ‘**wisdom of crowds**’. This will give readers access to a host of useful topics.

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