### Fun with Multi-Decade Charts

## Tuesday, April 03, 2007

Having seen more than my fair share of multi-decade charts showing one economic statistic or another, the same erroneous conclusions have oftentimes been observed based on a simple misinterpretation of the data.

Call it the "power of compounding" or whatever you wish, but many writers and commentators seem to mistake this very normal effect as a sign of impending calamity, sometimes building long and convoluted cases for or against one thing or another based on a simple chart which has been completely misread.

This same error had been made here on at least a few occasions long ago, before someone graciously pointed out the error of my ways.

Think of this post as a sort of Public Service Announcement for those of you who may feel compelled to make a case based on looking at a single data series on a multi-decade chart.

All the charts below show curves with a constant rate of change. That is, the data point for each year is calculated as "last year + last year x rate of change", where the rate of change is constant - either three, six, or nine percent.

Remember that - it is important.

The first chart shows a curve that increases at an annual rate of three percent since 1960 - a total of 46 years. There's not much there worth looking at although it does seem to be picking up pace there in the 1990s and into the new century.

Three percent over fifty years doesn't hold much intrigue, but what if you more than double the time period to get about a century's worth of data.

Now this starts to look threatening somehow - you see how it goes up there at the end? Some might claim that once the 1970s arrived, things got a little crazy, when in fact the annual rate of change is constant over the entire century at three percent. You see this sort of thing all the time when it comes to historical inflation or stock prices.

Going back to the original 46-year span and adding in a six percent curve shows just how puny that three percent growth looks by comparison.

With no other context, the three percent curves above can appear to be problematic, but when laid up next to a data set with twice the annual rate of change, it seems almost insignificant.

Now add in a nine percent curve over the same 46-year time period and both the three and six percent growth rates are pushed lower, tame by comparison, and the nine percent curve appears quite threatening.

Note how the scale has changed as well. Many times people just look at the shape of a curve without looking at the absolute magnitude which, in the chart below, is 12 times that of the first chart. Nine percent a year is a lot when sustained over many years.

Now expand back out to just over a century's worth of data and the once-menacing 100-year three percent growth (second chart above) is barely visible and the nine percent curve is nearly vertical.

Again, you could look at this chart and think that something really changed in 1980, but in fact the annual rate of change is constant and just look at the scale - 100,000!

Using a log-scale, which is probably the best way to look at all data series that span more than a decade or two, shows the real nature of the curves - the rate of change is unchanging and the lines are straight.

The lesson here?

Be careful with those multi-decade charts.

## 7 comments:

Hey, thanks for the clarification. I always thought there was something wrong with those charts that go back a hundred years.

I am getting 8.05% in my New Zealand CD, so looks like I just need to wait another 65 years and I will be rich!

Anon 12:01, I'm getting 8.5% on six month CDs in Hungary, it's only going to take me 58 to get there! Probably by then it's going to look like this: http://en.wikipedia.org/wiki/Image:Inflaci%C3%B3_utan_1946.jpg Tim, thanks for the great charts!

I remember back in the sixties and seventies there used to be a CPI index of some sort. Eventually, the index got so high the base year was changed. Then later the index went away altogher. Inflation sounds much better when it "only" rises .3 or .4% month after month.

Let me try that wiki link again…

Link

"The lesson here?

Be careful with those multi-decade charts."

I assume you are talking about Schiller's "Housing Bubble" chart going back to 1849 Amsterdam which "proves" the coming nuclear winter?

A little late for dinner, aren't you?

You've hit on an interesting point about compound growth. I think I draw a somewhat different conclusion than you: I think that, more often than not, continuing compound growth is ominous.

I am beginning to suspect that in the financial sphere, anything that compounds for a long time at rates grater than 3%

islikely to lead to a calamity. A rate of growth of 2-2.5% seems to somehow be more "natural". But even that may only apply to historically brief periods (for example, population growth of just 2.5% a year globally would still do us in within a few tens or hundreds of years).Soddy (and perhaps others) have pointed out that "permanent" returns at even "conservative" rates, like 6%, are an inherent paradox: if that growth rate was made concrete by converting it into ounces of gold, in a few hundred years, one would go from a ball of gold you could hold in your hand to one greater than the diameter of the earth. This is simply a statement about the permanence of such benefit.

I suspect it is indeed possible for wealth, in general, to grow at a reasonable compounded rate, forever. But in doing so, the method will likely continue to change entirely, and the benefit migrate from nation to nation (or between other sorts of groups). And the net rate might be quite "low"... 1-2%, maybe 2.5%, annually.

What seems to be "infinitely" compounded growth often turns out eventually to be an S-curve; or something that gets blown off periodically, leaving not much more than 1 or 2% (like the real growth in the Dow).

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