As we write this, the S&P 500 remains at (or near) an all-time high. When viewed on a linear scale over a long enough time period, the graph even begins to look vertical. Drops look crazily steep. The 1987 crash is nearly a blip on the map while the 2008 crash appears (correctly) to be the largest, quickest downfall in the past 40 years.

What incorrect decisions do investors make because of scale choice? What downside is there to picking a linear scale instead of an logarithmic scale?

### Stock Market Returns are Exponential

When looking at stock market returns, it’s important to consider the returns as *multiplicative* or *exponential*.

For example, if the stock market rises 10% and then decreases 10% it is **not** back to where it started! It will be 1% below where it started (the order in which it increases/decreases doesn’t matter, it will still be 1% lower.)

In DQYDJ’s own S&P 500 calculator, for example, all the reported numbers are *compounded annual growth rates* (CAGR). We multiply numbers together to produce the long term averages you see.

What is a natural consequence of this math? The largest increases come on the largest base. Therefore, the gains will look the largest when the price is high. AND the losses will look the largest when the price is high.

### Percentage Increase is the Best Metric for Investment Returns

To properly gauge returns look at the *percentage increases* because that is the impact gains or losses have on your contributions. This is especially true if you are periodically investing (perhaps investing in a 401(k) biweekly) because each decision point is a new entry to consider a percentage increase. Let’s take a look at an extreme example.

Comparing GE and AMZN’s returns since Mar 2007 reveals one of the major problems with linear scales: they lack the nuance to show when investments are impacted.

In this example, you literally cannot even see GE’s fluctuations register because the scale is so off the charts due to AMZN’s growth. Not only that, but it looks like a third of the returns occurred in the past 6 months alone. That’s not exactly the right way to evaluate an investment – consider that going from a 1500% to a 3100% increase is really just doubling (just like a 100% gain).

Now, let’s compare the same chart on a logarithmic scale. You can now see the massive decrease in AMZN value in late 2008 and the huge run-up they experienced from 2009-2011. Both periods are obscured on the linear scale.

Amazon’s 2015 returns look a bit more fantastic than their 2016 and 2017 returns when moved onto this logarithmic scale. That holds true in performance: AMZN returned ~110% in 2015 while returning ~15% and ~55% in 2016 and 2017, respectively.

### New Stock Market Records Every Month!

I went back and grabbed the first time the S&P hit 100 point intervals over time. Many benchmarks fell running up to the dot-com bubble (1995-1998 was especially prolific). The boom times were followed by a 13 year gap between 1500 and hitting 1600.

*Even with a consistent CAGR* round number benchmarks fall quickly and linear scales distort.

Take a look at how frequently new 100-point benchmarks fell the past few years due to the S&P 500’s fantastic returns. Another few years of even *modest* returns will yield a significant number of new highs.

This isn’t to say that this guarantees the rate of new records will *only* speed up. For example, reference the 13 year gap mentioned earlier. The stock market can wax and wane over long periods, but will quickly break round number price benchmarks when near a peak. (Let’s just hope that happens faster than every 13 years.)

Cheers,

Cameron Daniels

Clint says

I have to say, for long time horizons, I am a huge fan of the percent change bar chart, especially when you go back and add in dividend reinvestment.