# The Eight Wonder is an Anchor (Part 1)

Geometric progressions embody one of investors’ most ardent love-hate relationships, and like any self-respecting star-crossed lover, are wildly misunderstood.

On the one hand, they are the magic behind compound interest (the eighth wonder of the world, perhaps apocryphally, according to Einstein), but on the other, as Warren Buffet put it discussing Berkshire Hathaway’s returns:

You can be certain that this percentage will diminish in the future. Geometric progressions eventually forge their own anchors.

The human mind is generally terrible at understanding exponential growth, which leads to much confusion. For example, the real average annual compounded rate of return for an investment in the S&P500 between 1900 and June 2016 is approximately 6.5%, while that in constant 10-year treasuries about 2.5%. Most people will look at those numbers and think that the terminal value would differ around 10 times. The mental math would generally go something like this. The difference between the rates is 4%, and 4% times the number of years (116) equals 5.2 and then multiply that by some arbitrary number, let’s say 2, to account for some compounding, and you get a ratio of roughly 10 times. In reality, the ratio is about 100 times, so we were off by an order of magnitude (tsy = \$10, SP=\$1000).

Once we consider this example it is clear that a better mental model is necessary for considering long term growth rates. One useful trick, is that the time at which a doubling happens for a certain geometric rate is approximately 70 divided by the rate. Knowing this let’s take another crack at the previous example.

An investment growing at 2.5% per year (treasuries) will double in value every 70/2.5 = 28 Years. While one growing at 6.5% (S&P500) every 11 years (10.7). That means that treasuries will double about 4 times over 116 years (116/28) while equities 11 times (116/11). The equation we now have to solve is 2¹¹/2⁴, if we subtract the exponents we get a ratio of 2⁷ or 128 times, significantly closer to the correct number. Although this takes a little bit more work than our previous naive assumption, it can give us a much better understanding of the magnitudes involved.

Another issue for investors to consider is whether compounded rates of return represent the best heuristic for analyzing long term growth rates. As we just saw small differences in returns compound to giant differences in terminal wealth which is what investors are generally interested in.

For example, over a 30-year horizon using the previous long term averages a \$1 investment in the S&P and treasuries, would generate \$6.5 and \$2 dollars of terminal wealth respectively. Perhaps a better way of analyzing these returns is to think about them in terms of the average yearly increase relative to the original investment (Terminal Value/Initial Value/Years). In this case, we’d see that the S&P increased on average by 22% of the initial dollar every year, while treasuries increased at 7%. This format makes it easier to see the tripling of equities relative to treasuries over this time period.

As we have seen, it is unwise to examine compounded rates of return without taking into account the target investment period. This calls into question the validity of common risk adjusted return measures, such as the Sharpe Ratio, over differing investment horizons. For example Lin and Chou 2003 find that for shorter time periods, large cap equity portfolios dominate small cap, but over longer time periods small cap provides higher risk adjusted returns. Therefore, investors should calculate Sharpe Ratios with the periodicity of returns equal to their investment horizon.

In Part 1 we’ve seen the benefits of compounding (the Wonder) and a better way for investors to understand it. In Part 2 we’ll examine the drawbacks of compounding (the Anchor) and what investors can do about it.

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