Exponential!

The King and the Chessboard

You’ve probably heard the old story of the king that asked the inventor of chess what he wanted as a reward for his new game (or in other variants the king received a beautiful chessboard as a gift, or lost a match of chess).

The answer was to get him some rice (or wheat) in the following quantity: a single grain for the first square, two grains for the second one, four for the third and so on, always doubling the quantity until the last square of the chessboard.

The king agreed, wondering why the inventor only asked for a few grains or rice, but quickly discovered that the requested quantity of rice was more than he could possibly produce.

It’s fun to try this on a real chessboard: It starts easy, but after just a few iterations there is enough rice that it’s no longer possible to pile it inside a square. The total amount for the entire chessboard would be enough to carpet with rice the entire surface of the United States, Alaska included.

The story illustrates the phenomenon of exponential growth, where at every step either the quantity or its increment is multiplied by a constant factor.

Compounding of investments follows a similar multiplicative law.

Logarithmic Charts

Exponential processes are hard to grasp because in a limited number of steps the quantities become so huge that any fixed ‘stick’ we can use for comparison becomes quickly inadequate.

We are more used to observing and evaluating ‘linear’ processes, where a quantity increases or decreases by comparable amounts at each step.

Luckily, we can use logarithmic charts to better visualize exponential processes. For the y axis, a logarithmic chart doesn’t plot the quantity we are studying, but its logarithm.

Example: Market Index

The following graph shows the ‘value’ of the stock market over a period of 100 years.

The market value is represented here by the S&P 500 index since it was created in 1957, and by other representative indexes before that (scaled and spliced together to appear as a single index; Data prepared and shared by Robert Schiller: http://www.econ.yale.edu/~shiller/data.htm).

Judging from this graph, it would be tempting (and wrong!) to conclude that there was little market activity before 1980, that most of the gain came in the last 10 years and that the only two dramatic events in market history were the crashes of 2000 and 2008.

We know that the crash of 1929 was bigger than the one in 2008, so why is it not showing in the graph? The problem lies with the exponential growth of the index, causing the left side of the graph to appear squashed.

During the recession of 2008, the S&P 500 dropped from 1565 to 677 in five months. People investing in the S&P 500 at the top and selling at the bottom would have lost 57% of their investment.

People investing in the Dow in 1929 at the top and selling in 1932 at the bottom would have lost a staggering 89% of their investment.

In the graph above, the normalized index drops from 31.3 in 1929 to 4.8 in 1932, a difference of 26.5. 26.5 is 85% of the initial value 31.3 (we get 85% rather than 89% because the graph uses data averaged over a month rather than the daily values).

However, those same 26.5 points are only 1.7% of the 1565 value reached in 2008, and 0.8% of the 3278 at the end of the chart, making the 1929 drop barely visible.

A logarithmic chart of the same period and market index balances the exponential nature of the index and gives equal importance to old and new data:

Now the crash of 1929 is clearly visible, the index appears to grow with increased regularity over the entire time span, and we can notice more details in the graph.

The horizontal lines mark the index values of 10, 100 and 1000.

Logarithmic charts have the convenient property that the same relative drop or gain of an investment is represented by the same vertical distance on paper regardless of the investment’s starting value.

A Synthetic Example

Imagine I invest $1000 in each of three different accounts:

• An account with a fixed 3% Annual Percentage Yield (APY).
• An account with a fixed 7% APY.
• An account with an APY that alternates between 4% and 6%.

The following standard (linear scale) chart shows how my three investments grow over 40 years:

The same investment growth shown as a logarithmic graph (notice how the amounts on the vertical axis double each time):

This is an artificial example, but it shows that investments growing with a constant APR/APY are represented by straight lines in logarithmic charts.

The slopes of the lines are proportional to the APYs, making it much easier to compare them and in this example to spot the times when the orange line switches between the 4% and 6% APY.

When to Avoid Logarithmic Charts

Logarithmic charts are well suited to represent investments, given the multiplicative nature of losses and gains.

However, we are better at interpreting normal charts that have a linear scale, so we should not use logarithmic charts unless it is strictly necessary.

Clearly, we want to avoid logarithmic charts if the quantity doesn’t increase or decrease exponentially. But even for exponential curves, we should only use them if the timespan is long enough for the exponential behavior to dominate. For sufficiently short time spans (for example when plotting the S&P 500 over an interval of 1 year instead of 100 years), the logarithmic and linear chart scales produce similar graphs and we can choose the simpler linear scale.