## The shackle of time 1

##### Jan 16, 2009

I ran across this hugely successful chart on Dean Foster's home page (and noted that he and his Wharton colleagues have a nice blog picking apart statistical errors committed in public.)

This is a histogram plotting the historical year-on-year returns of the S&P 500 index, binned into 10%-levels. It succeeds on two levels: the innovation of printing the years inside little blocks provides extra information without distracting the overall picture; the key message of this plot, that the negative return of 2008 is a negative outlier in the history of returns, is extremely clear.

This, in my mind, is a superior presentation than the usual time-series line chart that we see in every economics publication. For some purposes, it is better to unshackle ourselves from the linear time dimension, and this is a good example.

One question/comment: within each 10% level, the years are arranged in reverse chronological order fro top to bottom. This facilitates searching for a particular year. The obvious alternative is to order by the actual level of return, so that the result is akin to a stem-and-leaf plot.

While I like the graphical aspect of the chart, I feel like it has limited function. This graph appears useful to anyone who has a one-year investment horizon. If I want to predict what next year's S&P 500 return is, I might take a random sample from this distribution. However, as a lazy investor, I never look at a one-year horizon so this creates two problems: if I am looking five years out, I can't take five samples from this distribution because there is serial correlation in this data for sure; even if I could take those five samples, it is difficult to compute the five-year return in my head.

So what I did was to take the data and replicate this histogram for 2-year, 3-year, 5-year, 10-year, etc. returns. The results are as follows. I decided to simplify further and use Tukey's boxplot instead of the histogram. The data are real compounded total returns from S&P 500 from 1910-2008.

The boxplot on the top right shows that there is about a 25% chance that an investment in the S&P 500 will return negative in real terms in any three-year period (below the green line). At the other end, there is a 25% chance of getting earning more than 50% on the principal during those three years.

The next set of boxplots compared 5-year returns to 10-year returns and 10-year returns to 20-year returns. If we have a 10-year horizon, there is still positive chance of reaching the end of the decade and finding the investment under water! The median 10-year return is approximately doubling the principal (about 8% per annum compounded).

In a twenty-year period, there is hardly any chance of not making money on the S&P. There were two positive outliers of over 1000% (about 13% per annum compounded over 20 years).

Reference: Data from Global Financial Data