A continuing theme in this blog is how the human brain is constantly evaluating probability and risk, and “acting” accordingly in controlling our bodies. Yet, most humans have a hard time visualizing what “risk” actually looks like. Even if we have a handle on the mathematics of probability, we can still have a hard time “seeing” it.
Below is the best example I have seen that puts investment risk data into an accessible picture. The scenario is, “What would happen if we plotted every possible outcome of returns from investing in the United States stock market in recent years?” First, we need to set some boundary qualifiers:
- We will invest in a mutual fund matching the Standard and Poors 500 index. This fund will consist of the top 500 available U.S. stocks, weighted by their relative total capitalization (the market value of total stockholders’ equity).
- We will buy shares every trading day and hold them, never selling, and reinvesting all dividends received in the same manner. This is a classic “buy-and-hold, income averaging” strategy. It minimizes the various risks of timing and lack of diversification, leaving only “non-diversifiable risk.”
- We will adjust the returns for inflation in order to take that factor out of the picture.
The green “cloud” in the graph below is actually thousands of lines tracking each investment’s total value if held for different lengths of time, from one day all the way up to 40 years. Think of the entire cloud as our total portfolio, balancing off the “winner” purchase days with the “loser” purchase days.
The line through the middle of the cloud is the regression, the “best fit” average if you owned all of these thousands of investments. Note that the line is straight only because the left axis is logarithmic, with each line representing a doubling (and halving, going down below 1) of the total cash we would have through both the increase of stock value and the reinvested dividends. Money, like many natural phenomena, tends to grow at a logarithmic rate. 
On this scale we start out with our initial investment, set to “1”, and hold it for a period of time up to 40 years. Note that the black regression line hits “2” at about 10 years, indicating how long it would take the average of our investments to double. Our money is just short of doubling again (to “4”) by year 20, and almost doubling again (to “8”) in year 30. So on average, for every $1,000 invested at time zero, we would have, by this investment strategy, nearly $8,000 thirty years later.
This doubling in about ten years is the growth we could expect from an investment with about a 7% annual compounded growth rate (in other words, we earning “interest on the interest” every year). In short, if we could really buy all of these investments every day and hold them, we would most likely see our money grow at about a 7% annual compounded rate.
But remember that the line down the middle is the average of our many investments. The green “cloud” is showing where some of our daily investments will wind up. Some track much higher than our 7% average line. In fact, on some days, our initial investment would have grown to eight times its original size in less than 10 years, which is an impressive annual return of over 20%.
On the downside, however, some of our daily investments will lose value, even possibly losing three quarters of our money in the first three years! Note that the left axis continues down from “1” to “0.5” (half) and “0.25”, (one-quarter of our original investment). Even if held for 40 years, some of those daily investments, invested on the “wrong day,” would grow at most four times, a long-term return of less than 4%. The problem here is that nobody (and I mean nobody) can reliably predict the “bad days.”
So what is this “7% investment risk” graph telling us? First, it is saying that, if we buy-and-hold the major U.S. stock market average and reinvest dividends for long periods of time, we can pretty reliably count on that 7% return.  But we have to “always be buying” (or at least investing every month) in order to build a reliable average.
What would this graph look like if we diversified our purchases with, say, a bond market index fund? First, you can envision the slope of the black line would flatten out. If our average return dropped to 3%, for instance, we would not expect to see a doubling of our investment until year 24 (72 divided by 3).  However, the “green cloud” range between the best case and worst case scenarios would narrow, indicating a portfolio with lower risk.
On the other hand, if we diversified our portfolio with investments more risky than the S&P 500 (say, the Russell 2000 index of smaller corporations), we should expect the slope of the line to steepen, and the “green cloud” would also get wider in both directions, indicating more risk.
And finally, what does this graph look like if we replaced the logarithmic scale on the left axis with a straight-line liner scale? It would look something like this, with the green line showing the approximate best case and the red line showing the approximate worst case for each day’s investment:
And in both cases, the graph is also telling us that, no matter how good an investment guru we think we are, we need to be prepared to lose money in the short term.
- To learn more about natural logarithmic growth, see this earlier two-part post.
- The caveat, as they normally say quickly in the advertisements, is that “past performance is no guarantee of future returns.”
- For more on the “Rule of 72,” see this earlier post.