Humans are captivated by stories, but largely oblivious to data. In addition, we really want certainty and conclusions when generally all that is available is uncertainty and probabilities.

For example, people frequently want a prediction of what the market will do this year, and I think there are two reasonable answers based on history:

- Most likely between a 29% loss and a 53% gain, but there is about a 1-in-20 chance it could be outside that range. (The average 12-month return from 1926-2018 for U.S. stocks was 12.05% with a standard deviation of 20.90%. 95% would be within 1.96 standard deviations so 12.05% +/- 40.96% is a range of -28.91% to +53.02%.)

- Most likely between a 20% loss and a 45% gain, but there is about a 1-in-20 chance it could be outside that range. (If you assume that the world is safer or different now so post-WWII numbers are a better estimate of the future, the average 12-month return from 1946-2018 for U.S. stocks was 12.21% with a standard deviation of 16.52%. 95% would be within 1.96 standard deviations so 12.21% +/- 32.37% is a range of -20.16% to +44.58%.

You could also argue that equity returns will be lower by some amount – maybe 1% lower because of lower inflation and another 2-3% lower from a lower ERP (Equity Risk Premium) going forward so the whole distribution is shifted down by that amount. If so you can adjust the ranges down by 3-4%. I also do think that starting post-WWII is too aggressive, but I can understand the logic of someone using it and I wouldn’t say they are wrong. I would point out though, if that is the correct distribution then 2008 was a huge outlier. If we use from 1926 it was fairly normal. (The worst 12-months in that debacle was March 2008 to February 2009, which had a 42.48% loss – a rare but reasonable 2.53 standard deviation event (1 in 175) if we use from 1926 to the month prior to that period, but an improbable 3.26 standard deviations (1 in 1795) if we start in 1946.) So, my *best* answer would be: “**Most likely between a 33% loss and a 50% gain, but there is about a 1-in-20 chance it could be outside that range.**”

Also, if you want to know the 100-year-flood number that would be 2.58 standard deviations. 12.05% minus a 3.5% adjustment for lower returns in the future is 8.55% minus 2.58*20.90% = -45.36%. (Of course, there is also a 1-in-100 chance of a positive 62.47%) Keep in mind, the worst-case scenario that has ever happened (in any area, not just market returns) was not the worst-case just prior to it happening. Think about that for a while.

I am anticipating some questions, here are the answers:

- You undoubtedly think those answers are wrong – you just
*really*don’t think the range is that high. I feel the same way,*but I know I’m wrong…*

- Clients must be profoundly unhappy with an answer like that. I know, but it is what it is. If I could improve on those figures I would be running a hedge fund engaged in market timing.

- I used the CRSP 1-10 figures, not the S&P 500 because the question is “what do you think the market will do?” not “what do you think the S&P 500 will do?” Most people think it is the same thing, and substantially they really are, the correlation is above 99%, the difference in geometric returns has been 25 basis points (advantage S&P500) and average annualized difference in standard deviation was 34 basis points (advantage CRSP 1-10). So, I wouldn’t really quibble if someone used the S&P500 to do these calculations, but I didn’t.

- I rounded off to a reasonable number of decimal places as I typed this up, but all the calculations used all the decimal points I had available – just in case you are following my math and find something slightly off.

- The correct returns to use for this exercise are arithmetic, not geometric. If you want to convert, the rough estimate (but it’s pretty good) is given by squaring the standard deviation (to get the variance), then subtracting half of that from the return. For example, I said, “The average 12-month return from 1926-2018 for U.S. stocks was 12.05% with a standard deviation of 20.90%.” 0.2090^2= 0.0437 That divided by 2 equals 0.0218. 12.05% minus 2.18% is 9.87% geometric return, which is the figure you are more accustomed to seeing. For more on this topic you can see my calculator here.

- I used 12-month periods, the maximum drawdown to expect is higher because it can go on for longer than 12-months. For example, from October 2007 through February 2009 was a 50.19% decline, but 2008 was just 36.71%, and as mentioned above, March 2008 to February 2009 had a 42.48% loss.

- I used a normal distribution rather than a log-normal one because for a one-year period they are trivially different. There was already more than enough math here to make most people’s heads hurt without introducing that complication.