Instead, we think of them as having fat tails (i.e. But if you look at the distribution of stock market returns over different time frames then you will find that returns aren’t even monomodal, i.e. So we can pretty confidently state (no hypothesis test needed!) So the 2 outlier dots represent a mere 0.237% of our observations. Finally, if one expands the time horizon to 10 years, the distribution of returns becomes trimodal, i.e. A -6.02 sigma (or worse) event occurs with 8.87*10^-8% frequency. We know that the current bull market is already the longest bull market in history so it is only reasonable to assume that it will end sometime in the next decade. Not great, but at least better than inflation. And the value on the Y-axis (Sample Quantiles, also in Z-scores) tells us how frequently we actually see it. Therefore we don’t have enough observations to be confident that our estimates of mean, standard deviation, etc. I wrote previously about how the finance industry models the risk of an investment. What we need is a distribution that is taller at the mean and that has fatter tails. With the normal distribution out of the way, let us find a distribution that better resembles the actual shape of equity returns. So we can use -20.4% to calculate our Z-score (since 2 out of the 842 observations are -20.4% or worse) along with the mean and standard deviation of the S&P 500’s monthly returns: Wow, a -20% monthly return is a 6 sigma event (6 standard deviations below the mean)! Now let’s calculate the Z-score of our actual data. Take a look, # Multiply by 2 to account for probabilities in right tail also, prob_left = norm.cdf(theoretical_z_score), Z-score = (observed - mean)/standard_deviation. Let’s first look at the annual returns of the S&P 500 index. Another way to check for normality is with a QQ plot (I also wrote a blog detailing how QQ plots work). It’s trying to tell us: It’s saying that we are observing 6 sigma events (massively improbably events) in our data at a much higher than expected frequency (approximately 3 sigma frequency). The -2.82 is a theoretical Z-score, a.k.a. So we expect it to happen once every 422 months, or once every 35 years. The X-axis location of the peak of the bell curve is the expected return and the width of the bell curve proxies its risk: But do risk estimates made with these assumptions actually make sense? Stock returns are roughly normal after all and a lot of the benefits of investment theory such as diversification hold true even in a world of less than normal stock returns and fat tails (perhaps even more so). the value below which we expect 0.237% of our observations to lie on a normal distribution. It’s very common in the investments industry to model the potential range of an investment’s future returns with a normal distribution. Why? Then there is a second peak, which corresponds to 10-year periods when investors experience both a secular bear and a secular bull market (or parts thereof). As you can see, on an annual scale, market returns are essentially random and follow the normal distribution relatively well. ... Asset returns … Investors who live through such a secular bear market have little to show for their investments at the end of the decade with a typical cumulative return in the single digits after ten years. Both of those represent S&P 500 returns of worse than -20%. The skinny middle and the fat tails imply that the normal distribution might not be the best describer of stock returns. The data is from Prof. Robert Shiller’s homepage.


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