If we want to be thorough, we should also record the investment’s correlation with our overall portfolio. Both of those represent S&P 500 returns of worse than -20%. The -2.82 is a theoretical Z-score, a.k.a. Why? At the end of the 10-year period, cumulative market returns are somewhere in the 60% to 80% range or about 5% to 6% per year. The 2 outlier dots represent disastrous monthly returns of -20.4% (2008 Financial Crisis) and -22.5% (this past month). Perhaps the finance industry can borrow a page or 2 from them. While most of the observations do fall more or less on the red line, we can see significant deviations on the left tail and smaller ones on the right tail. And then there is the third peak on the right side of the chart below, which corresponds to a secular bull market where markets rise practically without any interruption from nasty bear markets for a decade. As we can see, the last three years have delivered returns that are essentially in line with what can expect in a bull market environment. That’s 2 six sigma events (once in 90 million year type events) in a dataset that is only 70 years long. 18 You have been given this probability distribution for the holding-period return for KMP stock: Stock of the Economy Probability HPR Boom 0.30 Normal growth 0.50 12 % Recession 0.20 - 5 What is the expected standard deviation for KMP stock? But when we stress test our portfolios (as well as our own mental expectations of what the future might hold), we should definitely be cognizant of the supposed 4, 5, and 6 sigma events that actually seem to occur once every business cycle. extreme events happen more frequently than expected). 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). Make learning your daily ritual. Finally, if one expands the time horizon to 10 years, the distribution of returns becomes trimodal, i.e. 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)! Another way to check for normality is with a QQ plot (I also wrote a blog detailing how QQ plots work). 6.91% 7.25% 8.13% 8.85% 7.79% 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. Any time we can model something with normal distributions, it makes life a lot easier. And we observed 2 returns worse than -20%! This site uses cookies. The fat tails mean that extreme events occur more frequently in reality than what a normal distribution would predict. Click the link we sent to , or click here to log in. By using one of the common stock probability distribution methods of statistical calculations, an investor and analyst may determine the likelihood of profits from a holding. For your security, we need to re-authenticate you. So the 2 outlier dots represent a mere 0.237% of our observations. Previous Posts Referenced In This Article: Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. ... Asset returns … The first peak corresponds to decades that are in a secular bear market like the 1970s or the first decade of the 21stcentury. And does the assumption of normality understate, properly state, or overstate the frequency of market disasters (like what we experienced over the past few weeks)? Therefore we don’t have enough observations to be confident that our estimates of mean, standard deviation, etc. One issue is that financial markets have just not been around long enough. Larger returns (say, 3+ standard deviations away from the mean of approximately 0) were predicted with very low frequencies, while the returns closer to 0 were a good fit to the model. Instead, it is easy to identify different market regimes in the return distribution. As we can see, the last decade has been a typical secular bull market and not out of the ordinary at all. 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). the investment’s expected return) and the standard deviation (a.k.a. I wrote previously about how the finance industry models the risk of an investment. For some years, returns are abysmal, for others they are great. 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. Please. Another way to check for normality is with a QQ plot (I also wrote a blog detailing how QQ plots work). We all know that stock market returns are not normally distributed. The second peak corresponds to bull market environments where markets rise uninterrupted for three years in a row. The value on the X-axis (Theoretical Quantiles) tells us how frequently we expect to see an observation of that magnitude on a normal distribution (they are Z-scores, a.k.a. The data is from Prof. Robert Shiller’s homepage. I want to look at monthly returns so let’s translate these to monthly: Let’s overlay the actual returns on top of a theoretical normal distribution with a mean of 0.66% and a standard deviation of 3.5%: It looks approximately normal but if we look to the left of the distribution, we can see the famous fat tails. The distribution of stock returns is important for a variety of trading problems. Instead, we think of them as having fat tails (i.e. The data is from Prof. Robert Shiller’s homepage. That means we would expect it to happen once every 1/0.00237 months. I noticed a similar distribution for stock returns and similar results when fitting a gaussian distribution. As you can see, on an annual scale, market returns are essentially random and follow the normal distribution relatively well. Rather, there seem to be 2 regimes — a calm regime where we spend most of the time that is normally distributed (but with a lower volatility than 12%) and a regime with high volatility and terrible returns. it starts to have three peaks. 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? So we should acknowledge the possibility that the inferences we make (using the market data that we do have) will sometimes be woefully incorrect.

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