It’s formula is: A Z-score of -2.82 means the observed value was -2.82 standard deviations below the mean (the further it is from the mean in either direction, the less probable the observation). For your security, we need to re-authenticate you. If we want to be thorough, we should also record the investment’s correlation with our overall portfolio. 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)! 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. Both of those represent S&P 500 returns of worse than -20%. Now let’s calculate the Z-score of our actual data. Let’s check out the QQ plot for monthly S&P 500 returns: Deviations from the red 45 degree line represent differences from the normal distribution. We can confirm this via the cumulative density function (CDF method), which tells us, for a given distribution, the sum of the probabilities that lie to the left of the Z-score: The Z-score is a metric that connects magnitudes with probabilities. Yeah, that number doesn’t make sense to me either so let’s rephrase it. Please. ... Asset returns … For some years, returns are abysmal, for others they are great. I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021. If we look at rolling 3-year returns, we can see that the distribution of market returns become bimodal. Why? Another way to check for normality is with a QQ plot (I also wrote a blog detailing how QQ plots work). they don’t just have one peak in the middle of the distribution as predicted by the normal distribution. Because there has not been a single secular bull market in history that has lasted for two full decades. 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. I have plotted the price returns of the S&P 500 since 1871 together with the expected normal distribution of returns. Click the link we sent to , or click here to log in. 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. Put in this context, the year 2019 was one of the better years in the history of the S&P 500 but not an extreme year. 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 first peak corresponds to decades that are in a secular bear market like the 1970s or the first decade of the 21stcentury. While painful, the chaos in financial markets recently provides a good opportunity for us to question our assumptions. 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). are truly representative of the true distribution. Let’s first look at the annual returns of the S&P 500 index. So we expect it to happen once every 422 months, or once every 35 years. Previous Posts Referenced In This Article: Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. Object Oriented Programming Explained Simply for Data Scientists, Top 11 Github Repositories to Learn Python. I wrote previously about how the finance industry models the risk of an investment. 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. The distribution of stock returns is important for a variety of trading problems. the investment’s expected return) and the standard deviation (a.k.a. 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. What we need is a distribution that is taller at the mean and that has fatter tails. The 2 outlier dots represent disastrous monthly returns of -20.4% (2008 Financial Crisis) and -22.5% (this past month). 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. Perhaps the finance industry can borrow a page or 2 from them. 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. Another way to check for normality is with a QQ plot (I also wrote a blog detailing how QQ plots work). It reminds me a little of earthquake forecasting where scientists are trying to predict the magnitude and frequency of huge earthquakes (the likes of which might never have been recorded before) using a dataset dominated by quakes of small and medium magnitude. The fat tails mean that extreme events occur more frequently in reality than what a normal distribution would predict. The skinny middle and the fat tails imply that the normal distribution might not be the best describer of stock returns. 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? For example, the return of a portfolio consisting of many investments (each with normally distributed returns) is also normally distributed. 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’s trying to tell us: “Hey based on the mean and standard deviation of our data and most critically the assumption that our data is normally distributed, what we are observing here is super duper abnormal!”. A -6.02 sigma (or worse) event occurs with 8.87*10^-8% frequency. That’s 2 six sigma events (once in 90 million year type events) in a dataset that is only 70 years long. Finally, if one expands the time horizon to 10 years, the distribution of returns becomes trimodal, i.e. Do we scrap all our models and try to start again from scratch? 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). 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. 6.91% 7.25% 8.13% 8.85% 7.79% Therefore we don’t have enough observations to be confident that our estimates of mean, standard deviation, etc. Rewriting the relationship between the stock price and return shown in equation (5.2) we have, ln ST ln S0 RT. Instead, we think of them as having fat tails (i.e. Are stock returns actually normal? 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.

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