they don’t just have one peak in the middle of the distribution as predicted by the normal distribution. The correct distribution will tell you this. With the normal distribution out of the way, let us find a distribution that better resembles the actual shape of equity returns. And we observed 2 returns worse than -20%! 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. Transformers in Computer Vision: Farewell Convolutions! 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. Instead, it is easy to identify different market regimes in the return distribution. The second peak corresponds to bull market environments where markets rise uninterrupted for three years in a row. A -2.82 sigma (or worse) event occurs with 0.237% frequency. 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. The Z-score we just calculated is the X-axis position of the second-worst return on the QQ plot. 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 data is from Prof. Robert Shiller’s homepage. For option traders, the Black-Scholes option pricing model assumes lognormal asset price distributions. The 2 outlier dots represent disastrous monthly returns of -20.4% (2008 Financial Crisis) and -22.5% (this past month). The skinny middle and the fat tails imply that the normal distribution might not be the best describer of stock returns. 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. A More Accurate Probability Distribution of Stock Market Returns. More evidence of that is how the actual distribution of monthly S&P 500 returns is skinnier in its center than the normal distribution. I wrote previously about how the finance industry models the risk of an investment. the investment’s expected return) and the standard deviation (a.k.a. This site uses cookies. So we should acknowledge the possibility that the inferences we make (using the market data that we do have) will sometimes be woefully incorrect. I wrote previously about how the finance industry models the risk of an investment. In terms of years, if stock returns were truly normal, then we would expect a 6 sigma event like this one to occur once every 93,884,861 years. Instead, we think of them as having fat tails (i.e. The distribution of stock returns is important for a variety of trading problems. 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. However, we need to be aware that while the last decade was a run of the mill secular bull market, this also means that the next decade will likely be a mixed period where we experience the tail end of the current secular bull market and the front end of the next secular bear market. The probability distribution for the stock price is different from the distribution of returns in important ways. 6.91% 7.25% 8.13% 8.85% 7.79% 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. And the value on the Y-axis (Sample Quantiles, also in Z-scores) tells us how frequently we actually see it. Let’s first look at the annual returns of the S&P 500 index. And to describe an investment, we only need 2 values: the mean (a.k.a. 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). 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. 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 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)! that NO stock returns are not normal. If we want to be thorough, we should also record the investment’s correlation with our overall portfolio. The data is from Prof. Robert Shiller’s homepage. Object Oriented Programming Explained Simply for Data Scientists, Top 11 Github Repositories to Learn Python. 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. Because there has not been a single secular bull market in history that has lasted for two full decades. Finally, if one expands the time horizon to 10 years, the distribution of returns becomes trimodal, i.e. I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021. Make learning your daily ritual. As you can see, on an annual scale, market returns are essentially random and follow the normal distribution relatively well. standard deviations away from the mean, which implies a probability). For example, the return of a portfolio consisting of many investments (each with normally distributed returns) is also normally distributed. The scientific portion of risk management requires an estimate of the probability of more extreme price changes. 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. 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. For your security, we need to re-authenticate you. I have plotted the price returns of the S&P 500 since 1871 together with the expected normal distribution of returns. it starts to have three peaks. Let’s first look at the annual returns of the S&P 500 index. Do we scrap all our models and try to start again from scratch? To find out more, read our, This site requires JavaScript to run correctly. A -6.02 sigma (or worse) event occurs with 8.87*10^-8% frequency. The fat tails mean that extreme events occur more frequently in reality than what a normal distribution would predict. Any time we can model something with normal distributions, it makes life a lot easier. the investment’s risk). (5.7) Since the return is a normally distributed random variable, the … 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? As we can see, the last decade has been a typical secular bull market and not out of the ordinary at all.