Finally, if one expands the time horizon to 10 years, the distribution of returns becomes trimodal, i.e. Please. I wrote previously about how the finance industry models the risk of an investment. Any time we can model something with normal distributions, it makes life a lot easier. Because there has not been a single secular bull market in history that has lasted for two full decades. There is a first peak for cumulative 3-year returns of about 0% and a second peak for cumulative 3-year returns of about 30%. 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. We can also rethink specifically the way we estimate the frequency of disastrous market returns. 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. The distribution of stock returns is important for a variety of trading problems. The Z-score we just calculated is the X-axis position of the second-worst return on the QQ plot. It’s very common in the investments industry to model the potential range of an investment’s future returns with a normal distribution. More evidence of that is how the actual distribution of monthly S&P 500 returns is skinnier in its center than the normal distribution. Are stock returns actually normal? The 2 outlier dots represent disastrous monthly returns of -20.4% (2008 Financial Crisis) and -22.5% (this past month). 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. Perhaps the finance industry can borrow a page or 2 from them. For your security, we need to re-authenticate you. 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? 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!”. 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. The scientific portion of risk management requires an estimate of the probability of more extreme price changes. Therefore we don’t have enough observations to be confident that our estimates of mean, standard deviation, etc. 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? 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. If we want to be thorough, we should also record the investment’s correlation with our overall portfolio. 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. 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). I wrote previously about how the finance industry models the risk of an investment. Let’s first look at the annual returns of the S&P 500 index. So the 2 outlier dots represent a mere 0.237% of our observations. We all know that stock market returns are not normally distributed. In the dataset, there are 843 monthly observations in total. For option traders, the Black-Scholes option pricing model assumes lognormal asset price distributions. Previous Posts Referenced In This Article: Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. 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. I have plotted the price returns of the S&P 500 since 1871 together with the expected normal distribution of returns. (5.7) Since the return is a normally distributed random variable, the … 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)? Let’s first look at the annual returns of the S&P 500 index. The -2.82 is a theoretical Z-score, a.k.a. Moreover, assuming that they are causing us to understate the likelihood of disastrous market returns. I don’t think we need to go all the way there. 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. 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). While painful, the chaos in financial markets recently provides a good opportunity for us to question our assumptions. 6.91% 7.25% 8.13% 8.85% 7.79% 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. With the normal distribution out of the way, let us find a distribution that better resembles the actual shape of equity returns. I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021. are truly representative of the true 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. This site uses cookies. Instead, it is easy to identify different market regimes in the return distribution. Not great, but at least better than inflation. I have plotted the price returns of the S&P 500 since 1871 together with the expected normal distribution of returns. And we observed 2 returns worse than -20%! Rewriting the relationship between the stock price and return shown in equation (5.2) we have, ln ST ln S0 RT. they don’t just have one peak in the middle of the distribution as predicted by the normal distribution. A More Accurate Probability Distribution of Stock Market Returns. 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). As we can see, the last decade has been a typical secular bull market and not out of the ordinary at all. Make learning your daily ritual. Yeah, that number doesn’t make sense to me either so let’s rephrase it. it starts to have three peaks. the investment’s expected return) and the standard deviation (a.k.a. The skinny middle and the fat tails imply that the normal distribution might not be the best describer of stock returns. The first peak corresponds to bear market environments when 3-year market returns become essentially negative. 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.