Python for Finance: Are stock returns normally distributed?

Today we investigate whether stock returns are normally distributed! First, I show the difference between simple returns and log returns, highlighting the main reason log returns are preferred for financial analysis.
Then we explore whether CBA (ASX listed stock, Commonwealth Bank of Australia) log returns are normally distributed. We do this by considering several visual and statistical techniques; Quantile-Quantile Plots, Box Plots, Histograms, and numerically by Hypothesis Testing with Kolmogorov Smirnov and Shapiro Wilk tests.
00:00 Intro
00:27 Why use log returns?
02:18 Simple returns
07:28 Log returns
10:44 Are log returns normally distributed?
16:08 Quantile-Quantile Plots
18:00 Box Plots
19:20 Kolmogorov Smirnov test
22:10 Shapiro Wilk tests
As a high-level programming language, Python is a great tool for financial data analysis, with quick implementation and well documented API data sources, statistical modules and other frameworks related to the financial industry. We will be using Jupyter Lab as an interactive web browser editor for this series due to ease of use and presenting code in a live notebook is ideal for this tutorial series.
This is the third video of many on the topic of Python for Finance. The series will include general techniques used for financial analysis and act as an introduction for more in-depth tutorials that we may explore later (such as time series modelling, building financial dashboards, machine learning ect.).
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[@weirdcuteai]

Hi, thanks for the video! I just want to add that when you using stats.kstest, by default it compares with standard normal distribution (std=1 and mean =0), however your stock returns has different std & mean, so you might want to add args in your function, then the stock return might fit normal distribution better for example: ks_stat, p_value = kstest(log_returns, 'norm', args = (mean, std))

[@vladislavpyatnitskiy]

Thank you for a great content! I would like to add that there is one more test for normality which is Jarque-Bera test that should be familiar to students who have done econometrics. Therefore, I might suppose that you have completed non-economic major. However, I think it is nice to know other normality tests as well!

[@sahilharidas4747]

Great video! Just a bit of nit-picking though: at 13:00 when you subtract the mean and divide by standard deviation, you say that will 'normalise' the best and worst case scenarios when they are instead being 'standardised'

[@MA-oj8zk]

For those running into issues plotting the diagrams, replace df.Close.plot().update_layout(autosize=False,width=500,height=300).show(renderer="colab") with df.Close.plot().update_layout(autosize=False,width=500,height=300).show() Same for log_returns.plot(kind='hist').update_layout(autosize=False,width=500,height=300).show(renderer="colab") and log_returns.plot(kind = 'box').update_layout(autosize=False,width=350,height=500).show(renderer="colab")

[@yeuyang5551]

Before doing ks /sw test, did I need to standardize the log return?

[@konturgestalter]

excellent as always!

[@joaopedrorocha5693]

Wow, great class! Thanks! Was thinking that the mean of the simple return was a good to have a first idea of returns over time but it can overshoot a lot! Will be able to estimate better now! And learned what the heck a ks test is about hehe Thanks a lot! Just a tip ... I think that you could have just sum 1 to simple_returns instead of doing a list comprehension, it's cleaner and more efficient (on cell 57 of the notebook)

[@DimitarDobrinov]

Great explanation! I have a question. If arithmetic average of simple returns does not yield a correct measure of mean and we need to use geometric mean instead, what do we do with standard deviation? How will we compute standard deviation in a manner that deals with the fact that simple returns are not additive?

[@sdsagdfd117]

Perfect

[@kaiwang2924]

Simple and beautiful.

[@AkashChauhan-rw1qh]

Thanks for sharing with us........

[@joshuaGmartin2023]

Great stuff

[@ncheymbamalu4013]

Stock returns and log returns can be modeled more accurately with the t-distribution.

[@sahilharidas4747]

More generally, the t-distribution is better suited when the underlying population variance is unknown, as is usually the case in practice. However, the t-distribution becomes the standard normal distribution as the number of degrees of freedom approaches infinity.

[@FRANKWHITE1996]

Thanks 🙌

[@MA-oj8zk]

For those running into issues with the yahoofinance pandareader, here's the fix: import yfinance as yf yf.pdr_override()

[@omololaomotalade8105]

Hello, I got the same figure when adding log or simple return to the beginning price to get the ending price. My code: For log_return: data.Price[0] * np.exp(data['log_return'].mean() * len('log_return')) This code did not give me the price at the end of the timeseries For simple_return : data.Price[0] * (1 + data['simple_return'].mean()) ** len('simple_return') . Is there something I am doing wrong? Kindly respond.

[@jure4835]

Doesn't this basically show that log returns are normally distributed and therefore the stock returns have a lognormal distribution?

[@QuantPy]

Log returns could be normally distributed, depends on your data. Yes modelling stock returns with a lognormal distribution is a strong assumption, but one that is made often

[@modeocheng2190]

fantastic!

[@andrewbenson8842]

So what's the solution to this? Most all of theoretical finance assumes a normal distribution (Markowitz theory, CAPM, Black-Scholes). What do quants do in the real world when returns are not normally distributed?

[@QuantPy]

Excellent question, obviously there is no one answer that solves all problems. However things for you to research, a way to model different distributions for various assets is through copulas. This could be incorporated through Monte Carlo simulations for either “real” world risk or portfolio optimisation or “risk-neutral” world derivative valuation

[@andrewbenson8842]

@@QuantPy do you have any materials or information you could point me to that talk about how to deal with non normal data in finance? I'm aware of copulas but am definitely not an expert and haven't see how they can help with non normal data. My understanding was that copulas are used for risk management. When you say use monte carlo is it just a matter of making random draws from an empirical distribution rather than the normal distribution and seeing how that affects an option price for example?

[@wilsongomes3360]

good job. tks

[@gian_piano]

awesome content thank you so much!

[@carlosarrieta1145]

in which IDLE are you runing your code?

[@QuantPy]

This is Jupyter Lab (notebooks) for python. If you’re using straight IDE, I like to use Visual Studio Code 👍

[@vladk9152]

In a lot of engineering projects we ignore >3sigma events. Is it the same for finance projections? Let's say i want to make an ARIMA projection, or a monte carlo simulation. Should i keep these rare events as useful data?

[@TheMLPRyan]

Yes keep the rare events, have a look at what happened to LTCM when they didn’t include large deviations from their models.

[@ooddy24]

I would appreciate if you would move your picture outside of the script...it is hard enough to see the blurred script because its small scale. And your photo makes it worst...

[@QuantPy]

Thanks Manit, all code on my website in the link in the description if you’ve missed anything 👍

[@shaaf9817]

Do some videos on crypto