Modelling stock returns - the Cauchy distribution (Excel)

[@NEDLeducation]

You can find the spreadsheets for this video and some additional materials here: drive.google.com/drive/folders/1sP40IW0p0w5IETCgo464uhDFfdyR6rh7 Please consider supporting NEDL on Patreon: www.patreon.com/NEDLeducation

[@tatianapashkova7275]

Thanks a lot for the nice explanation! I was searching for a practical example with a Cauchy distribution, and found this super helpful.

[@corradoforza]

Thank you! Very clear and useful 💯💯💯

[@Bhavan71]

I believe you can estimate the mu with Best Linear Unbiased Estimators (BLUE) that order the data and usually ignore the smallest and largest data and gives more weight to the central data points. There is some formula to determine the weight of each data point, but I cannot remember. I remember this project at uni but it was 15 years ago or so. It ever really occurred to use the median but then that would be too easy..haha oh well. I think there is value in simplicity.

[@NEDLeducation]

Hi Bhavan, and thanks for the excellent question! There are approaches for Cauchy location parameter estimation with trimmed means, but explaining the trimmed mean and optimal trimming proportion would have been too much for this video :)

[@zaig7401]

Nice content. But I think maybe Cauchy distribution tail is a bit too fat. Can we pick a cut off point for stock returns to analysts tails in specific and make a log log plot and measure the slope of best fit to get the tail exponent? When the exponent is higher than two, we get reject some stable distribution like levy distribution. Is the right way to do it for analyzing tail or is there a better way to do it?

[@NEDLeducation]

Hi again, Dingxin, really appreciate your interest in the concepts we cover in our videos! :) Yes, you can by all means do the distribution fitting for the tail only. Alternatively, if you are concerned with goodness of fit at the tail, you can just apply Anderson-Darling test that has been developed pretty much to address such concerns. Levy distribution is not that easily applicable to stock returns though as it has a lower bound.

[@ivanklful]

Savva, another awesome video, that is stimulating me to think! So, Cauchy distribution is betting on much higher probabilities of extreme events (higher tails) and therefore we can model some alternative investments like cryptocurrencies, right? So FOREX is an example of alternative investments, and contrary of what people believe FOREX has less volatility than SP500, and such extreme events that predicted by Cauchy distribution are even less likely in FOREX trading than in SP500, so cryptocurrencies are more likely to apply to Cauchy distribution model and also Small Cap stocks and some commodities. So in extension to this video I would suggest you to analyze some Small Cap index (Russel 2000 the most known at least for me), Gold price, cryptocurrencies, and FOREX. So some time in the future consider to make videos about these investment instruments also. So your videos are pushing further my mind's boundaries, and also stimulate my imagination, so thanks for that a lot!😀

[@NEDLeducation]

Hi Ivan, and thanks so much for such kind words! These distributions are applicable to any assets that have sufficiently liquid trading to record enough return observations, and you are correct some of these functions have more applicability for some assets rather than others. All the best with your modelling and intellectual endeavours :)

[@hugolai8711]

Thanks for the video. I tried to transform the probability of an index ( Hong Kong Heng Sent Index, HSI) reaching certain price using normal distribution in Excel. Before watching your video, I try to transform the probability in normal distribution, say 85%, to Cauchy distribution. I assumed u is zero and find the x. Using this x and an arbitrary gamma, I find the 85% corresponding Cauchy distribution probability which is 70.62 % &69.27% with gamma in 1.7 and 2.0 respectively. After watching your video, I learnt I can use q (0.75-0.25)/2 as gamma. My question is:. 1. Can I assume the x for 50% in normal distribution is the same as that in Cauchy distribution? 2. Why median is used as u in your video? 3. How can I "verify " if my adopted gamma fit a particular stock apart from plotting all point on the cumulative graph?

[@NEDLeducation]

Hi Hugo, and glad you liked the video! As for your question, you can estimate the distribution parameters, including gamma, using maximum likelihood, I have got a video on that kzbin.info/www/bejne/rYKVgmx4hZ2Lf7M. To test how well a Cauchy distribution with particular location and scale parameters fits your data, you can use any goodness-of-fit test to your liking, for example, Kolmogorov-Smirnov kzbin.info/www/bejne/iF6wc3l5gJijb7M, Anderson-Darling kzbin.info/www/bejne/aHXLd4Vsf6mCY80, or Cramer-von Mises kzbin.info/www/bejne/pnTdaYqigKh9p5o (the examples in the videos use the normal distribution but the logic is the same for any distribution). Hope it helps!

[@Bhavan71]

I was wondering if you have even use normal and cauchy together. The normal distribution could simulate the market regime with low vol and average positive returns and the cauchy could simulate the more extreme moves? I have always been keen to try and identify market regimes in the data and fit their distributions well. Markov chains baby

[@NEDLeducation]

Hi Bhavan, and thanks for the excellent question! Normal-Cauchy mixtures are actually a thing, they are sometimes use in value-at-risk calculations (but not very often) - here is an example: www.sciencedirect.com/science/article/abs/pii/S0275531922000228. I use it for distribution fitting in one of my PhD subchapters as well (page 14).

[@Bhavan71]

@@NEDLeducation Thank you for the response. Fascinating. Having worked in risk departments in banks I am not aware of them doing this. However, I may not have been high up enough in rank to know. As far as I can tell they used normal distributions mostly. Thanks again.

[@aniketmathur792]

Is there a possibility to forecast such fatter tails. I mean the observations are historic in nature and if we were to assess the future extreme events, can we do it? like for 2022 and following 4-5 years ?

[@NEDLeducation]

Hi Aniket, and thanks for the question! Cauchy tails are obviously too fat for stock market indices, but they can be helpful in modelling more risky return distributions. If a Cauchy distribution is demonstrating a good fit in terms of various tests (like Kolmogorov-Smirnov) on historical data, you can be reasonably assured such risk patterns will continue into the future. Hope it heps!

[@michaelzumpano7318]

Let’s say a CDF of an exponential distribution has too thin a tail for the empirical data and the Cauchy CDF has too fat a tail - can you use a distribution ratio or difference to fit the data? Would the resulting coefficient be of value to track changes in the market over time? Could you create many categories for types of events (fed rate change, terrorist attack, etc) and come up with log coefficients that add together to create the distribution ratio between the two distributions?

[@NEDLeducation]

Hi Michael, and thanks for the excellent question! Mixture distributions are a thing indeed - most commonly you go for a normal-Cauchy mixture which has applications in risk management (for VaR calculations for example). However, if exponential distributions are too thin and Cauchy distributions are too thick, it might also be a sign that a generalised distribution family such as the error distribution or the Johnson SU distribution is appropriate. I have got videos on these here kzbin.info/www/bejne/ZmfGfYesZ85kmac and here kzbin.info/www/bejne/iqaqoJaGmcipgNU. I investigated quite a bit of theory and mathematics on distribution fitting for asset return modelling in one of my PhD chapters (albeit for cryptocurrencies, this is where the exponential-too-thin-Cauchy-too-thick problem is most apparent), and here is the chapter if you would like to have a read: papers.ssrn.com/sol3/papers.cfm?abstract_id=3847351

[@RustuYucel]

Any tut on Extreme Value Theory on excel? Wonderful cnntents by the way