Portfolio Optimisation with Higher Moments (Excel)
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- Опубликовано: 10 июл 2024
- How to optimise the portfolio when the returns are not normally distributed? In this video, we discuss an elegant extension to Markowitz portfolio theory and the constant absolute risk aversion (CARA) utility function, applying it to real-world data.
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NEDL have you thought of incorporating optimization using the Omega function (Score) - Omega by Keating and Shedwick (2002)
Ughh... I've been reading like 15 papers on this issue... and google finally let me find you.. thank you so much, you are a life saver!
Thank you for a great explanation of higher moments and portfolio optimization. Your channel is one of a kind.
Thank you! That's excellent. I believe this is the only video available on RUclips on the topic.
Your videos are amazing
thank you so muchhh
Thank you very much for all of those great videos!
Where do you have that formula from? Sometimes you mention a source next to the formula but this time there isn't. I'd love to quote it at a paper.
Hi Savva, thank you very much for the video! I noticed that you usually don't use the log returns in your videos, is that a reason behind it? Kind Regards, Hugo
Hi Ronald and thanks for your videos; can you explain me what is the origin of the "arbitrary distribution" formula and the usual value range of theta ? at the end... isn't it true that in a normal distribution Kurtosis K=0?
Hi Luca, that's Savva and it's me who has been recording the tutorials :)
Great questions!
The "arbitrary distribution" is one of the limitations for this portfolio optimisation technique with CARA as skewness and kurtosis alone as higher moments can theoretically take any values in this setting. To uniquely define a non-normal distribution function one actually needs infinitely many moments. Therefore, understanding what the theoretical distribution is that matches the data best is most of the time a very useful exercise to do.
The theoretically plausible range of theta is (0: +infinity), with investors tending to be closer to risk-neutral when theta is closer to zero and tending to be closer to absolutely risk averse when it tends to positive infinity. However, in practice theta=2 is generally used for "adventurous" or "aggressive" investors, theta=3.5 is used for "balanced" or "moderate" investors, and theta=5 is assigned to "cautious" investors.
The excess kurtosis of the normal distribution is indeed zero. A distinction between "kurtosis" and "excess kurtosis" is important, as excess kurtosis is equal to kurtosis minus 3. The kurtosis of the normal distribution is equal to 3, and the excess kurtosis of it is zero by definition. Subtracting 3 has the sole purpose of making it easier to compare the fatness of tails for a distribution to what it would have been if the distribution was normal.
Hope it helped and stay tuned for future videos!
Hi Savva,
Thanks again for this great tutorial.
I have a question. I estimated the parameters (ksi, lambda, delta, gamma) of the Johnson SU distribution using the "Percentile Matching Method." Then I used the estimated parameters to calculate the mu, sigma, skewness, and kurtosis. Then use your utility formula to optimize the asset allocation. I got different result if I started with different initial weights. Did I do anything wrong? or there I just found the local maximum instead of the global maximum? Thanks for your help.
Hi Jia, and glad you enjoyed the video! Excited to hear you have applied the techniques from my videos to your own project. It seems as the problem indeed has to do with the convergence of Solver and you have not done anything wrong. Hope this helps!
@@NEDLeducation Thanks for replying me. So how should I tackle this problem? Monte Carlo simulation? If I may ask, how do you optimize your portfolio for your investment?
Hi Nedl. Love the video. Is there a reason why the kurtosis term for the utility has 720 instead of 4!=24? I'm looking into the theory behind this and it's tripping me up. Thanks!
Hi Clayton, and many thanks for the feedback! As for your question, the kurtosis term is divided by 6! = 720 instead of 4! = 24. It comes from a general utility function for an arbitrary return distribution, where an infinite series of integrals gives you 6 factorial for the fourth moment. The best theoretical explanation I have stumbled across is given in Pierro and Mosevich (2008): www.witpress.com/Secure/elibrary/papers/CF08/CF08011FU1.pdf
@NEDL Ah I see now. It looks like the formula you're using is specific to CARA utility with arbitrary distribution which explains the constant theta. I was referring to equation (3) from this paper: www.hec.unil.ch/ejondeau/Publications/cEFM2006.pdf
It has a similar derivation but it's defined more in terms of an arbitrary utility function through it's derivatives evaluated at terminal wealth. If you were to plug in CARA utility for this I think it would give similar results but you would need to test it. Thanks for your help.
sorry quite technical question , what happens if theta set = 1 ? I mean in terms of overall performance historical statistics
Hi NEDL. Would it be wrong to use daily return-std dev instead of annual return-std dev in the calculation?
Hi Selim, and thanks for the question! No, this would be perfectly fine! Ultimately, you can choose the return and risk periodicity that is most convenient for you or that reflects your investment horizon the best. Hope it helps!
why do you not use covariance matrix in this case? is it because you are seeking maximum utility focusing only on skew and kurt and leaving the (mainstream) focus on variance/st.dev out of the picture? would it make sense to optimize portfolios focusing on higher moments using max or min solver function? like with markowitz portfolio optimization you try to maximise e(r) or minimise st.dev/volatility...does it make sense to look for highly negatively skewed portfolios using "minimize skew"?
btw thanks for the awesome video
Hi, and glad you liked the video! As for your question, this optimisation does focus on variance as well, alongside higher moments (skewness and kurtosis). In principle, the variance could be calculated using the covariance matrix but it is less important here as the optimal portfolio is determined numerically and not using the Merton algorithm I cover in this video: ruclips.net/video/fjEkkVwRl2A/видео.html. In terms of the skewness, all other things held equal, rational investors generally prefer positive skewness as it corresponds to heavier right tail (more upside than downside), unlike negative skewness (more downside than upside). Hope it helps!
very insightfull! thank you so muc!. I follow 15 US sector ETF and optimize return, omega and ulcer instead of std dev. Have you studied those theories?
Hi Alexander, and glad you liked the video! Will be definitely making videos on omega ratio and the ulcer index in the nearest future, these are by all means very interesting portfolio evaluation techniques.
plz explain value of theta =5 in the model ? thanks
Hi, and thanks for the great question! Theta stands for the risk aversion coefficient. Generally, you would have theta = 2 for less risk averse investors, theta = 3 (or 3.5) for moderate investors, and theta = 4 (or 5) for conservative investors.
please wrok on Portfolio Optimisation with indication both technical and fundemantal datas
how is the value of theta calculated?
This is a great methodology. If I need to add Cash as an asset class, what needs to be done?
Hi Deepak, and glad you found the video helpful! If you would like to add cash to the portfolio you can treat it as an asset with zero expected return, zero risk, and zero covariance with other assets. The optimisation will run as usual. Hope it helps!
@@NEDLeducation Thanks again. I tried with zero expected return, however, the formulae failed. I tried with small appreciation in cash (FD interest), and it worked. I am still not sure whether the procedure will advise me to shift to cash completely in case of a major fall.
@@deepaknidgalkar.2966 the model might advise to shift to cash completely only if theta is very large and all risky assets have positive kurtosis. Hope it helps!
@@NEDLeducation Thanks a lot.
I have a suggestion. The weightages calculated depend on the amount of past data used. To reduce sensitivity to older data, can something be worked out so that the calculated weightages are based more on recent data? Similar to EMA calculations.
thank you Erkan for this interesting moment , however , what is it that makes this additional algorithm useful to the end user investor , I mean rolling cagr as well as max DD and the corresponding recovery period relative to the other standard portfolio-based allocation algos
Can you do a video on resampling the efficient frontiers?
Hi William, and many thanks for the suggestion! I will certainly do a video on resampled efficient frontiers after I cover the basics of modern portfolio theory, so very soon!
An investor who is highly risk averse should consider a risk free govt security with return "r'" and no adjustment for volatility, skewness and kurtosis. Why would such an investor invest in equity based on adjusted negative returns after considering skewness and kurtosis?
Hi Aninda, and thanks for the excellent question! In this case, individual securities have negative utility when adjusting for skewness and kurtosis, whereas the portfolio of these has positive utility due to diversification (the effect of diversification on skewness and kurtosis is more complicated than that on variance, but generally they do tend towards zero or at least smaller values in diversified portfolios). Alternatively, a risk-averse investor might prefer a risky, yet right-skewed asset to a government bond with no upside or downside. Hope this helps!
How about a Risk Parity Portfolio?
Hi Ronald and thanks so much for your feedback. A risk parity portfolio is an interesting concept, we will definitely investigate it in future videos.
Hi I need help with an assignment in my finance subject, I want u use 4 different stocks and create an efficient portfolio frontier ? Could u help please?
Hi Soghraa, yes, I have got several videos on efficient portfolio frontiers, here is the one that uses Excel Solver you might find the easiest to follow for a four-stock case: ruclips.net/video/aBA3U1iNC7s/видео.html. There is also a video on the basic EPF with two stocks (ruclips.net/video/zjKbjG8D6xo/видео.html), and a more advanced video on how to actually build the frontier chart using matrix algebra (ruclips.net/video/fjEkkVwRl2A/видео.html). Hope it helps!
@@NEDLeducation thanx i am trying but getting some unavailable values, Iam using Bitcoin, s&p500, gold and United States 10 years bond yield, is it something that I do wrong ? Like following the video that u do with 5 stocks. Example I am as well using 252 days (market open) is it correct for the stocks I use??
@@sbatool6731 Hi again, that is normal as Bitcoin is traded 7 days a week and US stocks and bonds are traded at most 5 days a week. I would suggest downloading a US stock index and US bond index from a source like Yahoo finance and then map Bitcoin prices to the respective US trading days.
@@NEDLeducation i already did it as u say, there is not so many days left to the submission deadline. I could pay you to do it for me, how can I contact you ? Like trough fb, email etc?