Quantifying Odds in a Portfolio of Investments
Qualitative : Quantitative: A preparation of Qualitative data for Quantitative expressions
Simplicity & Elegance
Quantifying and predicting chaos with precision is a near-impossible proposition; modeling forward-looking irrationality requires quantifying near limitless branches of data from both macro and micro events and their resulting subsets of response towards those events, and so on and on. Massive amounts of computing power would be necessary to calculate any meaningful data.
Our solution to this problem was to minimize complexity and to condense probability density functions into singular metrics/variables. With these variables, further calculations of position-sizing and asset allocation techniques could be employed to maximize returns and minimize variability within the portfolio.
We briefly touched upon in A Turn-of-the-Decade Review of a Qualitative-Quantitative Approach to Investing: 2011-2021, that the Odds of Positive expected return was dependent on the Qualitative Fundamental Analysis and Outsized returns coincided with Quantitative approaches. In our context, quantitative techniques in question provided clarity and approximations to position-sizing within the context of the portfolio.
A pseudo-application of computing the odds of a bet (the position itself) meant a determination of potentiality and likelihood was needed. A Bayesian approach and the generous use of prior probabilities is the standard when interpreting qualitative data; this meant computing the following variables:
Potential ReturnPotential LossWinning Probability
In our view, the quantification of the variables set above are difficult and unwieldy, especially in a continuous distribution. The potential for human error is incredibly high; it requires significant estimates and expert opinions that cannot be quantified with absolute certainty. In a Bayesian approach, an update on the prior probabilities with new data mitigates this uncertainty. (although by all means, it does not eliminate all uncertainty)
Probable Finite States
While we may be informed with relevant historical data, we make the assumptions that we require forward-looking estimations. In the context of a continuous distribution, one possible solution is to define probable finite states. We define the infinite range of continuous random variables with established knowledge (expert opinion, predictive analysis, relevant prior data, etc.). The generous use of established knowledge to define prior probabilities is dependent on the quality of the knowledge in question.
Consider a thought experiment with the following dataset:
Starting Number: 1.00Possible Range of Outcomes: 0.00 to 2.00Duration: 1 Year
The thought experiment runs for a period of a year, of which the Starting Number of 1.00 will transition into a number that ranges from 0.00 to 2.00 at the end of the year. The number could be 1.5328, 0.02351, 0.82356, etc., a limitless range of possibilities. There is no certainty as to what the number ties its value with but there are analysts and experts that provide predictive analysis based on prior observations.
Immediately, there is a need to arrange the continuous distribution into a dataset that is easier to understand and manipulate. Determination of Finite States & Outcomes based on expert’s predictive analysis and established knowledge - most probable outcomes are consolidated, weighted with probabilities, and averaged into a singular metric/variable.
Probability Density Function:
https://en.wikipedia.org/wiki/Probability_density_function
https://www.youtube.com/watch?v=Fvi9A_tEmXQ
Weighted Averages:
https://sciencing.com/calculate-weighted-probabilities-5959518.html
https://en.wikipedia.org/wiki/Expected_value
https://www.investopedia.com/terms/w/weightedaverage.asp
In a real scenario, datasets would be sampled from each and every day and subsequently, with each sample, a change in probability.
There are many depths and complexity a model of such making can take, limited only by sheer computing power. Implementing Finite States however, remains a fundamental step in those calculations.
Black Swan Events & Tail Risk
Black Swan Events, as popularized by Nassim Nicholas Taleb in the book “The Black Swan: The Impact Of The Highly Improbable” (Random House Publishing Group, 2010) are events that are unpredictable and have potentially severe consequences.
Typically, Black Swan Events are interchangeably used with the term tail risk. In a normal distribution bell curve, black swan events can be found in the narrow far end of the bell curve, at least 6 standard deviations away - a 0.0000001% probability. For an event that occurs so frequently in financial markets, there is a clear deficiency in conventional approaches to Black Swan Events.
In his book, Nassim Taleb proposes that tail risk cannot be determined through any probabilistic methods and tools that employ normal statistical distribution. The argument posits that past sample sizes and observations do not sufficiently model nor predict for black swans, thereby creating a false sense of security. Examples of Black Swan Events:
Coronavirus in 20199/11 attacks in 2001
Black Swan Events are rarely, if not, accounted for in quantitative models. It requires an incredibly complex amount of data and it requires datasets of a forward-looking qualitative nature. A simple solution (not perfect) is to account these improbable events as a probable one in the list of Finite States and outcomes. An apocalyptic scenario of total loss, together with a low probability that is brought up to a meaningful value, would provide a margin of safety to the weighted averages.
Conclusion
In this article, we proposed finding the variables/metrics of Potential Return, Potential Loss and Winning Probability. In determining these same variables, datasets were found to be continuous and therefore needed further processing of data to prepare it for more manageable manipulation - Finite States were introduced and weighted probabilities averaged the most probable states/outcomes into singular variables/metrics.
In these same Finite States, Black Swan Events were accounted for and folded into the singular variable/metrics, creating a margin of safety by conservatively asserting odds of a particular bet.
Links & References:
Bayesian Statistics
http://www.columbia.edu/~md3405/BE_Risk_3_17.pdf
https://sciencing.com/calculate-weighted-probabilities-5959518.html
Normal Distribution
https://www.youtube.com/watch?v=rzFX5NWojp0
https://en.wikipedia.org/wiki/Probability_density_function
https://www.youtube.com/watch?v=Fvi9A_tEmXQ
https://www.investopedia.com/terms/p/pdf.asp
Black Swan Events
https://quantra.quantinsti.com/glossary/Six-Sigma-Event#
Weighted Averages
https://sciencing.com/calculate-weighted-probabilities-5959518.html
https://en.wikipedia.org/wiki/Expected_value
https://www.investopedia.com/terms/w/weightedaverage.asp
Related Articles
https://www.ray-kok.com/blog/qualitative-quantitative-approach-investing
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