Armed with a model of entropy that has long-term theoretical validity, and the ability to apply that model to daily time frames (since updated at the end to better model volatility) we can now get into some of the really interesting areas of entropy applicability.
“I’m a personal investor or sector analyst, I don’t give a damn about a portfolio of 500 stocks!” Truth be told, I think it’s interesting but it has slightly limited applicability. The holy grail (which I don’t really think I am close to yet) is the modeling of a single contract across this entropy framework, but here’s one potential path to it:
Thankfully, finance has developed Arbitrage Pricing Theory, APT, which empirically tracks the explanative power of different factors going into the pricing of a single security or portfolio of securities. Now, without adaptive individual-factor volatility to determine the smoothing width of the return distribution, an APT approach would require somewhere north of 50-100 factors depending on internal correlations to explain a security risk for the entropy model to be applicable. (Below these numbers a fix-width smoothing system is simply a glorified combinatorial exercise of how likely factors are to stack risk weights on each other.) With adaptive smoothing width however, the likely range of possible shapes of any return distribution can be started to be estimated to be essentially infinite as soon as at 10-20 components, but with greater versatility to accurately model distribution “quirks and kinks” around the tail ends after around 50 components.
Thus, with these adjustments, APT is a feasible way to adopt entropy modeling to single stocks, if enough different factors can be found! Yay! Building portfolios then becomes easy: multiply the factor weights by the portfolio weight of the modeled contract, remove circular references (inter-portfolio-contract factors, contract-portfolio factors) and simply re-balance to the sum of all weights being 1. Tada!
Some really interesting effects of portfolio construction start showing up if you extend the portfolio to 50 stocks with weak correlation: you can start to choose whether you want portfolio risk to be modeled from the portfolio components, or from their individual factors. For actively-managed portfolios, it might be beneficial to keep all contracts on “factor watch” since these portfolios might need to turnover individual stocks very often. For less actively managed portfolios, identifying (estimating) overall risk contribution in the portfolio of a section of the most risky stocks for the portfolio and then using entropy to fine-tune the decision of which components should be adjusted could potentially be a more interesting approach. If a principal components decomposition is kept of the factors, it becomes easy to determine which cluster of risk factors are adding to risk at any period and which stocks are heavily priced on acceptance of these risks.
Arbitrage pricing theory commonly uses 10-20 (commodities, domestic economic and financial data) factors factors for domestic companies, and allows expansion of all related currency rates if the company is heavily involved in cross-currency businesses, as well as data from the countries it operates in. This entropy approach through evaluating APT risk factors thus becomes a lot more powerful for companies that have diversified international businesses, and thus allows for large parts of Asia, the US large-cap indices, and European large indices alike to be rather well-modeled in theory through this framework, as long as there exist many factors that are quoted daily/weekly or could be modeled as such!