What are the terms of the benefit and how do you measure it? Given a definition and a way to calculate does anyone know what the results are?Apathizer wrote: ↑Mon Sep 26, 2022 3:55 pmI've made this point repeatedly. The cap weight TSM SV allocation is only nominal. A higher SV slant is necessary to provide significant diversification benefit.burritoLover wrote: ↑Mon Sep 26, 2022 9:40 am I'm not sure how anyone can argue that adding an SCV tilt is less diversified and more concentrated than holding only TSM. The top 20 stocks in TSM make up 30+% of the entire index and the top stock (AAPL) is double that of ALL the SCV stocks in TSM. Adding an SCV tilt reduces your concentration in the mega caps that overwhelm the TSM index even if you don't have a strong conviction with small/value factors.
And we know that anything outside of large caps is practically irrelevant in TSM - a difference in return in the hundredths of a percent over 50 years. So to say you already own SCV in TSM is just nice little meaningless anecdote.
I did a very naive calculation of that using diversification ratio as a measure for holding various allocations to VSIAX and VTSMX between 2012 and 2022 and got that the diversification ratio between 100/0 and 0/100 stayed between 1.00 at the ends and 1.03 maximum at 60% VSIAX, but I am not sure I know what I am doing. That is essentially no diversification at any level of VSIAX.
In this paper https://dial.uclouvain.be/memoire/ucl/e ... is%3A14352 the following definitions of diversification are explored:
1. The Shannon’s Entropy. Originally coming from the Information Theory and developed
by Claude Shannon (1948) to solve communication problems, Shannon’s Entropy has
later been applied in finance to measure the amount of information given by observing
the market.
2. The Diversification Delta. This measure introduced by Vermorken et al. (2012) is based
on empirical entropy. However, due to several drawbacks, we will use an alternative
version of it that has been developed by Salazar et al. (2014).
3. The Diversification Ratio. This measure proposed by Choueifaty (2006) is defined as
the ratio of the portfolio’s weighted average volatility to its overall volatility.
4. The Marginal Risk Contributions. This measure allows to decompose the total risk of
a portfolio into the contributions of each individual assets.
5. The Portfolio Diversification Index. This measure indicates the number of unique
investments in a portfolio and is useful to assess marginal and cumulative diversification
benefits across asset classes.
6. The Effective Number of Bets. This measure uses the entropy of some factors (of risk)
exposure distributions to indicate the diversification of a portfolio.
I used choice 3 on a very small data set. Choice 5 does not exactly mean what it appears to mean as it presumes that first of all one has done a principal components analysis, sometimes called factor analysis, but is not the same thing as factors in a Fama-French regression.
I admit to not being well grounded in this area.
As an example of entropy I tried a very simple minded example that might not actually be correct. The example is given 9 numbers between 1 and 9, which is more diversified 1,2,3,4,5,6,7,8,9 or 5,5,5,5,5,5,5,5,5 or 1,1,1,1,5,9,9,9,9? Note they all have a mean of 5. The respective entropies are 2.20, 0.00, and 0.96 so the first list is more "spread out" over values meaning more diversified.
Here is a paper that uses the number of bets analysis: https://portfoliooptimizer.io/blog/the- ... ification/ The example they cite in the paper is for a portfolio of bond funds and not for stock factor investing. Even in that case the author's conclusion might be unexpected.
Here is a typical Fama-French paper on risk factors but the terms diversify or diversification do not appear in the paper: https://www.bauer.uh.edu/rsusmel/phd/Fa ... _JFE93.pdf Indeed the term "risk" is not actually defined in the paper though it is clear that risk factor is a term that explains returns in a regression model.
In the paper originally quoted above risk is offered as either volatility (SD) or value at risk (VaR).