Historical and Fundamental Betas for Equity Risk Models

# Answer by Tal Fishman (score 17, accepted)


# Which approach to estimating fundamental factor models is better, cross-sectional (unobservable) factors or time-series (observable) factors?












There are many approaches to estimating fundamental factor equity models. I would like to focus on two traditional methods:

- The time-series regression approach of Fama and French. Factors are defined ex ante. Betas to the factors are estimated in the time-series.

- The Barra cross-sectional regression approach described in Menchero, Orr, and Wang (2011), Grinold and Kahn (2000) and Sheikh (1995). Factor realizations are derived ex post. Factors are estimated independently in each time period in the cross-section.

I'll sketch the methodology of each and the high-level pros/cons. I'm curious if anyone has experience or links to research regarding which approach is better for the purposes of hedging, optimal portfolio construction, and alpha generation.

Fama-French time-series regression approach:

- Build a design matrix where each column is a time-series of economic factor returns. These factors could be traditional economic factors but also may include "spread" returns such as Fama-French factors SMB, HML, MKT generated from portfolio sorts.

- Perform N time-series regressions (one per security). In particular, for each security regress the security returns on the economic factor returns and estimate the beta.

In this approach, the betas are constant and the factors are time-varying depending on the regression window. The advantage here is that estimation of beta is diversified away across securities, so it seems to be this approach would be superior for portfolio construction. The disadvantage is that betas are slower to respond to changes that change the risk profile of a firm (for example, in a sudden shift in Debt-to-Equity ratio).

Barra cross-sectional regression approach:

- Assume that fundamental factor characteristics are Betas. For example, create z-scores of the fundamental factor characteristics thereby generating Betas for each time slice for each security.

- Perform T cross-sectional regressions (one per factor). In particular, at each time slice regress the panel of security returns on the normalized Betas to estimate the un-observable factor realizations.

The advantage here is that the betas respond instantaneously to changes in firm characteristics. The disadvantage is that there is a potential errors-in-variable bias -- in fact, the errors from model mis-specification do not diversify away. Intuition suggests this approach may be better suited for alpha signal generation.

Is there research comparing the out-of-sample performance of these two methods for various applications (i.e. risk decomposition, portfolio construction, hedging, alpha signal generation)?

## Answer by Tal Fishman (score 17, accepted)

https://quant.stackexchange.com/a/1929

Jennifer Bender of MSCI Barra has a paper from 2007 entitled:

To Beta or Not to Beta: A Comparison of Historical Versus Fundamental Betas for Hedging Market Risk

She deals specifically and exclusively with which method is superior for hedging long-only portfolios. Not surprisingly, she finds that Barra's approach is better. She tests long-only and long-short portfolios separately, using random portfolios with asymmetric bets.

What you call "cross-sectional" or "unobservable" factors she calls "fundamental betas," and "time-series" or "observable" factors are termed "historical betas." Here is the abstract:

> Fundamental betas provide several conceptual advantages to historical betas--they reflect information on a timelier basis and are less likely to confuse noise for information. This paper revisits the advantages of using fundamental beta for hedging systematic risk in the U.S. Fundamental beta appears to be a more consistent measure for hedging market risk, particularly for investors who care about downside risk and tail risk.

BTW, I have met her IRL and she seems pretty smart, so I trust she did the research right, but of course you must always be skeptical of company research that makes their product look better than their competitors'.

Update after reading Bender's paper:

- For hedging, my impression is that there is not a great difference between the two approaches, although fundamental betas win out slightly over historical betas.

- For portfolio construction, where stability of betas is much more important, she makes a convincing case for fundamental betas. Fundamental betas are more stable during stable periods and adjust more quickly to large shifts in a stock's expected behavior due to M&A or spinoff activity. She also argues risk estimates will adapt more quickly to a regime change with fundamental versus historical betas, particularly as she finds the best historical beta overall to be a 60-month regression.

- Fundamental betas also provide a distinct advantage over historical for risk decomposition due to the rapidly changing nature of cross-correlations and the lower propensity for assigning idiosyncratic moves coincident with a particular risk factor to the beta for that factor.

- Alpha signal generation is not really the goal of either approach, so I'm not sure if it is fair to evaluate based on this.

The fact that the two major commercial risk model providers essentially use the same approach should also tell you something. Barclays Capital's US Equity Risk Model (for institutional POINT clients) also uses cross-sectional regressions. In fact, in the face of all this industry research, the real question is why do academics still use the historical time-series regression approach?

Gregory Connor published an article The Three Types of Factor Models: A Comparison of Their Explanatory Power in 1995 in the Financial Analysts Journal. However, his goal is to compare statistical and fundamental models, not the two types of fundamental models to each other (he also throws in macroeconomic models, which he finds to be close to useless). The research is somewhat dated but still interesting for those looking at risk models in general.

## Answer by demully (score 2)

https://quant.stackexchange.com/a/49196

A simple addendum, that doesn't seek to supplant either the learned question and answer above. The short answer is the initial distinction drawn between ex-ante and ex-post is critical. What do you then want to do with the analysis? Therein lies the answer.

Imagine a prosaic reality in which a stock's returns are driven by its beta to market, to value, to momentum, to quality (assuming this can be defined, let alone measured!), to stock-specific noise plus something else.

Implicitly, every factor approach assumes that stock-specific noise is random, and unexploitable. Equally implicitly, the other factors are then not fully random, and thus potentially exploitable. Or more modestly, they may be not exploitable; but nevertheless help describe the realities of risk. And thus allow the a manager to avoid taking unknown, unintended, and/or undesired risks in the portfolio.

Where they differ is that the ex-ante approach is grounded in economic intuition. Cheap vs Expensive stocks can and do exhibit group behaviour. Economists can then argue about why this happens; the degree to which this or should influence returns; and whether this is or is not consistent with observed market behaviour. Momentum is also "a thing". Its existence is, politely put, theoretically awkward. But if we can reject the null hypothesis that it's a fluke, then the economists can go back and argue about which of this or that behavioural anomaly is at work; and whether knowledge of its existence now means the inefficiency has become obsolete.

Ex-post doesn't care about the niceties. This group of stocks seem to behave like this en masse. Ditto that group like that; the n-th group like so; etc. The model doesn't care why. Nor does it suggest any reason why these patterns should persist. If any investor wants to go and try to concoct an economic narrative to underpin what the model describes, that's their privilege. But it's really not necessary: the model faithfully describes reality, albeit without explanation. The investor can equally run his portfolio through the model and see the risks he's running, even if he has no insight what these might mean.

So now consider my "something else". Let's call this factor "Quan". Some stocks have Quan; some do not. It's statistically significant, ie it has the potential to make a good or a bad year (or years) for an active manager. But there's no off-the-shelf economic story that explains it well. Let alone any intuition whether it should be biased to produce positive or negative returns, even conditional on economic or market conditions.

How you deal with Quan, I think, answers the original question!

If you take the view that it is irrational or imprudent to chase (or defy) some anomaly that makes little sense, and may be completely ephemeral going forward, then ex-ante wins. You've promised your investors that they're putting their capital at risk to try to collect risk premia that have a reasonable positive expected return even if they are not guaranteed. It's a reasonable and defensible position.

And vice versa if you take the view that Quan is a reality that must be managed even if it makes no sense. Your quant comes in and tells you that he thinks he has found a correlation, even if a little tenuous, between Quan and a particular set of economic catalysts about which you do have a view. When that starts to influence what your portfolio looks like, then ex-post has won.

The bottom line is that the choice is more revealing about how any investor chooses to invest; than about one being an inherently "better" market model.

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/78633

Let us first do away with jargon and think of how to get the right betas naturally. You can either use data to estimate betas, or if you believe you know the betas specific to today's conditions ("regime") better, just supply them yourself.

You will be rewarded if your estimates are better, and punished if you are overconfident. That is basically all you need to think about.

Long run, the regime specific betas should average out to the data-supported betas.

Both approaches have downsides of different kinds, and it isn't clear if one error is empirically worse than the other.

With cross section, you are exposed to the risk of mis-specifying some (say 50 out of 1000) betas, still getting a significant factor (you will because other betas are correct), and optimization then takes those betas and uses them as hedges.

With time series, you have the risk of making many estimations and some of them being way off, and optimization using exactly those to find the hedges.

To conclude, both have different upsides and downsides. I am not sure there is proof which downside is empirically worse (or manifests more often). Whether you choose to find betas from data or supply beta could very well depend on your specific expertise as well as properties of your portfolio.

Another point, if factors strongly exist, both approaches are worthwhile and similar. If factors don't have a massive effect, both approaches are producing a lot of noise and won't be useful for construction. So the main risk factor is how reliable your factors are v/s how you choose to estimate them.