Ang pricing kernel


New Member
There is something rather sloppy about saying price volatility being a macro-economic 'factor' parallel to growth and inflation. To me volatility is a proxy for risk, but it also encompasses market microstructure (liquidity, behavior of market makers who engage in what some might say manipulation, especially for more thinly traded securities-- algorithmic games certain dominant HFTs and MM's play). An aggressive seller or buyer with little regard for implementation shortfall will add to price volatility. I realize the CAPM model assumes price-takers, but we are talking about the real world.

Even in highly liquid markets, volatility encompasses differences of opinions (non-homogenous opinions), and uncertainty including uncertainty around inflation and growth. Inflation and growth then are more directly related to the economy and properly treated as macro-economic factors, whereas "volatility" is much more of a broad paintbrush, a derivative price-related statistic that may have rolled into it idiosyncratic uncertainties, microstructure and supply side dynamics, liquidity related dynamics (lumpiness), as well as proper "macroeconomic" considerations (what the Fed might do or say, inflation uncertainty, growth uncertainty).

Re: also inflation as "bad" for equity returns, if one regresses domestic equity returns to domestic inflation, there tends to be two regimes, one from 0 inflation to "modest inflation" where more inflation is positive for real returns, and much higher inflation where equity *real* returns decline with higher inflation, but not as negative as cash or fixed income. All these things also critically depend on the Fed/ECB responsiveness to tame inflation, making historical data useless were there a paradigm shift moving forward. (Would we look at the Volcker period anti-inflationary measures as representative of the Fed response looking forward? No-- so this is an anomalous period which wouldn't make sense to include in determining statistical relationships).

Finally, re: FF-3 or FF-4, I get they get a lot of respect in academia, but the fact remains that there are hundreds of anomalies now built atop of these 3 and 4 factor models, or atop of this APT model of growth/inflation/volatility. It's rather unimpressive and given too much credit if one understands numerical approximations/math. In the end, if you have enough factors or functions/vectors/tensors that are not fully parallel, you get a pretty good local approximation with 2-3 factors. It is analogous to just the first 2-3 terms in a Fourier/Lagrange/Taylor series. (Same can be said of Miller Modigliani). To someone versed in science/physics/engineering, none of this 'fit' is surprising. One could take three somewhat orthogonal "factors"/anomalies of the hundreds now in existence, and create an alternative FF3/FF4/FF5.

A deeper nuanced understanding in (slow quant) anomalies investing, then starts with understanding the interaction of factors (they have non-linear relationships), and in the linear approximation, understanding orthogonality (for example the greater degree of orthogonality among value-leaning metrics; growth/momentum leaning metrics; liquidity; and "quality" metrics.) An alternative, which is harder to explain to human beings would be taking in all factors, and using a technique like PCA to find 3 truly orthogonal (by design) "factors". The cutting-edge alpha generators, are probably are looking more at nonlinear relationships, global model parameters and tuning, alternative data-sources not well correlated with traditional factors, detecting regime/paradigm shifts.

Much too much time in how traditional finance is taught is centered around these stale finance ideas that are not useful to investment/asset managers (except in a marketing/credibility/signalling sense). Necessary? Perhaps-- one can argue these are fundamental building blocks. I tend to think there is a lot of fluff in finance. A lot of academic patting oneself on the back. But that opinion probably won't garner too much support.


New Member
Re: diminishing small-cap premia. This is merely conjecture; however, I feel the pickup in M&A activity for small and mid-sized businesses in the 80s/90s, and then the tendency for smaller firms to stay private post-SARBOX have contributed greatly to an environment unfavorable for US domestic small caps. In other words, adverse selection because of attractive small-cap firms being snapped up by larger companies, and quality small cap firms choosing NOT to list due to the onerous regulatory costs of SARBOX for smaller firms.

Of course there are other meaningful wrinkles. Small-cap stocks generally tilt more heavily in things like biotech, or real-estate within the healthcare of finance sectors. They often do not have the ability to capitalize on economies of scale/scope, and large network effects (or else they'd zoom past being small cap to being mid-cap then large-cap with very little time spent being 'small-cap'). They also have much less multinational exposure, though many do obtain inputs from overseas (such as equipment or inventory from China). They cannot capture multinational reg-arb, tax-arb, or rate-arb opportunities. They generally don't sell internationally (mega caps mostly import and export significantly more).

In the tech industry-- and this is anecdotal I admit-- some companies hit a hard wall at the small or mid-cap level due to a monopolistic canopy (take Twitter or Snapchat for example). Is this merely explained by stronger network effects of the giants? I think that is part of the story; however, I think a bigger part of the picture is monopolistic strategies pursued by the giants that has been allowed to grow rampant.

All-in-all, there is not enough good regulation in favor of small-cap (incubation/subsidies because incumbent network effects and scale scope economies can be impossible to overcome; better regulation and enforcement against antitrust/monopolistic practices such as with 3rd party repairs, walled-gardens or ecosystem governance monopolization, legal bullying/IP theft through having a deeper legal budget, intentional small-supplier IP diffusion facilitated by US multinationals, implicit collusion of oligopoly firms, product bundling to exclude competitors, predatory pricing strategies, etc.). And on the flip-side there is too much size favored, "bad regulation" working against smaller firms, especially small public firms.

America's political leaders should see all of there things as warning signs for America's future competitiveness, and as warning signs in terms of broad economic participation (equality in opportunity). Many small business owners are relegated to pursuing a TAM-capped micro-niche because of all these unfavorable factors. Scaling from SMID to large or megacap, I believe is harder than it ever was in most industries. That lack of competition hurts consumers and SMID shareholders. A wealth transfer to megacap firms' shareholders. (And for the really successful small-cap firms in sustainable niches-- I believe the alpha is in private equity/VC).


New Member
A comment on the momentum factor. To me I think there are two ways to think of momentum-- one is pure price momentum as an independent factor (as in the FF-Carhart Model). The other way to consider momentum is it being itself a bundle of factor momentum, and not distinct. The time series of most anomalies and factors show autocorrelation and volatility clustering. These feed into the momentum super-"factor". It's not a mainstream view, but a more granularly consistent one, I believe.

I've not read studies on it, but market frictions (microstructure, liquidity around sharp price reversals) cannot be ignored, generally, but especially with "momentum". That is, if frictionless assumptions are made, or implementation shortfall for momentum strategies underestimated, the factor return will be inflated. Bin-size, evaluation time frame, holding period (or exit threshold), etc. mean there are many parameters-- and my belief is academics have overfit some of these parameters thereby understating timing risk costs, and trading friction costs (and thereby exaggerating the performance of what is realistically exploitable momentum). A more robust/objective analysis of momentum would have to look at out-of-sample performance with different parameters.

In any case, if one takes the view that other factors exhibit autocorrelation, and that "momentum" is really a price phenomena that encapsulates other factor autocorrelations, a more consistent and rigorous study would look at the time series of these other true factors.


New Member
Re: Low volatility anomaly. I think it would help for there to be specification as to how the timeframes observed compare. For example, the video slide notes for "P2.T8. Risk Management & Investment Management Andrew Ang, Asset Management: A Systematic Approach to Factor Investing Alpha" pg 5, it's not at all clear if the side by side graphs are comparable and what exactly the differences are attributable to without knowing the study period and where the absolute scale vs. ranking scale align. On the left chart showing the volatility anomaly, it's clear that Sharpe (or a measure of risk-adjusted return) rises with lower volatility. This is consistent with how many in practice such as in AQR/Robeco describe the anonmaly-- not as a violation of CAPM or FF, but an improvement in risk or beta-adjusted performance for low-volatility stocks. I might call this a soft-form volatility anomaly. Some make the harder claim that isn't true depending on the timeframe studied, that absolute performance improves with lower volatility.

Ang's chart seems to suggest both risk-adjusted performance and absolute-performance improvement with lower beta. The curve on the left looks flattish (possibly slight improvement to no improvement in absolute performance) as you move toward higher beta quintiles until you get to the highest beta quintile whereby there is a drop in absolute performance. Anecdotally, I've seen this to be true with the most storied 'hot hand' stocks tending to be like the Hare in Aesop's fable-- losing the race. (You guys are the bionic turtles).

This would SEEM to contradict the chart on the right which shows an absolute not quintile scale on the X-axis, and a CAPM/FF relationship of higher return for higher beta (intuitively greater compensation for greater risk). But it's impossible to know if the difference between flattish-then-declining, to monotonously increasing is distorted by differences in x-axis scale, and if comparable periods are utilized, etc. I don't know what I'm looking at here and if CAPM/FF has strong evidence against it. My sense is, generally, except in the extreme high-beta end, the absolute returns of higher volatility stocks is generally higher than the absolute returns of lower volatility stocks as CAPM/FF would predict.

Finally, I have some thoughts as to why this anomaly might be present, and why it isn't fading away. The 'highest beta' absolute underperformance, and declining beta-adjusted performance of high beta-stocks, in my view has to do with a complex of phenomena that relate to over-confidence. It is conjecture, but I believe the anomaly to firstly be a manifestation of the Dunning-Kruger effect, and even professional investors under-estimating their competition, or incentivized to pretend they are superior to others as active managers (for marketing purposes).

Higher beta stocks generally have more cross-sectional dispersion. Many are big winners or big losers. Their high volatility often indicates deep value, or explosive growth, or uncertain quality. Therefore, there is more of a case to be made that if the losers can be separated from the winners, that the higher absolute returns can be captured through stock selection. But this belief being crowded, and competition being under-estimated (DK effect), results in an aggregate mis-pricing of CAPM/FF beta.

Focal points, and representativeness/accessibility, or what the younger people call FOMO also lead to positive feedbacks contributing to unproductive higher volatility. Yet price-momentum seems to be a positive factor. Why? As I've mentioned before, I believe momentum as a return-enhancing factor greatly depends on the parameters chosen-- and academics have conveniently chosen parameters when looking backwards 'prove' a strong momentum anomaly. My experience with momentum has been that the optimal parameters for determining 'momentum' vary dynamically over time in a complex way.

Focal points/representativeness biases such as from action cameras being 'hot' or 3D printing or nanotech being 'hot' are made crowded by non-expert amplification (news, influencers, lay gossip, hyperbole). Unlike the longer-period momentum timeframes, many of these 'hot' ideas tend to swing hot and cold more rapidly than your typical 4-18 month momentum factor periods. When these 'hot' periods last for a long time such as with dot-coms or cryptos, they may manifest as an illusory positive momentum factor return, and against the low-vol anomaly (aligned with FF/CAPM). 96-99 was one such period when dot coms could seem to do no wrong.

Growth, momentum, high beta were the winners. Dispersion and volatility of the benchmark were low. Low-vol underperformed. People trusted in a 'Fed put' even if that was not Greenspan's intention. People were unaware or apathetic to the high degree of complacency (in spite of Greenspan's 'irrational exuberance' comment which was a red flag to astute observers in Dec-1996). There is a dynamic (game theoretic) element to these race-to-the-bottom, or performance- or yield-chasing phenomena whereby active stock-selecting professionals cannot back down from 'trying to keep up' or 'catchup' with respect to their professional peers. This is professional "fear of missing out" or FOMO, the same as happens with retail investors and non-professionals calling their asset manager demanding why they don't have an allocation to metaverse tokens, Mars exploration stocks, or EV car companies. Even if they desire to be more objective, and they are not as easily swayed by news narratives and 'buzz', their desire to appease their client's FOMO, and their external and internal assessment being made relative to other professionals dictate they do fall into the same trap.

Sometimes this exuberance tends to peak and blow-off at the very end of a bull market (anecdotally-- small sample size).

When "reality hits" in these blow-offs, low vol represents some shelter. Boring potentially means safety, less pain. The tortoise trudges ahead.