Every year thousands of young people attend elite schools to learn about business. One of the core courses is Corporate Finance, and one of the key principles they learn is about risk and reward, and the standard theory is framework--not a model--that holds that the expected return of a financial asset is a function of risk, where you are paid to endure this unpleasant, irreducible characteristic. The quantity of risk is measured by a covariance with priced risk factors which are as-yet unidentified time series like the stock market, and there’s a linear relationship between this risk metric and expected returns. Risk measures the 'how much', and the price you receive for this is from risk premiums. Thus, expected returns are determined by this crucial, objective characteristic (Cliff Asness nicely describes the expected return as dominating the variance in the forward to Antti Ilmanen's Expected Returns, accurate because if you ever do a mean-variance optimization, those assumptions really drive the end result, but alas they have much greater uncertainty).

Yet as Mark Rubinstein said about the CAPM and its extensions, “More empirical effort may have been put into testing the CAPM equation than any other result in finance. The results are quite mixed and in many ways discouraging.” Eugene Fama and Kenneth French called the CAPM “empirically vacuous,” and APT creator Stephen Ross noted that “having a low, middle or high beta does not matter; the expected return is the same.” These are all major proponents of this approach, so I think it's fair to say that the standard model is a theory in search of validation. As the elusive risk factor is clearly not 'the market', but something like the market, new factors are proposed all the time.

Remember the CAPM, with its simple single factor model,

Supposedly, this worked great, but then, Fama and French showed that when you pre-sort data by size, there's no relationship with beta. Thus, they created the new 3-factor Fama-French portfolio

At this point, all bets were off. There were two ways to theoretically rationalize a factor. One could simply assert it as being intuitively risk, as Fama and French did, which via Arbitrage Pricing Theory logic should be priced. Alternatively, one could write down a Stochastic Discount Function, as Harvey and Siddique did in (2000) with their co-skewness paper:

'm' is sort of like super-string's M-theory: it can be whatever you want it to be, and the great elders of rigor have proven these are all kosher, so, once written down as above, you can simply append the resulting 'betas' to the above 3-factor Fama-French model and not explain exactly how they work together given one's earlier motivation that contained no value or growth factor (It reminds me of how Marx's totally wrong but complicated and rigorous Das Kapital allowed generations of theorists to talk about the Hegelian dialectic as if it were real, because nonbelievers simply didn't want to waste time on it, and insiders could all point to thoughtful people who believed and extended the great work).

Researchers Tobias Adrian of the Federal Reserve Bank of New York, Erkko Etula of the Federal Reserve Bank of New York (now at Goldman Sachs), and Tyler Muir of Northwestern University have a hot paper that is the latest best hope for the elusive risk factor that explains asset returns (Financial Intermediaries and the Cross-Section of Asset Returns). They argue that the key is broker-dealers, specifically, their leverage constraint. The nice thing is that one has data on their leverage going back to 1968 or so with quarterly data, so you can throw it against the wall, and guess what, "Our single-factor model prices size, book-to-market, momentum, and bond portfolios with an R2 of 77 percent and an average annual pricing error of 1 percent." That is, 25 size-value sorted portfolios, plus a momentum portfolio, and some US Tbond portfolios.

Considering that CAPM betas can't explain anything, how does this work? I'm not sure, but I suppose a lot of it is overfitting: there are only 25 portfolios targeted, and there are lots of potential SDFs (eg, consumption-labor-wealth VARs, consumption growth, Tobin's Q in various forms). As usual, they stress the deep theoretical roots to their metric:

Where LevFac is the seasonally adjusted change in Broker/Dealer leverage, there you go. As Frazzini and Pedersen's Betting Against Beta framework also included the market return with their leverage constraint, I'm not sure how they dropped that given their similar intuition.

I took their proxy of the SDF, and graph it next to the S&P for the past 10 years. You can see its correlated about 30% but catches the really big moves as in 2008 and 1987 (not pictured here).

Yet as Mark Rubinstein said about the CAPM and its extensions, “More empirical effort may have been put into testing the CAPM equation than any other result in finance. The results are quite mixed and in many ways discouraging.” Eugene Fama and Kenneth French called the CAPM “empirically vacuous,” and APT creator Stephen Ross noted that “having a low, middle or high beta does not matter; the expected return is the same.” These are all major proponents of this approach, so I think it's fair to say that the standard model is a theory in search of validation. As the elusive risk factor is clearly not 'the market', but something like the market, new factors are proposed all the time.

Remember the CAPM, with its simple single factor model,

E(R

_{i})=R_{f}+bE(R_{m}-R_{f})Supposedly, this worked great, but then, Fama and French showed that when you pre-sort data by size, there's no relationship with beta. Thus, they created the new 3-factor Fama-French portfolio

E(R

_{i})=R_{f}+bE(R_{m}-R_{f})+b_{size}(R_{small}-R_{big})+b_{value}(R_{value}-R_{growth})At this point, all bets were off. There were two ways to theoretically rationalize a factor. One could simply assert it as being intuitively risk, as Fama and French did, which via Arbitrage Pricing Theory logic should be priced. Alternatively, one could write down a Stochastic Discount Function, as Harvey and Siddique did in (2000) with their co-skewness paper:

Or as Jacobs and Wang did in their 2001 consumption growth volatility paper:

Researchers Tobias Adrian of the Federal Reserve Bank of New York, Erkko Etula of the Federal Reserve Bank of New York (now at Goldman Sachs), and Tyler Muir of Northwestern University have a hot paper that is the latest best hope for the elusive risk factor that explains asset returns (Financial Intermediaries and the Cross-Section of Asset Returns). They argue that the key is broker-dealers, specifically, their leverage constraint. The nice thing is that one has data on their leverage going back to 1968 or so with quarterly data, so you can throw it against the wall, and guess what, "Our single-factor model prices size, book-to-market, momentum, and bond portfolios with an R2 of 77 percent and an average annual pricing error of 1 percent." That is, 25 size-value sorted portfolios, plus a momentum portfolio, and some US Tbond portfolios.

Considering that CAPM betas can't explain anything, how does this work? I'm not sure, but I suppose a lot of it is overfitting: there are only 25 portfolios targeted, and there are lots of potential SDFs (eg, consumption-labor-wealth VARs, consumption growth, Tobin's Q in various forms). As usual, they stress the deep theoretical roots to their metric:

Guided by theory, we use shocks to the leverage of securities broker-dealers to construct an intermediate SDF.How does theory guide this? Well, remember, 'm' is the Stochastic Discount Factor,so, if you simply assert that

Where LevFac is the seasonally adjusted change in Broker/Dealer leverage, there you go. As Frazzini and Pedersen's Betting Against Beta framework also included the market return with their leverage constraint, I'm not sure how they dropped that given their similar intuition.

I took their proxy of the SDF, and graph it next to the S&P for the past 10 years. You can see its correlated about 30% but catches the really big moves as in 2008 and 1987 (not pictured here).

note: the BD factor is derived from a factor-mimicking portfolio from the 6 F-F size-value portfolios and the momentum portfolio as given in their paper

What I suspect, though I haven't done the experiment, is that if you regress individual stocks against this factor there will be a zero correlation with returns. That's the result of overfitting. You fit the target, in this case, some portfolios from Ken French's website, and you have a pub, especially if you write down an SDF, but it's just the flavor of the month, the latest potential solution to a perennial problem.

I shouldn't be too hard on it, it is intellectually honest work, very clear. Yet, these pop up all the time as one would expect with thousands of potential SDFs out there and the ease at which they can be rationalized. If one ever explained the cross-section of stock returns, I'd rethink my skepticism.

## 5 comments:

Why should there be a "factor" that explains returns? Perhaps the "factor" is whatever happens to be undervalued at that time? If a non-changing factor existed, wouldn't it prove markets where retarded?

You are right, its really very feasible that one country is exporter or importer of oil. The stock price of the same depends upon the country importing or exporting of oil.

Very Informative Article !

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Problem with this Eric is that B/D typically bring down leverage before it gets reported on Quarter-end dates. So the actual risk/leverage in the system is a lot higher if you account for this window-dressing. I am not trying to throw this hot-paper out with the bath-water but that is the problem with some of these papers.

The causation here seems like it isn't always in the direction of broker dealer leverage => asset prices. Asset prices => broker dealer leverage makes just as much sense in many scenarios which seems to lead to endogeneity issues.

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