Characteristics vs Covariances

Two students both ace the exam. One has a great tutor who drills the same material into every student they take on, the tutor is a shared, systematic influence. The other just has a good memory, a personal trait that travels with them regardless of who's teaching. If you only ever see students who both have that tutor and a good memory, you can't tell which one is actually earning the grade. Factor investing has the identical problem: a value stock both is cheap (a characteristic, like a low price-to-book ratio) and tends to move with other cheap stocks (a covariance, a loading on a "value" risk factor). Almost every value stock has both. Untangling which one the market actually pays you for needs a special kind of data, and the two most-cited studies in this literature disagree on the answer.

Setting up the horse race

Let $r_i$ be a stock's return, $\beta_i$ its loading on a risk factor (how much it moves per unit the factor moves), and $c_i$ its characteristic (say, book-to-market ratio). The covariance hypothesis says expected return depends only on $\beta_i$: you're compensated for bearing a risk shared with other stocks, the characteristic is just a proxy that happens to correlate with the loading. The characteristic hypothesis says expected return depends on $c_i$ directly: the market misprices cheap firms for reasons that have nothing to do with systematic risk, behavioral neglect, distress aversion, whatever, and the loading is just along for the ride because cheap firms tend to move together.

You can write this as one regression,

$E[r_i] = \lambda_0 + \lambda_1 \beta_i + \lambda_2 c_i$

In words: expected return is a baseline plus a reward for factor exposure ($\lambda_1$) plus a separate reward for the raw characteristic ($\lambda_2$). If $\lambda_2$ survives with $\beta_i$ held fixed, characteristics matter on their own; if $\lambda_1$ survives with $c_i$ held fixed, it's really about risk.

The trick is finding stocks where $\beta_i$ and $c_i$ come apart. Daniel and Titman (1997) built characteristic-balanced, covariance-varied portfolios: take stocks with the same book-to-market ratio but sort them further by their loading on the HML factor (their realized covariance with other value stocks), using loadings estimated from a period when those stocks weren't yet grouped that way.

Worked example 1. Suppose two portfolios both hold stocks at book-to-market $= 1.5$ (same characteristic), but Portfolio A's stocks happen to load heavily on HML ($\beta = 1.2$) while Portfolio B's stocks, despite being equally cheap, load weakly ($\beta = 0.3$) because they got grouped with cheap stocks only recently. If returns are covariance-driven, A should beat B by roughly $(1.2 - 0.3) \times \lambda_{HML}$; with a typical estimated HML premium of about 0.35% per month, that's a gap near 0.32% per month. Daniel and Titman instead found A and B earned almost the same return, close to zero gap, while portfolios sorted purely on the characteristic, holding loadings fixed, showed a large spread. Their conclusion: the characteristic did the work, not the covariance.

Worked example 2. Davis, Fama and French (2000) ran the mirror test over a longer 1929–1997 sample and reached the opposite verdict. Sorting stocks into six characteristic-and-loading buckets, they found a stock with book-to-market $= 2.0$ and HML loading $\beta = 1.0$ earned about 1.1% per month on average, while a stock with the same characteristic but $\beta = 0.4$ earned only about 0.6%, a 0.5% gap attributable to loading, not characteristic, in their longer sample. Same regression, same logic, opposite sign on which coefficient survives, the disagreement traces to sample period and portfolio construction, not to the underlying math.

Drag the noise up on this fitted line and watch how easily a real slope gets buried, or a spurious one appears, in a modest sample, a big part of why two careful studies on the same question, using different multi-decade windows, came back with different signs on $\lambda_1$ versus $\lambda_2$.

What this means in practice

If covariances price returns, the value premium should shrink or vanish exactly when the world needs insurance least, because it's compensation for risk, and risk pays worst in normal times. If characteristics price returns, the premium is closer to a persistent mispricing that should erode as more capital chases it, the way many market anomalies have. Practitioners who build multi-factor books quietly bet on the covariance story every time they size a factor tilt using a risk model's loadings rather than raw firm characteristics, the choice of construction method embeds an answer to this debate whether or not the desk has read either paper.