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.