Characteristics Are Covariances: Kelly, Pruitt and Su Reconcile Risk and Anomalies

There is a fight at the centre of asset pricing that has been running for thirty years, and it goes like this.

Cheap stocks (low price relative to book value) earn higher returns than expensive ones. Everyone agrees on the fact. The argument is about why.

The risk camp, led by Fama and French, says: cheap stocks are riskier. They are distressed, unglamorous, fragile companies, and they load more heavily on some underlying risk factor. Investors demand extra return for bearing that risk. The characteristic (cheapness) is a proxy for a risk exposure, and the exposure is what earns the premium.

The characteristics camp, most forcefully argued by Daniel and Titman, says: no. The extra return attaches to the characteristic itself, not to any risk exposure. Cheap stocks earn more because they are cheap, full stop, which really means the market is mispricing them.

For thirty years, these looked like mutually exclusive explanations, and enormous effort went into testing one against the other. Bryan Kelly, Seth Pruitt and Yinan Su came at it with a machine learning method and produced a result that reframes the fight: the characteristic and the risk exposure need not be different things at all.

The problem: you cannot see the risk factors, and betas move

The reason this argument has been so hard to settle is a measurement problem, and it has two halves.

First, the true risk factors are invisible. Fama and French did not discover the risk factors; they constructed portfolios (small minus big, cheap minus expensive) and proposed them as proxies for whatever the real risk factors are. If those proxies are imperfect, and they surely are, then any test using them is testing the proxy, not the theory.

You could instead use principal component analysis, a standard statistical method that extracts the dominant common movements from a panel of returns without you having to name them in advance. That gets you factors that are, by construction, the ones actually driving co-movement. But PCA has a crippling defect for this purpose.

Second, exposures change over time, and PCA assumes they do not. A company's exposure to a risk factor is not fixed. A firm that was expensive and stable in 2010 might be cheap and distressed by 2015. Its risk exposure has genuinely changed. Standard PCA estimates one fixed loading per stock over the whole sample, which is exactly wrong for the question at hand, because the entire interesting dynamic is the way exposures move as characteristics move.

So: PCA finds the right factors but cannot handle time-varying exposures. Traditional factor models handle exposures but have to guess the factors.

The key idea via analogy: let the characteristics tell you the beta

Kelly, Pruitt and Su's method, which they call Instrumented Principal Component Analysis, threads this needle with one move.

Do not estimate a stock's risk exposure as a free-floating number. Estimate it as a function of the stock's observable characteristics.

In plain terms: rather than asking "what is Apple's loading on factor 3?" and fitting a single number, you ask "what is the rule that turns a company's characteristics (its size, its valuation ratio, its momentum, its profitability, its volatility) into its loading on factor 3?" You estimate that rule once, from all the stocks together. Then any individual stock's loading at any moment is simply what the rule outputs when you plug in that stock's characteristics at that moment.

The analogy: instead of measuring every runner's speed individually, you learn the general relationship between a runner's height, weight, training load and their speed. Then to know how fast anyone is running today, you look up today's values of their attributes and apply the rule. When their attributes change, their predicted speed changes automatically, and you never had to re-measure them.

This solves both problems at once, and the elegance of it is the paper's real appeal.

The factors are latent, extracted from the data by a PCA-like procedure, so you are not stuck with somebody's hand-built proxies. The loadings are dynamic, because they are functions of characteristics that themselves change every month. The characteristics are not being used as predictors of return directly. They are being used as instruments for the unobservable risk exposures, which is where the method's name comes from.

The finding, and why it reframes the debate

The authors' headline empirical result is that a small number of IPCA factors, roughly five, explain the cross-section of average returns substantially better than the standard factor models.

And here is the crucial part. Once you account for exposures to those latent factors, the anomaly intercepts largely disappear. In the language of the field: the alphas associated with the characteristics shrink toward zero and lose their statistical significance.

Translate that. The extra return earned by cheap stocks is fully accounted for by the fact that cheapness implies a particular loading on a genuine common risk factor. There is no leftover, unexplained premium attaching to the characteristic itself.

That is a win for the risk camp. But it is a subtle one, and it explains why the debate was so intractable. The reason it was so hard to separate "characteristics" from "covariances" is that, in this framework, they are the same thing seen from different angles. The characteristic is the observable signature of the risk exposure. Asking whether the premium attaches to the characteristic or to the covariance is like asking whether a fever is caused by the illness or by the high temperature. The characteristic tells you what the exposure is. The exposure is what earns the premium.

Hence the title. Characteristics are covariances.

Why it mattered

• It offers a resolution to a thirty-year argument. Not a knockout blow (nothing in this field is), but a genuinely new framing in which the two sides are describing one mechanism rather than two competing ones.
• It is machine learning that produces economics, not just predictions. IPCA is a dimension-reduction technique with a clever twist, and its output is an interpretable factor structure with an economic claim attached. That is much harder, and much rarer, than getting a neural network to forecast returns.
• It handles the factor zoo differently. Rather than testing hundreds of characteristics one at a time for significance, IPCA absorbs them all as instruments for a small number of latent factors. The zoo is not tamed by culling it but by showing that its inhabitants are all reflections of a handful of underlying animals.
• It is used in practice. IPCA gives a risk model with dynamic, characteristic-driven betas, which is exactly what a portfolio manager wants, and it is more responsive than a static factor model.

The honest limitations

• The relationship between characteristics and loadings is assumed to be linear. IPCA's central rule maps characteristics to betas through a linear function. Gu, Kelly and Xiu's own later work suggests the truth is meaningfully non-linear, which means IPCA may be imposing a shape that the data does not have. (Kelly and co-authors went on to build autoencoder-based versions precisely to relax this.)
• The latent factors are statistical, not economic. IPCA extracts factors that explain co-movement. It cannot tell you what they are. Calling them "risk" is an interpretation, and a somewhat convenient one. A behavioural economist can look at the same latent factors and call them "common mispricing," and nothing in the statistics refutes them. The model reconciles characteristics with covariances, but it does not prove the covariances represent risk that anyone should be compensated for. This is the deepest objection, and the paper cannot fully answer it.
• The characteristics are themselves the products of a mined literature. The instruments fed into the model are the anomaly characteristics that decades of published research turned up, with all the survivorship and multiple-testing baggage that implies.
• It has been challenged. Subsequent commentary has questioned aspects of the specification and how much of the result is driven by modelling choices, which is entirely healthy and entirely normal for a paper making a claim this large.
• Explaining returns is not predicting them. IPCA's success is largely about accounting for average returns in-sample with a small factor structure. That is a different, and easier, task than forecasting which stocks will outperform next month.

The one-line takeaway

Kelly, Pruitt and Su built a version of principal component analysis in which a stock's exposure to the hidden risk factors is a function of its observable characteristics, allowing betas to move as the company changes, and found that once you do this, the anomaly premia largely vanish into risk exposures, suggesting that the thirty-year fight between "characteristics" and "covariances" was a fight between two descriptions of the same thing.