One Tent-Shaped Factor: Cochrane and Piazzesi on Bond Risk Premia

Fama and Bliss had shown that the forward-spot spread for a given maturity predicts that same bond's excess return. Campbell and Shiller had shown the yield spread predicts bond returns with the wrong sign relative to theory. Both papers used one number from the curve to predict one bond.

John Cochrane and Monika Piazzesi asked what happens if you use the whole curve at once. The answer was a substantially stronger predictor, a strange and beautiful shape, and a serious problem for the standard models.

The problem: one number is throwing information away

Fama and Bliss's regression is maturity-by-maturity. To predict next year's excess return on the four-year bond, use the four-year forward spread. To predict the five-year bond, use the five-year spread. Each bond gets its own single predictor.

That is tidy, but it is obviously wasteful. The yield curve is a rich object with information at every point. Why should the return on the four-year bond depend only on the four-year forward rate and be entirely unaffected by what the one-year and five-year forwards are doing? There is no reason to think the information is packaged that neatly. Cochrane and Piazzesi's move is simply to stop assuming it is.

The key idea via analogy: let the whole curve vote

The regression they run is straightforward. To predict the excess return on a bond over the coming year, use all the forward rates on the curve, the one-year, two-year, three-year, four-year and five-year forwards, together, as predictors.

Two things fall out, and both are remarkable.

First: the predictability is much stronger. Using the whole curve, they report R-squared values reaching as high as roughly 0.44 for one-year excess returns on bonds of one to five years' maturity. That is a large number for return predictability of any kind. Fama and Bliss's single-spread regressions produce a fraction of it. Most of the forecastable variation in bond returns had been sitting there unused.

Second, and this is the famous part: it is a single factor, and it has a shape.

When Cochrane and Piazzesi look at the regression coefficients on the five forward rates, they find that essentially the same pattern of weights predicts the return on the two-year bond, the three-year bond, the four-year bond and the five-year bond. The bonds differ only in how strongly they load on the pattern (longer bonds, more exposure, exactly as you would expect for a duration risk premium), not in the pattern itself.

So there is one combination of forward rates, one number you can compute each month, that predicts the excess return on every Treasury bond. They call it the return-forecasting factor.

And when you plot the weights against maturity, they trace out a tent: negative at the short end, rising to a peak in the middle of the curve, falling away at the long end. It looks like a single upside-down V. This is now known as the tent-shaped factor, and it is one of the more distinctive pictures in empirical finance.

The analogy: previous work asked one witness per case. Cochrane and Piazzesi asked the whole panel and discovered that the panel has a consistent voting rule, and that the same rule, weighted a bit differently, decides every case.

The factor is countercyclical. It rises in bad economic times, when investors are frightened and demand a lot to hold duration, and falls in good times. It also, intriguingly, predicts stock returns, hinting that whatever it is measuring is a broad, economy-wide swing in the price of risk rather than something peculiar to bonds.

The part that should worry a modeller

Here is the sting in the tail, and it is the reason this paper is more than a better regression.

Litterman and Scheinkman showed the yield curve is essentially three factors: level, slope and curvature. Every mainstream term structure model is built on that premise, using three (or a few) factors extracted from the curve.

Cochrane and Piazzesi report that an important component of their tent-shaped return-forecasting factor is not captured by level, slope and curvature. It contains information about future bond returns that the standard three factors do not see.

That is a direct problem for the entire modelling apparatus. A three-factor model, by construction, cannot express a risk premium that depends on a fourth thing. If the predictor of bond returns is partly invisible to the factors your model is built from, then your model cannot produce the risk premium dynamics that the data exhibits. This is a structural indictment, not a calibration complaint, and it sparked a long and unresolved argument about whether the tent factor is a real, "unspanned" source of risk premium information or a statistical mirage.

Why it mattered

It is the strongest evidence for bond return predictability. Whatever your priors, an R-squared reaching 0.44 for one-year returns is a striking number, and it made time-varying bond risk premia impossible to dismiss as a marginal curiosity.
The single-factor structure is the elegant part. One number predicting every bond's return is a far stronger and more disciplined claim than a pile of maturity-specific regressions. It says the bond risk premium is essentially one-dimensional, which is a real structural insight and not just a fitting exercise.
It broke the three-factor consensus. The finding that the return-forecasting factor is partly unspanned by level, slope and curvature launched the "unspanned risk premia" literature and forced modellers to confront the possibility that the object they were modelling was incomplete.
It is directly tradeable in principle. Compute the factor, go long duration when it is high, reduce when it is low. Systematic fixed income strategies built on curve-based timing are commercial descendants of this line of work.
The link to equities is provocative. A bond-market factor that also predicts stock returns suggests a single, common, countercyclical price of risk moving across asset classes, which is what macro-finance theories of the risk premium have always claimed.

The honest limitations

This paper has attracted more serious econometric criticism than almost any other in the field, and the criticism is not frivolous.

The statistics are exactly the kind that flatter themselves. Overlapping annual returns from a few decades of data. Highly persistent predictors. Five regressors chosen from a small, highly correlated set. This is textbook territory for overstated R-squared values and understated standard errors. Subsequent work has argued forcefully that the true out-of-sample predictability is far weaker than the in-sample numbers suggest, and some studies find the factor's real-time performance disappointing.
The tent shape may be less magical than it looks. With five highly collinear regressors, the estimated coefficient pattern is not very stable. Change the sample, change the data source, change the set of forwards used, and the tent can shift or lose its clean form. It is not clear the shape is a deep structural fact rather than an artefact of the particular regression on the particular sample.
In-sample is not out-of-sample. The factor was found by looking at the data. The honest test is whether it would have worked for someone standing in the past with only the data available then, and the evidence there is much more mixed. This is the standard, and standardly fatal, objection to any discovered predictor.
"Unspanned" is contested. Whether the return-forecasting factor genuinely contains information outside level, slope and curvature, or whether it is picking up measurement error and noise in the fitted yield curve (which, remember, is itself a smoothed estimate and not raw data), is an argument that is still live. If the extra information is noise, the most exciting part of the paper evaporates.
Predictability is not profit. The premium is compensation for genuine risk, earned by being long duration precisely when the world looks frightening. Harvesting it systematically means enduring the drawdowns it is paying you for, and no regression coefficient tells you how large those get.
One market, one era. US Treasuries over a period that included the great disinflation. A large part of the excess return to duration in that sample came from a historic, one-directional decline in yields, and it is fair to be sceptical about how the factor behaves in a world that does not repeat that.

The one-line takeaway

Cochrane and Piazzesi found that a single tent-shaped combination of forward rates predicts the excess return on every Treasury bond, with far more power than anyone had extracted before, and that this predictor is partly invisible to the level, slope and curvature factors on which nearly every term structure model is built.