The Zero-Beta Interest Rate

Q1. What is the zero-beta rate and how does the paper construct it?

The zero-beta rate is the expected return on a portfolio of stocks that is constructed to have zero covariance with all included asset-pricing factors; in a model where the SDF innovations are spanned by those factors, the expected return of this portfolio equals the intertemporal marginal rate of substitution — the correct price for rearranging consumption over time. The paper follows a three-step procedure. First, it estimates the betas of 130 CRSP stock portfolios with respect to seven factors (Fama-French 5-factor model augmented with a bond excess return factor and a default spread factor) using the Ledoit-Wolf (2017) shrinkage estimator for the factor covariance matrix, to mitigate the rank problem arising from 130 portfolios and 574 monthly observations. Second, it uses the betas to construct the minimum-variance zero-beta portfolio — the portfolio that minimizes return variance subject to having zero exposure to each factor, exploiting all 130 portfolios. Third, it regresses the return of this portfolio on five macro instruments (T-bill yield, lagged inflation, term spread, excess bond premium [EBP], and U6 unemployment) using GMM, in an exactly-identified system in which the same instruments used to predict the portfolio return are used as moment conditions. The fitted value of this regression is the zero-beta rate — the predictable component of the zero-beta portfolio return.

The key econometric property is that the GMM procedure simultaneously estimates the factor loadings and the predictive regression in a way that accounts for the estimation error in both. The standard errors for the predictive coefficients (γ) account for the fact that the betas used to construct the zero-beta portfolio are themselves estimated. Comparatively, an infeasible OLS regression predicting the zero-beta portfolio return has nearly identical point estimates and only slightly smaller standard errors, showing that the main estimation uncertainty comes from the predictive regression rather than from the beta estimation step.

Q2. What are the key properties of the estimated zero-beta rate?

The estimated zero-beta rate has three properties that distinguish it from safe rates and make it a plausible intertemporal price: it is predictable by macro instruments (with all predictors except the unemployment rate individually significant at 5%), it is high and volatile (8.3% annually on average, std dev 9.3%, compared to a low and stable Treasury bill yield), and it co-moves in the direction economic theory predicts with macro conditions. The spread between the zero-beta rate and the expected real Treasury bill yield averages roughly 7.6% per year. The average real zero-beta rate is similar to the average real return of the CRSP market index (which averages 11.8% annualized nominal, or about 8.1% real), consistent with earlier estimates of the average zero-beta return by Hong and Sraer (2016) and Bali et al. (2017). The standard deviation of the zero-beta portfolio’s excess return over its expected value is about 2.7% per month (9.4% annualized), substantially below the standard deviation of the market return, which is why the paper can reject predictability for the zero-beta portfolio even though predicting market returns is notoriously difficult.

In terms of time-series patterns: (1) the zero-beta rate increases more than one-for-one with the Treasury bill yield (consistent with Treasury bills having money-like qualities per Nagel [2016]); (2) it is decreasing in lagged inflation; (3) it falls when a recession is likely — specifically, the unemployment rate, term spread, and EBP collectively predict the zero-beta rate so that it is particularly low when U6 unemployment is low, the yield curve is inverted, and the EBP is high, conditions associated with elevated recession risk (Kiley [2022]).

Q3. How does the zero-beta rate fit the aggregate consumption Euler equation?

The central empirical finding is that the macro instruments that best predict the real return of the zero-beta portfolio are nearly proportional to the macro instruments that best predict real consumption growth — a non-mechanical result because the two regressions are estimated entirely separately, on different data series, with no consumption data used to construct the zero-beta rate. In the linearized Euler equation, the real zero-beta rate should predict real consumption growth in proportion to 1/σ (the IES). If the prediction coefficients on the instruments for the zero-beta rate are γ₀, and the corresponding coefficients for consumption growth are γc, then the vector Δ = γ₀ − σ × γc should be close to zero. Graphically, the expected real zero-beta rate and expected real consumption growth track each other closely over the full sample (Figure 1 in the paper), while the expected real Treasury bill return bears essentially no resemblance to expected consumption growth. With five instruments (L=5), the result is not mechanical: one would need L=1 to always find a σ that makes the result hold. The result is even stronger when the sample ends in December 2019, excluding the COVID episode, because the COVID consumption collapse introduces four-to-seventeen standard deviation consumption growth realizations that attenuate the fit.

Robustness: ridge-penalized estimation (using cross-validation to minimize out-of-sample squared forecast error) substantially attenuates both expected consumption growth and the zero-beta rate toward zero, but they remain approximately proportional. The result holds across alternative specifications (different factor models, different instruments) tested in Appendix Section G.

Q4. What do the statistical Euler equation tests find?

Using Stock-Wright (2000) weak-instrument-robust GMM tests of the non-linear consumption Euler equation, the paper finds: (a) the T-bill return fails the Euler equation test for all IES values (rejected); (b) the market return fails to reject the Euler equation for almost any IES values (weak-instruments problem, consistent with Yogo [2004]); and (c) the zero-beta rate fails to reject the Euler equation for IES values below 0.5 (σ above 2, i.e. risk aversion above 2 under CRRA), and rejects for IES above 0.5. The procedure tests the instrumented non-linear Euler equation: for a conjectured value of σ, it estimates δ (the discount factor) from the unconditional Euler moment, then tests the instrumented Euler moments for each of the five instruments. The test statistic is chi-square with 5 degrees of freedom; the confidence set is the set of σ values that cannot be rejected. The zero-beta portfolio’s intermediate predictability (between the easily-predicted T-bill and the hard-to-predict market return) provides the identification that gives this test meaningful power. The paper obtains its preferred IES estimate of approximately 0.2 (σ ≈ 5) by noting that scaling the zero-beta rate down by a factor of five makes it match expected consumption growth most closely.

The test faces a boundary problem: when σ is very large (above 10), the April 2020 consumption collapse creates a very large SDF realization that dwarfs all others, making the variance-covariance matrix nearly singular and the test uninformative. For this reason, the analysis is restricted to σ ≤ 10.

Q5. What happens to the zero-beta rate after a monetary policy shock?

Using the Romer-Romer (2004) and Nakamura-Steinsson (2018) identified monetary policy shocks, the paper finds that a surprise tightening raises the nominal and real Treasury bill yield but lowers the real zero-beta rate — opposite to what the standard Euler equation with the Treasury bill predicts. Both shocks are normalized to a 100-basis-point increase in the federal funds rate on impact. The estimated effect on the real Treasury bill yield is an immediate increase of roughly the same magnitude (slightly more transitory for the Romer-Romer shock). In contrast, the real zero-beta rate falls following the shock, and the effect is larger and more persistent for the Nakamura-Steinsson shock.

This result is consistent with the consumption Euler equation applied to the zero-beta rate, because the monetary shock also lowers expected consumption growth (well-established in the impulse-response literature). The decomposition in Appendix Section E shows the mechanism: a higher Treasury bill yield raises the zero-beta rate, but the monetary tightening also flattens the yield curve and widens credit spreads (raises the EBP); since both the term spread and EBP are predictors of the zero-beta rate with large coefficients, the indirect effects through these variables dominate and lower the zero-beta rate overall. This finding resolves a tension in structural macro models: while the standard New Keynesian model uses the Euler equation with the safe rate (as in Smets and Wouters [2003, 2007]) and requires habits or wedges to match the hump-shaped consumption response to monetary shocks, the Euler equation with the zero-beta rate is satisfied without additional mechanisms. A stylized three-period New Keynesian model in Appendix Section F shows a monetary tightening can simultaneously raise the safe rate and lower the zero-beta rate through endogenous changes in the convenience spread.

Q6. Can the zero-beta rate explain valuation ratio variation without time-varying risk premia?

The paper shows that the zero-beta rate is sufficiently high, volatile, and persistent to generate the observed variation in the price-dividend ratio of a consumption claim under the assumption of a constant and small equity risk premium, reproducing the finding of Campbell (1991) that discount rates must vary — but attributing the variation to the zero-beta rate rather than to time-varying excess returns. The Campbell-Shiller decomposition of the log price-dividend ratio implies that the expected price-dividend ratio must predict either future real zero-beta rates, future excess returns, or a combination. Under the hypothesis of constant expected excess returns, the price-dividend ratio variation is driven entirely by zero-beta rate variation. In a VAR that includes the five macro instruments plus the CAPE ratio, the implied standard deviation of expected log price-dividend ratio of a consumption claim is approximately 28% — comparable to the 27% standard deviation in the Campbell-Cochrane (1999) model calibration, which achieves this variation through habit formation and time-varying risk premia. The equivalent calculation using the Treasury bill yield rather than the zero-beta rate produces a standard deviation of only 9%, consistent with Campbell’s (1991) original finding that the risk-free rate variation is insufficient.

The paper interprets this as a resolution of both the equity premium puzzle (the average zero-beta return is approximately equal to the average market return, implying a roughly zero equity premium over the zero-beta rate) and the excess volatility puzzle (the zero-beta rate generates sufficient variation in the discount rate). The trade-off is that the unexplained spread between the zero-beta rate and the Treasury bill yield — averaging 7.6% annually — is instead left as a “convenience puzzle.” The paper argues this reframing is progress because the type of models required to explain a large convenience spread on safe assets (frictions, segmented markets, money-like demand for liquid assets) are quite different from those designed to explain large time-varying risk premia.

Key Concepts

zero-beta rate

the expected return on the minimum-variance portfolio of stocks that has zero covariance with each of the included asset-pricing factors; in a model where the SDF innovations are spanned by those factors, equals the conditional expectation of the intertemporal marginal rate of substitution, making it the correct intertemporal price of consumption; measured in this paper at 8.3% annually on average using GMM on 130 CRSP stock portfolios, January 1973–December 2020.

convenience spread

the gap between the zero-beta rate and the expected real Treasury bill yield, averaging approximately 7.6% per year in this paper’s estimates; interpreted as the non-pecuniary value that holders of safe assets (Treasury bills and equivalents) receive from liquidity, collateral, and money-like services — not an expected excess return relative to consumption risk but a departure from the risk-return tradeoff for investors who value safety and liquidity independently of consumption hedging.

SDF-orthogonal equity portfolio

the minimum-variance portfolio constructed by the paper to have zero covariance with all seven asset-pricing factors (Fama-French 5 plus bond and default factors); the portfolio whose expected return equals the zero-beta rate because, by construction, no factor risk premium enters its expected return; estimated using the Ledoit-Wolf (2017) shrinkage estimator applied to 130 stock portfolios to address the rank problem in beta estimation.

Euler equation failure with safe rates

the empirical finding that the expected real Treasury bill return does not co-move with expected real consumption growth — the standard failure documented by Hansen-Singleton (1983), Dunn-Singleton (1986), and Yogo (2004) — which the paper reinterprets not as a structural failure of the representative agent model but as a consequence of using the wrong interest rate; the same Euler equation holds when applied to the zero-beta rate, which the paper argues is the correct intertemporal price.