The Cross-Section and Time-Series of Stock and Bond Returns ∗

We propose a three-factor model that jointly prices the cross-section of returns on portfolios of stocks sorted on the book-to-market dimension, the cross-section of government bonds sorted by maturity, and time series variation in expected bond returns. The main insight is that innovations to the nominal bond risk premium price the book-to-market sorted stock portfolios. We argue that these innovations capture business cycle risk and show that dividends of the highest book-to-market portfolio fall substantially more than those of the low book-tomarket portfolio during NBER recessions. We propose a structural model that ties together the nominal bond risk premium, the cross-section of book-to-market sorted stock portfolios, and recessions. This model is quantitatively consistent with the observed value, equity, and nominal bond risk premia. First version: February 2009. For excellent research assistance, we thank Rustom Irani and Michelle Zemel. We thank Jaewon Choi and Ed Altman for sharing data with us. We thank the Editor, Jules van Binsbergen, John Campbell, John Cochrane, George Constantinides, Greg Duffee, Eugene Fama, Lars Hansen, John Heaton, Martin Lettau, Lars Lochstoer, Tobias Moskowitz, Stavros Panageas, Lubos Pastor, Monika Piazzesi, Maxim Ulrich, Pietro Veronesi, Bas Werker, Mungo Wilson, and seminar participants at Tilburg University, APG Investments, ULB ECARES, Temple University, University of Texas at Austin, New York University Stern finance, Boston University, Chicago Booth, University of Vienna, Erasmus University of Rotterdam, U.C. Berkeley Haas, New York University Stern’s macro lunch, Washington University at St. Louis, the Adam Smith Asset Pricing Conference in Oxford, the macro-finance conference at the University of Minnesota, the Amsterdam Asset Pricing Retreat, the Society for Economic Dynamics in Istanbul, the CEPR Financial Markets conference in Gerzensee, the European Finance Association conference in Bergen, and the American Finance Association conference in Atlanta for useful comments and suggestions. Booth School of Business, University of Chicago, Chicago, IL 60637; ralph.koijen@chicagobooth.edu; http://faculty.chicagobooth.edu/ralph.koijen. Koijen is also associated with Netspar (Tilburg University). Department of Finance, Anderson School of Management, University of California at Los Angeles, Box 951477, Los Angeles, CA 90095; hlustig@anderson.ucla.edu; http://www.econ.ucla.edu/people/faculty/Lustig.html. Department of Finance, Stern School of Business, New York University, 44 W. 4th Street, New York, NY 10012; svnieuwe@stern.nyu.edu; http://www.stern.nyu.edu/ svnieuwe. As long as some investors have access to both stock and bond markets, the absence of arbitrage opportunities imposes cross-market restrictions on the stochastic discount factor, henceforth SDF. Despite tremendous progress in the separate modeling of SDFs for bond markets and stock markets, the cross-market restrictions between stocks and bonds are typically not imposed. As a result, the state-of-the art bond pricing model does not price stocks and the state-of-the-art equity pricing model does not price bonds. We propose a parsimonious no-arbitrage SDF model that exploits these cross-market restrictions to learn about the pricing of risk across stock and bond markets. We find that three factors are able to account for most of the average return differences between book-to-market sorted equity portfolios, the aggregate stock market, and maturity-sorted government bond portfolios. The first factor is a stock market return factor familiar from the Capital Asset Pricing Model, the second factor is the level of the nominal term structure, and the third factor is a proxy for the nominal bond risk premium, the Cochrane and Piazzesi (2005) factor (CP factor). The first contribution of the paper is to document that value portfolio returns have a higher covariance with innovations in the nominal bond risk premium than growth portfolio returns; see Figure 1. Combined with the positive price of risk we estimate on innovations to the CP factor, this differential exposure results in a value premium. Differential exposure to level shocks accounts for the difference between returns on long-term and short-term bonds, consistent with Cochrane and Piazzesi (2008), while exposure to the market return accounts for the aggregate equity premium. The three-factor model reduces mean absolute pricing errors from 4.7% per year in a risk-neutral benchmark economy to 0.4% per year (Section II). In Section V, we present similar results for different sub-samples and for different equity and bond portfolios. In particular, our model does a very good job pricing corporate bond portfolios differing by credit rating, in ad-