Turning Alphas into Betas: Arbitrage and the Cross-section of Risk Thummim Cho∗ London School of Economics May 2018 Abstract What determines the cross-section of asset betas with a risk factor? Arbitrage activity plays an important role. I develop a model in which the cross-section of betas arises endogenously through the act of arbitrage. Testing the model’s predictions, I show empirically that arbitrage activity plays an important role in generating the cross-section of betas in multifactor and intermediary-based asset pricing models. The arbitrage channel for the betas can cause a cross-sectional asset pricing regression to suffer from endogeneity. ∗Department of Finance, London, UK. Email: [email protected]. I thank my advisors John Campbell, Jeremy Stein, Samuel Hanson, and Adi Sunderam for their outstanding guidance and support, and Andrei Shleifer for many insightful discussions. I also thank Tobias Adrian, Lauren Cohen, William Diamond, Erkko Etula, Wayne Ferson, Robin Greenwood, Valentin Haddad, Byoung-Hyoun Hwang, Christian Julliard, Yosub Jung, Dong Lou, Chris Malloy, Ian Martin, David McLean, Tyler Muir, Christopher Polk, Emil Siriwardane, Argyris Tsiaras, Dimitry Vayanos, Yao Zeng, and seminar participants at the Adam Smith Asset Pricing Workshop, Boston College, Columbia Business School, Cornell University, Dartmouth College, Harvard Business School, London School of Economics, University of British Columbia, University of Southern California, Hanyang University, Korea University, and Seoul National University for helpful discussions and comments. Robert Novy-Marx and Mihail Velikov generously allowed me to use their data on anomalies for preliminary empirical analyses. Jonathan Tan, Karamfil Todorov, and Yue Yuan provided superb research assistance. 1 Introduction What generates the cross-section of different riskiness of assets? That is, in the context of factor models in asset pricing, what makes an asset have a higher beta than others with respect to sys- tematic risk factors? Despite the obvious importance of these questions, most studies in empirical asset pricing treat betas as the exogenous right-hand variable that explains expected returns, stay- ing agnostic about where the betas come from. In contrast, this paper treats betas as the interesting left-hand variable—the very thing we need to explain. One obvious channel that can generate the cross-section of factor betas is the cash-flow funda- mentals: the stocks of firms whose businesses are riskier have higher factor betas.1 But what are the other important channels? And given those channels, is it valid to treat betas as an exogenous regressor in an asset pricing regression that explains expected returns? This paper explores a new channel for the cross-section of betas: the act of arbitrage that turns “alphas” into “betas.”2 This channel arises naturally from reinterpreting limits to arbitrage (Shleifer and Vishny, 1997) through the lens of a factor model: arbitrage with limited capital is endogenously risky because it causes a mispriced asset with an “alpha” (abnormal return) to attain an endogenous “beta” with the risk factors to which arbitrage capital is exposed. Applying this to the cross-section of betas, assets may have higher betas because they had higher alphas prior to arbitrage and attract correspondingly more arbitrage activity. A second motivation for studying the “alphas-into-betas” channel is its implications for asset pricing tests. A cross-sectional asset pricing regression attributes an asset’s expected return in excess of the risk-free rate to its betas with respect to risk factors, with alpha being the residual: e E [ri ] = λ1βi;1 + ::: + λJ βi;J + αi: Hence, identifying the price of risk λj requires βj to be “exogenous,” cross-sectionally uncorrelated with α. However, if the act of arbitrage on the test assets turns their αs into βj during the sample period of the regression, the exogeneity assumption ^ fails due to simultaneity, biasing the price of risk estimate λj upward. My key contribution is to flesh out this alphas-into-betas channel for the cross-section of betas and to highlight its empirical relevance for factor models suggested in the literature. 1Campbell and Mei (1993) and Campbell, Polk, and Vuolteenaho (2010). 2Andrew Lo first used this expression to refer to limits to arbitrage occurring in practice (Khandani and Lo, 2011). 1 To formalize my argument, I develop a model in which the cross-section of betas arises entirely through the act of arbitrage. In my three-period model, a representative arbitrageur trades a contin- uum of underpriced assets but faces exogenous wealth and funding shocks that generate variation in the capital she can deploy. The assets have a known terminal value, but in the early (time 1) and intermediate (time 2) trading periods, they are priced according to deterministic, downward- sloping demand curves formed by behavioral investors. Crucially, the severity of the underpricing in the absence of arbitrage capital—the “pre-arbitrage” alpha—differs across the assets. In this model, the cross-section of betas with respect to arbitrage capital shocks—the sole risk factor—arises entirely through the act of arbitrage. Furthermore, an asset with a higher pre- arbitrage alpha attains a higher endogenous “post-arbitrage” beta with respect to arbitrage capital shocks (Proposition 1). A higher pre-arbitrage alpha means that a larger fraction of the asset is held by the arbitrageur, since the arbitrageur plays a larger price-correcting role in that asset in equilib- rium. This, however, means that the asset’s price responds more to the variation in the arbitrage capital. From this explanation follows the prediction that the cross-section of average arbitrage positions in the assets should also explain their post-arbitrage betas (Proposition 2). Furthermore, this endogenous post-arbitrage beta arises only when the arbitrageur is capital- constrained (Proposition 3); an arbitrageur with does not generate endogenous betas in the assets when she has “deep pockets.” An asset pricing test that does not account for this endogenous generation of betas can lead to a false discovery of a pricing factor (Proposition 4). Using these predictions, I study the extent to which the alphas-into-betas channel explains the cross-section of betas in actual factor models. In particular, I study the beta exposures of 40 equity “anomalies”— the “long” and “short” portfolios (top and bottom deciles) of 20 anomaly characteristics known to predict abnormal returns—to the five factors of Fama and French (FF) (2015) and the funding-liquidity factor of Adrian, Etula, and Muir (2014). My main approach is to study the anomalies’ betas before and after 1993, the approximate year when arbitrageurs such as the long-short equity hedge funds began trading these anomalies according to short interest data, but I also exploit different sample years used in the academic papers that first studied each anomaly.3 3Chordia, Roll, and Subrahmanyam (2011) and Chordia, Subrahmanyam, and Tong (2014) were the first to suggest that anomaly trading grew rapidly around 1993. McLean and Pontiff (2016) were the first to exploit the cross-sectional differences in the publication sample years of anomalies. 2 2 .5 .5 0 0 0 -2 -.5 -.5 2 2 2 Change in CMA Beta -4 Change in RMW Beta R adj = 0.76 R adj = 0.31 R adj = 0.59 Change in Funding Beta -1 -1 -15 -10 -5 0 5 10 -15 -10 -5 0 5 10 -15 -10 -5 0 5 10 Pre-1993 FF 5-Factor Alpha Pre-1993 FF 5-Factor Alpha Pre-1993 FF 5-Factor Alpha Figure 1: Turning “Alphas” into “Betas” The figures show that, in the cross-section of 40 equity anomalies, the pre- to post-1993 change in the factor beta is predicted by the pre-1993 alpha. RMW and CMA are the profitability and investment factors of Fama and French (FF) (2015). The funding beta is with respect to the funding-liquidity factor of Adrian, Etula, and Muir (2014). The alpha is with respect to the five-factor model of FF. The change in the beta is estimated as the post-1993 beta minus the pre-1993 beta multiplied by the shrinkage factor of 0.6. Tables 5 and 9 show the regression counterparts that do not restrict the coefficient on the pre-1993 beta to be 0.6. The first two figures use monthly data over 1974-2016; the last figure uses quarterly data over the same period. The act of arbitrage plays a prominent role in determining the cross-section of factor betas. To show this for the five-factor model of FF, I first ask which of the five factors represents risks borne by arbitrageurs who trade the anomalies. I find that long-short equity hedge funds—the canonical equity anomaly arbitrageurs—are positively exposed to profitability and investment factors (RMW and CMA) but neutralize their exposures to market, size, and value factors (MKT, SMB, and HML). Intuitively, in a more familiar three-factor context (Fama and French, 1993), these arbitrageurs trade HML but not MKT and SMB. Since the new RMW and CMA factors are designed to replace HML, the arbitrageurs have positive exposures to RMW and CMA but not to the other factors.4 Consistent with this observation, an anomaly’s pre-1993 five-factor alpha predicts an increase in the post-1993 beta with respect to RMW and CMA (Figure 1; Proposition 1) but not with respect to MKT, SMB, and HML. That is, anomalies with positive (negative) pre-1993 alphas see their RMW and CMA betas increase (decrease) after arbitrage begins around 1993. For example, an anomaly with a positive (negative) pre-arbitrage alpha but no fundamental RMW beta attains a positive (negative) post-arbitrage endogenous RMW beta since a positive RMW shock increases the level of arbitrage capital and exerts positive (negative) price pressure on the anomaly.