Broker-Dealer Leverage and the Cross-Section of Stock Returns

Transcription

1 Broker-Dealer Leverage and the Cross-Section of Stock Returns Tobias Adrian Erkko Etula Tyler Muir January 2011 Abstract We document that average stock returns can be largely explained by their covariance with the aggregate leverage of security broker-dealers. Our single factor broker-dealer model compares favourably with standard multi-factor models in the cross-section size and book-to-market portfolios and outperforms such models when considering momentum and industry sorted portfolios. We show how to tie leverage to the rst order conditions of active investors portfolio choice and derive leverage as a state variable in the ICAPM. We interpret the risk captured by shocks to broker-dealer leverage as a re ection of unexpected changes in broader economic conditions that determine investors marginal utility of wealth. Keywords: cross sectional asset pricing, nancial intermediation, ICAPM JEl codes: G1, G12, G21 Capital Markets Function, Research and Statistics Group, Federal Reserve Bank of New York, 33 Liberty Street, New York, NY and Kellogg School of Management, Department of Finance. This paper is a revised combination of two previously circulated papers: Funding Liquidity and the Cross Section of Stock Returns (Adrian and Etula, 2010) and Intermediary Leverage and the Cross-Section of Expected Returns (Muir, 2010). We would like to thank Ariel Zucker for outstanding research assistance. We also thank Andrea Eisfeldt, Francesco Franzoni, Taejin Kim, Arvind Krishnamurthy, Ravi Jagannathan, Annette Vissing-Jorgensen, Jonathan Parker, Dimitris Papanikolaou, Stefan Nagel, Hans Dewachter, Wolfgang Lemke and seminar participants at Kellogg School of Management, the Bank of England, the European Central Bank, the Federal Reserve Bank of Boston, the Bank of Finland, HEC Paris, and Moody s KMV for useful comments and suggestions. The views expressed in this paper are those of the authors and do not necessarily re ect the position of the Federal Reserve Bank of New York or the Federal Reserve System.

2 1 Introduction Security brokers and dealers are active investors that aggressively adjust their risk exposures in response to changes in economic conditions. To the extent that the portfolio choice of broker-dealers satis es an optimality condition, broker-dealer aggregate balance sheets can be expected to contain information about economy-wide investment opportunities. By implication, it may be possible to use broker-dealer balance sheet aggregates to proxy for the stochastic discount factor that re ects the marginal utility of wealth in di erent states of the economy. In this paper, we present empirical evidence to support this hypothesis. Our crosssectional tests show that broker-dealer nancial leverage constitutes a state variable that passes important asset pricing tests. Speci cally, we show that shocks to brokerdealer leverage can alone explain the average returns on a wide variety of test assets, including portfolios sorted on size, book-to-market, momentum, and industries. The broker-dealer leverage factor is successful across all cross-sections in terms of high adjusted R-squared, low cross-sectional pricing errors (alphas), and prices of risk that are signi cant and remarkably consistent across portfolios. When taking all these criteria into account, our single factor outperforms standard multi-factor models tailored to price these cross-sections (see Fig. 1 for an example of our leverage factor s pricing performance in the cross section that spans 65 common test portfolios). However, we do not seek to argue that shocks to broker-dealer leverage drive aggregate asset prices. Rather, adjustments in broker-dealer leverage endogenously re ect changes in underlying economic state variables that may not be easily captured by an econometrician. While our strong empirical results are consistent with a number of rationalizations, our preferred theory builds on the intertemporal capital asset pricing model (ICAPM) 1

3 Realized Mean Return Factor Broker Dealer Model S1B5 Coal S3B5 Mom10 S2B5 S1B4 S2B4 S4B5 S2B3 S1B2 S1B3 S3B4 S4B4 ElcEq Beer S3B3 Oil S4B3 Food S2B2 S3B2 Carry Mom9 Mom Games Hlth Servs Rtail Mom8 Meals S5B5 Clths ChemsS4B1 S5B2 S5B4 S4B2 Fin FabPr Telcm Paper BusEq Mom6 Rm Rf Mom4Whlsl Mom3 S5B3 Books TxtlsHshld Mines Mom7 Util Cnstr Trans Autos S5B1 HML S3B1 Steel Mom5 S2B1 Mom2 SMB Other S1B1 Smoke Mom Predicted Expected Return Figure 1: Realized vs. Predicted Mean Returns. We plot the realized mean excess returns of 65 portfolios (25 Size and Book-to-Market Sorted Portfolios, 30 Industry Portfolios, and 10 Momentum Sorted Portfolios) and 4 factors (market, SMB, HML, MOM) against the mean excess returns predicted by a 1-factor broker-dealer model. The sample period is Q1/1968-Q4/2009. of Merton (1973). In the ICAPM, the endogenous portfolio choice of active, forwardlooking investors provides a window to expectations about economic conditions and investment opportunities and is therefore an important state variable for asset pricing. In our application of the ICAPM, active investors are risk-neutral leveraged nancial intermediaries subject to borrowing constraints related to the Value at Risk (VaR) of their balance sheet. The model shows that these risk constraints link economy-wide expectations of investment opportunities directly to the observable portfolio choice of 2

4 active investors. In this way, changes in leverage capture shifts in the marginal utility of wealth. By implication, investors require higher compensation for holding assets whose returns comove strongly with shocks to broker-dealer leverage. The remainder of the paper is organized as follows. Section 2 provides a discussion of the related literature, reviewing a number of alternative rationalizations for the link between nancial intermediary leverage and aggregate asset prices. Section 3 shows how such a link can be motivated by a two-agent extension of the ICAPM. Section 4 describes the data and section 5 conducts a number of asset pricing tests in the cross-section of stock returns. Section 6 concludes. 2 Financial Intermediaries and Asset Prices 2.1 Broker-Dealers and Procyclical Leverage Financial institutions serve an important economic role in the allocation of savings from ultimate savers to ultimate borrowers. As agents of savers, nancial institutions are subject to constraints that can be interpreted as outcome of agency problems between savers and their nancial institutions. The managers of the nancial institutions can be expected to take dynamically optimal, rational decisions to maximize the institution s value subject to such constraints, which in practice take the form of restrictions on risk management and compensation. The pro t-maximizing balance sheet management of nancial institutions is in stark contrast to the portfolio choice of households, which may be subject to many potential issues, such as slow adjustment (Jagannathan and Wang, 2006; Calvet, Campbell and Sodini, 2009), habits (Campbell and Cochrane, 1999), limited participation (Vissing-Jorgensen, 2002), and model uncertainty (Hansen and Sargent, 2007). Financial institutions, on the other hand, optimize their portfolio holdings frequently based on sophisticated models and extensive historical data on the 3

5 distribution of returns, and report timely and accurate data to their regulators. The aggregate portfolio choice of nancial institutions can thus be expected to provide a better empirical window to the time-varying investment opportunity set than aggregate measures based on household behavior. Empirically, Adrian and Shin (2010) show that security broker-dealers adjust their nancial leverage aggressively in response to changing economic conditions. Unlike the dynamics of household consumption, time-variation in U.S. broker-dealer aggregate balance sheet components is easily quanti ed using the Federal Reserve s Flow of Funds quarterly database. These data reveal that broker-dealers balance sheet management practices lead to high leverage in economic booms and low leverage in economic downturns: that is, broker-dealer leverage is procyclical. 1 Table 1: Broker-dealer leverage is pro-cyclical. The correlation of broker-dealer leverage growth with a selection of state variables, Q1/1968-Q4/2009. Correlation of Broker-Dealer Leverage Growth Market Pro t Financials Broker-Dealer Return Growth Return Asset Growth The procyclicality of broker-dealer leverage can be understood in terms of its comovement with other state variables, which we compute in the Table 1. We display the correlations of broker-dealer leverage growth with the value-weighted return on the U.S. equity market, pro t growth of the U.S. nancial sector (obtained from the Bureau of Economic Analysis), the value-weighted stock return on the U.S. nancial sector, and the asset growth of broker-dealers. Positive correlation in each case shows that leverage growth is pro-cyclical and related to wealth, compensation, and investment opportunities. The strong correlation of leverage growth with broker-dealer asset 1 We measure leverage by the the ratio of nancial assets to equity. 4

6 growth moreover suggests that the ability of broker-dealers to borrow is close to the amount actually borrowed, making leverage a proxy for funding liquidity. 2.2 Literature on Financial Intermediaries and Aggregate Asset Prices While our version of the ICAPM and the accompanying empirical tests are new, a number of other papers have derived theoretical models relating nancial intermediaries to asset prices. While none of these give the direct empirical implications that we test, they are broadly consistent with our ndings that low leverage states are states where the marginal utility of wealth is high and therefore assets that covary positively with leverage earn higher average returns. He and Krishnamurthy (2010) derive an intermediary CAPM in which intermediaries are marginal investors and the return on intermediary wealth can be used as a pricing kernel. Since our leverage factor is positively related to wealth and compensation in the intermediary sector, our results support this view as well. Geanakoplos (2009) shows how leverage can drive the business cycle and how asset prices can depend on the ability of optimists to use leverage. While Geanakoplos s model does not deliver cross-sectional asset pricing implications, it gives an intuitive justi cation for why times of growing broker-dealer leverage are good times with low marginal utility of wealth. Our results are also closely linked to the literature on limits of arbitrage (Shleifer and Vishny, 1997): broker-dealers are forced to decrease leverage precisely when prices are decreasing and expected returns are rising. Gromb and Vayanos (2002) show how tightening of funding constraints can lead to losses of liquidity. Again, while such theoretical models aid in our intuitive understanding of the empirical results, they do not give the explicit formal cross-sectional prediction that leverage should be priced in 5

7 the cross-section of returns. To the best of our knowledge, we are the rst to conduct a cross-sectional asset pricing test where intermediaries e ective risk aversion enters into the aggregate pricing kernel. More generally, our paper is related to the emerging strand of literature investigating the impact of balance sheet constraints on aggregate asset prices. In addition to the papers mentioned above, examples of such studies include Aiyagari and Gertler (1999), Basak and Croitoru (2000), and Caballero and Krishnamurthy (2004). The approach taken in this paper is closely related to the endogenous ampli cation mechanisms via the margin spiral of Brunnermeier and Pedersen (2009), where margin constraints are time-varying and can serve to amplify market uctuations through changes in riskbearing capacity. In this strand of literature, the empirical studies most relevant to ours are the investigations of foreign exchange markets by Adrian, Etula and Shin (2009) and of commodity markets by Etula (2009). Both papers introduce risk-based balance sheet constraints in a two-agent CAPM, generating time-varying e ective risk aversion that can be expressed in terms of observable state variables. Danielsson, Shin and Zigrand (2009) endogenize risk and e ective risk aversion simultaneously by solving for the equilibrium stochastic volatility function in a setting with value-at-risk constraints on nancial intermediaries. 2.3 Literature on Cross-Sectional Asset Pricing The idea that long-term investors care about shocks to investment opportunities originates in the ICAPM of Merton (1969, 1971, 1973). Kim and Omberg (1996) provide closed form solutions to a particular case of Merton s dynamic portfolio allocation behavior. Campbell (1993) solves a discrete-time empirical version of the ICAPM with a stochastic market premium, writing the solution in the form of a multifactor model. 6

8 Campbell (1996) tests this model on industry portfolios, but nds little improvement over the CAPM. Other empirical studies of the ICAPM include Li (1997), Hodrick, Ng, and Sengmueller (1999), Lynch (1999), Brennan, Wang, and Xia (2004), Guo (2002), Chen (2003), Ng (2004), Ang, Hodrick, Xing, Zhang (2006, 2009), Adrian and Rosenberg (2008), and Bali and Engle (2009). We also contribute to the vast literature of competing explanations for the size and value e ects (Fama and French, 1993), and the momentum e ect (Jegadeesh and Titman, 1993, 2001; Rouwenhorst, 1998, 1999; Chui, Titman, and Wei, 2000). It is well known that the Arbitrage Pricing Theory (APT) of Ross (1976) allows any pervasive source of common variation to be a priced risk factor. Fama and French (1993) follow the APT insight and describe the average returns on portfolios sorted by size and value using a three-factor speci cation, which complements the market model with a size factor and a value factor. However, since the APT is silent about the determinants of factor prices of risk, a model such as that of Fama and French cannot explain why the risk premia associated with certain factors are positive or negative, or the sources of the premia. The same caveat applies to other APT-motivated factor models constructed to explain asset pricing anomalies, including the the momentum factor of Carhart (1997). The failures of standard asset pricing models can also be interpreted in behavioral terms by arguing that the size, value, and momentum e ects are due to mispricing. Lakonishok, Shleifer, and Vishny (1994), for example, suggest that investors irrationally extrapolate past earnings growth and thereby overvalue companies that have performed well in the past. DeBondt and Thaler (1985, 1987), Barberis, Shleifer and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), Hong and Stein (1999), and Hong, Lim and Stein (2000) suggest that both momentum and long-term reversal are 7

9 the results of mispricing. In this paper, we seek to avoid these alternative explanations. Our preferred theoretical motivation combines insights from the literature on intermediaries and asset pricing with the literature on portfolio choice to develop a version of Merton s ICAPM based on the rst-order conditions of two rational investors, a long-horizon investor who is risk neutral but subject to a balance sheet risk constraint, and a myopic investor with constant relative risk aversion. Our empirical section may therefore be interpreted as an investigation of the extent to which cross-sectional asset pricing anomalies can be rationalized by intertemporal hedging considerations relevant for long-term investors. 3 ICAPM Motivation In the previous section, we considered a number of alternative explanations for why shocks to broker-dealer leverage might be priced in the cross-section of stocks. In this section, we show more formally how the insights from Merton s (1973) ICAPM may be used to yield some structure for our empirical tests. We make the basic conceptual assumption that equilibrium asset prices are determined by the interaction of two types of agents, A and P. Each agent j has a value function J j (w j ; x), where w j denotes wealth and x is a vector of state variables. Speci cally, x describes the stochastic evolution of the investment opportunity set. We denote by dr i the return on asset i in excess of the risk-free rate. Using this notation, the ICAPM Euler equations can be written as: i = A Cov dr i ; dwa + f A w A x Cov dr i ; dx ; i = P Cov dr i ; dwp + fx P Cov dr i ; dx ; w P where i is the expected excess returns on asset i, and j = J j www j J j w and f j x = J j wx J j w are 8

10 the sensitivities of the marginal utilities of wealth with respect to w j and x. Aggregation and market-clearing yields the ICAPM expression for equilibrium expected returns: i = Cov dr i ; dr M + Cov dr i ; dx F; (1) where dr M is the market excess return and = w A + w P w A = A + w P = P ; (2) F = f A x w A = A + f P x w P = P w A = A + w P = P ; (3) are the cross-sectional prices of risk. Note that and F are wealth-weighted combinations of the two agents preference parameters. 3.1 ICAPM with Financial Intermediaries The model in (1) is a completely generic two-agent ICAPM. In order to understand the equilibrium asset pricing implications of nancial intermediary portfolio choice, we next apply the model to an economy with Active nancial intermediaries (A) and P assive investors (P ). We let active investors be represented by leveraged nancial intermediaries, such as broker-dealers, who manage leverage in an active, procyclical fashion. Speci cally, we assume that active investors are risk neutral and maximize their expected portfolio 9

11 return subject to a risk constraint related to their Value at Risk (VaR): 2 J A t; w A ; x = max fy A i g i E e T w A (T ) (4) subject to : [V ar] : dw A 1 2 w A ; (5) [Budget] : dw A w A = iy A i dr i + r D dt; (6) where dw A is the quadratic variation of wealth and r D is the risk-free rate. The restriction on the VaR is a policy parameter times the instantanuous volatility of equity. Due to risk neutrality, the VaR constraint binds with equality. Appendix A.1 shows that the e ective risk aversion A = J A www A J A w of active investors is proportional to the Lagrange multiplier on the VaR constraint, which varies over time. We let passive investors represent the rest of the economy, including the remainder of the nancial rms, non- nancial rms and households. For simplicity, we assume that passive investors are myopic mean-variance optimizers with risk aversion given by the constant P. Appendix A.2 demonstrates that aggregation and market clearing yields the equilibrium ICAPM expressions (1) (3). In addition, we can nd reduced form expressions of the time varying e ective risk aversion A of active investors as a linear function of two state variables (x 1 ; x 2 ) 3 : A = A 1 x 1 + A 2 x 2 : (7) De ning the nancial leverage of active investors and passive investors as lev A = P i ya i and lev P = P i yp i, we nd: x 1 = 1 lev A ; x 2 = wa w P 1 1 : (8) lev A 2 A similar portfolio choice problem is examined in another context by Danielsson, Shin and Zigrand (2009). Adrian and Shin (2008) provide a microeconomic foundation for the Value-at-Risk constraint and show that it can be used to motivate the procyclical adjustment of leverage observed in the data. 3 See Equation (23) in the appendix. 10

12 Notably, this simple example shows how state variables x related to nancial intermediary leverage enter the economy s pricing kernel. Shocks to intermediary leverage may thus be priced in the cross-section of asset returns. 3.2 Empirical Implications The ICAPM equilibrium (1) along with state variables (8) motivate a linear factor model, which can be tested via the Fama and MacBeth (1973) cross-sectional regression for average excess returns Ert+1: i Note that im = Cov(ri t+1 ;rm t+1) V ar(r M t+1) Er i t+1 = + im M + 0 ix x + i ; (9) is the market beta with risk premium M = V ar r M t+1 and ix = Cov(ri t+1 ;x t+1 E tx t+1) V ar(x t+1 E tx t+1 ) are additional risk exposures with risk premia x = V ar (~x t+1 ) F, where ~x t+1 = x t+1 E t x t+1 are innovations to the state variables. As usual, the betas can be estimated from the time-series regression: r i t+1 = a i + im r M t ix~x t+1 + i t+1: (10) 4 Data and Construction of State Variables Our simple theoretical framework identi es two new potential risk factors for the pricing kernel. In this section, we construct proxies for these state variables using data on the aggregate balance sheets of U.S. security broker-dealers (active investors) and the rest of the U.S. economy (passive investors). Guided by our theoretical speci cation (8), we construct the following state vari- 11

13 ables (BD abbreviates Broker-Dealer ): x 1 t = 1 lev A t x 2 t = wa t w P t = EquityBD t Assets BD 1 1 lev A t t = CapitalRatio BD t (11) = EquityBD t Equity Non-BD t (1 CapitalRatio BD t ) (12) That is, our rst state variable is simply the capital ratio (inverse of nancial leverage) of broker-dealers. The second state variable is ratio of broker-dealer equity to nonbroker-dealer equity, multiplied by one minus broker-dealer capital ratio, which we will henceforth call the scaled wealth ratio. We construct quarterly series of these variables using data on the book values of total nancial assets and total nancial liabilities of broker-dealers and the rest of the U.S. economy as captured in the Federal Reserve Flow of Funds. 4 While the Flow of Funds data begins in the rst quarter of 1952, the data from the broker-dealer sector prior to 1968 raises suspicions. In particular, broker-dealer equity is negative over the period Q1/1952-Q4/1960 and extremely low for most of the 1960s, resulting in unreasonably low capital ratios. As a result, we begin our sample in the rst quarter of The state variables are plotted in Figure 2. To estimate the unconditional model (10), we construct shocks ~x t+1 to the log state variables as residuals from a vector autoregression conditioned on information available at time t. We incorporate a one-quarter announcement lag for the Flow of Funds variables. 5 As there is substantial multicollinearity between the two state variables, we orthogonalize the innovations of the scaled wealth ratio with respect to the innovations of the capital ratio. We obtain all data on equity portfolios and risk factors from Kenneth French s data library and cumulate these variables to quarterly frequency. 4 Note that equity t = (total nancial assets t - total nancial liabilities t ). 5 For instance, the conditional expectation at the end of March 2000 uses data from the most recent Flow of Funds release, which corresponds to December

14 Figure 2: Broker-dealer state variables. We plot the levels of broker-dealer capital ratio and the scaled broker-dealer wealth ratio (ratio of broker-dealer equity to nonbroker-dealer equity scaled by one minus broker-dealer capital ratio), as reported in the Federal Reserve s Flow of Funds Database. 5 Empirical Approach and Results We test speci cations of the linear factor model (9) in the cross-section of stock returns. The model predicts that the average cross-sectional pricing error () is zero such that all returns in excess of the risk-free rate are compensation for systematic risk. As test assets, we consider the following portfolios that address well-known shortcomings of the CAPM: 25 size and book-to-market portfolios, 25 size and momentum portfolios, and 30 industry portfolios. We compare the performance of our new pricing factors to existing benchmark models in each cross-section of test assets. Whenever a factor is a return, we include it also as a test asset since the model should apply to it as well. For instance, when pricing the portfolios sorted on size and book-to-market, we include the Fama-French (1993) factors market, SMB and HML also as test assets. This forces traded fac- 13

15 tors to price themselves and also allows us to evaluate how our model prices these important benchmark factors. A good pricing model features an economically small and statistically insigni cant alpha, statistically signi cant and stable prices of risk across di erent cross-sections of test assets, and high explanatory power as measured by the adjusted R-squared statistic. In order to correct the standard errors for the preestimation of betas, we report t-statistics of Jagannathan and Wang (1998) in addition to the t-statistics of Fama and MacBeth (1973). Following the above evaluation criteria, and by applying our model to a wide range of test assets, we seek to sidestep the criticism of traditional asset pricing tests of Lewellen, Nagel and Shanken (2010). Importantly, we show success of the model beyond the highly correlated size and book-to-market portfolios: Since the three Fama- French factors explain almost all time-series variation in these returns, the 25 portfolios essentially have only 3 degrees of freedom. As Lewellen, Nagel and Shanken point out, choosing factors that are even weakly correlated with the three Fama-French factors can give success in this cross-section and many existing models that show success in this cross-section have little or no power when other test portfolios are considered. We avoid this criticism by also including the more challenging momentum portfolios and industry portfolios as test assets. The sample considered in the main text is Q1/1968-Q4/2009. We display the results for the subsample that excludes the nancial crisis in the Appendix. 6 The results for the pre-crisis subsample, Q1/1968-Q4/2006, are similar to the results for the full sample. We also include the Pastor-Stambaugh market liquidity factor as a control in the pre-crisis sample and show that it does not a ect our comparisons. 7 6 See Tables A1-A4. The appendix also show the performance of our model in the cross-section sorted on size and long-term reversal (Table A5). 7 The Pastor-Stambaugh factor is included only in the shorter sample because it is not available 14

16 5.1 Size and Book-to-Market Portfolios We begin our asset pricing exercises with the the cross-section of 25 size and book-tomarket sorted portfolios, which since the seminal work of Fama and French (1993) has become a standard test benchmark for factor models. This cross-section highlights the inability of the CAPM to account for the over-performance of small and value stocks, a result that we con rm in column (i) of Table 1: The single market factor has no explanatory power for the average returns, yielding a large, highly statistically signi cant cross-sectional alpha of 1:54% per quarter. The cross-sectional price of risk associated with the market factor is economically small and statistically insigni cant. We contrast this failure of the CAPM with the performance of our single-factor brokerdealer model. Column (iii) shows that the broker-dealer capital ratio is able to explain 55% of the cross-sectional variation in average returns as measured by the adjusted R-squared statistic. Moreover, the cross-sectional alpha is only 0:3% per quarter and statistically indistinguishable from zero. As expected, the price of risk associated with the capital ratio is negative and statistically signi cant assets that hedge against adverse leverage shocks earn lower average returns. Column (iv) shows that including our second broker-dealer variable, the scaled wealth ratio, as an additional risk factor increases the explanatory power of the model to 64% while the alpha remains small and statistically insigni cant. As predicted by our theory, 8 the cross-sectional price of risk associated with shocks to the scaled wealth ratio is positive assets whose returns comove positively with broker-dealer wealth shocks are expected to earn higher average returns but statistically we cannot reject the hypothesis that the estimate is zero. Both broker-dealer factors are signi cant in past We thank Robert Stambaugh for posting the factor data on his website. 8 See Appendix A.2. 15

17 column (v), which adds the market factor to the speci cation. However, the adjusted explanatory power of the model remains unchanged. In the exercises that follow, we will nd that the price of risk associated with the scaled wealth ratio is often statistically insigni cant. Since our goal is to nd a robust, theoretically motivated pricing model that is able to explain average returns with consistent prices of risk across di erent sets of test assets and speci cations, our analysis will focus primarily on the rst brokerdealer factor, the capital ratio. We next compare the performance of our broker-dealer factor to the Fama-French three-factor benchmark. Column (ii) demonstrates that the Fama-French model tailored to price this cross-section produces an adjusted R-squared of 57%. Yet, this is only two percentage points greater than the explanatory power of our single-factor broker-dealer model. Note also that only the market factor and the HML factor have signi cant prices of risk. To complete the picture, column (vi) combines our brokerdealer factor with the three Fama-French factors in a four-factor speci cation. Stunningly, our broker-dealer factor drives out the value factor: the price of risk associated with HML nearly halves and the factor is no longer signi cant at 5% level. The additional explanatory power of the combined model is limited to about ten percentage points and the alpha remains small and statistically insigni cant. These observations suggest that the information content of capital ratio shocks overlap to an important extent with the information content of the Fama-French factors. The four panels of Figure 5 provide a graphical illustration of the performance of our broker-dealer models relative to the single and multi-factor benchmarks. 16

18 5.2 Size and Momentum Portfolios Table 2 and Figure 6 report the pricing results for the 25 size and momentum sorted portfolios. The format follows that of Table 1 but now the momentum factor of Carhart (1997) replaces the HML factor in the three-factor benchmark speci cation. Column (i) again con rms that the market model has no explanatory power for this cross-section. This is contrasted with column (iii), which shows that the brokerdealer capital ratio is alone capable of explaining as much as 75% of the cross-sectional returns with a small and statistically insigni cant alpha of 0:35%. As before, the price of risk associated with shocks to the capital ratio is negative and statistically signi cant. Columns (iv)-(v) demonstrate that the price of risk associated with the scaled broker-dealer wealth ratio is statistically insigni cant and neither it nor the market adds further explanatory power to the model. The alphas remain statistically insigni cant. The three-factor benchmark in column (ii) explains 77% of the crosssectional returns but produces a statistically signi cant alpha of 0:36%. Thus, our single-factor broker-dealer model again rivals the multi-factor benchmark on its very own turf. Combining our broker-dealer factor with the three-factor benchmark in column (vi) increases the explanatory power of the model to 87% and further decreases the magnitude of the alpha. In this combined speci cation, the magnitude of our leverage factor decreases, suggesting that its information content overlaps somewhat with that of the momentum factor. 5.3 Industry Portfolios Table 3 and Figure 7 display our pricing results for the 30 industry portfolios, which have posed a challenge to existing asset pricing models. Column (i) once again con- 17

19 rms the well-known result that the CAPM cannot price this simple cross-section, leaving as much as 1:44% of quarterly average returns unexplained. The results for our broker-dealer pricing models in columns (iii)-(v) tell a di erent story. In contrast to the CAPM, our single-factor broker-dealer model is able to explain 24% of the crosssectional variation with an economically smaller and statistically insigni cant alpha of 1:01%. Adding the scaled wealth ratio brings the explanatory power up to 44%, while our three-factor broker-dealer model explains 52% of the cross-section. However, consistent with our previous results, only shocks to the capital ratio receive a statistically signi cant price of risk. As multi-factor benchmark, we use the Fama-French model. The results in column (ii) show that the three Fama-French factors are able to explain only 6% of the industry cross-section with a larger and statistically signi cant alpha of 1:39%. Also, the prices of risk associated with the market, SMB and HML factors are statistically insigni cant. Thus, our single broker-dealer factor greatly outperforms the three-factor benchmark in this cross-section. The speci cation in column (vi) combines our broker-dealer factor with this benchmark to show that the price of risk associated with shocks to the capital ratio is hardly a ected by the Fama-French factors. The adjusted R-squared only increases by a few percentage points to 27% but the alpha becomes statistically signi cant, suggesting that the Fama-French factors may add more harm than good to our single-factor broker-dealer model. 5.4 Simultaneous Testing of Portfolios We have seen that our preferred broker-dealer factor, the capital ratio, has broad explanatory power with consistent prices of risk over a number of cross-sections. To further illustrate this property, we next conduct an exercise where the 25 size and book- 18

20 to-market portfolios, 10 momentum portfolios, and 30 industry portfolios are included simultaneously as test assets. These results are reported in Table 4 with graphical illustrations in Figure 8. This set of 65 test assets presents a fair test of a model s ability to t the cross-section of average returns, avoiding the critiques of Lewellen, Nagel, and Shanken (2010), and captures simultaneously four well-known asset pricing anomalies: size, book-to-market, momentum, and industry. Column (ii) shows that the Fama-French three-factor model has no explanatory power in this cross-section, with a negative adjusted R-squared statistic. The fourfactor benchmark in column (iii) which appends the Fama-French model with the momentum factor explains 42% of the cross-sectional variation in average returns but produces an alpha of 0:80% per quarter that is highly statistically signi cant. We contrast this result with column (iv), which demonstrates that our single broker-dealer factor alone outperforms the four-factor benchmark with an adjusted R-squared of 45%. Importantly, the alpha for the single broker-dealer factor is 0:67% and statistically insigni cant while the price of risk associated with shocks to broker-dealer capital ratio is 0:23% and highly statistically signi cant. These results for the combined crosssection re ect the consistent price of risk that shocks to the broker-dealer capital ratio bear across di erent cross-sections of test portfolios. In order to understand the successes and failures of the above models at a portfolio level, Figures 1, 3, and 4 plot the realized average returns against predicted average returns with labels on the individual portfolios. Figure 1 shows that the strong performance of the single broker-dealer factor stems mainly from the correct pricing of the industry portfolios and the momentum portfolios, except for the lowest momentum portfolio (Mom1) where even the four-factor model fails (see Fig. 4). The value/size cross-section is also priced well apart from the extreme small growth and extreme small 19

21 value portfolios (S1B1 and S1B5). Yet, and perhaps most notably, the broker-dealer factor is able to correctly price the value factor (HML) and size factor (SMB) a dimension where the Fama-French model itself fails miserably (see Fig. 3). 5.5 Discussion of Pricing Results The results in Tables 1-4 demonstrate that our single broker-dealer factor, the capital ratio, does remarkably well in pricing well-known asset pricing anomalies. The single factor model exhibits consistently strong pricing performance across all cross-sections of test assets, as judged by the explanatory power, the pricing error, and the economic magnitude and signi cance of the prices of risk. The performance of our model rivals and in some cases even exceeds that of the portfolio-based benchmarks that were speci cally tailored to explain each anomaly. Yet, what we nd most notable is that the prices of risk associated with our preferred broker-dealer factor are not only statistically signi cant across di erent sets of test assets, but are also relatively stable in magnitude across all four cross-sections, which was con rmed above in the combined cross-section of 65 test assets. In the singlefactor broker-dealer model (column (iii) of Tables 1-3) the price of risk associated with shocks to the broker-dealer capital ratio varies from 0:33% (size/book-to-market) to 0:36% (size/momentum) to 0:13% per quarter (industries). Including many portfolios at once (size/book-to-market, momentum, and industries) gives a price of risk of 0:23%. These ndings lend additional support to the broad-based performance of our broker-dealer model. 20

22 5.6 Further Tests In order to better understand the commonality between our two state variables and existing benchmarks, including both portfolio-based and macroeconomic models, we next examine how the factor prices of risk implied by our broker-dealer model relate to the factor prices of risk implied by such benchmarks. Table 5 conducts this comparison for three benchmark speci cations: the Fama-French three-factor model, the Lettau and Ludvigson (2001) conditional consumption CAPM model, and a three-factor macro model adapted from the speci cation of Chen, Roll and Ross (1986). 9 The results in the rst panel show that the price of risk of the capital ratio is negatively correlated with the price of HML risk, while the price of risk of the scaled wealth ratio is positively correlated with HML risk, each with a correlation parameter around 0:75. Both correlations are statistically signi cant at the 1% level. This suggests that a high value premium is associated with a high price of insurance against adverse leverage shocks (recall that capital ratio is the inverse of leverage) as well as adverse wealth shocks. Because the price of risk of the capital ratio is negative, and HML correlates negatively with the price of risk of the capital ratio, it follows that the broker-dealer leverage premium correlates positively with the growth premium, i.e. it tends to increase when growth stocks are doing well relative to value stocks. While our tests certainly do not show that broker-dealers are driving the value premium, the ndings are however consistent with the notion that a common source of risk may be driving both the value premium and the uctuations in broker-dealer leverage. For the size factor SMB, its price of risk correlates negatively with that of the capital ratio; it is not signi cantly related to the price of risk associated with shocks to the scaled 9 We thank Martin Lettau for making the factors used in Lettau and Ludvigson (2001) available on his website. All other macroeconomic data are obtained from Haver Analytics. 21

23 wealth ratio. In the second panel, we show that the prices of risk of our state variables correlate positively with the price of risk of Lettau and Ludvigson s cay factor. The capital ratio is negatively correlated with consumption growth c and the consumption growth interaction cay c, whereas the scaled wealth ratio is positively correlated with each. This suggests that adverse shocks to investment opportunities as proxied by positive shocks to broker-dealer capital ratio or negative shocks to broker-dealer wealth tend to coincide with adverse shocks to consumption growth. Finally, the third panel shows that the prices of risk associated with shocks to the broker-dealer capital ratio is also highly negatively correlated with the compensation for shocks to industrial production and for in ation risk. Conversely, the scaled wealth ratio is highly positively correlated with compensation for industrial production risk and highly negatively correlated with the compensation for con dence risk. Intuitively, adverse shocks to investment opportunities tend to coincide with lower-than-expected industrial production and higher unexpected in ation and default spreads. Taken together, the economically meaningful and statistically signi cant correlations between the prices of risk of our states variables and other common risk factors lend support to the view that shocks to our broker-dealer factors re ect unexpected changes in underlying economic and nancial fundamentals. It is in this light that we interpret the robust pricing performance of our broker-dealer model across a wide range of test assets. 6 Conclusion Broker-dealers are active investors that aggressively adjust their portfolios and risk exposure in response to economic conditions. In this paper, we ask whether broker- 22

24 dealer leverage can capture the marginal utility of wealth of active investors and hence be used as reduced form representation of the pricing kernel. We show that our brokerdealer model, with the broker-dealer capital ratio as the single pricing factor, does remarkably well in pricing the challenging cross-section of industry portfolios, clearly dominating benchmark models. Furthermore, the single factor compares favourably to the Fama-French model in the cross section of size and book-to-market sorted portfolios and it rivals the benchmark tailored to explain the cross section of size and momentum sorted portfolios. Our factor is successful across all these cross-sections in terms of high adjusted R-squared statistics, small and statistically insigni cant cross-sectional pricing errors (alphas), and prices of risk that are signi cant and consistent across portfolios. When taking all these criteria into account, our single factor outperforms standard multi-factor models tailored to price the cross-sections considered. As a theoretical motivation to our approach, we demonstrate how leverage enters the pricing kernel in an intertemporal asset pricing model through the investment decisions of long-term active investors subject to a balance sheet risk constraint (VaR). While our representative active investors the security broker-dealers have been studied extensively in the context of market making literature, the study of the information content of aggregate broker-dealer balance sheets in pricing the cross-section of stock returns is new. We regard our study as a rst step in understanding the aggregate asset pricing implications of active intermediaries in the context of long-term portfolio choice. Our results lend support to the view that the portfolio choice of active forward-looking investors provides a window to economy-wide expectations of future investment opportunities. 23

25 A Appendix A.1 Portfolio Choice of Active Investors In order to solve the portfolio choice problem (4) (6), we write the Hamilton-Jacobi- Bellman equation: E t dj A 0 = max fy A g i dt dw F w F! 2 1 ; (13) 1 where is the Lagrange multiplier on the value-at-risk constraint. We make the following guess for the value function (see Merton, 1973): J A t; x; w A = e f(t;x) w A f (T; x) = T, which implies E t dj A = f J A t +fx 0 dt E t [dx] dw A dw A dx 0 +E t + f dt w A dt w A x + 1 hdx 0 dxi f xx + fx 0 dt 2 dt hdx 0 dxi f x, dt where partial derivatives are denoted by subscripts. De ning A = J A, one obtains the rst order condition: By the binding VaR constraint, dw A 1 2 dw A 1 2 y A = 1 A (0 ) 1 ( + 0 xf x ). (14) = wa, such that: = w Ap y A0 ( 0 ) y A = wa A q ( + 0 xf x ) 0 ( 0 ) 1 ( + 0 xf x ) = wa, which implies that the e ective risk aversion is given by: q A = ( + 0 xf x ) 0 ( 0 ) 1 ( + 0 xf x ). (15) From (14), we see that the asset demands of the active investors are identical to the standard ICAPM choices, but where the risk-aversion parameter A is the scaled 24

26 Lagrange multiplier associated with the risk constraint. Even though the active investor is risk-neutral, it behaves as if it were risk-averse. In other words, the risk-aversion of the active investor uctuates with shifts in funding conditions. As the risk constraint binds more strongly, A increases and leverage must be reduced. Note that A is proportional to the generalized Sharpe ratio (adjusted for hedging costs) for the set of risky securities traded in the market as a whole. A.2 Equilibrium Returns To close the model, we assume that there is a second, passive (P ) group of investors that are non nancial corporations or households with constant relative risk aversion P. For expositional simplicity, we assume that their demands are myopic: 10 Market clearing implies: y A y P = 1 P (0 ) 1. (16) w A w A + w P + yp where s is a value-weighted aggregate supply of assets. w P = s; (17) w A + wp Plugging the asset demands (14) and (16) of the two investor types in the market clearing condition gives: w P P + wa ( 0 ) 1 + A w A A ( 0 ) 1 0 xf x = w A + w P s; or = 0 S w P = P + w A = A w A = A w P = P + w A = A 0 xf x : (18) 10 Allowing for intertemporal asset choice of passive investors is straightforward. However, for the purposes of this paper there is little value added to justify the cost of additional complexity in the equilibrium expressions. 25

27 Denote the covariance matrix of individual asset returns with the market portfolio by 0 M = ( 0 ) s; and the wealth-weighted risk aversion and the prices of risk of the state variables by = F x = w P + w A w P = P + w A = ; A w A = A w P = P + w A = f x; A such that the expected returns (18) can be written in the usual ICAPM form: = 0 M 0 xf x A.2.1 Equilibrium, F x, and A = Cov t (dr; dr M ) Cov t (dr; dx) F x : (19) We can now solve for the equilibrium prices of risk and F x, and for the scaled Lagrange multiplier A in terms of observable variables. Plugging (19) into the two investors rst order conditions gives: y A = s 1 A A (0 ) 1 0 x (F x f x ) ; (20) y P = s 1 P P (0 ) 1 0 xf x : (21) De ning the nancial leverage of active investors and passive investors as lev A = P i ya i and lev P = P i yp i, and normalizing P i s i = 1, we can rewrite (17) as: w P + w A w P lev A wa w P = levp : Using (21), it follows that: w P + w A w P lev A wa w P = P 1 P Q xf x ; 26

28 where we have de ned Q x = 1 0 ( 0 ) 1 0 x. We can rewrite the above as: = P 1 + wa w 1 leva + Q P x F x = P 1 + wa w 1 leva w A = A + Q P x w P = P + w A = f x: A On the other hand, we know that = w P +w A w P = P +w A = A, which allows us to solve for A : Since F x = = P wa w P A = P + Q x f x 1 lev A P wa w P w A = A w P = P +w A = A f x, we use the latter to obtain: F x = and wa lev A w P A5 (22) 1 + wa w P + Q P x f x 1 ; (23) 1 lev A w A w P lev A f x 1 + wa w P + Q x f x = P : (24) To gain intuition in (22) reduce to: (24), note that if both investors are myopic, the solutions = P 1 + wa w 1 leva ; P A = lev A ; F x = 0: That is, the e ective risk aversion of the economy,, decreases in the leverage of the active investors. The greater the wealth share of active investors, the greater the impact of their leverage on. 27

29 A.2.2 State Variables By inspection of (22) (24), we nominate the following two state variables: It follows that: Fx1 (x) F x2 (x) = P 2 x 1 = x 2 = wa w P 41 + x 2 1 x 1 1 lev ; A (25) 1 : (26) 1 lev A 1 x 1 fx1 1 + x 2 1 x 1 P 13 A5 ; (27) 1 + x 2 1 x 1 P + Q x f x ; (28) = 1 x 2 1 x 1 x x 2 1 x 1 + Q x f x = P f x2 A (x) = P + Q x f x x1 P x 2 : (29) Note that we can use (29) to solve for the value function of active investors. This is what we do next. A.2.3 Solving for the Value Function of Active Investors Plugging the optimal portfolio choice of active investors (14) back into the Hamilton- Jacobi-Bellman equation (13) gives: 0 = f t + f 0 x x + y A0 + r D + y A0 hdrdx 0 i f x (f xx x 0 x + f 0 x x 0 xf x ) = f t + f 0 x x + 1 A ( + 0 xf x ) 0 ( 0 ) 1 ( + 0 xf x ) + r D (f xx x 0 x + f 0 x x 0 xf x ) : Using the expression for A from (15), we obtain: 0 = f t + f 0 x x + A 2 + rd (f xx x 0 x + f 0 x x 0 xf x ) : (30) In order to solve the PDE in (30), we make the simplifying assumption that all second moments are constant. Using the equilibrium expression (29) for the scaled Lagrange 28

30 multiplier, the PDE becomes a ne in x 1 and x 2. Hence, we make the following guess for the value function: f (t; x) = A (T t) + B 1 (T t) x 1 + B 2 (T t) x 2 ; which implies: f x1 = B 1 (T t) ; f x2 = B 2 (T t) ; f xx = 0; f t = A 0 B 0 1x 1 B 0 2x 2 : Under the assumption that x (x) = k (x x), the PDE (30) simpli es to: A 0 + B 0 1x 1 + B 0 2x 2 = B 1 k 1 (x 1 x 1 ) + B 2 k 2 (x 2 x 2 ) + P + Q x1 B 1 + Q x2 B 2 P x 2 1 x 2 2 +r D B B with boundary conditions A (0) = and B (0) = 0. Thus, the problem can be expressed as a system of four equations: A 0 = B 1 k 1 x 1 B 2 k 2 x 2 + r D B B ; B 0 1 = B 1 k 1 + P + Q x1 B 1 + Q x2 B 2 2 ; B 0 2 = B 2 k 2 P 2 ; all of which have straightforward analytical solutions. Steady State Value Function. In steady states where the time derivatives are 29

31 zero, we obtain: f x1 = P + Q x2 f x2 2 k 1 Q x1 ; (31) f x2 = P k 2 2 ; (32) Note that f x2 < 0. Recall also that Q x = 1 0 ( 0 ) 1 0 x; in other words, Q x1 and Q x2 are sums of OLS regression coe cients from time-series regressions of each state variable on the set of test assets. Estimated from quarterly data, Q x1 and Q x2 are of similar magnitudes and lie between 0:05 and 0:03 (depending on the set of test assets), implying that the denominator of (31) is positive. It follows that f x1 is positive if: P + Q x2 f x2 > 0 which holds since Q x2 < 0., 2 Q x2 k 2 > 0; Steady-State Prices of Risk. The prices of risk F x associated with the state variables are given by (24) as: Fx1 = F x2 w A lev A w P 1 + wa + Q w P x f x = P fx1 Thus, the signs of F x are the same as the signs of f x if the common multiplier w A lev A = 1 + wa + Q w P w P x f x = P is positive. Since the numerator of the expression is always positive, this condition holds if: P 1 + wa + Q w P x f x > 0: A su cient (but not necessary) condition is P +Q x f x > 0, which by (29) is equivalent to requiring that the e ective risk aversion of broker-dealers A is positively related to inverse of broker-dealer leverage x 1. Thus, we may expect F x1 > 0 and F x2 < 0, which implies that the expected factor risk premia are x1 < 0 and x2 > 0. f x2 : 30