Information Factor: A One Factor Benchmark Model for Asset Pricing

Transcription

1 Information Factor: A One Factor Benchmark Model for Asset Pricing Anisha Ghosh y Christian Julliard z Alex P. Taylor x April 24, 2015 Abstract Given a set of test assets, a relative entropy minimization approach can be used to estimate the most likely pricing kernel to price the given cross-section out-of-sample. Compared to leading empirical asset pricing models, such as the Fama-French 3-factor and Carhart models, our out-of-sample pricing kernel delivers smaller pricing errors and better cross-sectional t. Moreover, a tradable information portfolio that mimics this kernel: a) has a high Shape ratio that consistently outperforms the 1=N benchmark out-of-sample; b) extracts pricing information not captured by the Fama-French and momentum factors it leads to an information anomaly, generating high s of 3:5%- 22:0% per annum. These results hold for a wide cross section of assets consisting of size, book-to-market-equity, momentum, industry, and long term reversal sorted portfolios. Keywords: Pricing Kernel, Relative Entropy Minimization, Factor Mimicking Portfolios, Factor Models, Alpha. JEL Classi cation Codes: G11, G12, G13, C13, C53 We bene ted from helpful comments from Andrew Ang, George Constantinides, Francis Diebold, Christopher Polk, Robert Stambaugh, Jessica Wachter and seminar participants at Wharton, ICEF, and LSE. Any errors or omissions are the responsibility of the authors. y Tepper School of Business, Carnegie Mellon University; anishagh@andrew.cmu.edu. z Department of Finance and FMG, London School of Economics, and CEPR; c.julliard@lse.ac.uk. x Department of Finance, Manchester Business School; alex.taylor@mbs.ac.uk.

2 I Introduction Asset prices contain information about the underlying risk neutral probability measure, the latter being such that the expected return on all traded assets, under the risk neutral measure, is equal to the risk free rate. The risk neutral measure is the product of the pricing kernel, that re ects the stochastic discounting of possible future states, and the physical probabilities of these states. Therefore, without additional restrictions, asset prices alone are not su cient to separately identify the pricing kernel and the physical probabilities (see e.g., Borovicka, Hansen, and Scheinkman (2014)). Consequently, the following two strands of literature in nancial economics have evolved with relatively little intersection. The rst focuses on using asset prices, particularly the prices of options, to recover the risk neutral probability measure. Prominent examples in this category include Cox and Ross (1976b). The second strand of literature, motivated by the shortcomings of the traditional one-factor CAPM, focuses on identifying priced risk factors that can help explain better the cross-section of returns for di erent classes of nancial assets. Both, dynamic equilibrium models as well as reduced-form models, have been proposed that identify new risk factors. Prominent examples of risk factors motivated by equilibrium theories include the returns on physical investment (Cochrane (1996)), the consumptionto-wealth ratio (Lettau and Ludvigson (2001a, 2001b)), the housing collateral ratio (Lustig and Nieuwerburgh (2005)), cumulated consumption growth (Parker and Julliard (2005)), the relative share of housing in total consumption (Piazzesi, Schneider, and Tuzel (2007)), and the labor income to aggregate consumption ratio (Santos and Veronesi (2006)). Empirically motivated risk factors include the three Fama-French factors (Fama and French (1993)) and the elevan factors summarized in Stambaugh, Yu, and Yuan (2012). We bridge these two literatures and show that we can jointly estimate, using asset returns data, the in-sample risk neutral measure and the sources of priced risk (the pricing kernel). Given time series data on a cross-section of test assets, we rely on a model free, information-theoretic (or relative entropy minimization) approach to estimate the most likely (in a maximum likelihood sense) risk neutral probability measure that prices the given cross section. This involves estimating, in sample, a risk neutral probability measure that minimizes the Kullback-Leibler Information Criterion (or relative entropy) between the physical and the risk neutral measures. The solution to this problem is a non-linear function of asset returns and the Lagrange multipliers assosciated with assets cross-sectional pricing restrictions i.e. the shadow value of slacking the Euler equation restrictions. Using the estimated risk neutral measure and approximating the physical measure with an empirical distribution 2

3 that assigns probability 1=T to each realized state (observation) in the sample, the in-sample SDF is obtained as the Radon-Nikodym derivative of the risk neutral measure with respect to the physical measure. This appproach to estimating the SDF delivers a non-parametric maximum likelihood estimate of the SDF and, therefore, can be interpreted as the most likely estimate of a one-factor pricing model for the cross section used for its construction. We project the SDF out-of-sample for the purposes of cross-sectional pricing and optimal asset allocation. In particular, using the in-sample estimated Lagrange multipliers, we construct the out-of-sample SDF in a rolling fashion, and use it as the single factor to price the cross-section of test assets. Our approach does not require taking a stance on either the number or the identity of the underlying risk factors or on the functional form of the pricing kernel. Instead, the approach allows us to conveniently summarize all the relevant information contained in, possibly multiple, priced risk factors in the form of a single time series for the SDF. We refer to the out-of-sample SDF as the Information SDF (I-SDF). The estimated SDF, being a nonlinear function of the asset returns used in its construction, is not a traded factor. Therefore, we also construct a tradable portfolio that mimics the estimated kernel by projecting the SDF on to the set of test assets in sample, and using the projection coe cients (normalized to be in the scale of portfolio weights) to construct, out-of-sample, what we refer to as the Information Factor or the Information Portfolio (I-P). If the in-sample SDF successfully prices the cross section of assets used for its construction, i.e. satis es exactly the Euler equations, then the traded portfolio obtained by a projection of the SDF onto the set of test assets is identically equal to the tangency portfolio of the assets. In other words, in this case the in-sample information portfolio is an optimally diversi ed portfolio of the test assets with the highest possible Sharpe ratio. We estimate the I-SDF and I-P for diverse sets of equity portfolios including portfolios sorted on the basis of size, book-to-market-equity, momentum, industry, and long-term reversals and analyze their ability to explain the cross-section of returns as well deliver optimally diversi ed portfolios of the test assets. Our main results can be summarized as follows. First, we show that, compared to leading multifactor models, such as the Fama-French 3 (FF3) or 4 (including momentum) factor models, the I-SDF and I-P deliver smaller pricing errors on all the di erent sets of test assets over , despite being only a one-factor model. Moreover, they explain a larger fraction of the cross-sectional variation of returns. These results hold for a variety of measures of cross-sectional t, in addition to the standard OLS R 2. Second, the I-Ps consistently outperform a number of standard benchmarks out-ofsample, in terms of Sharpe ratios and certainty-equivalent (CEQ) returns. For example, 3

4 when the 25 size and book-to-market-equity sorted portfolios are used as test assets, the I-P delivers an annualized Sharpe Ratio of 1:0 at the monthly frequency, and 0:83 at the quarterly frequency. The Sharpe Ratios produced by the naive 1=N diversi cation strategy are less than half at 0:44 and 0:41, respectively, at the monthly and quarterly frequencies. These results are in stark contrast to those in DeMiguel, Garlappi, and Uppal (2009) who show that the out-of-sample performance of the sample based mean-variance model, as well as its various extensions speci cally designed to reduce the estimation error in the inputs to the mean-variance problem, is typically worse than that of the 1=N rule in terms of the Sharpe ratio and CEQ returns. Not only do the I-Ps outperform the 1=N rule in terms of the Sharpe ratio, they also have higher Sharpe ratios compared to other standard benchmarks. For example, the HML factor has annualized Sharpe ratios of 0:48 and 0:40, respectively, at the monthly and quarterly frequencies. These are less than half of those delivered by the I-Ps, even when only the 25 size and book-to-market-equity sorted portfolios are used in their construction. Similar results are obtained by comparing the Sharpe ratio of the I-P constructed from the ten momentumsorted portfolios with the Sharpe ratio of the momentum factor of Carhart (1997). In fact, our empirical tests show that the I-Ps are statistically indistinguishable from the maximum Sharpe ratio portfolio of the test assets out-of-sample. This suggests that the I-P provides a better encoding of an anomaly, compared to the standard method of characterizing an anomaly as the di erence in the portfolio returns between the smallest and largest deciles of stocks sorted on the basis of the anomaly variable. To our knowledge, this is the rst paper that tries to bridge the gap between cross-sectional asset pricing and optimal asset allocation literatures. Finally, we show that the I-Ps extract novel pricing information not captured by the FF3 or 4 factor models. They leads to an information anomaly, generating high s of 3:5% 23:8% per annum relative to the FF3 and 4 factor models. While the analysis in this paper focuses on equity portfolios, note that our methodology is very general and may be applied to other asset classes including bonds, derivatives, and currencies. Our paper is close in spirit to, and innovates upon, the long tradition of using asset prices to estimate the risk neutral probability measure (see e.g. Jackwerth and Rubinstein (1996), and Ait-Sahalia and Lo (1998)) and use this information to extract an implied pricing kernel (see e.g. Ait-Sahalia and Lo (2000), Hansen (2014), Rosenberg and Engle (2002), and Ross (2011)). Ghosh, Julliard, and Taylor (2014) used the relative entropy minimization approach described above to derive entropy bounds on the pricing kernel and its components that are 4

5 typically tighter and more exible than the seminal Hansen-Jagannathan bounds. They also estimated the most likely pricing kernel and compared it to the kernels implied by popular equilibrium asset pricing models, thereby o ering insights on their empirical performance. The use of an entropy metric is also closely related to the works of Stutzer (1995, 1996) and Kitamura and Stutzer (2002) that rst suggested using this information-theoretic alternative to the standard GMM approach to conduct inference in asset pricing models. Julliard and Ghosh (2012) relied on this entropy based inference approach to assess the empirical plausibility of the rare events hypothesis to explain the equity premium puzzle. Our paper also contributes to the extensive cross-sectional asset pricing literature that seeks to identify priced risk factors to explain the cross section of returns of di erent classes of nancial assets. Prominent examples include the risk factors identi ed in Chen, Novy- Marz, and Zhang (2011), Fama and French (1992), and Carhart (1997). Lewellen, Nagel, and Shanken (2010) o er a critical assessment of asset pricing tests and conclude that, although many of the proposed factors seem to perform well in terms of producing high cross-sectional R 2 and small pricing errors, this result is largely driven by the strong factor structure of the size and book-to-market-equity sorted portfolio returns (that are often used as the only test assets) that makes it quite likely for arbitrarily chosen two or three factors, that have little correlation with the returns, to produce these results (see also Bryzgalova (2014)). Our results, on the other hand, hold not only for size and book-to-market-equity sorted portfolios, but also for portfolios sorted on a variety of other characteristics such as momentum, industry, and long term reversals. Our paper contributes to the strategic asset allocation literature. While Markowitz (1952) derived the optimal portfolio rule in a static mean-variance setting, implementation of the approach requires the estimation of the inputs, namely the expected returns and the variancecovariance matrix of the risky assets to be included in the portfolio. While extensive research e ort has been dedicated to proposing approaches to reduce the estimation error in the inputs, DeMiguel, Garlappi, and Uppal (2009) show that the out-of-sample performance of the sample based mean-variance model, as well as its various extensions speci cally designed to reduce the estimation error, is typically worse than that of the 1=N rule in terms of the Sharpe ratio and CEQ returns. Our results auggest that, at least for the quite diverse set of equity portfolios considered, the information portfolios typically have substantially higher Sharpe ratios and certainty equivalent returns than the naive 1=N portfolio. Finally, the nding that our model free, most likely SDF successfully prices a broad cross section of assets suggests that it is useful in addressing several important problems in macroeconomics and nance. For instance, it can be used to estimate, in a model free way, 5

6 the cost of business cycles. Most existing work on this topic, originating with the seminal work of Lucas (1987), has relied on parametric speci cations of preferences and the dynamics of the state variables in the economy. Not only do the conclusions depend critically on the underlying parametric structure, but the ability of the SDFs identi ed in their procedures to successfully price assets have not been examined, suggesting that their conclusions regarding the cost of business cycles may be potentially misleading. The remainder of the paper is organized as follows. Section II describes our methodology of extracting the most likely kernel as well as the information portfolio from a vector of asset returns. In Section III, we discuss the di erent performance evaluation measures used. Section IV describes the data. The empirical results are presented in Section V. Section VI concludes with suggestions for future research. II Recovery of the SDF and Information Factor The absence of arbitrage opportunities implies the existence of a strictly positive pricing kernel (also known as the stochastic discount factor), M, such that the expectation of the product of the kernel and a vector of excess returns, R e 2 R N, is zero under the physical probability measure, P: Z 0 = E P [M t R e t] = M t R e tdp. Under weak regularity conditions, the above pricing restrictions for the SDF can be rewritten as 0 = Z Z Mt M Re t dp = R e t dq E Q [R e t], where x E [x t ], and Mt is the Radon-Nikodym derivative of Q with respect to P. For M the above change of measure to be legitimate, we need absolute continuity of the measures Q and P. = dq dp We estimate the risk neutral probability measure Q b as Z dq dq bq = arg mind (QjjP) arg min Q Q dp ln dp dp s.t. Z R e t dq = 0, (1) where D (AjjB) := R ln da da R da da ln db denotes the relative entropy of A with respect to db db db B, i.e. the Kullback-Leibler Information Criterion (KLIC) divergence between the measures A and B (White (1982)). Note that D (AjjB) is always non negative, and has a minimum at zero that is reached when A is identical to B. This divergence measures the additional 6

7 information content of A relative to B and, as pointed out by Robinson (1991), it is very sensitive to any deviation of one probability measure from another. Therefore, the above equation is a relative entropy minimization under the asset pricing restrictions coming from the Euler equations. 1 The de nition of relative entropy, or KLIC, implies that this discrepancy metric is not symmetric, that is generally D (AjjB) 6= D (BjjA) unless A and B are identical (hence their divergence is always zero). 2 This implies that for measuring the information divergence between Q and P, we can also invert the roles of Q and P in equation (1) to recover Q as Z bq arg min D (PjjQ) arg min ln dp Z dp s.t. R e t dq = 0: (2) Q Q dq Having estimated Q, the pricing kernel can be recovered (up to a positive constant scale) M via the Radon-Nikodym derivative: t = dq. In doing so, we approximate the physical M dp probability distribution P with the empirical distribution over the observed data sample, i.e. bp t = 1=T, 8t = 1; 2; :::; T. 3 But why should relative entropy minimization be an appropriate criterion for recovering the pricing kernel? There are several reasons for this choice. First, the approaches in equations (1) and (2) deliver non-parametric maximum likelihood estimates of the risk neutral measure and, therefore, the pricing kernel. That is, the above KLIC minimization is equivalent to maximizing the likelihood in an unbiased procedure for nding the risk neutral measure (see e.g. Kitamura (2006)). Note that this is also the rationale behind the principle of maximum entropy (see e.g. Jaynes (1957a, 1957b)) 1 This approach was rst suggested by Stutzer (1995) to recover the risk neutral probability measure. Ghosh, Julliard, and Taylor (2014) extended the methodology to recover the missing component of the SDF for a broad class of consumption-based asset pricing models as well as construct entropy bounds on the SDF and its components that are tighter and more exible than the seminal Hansen-Jagannathan bounds. 2 Information theory provides an intuitive way of understanding the asymmetry of the KLIC: D (AjjB) can be thought of as the expected minimum amount of extra information bits necessary to encode samples generated from A when using a code based on B (rather than using a code based on A). Hence generally D (AjjB) 6= D (BjjA) since the latter, by the same logic, is the expected information gain necessary to encode a sample generated from B using a code based on A: 3 Note that, in the absence of any assumptions regarding the state space, bp t = 1=T, 8t = 1; 2; :::; T, is the non-parametric maximum likelihood estimator of the physical probability measure P, in the sense that it is the solution to the following maximization problem: ( ) TX TX max log (p t ) s.t. (p 1 ; p 1 ; :::; p T ) : p t = 1, p t > 0, t = 1; 2; :::; T. t=1 t=1 7

8 in physical sciences and Bayesian probability that states that, subject to known testable constraints the asset pricing Euler restrictions in our case the probability distribution that best represent our knowledge is the one with maximum entropy, or minimum relative entropy in our notation. Appendix A.1 provides a derivation of this result. Second, the use of relative entropy, due to the presence of the logarithm in the objective functions in equations (1)-(2), naturally imposes the non negativity of the pricing kernel. Third, our approach to recover M t satis es the Occam s razor, or law of parsimony, since it adds the minimum amount of information needed for the pricing kernel to price assets. This is due to the fact that the relative entropy is measured in units of information. Fourth, it is straightforward to add conditioning information: given a vector of conditioning variables Z t 1, one simply has to multiply (element by element) the argument of the integral constraints in equations (1) and (2) by the conditioning variables in Z t 1. Fifth, there is no ex-ante restriction on the number of assets that can be used in constructing M. The approach does not require a decomposition of M into short and long run components (cf. Alvarez and Jermann (2005)). Nor does it rely on the existence of a continuum of options price data (c.f. Ross (2014)). Sixth, as implied by the work of Brown and Smith (1990), the use of entropy is desirable if we think that tail events are an important component of the risk measure. 4 Finally, this approach is numerically simple when implemented via duality (see e.g. Csiszar (1975)). That is, when implementing the entropy minimization in equation (1), each element of the series fm t g T t=1 can be estimated, up to a positive constant scale factor, as cm t M t b; R e t = eb 0 R e t TX t=1 e b 0 R e t, 8t (3) where b 2 R N is the vector of Lagrange multipliers that solve the following unconstrained convex problem 1 b arg min T TX e 0 R e t ; t=1 (4) 4 Brown and Smith (1990) develop what they call a Weak Law of Large Numbers for rare events, that is they show that the empirical distribution that would be observed in a very large sample converges to the distribution that minimizes the relative entropy. Relying on this inslight, Ghosh, Julliard, and Taylor (2011) used this relative entropy minimization approach to analyze the empirical plausibility of the rare events hypothesis to explain a host of asset pricing puzzles. 8

9 and this last expression is the dual formulation of the entropy minimization problem in equation (1). Similarly, the entropy minimization in equation (2) is solved by cm t M t b; R e t = 1 T (1 + b, 8t (5) 0 R e t) where b 2 R N is the solution to b arg min TX log(1 + 0 R e t); (6) t=1 and this last expression is the dual formulation of the entropy minimization problem in equation (2). Note that the above duality results imply that the number of free parameters available in estimating fm t g T t=1 is equal to the dimension of (the Lagrange multiplier) that is, it is simply equal to the number of assets considered in the Euler equation. Also, note that, since the b s in equations (4) and (6) are akin to Extremum Estimators (see e.g. Hayashi (2000, Ch. 7)), under standard regularity conditions (see e.g. Amemiya n o T (1985, Theorem 4.1.3)), one can construct asymptotic con dence intervals for cmt. We use the above methodology to recover the time series of the SDF in a rolling out-ofsample fashion. In particular, for a given cross section of asset returns, we divide the time series of returns into rolling subsamples of length T, nal date T i, and constant s = T i+1 T i, i = 1; 2; :::. In subsample i, we estimate the vector of Lagrange multipliers Tj t=1 by solving the minimizations in equations (4) and (6) for the two alternative de nitions of relative entropy. Using the estimates of the Lagrange multipliers, b Tj, the out-of-sample Information-SDF (I-SDF) M btj ; Re t is obtained for the subsequent s periods, t = T i + 1, T i + 2,..., T i+1, using equations (3) and (5). This process is repeated in each subsample to obtain the time series of the estimated kernel over the out-of-sample evaluation period. Note that the relative entropy minimizing pricing kernel, while being a function of asset returns, is not directly a traded asset or portfolio of assets. We create a mimicking portfolio, maximally correlated with the kernel, in a rolling out-of-sample fashion. We refer to this portfolio as the Information Portfolio (I-P) or the Information Factor. The I-P is constructed as follows. In subsample i, the estimates of the Lagrange multipliers, b Tj, are used to construct the in-sample SDF M c i;t M btj ; Re t, t = T i T + 1, T i T + 2,..., T i. We then project c M i;t on the space of excess returns to obtain the vector of portfolio weights 9

10 ! Ti 2 R N (normalized to sum to unity): i 1 h^a Tj ; ^b Tj = arg min T (a Tj ;b Tj )! Ti = ^b Tj ^b Tj 1, XT i t=t i T +1 2 ^Mi;t a Tj b Tj t Re ; where 1 denotes a column vector of ones. Using the portfolio weights vector, the out-ofsample I-P is obtained as R IP t =! 0 T i R e t for the subsequent s periods, t = T i + 1, T i + 2,..., T i+1. This process is repeated for each subsample to obtain the time series of the I-P over the out-of-sample evaluation period. In our empirical analysis, the out-of-sample evaluation period starts from 1963:07 to coincide with the analysis Fama and French (1993) and Lewellen, Nagel, and Shanken (2010). We set s = 12 months (4 quarters) for monthly (quarterly) data. This corresponds to an annual rebalancing of the portfolio. III Cross-Sectional Performance Evaluation Measures For a given cross-section of test assets, we construct the out-of-sample I-SDF and I-P using the relative entropy minimizing procedure described in Section II. We evaluate the empirical performance of the I-SDF and I-P for the test assets at the monthly and quarterly frequencies. We compare the performance of these factors to that of the one-factor CAPM and the three factor Fama-French model. We use the two-step methodology of Fama and MacBeth (1973) to assess the ability of the each factor model to price the cross-section of test assets. In the rst step, the factor loadings for the test assets are estimated from a time series regression of the excess returns on the factors: R e t = a + BF t + " t. In the second step, the factor risk premia are obtained from a cross-sectional regression of the average excess asset returns,, on the factor loadings estimated from the rst stage: = z + B +, = C + ; C [ B], 0 [z 0 ] Following the suggestions of Lewellen, Nagel, and Shanken (2010), we report several statistics from the above cross-sectional regression as alternative measures of performance. 10

11 First, we report the standard OLS cross-sectional adjusted-r 2 (hereafter referred to as the R 2 OLS). This measure su ers from the shortcoming that if returns have a strong factor structure (e.g., the size and book-to-market-equity sorted portfolio returns), then arbitrarily chosen two or three factors, that have little correlation with the returns, would be quite likely to produce large values of this statistic (see e.g., Lewellen, Nagel, and Shanken (2010), Bryzgalova (2014)). Second, we also report the GLS adjusted-r 2 (hereafter referred to as the R 2 GLS), that is obtained from the cross-sectional regression of b V 1=2 on b V 1=2 [ B], where V V ar (R e ). The R 2 GLS for a model, unlike the R 2 OLS, is completely determined by the model-implied factor s proximity to the minimum variance frontier and, in general, presents a more stringent hurdle for models. Third, we report the cross-sectional T 2 statistic of Shanken (1985) given by T 2 b 0 S + a b; where S + a = pseudoinverse of estimated a = F yy T, x = [; B], y I x (x 0 x) 1 x 0. T 2 has an asymptotic 2 distribution with degrees of freedom N K 1, where K denotes the number of factors, and noncentrality parameter 0 + a = 0 (yy) + T ( F ). We compute the asymptotic p-value of this statistics under the null hypothesis that the model explains the vector of expected returns perfectly, i.e., the vector of pricing errors = 0. Fourth, we report the quadratic q 0 (yy) +, that measures how far the factor is from the mean-variance frontier. In particular, it is equal to the di erence between the squared Sharpe ratio of the tangency portfolio of the test assets and the maximum squared Sharpe ratio attainable from the model-implied factors (or their mimicking portfolios in the case of non traded factors). Finally, we report 90% con dence intervals for the statistics. The simulated con dence intervals are obtained using the approach suggested by Stock (1991). Consider rst the construction of the con dence intervals for the R 2 OLS. The simulations have two steps. First, we x a true (population) cross-sectional R 2 that we want the model to have and alter the (N 1) vector of expected returns,, as = h (C) + ", where C [ B], B denotes the vector of factor loadings in the historical sample, and " N (0; 2 "). The constants, h and 2 ", are chosen to produce the right cross-sectional R 2 and maintain the historical cross-sectional dispersion in average returns. Second, we jointly simulate arti cial time series of the factor and the returns of the same length as the historical data by sampling, with replacement, 11

12 from the historical time series. We then perform the two-pass regression methodology to estimate the sample cross-sectional R 2 in the simulated sample. We repeat the second step 1; 000 times to construct a sampling distribution of the R 2 statistic conditional on the given population R 2. This procedure is repeated for all values of the population R 2 between 0 and 1. The con dence interval for the true R 2 represents all values of the population R 2 for which the estimated R 2 in the historical sample falls within the 5th and 95th percentiles of the sample distribution. A con dence interval for q is found using a method similar to that used to obtain the con dence interval for the true (population) cross-sectional R 2. Speci cally, a given population R 2 implies a speci c value of q. We plot the sample distribution of the T 2 statistic as a function of q. The con dence interval for the true q represents all values of the q for which the estimated T 2 in the historical sample falls within the 5th and 95th percentiles of the sample distribution. For the T 2 statistic, we report its nite-sample p-value, obtained from the above simulations, as the probability that the T 2 statistics in the simulated samples exceeds the value of the statistic in the historical data for q = 0. IV Data For our empirical analysis, we focus on monthly returns data on several cross sections of equity portfolios: the 25 size and book-to-market-equity sorted portfolios of Fama-French, the 10 momentum-sorted portfolios, the 30 industry-sorted portfolios, and the 25 portfolios formed on long term reversal and size. The proxy for the risk-free rate is the one-month Treasury Bill rate. The returns data are obtained from Kenneth French s data library. We apply our methodology to estimate the I-SDF and I-P at the monthly and quarterly frequencies. The quarterly returns on the above assets are obtained by compounding the monthly returns within each quarter. Excess returns on the assets are then computed by subtracting the risk free rate. V Empirical Results We evaluate the ability of the I-SDF and I-P to (a) explain the cross section of returns out-of-sample and (b) deliver optimally diversi ed portfolios of the test assets out-of-sample. Our out-of-sample evaluation period starts in 1963:07. This start date coincides with that in Fama and French (1993), Lewellen, Nagel, and Shanken (2010), as well as DeMiguel, 12

13 Garlappi, and Uppal (2009). For a given set of test assets, we construct the I-SDF and I-P in a rolling out-of-sample fashion as described in Section II. Note that the I-SDF and, therefore, the I-P depend on the set of test assets used for their construction. Given the concern that it may be relatively easy to nd factors that seem to do a good job at explaining the cross section of returns (i.e., produce high cross-sectional OLS R 2 and small pricing errors) of especially the size and B/M sorted portfolios because of their strong factor structure, we use our methodology to not only price the size and B/M sorted portfolios, but also portfolios sorted on the basis of other characteristics including momentum, industry, and long term reversals. In Section V.1, we present cross-sectional regression results for the di erent sets of test assets while V.2 describes the properties of the I-Ps. V.1 Cross-Sectional Tests Table 1 presents results when the test assets consist of the 25 size and book-to-market-equity sorted portfolios of FF. Consider rst Panel A that reports results at the monthly frequency. Row 1 shows that, when the I-SDF is used as the sole factor, its estimated price of risk is strongly statistically signi cant. Since the regression uses the monthly excess returns as the dependent variable, the intercept can be interpreted as the estimated monthly zero beta rate over and above the risk free rate. Therefore, the annualized zero beta rate is 3:6%. Although this is statistically signi cant, part of it may be attributable to the di erences in lending and borrowing rates (1-2%). Moreover, Rows 3 and 4 show that the CAPM and the FF3 model produce substantially higher annualized intercepts of 13:2%. The I-SDF produces an R 2 OLS of 68:6% and, more importantly, the R 2 GLS is very similar to the R 2 OLS at 59:6%. Note that, the GLS R 2 is high if and only if the factor is close to the mean-variance frontier and, in general, provides a more stringent hurdle for asset pricing models. The T 2 statistic shows that the model is not rejected at the 10% level of signi cance. Finally, the q statistic, that equals the di erence between the squared Sharpe ratio of the tangency portfolio of the test assets and the squared Sharpe ratio of the factor-mimicking portfolio, is 0:077 and its 90% con dence interval includes 0, i.e., the I-SDF mimicking portfolio is statistically indistinguishable from the maximum Sharpe ratio portfolio of the test assets. Row 2 shows that very similar results are obtained when the I-P is used as the single factor in the cross-sectional regression. Note that while a factor and its mimicking portfolio produces the same intercept, R 2, and pricing errors in a cross-sectional regression in-sample, the same does not hold out-of-sample. The small di erences between Rows 1 and 2 are 13

14 because of the out-of-sample nature of the construction of the I-SDF and I-P. Figure 1 plots the time series of the I-SDF (Panel A) and I-P (Panel B), along with the NBER designated recession periods (shaded areas) and the major stock market downturns (vertical dasheddotted lines) identi ed in Mishkin and White (2002). In Row 3, we present the results for the unconditional CAPM. The market risk premium has the wrong sign and is not statistically signi cantly di erent from zero. The intercept, on the other hand, is strongly signi cant with an annualized value of 13:2%.The OLS and GLS R 2 are much smaller at 3:97% and 28:8%, respectively, compared to those obtained with the I-SDF and I-P. The T 2 statistic is double those obtained with the I-SDF and I-P, and has a p-value of zero. The q statistic is closely related to the R 2 GLS and the T 2 statistics and, therefore, not surprisingly, provides similar conclusions. Row 4 presents results for the FF 3-factor model. The results show that the market risk premium is not statistically signi cant but the risk premia associated with the factors proxying for risks related to size and book-to-market-equity are both signi cantly positive. However, the intercept is statistically and economically large with an annualized value of 13:2%, the same as that obtained with the market risk factor alone in Row 3. The R 2 OLS is high at 71:3%, consistent with existing empirical evidence that the 3 FF factors explain a large fraction of the time series and cross-sectional variation in the returns of the 25 FF portfolios. However, moving to a GLS cross-sectional regression, the R 2 drops sharply to 40:9%. This is in stark contrast to the I-SDF and I-P that deliver very similar R 2 using OLS and GLS procedures. The T 2 statistic is larger than those obtained with the I-SDF (51:5 vs 37:5) and I-P (51:5 vs 37:1), and has a p-value of zero. The q statistic is also larger than those obtained with the I-SDF (0:096 vs 0:077) and I-P (0:096 vs 0:072). Moreover, the 90% con dence interval of the q statistic does not include 0 i.e., the maximum Sharpe ratio obtainable from the 3 FF factors is statistically di erent from the Sharpe ratio of the tangency portfolio of the test assets. Finally, Row 5 presents results when the I-P is used in conjunction with the 3 FF factors in the cross sectional regression. Note that the risk premium for the I-P remains strongly statistically signi cant even in the presence of the 3 FF factors and its magnitude is almost identical to that obtained when it is used as the sole factor in Row 2. Although the R 2 OLS is higher at 86:1% compared to 68:6% in Row 2, the R 2 GLS is very similar between the two rows (62:2% vs 59:6%). Even stronger results are obtained at the quarterly frequency in Panel B. The I-P delivers a statistically and economically insigni cant annualized intercept of 0:8%, a strongly significant risk premium, and R 2 OLS and R 2 GLS of 83:7% and 51:6%, respectively. The T 2 statistic 14

15 is 28:8 and has a p-value of 0:535. The CAPM, on the other hand, produces a large intercept with annualized value of 9:6%, a negative R 2 OLS, an R 2 GLS of 8:8%, and a T 2 statistic with p-value of 0%. The FF 3-factor model also produces a large intercept of 11:2%. Although the R 2 OLS is high at 74:7%, the GLS R 2 drops sharply to 17:7%. Note that, as noted in Lewellen, Nagel, and Shanken (2010), it is relatively easy to nd factors that produce large R 2 OLS for the 25 FF portfolios because of their strong factor structure. What is more impressive is that a single factor, namely the I-SDF or the I-P, does even better than the 3 FF factors. Moreover, similar conclusions are obtained if, rather than relying on the R 2 OLS alone, more stringent hurdles are imposed on the model via the R 2 GLS, T 2, and q statistics, and their con dence bands. We next show that the superior performance of our model holds not only for the size and book-to-market-equity sorted portfolios, but also for portfolios formed by sorting stocks on the basis of other characteristics such as prior returns, industry, etc. Tables 2-5 report the cross-sectional regression results when the set of test assets consists of (a) the 10 momentum sorted portfolios, (b) the 25 portfolios formed on the basis of size and long-term reversal, (c) the 10 industry portfolios and the smallest and largest deciles of portfolios formed on the basis of size, B/M, and prior returns, and (d) the 30 industry portfolios, respectively. The results, in each case, are very similar to those obtained with the 25 FF portfolios in Table 1. Overall, we nd that the I-SDF and I-P produce smaller pricing errors and larger cross sectional R 2 s compared to leading empirical asset pricing models, such as the Fama-French 3-factor and the Carhart 4-factor models, despite being only a one factor model. Moreover, the approach delivers successful one-factor models for diverse sets of asset classes. V.2 Information Portfolio: An Optimally Diversi ed Portfolio The results of the previous subsection show that the I-P o ers a good one-factor benchmark model for pricing broad-cross sections of equity portfolios. We next investigate the implications of the I-P for strategic asset allocation. When investors utility functions depend only on the mean and variance of a portfolio s return, Markowitz (1952) derived the optimal rule for allocating wealth across a set of risky assets. However, the practical implementation of the approach requires the estimation of the inputs, namely the expected returns of the assets and their variance-covariance matrix. For a set of N = 25 risky assets, the estimation of these moments as their sample analogs requires the estimation of N + N(N+1) = parameters. Not surprisingly, these optimal portfolios often have extreme weights on its constituent assets that uctuate substantially over time, and perform poorly out-of-sample. 15

16 Given the widespread use of the mean-variance approach to asset allocation among both academics and practitioners, substantial research e ort has been devoted to trying to reduce the estimation error and improving the performance of the model. DeMiguel, Garlappi, and Uppal (2009) evaluate the out-of-sample performance of the sample based mean-variance approach, as well as a broad set of its extensions designed to reduce the e ect of estimation error, using several di erent sets of test assets. They conclude that the optimally diversi ed portfolio constructed using each of the above approaches typically underperforms a naive diversi cation strategy consisting of an equally-weighted (1=N) portfolio of the test assets, using several performance evaluation measures. We evaluate the out-of-sample performance of the I-P using the performance measures in DeMiguel, Garlappi, and Uppal (2009), namely (i) the Sharpe ratio and (ii) the certaintyequivalent (CEQ) return for the expected utility of a mean-variance investor. The Sharpe ratio is de ned as csr I-P = b I-P b I-P, where b I-P and b I-P are sample mean and standard deviation, respectively, of the out-ofsample excess returns on the I-P. The CEQ return is de ned as the risk free rate that would make an investor, with mean-variance preferences and coe cient of risk aversion = 1, indi erent between the risky I-P and the risk free rate: [CEQ I-P = b I-P 2 b2 I-P. For each set of test assets, we compute the Sharpe Ratio, the CEQ return, as well as the rst four moments of the I-P. As a benchmark to facilitate comparison, we also compute the corresponding statistics for the 1=N portfolio of the test assets. In addition to the equally-weighted portfolio, we also compare the performance of the I-P to other standard benchmarks including the market portfolio, the HML factor of Fama and French (1993), and the momentum factor of Carhart (1997). The results are reported in Table 6. Panel A presents the results at the monthly frequency. Consider rst Row 4 where the I-P is constructed from the 25 size and book-to-market sorted portfolios. The annualized Sharpe ratio of the I-P is 1:0. The Sharpe ratio of the corresponding 1=N benchmark (reported in parentheses below) less than half at 0:44. The I-P not only outperforms the 1=N benchmark, but also the market portfolio that has a Sharpe ratio of 0:32 (Row 1), the HML factor that has a Sharpe ratio of 0:48 (Row 2), as well as the momentum factor whose Sharpe ratio is 0:57 (Row 3). Similar conclusions are obtained using the CEQ return as the measure of performance. A mean-variance investor 16

17 with = 1 would need an annualized risk free rate of 22:0% in order to not invest in the I-P, whereas a risk free rate of only 7:2% is required for him to not invest in the 1=N portfolio. Similarly, risk free rates of only 3:6%, 4:8%, and 7:2%, respectively, are required for him to be indi erent between the risk free rate and the market, the HML, and the momentum portfolios. Figure 2 plots the path of $1 invested in the I-P over the entire out-of-sample evaluation period 1963: :12. Note that, because we use excess returns in the construction of the I-P, this corresponds to a long-short strategy that is short $1 in the risk free rate and uses the proceeds to invest in the optimal portfolio of the risky assets. To facilitate comparison, we also plot the path of $1 invested in several alternative benchmarks, including the excess return on the market portfolio, the SMB, and the HML portfolios. As is evident from the gure, the I-P outperforms, by a wide margin, each of the benchmarks. Moreover, the performance is robust across all subperiods. Also, the portfolio has low turnover and, therefore, low trading costs since the rebalancing is done once a year in the month of June. The I-P is an optimally weighted portfolio of the 25 size and book-to-market sorted portfolios. Therefore, the question arises as to whether our approach relies on extreme weights on the constituent portfolios that also uctuate wildly over time. Figure 3 plots the time series of weights on each of the 25 portfolios in the I-P. The gure makes clear that the vast majority of the weights lie in the [ 2; 2] interval and, therefore, are not extreme. In order to provide more intuition regarding the composition of the I-P, Figure 4 plots the aggregate weights on portfolios of small, big, growth, and value stocks in the I-P. For instance, denoting (1; 5) as the portfolio with stocks in the smallest size quintile and the largest book-to-market-equity quintile, the line labeled Small in the gure plots the sum of the weights on portfolios (1; 1), (1; 2), (1; 3), (1; 4), and (1; 5) at each date. The Big, Growth, and Value curves are similarly de ned. The Growth and Value curves reveal that the I-P typically takes a long position in value stocks and a short position in growth stocks, much like the HML factor of Fama-French. However, unlike the latter, the weights on the long and short ends are not constant in the I-P. Although the weights almost always lie between 2 and +2, they do vary over time. The Small and Big curves o er a less clean interpretation of a long-short strategy and resemble less the SMB factor. Overall, the weights on the small, large, growth, and value stocks in the I-P are quite di erent from those implied in the SMB and HML factors. Moreover, our results suggest that this alternative weighting scheme leads to substantially better performance, both in terms of out-of-sample pricing as well as constructing optimally diversi ed portfolios. Row 5 presents results when the I-P is constructed from the 10 momentum sorted port- 17

18 folios. Once again, the I-P has a Sharpe ratio almost triple that of the 1=N portfolio (0:81 vs 0:29) and a CEQ return more than 7 times higher that of the 1=N portfolio (26:4% vs 3:6%). Similar results are obtained when the I-P is constructed from the 25 long-term reversal and size sorted portfolios (Row 6) and the small, big, growth, value, winners, and losers portfolios and the 10 industry sorted portfolios (Row 7). Row 8 shows that the I-P underperforms the 1=N portfolio in terms of Sharpe ratio and CEQ return when the test assets consist of the 30 industry portfolios. Similar results are obtained at the quarterly frequency in Panel B. Overall, our results show that the I-P typically outperforms the naive 1=N portfolio as well as other standard benchmarks out-of-sample, in terms Sharpe ratio and CEQ return. Moreover, this results seems quite robust to the set of risky assets used for its construction as well as the data frequency. This is consistent with the nding in Section V.1 that the I-P is statistically indistinguishable from the maximum Sharpe ratio portfolio of the test assets out-of-sample. Therefore, the I-P o ers an attractive procedure for optimal asset allocation across risky assets. Why does the I-P have these superior properties relative to other approaches to implementing mean-variance optimal portfolios? Note that the maximum Sharpe ratio portfolio of a set of risky assets must price the cross section of assets because it identi es the Capital Allocation Line (CAL). Existing methods for constructing this portfolio focus on the asset side i.e., the tangency point between the mean-variance e cient frontier and the CAL. Implementation of sample-based approach, therefore, require estimation of a large number of parameters (350 for N = 25). DeMiguel, Garlappi, and Uppal (2009) show, via simulations, that for a sample-based mean-variance strategy to achieve a higher CEQ return than the 1=N portfolio, would require an estimation window of 3000 months (250 years) when N = 25. They also show that various extensions of the sample-based approach, developed explicitly to mitigate the e ects of estimation error, also require very long estimation window. Our approach to constructing the I-P focuses, instead, on the SDF side i.e., the tangency of the investors indi erence curves to the CAL. This requires the estimation of only N (the number of risky assets) parameters of the dual solution as described in II. V.3 Alpha of Information Portfolio We show that the I-Ps contain novel pricing information not captured by standard multifactor asset pricing models, such as the FF 3-factor and the Carhart 4-factor models. Table 7 reports time series regressions of the I-P, constructed from each set of test assets, on the 3 18

19 FF factors. Whenever the test assets include momentum-sorted portfolios, we also include the momentum factor as a regressor in addition to the 3 FF factors. Panel A shows that, at the monthly frequency, the I-Ps generate statistically and economically large s, varying from 3:5%-17:7% per annum, relative to these factor models. The R 2 OLS from the regressions vary from 20:7%-46:6%, showing that a substantial proportion of the variability in the I-Ps cannot be explained by the movements in the FF3 or momentum factors. The last column reports the Information Ratio, de ned as the estimated alpha divided by the standard deviation of the residual from each time series regression. The Information Ratio, therefore, measures the Sharpe ratio of a hedged strategy that has an alpha equal to the estimated alpha and that has no systematic risk with respect to the FF3 or momentum factors (i.e., its beta with respect to each of these factors is zero). The results reveal that the Information Ratios are economically large, varying from 0:14-0:73 per annum. Similar results are obtained at the quarterly frequency in Panel B, where the s vary from 4:7%-23:8%, the R 2 OLS from 15:4%-42:6%, and the Information Ratios from 0:15-0:62. VI Conclusion and Extensions Given a set of test assets, we show how an information-theoretic (relative entropy minimization) approach can be used to estimate the most likely pricing kernel that prices the given cross section. A portfolio that mimics this kernel, the so-called information portfolio, has several interesting properties. First, it can be interpreted as the most likely estimate of a one-factor pricing model for the cross section. Compared to the leading empirical asset pricing models, such as the Fama-French 3-factor or 4-factor (including momentum) models, it delivers smaller pricing errors despite being only a one-factor model. Second, the information portfolio has a high Sharpe ratio, consistently beating the 1/N benchmark out-of-sample. Third, we show that the portfolio extracts pricing information not captured in standard asset pricing models. Our portfolio leads to an information anomaly, generating high alphas of around 3:5% 23:8% per annum relative to the FF model. Finally, these results hold for a wide cross section of assets consisting of size, book-to-market-equity, momentum, industry, and long term reversal sorted portfolios. The analysis so far has focused on the construction of the most likely pricing kernel and the mimicking information portfolio for a given set of assets. While this is undoubtedly an important step, the broader economic question is whether there exists a pricing kernel that can successfully prices all the assets. While the absence of arbitrage opportunities implies the existence of an SDF, the SDF is unique only under the additional condition of 19

20 market completeness. Our information-theoretic methodology can help shed light on how the pricing kernels constructed from di erent asset classes di er from one another thereby o ering guidance regarding the reasons (if any) for market incompleteness. Finally, the current paper focuses on common stocks. However, our methodology is very general and may be applied to other asset classes including bonds, derivatives, and currencies. 20

21 A Appendix A.1 Maximum Likelihood Analogy Let the vector z t be a su cient statistic for the state of the economy at time t. That is, z t can be thought of as an augmented stated vector containing the beginning of period state variables, as well as time t shocks realisations and expectations about the future. Given z t, the equilibrium quantities, e.g. returns R e and the sdf M, are just a mapping from z on to the real line i.e., M (z) : z! R + ; R e (z) : z! R N ; M t M (z t ) ; R e t R e (z t ) where z t is the time t realisation of z. Equipped with the above de nition, we can rewrite the Euler equations as Z Z 0 = E P [R e tm t ] R e tm t dp = R e (z) M (z) p (z) dz (7) where p (z) is the pdf associated with the physical measure P. Following Owen (1988, 1990, 1991), approximating the continous distribution p (z) with a multinomial distribution fp t g T t=1 that assigns probability weight p t to the time t realizations of z, a non-parametric maximum likelihood estimator (NPMLE) of fp t g T t=1 can be obtained as f^p t g = arg max 1 T TX ln p t s.t. p t 2 and (7) holds. t=1 Note that, under the regularity conditions required by the above estimator (ergodicity in particular), we have 1 T TX t=1 ln p t p! T!1 E P [ln p (z)] i.e. the NPMLE of P maximises the expected log likelihood. Moving to the risk neutral measure, from equation (7), we have Z 0 = E P [R e tm t ] = E Q [R e t] = R e (z) q (z) dz (8) where q (z) is the pdf associated with the risk neutral measure Q. Note that Z D (PjjQ) = ln dp Z Z dq dp = p (z) ln p (z) dz p (z) ln q (z) dz Hence bq arg mind (PjjQ) = arg max E P [ln q (z)] s.t. E Q [R e t] = 0, Q Q 21

22 that is, the minimum entropy estimator maximises the expected risk neutral log likelihood. Note that ML with risk neutral likelihood is not uncommon, for instance this is one of the approaches used for term structure modelling (see e.g. Hamilton and Wu (2012)). From the above, the feasible NPMLE can be obtained as fbq t g T t=1 = arg max 1 provided that 1 T TX t=1 T TX ln q t s.t. q t 2 and (8) holds t=1 ln q t p! T!1 E P [ln q (z)]. Note also that the NPMLE of p(z) is is simply p t = 1=T 8t (see e.g. Owen (1991, 2001)) i.e., the maximum entropy distribution, hence bq contains all the necessary information to fully characterise the state-price density. Hence, the estimate of Q maximises the non-parametric log likelihood the data. However, instead of the physical one, it maximizes the risk neutral log likelihood. Note also that b Q is an extremum estimator in the sense of Hayashi (2000, Ch. 7). Note that, for any equilibrium quantity A t, we have that A t A (z t ) by the de nition of the augmented state vector z. Hence, given any function f (:), the risk neutral expectation Z E Q [f (A t )] f (A (z)) q (z) dz can be estimated (see e.g. Kitamura (2006)) as \ E Q [f (A t )] = TX f (A t ) bq t : t=1 A.2 Relative Entropy in Terms of Cumulants Denoting by ^q t the minimum entropy risk neutral measure that solves Equation (2), we have Z Z ln ^q t dp = ln ^q t dp ln E e ln ^qt ; where we have imposed the normalization E [^q t ] = 1. Note also that, by construction, ^q 2 M +, that is the relative entropy minimization identi es an admissible SDF in the Hansen and Jagannathan (1997) sense. Recall that the cumulant generating function (i.e. function) of a random variable ln x t is k x (s) = ln E s ln e xt 22 the log of the moment generating

23 and, with appropriate regularity conditions, it admits the power series expansion k x (s) = 1X j=1 x j s j j! ; (9) where the j-th cumulant, j, is the j-th derivative of k x (s) evaluated at s = 0. That is, x j capture the j-th moment of the variable ln x t i.e. x 1 re ects the mean of the variable, x 2 the variance, x 3 the skewness, x 4 the kurtosis, and so on. For instance, if ln x t N ( x ; x), 2 we have x 1 = x, x 2 = x, 2 x j>2 = 0. We can, therefore, rewrite D P jjq b as D P jj ^Q ^q ^q ^q 2 = 2! + 3 3! + 4 4! + :: 23

24 References Ait-Sahalia, Y., and A. W. Lo (1998): Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices, Journal of Finance, 53(2), (2000): Nonparametric risk management and implied risk aversion, Journal of Econometrics, 94(1-2), Alvarez, F., and U. J. Jermann (2005): Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth, Econometrica, 73(6), Amemiya, T. (1985): Advanced econometrics. Harvard University Press, Cambridge: Mass. Borovicka, J., L. P. Hansen, and J. A. Scheinkman (2014): Misspeci ed Recovery, mimeo. Brown, D. E., and R. L. Smith (1990): A Correspondence Principle for Relative Entropy Minimization, Naval Research Logistics, 37(2). Carhart, M. M. (1997): On Persistence in Mutual Fund Performance, Journal of Finance, 52(1), Chen, L., R. Novy-Marz, and L. Zhang (2011): An Alternative Three-Factor Model, University of Rochester manuscript. Cochrane, J. H. (1996): A Cross-Sectional Test of an Investment-Based Asset Pricing Model, Journal of Political Economy, 104(3), Cox, J. C., and S. A. Ross (1976a): A Survey of Some New Results in Financial Option Pricing Theory, Journal of Finance, 31, (1976b): The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics, 3, Csiszar, I. (1975): I-Divergence Geometry of Probability Distributions and Minimization Problems, Annals of Probability, 3, DeMiguel, V., L. Garlappi, and R. Uppal (2009): Optimal Versus Naive Diversi cation: How Ine cient is the 1/N Portfolio Strategy?, Review of Financial Studies, 22(5), Fama, E. F., and K. R. French (1992): The Cross-Section of Expected Stock Returns, The Journal of Finance, 47, (1993): Common Risk Factors in the Returns on Stocks and Bonds, The Journal of Financial Economics, 33,

25 Fama, E. F., and J. MacBeth (1973): Risk, Return and Equilibrium: Empirical Tests, Journal of Political Economy, 81, Ghosh, A., C. Julliard, and A. Taylor (2011): What is the Consumption-CAPM missing? An informative-theoretic Framework for the Analysis of Asset Pricing Models, FMG Discussion Papers dp691, Financial Markets Group. Ghosh, A., C. Julliard, and A. P. Taylor (2014): What is the Consumption-CAPM Missing? An Information-Theoretic Framework for the Analysis of Asset Pricing Models, Working paper. Hansen, L. P. (2014): Nobel Lecture: Uncertainty Outside and Inside Economic Models, Forthcoming in the Journal of Political Economy. Hansen, L. P., and R. Jagannathan (1997): Asssessing Speci cation Errors in Stochastic Discount Factor Models, The Journal of Finance, 52, Hayashi, F. (2000): Econometrics. Princeton University Press: Princeton. Jackwerth, J. C., and M. Rubinstein (1996): Recovering Probability Distributions from Option Prices, Journal of Finance, 51(5), Jaynes, E. T. (1957a): Information Theory and Statistical Mechanics, The Physics Review, 106(4), (1957b): Information theory and statistical mechanics II, The Physical Review, 108(2), Julliard, C., and A. Ghosh (2012): Can Rare Events Explain the Equity Premium Puzzle?, Review of Financial Studies, 25(10), Kitamura, Y. (2006): Empirical Likelihood Methods in Econometrics: Theory and Practice, Cowles Foundation Discussion Papers 1569, Cowles Foundation, Yale University. Kitamura, Y., and M. Stutzer (2002): Connections between entropic and linear projections in asset pricing estimation, Journal of Econometrics, 107(1-2), Lettau, M., and S. Ludvigson (2001a): Consumption, Aggregate Wealth, and Expected Stock Returns, Journal of Finance, 56(3), (2001b): Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia Are Time-Varying, Journal of Political Economy, 109, Lewellen, J., S. Nagel, and J. Shanken (2010): A skeptical appraisal of asset pricing tests, Journal of Financial Economics, 96(2),

26 Lustig, H. N., and S. G. V. Nieuwerburgh (2005): Housing Collateral, Consumption Insurance, and Risk Premia: An Empirical Perspective, The Journal of Finance, 60(3), pp Markowitz, H. M. (1952): Portfolio Selection, Journal of Finance, 7, Owen, A. B. (1988): Empirical Likelihood Ratio Con dence Intervals for a Single Functional, Biometrika, 75(2), Parker, J. A., and C. Julliard (2005): Consumption Risk and the Cross-Section of Expected Returns, Journal of Political Economy, 113(1). Piazzesi, M., M. Schneider, and S. Tuzel (2007): Housing, consumption and asset pricing, Journal of Financial Economics, 83(3), Robinson, P. M. (1991): Consistent Nonparametric Entropy-based Testing, Review of Economic Studies, 58(3), Rosenberg, J. V., and R. F. Engle (2002): Empirical pricing kernels, Journal of Financial Economics, 64(3), Ross, S. A. (2011): The Recovery Theorem, Working Paper 17323, National Bureau of Economic Research. (2014): Recovery Theorem, Journal of Finance, Forthcoming in Journal of Finance. Santos, T., and P. Veronesi (2006): Labor Income and Predictable Stock Returns, Review of Financial Studies, 19(1), Shanken, J. (1985): Multivariate Tests of the Zero-Beta CAPM, Journal of Financial Economics, 14, Stambaugh, R. F., J. Yu, and Y. Yuan (2012): The Short of It: Investor Sentiment and Anomalies, Journal of Financial Economics, pp (forthcoming): Arbitrage Asymmetry and the Idiosyncratic Volatility Puzzle, Journal of Finance. Stock, J. (1991): Con dence Intervals for the Largest Autoregressive Root in US Macroeconomic Time Series, Journal of Monetary Economics, 28, Stutzer, M. (1995): A Bayesian approach to diagnosis of asset pricing models, Journal of Econometrics, 68(2),

27 (1996): A simple Nonparametric Approach to Derivative Security Valuation, Journal of Finance, LI(5), White, H. (1982): Maximum Likelihood of Misspeci ed Models, Econometrica, 50,

28 Table 1: 25 FF Portfolios, 1963: :12 const: IP sdf Rm SMB HML R 2 OLS (%) R 2 GLS (%) T 2 q 0:003 (5:73) 0:003 (5:70) 0:011 (3:40) 0:011 (2:50) 0:004 (1:20) 0:028 (11:33) (1:29) 0:024 (2:79) 0:028 (2:19) 0:005 (0:403) 0:023 (7:32) 0:025 (5:26) 0:135 (11:17) 0:108 (3:71) 0:341 ( 7:06) 5:46 ( 3:13) 0:004 ( 1:41) 0:006 ( 1:53) 0:0004 (0:132) ( 0:308) 0:015 ( 1:15) 0:010 (0:768) Panel A: Monthly (3:86) 0:003 (6:75) 0:004 (6:87) 0:004 (8:48) Panel B: Quarterly 0:007 (4:95) 0:008 (6:60) 0:013 (7:20) 0:012 (7:85) 67:0 [39:5;100] 68:6 [45:7;100] 3:97 [ 4:35;61:4] 71:3 [21:1;90:9] 86:1 26:8 [ 1:22;100] 83:7 [83:3;100] 3:92 [ 4:35;25:9] 74:7 [30:3;93:1] 56:6 [52:4;100] 59:6 [41:3;100] 28:8 [6:43;59:9] 40:9 [20:5;90:8] 62:2 30:8 [12:3;70:5] 51:6 [46:9;100] 8:50 [ 0:57;43:5] 17:7 [ 7:50;66:3] 37:5 (0:207) 37:1 (0:099) 71:6 (0:000) 51:5 () 29:3 (0:083) 41:3 (0:451) 28:8 (0:535) 80:9 (0:000) 59:3 (0:003) 0:077 [0:00;0:09] 0:072 [0:00;0:08] 0:128 [0:04;0:34] 0:096 [0:03;0:16] 0:058 0:332 [0:00;0:60] 0:227 [0:00;0:13] 0:431 [0:08;0:97] The table reports several statistics from cross-sectional regressions of average excess returns of the 25 Fama-French portfolios on the estimated factor loadings for di erent asset pricing models. Panels A and 83:4 [46:0;100] 46:5 [19:6;100] 31:1 (0:309) 0:351 [0:08;0:71] 0:217 [0:00;0:28] B report results at the monthly and quarterly frequencies, respectively. Rows 1 and 2 in each panel present results when the factor consists of the information portfolio (the factor-mimicking portfolio for the non-param etrically extracted SD F ) and the non-param etrically extracted SD F, resp ectively. T he non-param etric SD F is extracted from the 25 Fam a-french p ortfolios using a relative entropy m inim izing pro cedure, in a rolling out-of-sam ple fashion starting 1963:07. R ow s 3 and 4 present results for the C A P M and the Fam a-french 3-factor m o del, resp ectively. F inally, in R ow 5, the factors include the 3 Fam a-french factors in conjunction w ith the inform ation p ortfolio. For each m o del, the table rep orts the intercept and slop es, along w ith t-statistics in parentheses. It also rep orts the O L S adjusted-r 2 and the G L S adjusted-r 2, along w ith the 90% con dence intervals for the true underlying p opulation adjusted-r 2 in square brackets b elow. T he con dence intervals are constructed via sim ulations using the approach suggested by Sto ck (1991) and used by L ewellen, N agel, and Shanken (2010). T he last two colum ns rep ort, resp ectively, Shanken s (1985) cross-sectional T 2 statistic, along w ith its asym ptotic p-value in parentheses, and the q statistic that m easures how far the factor-m im icking p ortfolios are from the m ean-variance frontier. 28

29 Table 2: 10 Momentum Portfolios, 1963: :12 const: IP sdf Rm SMB HML MOM R 2 OLS R 2 GLS T 2 q 0:004 (9:99) (7:80) 0:014 (2:04) 0:022 (1:27) 0:003 (0:42) 0:008 (6:39) 0:006 (5:60) 0:038 (2:40) 0:061 (1:08) 0:036 (2:43) 0:034 (13:66) 0:039 (4:36) 0:107 (9:90) 0:084 (3:69) 0:27 ( 9:16) 1:07 ( 7:93) 0:009 ( 1:45) 0:013 ( 0:75) 0:004 (0:60) 0:024 ( 1:59) 0:050 ( 0:83) 0:024 ( 1:54) 0:011 ( 0:61) 0:005 ( 0:80) 0:038 (1:03) 0:032 (3:28) Panel A: Monthly 0:032 ( 1:18) 0:015 ( 1:60) 0:007 (6:02) 0:006 (14:64) Panel B: Quarterly 0:021 (0:63) 0:0005 (0:05) 0:022 (5:46) 90:2 [66:3;100] 95:4 [55:0;100] 10:9 [ 12:5;78:6] 78:9 [ 78:2;100] 97:7 [14:5;100] 87:3 [16:8;100] 95:4 [73:0;100] 14:4 [ 12:5;80:9] 75:4 [ 72:8;94:6] 68:1 [16:0;100] 83:6 [74:5;100] 2:0 [ 7:85;42:4] 2:59 [ 25:5;92:2] 82:1 [ 30:9;100] 75:2 [63:2;100] 78:6 [70:4;100] 6:49 [ 6:8;31:2] 1:56 [ 23:6;77:3] 12:37 (0:325) 6:37 (0:665) 40:15 (0:000) 8:81 (0:386) 2:61 (0:605) 8:12 (0:529) 7:36 (0:589) 39:55 (0:000) 9:55 (0:188) 0:024 [0:00;0:05] 0:012 [0:00;0:018] 0:074 [0:03;0:43] 0:044 [0:00;0:34] 0:006 [0:00;0:30] 0:056 [0:00;0:19] 0:047 [0:00;0:07] 0:226 [0:05;1:15] The table reports several statistics from cross-sectional regressions of average excess returns of the 10 momentum -sorted portfolios on the estimated factor loadings for di erent asset pricing models. Panels 0:021 (20:80) 98:4 [84:3;100] 86:7 [39:7;100] 1:35 (0:826) 0:131 [0:00;0:96] 0:014 [0:00;0:14] A and B report results at the monthly and quarterly frequencies, respectively. Rows 1 and 2 in each panel present results when the factor consists of the information portfolio (the factor-mimicking portfolio for the non-param etrically extracted SD F ) and the non-param etrically extracted SD F, resp ectively. T he non-param etric SD F is extracted from the 10 m om entum -sorted p ortfolios using a relative entropy m inim izing pro cedure, in a rolling out-of-sam ple fashion starting 1963:07. R ow s 3 and 4 present results for the C A P M and the Fam a-french 4-factor m o del (including the m om entum factor), resp ectively. Finally, in Row 5, the factors include the 4 Fama-French factors in conjunction with the information portfolio. For each model, the table reports the intercept and slopes, along with t-statistics in parentheses. It also rep orts the O L S adjusted-r 2 and the G L S adjusted-r 2, along w ith the 90% con dence intervals for the true underlying p opulation adjusted-r 2 in square brackets b elow. T he con dence intervals are constructed via simulations using the approach suggested by Stock (1991) and used by Lewellen, Nagel, and Shanken (2010). The last two columns report, respectively, Shanken s (1985) cross-sectional T 2 statistic, along w ith its asym ptotic p-value in parentheses, and the q statistic that m easures how far the factor-m im icking p ortfolios are from the m ean-variance frontier. 29

30 Table 3: 25 Portfolios Formed on Long-Term Reversal and Size, 1963: :12 const: IP sdf Rm SMB HML R 2 OLS (%) R 2 GLS (%) T 2 q 0:006 (7:06) (2:13) 0:005 (1:38) (0:68) ( 0:86) 0:023 (11:80) 0:008 (3:13) 0:008 (1:04) 0:006 (0:651) (0:329) 0:024 (7:98) 0:022 (5:50) 0:075 (5:71) 0:070 (5:28) 0:22 ( 2:18) 0:33 ( 0:300) (0:78) (0:93) 0:006 (2:77) 0:013 (1:81) 0:009 (1:03) 0:012 (1:80) 0:001 (1:62) 0:003 (3:99) Panel A: Monthly 0:007 (4:96) 0:004 (3:39) Panel B: Quarterly 0:005 (2:59) 0:011 (5:13) 0:020 (4:76) 0:007 (1:47) 13:5 [ 4:35;100] 72:3 [48:9;100] 1:6 [ 4:35;37:4] 74:3 [34:9;100] 84:5 [59:2;100] 3:94 [ 4:35;36:3] 56:8 [23:8;100] 8:7 [ 4:35;70:8] 77:8 [18:9;100] 61:5 [47:1;90:2] 68:0 [66:4;100] 10:1 [0:15;33:1] 26:1 [1:66;100] 66:4 [78:5;100] 27:2 [21:1;45:3] 56:5 [48:7;100] 1:33 [ 2:26;31:0] 11:96 [ 7:61;84:3] 25:07 (0:603) 18:35 (0:830) 58:14 (0:001) 40:37 (0:019) 16:69 (0:723) 54:50 (0:353) 23:96 (0:732) 68:46 () 48:86 (0:018) 0:047 [0:00;0:03] 0:037 [0:00;0:01] 0:103 [0:03;0:18] 0:077 [0:01;0:10] 0:033 [0:00;0:02] 0:291 [0:00;0:44] 0:167 [0:00;0:07] 0:372 [0:09;0:68] T he table rep orts several statistics from cross-sectional regressions of average excess returns of the 25 long term reversal and size sorted p ortfolios on the estim ated factor loadings for di erent asset 86:7 53:0 22:61 (0:309) 0:301 [0:04;0:49] 0:153 pricing m o dels. Panels A and B rep ort results at the m onthly and quarterly frequencies, resp ectively. R ow s 1 and 2 in each panel present results w hen the factor consists of the inform ation p ortfolio (the factor-m im icking p ortfolio for the non-param etrically extracted SD F ) and the non-param etrically extracted SD F, resp ectively. T he non-param etric SD F is extracted from the 25 long term reversal and size sorted p ortfolios using a relative entropy m inim izing pro cedure, in a rolling out-of-sam ple fashion starting 1963:07. R ow s 3 and 4 present results for the C A P M and the Fam a-french 3-factor m o del, respectively. Finally, in Row 5, the factors include the 3 Fama-French factors in conjunction with the information portfolio. For each model, the table reports the intercept and slopes, along with t-statistics in parentheses. It also rep orts the O L S adjusted-r 2 and the G L S adjusted-r 2, along w ith the 90% con dence intervals for the true underlying p opulation adjusted-r 2 in square brackets b elow. T he con dence intervals are constructed via sim ulations using the approach suggested by Sto ck (1991) and used by L ewellen, N agel, and Shanken (2010). T he last two colum ns rep ort, resp ectively, Shanken s (1985) cross-sectional T 2 statistic, along w ith its asym ptotic p-value in parentheses, and the q statistic that m easures how far the factor-m im icking p ortfolios are from the m ean-variance frontier. 30

31 Table 4: Small, Large, Growth, Value, Winners, Losers, 10 Industry, 1963: :12 const: IP sdf Rm SMB HML MOM R 2 OLS R 2 GLS T 2 q 0:001 (2:49) 0:003 (7:45) 0:007 (2:02) 0:003 (1:25) 0:003 ( 1:58) 0:014 (12:44) 0:009 (5:81) 0:021 (2:10) 0:012 (1:47) ( 0:26) 0:027 (8:62) 0:035 (8:10) 0:100 (6:61) 0:107 (4:62) 1:12 ( 16:7) 3:88 ( 6:92) ( 0:70) (1:09) 0:007 (4:45) 0:005 ( 0:52) 0:004 (0:45) 0:017 (2:16) Panel A: Monthly (2:42) 0:003 (5:44) (1:92) 0:0003 (0:51) Panel B: Quarterly 0:006 (2:50) 0:008 (3:89) 0:005 (2:01) (1:10) 0:009 (8:78) 0:008 (10:65) 0:027 (7:97) 0:024 (8:43) 94:9 83:0 3:5 84:8 94:7 75:8 74:0 5:1 82:9 89:6 88:0 84:2 1:17 40:3 84:8 60:9 79:9 0:73 37:3 75:8 5:39 (0:980) 9:89 (0:770) 62:76 (0:000) 27:23 (0:004) 5:73 (0:838) 17:15 (0:248) 10:85 (0:698) 67:17 (0:000) 27:86 (0:003) 8:27 (0:602) 0:013 0:019 0:112 0:052 0:012 0:145 0:077 0:358 0:178 0:062 T he table rep orts several statistics from cross-sectional regressions of average excess returns of the 10 industry p ortfolios and the top and b ottom deciles of p ortfolios sorted on the basis of size, b o ok-tom arket-equity, and m om entum on the estim ated factor loadings for di erent asset pricing m o dels. Panels A and B rep ort results at the m onthly and quarterly frequencies, resp ectively. R ow s 1 and 2 in each panel present results when the factor consists of the information portfolio and the non-parametrically extracted SDF, respectively. The non-parametric SDF is extracted from the 10 industry portfolios and the top and b ottom deciles of p ortfolios sorted on the basis of size, b o ok-to-m arket-equity, and m om entum, using a relative entropy m inim izing pro cedure, in a rolling out-of-sam ple fashion starting 1963:07. Rows 3 and 4 present results for the CAPM and the Fama-French 4-factor model (including the momentum factor), respectively. Finally, in Row 5, the factors include the 4 Fama-French factors in conjunction w ith the inform ation p ortfolio. For each m o del, the table rep orts the intercept and slop es, along w ith t-statistics in parentheses. It also rep orts the O L S adjusted-r 2 and the G L S adjusted-r 2, along w ith the 90% con dence intervals for the true underlying p opulation adjusted-r 2 in square brackets b elow. T he con dence intervals are constructed via sim ulations using the approach suggested by Sto ck (1991) and used by L ewellen, N agel, and Shanken (2010). T he last two colum ns rep ort, resp ectively, Shanken s (1985) cross-sectional T 2 statistic, along w ith its asym ptotic p-value in parentheses, and the q statistic. 31

32 Table 5: 30 Industry Portfolios, 1963: :12 const: IP sdf Rm SMB HML R 2 OLS (%) R 2 GLS (%) T 2 q (2:58) 0:001 (1:42) 0:006 (3:55) 0:006 (2:46) 0:000 (0:023) 0:011 (2:93) 0:010 (3:48) 0:017 (3:55) 0:023 (2:51) 0:020 (2:58) 0:022 (5:74) 0:025 (5:79) 0:043 (2:66) 0:042 (2:42) 0:15 ( 3:97) 1:72 ( 1:64) 0:0001 ( 0:12) 0:001 ( 0:36) 0:005 (2:93) 0:001 (0:127) 0:006 ( 0:62) 0:003 ( 0:350) Panel A: Monthly 0:0004 (0:26) 0:0001 ( 0:14) 0:001 ( 0:75) 0:001 ( 1:07) Panel B: Quarterly 0:003 (0:71) 0:006 (1:81) ( 0:43) ( 0:562) T he table rep orts several statistics from cross-sectional regressions of average excess returns of the 30 Industry p ortfolios on the estim ated factor loadings for di erent asset pricing m o dels. Panels A and 33:8 52:5 3:52 8:33 60:4 5:55 17:3 3:51 8:40 26:6 50:1 65:6 2:30 7:32 64:6 31:4 39:9 2:68 1:64 42:9 12:61 (0:994) 8:56 (1:00) 26:69 (0:535) 26:00 (0:463) 7:55 (1:00) 23:49 (0:708) 20:66 (0:839) 35:16 (0:165) 33:57 (0:146) 16:29 (0:906) 0:023 0:016 0:047 0:046 0:015 0:133 0:116 0:187 0:181 0:098 B report results at the monthly and quarterly frequencies, respectively. Rows 1 and 2 in each panel present results when the factor consists of the information portfolio (the factor-mimicking portfolio for the non-param etrically extracted SD F ) and the non-param etrically extracted SD F, resp ectively. T he non-param etric SD F is extracted from the 30 Industry p ortfolios using a relative entropy m inim izing pro cedure, in a rolling out-of-sam ple fashion starting 1963:07. R ow s 3 and 4 present results for the C A P M and the Fam a-french 3-factor m o del, resp ectively. F inally, in R ow 5, the factors include the 3 Fam a-french factors in conjunction w ith the inform ation p ortfolio. For each m o del, the table rep orts the intercept and slop es, along w ith t-statistics in parentheses. It also rep orts the O L S adjusted-r 2 and the G L S adjusted-r 2, along w ith the 90% con dence intervals for the true underlying p opulation adjusted-r 2 in square brackets b elow. T he con dence intervals are constructed via sim ulations using the approach suggested by Sto ck (1991) and used by L ewellen, N agel, and Shanken (2010). T he last two colum ns rep ort, resp ectively, Shanken s (1985) cross-sectional T 2 statistic, along w ith its asym ptotic p-value in parentheses, and the q statistic that m easures how far the factor-m im icking p ortfolios are from the m ean-variance frontier. 32

33 Table 6: Summary Statistics of Information Portfolio & Returns Assets Mean Volatility Sharpe Ratio Skewness Kurtosis CEQ Panel A: Monthly Market - Risk Free 0:004 0:045 0:091 0:567 5:028 0:003 HML 0:004 0:029 0:139 0:034 5:440 0:004 Momentum Factor 0:007 0:044 0:164 1:419 13:65 0:006 R IP (F F 25) R IP (10Momentum) R IP (25Long T ermreversal&size) R IP (S;B;G;V;W;L;10Industry) R IP (30Industry) 0:021 (0:007) 0:030 (0:004) 0:013 (0:007) 0:027 (0:005) (0:005) 0:073 (0:051) 0:127 (0:048) 0:064 (0:051) 0:088 (0:046) 0:083 (0:048) Panel B: Quarterly 0:288 (0:128) 0:235 (0:085) 0:206 (0:137) 0:306 (0:106) 0:018 (0:112) 0:384 ( 0:575) 0:352 ( 0:326) 0:212 ( 0:444) 0:679 ( 0:490) 0:040 ( 0:522) 5:541 (5:589) 8:022 (4:793) 5:111 (5:865) 6:180 (4:953) 6:318 (5:708) 0:018 (0:006) 0:022 (0:003) 0:011 (0:006) 0:023 (0:004) 0:001 (0:004) Market - Risk Free 0:013 0:087 0:150 0:435 3:635 0:009 HML 0:012 0:060 0:204 0:109 4:754 0:010 Momentum Factor 0:020 0:081 0:254 1:411 10:13 0:017 R IP (F F 25) R IP (10Momentum) R IP (25Long T ermreversal&size) R IP (S;B;G;V;W;L;10Industry) R IP (30Industry) 0:080 (0:021) 0:085 (0:013) 0:042 (0:023) 0:083 (0:016) 0:007 (0:017) 0:194 (0:103) 0:239 (0:093) 0:134 (0:104) 0:173 (0:090) 0:180 (0:093) The table reports the mean, volatility, Sharpe ratio, skewness, and kurtosis of the information portfolio constructed using di erent sets of test assets, with the corresponding statistics for an equally-weighted 0:413 (0:207) 0:354 (0:143) 0:313 (0:220) 0:480 (0:175) 0:041 (0:186) 0:410 ( 0:183) 0:090 ( 0:231) 0:168 ( 0:057) 0:181 ( 0:315) 0:029 ( 0:298) 3:955 (3:576) 5:295 (3:805) 3:833 (3:865) 3:463 (3:794) 2:934 (3:942) 0:061 (0:016) 0:056 (0:009) 0:033 (0:018) 0:068 (0:012) 0:009 (0:013) portfolio of the test assets reported in parentheses below. Panels A and B report results at the monthly and quarterly frequencies, respectively. Each row presents results when the information portfolio is constructed from the cross-section of test assets mentioned in Column 1. 33

34 Table 7: With Respect to FF Factors, 1963: :12 Assets (%) Rm SMB HML MOM R 2 OLS (%) Info Ratio FF25 1:31 (4:89) 10 Momentum 1:37 (3:06) 25 Long-Term Reversal & Size 0:69 (2:79) S, B, G, V, W, L, 10 Industry 0:93 (3:28) 30 Industry 0:29 (0:96) FF25 5:49 (4:01) 10 Momentum 2:63 (1:77) 25 Long-Term Reversal & Size 2:39 (2:57) S, B, G, V, W, L, 10 Industry 3:21 (2:77) 30 Industry 1:16 (1:00) Panel A: Monthly 0:47 (7:44) 0:94 (9:03) 0:40 (6:92) 0:59 (9:00) 0:81 (11:43) 0:21 (2:41) 0:30 ( 2:09) 0:35 (4:39) 0:35 (3:90) 0:87 ( 8:79) Panel B: Quarterly 0:59 (3:29) 1:20 (6:52) 0:50 (4:16) 0:67 (4:70) 1:02 (6:73) 0:43 (1:72) 0:21 ( 0:82) 0:01 (0:06) 0:29 (1:42) 1:01 ( 4:72) 1:30 (13:82) 0:26 (1:67) 0:92 (10:56) 1:28 (12:87) 0:59 ( 5:57) 1:12 (4:85) 0:43 (1:76) 0:93 (5:94) 1:16 (6:15) 0:66 ( 3:37) 1:67 (16:52) 1:25 (19:70) 1:92 (10:76) 1:23 (8:89) 26:9 0:211 36:1 0:135 20:7 0:121 46:6 0:145 28:3 0:041 15:4 0:310 42:6 0:147 18:0 0:199 33:6 0:230 29:1 0:077 T he table rep orts the intercept and slop e co e cients, along w ith the t-statistics in parentheses, as well as the O L S adjusted-r 2 from tim e series regressions of the return on the inform ation p ortfolio on the Fam a-french factors. Panels A and B rep ort results at the m onthly and quarterly frequencies, resp ectively. E ach row presents results w hen the inform ation p ortfolio is constructed from the cross-section of test assets mentioned in Column 1. 34

35 Figure 1: The gure plots the time series of the non-parametrically extracted SDF (Panel B) and its corresponding factor-mimicking information portfolio (Panel A) at the monthly frequency over 1963: :12 when the cross-section of test assets consists of the 25 Fama-French portfolios. The grey shaded areas denote the NBER-designated recession periods and the dashed-dotted vertical lines denote the major stock market crashed identi ed in Mishkin and White (2002). 35

36 Figure 2: The gure plots the path of $1 invested in the information portfolio (red line) and the market portfolio (green line) over 1963: :10. The information portfolio is non-parametrically extracted at the monthly frequency starting 1963:07 from the 25 Fama-French portfolios using a relative entropy minimization procedure in a rollinf out-of-sample fashion. 36