Portfolio-Based Tests of Conditional Factor Models 1

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1 Portfolio-Based Tests of Conditional Factor Models 1 Abhay Abhyankar Devraj Basu Alexander Stremme Warwick Business School, University of Warwick November 2002 Preliminary; please do not Quote or Distribute address for correspondence: Alexander Stremme Wariwck Business School University of Warwick Coventry, CV4 7AL United Kingdom a.stremme@wbs.warwick.ac.uk 1 We thank Keng-Yu Ho for research assistance. We would also like to thank Wayne Ferson for helpful discussions. We also thank Kenneth French and Martin Lettau for making available the data used in this study. All remaining errors are ours. 1

2 Portfolio-Based Tests of Conditional Factor Models November 2002 Preliminary; please do not Quote or Distribute Abstract In this paper, we construct portfolio-based tests of conditional factor models, developed by exploiting the close link between the stochastic discount factor framework and unconditional mean-variance efficiency. We show that an asset pricing model admits a conditional factor representation if and only if there is some combination of the factor mimicking portfolios that is unconditionally mean-variance efficient. Conversely, for a given set of factors, our analysis allows us to construct a managed portfolio of the factor mimicking portfolios that attains the maximum Sharpe ratio generated by the factors. We thus obtain the factor loadings, as functions of the conditioning information, of the optimal conditional factor model for given choice of factors. We use our results to empirically test several factor models proposed in the literature. We find that the Consumption Capital Asset Pricing Model, scaled by lagged consumptionwealth ratio, as well as the model of Harvey and Siddique (2000) perform comparatively well as conditional asset pricing models, while the Fama and French (1992) model does less well. JEL Classification: C31, G11, G12 2

3 1 Introduction The objective of this paper is to develop portfolio-based tests of conditional factor models. These tests are constructed by exploiting the close links between the stochastic discount factor framework and mean-variance efficiency. The concept of mean-variance efficiency has important implications for tests of asset pricing models. In the presence of conditioning information, one can construct managed portfolios that expand the mean-variance efficient frontier. This places tighter restrictions on conditional asset pricing models, since the set of attainable pay-offs that the model must price correctly is larger. In the presence of conditioning information, the study of conditional efficiency is not always appropriate since it focuses on the ex-post rather than ex-ante optimal use of this information, as highlighted also by Dybvig and Ross (1985). Moreover, tests based on conditional efficiency require the evaluation of conditional moments, which poses problems in practical implementations. Therefore, we focus our attention on unconditional efficiency, as considered also in Ferson and Siegel (2002b). We develop techniques to construct and test conditional factor models, by relating them to the unconditional mean-variance efficiency of the associated factor-mimicking portfolios. A similar approach, without conditioning information, is developed in Huberman, Kandel, and Stambaugh (1987). We show that, if a model admits a conditional factor representation in the sense that a linear combination of factors constitutes an admissible stochastic discount factor for the model, then a combination of the factor mimicking portfolios is in fact unconditionally mean-variance efficient. This provides a necessary condition for a factor model to be viable. Conversely, for a given set of factors we construct a managed combination of the factor mimicking portfolios that attains the maximum Sharpe ratio generated by the factors. If 3

4 this Sharpe ratio is equal to the maximum Sharpe ratio generated by the base assets, then this portfolio can be lifted to a conditional linear factor pricing model. Moreover, even if exact factor pricing cannot be achieved, our methodology allows us to construct the loadings of the best possible conditional factor model for a given choice of factors. We use our results to test various conditional factor models. We test the three-factor Fama and French (1992) model, the augmented Fama-French model with a skewness factor similar to the one proposed by Harvey and Siddique (2000), and the Consumption CAPM. Using dividend yield and consumption-wealth ratio (Lettau and Ludvigson 2001a) as conditioning variables we find that, while the skewness factor clearly improves the performance of the Fama-French model, the augmented model still only accounts for at most 82% of the Sharpe ratio generated by the base assets. In contrast, for the Consumption CAPM, with aggregate consumption growth rate as the factor, this figure rises to 90%. Our paper is most closely related to Ferson and Harvey (1999), who study the performance of the Fama-French model on conditional expected returns. They test the ability of the model to capture common dynamic patterns in returns, modelled using a set of lagged predictor variables, and find that the model does not perform well as a conditional asset pricing model. The same is true of the four-factor model of Carhart (1997). Ferson and Harvey (1999) allow both models to have time-varying coefficients which are linear in the predictive variables. Our analysis extends theirs to allow for non-traded factors. Moreover, we show that the coefficients of the optimal factor model for given choice of factors are in fact non-linear functions of the conditioning variable, in contrast to their analysis. Ferson and Siegel (2002b) develop tests for stochastic discount factors and portfolio efficiency with conditioning information. They refine tests of portfolio efficiency to include conditioning 4

5 information and find that these tests reject the Fama-French model in any fixed-weight combination. Our analysis focuses on testing for the unconditional mean-variance efficiency of factor mimicking portfolios by deriving an explicit expression for the maximum achievable Sharpe ratio from a set of factors. We extend the analysis of Ferson and Siegel (2002b) by considering factor models with coefficients that are non-linear functions of the conditioning variables. Moreover, our results yield necessary and sufficient conditions for a given set of factors to constitute a viable conditional asset pricing model. Breeden, Gibbons, and Litzenberger (1989) construct factor-mimicking portfolios for consumption growth in their tests of the consumption-based Capital Asset Pricing Model (C- CAPM). We extend their analysis to include conditioning information and find that the conditional C-CAPM seems to perform comparatively well when consumption-wealth ratio is used as conditioning variable. Our results thus broadly agree with those of Lettau and Ludvigson (2001b), while our analysis differs from theirs in that we use the traded mimicking portfolio for consumption growth rather than consumption growth itself. Kraus and Litzenberger (1976) and Harvey and Siddique (2000) consider asset pricing models which allow the skewness risk in the return distribution to be priced. We extend these models to incorporate conditioning information and find that the inclusion of a skewness factor considerably enhances their performance as conditional asset pricing models. The remainder of this paper is organized as follows; Section 2, briefly reviews the theoretical background and establishes our notation. In Section 3, we discuss conditional factor models and factor mimicking portfolios, and state our main results. In Section 4, we develop tests for conditional factor models based on their maximum Sharpe ratio. Our empirical results are reported in Section 5. Section 6 concludes. Most proofs are given in the mathematical appendix. 5

6 2 Theoretical Background In this section, we provide a brief outline of the underlying asset pricing theory, and establish our notation. We first construct the space of state-contingent pay-offs, and within it the space of traded pay-offs, augmented by the use of conditioning information. Next, we define a generalized notion of return within this space. 2.1 Set-Up and Notation We begin by constructing the space of state-contingent pay-offs. We fix a probability space (Ω, F, P ), endowed with a discrete-time filtration (F t ) t. We fix t > 0, and consider the period beginning at time t 1 and ending at t. Denote by L 2 t the Hilbert space of all F t - measurable random variables that are square-integrable with respect to P, endowed with the canonical inner product E ( x t y t ). We interpret Ω as the set of states of nature, and L 2 t as the space of all (not necessarily attainable) state-contingent pay-off claims. Conditioning Information: To incorporate conditioning information, we take as given a sub-σ-field G t 1 F t. We think of G t 1 as summarizing all information on which investors base their portfolio decisions at time t 1. In particular, asset prices at time t 1 will typically depend on G t 1. In most practical applications, G t 1 will be chosen as the σ-field generated by a set of conditioning instruments, variables observable at time t 1 that contain information about the distribution 6

7 of future asset returns 1. By replacing the inner product on L 2 t by its conditional version, E ( x t y t G t 1 ), and completing the space with respect to the implied norm if necessary, we turn L 2 t into a conditional Hilbert space (Hansen and Richard 1987). To emphasize the conditional structure, we denote this space by L 2 t ( G t 1 ). Traded Base Assets: Throughout this paper, we focus on the case in which the space of traded pay-offs is generated by a finite 2 set of base assets. Specifically, we assume there are n traded risky assets, indexed k = 1... n. We denote the gross return (per dollar invested) of the k-th asset by rt k L 2 t, and by R t := ( rt 1... rt n ) the n-vector of risky asset returns. In addition to the risky assets, we assume that a risk-free is traded with gross return rt 0 = r f. We now define X t as the space of all elements x t L 2 t that can be written in the form, n x t = θt 1r 0 f + ( rt k r f )θt 1, k (1) k=1 for G t 1 -measurable functions θt 1. k To simplify notation, we write this in vector form as x t = θt 1r 0 f + ( R t r f e ) θ t 1, where θ t 1 = ( θt θt 1 n ) is the vector of weights on the risky assets, and e is an n-vector of ones. We interpret X t as the space of managed pay-offs, obtained by forming combinations of the base assets with weights θt 1 k that are functions of the conditioning information. Throughout this paper, we will often refer to X t as the augmented pay-off space. 1 Examples of such variables considered in the literature include dividend yield (Fama and French 1988), interest rate spreads (Campbell 1987), or consumption-wealth ratio (Lettau and Ludvigson 2001a). 2 Note that, unlike in the fixed-weight case, the space of managed pay-offs is infinite-dimensional even when there is only a finite number of base assets. 7

8 Pricing Function: Since the base assets are characterized by their returns, we set Π t 1 ( rt k ) = 1, and extend the function Π t 1 to all of X t by conditional linearity. In other words, the cost of a portfolio of base assets is given by the sum of the weights it assigns to the base assets. Since the weights on the risky assets in (1) are applied to their excess returns, we simply obtain, n [ Π t 1 ( x t ) = θt 1Π 0 t 1 ( r f ) + Πt 1 ( rt k ) Π t 1 ( r f ) ] θt 1 k = θt 1. 0 (2) k=1 By construction, X t is a conditionally linear subspace of L 2 t ( G t 1 ). Replacing X t by its closure if necessary, we can also assume that X t is complete and hence itself a conditional Hilbert space. By definition, the function Π t 1 is conditionally linear on X t, and continuous by Cauchy-Schwartz. In other words, the pricing rule satisfies the law of one price, a weak from of no-arbitrage condition. Throughout the remainder of this paper, we will refer to the pair ( X t, Π t 1 ) as the conditional market model generated by the given set of base assets. 2.2 A Generalized Notion of Return By an admissible stochastic discount factor (SDF), we mean an element m t L 2 t that prices all base assets conditionally correctly, that is E t 1 ( m t rt k ) = 1 for all k = 0, 1... n. Note that, for an arbitrary managed pay-off x t X t, conditional linearity then implies, n [ E t 1 ( m t x t ) = θt 1E 0 t 1 ( m t r f ) + Et 1 ( m t rt k ) E t 1 ( m t r f ) ] θt 1 k = θt 1 0 (3) k=1 In other words, an SDF that prices all base assets correctly is necessarily compatible with the pricing function Π t 1 for managed pay-offs in the sense that E t 1 ( m t x t ) = Π t 1 ( x t ). 8

9 Finally, taking unconditional expectations in (3), one obtains the unconditional restriction, E( m t x t ) = E( Π t 1 ( x t ) ) =: Π 0 ( x t ). (4) Using an unconditional orthogonality argument, it is easy to show that a candidate m t is an admissible SDF if and only if (4) folds for all x t X t. In a slight abuse of terminology, we refer to Π 0 as the generalized pricing function induced by the base assets on the space of managed payoffs. We now define the space of generalized returns as 3, R t = { r t X t Π 0 ( r t ) = 1 }. (5) An element r t R t is called unconditionally efficient if it has minimum unconditional variance among all elements r t R t with the same unconditional mean. Note that, in the presence of conditioning information, one could also consider the concept of conditional efficiency. However, we are interested in the ex-ante efficiency of managed portfolios, rather than the ex-post efficiency once conditioning information is known 4. To conclude this section, we define the generalized Sharpe ratio in the space R t, E( r t ) r f λ = sup. (6) r t R t σ( r t ) In Section 4, we derive an explicit expression for λ in terms of the conditional moments of the base asset returns. The maximum generalized Sharpe ratio constitutes the benchmark 3 The Gallant, Hansen, and Tauchen (1990) (GHT) discount factor bounds, and their implementation by Bekaert and Liu (2001), are implicitly based on this space of generalized returns. 4 As Dybvig and Ross (1985) show, when portfolio managers possess information not known to outside investors, their (conditionally efficient) strategies may seem conditionally inefficient to outsiders. Hence, the notion of unconditional efficiency is a more appropriate measure of performance in this case. 9

10 against which factor models will be tested. We use the fact that a generalized return is unconditionally efficient if and only if its implied generalized Sharpe ratio equals λ. Remark: Ferson and Siegel (2001) extensively study portfolio efficiency in the space of conditional returns, defined by the more restrictive constraint Π t 1 ( r t ) 1. While this notion of return may be more intuitive, the resulting return space is much smaller 5. In the context of asset pricing model tests, it is reasonable to focus on the largest possible return space, since it delivers the tightest testable restrictions. It should be noted, however, that all our results hold equally well in the case of conditional returns, and even in the fixed-weight case without conditioning information. In particular, our results applied to conditional returns, extend those of Ferson and Siegel (2002b). 3 Factor Models and Mimicking Portfolios The purpose of this section is twofold. First, we show that a given market model admits a factor representation if and only if there is a linear combination of factor mimicking portfolios that is unconditionally efficient. This result is at the heart of the factor model tests developed in Section 4. Second, for a given choice of factors, we show how to construct an optimal factor representation that comes as close as possible to pricing the assets correctly. It turns out that the optimal factor loadings are non-linear functions of the conditioning information, which is in contrast to Ferson and Harvey (1999). 5 In fact, while the space R t of generalized returns has co-dimension one in the augmented pay-off space, the space of conditional returns is of infinite co-dimension. 10

11 3.1 Conditional Factor Models Our focus here is not the selection of factors, but rather the construction and testing of models for a given set of factors. Therefore, we take as given m factors, F i t L 2 t, indexed i = 1... m. Note that in general we do not assume that the factors are traded, we may have Ft i X t. Denote by F t = ( Ft 1,..., Ft m ) the m-vector of factors. To define what we mean by a conditional factor model, we use the characterization in terms of SDFs 6. Definition 3.1 We say that the model ( X t, Π t 1 ) admits a conditional factor structure, if there exist G t 1 -measurable functions a t 1 and b i t 1 such that, m t = α t 1 + m i=1 F i t b i t 1 (7) is an admissible SDF for the model, that is E t 1 ( x t m t ) = Π t 1 ( x t ) for all x t X t. In vector notation, we write (7) as m t = α t 1 + F t b t 1, where b t 1 = ( b 1 t 1,..., b m t 1 ) denotes the vector of factor loadings. It is important to emphasize that the above specification defines a conditional factor model, in that the coefficients a t 1 and b i t 1 are allowed to be functions of the conditioning information. In the case where an admissible stochastic discount factor of the form (7) can be found with unconditionally constant coefficients, we say that the model admits a fixed-weight factor structure. 6 It is easy to show (Cochrane 2001) that this characterization is equivalent to the more traditional formulation in terms of expected base asset returns. 11

12 3.2 Factor-Mimicking Portfolios Since in general, the factors cannot be assumed to be traded assets, we must first construct factor-mimicking portfolios in order to relate factor models to portfolio efficiency. Definition 3.2 An element f i t factor F i t L 2 t if and only if Π t 1 ( f i t ) = 1, and ρ 2( f i t, F i t R t is called a factor-mimicking portfolio (FMP) for the ) ( ) ρ 2 r t, Ft i for all r t R t with Π t 1 ( r t ) = 1. (8) Note that we require the FMP to be a conditional return. This is necessary because we wish to study the unconditionally efficient frontier spanned by the FMPs as base assets. Denote by f t = ( ft 1,..., ft m ) the m-vector of FMPs for the given factors. In the following section, we show (Theorem 4.1) that the model admits a conditional factor representation if and only if there is a portfolio of FMPs that is unconditionally efficient. We now give an explicit characterization of the FMPs in terms of their base asset weights. To simplify notation, we write E t 1 ( ) for the conditional expectation operator with respect to G t 1. We define the conditional moments, µ t 1 = E t 1 ( R ( t r f e ), and Λ t 1 = E t 1 ( Rt r f e )( R t r f e ) ) (9) In other words, excess returns can be written as R t r f e = µ t 1 + ε t, where µ t 1 is the conditional expectation given conditioning information, and ε t is the residual term with zero conditional mean and variance-covariance matrix Σ t 1 = Λ t 1 µ t 1 µ t 1. This is the formulation of the asset pricing model with conditioning information used in Ferson and 12

13 Siegel (2001) 7. Similarly, we denote the mixed conditional moments of the factors by ν t 1 = E t 1 ( Ft ), and Qt 1 = E t 1 ( ( Rt r f e ) F t ) (10) Remark: Note that, if an admissible SDF of the form (7) exists, this implies, ( ) 0 E t 1 ( Rt r f e )m t = at 1 µ t 1 + Q t 1 b t 1. Conversely, if a t 1 and b t 1 exist so that a t 1 µ t 1 + Q t 1 b t 1 = 0, then m t in (7) prices all excess returns correctly and can hence be modified to be an admissible SDF. In other words, the model admits a conditional factor structure if and only if the image of the conditional linear operator Q t 1 contains µ t 1. Proposition 3.3 For a given factor F i t, the factor-mimicking portfolio can be written as, ft i = r f + ( Rt r f e ) ( ) θ i t 1 with θt 1 i = Λ 1 t 1 q i t 1 κ i µ t 1 where q i t 1 is the column of Q t 1 corresponding to factor i, and κ i is a constant. (11) Proof: Appendix A.1. Note that the constant κ i in the above expression is directly related to the unconditional mean of the FMP. In the case where a risk-free asset is present, this constant is not uniquely determined, since the first-order condition arising from maximizing the correlation in (8) is independent of that mean. It will be convenient to write yt 1 i = qt 1 i κ i µ t 1, and define Y t 1 as the matrix whose columns are the yt 1. i With this notation, we obtain, 7 Note however that our notation differs slightly from that used in Ferson and Siegel (2001), who define Λ t 1 to be the inverse of the conditional second-moment matrix. 13

14 Corollary 3.4 The conditional moments of the factor mimicking portfolios are given by, E t 1 ( ft r f e ) = Y t 1Λ 1 t 1µ t 1, and E t 1 ( ( ft r f e )( f t r f e ) ) = Y t 1Λ 1 t 1Y t 1. In the following sections, we use these results to obtain an explicit expression for the maximum generalized Sharpe ratio generated by the factor mimicking portfolios. 4 Tests of Conditional Linear Factor Models In this section, we develop a Sharpe ratio based test for conditional factor models. We show that the model admits a conditional factor representation if and only if the maximum generalized Sharpe ratio generated by the factor mimicking portfolios equals that generated by the base assets. Obviously, this is the case if and only if there exists a portfolio of FMPs that is unconditionally efficient in the augmented pay-off space. In order to study the efficiency of the factor mimicking portfolios, we treat them as base assets of a hypothetical pay-off space. More specifically, denote by Xt F the space of managed pay-offs x t L 2 t that can be written in the form, x t = φ 0 t 1r f + m ( ft i r f )φ i t 1, (12) i=1 for G t 1 -measurable functions φ i t 1. In other words, Xt F is the space of pay-offs that are attainable by forming managed portfolios of the factor mimicking portfolios. Since the mimicking portfolios are by construction conditional returns, we have Π t 1 ( x t ) = φ 0 t 1 for any x t Xt F of the form (12). Mimicking the construction in Section 2.2, we define the space of generalized returns spanned by the FMPs as, Rt F = R t Xt F. 14

15 Denote by λ F the maximum generalized Sharpe ratio in the space R F t, λ F = sup r t R F t E( r t ) r f σ( r t ). (13) In other words, λ F is the maximum generalized Sharpe ratio that can be achieved by forming portfolios of factor-mimicking portfolios. We now formulate the main result of this section, which builds the basis for the factor model tests estimated in the following section. Theorem 4.1 The model ( X t, Π t 1 ) admits a conditional factor structure if and only if λ F = λ Proof: The proof of Theorem 4.1 is based on a series of propositions and lemmas which are of independent interest and therefore are stated explicitly in the subsequent sections. The proof of the theorem is concluded in Appendix A.4. As a consequence of the above result, the difference λ λ F can be interpreted as a test of whether it is possible to construct a conditional linear asset pricing model from a given set of factors. Since by construction R F t R t, we always have λ F λ, with equality if and only if there exists a portfolio r t R F t that is unconditionally efficient in the space R t. Corollary 4.2 The model ( X t, Π t 1 ) admits a conditional factor structure if and only if there exist G t 1 -measurable functions φ i t 1 with E( φ 0 t 1 ) = 1, such that φ 0 t 1r f + m i=1 ( ) φ i t 1 f i t r f is unconditionally efficient in the space R t of generalized returns. (14) Proof: Follows directly from Theorem

16 4.1 Maximum Generalized Sharpe Ratios The following proposition provides an explicit characterization of the maximum Sharpe ratio in terms of the conditional moments of the base asset returns. Proposition 4.3 The maximum generalized Sharpe ratio in the space R t is given by λ = h, where h 2 = E( H 2 t 1 ), and H 2 t 1 = µ t 1 Σ 1 t 1 µ t 1, (15) Proof: The proof is given in Appendix A.2. Expression (15) for the maximum Sharpe ratio has many interesting features; first, it extends the expression given in Equation (16) of Jagannathan (1996) to the case with conditioning information. It is well-known (Cochrane 2001) that in the fixed-weight case without conditioning information the maximum (squared) Sharpe ratio is given by an expression of the form (15), with conditional moments replaced by unconditional ones. In other words, H t 1 represents the maximum conditional Sharpe ratio, once the realization of the conditioning information is known. Hence, the maximum squared unconditional Sharpe ratio is simply given by the expectation of the maximum squared conditional Sharpe ratio. For the case of only one risky asset, this result was also shown in Cochrane (1999). Ferson and Siegel (2001) calculate the maximum Sharpe ratio in their setting. Unfortunately, their expression does not admit an interpretation of this sort. To see this, we note the in our notation their maximum squared Sharpe ratio can be written as, λ 2 F S = E ( H 2 t H 2 t 1 )/ ( E 1 ). (16) 1 + Ht

17 Moreover, since the space of conditional returns is smaller than the space of generalized returns considered here, the Sharpe ratio in their setting will be lower than ours 8. As a consequence, in contrast to the tests developed in this paper, tests based on the Ferson and Siegel (2001) Sharpe ratio will in general yield necessary but not sufficient conditions. Corollary 4.4 The maximum generalized Sharpe ratio in the space R F t is given by λ F = h F, where h 2 F = E( H 2 F,t 1 ), and H 2 F,t 1 = µ t 1Λ 1 t 1Y t 1 [ Y t 1Λ 1 t 1Σ t 1 Λ 1 t 1Y t 1 ] 1Y t 1Λ 1 t 1µ t 1, (17) Proof: We apply Proposition 4.3, using the factor-mimicking portfolios as base assets. Using the conditional moments from Corollary 3.4 to substitute for µ t 1 and Σ t 1 in (15) gives the desired result. Finally, we characterize the weights on the mimicking portfolios of the portfolio that attains the maximum Sharpe ratio in (17). These weights are in fact proportional to the factor loadings in the optimal conditional factor model for given choice of factors. Proposition 4.5 The maximum generalized Sharpe ratio in (17) is attained by, rt = φ 0 t 1 r f + ( f t r f e ) φ t 1 with φ 0 t 1 = 1 + H2 F,t h 2 F (18) and φ t 1 = r f 1 + h 2 F Proof: The proof is given in Appendix A.3. [ Y t 1Λ 1 t 1Σ t 1 Λ 1 t 1Y t 1 ] 1 Y t 1Λ 1 t 1µ t 1 8 This can also be shown directly using Jensen s inequality. 17

18 5 Empirical Analysis In this section, we use the theory developed in Sections 3 and 4 to test several conditional factor models. The models considered are the Consumption CAPM, the Fama-French threefactor model, and finally the Fama-French model augmented with a skewness-based term following Harvey and Siddique (2000). We specialize the set-up of the preceding sections to the case of a single instrument. Specifically, let y t 1 be a given F t 1 -measurable conditioning variable, and set G t 1 = σ( y t 1 ). For the estimation, we will use the de-meaned variable yt 1 0 = y t 1 E ( y t 1 ). To estimate the conditional moments, we postulate that the base asset returns and the factors depend on the conditioning instrument via a linear relationship of the form, ( Rt r f e F t ) ( µ = ν ) ( β + γ ) yt ( εt η t ) (19) where ε t and η t are independent of y 0 t 1 with E t 1 ( ε t ) = E t 1 ( η t ) = 0. In the notation of Section 3, we can then calculate the conditional moments as, µ t 1 = µ + βy 0 t 1 and Λ t 1 = ( µ + βy 0 t 1 )( µ + βy 0 t 1 ) + E t 1 ( ε t ε t ) ν t 1 = ν + γy 0 t 1 and Q t 1 = ( µ + βy 0 t 1 )( ν + γy 0 t 1 ) + E t 1 ( ε t η t ) The maximum generalized Sharpe ratios generated by the base assets and the factor mimicking portfolios, respectively, are then calculated using (15) and (17). 18

19 5.1 Results As base assets, we use monthly returns on the twelve industry portfolios of Kenneth French 9, for the period beginning in 1959:01 and ending in 2001:07. The conditioning variables used are dividend yield (DY) and the consumption-wealth ratio 10 (CAY) as considered in Lettau and Ludvigson (2001a), for the same sampling period. For a given set of factors, we estimate a joint regression of the form (19). As a benchmark, we use Proposition 4.3 to compute the maximum generalized Sharpe ratio generated by the base assets. The results are reported in Table 1. For monthly returns on the twelve industry portfolios, we obtain an annualized Sharpe ratio of 1.16 and 1.09, using CAY and DY as conditioning variables, respectively. We also report the corresponding Sharpe ratios using quarterly data. As observed in Ferson and Siegel (2002a), the (annualized) quarterly Sharpe ratios are considerably higher (1.28) than those obtained from monthly data. In order for any of the factor models considered below to be viable, the Sharpe ratio generated by the factor mimicking portfolios must equal this benchmark. Note that in Table 1, we also report the fixed-weight Sharpe ratios, reflecting the case where conditioning information is not used. The difference between the generalized and the fixed-weight Sharpe ratios provides a diagnostic tool to assess the extent to which conditioning information expands the efficient frontier 11. While the use of DY increases annual Sharpe ratios by about 40%, CAY at monthly frequency has a much stronger effect, raising the annual Sharpe ratio by about 48%. In other words, when CAY is used 9 Available at 10 Available at mlettau. 11 In Abhyankar, Basu, and Stremme (2002) the difference between the optimal and the fixed-weight Sharpe ratio is used to construct a test that measures the economic value of return predictability. 19

20 as conditioning variable, a conditional asset pricing model has to work harder to price the assets correctly. a) Fama-French Model: The Fama-French model augments the classic CAPM with two zero-cost portfolios, based on firm size (factor SMB, small minus big ), and book-to-market ratio (factor HML, high minus low ). We estimate (19), using monthly excess returns on the base assets and the three factors as dependent variables, and lagged CAY or DY as explanatory variables, and construct the corresponding factor mimicking portfolios. Following the results of Section 3, for the Fama- French model to be a viable conditional asset pricing model, the maximum Sharpe ratio generated by the factors must equal that of the base assets. The corresponding results are reported in Table 1. We find that the maximum (annualized) Sharpe ratios generated by the factors are 0.85 and 0.83, respectively, using CAY and DY as conditioning variables. These account for only 73% respectively 76% of the maximum Sharpe ratios generated by the base assets. Our results broadly confirm those of Ferson and Harvey (1999) and Ferson and Siegel (2002b), who find that the Fama-French model does not work well conditionally. Panel A of Figures 1 and 2 plots the frontiers spanned by the base assets (bold-faced line) and the factor mimicking portfolios (light-weighted line). As a benchmark, we also plot the fixed-weight frontier (dashed line) spanned by the base assets. The factor mimicking portfolios themselves are shown as dots in the graph. As it displays the poorest performance of the models considered in this study, we chose this model to construct the optimal factor structure for the given choice of factors. The optimal factor loadings, as functions of the conditioning variable CAY, are shown in Figure 3. 20

21 b) Fama-French Model with Skewness Factor: Harvey and Siddique (2000) propose augmenting the the Fama-French model with a factor that captures the skewness risk of returns. They construct this additional factor by orthogonalizing the squared excess return on the market portfolio with respect to the market portfolio. In contrast, we consider a model in which the skewness factor is given directly by the square of the excess market return. This is because in our setting, factor correlations are accounted for in the construction of the conditional factor loadings, so that no orthogonalization is necessary. The model considered here is also closely related to Kraus and Litzenberger (1976). The Sharpe ratios generated by the augmented model are 0.93 and 0.90, respectively (see also Table 1), which account for 81% and 82% of the respective maximum Sharpe ratios generated by the base assets. The corresponding efficient frontiers are plotted in Panel B of Figures 1 and 2. Note that the factor mimicking portfolio for the skewness factor is very close to the efficient frontier, which may account for the superior performance of this model. c) The Consumption CAPM For the Consumption CAPM, the single factor is the growth rate in aggregate consumption. Since consumption is measured only quarterly, we use quarterly data for both the base asset returns and the conditioning variable CAY. Note first that the effect of CAY on the efficient frontier is much weaker at quarterly frequency, increasing the annual Sharpe ratio by only 26% compared with the fixed-weight case. We then construct the mimicking portfolio for this factor, and compute the implied maximum Sharpe ratio. The results, reported in Panel B of Table 1, indicate that the Consumption CAPM, augmented by the optimal use of the lagged CAY, seems to work relatively well as a conditional factor model (the factor accounts 21

22 for about 89% of the maximum asset Sharpe ratio). Our results are thus in broad agreement with those of Lettau and Ludvigson (2001b), but our methodology differs from theirs in that we use a traded portfolio (the factor mimicking portfolio). Figure 4 plots the corresponding efficient frontiers in this case. In fact, the factor mimicking portfolio itself (represented by the dot in the figure) appears to be close to being unconditionally mean-variance efficient. 6 Conclusion This paper develops portfolio based tests of conditional factor models both with and without conditioning information. These tests are developed by exploiting the close links between the stochastic discount factor (SDF) framework developed in Hansen and Jagannathan (1991) and mean-variance efficiency. We show that if m = a+b F is a true SDF, then a combination of the factor mimicking portfolios of m is in fact unconditionally mean-variance efficient. This provides a necessary condition for a factor model a + b F to be an SDF. Conversely given a set of factors we construct a managed portfolio of the factor-mimicking portfolios which a attains the maximum Sharpe ratio generated by the factors. If this Sharpe ratio is equal to the maximum Sharpe ratio of the assets, then this portfolio can be lifted to a conditional asset pricing model. We use these results to test various conditional factor models and to construct the optimal managed combination of the factors. We find that the Consumption Capital Asset Pricing Model scaled by consumption-wealth ratio as well as the model of Harvey and Siddique (2000) perform well as conditional asset pricing models 22

23 A Mathematical Appendix A.1 Proof of Proposition 3.3: [TO BE DONE] A.2 Proof of Proposition 4.3: We use Lemma 3.3 in Hansen and Richard (1987), applied to the trivial σ-field. It implies that the unconditionally efficient generalized return with mean m can be written as rt + w(m) zt, with w(m) = m E( r t ). (20) E( zt ) Here, r t R t is the unique generalized return with minimal second moment, and z t is the Riesz representation of the expectation functional on the space Z t of excess returns, see (3.8) in Hansen and Richard (1987). From (20) it follows by orthogonality that the variance of the efficient return with mean m is given by, σ 2 ( m ) = E( r t ) E( r t ) r f ( m r f ) 2. Hence, the frontier even in the case of generalized returns has the familiar wedge shape, and the maximum (squared) unconditional Sharpe ratio is given by, λ 2 = r f E( rt ). (21) E( rt ) From Lemma A.1 below we obtain the weights for the generalized return r t. A straightforward calculation then yields E( r t ) = r f / ( 1 + h 2 ). This, together with (21) gives the desired result. 23

24 Lemma A.1 The generalized return r t with minimum unconditional second moment is, r t = θ 0 t 1r f + ( Rt r f e ) θt 1 with θ t 1 = r f 1 + h 2 Σ 1 t 1µ t 1 and θt 1 0 = 1 + H2 t h 2 Proof: Throughout the proof, we will omit the time subscript. To characterize the weights of the minimum second moment return, we use calculus of variation. Let θ 0 (ε) and θ(ε) be differentiable one-parameter families of admissible weights, and define r(ε) = θ 0 (ε)r f + ( R r f e ) θ(ε). We assume E( θ 0 (ε) ) = 1 for all ε, so that the r(ε) are indeed generalized returns for all ε. Suppose that the minimum second moment is attained at ε = 0, then we must have, d dε E ( r(ε) ) 2 = 0 (22) ε=0 We will use this condition to determine the solutions θ 0 and θ separately. Part (i): Characterizing θ 0 Let φ 0 be an arbitrary weight function, and set a = 1/E( φ 0 ). Define θ 0 (ε) = (1 ε)θ 0 + εaφ 0, θ(ε) θ By construction, E( θ 0 (ε) ) = 1 for all ε, so that this specification indeed generates a family of generalized returns. In this case, the first-order condition (22) becomes, 0 = 2r 2 fe ( θ 0 (aφ 0 θ 0 ) ) + 2r f E ( (aφ 0 θ 0 )µ θ ) 24

25 Multiplying this equation by E( φ 0 ) = 1/a, we obtain, E ( [ r f θ 0 + µ θ ]φ ) 0 = γe ( φ ) 0 for some unconditional constant γ. Since this equation must hold for all φ 0, we obtain r f θ 0 + µ θ = γ (23) Finally, since E( θ 0 ) = 1, taking unconditional expectations implies, γ = r f + E ( µ θ ) (24) Part (ii): Characterizing θ Let φ be an arbitrary vector of weights for the risky assets, and set θ 0 (ε) θ 0, θ(ε) = θ + εφ Since E( θ 0 (ε) ) = E( θ 0 ) = 1, this specification generates a family of generalized returns. Hence, we can apply the first-order condition (22) to obtain, 0 = 2r f E ( θ 0 µ φ ) + 2E ( θ Λφ ) Since this equation must hold for all φ, we obtain, r f θ 0 µ = Λθ = [ Σ + µµ ] θ Substituting from (23) into the left-hand side of this equation, we can write, θ = γσ 1 µ, Multiplying this by µ and taking unconditional expectations, we obtain, E ( µ θ ) = γe ( µ Σ 1 µ ) = γe( H 2 ) =: γh 2 (25) 25

26 Finally, equating (24) and (25), γh 2 = E ( µ θ ) = γ r f, which implies γ = r f 1 + h 2 Hence, the weights for the risky assets are given by, r f θ = γσ 1 µ = 1 + h 2 Σ 1 µ Finally, using (23), we obtain the weight for the risk-free asset, r f θ 0 = γ µ θ = γ + r fh 2 This completes the proof of Lemma A h 2, which implies θ0 = 1 + H2 1 + h 2 A.3 Proof of Proposition 4.5: [TO BE DONE] A.4 Proof of Theorem 4.1 We first show the only if part. Suppose that there exist conditional coefficients a t 1 and b i t 1 such that (7) indeed is a viable SDF for the model. [TO BE DONE] 26

27 References Abhyankar, A., D. Basu, and A. Stremme (2002): The Economic Value of Asset Return Predictability: A Sharpe Ratio Based Test, working paper, Warwick Business School. Bekaert, G., and J. Liu (2001): Conditioning Information and Variance Bounds on Pricing Kernels, working paper, Columbia Business School. Breeden, D., M. Gibbons, and R. Litzenberger (1989): Empirical Test of the Consumption-Oriented CAPM, Journal of Finance, 44(2), Campbell, J. (1987): Stock Returns and the Term Structure, Journal of Financial Economics, 18, Carhart, M. (1997): On Persistence in Mutual Fund Performance, Journal of Finance, 52(1), Cochrane, J. (1999): Portfolio Advice for a Multi-Factor World, Economic Perspectives, Federal Reserve Bank of Chicago, 23(3), Cochrane, J. (2001): Asset Pricing. Princeton University Press, Princeton, New Jersey. Dybvig, P., and S. Ross (1985): Differential Information and Performance Measurement using a Security Market Line, Journal of Finance, 60, Fama, E., and K. French (1988): Dividend Yields and Expected Stock Returns, Journal of Financial Economics, 22,

28 Fama, E., and K. French (1992): The Cross-Section of Expected Returns, Journal of Finance, 47, Ferson, W., and C. Harvey (1999): Conditioning Variables and the Cross-Section of Stock Returns, Journal of Finance, 54, Ferson, W., and A. Siegel (2001): The Efficient Use of Conditioning Information in Portfolios, Journal of Finance, 56(3), Ferson, W., and A. Siegel (2002a): Stochastic Discount Factor Bounds with Conditioning Information, forthcoming, Review of Financial Studies. Ferson, W., and A. Siegel (2002b): Testing Portfolio Efficiency with Conditioning Information, working paper, University of Washington. Gallant, R., L. Hansen, and G. Tauchen (1990): Using Conditional Moments of Asset Payoffs to Infer the Volatility of Intertemporal Marginal Rates of Substitution, Journal of Econometrics, 45(1), Hansen, L., and R. Jagannathan (1991): Implications of Security Markets Data for Models of Dynamic Economies, Journal of Political Economy, 99(2), Hansen, L., and S. Richard (1987): The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models, Econometrica, 55(3), Harvey, C., and A. Siddique (2000): Conditional Skewness in Asset Pricing Tests, Journal of Finance, 55(3),

29 Huberman, G., S. Kandel, and R. Stambaugh (1987): Mimicking Portfolios and Exact Arbitrage Pricing, Journal of Finance, 42(1), 1 9. Jagannathan, R. (1996): Relation between the Slopes of the Conditional and Unconditional Mean-Standard Deviation Frontier of Asset Returns, in Modern Portfolio Theory and its Applications: Inquiries into Asset Valuation Problems, ed. by S. Saito et al. Center for Academic Societies, Osaka, Japan. Kraus, A., and R. Litzenberger (1976): Skewness Preference and the Valuation of Risky Assets, Journal of Finance, 31, Lettau, M., and S. Ludvigson (2001a): Consumption, Aggregate Wealth and Expected Stock Returns, Journal of Finance, 56(3), Lettau, M., and S. Ludvigson (2001b): Resurrecting the (C)CAPM: A Cross-Sectional Test when Risk Premia are Time-Varying, Journal of Political Economy, 109(6),

30 Panel A: Fama-French Model Factor Frontier Asset Frontier Fixed-Weight Expected Return Standard Deviation Panel B: Fama-French Model with Skewness Factor Factor Frontier Asset Frontier Fixed-Weight Expected Return Standard Deviation Figure 1: Using CAY as Predictor Variable These figures show the estimated efficient frontier generated by the base assets, both in the fixed-weight case (dashed line) and with conditioning information (bold-faced line), and the frontier generated by the factor mimicking portfolios (light-weighted line), for the three-factor Fama-French model (Panel A) and the Fama-French model augmented by a skewness factor (Panel B). The predictor variable used is CAY. The factor mimicking portfolios themselves are shown as dots. 30

31 Panel A: Fama-French Model Factor Frontier Asset Frontier Fixed-Weight Expected Return Standard Deviation Panel B: Fama-French Model with Skewness Factor Factor Frontier Asset Frontier Fixed-Weight Expected Return Standard Deviation Figure 2: Using DY as Predictor Variable These figures show the estimated efficient frontier generated by the base assets, both in the fixed-weight case (dashed line) and with conditioning information (bold-faced line), and the frontier generated by the factor mimicking portfolios (light-weighted line), for the three-factor Fama-French model (Panel A) and the Fama-French model augmented by a skewness factor (Panel B). The predictor variable used is DY. The factor mimicking portfolios themselves are shown as dots. 31

32 Factor Weight 0-10 Mkt-Rf -20 SMB HML Rf Realization of CAY Figure 3: Optimal Factor Model This graph shows the optimal weights, as functions of the predictor variable CAY, on the factor mimicking portfolios and the risk-free asset, for the three-factor Fama-French model. 32

33 Predictor (annualized) Factors Variable Sharpe ratios (fw) assets fmp s Panel A: Monthly Returns RMF HML SMB CAY RMF HML SMB SKEW CAY RMF HML SMB DY RMF HML SMB SKEW DY Panel B: Quarterly Returns CONSGR CAY Table 1: Test Results This table reports the maximum (annualized) Sharpe ratios generated by the base assets and the factor mimicking portfolios, respectively. The factor models tested are the three factor (RMF, HML and SMB) Fama-French model, the Fama-French model augmented by the skewness factor (SKEW), and the consumption growth (CONSGR) model. The predictor variables used are CAY and DY as indicated. 33

34 0.030 Factor Frontier Asset Frontier Fixed-Weight Expected Return Standard Deviation Figure 4: Consumption CAPM This figures show the estimated efficient frontier generated by the base assets, both in the fixed-weight case (dashed line) and with conditioning information (bold-faced line), and the frontier generated by the consumption growth factor (light-weighted line). The predictor variable used is CAY. The factor mimicking portfolio is shown as a dot. 34