Evaluating asset pricing models in a Fama-French framework

Transcription

1 Evaluating asset pricing models in a Fama-French framework Carlos Enrique Carrasco Gutierrez Wagner Piazza Gaglianone y Preliminary version: March This version: April Abstract In this work we propose a new methodology to compare di erent stochastic discount factor (SDF) proxies based on relevant market information. The starting point is the work of Fama and French, which evidenced that the asset returns of the U.S. economy could be explained by relative factors linked to characteristics of the rms. In this sense, we construct a Monte Carlo simulation to generate a set of returns perfectly compatible with the Fama and French factors and, then, investigate the performance of di erent SDF proxies. We use some goodnessof- t statistics and the Hansen Jagannathan distance as formal criteria to compare asset-pricing models. An empirical application of our setup is also provided, revealing that factor models with a principal-component technique exhibit (by far) the best performance among several traditional SDF proxies. Keywords: Asset Pricing, Fama and French model, Stochastic Discount Factor, Hansen- Jagannathan distance. JEL Codes: G12, C15, C22. Corresponding author. Graduate School of Economics, Getulio Vargas Foundation, Praia de Botafogo 190, s.1104, Rio de Janeiro, RJ , Brazil ( cgutierrez@fgvmail.br). y Research Department, Central Bank of Brazil ( wagner.gaglianone@bcb.gov.br). The usual disclaimer applies. 1

2 1 Introduction In this work, we propose a new methodology to compare di erent stochastic discount factor or pricing kernel proxies. 1 In asset pricing theory, one of the major interests for empirical researchers is oriented by testing whether a particular asset pricing model is (indeed) supported by the data. In addition, a formal procedure to compare the performance of competing asset pricing models is also of great importance in empirical applications. In both cases, it is of utmost relevance to establish an objective measure of model misspeci cation, to evaluate how wrong a given model might be, or to formally compare a set of competing models. The most useful measure is the well-known Hansen and Jagannathan(1997) distance (so-called HJ-distance), which has been used both as a model diagnostic tool and also as formal criteria to compare asset pricing models. This type of comparison has been employed in many recent papers. 2 As argued by Hansen and Richard (1987), observable implications of candidate models of asset markets are conveniently summarized in terms of their implied stochastic discount factors. a result, some recent studies of the asset pricing literature have been focused on proposing an estimator for the SDF and also on comparing competing pricing models in terms of the SDF model. For instance, see Lettau and Ludvigson (2001b), Chen and Ludvigson (2008), Araujo, Issler and Fernandes (2006). A di erent route to investigate and compare asset pricing models has also been suggested in the literature. The main idea is to assume a data generation process (DGP) for a set of asset returns, based on some assumptions about the asset prices and, then, create a controlled framework, which is used to evaluate and compare the asset pricing models. In this sense, Fernandes and Vieira (2006) study, through Monte Carlo simulations, the performance of di erent SDF estimatives at di erent environments. Firstly, the authors consider that all asset prices follow a geometric Brownian motion. In this case, one should expect that a SDF proxy based on a geometric Brownian motion assumption would have a better performance, in comparison to an asset pricing model that does not assume this hypothesis. In a second setting, they generate asset prices that vary according to a stationary 1 We use the term "stochastic discount factor" as a label for a state-contingent discount factor. 2 For instance, by using the HJ-distance, Campbell and Cochrane (2000) explain why CAPM and its extensions better approximate asset pricing models than the standard consumption based model; Jagannathan and Wang (2002) compare the SDF method with Beta method in estimating risk premium; Dittmar (2002) uses the HJ-distance to estimate the nonlinear pricing kernels in which the risk factor is endogenously determined and preferences restrict the de nition of the pricing kernel. Other examples in the literature include Jagannathan, Kubota and Takehara (1998), Farnsworth, Ferson, Jackson, and Todd (2002), Lettau and Ludvigson (2001a) and Chen and Ludvigson (2008). As 2

3 Ornstein-Uhlenbeck process as done in Vasicek (1977). However, a critical issue of this procedure is that the best asset pricing model inside this particular environments (i.e., when the asset prices are supposed to follow a geometric Brownian motion or a stationary Ornstein-Uhlenbeck process), might not be a good model in the real world. In other words, the best estimator of each controlled framework might not necessarily exhibit the same performance for observed stock market prices of a real economy. In this paper, we use the controlled approach of Fernandes and Vieira (2006), but instead of generating the asset returns from an ad-hoc assumption about the DGP of returns, we use related market information from the real economy. Our starting point is the work of Fama and French, which evidenced that asset returns of the U.S. economy could be explained by relative factors linked to characteristics of the rms. 3 Based on the Fama and French factors, we rstly construct a Monte Carlo simulation to generate a set of returns that is perfectly compatible with these factors. The next step is to compare the performance of di erent SDF proxies documented in the literature, and then investigate their relative performances. Although we do not directly use market returns data, we are able to compare di erent SDFs by using important market information provided by those factors. Furthermore, because our approach enables us to construct a data generation process of the SDF provided by the Fama and French speci cation, it is possible to compare competing proxies through some goodness-of- t statistics. In addition, it is relevant to test if a set of SDF candidates satisfy the law of one price, such that 1 = E t (m t+1 R i;t+1 ), where m t+1 is referred to the investigated stochastic discount factor. Thus, we say that a SDF "prices" the assets if this equation is in fact satis ed. In this sense, we test the previous restriction by evaluating the Hansen Jagannathan distance of each SDF candidate model. Even though the considered SDF models are misspeci ed, in practical terms we are interested in those models with the lowest HJ distance. Hansen and Jagannathan show that the HJ-distance is the pricing error for the portfolio that is most misspriced by the model. Therefore, we propose a new methodology to compare di erent stochastic discount factors by using the market information in a Fama and French (1992, 1993) environment. The main objective here is not to propose a DGP process of actual market returns, but to provide a controlled environment that allows us to properly compare and evaluate di erent SDF proxies. This work follows the idea of Farnsworth et al. (2002), which study di erent SDFs by constructing arti cial mutual funds using real stock returns from the CRSP data. 3 Fama and French (1993, 1995) argue that a three-factor model is successful because it proxies for unobserved common risk in portfolio returns. 3

4 To illustrate our methodology, we present an empirical application, in which several SDF models are compared: a) Araujo, Issler and Fernandes (2006); b) Brandt, Cochrane and Saint-Clara (2006), c) Hansen and Jagannathan (1991); d) the unconditional CAPM; e) Principal Component Factors (PCF); and f) the Fama and French proxy itself. An important feature of our results (summarized in tables 4, 5 and 6) is that Principal Component Factor models exhibit (by far) the best performance in comparison to some traditional SDF proxies. This work is organized as follows: Section 2 presents the Fama and French model and describes the Monte Carlo simulation strategy; Section 3 presents the results of the empirical application; and Section 4 shows the main conclusions. 2 The stochastic discount factor and the Fama and French model A general framework to asset pricing is well describe in Harrison and Kreps (1979), Hansen and Richard (1987) and Hansen and Jagannathan (1991), associated to the stochastic discount factor (SDF), which relies on the pricing equation: p t = E t (m t+1 x i;t+1 ) ; (1) where E t () denotes the conditional expectation given the information available at time t, p t is the asset price, m t+1 the stochastic discount factor, x i;t+1 the asset payo of the i-th asset in t+1. This pricing equation means that the market value today of an uncertain payo tomorrow is represented by the payo multiplied by the discount factor, also taking into account di erent states of nature by using the underlying probabilities. The stochastic discount factor model provides a general framework for pricing assets. As documented by Cochrane (2001), asset pricing can be summarized by two equations: p t = E t [m t+1 x t+1 ] ; (2) m t+1 = f (data, parameters) : (3) This way, one can conveniently separate the task of specifying economic assumptions of the model given by equation (3), from the task of deciding which kind of empirical representation to pursue or understand. Based on a determinate model representing by the function f (), we next show how the pricing equation (2) can lead to di erent predictions stated in terms of returns. For instance, in the Consumption-based Capital Asset Pricing Model (CCAPM) context, the SDF corresponds to the intertemporal marginal rate of substitution. Hence, the pricing equation 4

5 (2) mainly illustrates the fact that consumers equate marginal rates of substitution to prices, i.e. m t+1 = u0 (c t+1 ) u 0 (c, where is the discount factor for the future, c t) t is consumption and u () is a given utility function. The speci cation of m t+1 comes from the rst-order conditions h of the i consumption-based model, summarized by the well-known Euler equation: p t = E t u0 (c t+1 ) u 0 (c x t) t Fama and French framework Fama and French (1992) show that, besides the market risk, there are other important factors that help explain the average return in the stock market. This evidence has been demonstrated in several works for di erent stock markets (see Gaunt (2004) and Gri n (2005) for a good review). Although there is not a clear link between these factors and the economic theory (e.g., CAPM model), these evidences show that some additional factors might (quite well) help to understand the dynamics of the average return. These factors are known as the size and the book-to-market equity and represent special features about rms. Fama and French (1992) argue that size and book-to-market equity are indeed related to economic fundamentals. Although they appear to be ad hoc variables in an average stock returns regression, these authors justify them as expected and natural proxies for common risk factors in stock returns. The factors (i) The SMB (Small Minus Big) factor is constructed to measure the size premium. In fact, it is designed to track the additional return that investors have historically received by investing in stocks of companies with relatively small market capitalization. A positive SMB in a given month indicates that small cap stocks have outperformed the large cap stocks in that month. On the other hand, a negative SMB suggests that large caps have outperformed. (ii) The HML (High Minus Low) factor is constructed to measure the premium-value provided to investors for investing in companies with high book-to-market values (essentially, the value placed on the company by accountants as a ratio relative to its market value, commonly expressed as B/M). A positive HML in a given month suggests that value stocks (i.e., high B/M) have outperformed the growth stocks (low B/M) in that month, whereas a negative HML indicates that growth stocks have outperformed. 4 4 Notice that, in respect to SMB, small companies logically are expected to be more sensitive to many risk factors, as a result of their relatively undiversi ed nature, and also their reduced ability to absorb negative 5

6 (iii) The Market factor is the market excess return in comparison to the risk-free rate. We proxy the excess return on the market (R m R f ), by the value-weighted portfolio of all stocks listed on the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX) and NASDAQ stocks (from CRSP) minus the one-month Treasury Bill rate. The Model Fama and French (1993, 1996) propose a three-factor model for expected returns (see also Fama and French (2004) for a good survey). E(R it ) R ft = im [E(R Mt ) R ft ] + is E(SMB t ) + ih E(HML t ); (4) where the betas im, is and ih are slopes in the multiple regression (4). Hence, one implication of the expected return equation of the three-factor model is that the intercept in the time-series regression (5), is zero for all assets i: R it R ft = im (R Mt R ft ) + is SMB t + ih HML t + " it : (5) Using this criterion, Fama and French (1993, 1996) nd that the model captures much of the variation in average return for portfolios formed on size, book-to-market equity and other price ratios, which exactly represent some of the problems of the CAPM model. Expected return - beta representation The Fama and French approach is (in fact) a multifactor model, that can be seen as an expectedbeta 5 representation of linear factor pricing models of the form: E(R i ) = + im m + is s + ih h + i i 2 f1; :::; Ng : (6) If we run this cross sectional regression of average returns on betas, one can estimate the parameters (, m, s, h ). Notice that is the intercept and m, s and h the slope in this cross-sectional relation. In addition, the im, is and ih are the unconditional sensitivities of the i-th asset to the factors 6. Moreover, ij, for some j 2 fm; s; hg, can be interpreted as the amount nancial events. On the other hand, the HML factor suggests higher risk exposure for typical value stocks in comparison to growth stocks. 5 The main objective of the beta model is to explain the variation in terms of average returns across assets. 6 An unconditional time-series approach is used here. The conditional approaches to test for international pricing models include those by Ferson & Harvey (1994, 1999) and Chan Karolyi & Stulz (1992). 6

7 of risk exposure of asset i to factor j, and j as the price of such risk exposure. Hence, the betas are de ned as the coe cients in a multiple regression of returns on factors: R it R ft = im RMt ex + is SMB t + ih HML t + " it t 2 f1; :::; T g ; (7) where RMt ex = (R Mt R ft ). Following the equivalence between this beta-pricing model and the linear model for the discount factor M, in an unconditional setting (see Cochrane, 2001) we can estimate M as: M = a + b 0 f; (8) where f = [RM ex; SMB; HML]0 and the relations between e, and a and b, are given by: a = 1 and b = E ff 0 1 : (9) Constructing the Fama and French environment Based on the assumption that R Mt, SMB t and HML t are known variables, we can reproduce a Fama and French environment following the three factors Fama and French model: R i;t R ft = im (R Mt R ft ) + is SMB t + ih HML t + " it ; (10) To do so in practical terms, we propose the following steps of a Monte Carlo simulation: 1. Firstly, calibrate each parameter k ij, for j 2 fm; s; hg and i 2 f1; ::::Ng according to previous estimations in Fama and French (1992,1993). Therefore, we will generate for each j a N-dimensional vector of asset returns. 2. By considering k ij created in step 1 for some i 2 f1; ::::Ng and using the known factors R Mt, SMB t and HML t, we generate a returns vector along the time dimension, through equation (7). The iid shock " it is assumed to be a white noise with zero mean and constant variance. 3. Repeating step 2 for each i 2 f1; ::::Ng, we create the matrix R k of asset returns, in which rows are formed by di erent returns and columns represent the time dimension. 4. Evaluate the mean of R k across each row to generate a cross-section vector. Now, it is possible to estimate the parameters k and k through the equation (6). 5. Estimate parameters a k and b k from the equivalence relation shown in equation (9). Finally, the stochastic discount factor can be estimated by using equation (8). 6. Repeat steps 1 to 5 for an amount of K replications in order to construct the Monte Carlo simulation. 7

8 2.2 Evaluating the performance of competing models In the asset pricing literature, some measures are suggested to compare competing asset pricing models. The most famous measure is the Hansen and Jagannathan distance. However, as long as the data generation process (DGP) is known in each speci cation of the Fama and French model, it is also possible to use some simple sample statistics. This route is adopted by Fernandes and Vieira (2006) to compare the relative performance among SDF proxies, based on goodness-of- t statistics to assess whether the estimators correctly proxy the DGP. In addition, we use the Hansen and Jagannathan distance to test how wrong a model is and to compare the performance of di erent asset pricing models. The Hansen-Jagannathan (1997) distance measure is a summary of the mean pricing errors across a group of assets. The HJ measure may also be interpreted as the distance between the SDF candidate and one that would correctly price the primitive assets. The pricing error can be written by t = E t (m t+1 R i;t+1 ) 1. Notice, in particular, that t depends on the considered SDF, and the SDF is not unique (unless markets are complete). Thus, di erent SDF proxies can produce di erent HJ measures. Goodness-of- t statistics We use two goodness-of- t statistics. The rst suggested statistic is merely a standardized version of the mean squared error of the SDF proxies. The other one just compares the sample correlation between the actual and estimated stochastic discount factors. Let M t be the DGP stochastic discount factor generated by the Fama and French speci cation, and M c t s the SDF proxy provided by the model s in a family S of asset pricing models. The standardized mean squared error of the estimators is computed as: P 2 T cm s i=1 t M t \MSE s = P T i=1 M 2 t ; for s 2 S; (11) and the sample correlation between the actual and estimated SDF is given by: b s = corr( c M s t ; M t ); for s 2 S: (12) 8

9 3 Empirical Application In this section, we present an empirical exercise of our proposed framework to compare alternative models of the stochastic discount factor, commonly discussed in the literature. A. Hansen and Jagannathan (Primitive-E cient Stochastic Discount Factors) Hansen and Jagannathan (1991) proposed an identi cation of M t+1 by the projection of M t+1 onto the space of payo s, which makes it straightforward to express Mt+1, the mimicking portfolio, only as a function of observables. They describe a general framework to asset pricing, associated to the stochastic discount factor, which relies on the pricing equation: 1 = E t [M t+1 R t+1 ] : (13) By considering the stochastic discount factor strictly positive and some additional hypotheses in the previous pricing equation (13), these authors show that the mimicked discount factor M t+1 has a direct relation to the minimal conditional variance portfolio. Moreover, they exploit the fact that it is always possible to project the SDF onto the space of payo s, which makes it straightforward to express the mimicking portfolio as a function of only observable values: M t+1 = { 0 N Et R t+1 Rt Rt+1 ; (14) where { N is a N 1 vector of ones, and R t+1 is a N 1 vector stacking all asset returns. Equation (14) delivers a nonparametric estimate of the SDF that is solely a function of asset returns. However, one disadvantage of this procedure consists in evaluating the conditional moment E t R t+1 R 0 t+1. Notice that, as long as the number of assets increases, it becomes more di cult to estimate M t+1 through (14). In the limit case, i.e. N! 1, the matrix E t R t+1 R 0 t+1 will be of in nite order. Even for a nite but large number of assets, there will be possible singularities in that matrix, since the correlation between some assets may be very close to unity. Therefore, equation (14) re ects the fact that M t+1 is a linear function of a conditional minimum-variance e cient portfolio. We also refer to this estimative as a Primitive-E cient SDF. B. Brandt, Cochrane and Santa-Clara (2006) Brandt, Cochrane and Santa-Clara (2006) consider that the asset prices follow a geometric Brownian motion (GBM). Such hypothesis is de ned by the following partial di erential equation: dp P = R f + dt db; (15) 9

10 where, dp P = dp1 0, P 1 +; :::; dp N P N = (1 ; :::; n ) 0, is a N N positive de nite matrix, P i is the price of the asset i, the risk premium vector, R f the risk free rate, and B a standard GBM of dimension N. Using Itô theorem, is possible to show that: R i t+t = P i t+t P i t = e (Rf + i 1 2 i;i)t+ p t i Z t; (16) where, Z t is a vector of N independent variables with Gaussian distribution. Therefore, the SDF proposed by these authors is calculated as M t+t = e (Rf )t p t 2 1 0Zt : (17) Thus, Brandt, Cochrane and Santa-Clara (2006) suggest the following SDF estimator: cm t = e (Rf b0 b 1 b)t b 0 b 1 R t R ; (18) where, b; R and b are estimated by: such that, R t = R 1 t ; :::; R N t b = 1 1 t T R b = R f ; (19) t TX R t R R t R ; (20) t=1 0 and R = P T t=1 R t. C. Araujo, Issler and Fernandes (2006) A novel estimator for the stochastic discount factor (within a panel data context) is proposed by Araujo, Issler and Fernandes (2006). This setting is slightly more general than the GBM setup put forth by Brandt, Cochrane and Santa-Clara (2006). In fact, this estimator assumes that, for every asset i 2 f1; :::; Ng, M t+1 Rt+1 i is conditionally homoskedastic and has a lognormal distribution. In addition, under asset pricing equation (13) and some mild additional conditions, they show that a consistent estimator for M t is given by: cm t = R G t 1 T P T t=1 R A t R G t! ; (21) where R A t = N i=1 R 1 N i;t and R G t = 1 N P N t=1 R i;t are respectively the cross-sectional arithmetic and geometric average of all gross returns. Therefore, this nonparametric estimator depends exclusively on appropriate averages of asset returns that can easily be implemented. 10

11 D. Capital Asset Pricing Model - CAPM Assuming the unconditional CAPM, the SDF is a linear function of market returns calculated as: m t+1 = a + br w;t+1 ; where R w;t+1 is the gross return on the market portfolio of all assets. In order to implement the static CAPM, for practical purposes, it is commonly assumed that the return on the value-weighted portfolio of all stocks listed on NYSE, AMEX, and NASDAQ is a reasonable proxy for the return on the market portfolio of all assets of the U.S. economy. E. Principal Component Factors (PCF) Model The multi-factor model The CAPM and the three factor model of Fama and French are linear factor models because their SDF are linear functions of factors. We can generalize the factor models through the equation: KX R i;t = i + i;k f i;t + " i;t ; (22) k=1 where R i;t are the gross returns and f i;t ; i = 1; :::; K the factors that summarize the systematic variation of the stock market. Therefore, their expected-beta representation is of the form: KX E(R i ) = + i;k i;t + i i 2 f1; :::; Ng : (23) k=1 The factor loadings can be consistently estimated by simple OLS regressions of the form (23). Following the equivalence between this beta-pricing model and the linear model for the discount factor M t (in unconditional setting), we can estimate M t in the following way: KX ^M t = a + b k f k;t (24) k=1 where a and b, are given by: a = 1 and b = cov ff 0 1 : (25) In empirical analysis, a critical issue is to identify the number of factors K to be used. In this sense, we use the principal component analysis to identify the number of factors to be used in the multi-factor models. The idea is to employ a large dataset to construct the factors in this model, as next explained. 11

12 Principal Components Analysis (PCA) The CAPM explains the cross-section variability of data by the market return and also the SMB, HML factor of the Fama and French model, which are informative about size and book-tomarket of the rms. The main idea of using principal component analysis is to use few factors to resume most of the variance in the primitive dataset of asset returns. This way, the number of factors K to be used in the factor model will be the number of principal components. There are few conventions in order to select the numbers of components. In this work, such a number is based on the fact that the amount of variation accounted for by k eigenvectors can exactly be determined by summing up the eigenvalues of those k eigenvectors, and dividing it by the sum of all eigenvalues. 7 P k n=1 n ::: + N Cuto Fraction (26) Therefore, if one chooses an acceptable cuto value, it follows that an acceptable percentage of the total variation is accounted for. Thus, this procedure can form the basis of a stopping rule to identify the desired number of principal components. Moreover, we also take into account the frequency of time observations. For instance, if we have a monthly dataset (T = 200) and the number of assets is equal to N = 25, then, the PCA will produce an amount of 25 eigenvalues and 25 eigenvectors (on a monthly basis). In this case, the total number of computed eigenvalues is equal to = 5; 000. To nd monthly eigenvectors, we use the previous year s returns, and in each month the whole process is repeated revealing new results. The time-dependence nature allows us to compute equation (26) in each period t and, given an acceptable cuto value, we can nd the number of principal components that satisfy equation (26) for all periods t 2 f1; :::; T g. Therefore, the number of principal components is endogenously determined based on a cuto value. For instance, if one sets the Cuto Fraction =90% for all time periods, then, the PCA accounts for the 90% of the whole variability of the dataset. A multi-factor model with principal component analysis will be here denominated as a Principal Component Factors (PCF) model. 3.1 Monte Carlo design Our strategy in the numerical simulation is aimed to simulate the real econometricians behavior, when they compute a SDF model. In this sense, we construct a DGP from the Fama and French setup, by considering N = 80 as our set of primitive assets. 7 See appendix A for further details. 12

13 A rst restriction imposed in our setup consists in limiting the amount of information available to econometricians. We represent this situation by considering that only N = 70 assets (out of the total amount of N = 80) are in fact observed by econometricians. In addition, we assume that only N = 25 asset are indeed used by these econometricians. The main reason is the natural restriction in modeling a SDF proxy: For instance, if two assets have a linear correlation close to one, then, it leads to a restriction in using all information. In particular, consider the estimation of the inverse of the conditional moment E t R t+1 Rt+1 0 of the Hansen-Jagannathan SDF estimator. To account for the mentioned restriction, we consider a maximum number of N = 25 assets to estimate all SDF models studied in this work. The only exception is the SDF proxy generated from the PCF model, which is assumed to use all available information (i.e., N = 70). In order to estimate the stochastic discount factors, we rstly specify an adequate setup for the Fama and French environment. To do so, we study the SDF proxies under the speci cations, represented by equations (10). The rst step is to consider the factors (R Mt R ft ), SMB t and HML t of the U.S. economy, which are extracted from the Kenneth R. French website. 8 Next, we calibrate the parameters im ; is and ih according to previous estimations of Fama and French (1992,1993) and estimate the parameters (, m, s, h ) from the cross-sectional regression (6), observing their signi cance through the F -statistic or the t-statistic for individual parameters. We set N = 80 as our set of primitive assets and, then, N = 25 or N = 70 as the observed assets. We also consider three sample sizes T = f100; 200; 400g. Thus, we estimate the stochastic discount factors for the three-factor model of Fama and French, and repeat the mentioned procedure for an amount of K = 1; 000 replications. Some descriptive statistics of the generated SDFs are presented in appendix B. Finally, the evaluation of the SDF proxies is conducted and the Monte Carlo results are summarized by the two goodness-of- t statistics and the HJ distance, which are averaged across all replications. Moreover, the robustness of the results is analyzed through the di erent sample sizes. In each replication of the Monte Carlo simulation, the following SDF proxies are estimated: cm a t : Stochastic discount factor of Araujo, Issler and Fernandes (2006); cm b t : Stochastic discount factor of Brandt, Cochrane and Santa-Clara (2006); cm c t : Stochastic discount factor of Hansen and Jagannathan (1991); 8 More information about data can be found in: 13

14 cm d t : Stochastic discount factor of the CAPM; cm e t : Stochastic discount factor of the Principal Component Factors (PCF) model, in which the percentage of variability ranges from 80% to 95%; cm f t : Stochastic discount factor implied by the Fama and French setup Results In Figure 1, the estimates of the SDF proxies are shown (for illustrative purpose) for one replication of the Monte Carlo simulation, with a sample size T = 100. This gure also shows that the estimative of Brandt et al. (2006), M c t b, and of Araujo et al. (2006), M c t a, are respectively the most and less volatile. The PCF model with a cuto value of 85% (and the three principal components as factors, i.e., PC=3) exhibits a dynamic behavior similar to the Fama & French DGP. Figure 1 - Three factors, with a sample size T = SDF Fama and French (DGP) SDF Araujo, Issler and Fernandes SDF Brandt, Cochrane and S.C SDF Hansen and Jagannathan C APM Principal Component Factors (85%,PC = 3) Notes: a) Figure 1 shows one replication out of the total 1,000 replications. b) We adopt N=25 assets and T=100 observations. 9 The c M f t is estimated by using N=25 assets, whereas the DGP is generated from N=80 assets. 14

15 Percent of Total Variation Figure 2 shows the ratio P k n=1 n = ( ::: + N ) for the rst twenty eigenvectors (time series for k=1,2,...,20). This picture suggests that the rst fourteen eigenvectors account for a cuto fraction of 90%. Thus, by retaining only the top fourteen eigenvectors, it follows that the dimensionality is sharply reduced by 82.5% (while retaining 90% of the original information). Figure 2 - Fraction of total variation, with a sample size T = Fraction of Total Variation vs. Time Time (Months) Regarding the performance of the SDF proxies, tables 4, 5 and 6 (presented in appendix C) report the evaluation statistics provided by the Monte Carlo simulation. Note that the results presented in all tables are very similar, suggesting that results are robust to sample size. Firstly, note that the mean square error satis es the following inequalities (T = 100): \MSE f < \MSE e2 < \MSE e3 < \MSE e1 < \MSE a < \MSE e4 < \MSE d < \MSE c < \MSE b. 10 Notice that, as already expected, the Fama and French SDF proxy has the best performance, which is a natural result given that the DGP is entirely based on the Fama & French setup. In addition, note the relative superiority of the Principal Component Factors (PCF) model, in comparison to the other estimators. The Araujo et al. (2006) SDF proxy shows quite a good performance, followed by the CAPM model, whereas the Hansen and Jagannathan (1991) and Brandt, Cochrane and Santa-Clara (2006) seem to exhibit the worst performance. 10 For the sample size T = 200 the "ranking order" is the following: \MSE f < \MSE e1 < \MSE e2 < \MSE e3 < \MSE a < \MSE e4 < \MSE d < \MSE b < \MSE c. In a similar result, the ranking for T = 400 is the following: \MSE f < \MSE e2 < \MSE e1 < \MSE e3 < \MSE a < \MSE e4 < \MSE b < \MSE c < \MSE d. 15

16 In respect to the correlation of the true SDF with the considered SDF proxies, we have obtained the following ranking order (T = 100): M ce2 t = M c t e3 M cf t M c t e1 M c t a M c t e4 M c t b M c t c cm t d. 11 This implies that the Fama & French proxy and the PCF models (in general) best track the dynamic path of the true SDF. On the other hand, the other estimators show a correlation value always below 0.9, in which the Araujo et al. (2006) SDF proxy is the best among them, followed by the Brandt, Cochrane and Santa-Clara (2006), and the Hansen and Jagannathan (1991). The CAPM model exhibits the worst performance (with a negative correlation!) Finally, in respect to the HJ distance 12 (which for the corrected-speci ed model should be as close as possible to zero), notice that the PCF estimators are (again) the best ones in all sample sizes. In respect to the other SDF estimators, note that (in all samples) all of these proxies have a similar HJ distance, possibly due to nite sample distortions. Putting all together, the numerical results show that (in fact) the SDF proxy of Fama and French has the best performance (as already expected), followed by the Principal Component Factors model, which is showed to be a powerful tool to estimate SDFs. The Araujo et al. (2006) proxy has the best MSE and correlation responses among the other estimators, possibly due to its nonparametric nature and quite weak assumptions. The CAPM model shows a good MSE response, but a negative correlation with the true SDF, revealing its weakness in tracking the real dynamic of the true SDF. Moreover, the Hansen and Jagannathan (1991) and Brandt et al. (2006) SDFs do not have a good performance in the Fama and French setup. This result for the Hansen and Jagannathan proxy is possibly due to the fact that we are only using N = 25=70 = 35:71% of the available information. An interesting question to ask at this point is: How the performance of this model would be a ected by considering high values like N = 50 or N = 70 (i.e., all available information)? 13 Recall that the estimation of the inverse of the matrix E t R t+1 R 0 t+1 may exhibit near singularities for a nite but large number of assets. On the other hand, in the later proxy, the geometric Brownian motion hypothesis is possibly a strong assumption for actual asset returns. 11 For the sample size T = 200, it follows that: M cf t M c t e1 M c t e2 M c t a M c t e3 M c t e4 M c t b M c t c M c t d, and for T = 400 we have that: M cf t M c t e2 M c t e1 M c t e3 M c t a M c t e4 M c t b M c t c M c t d. 12 We compute the HJ distance based on the MatLab codes of Mike Cli, available at: 13 The Hansen and Jagannathan model is quite an interesting SDF estimator, since it naturally satis es the law of one price. Notice that, for N=80, one should expect the HJ distance of this estimator to be zero. 16

17 4 Conclusions In the present work, we propose a new methodology to compare di erent stochastic discount factor (SDF) proxies based on relevant market information. The starting point is the work of Fama and French, which evidenced that the asset returns of the U.S. economy could be explained by relative factors linked to characteristics of the rms. Thus, we construct a "Fama and French world", through a Monte Carlo simulation, to generate a set of returns that is perfectly compatible with those factors. Through numerical simulations, we produce returns time series according to the Fama and French environment, based on factors such as the market portfolio return, size and book-to-market equity of the U.S. economy. Then, we present an empirical application, in which several SDF proxies are compared: a) Araujo, Issler and Fernandes (2006); b) Brandt, Cochrane and Saint-Clara (2006); c) Hansen and Jagannathan (1991); d) CAPM; e) Principal Component Factors (PCF); and f) the Fama and French proxy itself. This controlled framework allows us to use simple sample statistics to compare the SDF candidate models with the true SDF implied by the Fama and French DGP. In addition, we use the Hansen and Jagannathan distance as a formal measure of model misspeci cation. The overall results indicate that the SDF proxy suggested by the Principal Component Factor model dominates some traditional SDF estimators. Furthermore, several important topics remain for future research. For instance, one might adopt a nite sample correction in the HJ distance, in order to properly test for the equality between the SDF proxies and the implied Fama and French SDF (e.g., Ren & Shimotsu (2006) or Kan & Robotti (2008)). The details of this route remain an issue to be further explored. References [1] Araujo, F., Issler, J. V., Fernandes, M., Estimating the stochastic discount factor without a utility function, Princeton University, Getulio Vargas Foundation, and Queen Mary, University of London. [2] Brandt, M. W., Cochrane, J. H., Santa-Clara, P., International risk sharing is better than you think, or exchange rates are too smooth, forthcoming in Journal of Monetary Economics. 17

18 [3] Campbell, J.Y., Cochrane, J.H Explaining the poor performance of consumption-based "asset pricing models". Journal of Finance 55, [4] Chan, K.C., Karolyi, G.A., Stulz, R.M., Global Financial Markets and the Risk Premium on U.S. Equity. NBER Working Paper n [5] Chen and Ludvigson, Land of Addicts? An Empirical Investigation of Habit-Based Asset Pricing Models. Working paper, New York University. [6] Cochrane, J. H., Asset Pricing, Princeton University Press, Princeton. [7] Dittmar, Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross Section of Equity Returns. Journal of Financial Economics 33, [8] Fama, E. F. and K.R. French, The Cross Section of Expected Stock Returns, Journal of Finance 47, [9] Fama, E. F. and K.R. French, Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33, [10] Fama, E. F. and K.R. French, Size and Book-to-Market Factors in Earnings and Returns, Journal of Finance 50, [11] Fama, E. F. and K.R. French, Multifactor Explanations of Asset Pricing Anomalies, Journal of Finance 51(1), [12] Fama, E. F. and K.R. French, Value versus growth: the international evidence, Journal of Finance 53, [13] Fama, E. F. and K.R. French, The Capital Asset Pricing Model: Theory and Evidence. The Journal of Economic Perspectives 18(3), [14] Farnsworth, H., Ferson, W.E., Jackson, D., Todd, S., Performance evaluation with stochastic discount factors, Journal of Business 75, [15] Fernandes, M., Vieira, G., Revisiting the e ciency of risk sharing between UK and US: Robust estimation and calibration under market incompleteness. Mimeo. [16] Ferson, W.E., Harvey, C.R., Sources of Risk and Expected Returns in Global Equity Markets. Journal of Banking and Finance 18,

19 [17] Ferson, W.E., Harvey, C.R., Conditioning Variables and the Cross Section of Stock Returns. Journal of Finance 54(4), [18] Gaunt, C., Size and book to market e ects and the Fama French three factor asset pricing model: evidence from the Australian stock market. Accounting and Finance 44, [19] Gri n, J.M., Are the Fama and French Factors Global or Country-Speci c? The Review of Financial Studies 15(3), [20] Hansen, L.P., Jagannathan, R., Implications of security market data for models of dynamic economies, Journal of Political Economy 99, [21] Hansen, L. P., Jagannathan, R., Assessing Speci cation Errors in Stochastic Discount Factor Models. Journal of Finance 52(2), [22] Jagannathan, R., Kubota, K., Takehara, H., Relationship between Labor-Income Risk and Average Return: Empirical Evidence from the Japanese Stock Market. Jounal of Business 71, [23] Jagannthan, R., Wang, Z., Empirical evaluation of asset-pricing models: A comparison of the SDF and beta methods" Journal of Finance 57, [24] Johnson, R.A., Wichern, D.W., Applied multivariate statistical analysis. Third edition. New Jersey: Prentice-Hall. [25] Kan, R., Robotti, C., Model Comparison Using the Hansen-Jagannathan Distance. Working paper, University of Toronto. [26] Lettau, M., Ludvigson, S., 2001a. Consumption, Aggregate Wealth, and Expected Stock Returns. Journal of Finance 56(3), [27] Lettau, M., Ludvigson, S., 2001b. Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia are Time-Varying. Journal of Political Economy 109(6), [28] Lintner, J., The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics 47(1), [29] Sharpe,W.F., Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance 19(3), [30] Vasicek, O., An equilibrium characterization of the term structure, Journal of Financial Economics 5,

20 Appendix A. The Principal Component analysis A Principal Component Analysis (PCA) is a method of statistical analysis useful in data reduction and interpretation of multivariate datasets. It is mainly concerned in explaining the variancecovariance structure of a set of variables through a few linear combinations of these variables. Consider the random returns Rt 0 = (R 1;t :::::R N;t ) and the respective covariance matrix with eigenvalues 1 2 ::: N 0. De ne the linear combinations: Y n = b 0 nr t n = 1; :::; N (27) where b 0 n is a vector of coe cients. From (27) we obtain: V ar(y n ) = b 0 nb 0 n and Cov(Y n ; Y k ) = b 0 nb 0 k for n; k = 1; :::; N. The rst principal component is Y 1 = b 0 1 R t that maximizes V ar(b 0 1 R t ) subject to b 0 1 b 1 = 1. The second principal component is Y 2 = b 0 2 R t that maximizes V ar(b 0 2 R t ) subject to b 0 2 b 2 = 1 and Cov(b 0 1 R t; b 0 2 R t) = 0. At the n-th step, the n-th principal component is Y n = b 0 nr t that maximizes V ar(b 0 nr t ) subject to b 0 nb n = 1 and Cov(b 0 nr t ; b 0 k R t) = 0 for all k < n. Now de ne ( n ; e n ), for n = 1; :::; N, as the eigenvalue-eigenvector pairs of. Therefore, it can be shown (see Johnson and Wichern, 1998) that the n-th principal component is given by: Y n = e 0 nr t n = 1; :::; N; (28) where V ar(y n ) = e 0 ne 0 n = n n = 1; :::; N; (29) Cov(Y n ; Y k ) = e 0 ne 0 k = 0 n 6= k; (30) and NX NX V ar(r n;t ) = ::: + N = V ar(y n ): (31) n=1 n=1 20

21 Appendix B. Descriptive statistics In this section, we present some descriptive statistics of the generated stochastic discount factors, which are averaged across the K = 1; 000 replications based on the sample sizes T = f100; 200; 400g. For instance, for T = 100, the Jarque-Bera statistic indicates the frequency of rejection of the normality hypothesis across the 1,000 replications (based on a 5% signi cance level). For example, in Table 1 for Hansen & Jagannathan proxy, the statistic Freq. Jarque-Bera is equal to , which means that in 96.10% of the replications the normality hypothesis is rejected at a 5% signi cance level. In contrast, the Araujo et al. (2006) proxy, in the same table, is never rejected in the normality hypothesis test. Table 1 - Descriptive statistics of the SDF proxies (T = 100) Fama and French model with three factors sample size = 100 DGP (FF) Araujo Saint Clara H & J CAPM PCF PCF PCF PCF FF 80% 85% 90% 95% Number of assets N = 80 N = 25 N = 25 N = 25 N = 25 N = 70 N = 70 N = 70 N = 70 N = 25 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Freq. Jarque Bera Notes: DGP (FF) means Data-Generating Process of the Fama & French model; H&J means Hansen and Jagannathan SDF; and PCF means Principal Component Factors SDF and its % of variability. 21

22 Table 2 - Descriptive statistics of the SDF proxies (T = 200) Fama and French model with three factors sample size = 200 DGP (FF) Araujo Saint Clara H & J CAPM PCF PCF PCF PCF FF 80% 85% 90% 95% Number of assets N = 80 N = 25 N = 25 N = 25 N = 25 N = 70 N = 70 N = 70 N = 70 N = 25 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Freq. Jarque Bera Notes: DGP (FF) means Data-Generating Process of the Fama & French model; H&J means Hansen and Jagannathan SDF; and PCF means Principal Component Factors SDF and its % of variability. Table 3 - Descriptive statistics of the SDF proxies (T = 400) Fama and French model with three factors sample size = 400 DGP (FF) Araujo Saint Clara H & J CAPM PCF PCF PCF PCF FF 80% 85% 90% 95% Number of assets N = 80 N = 25 N = 25 N = 25 N = 25 N = 70 N = 70 N = 70 N = 70 N = 25 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Freq. Jarque Bera Notes: DGP (FF) means Data-Generating Process of the Fama & French model; H&J means Hansen and Jagannathan SDF; and PCF means Principal Component Factors SDF and its % of variability. Appendix C. Empirical Results In the following tables, we present the Monte Carlo results, which are averaged across the 1,000 replications. 22

23 Table 4 - Fama and French model with (RMt Rft), SMB and HML factors (sample size T = 100) Fama and French model with three factors (N=80) (Over the time period from 09/1999 to 12/2007) Sample size = 100 Principal Component Factors (PCF) and % of variability Araujo Saint Clara H & J CAPM PCF PCF PCF PCF Fama French 80% 85% 90% 95% SDFa SDFb SDFc SDFd SDFe1 SDFe2 SDFe3 SDFe4 SDFf (npc = 2) (npc = 3) (npc = 3) (npc = 15) Sample Size N = 25 N = 25 N = 25 N = 25 N = 70 N = 70 N = 70 N = 70 N = 25 MSEa MSEb MSEc MSEd MSEe1 MSEe2 MSEe3 MSEe4 MSEf Mean Std. Dev corr_a corr_b corr_c corr_d corr_e1 corr_e2 corr_e3 corr_e4 corr_f Mean Std. Dev HJ dist. SDFa HJ dist. SDFb HJ dist. SDFc HJ dist. SDFd HJ dist. SDFe1 HJ dist. SDFe2 HJ dist. SDFe3 HJ dist. SDFe4 HJ dist. SDFf Mean Std. Dev E E E E Notes: a) Bold numbers show the best SDF for each statistic and the underlined number the Fama and French proxy with 25 observations. b) All results are averaged across the 1,000 replications. c) The parameters varies from im 2 [0:2; 1:0] ; is 2 [ 0:7; 2:3] ; ih 2 [ 0:7; 2:3] 23

24 Table 5 - Fama and French model with (RMt Rft), SMB and HML factors (sample size T = 200) Fama and French model with three factors (N=80) (Over the time period from 05/1991 to 12/2007) Sample size = 200 Principal Component Factors (PCF) and % of variability Araujo Saint Clara H & J CAPM PCF PCF PCF PCF Fama French 80% 85% 90% 95% SDFa SDFb SDFc SDFd SDFe1 SDFe2 SDFe3 SDFe4 SDFf (npc = 4) (npc = 8) (npc = 14) (npc = 26) Sample Size N = 25 N = 25 N = 25 N = 25 N = 70 N = 70 N = 70 N = 70 N = 25 MSEa MSEb MSEc MSEd MSEe1 MSEe2 MSEe3 MSEe4 MSEf Mean Std. Dev corr_a corr_b corr_c corr_d corr_e1 corr_e2 corr_e3 corr_e4 corr_f Mean Std. Dev HJ dist. SDFa HJ dist. SDFb HJ dist. SDFc HJ dist. SDFd HJ dist. SDFe1 HJ dist. SDFe2 HJ dist. SDFe3 HJ dist. SDFe4 HJ dist. SDFf Mean E E E E Std. Dev E E E E Notes: a) Bold numbers show the best SDF for each statistic and the underlined number the Fama and French proxy with 25 observations. b) All results are averaged across the 1,000 replications. c) The parameters varies from im 2 [0:2; 1:0] ; is 2 [ 0:7; 2:3] ; ih 2 [ 0:7; 2:3] 24