2 The stochastic discount factor and the Fama and French model

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Evaluating asset pricing models in a Fama-French framework

Carlos Enrique Carrasco Gutierrez Wagner Piazza Gaglianone^y

Preliminary version: March 2008.

This version: April 2008.

Abstract

In this work we propose a new methodology to compare di¤erentstochastic 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 goodness- of-…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 22.253-900, Brazil (e-mail: [email protected]).

yResearch Department, Central Bank of Brazil (e-mail: [email protected]). The usual disclaimer applies.

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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. As 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

1We use the term "stochastic discount factor" as a label for a state-contingent discount factor.

2For 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; Jagan- nathan 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).

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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 that1 =E_t(m_t+1R_i;t+1), wherem_t+1is 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.

3Fama and French (1993, 1995) argue that a three-factor model is successful because it proxies for unob- served common risk in portfolio returns.

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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+1x_i;t+1); (1)

whereEt( )denotes the conditional expectation given the information available at time t,ptis the asset price,m_t+1 the stochastic discount factor,x_i;t+1the asset payo¤ of thei-th asset int+ 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+1x_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

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(2) mainly illustrates the fact that consumers equate marginal rates of substitution to prices, i.e.

m_t+1 = ^u_u⁰^(c₀_(c^t+1⁾

t) , where is the discount factor for the future, c_t is consumption and u( ) is a given utility function. The speci…cation of mt+1 comes from the …rst-order conditions of the consumption-based model, summarized by the well-known Euler equation: p_t=E_th _u₀_(c

t+1) u⁰(ct) x_t+1i

.

2.1 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 thesizeand thebook-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 thepremium-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

4Notice 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

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(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(Rit) R_{f t}= _im[E(RM t) R_{f t}] + _isE(SM Bt) + _ihE(HM Lt); (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 assetsi:

R_it R_{f t}= _im(R_{M t} R_{f t}) + _isSM B_t+ _ihHM L_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 expected- beta⁵ representation of linear factor pricing models of the form:

E(Ri) = + _{im m}+ _{is s}+ _{ih h}+ i i2 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 factors⁶. Moreover, _ij, for somej 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.

5The main objective of the beta model is to explain the variation in terms of average returns across assets.

6An 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).

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of risk exposure of asset ito 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:

Rit Rf t= _imR^ex_{M t}+ _isSM Bt+ _ihHM Lt+"it t2 f1; :::; Tg; (7) where R^ex_{M t} = (R_{M t} R_{f t}). 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+b⁰f; (8)

where f = [R^ex_M; SM B; HM L]⁰and the relations between e , and aand b, are given by:

a= 1

and b= E f f⁰ ¹ : (9)

Constructing the Fama and French environment

Based on the assumption thatRM t,SM Bt and HM Lt are known variables, we can reproduce a Fama and French environment following the three factors Fama and French model:

Ri;t Rf t= _im(RM t Rf t) + _isSM Bt+ _ihHM Lt+"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_{M t},SM B_t andHM L_t, we generate a returns vector along the time dimension, through equation (7). The iidshock"it is assumed to be a white noise with zero mean and constant variance.

3. Repeating step 2 for eachi 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 parametersa^k andb^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.

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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+1R_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.

LetMt be the DGP stochastic discount factor generated by the Fama and French speci…cation, and Mc_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:

\ M SE_s=

PT

i=1 Mc_t^s M_t ² PT

i=1M_t² ; f or s2S; (11)

and the sample correlation between the actual and estimated SDF is given by:

bs=corr(Mc_t^s; M_t); f or s2S: (12)

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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 ofMt+1 by the projection of Mt+1

onto the space of payo¤s, which makes it straightforward to expressM_t+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 =Et[Mt+1Rt+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 factorM_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 E_t R_t+1R_t+1⁰ ¹R_t+1; (14) where{_N is aN 1vector of ones, andR_t+1is aN 1vector 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 Et Rt+1R⁰_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 Et Rt+1R⁰_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 Mt+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 + ¹²dB; (15)

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where, ^dP_P = ^dP_P¹

1 +; :::;^dP_P^N

N

0, = ( 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:

Rⁱ_t+ _t= P_t+ⁱ _t

P_tⁱ =e(^R^f⁺ i 1

2 i;i) ^t+^p ^t i¹² 0

Zt

; (16)

where, Zt is a vector of N independent variables with Gaussian distribution. Therefore, the SDF proposed by these authors is calculated as

M_t+ _t=e (^R^f⁺¹₂ ⁰ ¹ ) ^t ^p ^t ¹² ⁰^Z^t: (17) Thus, Brandt, Cochrane and Santa-Clara (2006) suggest the following SDF estimator:

Mct=e (^R^f+¹₂b⁰b ¹b) ^t b⁰b ¹ Rt R ;

(18) where, b; Rand b are estimated by:

b= R R^f

t ; (19)

b = 1 t

1 T

XT t=1

R_t R R_t R ; (20)

such that, R_t= R¹_t; :::; R^N_t ⁰ and R=PT t=1R_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 i2 f1; :::; Ng,Mt+1Rⁱ_t+1is 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 forM_t is given by:

Mc_t= R^G_t

1 T

PT

t=1R^A_t R^G_t

!

; (21)

where R^A_t = ^N_i=1R

1 N

i;t and R^G_t = _N¹ PN

t=1R_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.

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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:

R_i;t = _i+ XK

k=1

i;kf_i;t+"_i;t; (22) 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:

E(R_i) = + XK k=1

i;k i;t+ _i i2 f1; :::; Ng: (23)

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 Mt (in unconditional setting), we can estimateMt in the following way:

M^_t=a+ XK k=1

b_kf_k;t (24)

where aand b, are given by:

a= 1

and b= cov f f⁰ ¹ : (25)

In empirical analysis, a critical issue is to identify the number of factorsK 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.

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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-to- market 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 factorsK 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 keigenvectors, and dividing it by the sum of all eigenvalues.⁷

Pk

n=1 n

1+ 2+:::+ 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 25 200 = 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; :::; Tg. Therefore, the number of principal components is endogenously determined based on a cuto¤ value.

For instance, if one sets theCuto¤ 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.

7See appendix A for further details.

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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 = 70assets (out of the total amount ofN = 80) are in fact observed by econometricians. In addition, we assume that only N = 25asset 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+1R⁰_t+1 of the Hansen-Jagannathan SDF estimator.

To account for the mentioned restriction, we consider a maximum number ofN = 25assets 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 (RM t Rf t), SM Bt and HM Lt 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 thet-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 sizesT =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;000replications. 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:

Mc_t^a : Stochastic discount factor of Araujo, Issler and Fernandes (2006);

Mc_t^b : Stochastic discount factor of Brandt, Cochrane and Santa-Clara (2006);

Mc_t^c : Stochastic discount factor of Hansen and Jagannathan (1991);

8More information about data can be found in:

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

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Mc_t^d : Stochastic discount factor of the CAPM;

Mc_t^e : Stochastic discount factor of the Principal Component Factors (PCF) model, in which the percentage of variability ranges from 80% to 95%;

Mc_t^f : Stochastic discount factor implied by the Fama and French setup.⁹

3.2 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), Mc_t^b, and of Araujo et al. (2006), Mc_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 sizeT = 100

0 10 20 30 40 50 60 70 80 90 100

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

SD F Fama and Fr enc h ( D GP) SD F Araujo, Is s ler and Fernandes SD F Brandt, C oc hrane and S.C SD F H ans en and J agannathan C APM

Princ ipal C omponent Fac tors (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.

9TheMc_t^f is estimated by using N=25 assets, whereas the DGP is generated from N=80 assets.

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Figure 2 shows the ratioPk

n=1 n =( 1+ 2+:::+ 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 sizeT = 200

0 20 40 60 80 100 120 140 160 180 200

50 55 60 65 70 75 80 85 90 95 100

Fraction of Total Variation vs. Time

Time (Months)

Percent of Total Variation

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): M SE\f <

M SE\_e2 <M SE\_e3 <M SE\_e1 <M SE\_a<M SE\_e4 <M SE\_d <M SE\_c <M SE\_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.

1 0For the sample size T = 200 the "ranking order" is the following: M SE\f <M SE\e1 <M SE\e2 <M SE\e3

<M SE\a<M SE\e4 <M SE\d<M SE\b<M SE\c. In a similar result, the ranking forT = 400is the following:

\

M SEf <M SE\e2 <M SE\e1<M SE\e3 <M SE\a<M SE\e4 <M SE\b<M SE\c<M SE\d.

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In respect to the correlation of the true SDF with the considered SDF proxies, we have obtained the following ranking order (T = 100): Mc_tê2 = Mc_tê3 Mc_t^f Mc_tê1 Mc_tâ Mc_tê4 Mc_t^b Mc_t^c Mc_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 distance¹² (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 usingN = 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 orN = 70 (i.e., all available information)?¹³ Recall that the estimation of the inverse of the matrixEt Rt+1R⁰_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.

1 1For the sample sizeT = 200, it follows that: Mc_t^f Mc_tê1 Mc_tê2 Mc_tâ Mc_tê3 Mc_tê4 Mc_t^b Mc_t^c Mc_t^d, and forT = 400we have that: Mc_t^f Mctê2 Mctê1 Mctê3 Mctâ Mctê4 Mct^b Mct^c Mct^d.

1 2We compute the HJ distance based on the MatLab codes of Mike Cli¤, available at:

http://mcli¤.cob.vt.edu/

1 3The 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.

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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.

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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 variance- covariance structure of a set of variables through a few linear combinations of these variables.

Consider the random returns R⁰_t = (R1;t :::::RN;t) and the respective covariance matrix with eigenvalues ₁ ₂ ::: _N 0. De…ne the linear combinations:

Yn=b⁰_nRt n= 1; :::; N (27)

where b⁰_n is a vector of coe¢ cients. From (27) we obtain: V ar(Yn) = b⁰_n b⁰_n and Cov(Yn; Yk) = b⁰_n b⁰_k for n; k = 1; :::; N. The …rst principal component is Y₁ =b⁰₁R_t that maximizesV ar(b⁰₁R_t ) subject to b⁰₁b₁ = 1. The second principal component is Y₂ = b⁰₂R_t that maximizes V ar(b⁰₂R_t ) subject to b⁰₂b2 = 1 and Cov(b⁰₁Rt; b⁰₂Rt) = 0. At the n-th step, the n-th principal component is Y_n=b⁰_nR_t that maximizesV ar(b⁰_nR_t )subject to b⁰_nb_n= 1 and Cov(b⁰_nR_t; b⁰_kR_t) = 0 for all k < n.

Now de…ne( n; en), forn= 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:

Yn=e⁰_nRt n= 1; :::; N; (28)

where

V ar(Yn) = e⁰_n e⁰_n= n n= 1; :::; N; (29) Cov(Y_n; Y_k) = e⁰_n e⁰_k= 0 n6=k; (30) and

XN n=1

V ar(R_n;t) = ₁+ ₂+:::+ _N = XN n=1

V ar(Y_n): (31)

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Appendix B. Descriptive statistics

In this section, we present some descriptive statistics of the generated stochastic discount factors, which are averaged across theK= 1;000replications based on the sample sizesT =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 statisticFreq. Jarque-Berais equal to 0.9610, 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 0.9094 0.9846 0.9216 0.9941 0.9856 1.0021 0.9966 0.9966 0.9966 0.9109

Median 0.9194 0.9843 0.7399 0.9885 0.9981 1.0188 1.0068 1.0068 0.9903 0.9221

Maximum 1.9050 1.2865 4.2011 2.5987 1.1375 2.4390 2.0535 2.0535 2.1373 1.8771

Minimum 0.2144 0.8581 0.1557 -0.5760 0.7826 0.2045 0.2411 0.2411 0.0250 0.2009

Std. Dev 0.2854 0.0697 0.6786 0.6290 0.0806 0.3763 0.3050 0.3050 0.3987 0.2834

Skewness 0.4095 1.2237 2.1603 0.0365 -0.3906 0.6318 0.3938 0.3938 0.1844 0.3676

Kurtosis 4.0003 6.4074 10.2932 2.9695 2.9854 4.5498 4.0033 4.0033 3.2720 4.0262

Freq. Jarque-Bera 0.4560 0.0000 0.0000 0.9610 1.0000 0.0000 0.5030 0.5030 0.8750 0.5460 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.

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80% 85% 90% 95%

Mean 0.9079 0.9849 0.9225 0.9922 0.9992 0.9963 0.9963 0.9963 0.9963 0.9107

Median 0.9055 0.9800 0.8130 0.9905 1.0086 0.9891 0.9885 0.9887 0.9899 0.9087

Maximum 2.0786 1.2878 3.7528 2.2771 1.2141 2.2464 2.2531 2.2814 2.3684 2.0388

Minimum 0.2696 0.8450 0.2285 -0.2284 0.6297 0.2715 0.1826 0.0687 -0.1331 0.2848

Std. Dev 0.2858 0.0626 0.4989 0.4479 0.0880 0.3075 0.3277 0.3592 0.4269 0.2767

Skewness 0.5005 1.2730 2.0933 0.0358 -0.6821 0.4759 0.3972 0.3026 0.1789 0.4641

Kurtosis 4.3983 7.2203 11.1584 3.0501 4.2628 4.3360 4.0266 3.7131 3.3264 4.3572

Freq. Jarque-Bera 0.0000 0.0000 0.0000 0.9400 0.0000 0.0220 0.1920 0.4640 0.7890 0.0000

Notes: DGP (FF) means Data-Generating Process of the Fama & French model; H&J means Hansen and Jagannathan SDF;

80% 85% 90% 95%

Mean 0.9246 0.9836 0.9196 0.9912 0.9860 0.9937 0.9950 0.9950 0.9948 0.9256

Median 0.9284 0.9778 0.8581 0.9931 0.9885 0.9993 0.9982 0.9965 0.9949 0.9293

Maximum 2.0400 1.5179 3.2699 2.0964 1.1510 2.0315 2.1765 2.1851 2.2532 2.0132

Minimum -0.6473 0.6969 0.1525 -0.4386 0.7561 -0.5770 -0.6857 -0.6898 -0.7080 -0.6183

Std. Dev 0.2560 0.0721 0.3620 0.3437 0.0453 0.2519 0.2726 0.2788 0.3389 0.2517

Skewness -0.4025 1.9781 1.8771 -0.1364 -0.3283 -0.5009 -0.4141 -0.3956 -0.2298 -0.4126

Kurtosis 9.2783 14.7780 10.8551 3.9930 5.4366 9.2860 9.2902 8.7783 5.5965 9.2337

Freq. Jarque-Bera 0.0000 0.0000 0.0000 0.2800 0.0000 0.0000 0.0000 0.0000 0.0067 0.0000

Notes: DGP (FF) means Data-Generating Process of the Fama & French model; H&J means Hansen and Jagannathan SDF;

Appendix C. Empirical Results

In the following tables, we present the Monte Carlo results, which are averaged across the 1,000 replications.