Title

The time-varying pricing of value and size
premia in Japan

Author(s)Tsuji, Chikashi

Citation

大阪大学経済学. 2009, 58(4), p. 68-87

Version Type VoR
URL

https://doi.org/10.18910/24469

rights
Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/
Osaka University

tsuji :2009/3/5(2:36)

OSAKA ECONOMIC PAPERS

Vol.58 No.4

March 2009

The time-varying pricing of value and size

premia in Japan∗

Chikashi TSUJI †Abstract
Following Fama and French’s (1996) suggestion of the equivalence of their three-factor
model and established equilibrium pricing models, this paper examines whether the
Fama–French small-minus-big (SMB) and high-minus-low (HML) factors mimic thestate variables in Merton’s (1973) intertemporal capital asset pricing model (ICAPM).
We test this equivalence for the Japanese stock market over the period 1982 to 2003 using a multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
model. Our direct test of the time-varying pricing of the Fama–French factors in the
ICAPM finds that both SMB and HML play the role of well-priced state variables. Ourfindings thus support the suggestion that SMB and HML factors indeed mimic the statevariables in Merton’s ICAPM in Japan.
Key Words: Conditional CAPM; Fama–French model; Multivariate GARCH Model;
Panel data analysis; Time-varying price of risk.
JEL Classification: G12; G15.

1 Introduction
What is the role of Fama and French’s (1993) small-minus-big (SMB) and high-minus-low(HML) factors in equity markets? Fama and French (1996, p.57) suggested that “…the empirical
successes of the Fama–French model suggest that it is an equilibrium pricing model, a three-factor
version of Merton’s (1973) intertemporal capital asset pricing model (ICAPM) or Ross’s (1976)
arbitrage pricing theory (APT). In this view, SMB and HML mimic combinations of two underlying
risk factors or state variables of special hedging concern to investors”. Subsequent to Fama andFrench (1996), several studies, including Liew and Vassalou (2000) and Petkova (2006), investi-∗†

The author is grateful to Professor Dr. Kazuhiko Nishina for his kind recommendation of this paper to this volume.
Correspondence: Chikashi TSUJI, Associate Professor, Department of Social Systems and Management, Graduate Schoolof Systems and Information Engineering, University of Tsukuba. Address: 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573,
Japan.

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The time-varying pricing of value and size premia in Japan− 69 −

gated the role of the SMB and HML factors.1 However, as far as the authors are aware, there is no
known research that directly tests whether SMB and HML mimic the state variables in the ICAPM
strictly using Merton’s (1973) framework. Therefore, the objective of this paper is to empiricallytest this proposition by examining the pricing of SMB and HML as state variables strictly in thecontext of Merton’s (1973) ICAPM. The multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) approach employed for this purpose enables us to perform the test more
directly because with it we obtain the conditional covariance risk of SMB and HML. If SMB and
HML mimic the state variables in the ICAPM, they should be priced in the ICAPM framework.
Moreover, despite the eﬀectiveness of the multivariate GARCH model in asset pricing, its application to asset pricing models in the existing literature is limited. Thus, our approach is relatively
novel.2
From a technical point of view, we incorporate time-varying risk prices into our analysis. Thisis another interesting characteristic of our approach since many existing studies, such as Harvey(1989), Ng (1991), and Zhou (1994), take into account only time-varying risk.3
The contribution of this paper is as follows. First, both our monthly and panel data analysisof time-varying risk pricing reveals that the SMB is a well priced state variable in the context
of Merton’s (1973) ICAPM. Second, both our monthly and panel data analysis of time-varying
risk pricing also finds that the HML is also generally a priced state variable in the frameworkof Merton’s (1973) ICAPM. Third, from the viewpoint of model evaluation, our formal F-tests
indicate that the conditional Fama–French model is indeed better than the conditional CAPM. This
also supports Fama and French (1996) because it shows the eﬀectiveness of SMB and HML in the
ICAPM. Finally, and as an ancillary finding, we find that the conditional covariance market risk in
the CAPM, as derived by the multivariate GARCH model, is strongly priced in Japan. This result
is also interesting because in many studies, including Fama and French (2006), and Lewellen and1

Petkova (2006) found that the SMB and HML are correlated with innovations in variables that would describe investment
opportunity in the context of Merton’s ICAPM. Liew and Vassalou (2000) considered that GDP growth is representativeof the investment opportunities in Merton’s ICAPM, and found that HML and SMB are linked to future GDP growth inten major international countries. That is to say, both studies tested whether SMB and HML are related with variables thatwould describe investment opportunities in Merton’s (1973) ICAPM. We consider these as indirect tests of the proposition
suggested by Fama and French (1996).

2Studies using multivariate GARCH models include, for example, Beine (2004), Tai (2004), Audrino and Barone-Adesi(2005), Christiansen (2005), Ewing and Malik (2005), Giamouridis and Vrontos (2007), and Leon et al. (2007). However,
the multivariate GARCH models in this body of work, with the exception of Leon et al. (2007), are generally not
employed in asset pricing models of stock returns. As for the remaining studies, Beine (2004) examines foreign exchange
markets, Tai (2004) addresses contagion during the 1997 Asian crisis, Audrino and Barone-Adesi (2005) focus on theestimation and forecasting of volatility, Christiansen (2005) analyses term structure models, Ewing and Malik (2005)
examine ‘spillover eﬀects’ in financial markets, and Giamouridis and Vrontos (2007) consider hedge fund portfolios.

3

The time-varying characteristics of both covariance risk and the price of risk are clearly crucial for asset pricing. Thereis substantial empirical evidence that the level of risk varies over time (see Bollerslev et al. (1988), Harvey (1989),
Ng (1991), and Zhou (1994), amongst others). However, in many earlier studies, including Harvey (1989), Campbell(1996), Hansson and Hördahl (1998), and Guo (2006), covariance risk is regarded as time varying, but the price of risk is
considered constant for a particular period. Other analyses that deal with the price of risk include Friend and Westerfield(1981), Sauer and Murphy (1992), Ferson and Harvey (1994), Priestley (1996), Flannery et al. (1997), Doukas et al.
(1999), Gibson and Mougeot (2004), and Leon et al. (2007). These studies also generally assume that the price of risk
is constant for a specific period. While Polk et al. (2006) undertake an empirical analysis that exploits the price of risk,
their aim and approach diﬀer from our study.

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Vol.58 No.4

Nagel (2006), the applicability of CAPM is often in refute.
The remainder of the paper is organized as follows. Section 2 describes the conditional asset
pricing models to be tested. Section 3 details the methodology and Section 4 describes the data.
The empirical results and their interpretation are supplied in Sections 5 and 6. Section 7 presents
some conclusions.
2 ModelTo start with, Cochrane (2005) expresses Merton’s (1973) ICAPM as:f

Et (Rit+1 ) − Rt ≈ rrat covt (Rit+1 , ΔWt+1 /Wt ) + λzt covt (Rit+1 , Δzt+1 ),(1)

where Et (Rit+1 ) is the conditional expected return of asset i, Rtf is the risk-free rate, rrat is therelative risk aversion coeﬃcient,4 covt (Rit+1 , ΔWt+1 /Wt ) is the conditional covariance between the
return of asset i and the change in wealth (or return on the market portfolio), covt (Rit+1 , Δzt+1 ) is theconditional covariance between the return of asset i and the change in state variable z, and λzt is therisk price for the state variable.
Thus, following Cochrane’s (2005) expression (1), we test the pricing of the market factor, SMB
and HML in the framework of Merton’s ICAPM. First, we begin by setting our first model as thefollowing conditional CAPM. The model for period t is an equilibrium relation for the conditional
expected return of an asset in excess of the risk-free rate when agents use the information available
at the end of period t − 1:
E (ri,t − r f,t )|Ωt−1 =
=
βi,t

≡



E (rm,t − r f,t )|Ωt−1 Cov ri,t , rm,t |Ωt−1
Var rm,t |Ωt−1
βi,t E (rm,t − r f,t )|Ωt−1 .

Cov ri,t , rm,t |Ωt−1 ,Var rm,t |Ωt−1

(2)

(3)

where ri,t and rm,t are the one-period returns on an asset i and the market portfolio, respectively, r f,t
is the one-period risk-free rate, and Ωt−1 is the information available to the market at time t − 1.
From a cross-sectional perspective, this model implies that the conditional expected excess returns
vary with the diﬀerent conditional beta values or diﬀerent conditional covariances. From a timeseries perspective, the model has an implication that the conditional expected excess returns changeover time with three time-varying components: the conditional market risk premium, the market
conditional variance, and the conditional covariance between the asset’s return and the market’s
4

As shown in equation (1), the time-varying characteristics of the risk price are important. Time-varying risk aversion isparticularly focused upon in Able (1990), Constantinides (1990), Campbell and Cochrane (1999), Chen and Pakoš (2006),
Coudert and Gex (2006), Li (2007), and Maio (2007).

tsuji :2009/3/5(2:36)

The time-varying pricing of value and size premia in Japan

March 2009

− 71 −

return. Unlike Harvey (1989), Engel et al. (1995), and Hansson and Hördahl (1998), among others,
we assume that the market price of risk is time varying rather than stable:E (rm,t − r f,t )|Ωt−1 .
δm,t ≡
Var rm,t |Ωt−1(4)

Thus, and diﬀerently to other studies, we use the following conditional ICAPM for a single asset ifor our first model, referred to as Model 1:5E (ri,t − r f,t )|Ωt−1 = δm,tCov ri,t , rm,t |Ωt−1 .

(5)

In this formulation, the estimation of the time-varying covariance, Cov[ri,t , rm,t |Ωt−1 ], is necessaryfor evaluating the model. By inspecting the statistical significance of δm,t using these covariances,
we can judge whether the covariance market risk is priced. Thus, tests using Model 1 are the first
method of analysis for our empirical work. We also use models (6) and (7) below to test our focus,
namely the pricing of the Fama–French SMB and HML factors, respectively.

E (ri,t − r f,t )|Ωt−1 =E (ri,t − r f,t )|Ωt−1 =δ smb,tCov ri,t , r smb,t |Ωt−1 ,δhml,t Cov ri,t , rhml,t |Ωt−1 ,

(6)
(7)

where δ smb,t : the time-varying risk price of SMB factor returns, Cov[ri,t , r smb,t |Ωt−1 ]: the conditional
covariance between asset returns and SMB factor returns, r smb,t , δhml,t : the time-varying risk priceof HML factor returns, Cov[ri,t , rhml,t |Ωt−1 ]: the conditional covariance between asset returns and
HML factor returns, rhml,t .
For our second model, Model 2, and in reference to (8) below, we add Cov[ri,t , r smb,t |Ωt−1 ] toModel 1:






E (ri,t − r f,t )|Ωt−1 = δm,t Cov ri,t , rm,t |Ωt−1 + δ smb,t Cov ri,t , r smb,t |Ωt−1 .

(8)For our third model, Model 3, and as described in (9) below, we add Cov[ri,t , rhml,t |Ωt−1 ] to Model
1:






E (ri,t − r f,t )|Ωt−1 = δm,tCov ri,t , rm,t |Ωt−1 + δhml,t Cov ri,t , rhml,t |Ωt−1 .

(9)Finally, our last model, Model 4, yields the following three-factor model (10) which can be interpreted as a version of Merton’s (1973) ICAPM including Fama–French factors:
5

Model 1 is also considered as a generalization of the one-period CAPM developed by Sharp (1964)–Lintner (1965)–
Mossin (1966). Other studies concerning the conditional CAPM include Jagannathan and Wang (1996), Lettau andLudvigson (2001), Ang and Chen (2007), and Petkova and Zhang (2005).

tsuji :2009/3/5(2:36)

OSAKA ECONOMIC PAPERS− 72 −E (ri,t − r f,t )|Ωt−1=
δm,tCov ri,t , rm,t |Ωt−1+δ smb,t Cov ri,t , r smb,t |Ωt−1 + δhml,t Cov ri,t , rhml,t |Ωt−1 .

Vol.58 No.4

(10)

3 Methodology
As argued in the excellent survey by Bauwens et al. (2006), the multivariate GARCH model iscrucially important in the context of asset pricing because the model is very useful for calculating
the time-varying covariances or factor loadings.
To evaluate the time-varying risk prices, δm,t , δ smb,t , and δhml,t detailed earlier, we first estimate
the time-varying covariances, Cov[ri,t , rm,t |Ωt−1 ], Cov[ri,t , r smb,t |Ωt−1 ], and Cov[ri,t , rhml,t |Ωt−1 ], withthe multivariate BEKK (Engle and Kroner (1995)) GARCH model. This particular model ensuresthat the H matrix is always positive definite, and is specified by:
H t = W + B H t−1 B + A Ξ t−1 Ξt−1 A,(11)where W, A, and B are 2 × 2 matrices of parameters, and W is assumed to be symmetric andpositive definite.
For the purpose of clarity, in the case of two assets, we define the matrices as follows:⎡⎢⎢⎢ h11,t
H t = ⎢⎢⎢⎣
h12,t

⎤
⎡
⎢⎢ w11h12,t ⎥⎥⎥⎥⎥⎥⎦ , W = ⎢⎢⎢⎢⎣
h
w
22,t

⎤
⎤
⎡
⎢⎢⎢ a11 a12 ⎥⎥⎥w12 ⎥⎥⎥⎥⎥⎥⎥ ,⎢⎥⎥ , A = ⎢⎢⎣w22 ⎦a21 a22 ⎦
12
⎤
⎤
⎡
⎡
⎢⎢⎢ u1,t ⎥⎥⎥⎢⎢⎢ b11 b12 ⎥⎥⎥
⎥
⎥⎥⎥ .
⎢
⎢
⎥⎥ , Ξ t = ⎢⎢⎣B = ⎢⎢⎣
b21 b22 ⎦u2,t ⎦The model is then written in full as:
h11,t

=

w11 + a211 u21,t−1 + a221 u22,t−1 + 2a11 a21 u1,t−1 u2,t−1+b211 h11,t−1 + b221 h22,t−1 + 2b11 b21 h12,t−1 ,

h22,t

=

w22 + a212 u21,t−1 + a222 u22,t−1 + 2a12 a22 u1,t−1 u2,t−1+b212 h11,t−1 + b222 h22,t−1 + 2b12 b22 h12,t−1 ,

h12,t

=

w12 + a11 a12 u21,t−1 + a21 a22 u22,t−1 + (a12 a21 + a11 a22 )u1,t−1 u2,t−1+b11 b12 h11,t−1 + b21 b22 h22,t−1 + (b11 b22 + b12 b21 )h12,t−1 .

With regard to model estimation, the parameters of the multivariate GARCH models for any of theabove specifications can be estimated by maximizing the log-likelihood function:Tl(θ) = −1
TN
log 2π −(log |H t | + Ξt H−1
Ξ t ),t2
2 t=1

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The time-varying pricing of value and size premia in Japan− 73 −

where θ denotes all of the unknown parameters to be estimated, N is the number of assets, T is thenumber of observations, and H t and Ξ t are as previously defined.
After deriving the time-varying covariances, Cov[ri,t , rm,t |Ωt−1 ], Cov[ri,t , r smb,t |Ωt−1 ], and Cov[ri,t ,rhml,t |Ωt−1 ], from the multivariate GARCH model, we first estimate regressions (5), (6), and (7)using monthly cross-sections. Then, the time-varying prices of risk δm,t , δ smb,t , and δhml,t can beevaluated monthly. If SMB and HML mimic the state variables of Merton’s (1973) ICAPM, the covariance risks of Cov[ri,t , r smb,t |Ωt−1 ] and Cov[ri,t , rhml,t |Ωt−1 ] should be priced. Further, to evaluate
the degree of pricing for the risk factors in Models 1 to 4 for periods longer than a month, and to
take both cross-sectional and time-series aspects into account, we undertake panel data analysis. By
pooling the monthly data on 25 size-ranked portfolios in Japan, we can conduct a balanced panel
data analysis in arbitrary time spans. That is, we analyse models (5), (8), (9) and (10) for the entire
sample period from January 1982 to December 2003 and for four sub-sample periods using pooled
regression. The details of the empirical results are provided in Sections 5 and 6. As argued, themultivariate GARCH model enables us to directly test time-varying risk pricing in the context ofMerton’s (1973) ICAPM.
4 DataThe sample period of the data analysed in this paper is from January 1982 to December 2003.
The individual data series are the risk-free percentage rate, r f,t , the market portfolio percentage
return, rm,t , the Fama–French SMB factor percentage return, r smb,t , the Fama–French HML factor
percentage return, rhml,t , and ri,t are the returns on 25 size-ranked portfolios constructed using thereturns of stocks listed on the Tokyo Stock Exchange (TSE) First Section.
In greater detail, r f,t is the gensaki rate from the Japan Securities Dealers Association from January 1982 to May 1984 and the one-month median rate on negotiable-time certificates of deposit(CD) from the Bank of Japan from June 1984 to December 2003.6 The market return rm,t is thevalue-weighted return on all stocks in the TSE First Section: this data is provided by the Japan Securities Research Institute (JSRI). The Fama–French SMB and HML factor returns are constructedfollowing the procedures in Fama and French (1993) using the return data of the TSE First Section stocks obtained from the JSRI.7 We also constructed 25 size-ranked portfolio returns ri,t , also
following Fama and French (1993).8

6

One-month CD rates are not available before June 1984. Accordingly, following Hamao (1988), we specify the gensaki
rate as the risk-free rate before June 1984.

7

More details of the constructions of SMB and HML in Japan, see Tsuji (2008).

8

To construct the size-ranked portfolios, all TSE First Section stocks are allocated to one of 25 groups based on theirmarket equity (stock price times shares outstanding) at the end of September of each year t (1981−2003). The valueweighted monthly returns on the portfolios are then calculated from the following October to the next September. Only
firms with ordinary common equity are included in our analysis. This means that REITs (Real Estate Investment Trusts)
and beneficial interest units are excluded (See also Tsuji (2008).).

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OSAKA ECONOMIC PAPERS− 74 −

Vol.58 No.4

Table 1
Sample statistics of the value-weighted returns on 25 portfolios
formed on the basis of size: The case in Japan from January 1982 to December 2003
Mean

MedianStd. Dev.

Skewness

Kurtosis

Small

3.663

3.058

Size 2

2.721

2.310

10.537

1.047

5.931

9.006

0.592

Size 3

2.480

4.803

2.108

8.560

0.384

Size 4

3.538

2.017

1.674

8.238

0.312

4.611

Size 5

2.065

1.553

7.794

0.261

3.874

Size 6

1.608

1.422

7.704

0.542

5.356

Size 7

2.118

2.253

8.153

0.905

7.051

Size 8

1.824

1.827

7.556

0.377

4.376

Size 9

1.527

1.951

7.349

0.165

4.154

Size 10

1.761

1.380

7.474

0.255

4.072

Size 11

1.741

1.609

7.238

0.297

4.555

Size 12

1.522

1.537

6.819

0.147

3.806

Size 13

1.532

1.302

6.837

0.072

3.910

Size 14

1.720

1.535

6.906

0.334

4.390

Size 15

1.420

1.833

6.549

0.142

4.828

Size 16

1.519

1.459

6.633

0.072

4.222

Size 17

1.275

0.993

6.510

0.106

4.784

Size 18

1.396

1.296

6.292

0.107

4.717

Size 19

1.480

1.496

5.833

0.251

5.860

Size 20

1.314

1.323

6.131

0.102

4.704

Size 21

1.243

1.304

5.993

0.166

4.715

Size 22

1.367

1.466

5.865

-0.0114.489

Size 23

1.320

1.257

5.524

0.174

4.594

Size 24

1.233

1.050

5.564

0.216

4.078

Big

0.867

0.867

5.982

0.337

3.721

Portfolio

Notes: The sample statistics of the value-weighted returns of 25 portfolios formed on the basis of size are displayed. The sample
period is from January 1982 to December 2003. The size-ranked portfolios are constructed following Fama and French (1993). In
constructing the size-ranked portfolios, TSE (Tokyo Stock Exchange) First Section stocks were allocated to 25 groups based on theirmarket equity (stock price times shares outstanding) at the end of September of each year t (1981-2003). Value-weighted monthly
returns on the portfolios were then calculated from October to the following September. Only firms with ordinary common equity areincluded. REITs (Real Estate Investment Trusts) and units of beneficial interest are excluded.

5 Time-varying pricing of market risk and Fama–French factors
This section provides the empirical results for the pricing of the time-varying covariance risks ofthe market, SMB, and HML factors. Table 1 provides sample statistics of the value-weighted returnsof the 25 size-ranked portfolios over the period from January 1982 to December 2003. The mean
returns of the size-ranked portfolios show the rather clear pattern of a monotonic increase from thelargest portfolio to the smallest portfolio. This gives clear evidence of a size-eﬀect in Japan. Figure
1 depicts the characteristics of the three covariance risks. From the three graphs in Figure 1, we
can see that market covariance risk gradually becomes higher in smaller portfolios (Panel A), SMB
covariance risk is clearly higher in smaller portfolios (Panel B), and HML covariance risk is also
generally higher in smaller portfolios (Panel C). The increasing pattern of covariance risk is mosttsuji :2009/3/5(2:36)

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The time-varying pricing of value and size premia in Japan− 75 −

Figure 1
Time-varying covariances between market return, SMB, and HML and monthly percentage returns on portfolios
formed on size: The case of Japan from January 1982 to December 2003

Panel A Average time-varying covariances between market
return and the size-ranked portfolio returns

Panel B Average time-varying covariances between SMB andthe size-ranked portfolio returns

Panel C Average time-varying covariances between HML
and the size-ranked portfolio returns

distinct for the SMB factor.
Next, applying equations (5), (6), and (7) to the cross-sections, we obtain the monthly timevarying price of risk for the three risk factors in the 25 size-ranked portfolios. Because each regression comprises a cross-section, White’s (1980) heteroskedasticity-consistent covariance matrix
is used to calculate the p-values. First, Table 2 displays the monthly time-varying price of market
risk from January 1982 to December 2003. As shown in Table 2, the price of market risk using theconditional CAPM (5) is generally statistically significant. Statistically significant risk prices withtheoretically consistent positive signs (the bold figures in Table 2) are found in 136 of the 264 cases.
Second, Table 3 displays the monthly time-varying price of the SMB covariance risk for the same
portfolios and sample period as in Table 2. The observed trends in pricing are very similar to thosefor market risk: the monthly time-varying price of SMB covariance risk is generally statisticallysignificant, and the number of significant risk prices with positive signs (the bold figures in Tabletsuji :2009/3/5(2:36)

OSAKA ECONOMIC PAPERS− 76 −

Vol.58 No.4

Table 2Monthly time-varying price of market risk on 25 portfolios formed on the basis of size:
The case of Japan from January 1982 to December 2003
January
1982

Risk pricep-value

February

March

April

May

June

July

AugustSeptember

OctoberNovember December

0.110 ** -0.083 ** -0.094 **

0.036

0.008

0.043

-0.019-0.125 ** -0.091 **0.090 **

0.314 ** 0.062 **

0.000

0.005

0.127

0.763

0.285

0.432

0.000

0.000

0.010

0.000

0.003

0.138 **

0.092 ** 0.290 **0.242 ** 0.170 **0.103 *

0.312 ** 0.164 **0.027

0.066 *

0.095 **

0.328 **

0.002

0.000

0.017

0.000

0.000

0.000

1983

Risk pricep-value

0.003

0.000

0.000

0.343

0.048

0.000

1984

Risk pricep-value

0.288 **

0.051

0.248 ** -0.001-0.274 ** 0.123 ** -0.009

0.254 **

0.029

0.231 **

0.149 **

0.011

0.000

0.150

0.000

0.000

0.006

0.214

0.000

0.000

0.626

Risk pricep-value

0.187 **

0.137 *

0.083 **

0.013

0.221 ** 0.154 **0.002

0.258 **

0.187 ** 0.120 ** 0.141 ** 0.102 **

0.000

0.021

0.000

0.565

0.000

0.975

0.000

0.000

0.173 **

0.410 **

0.376 **

0.088

0.149 ** 0.323 **

0.009

0.063

-0.233 ** -0.032

0.405 ** -0.039 *

0.000

0.000

0.798

0.335

0.009

0.000

1985
1986

Risk pricep-value

1987

Risk pricep-value

0.143 ** 0.137 **1988

Risk pricep-value

0.446 ** 0.193 **1989

Risk pricep-value

1990

0.000
0.000

0.000

0.000
0.000

0.839

0.001

0.000

0.000

0.051

0.001

0.000

0.070 **

0.136 **

0.633 **

0.143 ** 0.306 ** 0.318 **0.086 ** -0.245 ** -0.057 ** -0.026

0.006

0.000

0.000

0.075 ** 0.242 **

0.000

0.001

0.140 **

0.113 ** -0.123 ** -0.084 ** -0.026

0.005

0.000

0.005

0.000

0.155

-0.063 ** 0.328 ** 0.062 **

0.000

0.000

0.000

0.005

0.000

0.215 **

0.177 ** -0.088 ** 0.276 ** 0.107 ** 0.357 **

0.012

0.211 **

0.179 **

0.000

0.000

0.000

0.000

0.454

0.000

0.000

0.000

0.000

0.005

-0.121 ** -0.468 ** -0.101 ** 0.400 **
0.001

0.248

0.009

0.029

0.167 **

0.090

0.002

0.000

0.123

0.001

0.430 ** -0.033

Risk price -0.027
p-value
0.526

0.000

0.976

0.000

0.000

0.002

0.039 *

0.018

-0.507 ** -0.392 ** 0.260 ** -0.141 **

0.003

0.008

0.000

0.017

0.311

0.000

0.574

1991

Risk price -0.072 ** 0.320 ** 0.036 ** 0.003
p-value
0.000
0.000
0.000
0.404

-0.001-0.107 ** -0.007-0.222 ** 0.182 **0.086 ** -0.240 ** -0.007

0.795

0.000

0.000

0.000

1992

Risk price -0.086 ** -0.040 ** -0.238 ** -0.178 ** 0.184 ** -0.240 ** -0.112 ** 0.233 ** -0.060 ** -0.066 ** 0.064 ** 0.004
p-value
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.776

1993

Risk price -0.053 **p-value
0.000

0.002

0.507 **

0.349 ** 0.159 ** -0.130 ** 0.067 **0.050 ** -0.075 ** -0.167 ** -0.598 ** 0.101 **

0.816

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1994

Risk pricep-value

0.326 **

0.024 **

0.014

0.079 **

0.099 ** 0.084 ** -0.053 ** -0.021 *-0.181 **0.017-0.205 ** 0.109 **

0.000

0.000

0.313

0.000

0.000

1995

Risk price -0.143 ** -0.317 ** -0.121 ** 0.015
p-value
0.000
0.000
0.000
0.052

1996

Risk pricep-value

1997

Risk price -0.129 ** 0.006
p-value
0.000
0.657

1998

Risk pricep-value

0.640 ** 0.128 ** -0.050 ** -0.108 ** 0.039 **1999

Risk pricep-value

0.085 ** 0.035 *
0.000

2000

Risk pricep-value

0.262 **
0.000

0.343

2001

Risk pricep-value

0.064 ** 0.101 **
0.001

2002

Risk price -0.064 *p-value
0.028

0.290 **
0.000

0.000

0.000

0.000

0.005

0.000

2003

Risk pricep-value

0.073 *

0.261 **

0.059 *

0.296 **

0.434 ** 0.532 **0.049

0.285 ** 0.137 **0.144 ** -0.221 ** 0.159 **

0.016

0.000

0.020

0.000

0.000

0.096

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.073

0.000

0.197

0.000

0.004

0.191 ** 0.261 **

0.000

0.651

0.000

0.000

0.050

0.000

0.000

0.000

0.000

0.000

0.000

-0.303 ** -0.128 ** 0.407 ** 0.282 ** -0.010

0.195 ** -0.102 ** 0.194 ** 0.302 ** -0.025 *
0.000

0.000

0.338

0.000

0.000

0.220

0.000
0.000

0.044 ** -0.349 ** -0.044 ** 0.153 ** -0.157 ** -0.062 ** -0.331 **

0.000

0.000

0.047

0.006

-0.121 **0.112 **

0.298 **

0.074 ** -0.156 ** -0.353 ** -0.307 ** 0.086 *-0.344 ** -0.388 **

0.000

0.000

0.000

0.000

0.001

0.000

0.110 **

0.114 ** -0.380 ** -0.149 ** -0.026 *0.411 ** -0.142 **

0.001

0.000

0.001

0.000

0.005
0.000

0.000
0.000

0.000
0.013

0.001

0.000
0.000

0.000

0.000

0.000

0.000

0.024

0.000

0.000

0.434 **

0.249 ** -0.014

0.355 ** 0.070 ** -0.009

0.014-0.074 ** -0.075-0.139

0.016

0.000

0.000

0.000

0.507

0.000

0.053

0.034

0.255 ** -0.114 ** 0.196 ** 0.694 ** -0.212 ** 0.153 ** -0.065 ** -0.268 **0.105 ** -0.139 **

0.000

0.008

0.000

0.310 **

0.410 ** -0.077 ** 0.143 ** -0.249 ** -0.048-0.239 ** 0.198 ** -0.085 ** -0.153 **

0.000

0.000

0.000

0.153 **

0.136 ** 0.280 ** -0.292 ** -0.054 ** -0.063 ** -0.023

0.592
0.000
0.000

0.000
0.000
0.000
0.000

0.000
0.001
0.000

0.480
0.000
0.078

0.000

0.056
0.000

0.000

0.143

0.000

0.000

0.000

0.000

-0.269 **0.022-0.127 **

0.000

0.334

0.000

0.000

0.000

Notes: The monthly time-varying price of market risk on 25 size-ranked portfolios is displayed for the sample period from January 1982 to December 2003. The
risk price of the conditional CAPM is calculated using the conditional time-varying covariances from a multivariate GARCH model. The portfolios are formed
following the procedures in Fama and French (1993). At the end of September of each year t (1981-2003), TSE (Tokyo Stock Exchange) First Section stocks areallocated to one of 25 groups based on their September market equity (stock price times shares outstanding). Value-weighted monthly returns on the portfolios arethen calculated from October to the following September. Only firms with ordinary common equity are included. REITs (Real Estate Investment Trusts) and unitsof beneficial interest are excluded. p-values are calculated using White's (1980) heteroskedasticity consistent covariance matrix. ** and * denotes statisticalsignificance at the 1% and 5% level, respectively. Figures in bold denote statistical significance with a theoretically consistent sign.

3) represent 137 of the 264 cases.
Third, Table 4 presents the monthly time-varying price of the HML covariance risk for the same
portfolios and sample period as in Tables 2 and 3. However, the pricing trends are very diﬀerent
from those found for the market and SMB factors. In particular, the monthly time-varying pricetsuji :2009/3/5(2:36)

The time-varying pricing of value and size premia in Japan

March 2009

− 77 −

Table 3Monthly time-varying price of risk for the SMB factor on 25 portfolios formed on the basis of size:
The case of Japan from January 1982 to December 2003
January
1982

February

March

April

May

June

July

AugustSeptember

OctoberNovember December

Risk pricep-value

0.146 ** -0.020-0.059-0.0530.042

0.178 **

0.026

-0.242 ** -0.153 **0.024

0.264 **

0.001

0.120

0.053

0.367

0.003

0.583

0.000

0.000

0.664

0.000

0.584

1983

Risk pricep-value

0.227 ** 0.147 **0.267 **

0.427 ** 0.247 **0.086

0.673 ** 0.163 **0.036

0.198 **

0.174 **

0.363 **

0.000

0.000

0.000

0.238

0.000

1984

Risk pricep-value

0.171

0.044

-0.232 ** 0.155 **1985

Risk pricep-value

1986

Risk pricep-value

0.312 ** 0.470 ** 0.258 **1987

Risk pricep-value

0.103 ** 0.087 **
0.000

0.000

0.137

1988

Risk pricep-value

0.819 **

0.337 ** 0.184 **

0.000

0.000

0.002

0.000

1989

Risk pricep-value

0.604 ** -0.053 ** 0.165 **0.325 ** 0.304 ** -0.069 ** 0.376 ** 0.340 ** 0.471 **0.036

0.284 ** 0.318 **

0.000

0.001

0.003

0.000

0.097

0.000

1990

Risk pricep-value

0.146 *

0.024

-0.390 ** -0.422 ** 0.340 ** 0.076 ** 0.079 ** -0.668 ** -0.668 ** 0.805 ** -0.463 ** -0.015

0.022

0.573

0.000

1991

Risk price -0.217 ** 0.633 ** 0.065 **p-value
0.000
0.000
0.000

1992

Risk price -0.199 ** -0.071 ** -0.529 ** -0.404 ** 0.292 ** -0.382 ** -0.241 ** 0.390 ** -0.102 *p-value
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.027

1993

Risk price -0.165 ** 0.030
p-value
0.000
0.134

1994

Risk pricep-value

1995

Risk price -0.214 ** -0.590 ** -0.232 **p-value
0.000
0.000
0.000

1996

Risk pricep-value

1997

Risk price -0.112 ** -0.001
p-value
0.000
0.965

1998

Risk pricep-value

0.447 ** 0.058 ** -0.040 ** -0.080 ** 0.026 **1999

Risk pricep-value

0.127 ** 0.058 *2000

Risk pricep-value

0.093 ** -0.001

0.185 ** -0.112 ** 0.163 **0.444 ** -0.236 ** 0.200 ** -0.095 ** -0.446 **0.136 ** -0.219 **

0.000

0.000

0.000

0.000

2001

Risk pricep-value

0.116 ** 0.270 ** 0.406 **2002

Risk price -0.045
p-value
0.231

2003

Risk pricep-value

0.000

0.405
0.009

0.584 ** 0.145 *
0.000

0.000

0.616

0.000

0.000

0.326 **

0.007

0.399 **

0.316 ** -0.023

0.234

0.000

0.891

0.000

0.067

0.082

0.004

0.000

0.324

0.123 **

0.093 *

0.382 ** 0.309 **0.193 ** 0.223 ** 0.146 ** 0.140 **0.285 **

0.066

0.000

0.004

0.027

0.000

0.001

0.093

0.018
0.000

0.005

0.000
0.309

0.246 ** 0.238 *

0.000

0.016

0.020

0.003

0.000

0.118 ** 0.165 **0.283 ** -0.048-0.072 ** -0.175 **0.007

0.171 ** -0.062 **

0.004

0.006

0.000

0.000

0.005

0.354

0.000

0.001

0.061

0.072

0.702 ** 0.186 **0.333 ** 0.377 ** 0.091 ** -0.251 *-0.005

0.044

0.001

0.902

0.096

0.000

1.125 **
0.000

0.000

0.000

0.056

0.000

0.334 ** 0.085 ** -0.280 ** -0.027 *-0.057 ** -0.059 **

0.000

0.000

0.000

0.005
0.005

0.000
0.000
0.000

0.000

0.000

0.593 **

0.000

0.000

0.129

0.000

0.043
0.000

0.000

0.000

0.000

0.000

0.014

0.001

-0.210 ** -0.031-0.447 ** 0.330 **

0.135

0.932

0.000

0.000

0.123

0.000

0.000

0.000
0.000

0.036

0.000

0.345 ** 0.041 *

0.000

0.047
0.000
0.189

0.211 ** -0.457 ** -0.015
0.000

0.000

0.110

-0.163 **0.116 **

0.068

0.000

0.000

0.069

1.221 ** 0.561 ** -0.120 ** 0.076 ** 0.061 ** -0.145 ** -0.364 ** -0.647 ** 0.139 **
0.000

0.000

0.668 **

0.069 ** 0.095 **0.192 **

0.188 ** 0.247 ** -0.094 ** -0.085 ** -0.345 **

0.000

0.000

0.000

0.000

0.111

0.000

0.000

0.016

-0.548 ** -0.196 ** 0.678 **0.594 ** -0.049 *0.029

0.504 **

0.542 **

0.214

0.000

0.000

0.000

0.000

0.259

0.000

0.000

0.736 ** -0.021

0.054-0.676 ** -0.044

0.263 ** -0.279 ** -0.213 ** -0.484 **

0.000

0.002

0.002

0.429 ** -0.210 ** 0.390 **
0.000

0.000
0.000

0.000

0.000

0.000
0.020
0.955

0.000
0.000

0.000
0.000

0.000
0.000

0.000
0.025

0.000

0.000

0.043

-0.425 ** 0.195 **

0.000

0.000

0.451

0.161

0.081 *

0.348 **

0.060 ** -0.298 ** -0.218 ** -0.457 ** 0.067 ** -0.293 ** -0.220 **

0.000

0.016

0.000

0.000

0.000

0.707 **

0.628 ** -0.022

0.000

0.000
0.000

0.001
0.621
0.000

0.000

0.000

0.000

0.000
0.000

0.000
0.000

0.105 ** 0.093 ** -0.370 ** -0.246 ** -0.037 *

0.680 ** -0.206 **

0.004

0.000

0.000

0.000

0.028

0.821 ** 0.101 ** -0.012

0.002-0.004-0.148 ** -0.179 ** -0.179 **

0.000

0.923

0.000

0.002
0.000

0.603 ** -0.142 ** 0.303 ** -0.401 **
0.000

0.000

0.000

0.000

-0.212 **

0.003

0.194

0.000

0.000

0.001

0.000

0.000

-0.391 ** 0.314 ** -0.220 ** -0.258 **

0.000

0.000

0.991

0.000

0.000

0.188 **

0.230 ** 0.523 ** -0.526 ** -0.026-0.094 *-0.028-0.623 ** -0.032-0.144 **

0.000

0.000

0.000

0.018

0.187

0.000

0.000

0.367

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.545 **

0.000

0.000

0.663

0.000

0.311

0.000

0.261 ** 0.535 ** 0.106 **0.470 ** 0.541 **0.711 ** 0.057 *0.515 ** 0.283 ** 0.265 ** -0.503 ** 0.264 **

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.044

0.000

0.000

0.000

0.000

Notes: The monthly time-varying price of risk for the SMB factor on 25 size-ranked portfolios is displayed for the sample period from January 1982 to December
2003. The risk price is calculated using the conditional time-varying covariances from a multivariate GARCH model. The portfolios are formed following theprocedures in Fama and French (1993). At the end of September of each year t (1981-2003), TSE (Tokyo Stock Exchange) First Section stocks are allocated to oneof 25 groups based on their September market equity (stock price times shares outstanding). Value-weighted monthly returns on the portfolios are then calculated
from October to the following September. Only firms with ordinary common equity are included. REITs (Real Estate Investment Trusts) and units of beneficial
interest are excluded. p-values are calculated using White's (1980) heteroskedasticity consistent covariance matrix. ** and * denotes statistical significance at the
1% and 5% level, respectively. Figures in bold denote statistical significance with a theoretically consistent sign.

of HML covariance risk is less statistically significant than either the market or SMB factors, andthe number of statistically significant risk prices with positive signs (the bold figures in Table 4) isequal to only 86 of the 264 cases.
The similarities and diﬀerences in the pricing of three types of risk are well presented in Figure
2. This figure provides the degree of year-by-year pricing of the three factors. The yearly values

tsuji :2009/3/5(2:36)

OSAKA ECONOMIC PAPERS− 78 −

Vol.58 No.4

Table 4Monthly time-varying price of risk for the HML factor on 25 portfolios formed on the basis of size:
The case of Japan from January 1982 to December 2003
January

February

March

April

May

June

July

Risk pricep-value

0.745 **

0.187

0.313

-0.709 **

0.005

0.361

-0.224

0.009

0.615

0.184

0.001

0.972

0.456

0.260

1983

Risk pricep-value

1.051 **

0.370

0.508

2.297 **

1.770 **

0.189

1.319 ** -0.515 **

0.308

0.438

0.000

0.000

0.544

0.007

1984

Risk price -0.341
p-value
0.670-0.506 *-0.2640.104

1.734 ** -0.658 ** 0.567 *

0.548

0.448

1985

Risk price -0.455 ** -0.328 *p-value
0.000
0.032

-0.269 ** -0.034
0.005

1986

Risk price -0.147
p-value
0.143-0.446

0.030

0.098

0.908

1982

1987
1988

0.001

0.031

Risk price -0.471 ** -0.433 ** -0.327 *p-value
0.000
0.000
0.041
Risk pricep-value

0.006

0.000

September

OctoberNovember December-0.712 ** -0.389 **

0.470

0.231

0.001

0.334

0.059

0.402

0.751

0.042

-0.143

0.159

0.454

0.434

0.526

0.069

0.001

0.720

0.460

-0.009-0.059-0.738 ** -0.455 ** -0.023

0.965

0.507

0.000

0.001

0.834

-1.137 ** 0.309 ** 0.332 **0.019-0.083-0.0430.017-0.085

0.772

0.000

0.000

0.001

0.730

0.144

0.615

0.690

0.084

0.354

-0.792 ** -0.036-0.0660.068-0.314 *-0.062-0.777 **0.241 **

0.121

0.000

0.241

0.505

0.037

0.231

0.005

0.000

0.928

0.022

August-0.545 ** -1.733

0.803 ** 1.104 ** 0.621 **0.296 ** -0.127-0.123 **

0.182

0.000

0.057

0.002

0.003

0.000

0.002

0.724

0.002

0.071

0.720

-0.975 **

0.126

0.366

0.963

-0.112-0.7580.148 *

0.157

0.037

-1.072-0.106

0.169

0.000

0.443

0.546

0.067

0.769

0.063

0.048

0.254

0.900

0.060

0.599

1.376 **

1.868 **

1.496 ** -0.163

1.133 ** 1.212 **2.408 **

0.498

1.649 ** -0.619

0.000

0.000

0.000

0.000

0.081

0.000

-0.0081.295 ** 1.236 ** -1.292 ** 0.294 ** -0.008

1989

Risk price -1.720
p-value
0.067-0.223 *
0.020

0.000

0.000

1990

Risk pricep-value

0.396

0.665 **

1.256 *

0.207 ** -0.790 ** -0.083 *

0.150

0.000

0.016

0.010

0.000

0.020

0.817

0.000

1991

Risk pricep-value

0.207 ** -1.015 ** -0.117 ** -0.021

0.003

0.555 **

0.051

1.240 ** -1.084 ** -0.446 ** 1.150 **0.068 *

0.000

0.836

0.000

0.116

0.000

0.025

1992

Risk pricep-value

0.728 ** 0.307 **1993

Risk pricep-value

0.287 ** -0.008-6.054 ** 2.404 **1.134 ** -0.757 ** 0.526 ** 0.338 ** -0.561 ** -1.401 ** -1.452-0.837 *

0.000

0.000

0.000

1994

Risk price -1.972 *p-value
0.021

0.000

0.000
0.000

0.000

3.025 **

1.045 ** -0.596 ** 0.618 **0.341 ** -0.819 ** 0.206 **0.305 ** -0.303 ** -0.076

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.451

0.000

0.000

0.000

0.000

0.517

0.022

-0.489 ** -0.872 ** -0.071-0.067-0.572 ** -2.043 **

0.350-3.418 **

1.217

0.000

0.127

0.000

0.000

0.542

0.000

0.000

0.506

0.000

0.056

1995

Risk price -2.296 ** -1.968 ** -0.483
p-value
0.005
0.000
0.171

-0.008-2.391 ** 1.236 **3.628 ** -2.177 **

0.043

0.028

-2.490 ** 2.626 **

0.952

0.471

0.809

0.000

1996

Risk pricep-value

1997

Risk price -0.961 ** -0.021
p-value
0.000
0.912

1998

Risk pricep-value

1999
2000

Risk pricep-value

0.811

0.002

0.001

0.000

1.198 ** -0.700 ** 1.089 ** 2.302 ** -0.094

0.168-2.178 ** -0.191

0.988 ** -1.079 ** -0.578 ** -2.275 **

0.000

0.110

0.000

0.000

0.000

0.000

0.000

0.562

0.191

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.061

-0.138 ** -0.148

2001

0.923

0.053

0.000

0.000

0.000

0.186

0.000

0.128

0.000

0.000

0.000

-1.453 **0.239

3.676 ** -0.300 **1.112 **

2.379 ** -3.277 ** -0.222-0.149-0.727 **

0.000

0.699

0.000

0.000

0.000

0.891

0.000

0.002

0.000

0.521

1.319 **

0.225 ** -0.110 ** -0.304 ** 0.107 ** 0.283 ** 0.364 ** -1.299 ** -1.108 ** -0.197

2.933 ** -1.095

0.000

0.000

0.000

0.000

0.001

0.001

0.000

0.000

0.000

0.408

0.003

Risk price -0.298
p-value
0.496

0.328

2.558

-1.605 *0.018

1.282 **

0.145

-0.165-0.092-0.642 ** -0.439 ** -0.335 **

0.280

0.035

0.905

0.003

0.744

0.174

0.703

0.000

0.532

0.000

0.071
0.000

0.153 ** -0.047

0.233 ** -0.179 ** 0.353 **0.697 ** -0.308 ** 0.242 ** -0.151 ** -0.626 ** -0.110

0.080

0.000

0.145

0.000

0.001

0.000

0.003

0.594

0.782

Risk price -0.334 ** -0.203
p-value
0.000
0.397

0.074

-0.266-0.221 ** 0.509 ** -0.704 ** 0.580 **0.479 ** -0.076

0.283 *

-0.393 **

0.837

0.696

0.000

0.000

0.005

0.026

0.006

0.547 ** -0.115-0.475-1.189 ** 0.783 ** 0.471 ** 0.295 *1.547 ** -0.294

0.167

0.237

0.000

0.299

2002

Risk pricep-value

2003

Risk pricep-value

0.365 *

0.752

0.529

0.000
0.000

0.001
0.001

0.000

0.006
0.830

0.018

0.146

0.007

0.004

0.000

0.000

0.022

0.063

-0.0620.430 ** 1.396 ** 2.347 **2.772 **

0.014

0.672 **

0.666 ** 0.507 ** -1.289 ** 0.593 **

0.613

0.866

0.000

0.000

0.838

0.000

0.000

0.001

0.000

0.004

0.125
0.000

0.000

Notes: The monthly time-varying price of risk for the HML factor on 25 size-ranked portfolios is displayed for the sample period from January 1982 to December
2003. The risk price is calculated using the conditional time-varying covariances from a multivariate GARCH model. The portfolios are formed following theprocedures in Fama and French (1993). At the end of September of each year t (1981-2003), TSE (Tokyo Stock Exchange) First Section stocks are allocated to oneof 25 groups based on their September market equity (stock price times shares outstanding). Value-weighted monthly returns on the portfolios are then calculated
from October to the following September. Only firms with ordinary common equity are included. REITs (Real Estate Investment Trusts) and units of beneficial
interest are excluded. p-values are calculated using White's (1980) heteroskedasticity consistent covariance matrix. ** and * denotes statistical significance at the
1% and 5% level, respectively. Figures in bold denote statistical significance with a theoretically consistent sign.

are averages of the monthly values of the risk prices. Figure 2 suggests that the pattern of the timevarying pricing of the market and SMB factors is similar because both are priced at a similar time
point and with comparable signs. However, the magnitude of pricing for SMB is a little larger thanthe market factor. While Figure 2 also shows that the pattern of the time-varying pricing between

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The time-varying pricing of value and size premia in Japan− 79 −

Figure 2
Yearly average of the time-varying price of risk for the market, SMB, and HML factors on 25 portfolios
formed on the basis of size: The case of Japan from January 1982 to December 2003

HML and either the market factor or SMB diﬀers, both the timing and magnitude of pricing are very
diﬀerent between HML and either the market or SMB. It is also interesting that the results in Tables
2, 3, 4 suggest that from a risk pricing viewpoint, the conditional CAPM generally demonstrates
better performance than that suggested by Chen et al. (1986), Hamao (1988), and Fama and French(2006), amongst others.
Next, we further compare the pricing of each risk by implementing the panel data analysis. Table
5 presents the results. First describing the results for market risk, this is shown to be well priced
in all periods except for Model 2 in the sub period January 1987 to December 1992 (Panel C).
Second, for the SMB covariance risk, this is also well priced in all periods except for Models 2
and 4 in the sub period January 1993 to December 1998 (Panel D). Thus, we appreciate that themarket and SMB covariance risks are also well priced in the panel data analysis. Third, regarding
the HML covariance risk, this is also well priced except for Models 3 and 4 in the sub period fromJanuary 1982 to December 1986 (Panel B) and Models 3 and 4 in the sub period from January 1999
to December 2003 (Panel E). Thus, we can again see that the pricing of HML is diﬀerent from thetsuji :2009/3/5(2:36)

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Table 5
Panel data analysis of the time-varying price of risk on 25 portfolios
formed on the basis of size: The case of Japan from January 1982 to December 2003
Market risk pricep-value

Risk price of SMB

p-value

0.0306 **0.00000.0119 *0.03320.0270 **0.00030.0273 **0.00030.1005 **0.00000.0823 **0.0000-0.0137

0.1315-0.0404 **0.00020.0379 **0.00130.0720 **0.0000

Risk price of HML

p-value

0.0972 **0.00000.0879 **0.0000-0.0018

0.9264-0.0082

0.6729Panel A From January 1982 to December 2003
Model 10.0361 **0.0000Model 20.0209 **0.0000Model 30.0419 **0.0000Model 40.0354 **0.0000Panel B From January 1982 to December 1986
Model 10.1017 **0.0000Model 20.0884 **0.0000Model 30.1016 **0.0000Model 40.0879 **0.0000Panel C From January 1987 to December 1992
Model 10.0142 **0.0017

Model 2-0.0258 **

0.0002Model 30.0622 **0.0000Model 40.0254 **0.00270.2411 **0.00000.2205 **0.00000.0727 **0.00700.1407 **0.0000Panel D From January 1993 to December 1998
Model 10.0276 **0.0000Model 20.0364 **0.0000Model 30.0223 **0.0001Model 40.0432 **0.0000Panel E From January 1999 to December 2003
Model 1

0.0696 **0.0000Model 20.0496 **0.0000Model 30.0698 **0.0000Model 40.0375 **0.0001-0.00240.8989-0.0818 **0.0016

Notes: Four asset pricing models are evaluated using the panel data analysis. ** and * attached to the coefficients denotes the values are
significant at the 1% and 5% level, respectively. Model 1 is E[(ri,t rf,t) | t 1] = m,t Cov[ri,t, rm,t | t 1]; Model 2 is E[(ri,t rf,t) | t 1] = m,t Cov[ri,t,
rm,t |

t 1

]+

smb,t

Cov[ri,t, rsmb,t |t 1]; Model 3 is E[(ri,t rf,t) |

t 1

]=

m,t

Cov[ri,t, rm,t |

t 1

]+

hml,t

Cov[ri,t, rhml,t |t 1]; Model 4 is E[(ri,t rf,t) |]t 1= m,t Cov[ri,t, rm,t | t 1] + smb,t Cov[ri,t, rsmb,t | t 1] + hml,t Cov[ri,t, rhml,t | t 1]. The periods tested are the whole sample period of January 1982 to
December 2003 and four sub-sample periods. All conditional time-varying covariances used in the tests are derived from the multivariate GARCH
model. Figures in bold denote that the values are statistically significant with theoretically consistent signs.

market factor and SMB in the panel data analysis. However, HML covariance risk is as generally
well priced as the market and SMB covariance risks in the ICAPM framework. On the basis ofthese pricing results, we believe that SMB and HML mimic the state variables in Merton’s (1973)
ICAPM as suggested by Fama and French (1996).
6 Comparisons of the conditional asset pricing models
In this section, we compare and more formally evaluate the empirical performance of the four
conditional asset pricing models: namely, (5), (8), (9) and (10). By doing so, we attempt to judge thepricing degree of the SMB and HML covariance risks more formally in the context of the ICAPM.
More specifically, we perform F-tests using the F-test statistic in (12). We compare the fourtsuji :2009/3/5(2:36)

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The time-varying pricing of value and size premia in Japan− 81 −

models with five tests as follows: Test 1: Model 1 versus Model 2 (null hypothesis H0 is δ smb,t = 0),
Test 2: Model 1 versus Model 3 (null hypothesis H0 is δhml,t = 0), Test 3: Model 2 versus Model 4(null hypothesis H0 is δhml,t = 0), Test 4: Model 3 versus Model 4 (null hypothesis H0 is δ smb,t = 0),
and Test 5: Model 1 versus Model 4 (null hypothesis H0 is δ smb,t = 0 and δhml,t = 0). Since these arenested tests, under the null hypothesis the F-statistic (12) follows the F-distribution with degrees
of freedom r and nT − k (Maddala, 1992).
F=(RRS S − URS S )/r,URS S /(nT − k)(12)where RRS S is the sum of squared residuals of the restricted model, URS S is the sum of squared
residuals of the unrestricted model, r is the number of restrictions, n is the number of samples
in each month, T is the number of months in the test period, and k is the number of explanatory
variables in the unrestricted model.
We now describe the results shown in Table 6 in order. First, in Test 1 Model 1 is not rejectedagainst Model 2 except for the sub period from January 1987 to December 1992 (Panel C). Second,in Test 2 Model 1 is rejected against Model 3 for the whole sample period (Panel A) and for the
sub period from January 1987 to December 1992 (Panel C). The results for Tests 1 and 2 imply that
HML adds explanatory power to Model 1, while SMB does not add explanatory power to Model 1.
Third, in Test 3 Model 2 is rejected against Model 4 for the whole sample period (Panel A) andfor the sub period from January 1987 to December 1992 (Panel C). Fourth, in Test 4 Model 3 is
not rejected against Model 4 except for the sub period from January 1987 to December 1992 (Panel
C). The results in Tests 3 and 4 are similar to Tests 1 and 2: namely, Tests 3 and 4 indicate that
HML adds explanatory power to Model 2, while SMB does not add explanatory power to Model 3.
However, taking into consideration the strong monthly pricing of SMB shown in Table 3, we argue
that the weak contribution of SMB in Models 2 and 4 is not because of the weak pricing of SMB
itself, rather the similar pricing pattern of the market factor and SMB first shown in Figure 2. On
the other hand, because the pricing pattern diﬀers for market and HML, as also shown in Figure 2,
we consider that HML contributes well to Models 3 and 4.
Finally, in Test 5, Model 1 is rejected against Model 4 except for the sub period from January
1982 to December 1986 in Panel B. Therefore, we consider the conditional Fama–French model,
constructed by incorporating the time-varying risk of SMB and HML in Merton’s (1973) ICAPM, isa better pricing model than the simple conditional CAPM. Therefore, as the above evidence demonstrates, SMB and HML are priced and contribute to improvements in the ICAPM model. Consequently, SMB and HML mimic state variables in Merton’s (1973) ICAPM as Fama and French(1996) insisted.

p-value

0.0871

0.98100.00031.00000.99640.00060.28404.4966 **0.00031.00000.99980.00001.00000.0000H0:hml,t= 0 (Model 2 holds)

0.37870.71893.7104 **0.00682.6273 **F-statistic

0.99780.84240.00001.00000.0000p-valuep-value

0.77540.52511.5689 *0.50040.17950.77710.97440.03650.98151.0000

smb,t

hml,t

1.5520 **1.6143 **12.2040 **1.00140.00870.00450.00000.47200.0000p-value= 0 (Model 1 holds)

8.0795 **F-statistic= 0 andH0:

F-statistic= 0 (Model 3 holds)

H0:
smb,t

Test 5Model 1 versus Model 4

Test 4Model 3 versus Model 4

t 1

]=

E[(ri,t rf,t) |

m,t

]+

hml,t

t 1

]+

Cov[ri,t, rhml,t |Cov[ri,t, rm,t |

]=
t 1

t 1

]=
t 1

Cov[ri,t, rm,t |m,t0 in Model 4: E[(ri,t rf,t) |Cov[ri,t, rsmb,t |
t 1

]+
hml,t

Cov[ri,t, rhml,t |t 1]. When H0 is rejected, Model 4 is supported against Model 3. In Test 5, the null hypothesis is H0:

smb,t

smb,t

].
smb,t

t 1

= 0 andhml,t=

0 in Model 4:]+
t 1

Cov[ri,t, rhml,t |hml,tCov[ri,t, rm,t |m,t]+]. When H0 is rejected, Model 4 is supported against Model 2. In Test 4, the null hypothesis is H0: smb,t = 0 and the alternative hypothesis is H1:

smb,t

t 1

hml,t

0 in Model 3: E[(ri,t rf,t) |0, and the alternative hypothesis is H1: smb,t 0 and hml,t 0 in Model 4: E[(ri,t rf,t) | t 1] = m,t Cov[ri,t, rm,t | t 1] + smb,t Cov[ri,t, rsmb,t | t 1] + hml,t Cov[ri,t, rhml,t | t 1]. When H0 is rejected, Model 4 is supported
against Model 1. The tests are performed for the entire sample period from January 1982 to December 2003 and for the four sub-sample periods. All conditional time-varying covariances used in the tests arederived from the multivariate GARCH model.

t 1

Cov[ri,t, rsmb,t |hml,t= 0 and the alternative hypothesis is H1:hml,t= 0 and the alternative hypothesis is H1:hml,tWhen H0 is rejected, Model 3 is supported against Model 1. In Test 3, the null hypothesis is H0:

Model 2 is supported against Model 1. In Test 2, the null hypothesis is H0:

Notes: F-tests are performed for model evaluation using the panel data analysis. ** and * attached to the F-statistic denotes that the null hypothesis is rejected at the 1% and 5% level of statistical significance,
respectively. In Test 1, the null hypothesis is H0: smb,t = 0 and the alternative hypothesis is H1: smb,t 0 in Model 2: E[(ri,t rf,t) | t 1] = m,t Cov[ri,t, rm,t | t 1] + smb,t Cov[ri,t, rsmb,t | t 1]. When H0 is rejected,

0.4016

Panel E From January 1999 to December 2003

0.0886

Panel D From January 1993 to December 1998

2.3025 **Panel C From January 1987 to December 1992

0.50253.8724 **p-value

Test 3Model 2 versus Model 4− 82 −

Panel B From January 1982 to December 1986

1.4038= 0 (Model 1 holds)

F-statistichml,tPanel A From January 1982 to December 2003

F-statistic= 0 (Model 1 holds)

H0:

smb,t

H0:

Test 2Model 1 versus Model 3

Test 1Model 1 versus Model 2

Table 6
Panel data F-tests for model evaluation using 25 portfolios formed on the basis of size:
The case of Japan from January 1982 to December 2003

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− 83 −

7 Conclusions
This paper investigated whether Fama–French factors, as suggested by Fama and French (1996),
mimic state variables in Merton’s (1973) ICAPM framework. To test this particular interpretationof the Fama–French factors, we examined the degree of pricing of the market, SMB, and HML
covariance risks for the Japanese stock market in the context of Merton’s (1973) ICAPM, while
taking the time-varying characteristic of risk prices into account. Our conclusions on Fama andFrench’s (1996) suggestion are summarized as follows.
• First, both our monthly and panel data analysis of time-varying risk pricing reveals that the
SMB factor is well priced in the context of Merton’s (1973) ICAPM. This provides supportive
evidence for Fama and French’s (1996) suggestion.
• Second, both our monthly and panel data analysis of time-varying risk pricing also finds thatthe HML factor is generally priced in the framework of Merton’s (1973) ICAPM although
the degree of pricing is a little weaker than the SMB factor. This finding also supports Fama
and French’s (1996) suggestion.
• Third, from the viewpoint of model evaluation using formal F-tests, the conditional CAPM
is rejected against the conditional Fama–French model. This suggests that the conditional
Fama–French model is superior to the conditional CAPM. This finding again supports Fama
and French’s (1996) suggestion.
In addition, in the process of our analysis, and as an ancillary result, we find that the conditional covariance market risk in the CAPM derived with a multivariate GARCH model is generally
strongly priced in Japan. This finding is interesting because in many studies the simple CAPM isvery often refuted.
Furthermore, from a technical point of view, our paper has several noteworthy characteristics.
These include: (1) a detailed examination of risk pricing and model evaluation; (2) the incorporationof both time-varying risks and the time-varying price of risk in the ICAPM; (3) consideration ofboth time-series and cross-sectional aspects in panel data analysis when evaluating the ICAPM;(4) the implementation of more direct tests of the eﬀectiveness of the time-varying covariance risk
using a multivariate GARCH model.
We consider that these technically advanced aspects of our approach enable more direct anddetailed tests of Fama and French’s (1996) suggestion. As a result, we obtain evidence in Japanthat supports Fama and French’s (1996) assertion that SMB and HML mimic the state variables in
Merton’s (1973) ICAPM. Nevertheless, international research along similar lines of inquiry is stillrequired.

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Acknowledgements
The authors acknowledge the generous financial assistance of the Japan Society for the Promotionof Science and the Zengin Foundation for Studies on Economics and Finance. We would also liketo thank Jason McQueen, Takao Kobayashi, Giorgio Szego, and Geert Bekaert for providing very
helpful information.

References[1] Abel, A. (1990) Asset prices under habit formation and keeping up with the Joneses, American
Economic Review Papers and Proceedings 80, 38-42.
[2] Ang, A. and Chen, J. (2007) CAPM over the long-run: 1926-2001, Journal of Empirical Finance 14, 1-40.
[3] Audrino, F. and Barone-Adesi, G. (2005) Functional gradient descent for financial time series
with an application to the measurement of market risk, Journal of Banking & Finance 29, 959977.
[4] Bauwens, L., Laurent, S. and Rombouts, J.V.K. (2006) Multivariate GARCH models: A survey,
Journal of Applied Econometrics 21, 79-110.
[5] Beine, M. (2004) Conditional covariances and direct central bank interventions in the foreignexchange markets, Journal of Banking & Finance 28, 1385-1411.
[6] Bollerslev, T., Engle, R.F. and Wooldridge, J.M. (1988) A capital asset pricing model with
time-varying covariances, Journal of Political Economy 96, 116-31.
[7] Campbell, J.Y. (1996) Understanding risk and return Journal of Political Economy 104, 298345.
[8] Campbell, J.Y. and Cochrane, J.H. (1999) By forth of habit: A consumption-based explanationof aggregate stock market behavior, Journal of Political Economy 107, 205-251.
[9] Chen, H. and Pakoš, M. (2006) Habit formation, time-varying risk aversion and cross-sectionof expected returns, Working Paper, University of British Columbia.
[10] Chen, N., Roll, R. and Ross, S. (1986) Economic forces and the stock market, Journal ofBusiness 59, 383-403.
[11] Christiansen, C. (2005) Multivariate term structure models with level and heteroskedasticity
eﬀects, Journal of Banking & Finance 29, 1037-1057.
[12] Cochrane, J. H. (2005) Asset Pricing, Princeton University Press, New Jersey, USA.

tsuji :2009/3/5(2:36)

March 2009

The time-varying pricing of value and size premia in Japan− 85 −[13] Constantinides, G.M. (1990) Habit formation: A resolution of the equity premium puzzle,
Journal of Political Economy 98, 519-543.
[14] Coudert, V. and Gex, M. (2006) Does risk aversion drive financial crises? Testing thepredictive power of empirical indicators, Working Paper, Center d’etudes prospectives etd’informations internationales.
[15] Doukas, J. Hall, P.H. and Lang, L.H.P. (1999) The pricing of currency risk in Japan, Journalof Banking & Finance 23, 1-20.
[16] Engel, C., Frankel, J.A., Froot, K.A. and Rodrigues, A.P. (1995) Tests of conditional meanvariance eﬃciency on the U.S. stock market, Journal of Empirical Finance 2, 3-18.
[17] Engle, R.F. and Kroner, K.F. (1995) Multivariate simultaneous generalized ARCH, Econometric Theory 11, 122-50.
[18] Ewing, B.T. and Malik, F. (2005) Re-examining the asymmetric predictability of conditionalvariances: The role of sudden changes in variance, Journal of Banking & Finance 29, 26552673.
[19] Fama, E. and French, K. (1993) Common risk factors in the returns on stocks and bonds,
Journal of Financial Economics 33, 3-56.
[20] Fama, E. and French, K. (1996) Multifactor explanations of asset pricing anomalies, Journalof Finance 51, 55-84.
[21] Fama, E. and French, K. (2006) The Value Premium and the CAPM, Journal of Finance 61,
2163-2185.
[22] Ferson, W.E. and Harvey, C.R. (1994) Sources of risk and expected returns in global equity
markets, Journal of Banking & Finance 18, 775-803.
[23] Flannery, M.J., Hameed, A.S. and Harjes, R.H. (1997) Asset pricing, time-varying risk premia
and interest rate risk, Journal of Banking & Finance 21, 315-335.
[24] Friend, I. and Westerfield, R. (1981) Risk and capital asset prices, Journal of Banking &Finance 5, 291-315.
[25] Giamouridis, D. and Vrontos, I. D. (2007) Hedge fund portfolio construction: A comparisonof static and dynamic approaches, Journal of Banking & Finance 31, 199-217.
[26] Gibson, R. and Mougeot, N. (2004) The pricing of systematic liquidity risk: Empirical evidence from the US stock market, Journal of Banking & Finance 28, 157-178.
[27] Guo, H. (2006) Time-varying risk premia and the cross section of stock returns, Journal of
Banking & Finance 30, 2087-2107.

tsuji :2009/3/5(2:36)

OSAKA ECONOMIC PAPERS− 86 −

Vol.58 No.4

[28] Hamao, Y. (1988) An empirical examination of the arbitrage pricing theory, Japan and theWorld Economy, 1, 45-61.
[29] Hansson, B. and Hördahl, P. (1998) Testing the conditional CAPM using multivariate
GARCH-M, Applied Financial Economics 8, 377-388.
[30] Harvey, C.R. (1989) Time-varying conditional covariances in tests of asset pricing models,
Journal of Financial Economics 24, 289-317.
[31] Jagannathan, R. and Wang, Z. (1996) The Conditional CAPM and the Cross-Section of Expected Returns, Journal of Finance 51, 3-53.
[32] Leon, A., Nave, J.M. and Rubio, G. (2007) The relationship between risk and expected returnin Europe, Journal of Banking & Finance 31, 495-512.
[33] Lettau, M. and Ludvigson, S. (2001) Resurrecting the (C)CAPM: A Cross-Sectional Test
When Risk Premia Are Time-Varying, Journal of Political Economy 109, 1238-1287.
[34] Lewellen, J. and Nagel, S. (2006) The conditional CAPM does not explain asset-pricing
anomalies, Journal of Financial Economics 82, 289-314.
[35] Li, G. (2007) Time-varying risk aversion and asset price, Journal of Banking & Finance 31,
243-257.
[36] Liew, J. and Vassalou, M. (2000) Can book-to-market, size and momentum be risk factors thatpredict economic growth?, Journal of Financial Economics 57, 221-245.
[37] Lintner, J. (1965) The valuation of risky assets and the selection of risky investments in stock
portfolios and capital budgets, Review of Economics and Statistics 47, 13-37.
[38] Maddala, G.S. (1992) Introduction to econometrics, Prentice-Hall, Inc, New Jersey, USA.
[39] Maio, P. (2007) ICAPM with time-varying risk aversion, Working Paper, Universidade Novade Lisboa.
[40] Merton, R.C. (1973) An intertemporal capital asset pricing model, Econometrica 41, 867-887.
[41] Mossin, J. (1966) Equilibrium in a Capital Asset Market, Econometrica 34, 768-783.
[42] Ng, L. (1991) Tests of the CAPM with time-varying covariances: A multivariate GARCH
approach, Journal of Finance 46, 1507-21.
[43] Petkova, R. (2006) Do the Fama and French Factors Proxy for Innovations in Predictive Variables?, Journal of Finance 61, 581-611.
[44] Petkova, R. and Zhang, L. (2005) Is value riskier than growth?, Journal of Financial Economics 78, 187-202.

tsuji :2009/3/5(2:36)

March 2009

The time-varying pricing of value and size premia in Japan− 87 −[45] Polk, C., Thompson, S. and Vuolteenaho, T. (2006) Cross-sectional forecasts of the equity
premium, Journal of Financial Economics 81, 101-141.
[46] Priestley, R. (1996) The arbitrage pricing theory, macroeconomic and financial factors, andexpectations generating processes, Journal of Banking & Finance 20, 869-890.
[47] Ross, S. A. (1976) The arbitrage theory of capital asset pricing, Journal of Economic Theory
13, 341-360.
[48] Sauer, A. and Murphy, A. (1992) An empirical comparison of alternative models of capital
asset pricing in Germany, Journal of Banking & Finance 16, 183-196.
[49] Sharpe, W. (1964) Capital asset prices: A theory of market equilibrium under conditions ofrisk, Journal of Finance 19, 425-42.
[50] Tai, C. (2004) Can bank be a source of contagion during the 1997 Asian crisis? Journal of
Banking & Finance 28, 399-421.
[51] Tsuji, C. (2008) On the Alphas in Japan, Osaka Economic Papers 57, 143-163.
[52] White, H. (1980) A Heteroskedasticity-Consistent Covariance Matrix Estimator and Direct
Test for Heteroskedasticity, Econometrica 48, 817-838.
[53] Zhou, G. (1994) Analytical GMM tests: asset pricing with time-varying risk premiums, Review of Financial Studies 7, 687-709.