Multi-Factor Asset-Pricing Models Under Markov Regime Switches: Evidence from the Chinese Stock Market

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Chen, Jieting; Kawaguchi, Yuichiro

Article Multi-factor asset-pricing models under Markov regime switches: Evidence from the Chinese stock market

International Journal of Financial Studies

Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel

Suggested Citation: Chen, Jieting; Kawaguchi, Yuichiro (2018) : Multi-factor asset-pricing models under Markov regime switches: Evidence from the Chinese stock market, International Journal of Financial Studies, ISSN 2227-7072, MDPI, Basel, Vol. 6, Iss. 2, pp. 1-19, http://dx.doi.org/10.3390/ijfs6020054

This Version is available at: http://hdl.handle.net/10419/195705

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https://creativecommons.org/licenses/by/4.0/ www.econstor.eu International Journal of Financial Studies

Article Multi-Factor Asset-Pricing Models under Markov Regime Switches: Evidence from the Chinese Stock Market

Jieting Chen 1,* ID and Yuichiro Kawaguchi 2 1 Graduate School of Commerce, Waseda University, Tokyo 169-8050, Japan 2 Graduate School of Business and Finance, Waseda University, Tokyo 169-8050, Japan; [email protected] * Correspondence: [email protected]; Tel.: +81-9099480351

Received: 21 February 2018; Accepted: 16 May 2018; Published: 20 May 2018

Abstract: This paper proposes a Markov regime-switching asset-pricing model and investigates the asymmetric risk-return relationship under different regimes for the Chinese stock market. It was found that the Chinese stock market has two signiﬁcant regimes: a persistent bear market and a bull market. In regime 1, the risk premiums on common risk factors were relatively higher and consistent with the hypothesis that investors require more compensation for taking the same amount of risks in a bear regime when there is a higher risk-aversion level. Moreover, return dispersions among the Fama–French 25 portfolios were captured by the beta patterns from our proposed Markov regime-switching Fama–French three-factor model, implying that a positive risk-return relationship holds in regime 1. On the contrary, in regime 2, when lower risk premiums could be observed, portfolios with a big size or low book-to-market ratio undertook higher risk loadings, implying that the stocks that used to be known as “good” stocks were much riskier in a bull market. Thus, a risk-return relationship followed other patterns in this period.

Keywords: Markov regime-switching; anomaly; Chinese stock market; risk-return relationship

JEL Classiﬁcation: G10; G12; C32; G15

1. Introduction The Modern Portfolio Theory (MPT), first introduced by Markowitz(1952), describes the relationship between risk and expected return statistically using mean-variance optimization. Modern finance theory, therefore, stepped onto a new stage. Based on the framework of MPT, the Capital Asset Pricing Model (CAPM) was proposed to promote the study of asset-pricing and was developed by Treynor(1961, 1962), Sharpe(1964), Lintner(1965a, 1965b), and Mossin(1966). However, as more and more abnormal cross-sectional returns were found to be persistent and were unable to be explained by traditional asset-pricing models, Fama and French(1992, 1993) proposed a three-factor model in which the risk factors, such as MKT(Market), SMB (Small Minus Big), HML (High Minus Low), had an explanatory power on abnormal returns. Many researchers try to use multi-factor models to interpret anomalies in areas such as momentum and investment. By constructing several characteristic-based factors, such as the momentum-based factor UMD (Up Minus Down, Carhart 1997), the profitability-based factor RMW (Robust Minus Weak, Fama and French 2015, 2016), and the investment-based factor AGR (Asset Growth Return, Chen 2017), the multi-factor asset-pricing models are improved and can explain most of the characteristics related to abnormal returns. Although the mean-variance model, CAPM, and the multi-factor model are logically simple and useful in practice, they are static, single-period linear models, which can hardly fit the real world

Int. J. Financial Stud. 2018, 6, 54; doi:10.3390/ijfs6020054 www.mdpi.com/journal/ijfs Int. J. Financial Stud. 2018, 6, 54 2 of 19 risk-return relationship in the long run (Rossi and Timmermann 2010; Cotter and Salvador 2014). Using dynamic programming and stochastic analysis, Merton(1969, 1971) takes the investment opportunity set into consideration and expands the CAPM from single-period to multi-period, establishing the dynamic portfolio theory, known as the Intertemporal Capital Asset Pricing Model (ICAPM). Nevertheless, as Merton(1973) predicted in his ICAPM, the investment opportunities may vary with time, suggesting a conditional risk-return relationship. However, as He et al.(1996), Ferson and Harvey(1999) , and Ghysels(1998) have pointed out, a lot of conditional asset-pricing models fail in their empirical performance and hardly capture the beta risk dynamics. We address this problem by putting multi-factor asset-pricing models under the Markov regime switching (MRS) framework, in which regime switches follow a Markov chain rule, and observations will be continuous in the time horizon. Since an MRS model can catch structure changes in economic dynamics, market structure, or other unexpected shocks in the time-series, an MRS adjusted multi-factor model combines the superiorities of linear and non-linear models. Many studies have attempted this method. Coggi and Manescu(2004) presented a state-dependent version of the Fama–French three-factor model (1993). They found two separate regimes coexisting in the US market, one of which had higher factor loadings regarding the HML factor. Guidolin and Timmermann(2007, 2008) showed that size effect and book-to-market ratio (B/M) effect are strong in a bull state and diminished around a bear regime. Ozoguz(2008) also used a two-state regime-switching model and found a negative relationship between the level of uncertainty and asset valuations. The conditional model is successful in explaining part of the cross-sectional variation in average portfolio returns. Ammann and Verhofen(2009) adopted a Markov Chain Monte Carlo method to modify the Carhart(1997) four-factor model. They found that in a high-variance regime, only value stocks perform well, while in low-variance regimes, market portfolio and momentum stocks have higher returns. Abdymomunov and Morley(2011) investigated the time-variations in CAPM betas for B/M and momentum sorted portfolios under different market volatility regimes, and found that regime switching conditional CAPM can better explain stock returns than an unconditional CAPM. Mulvey and Zhao(2011) explored a non-linear factor model for allocating assets over the short run. Cotter and Salvador(2014) also developed a regime switching multi-factor framework to explore the risk-return relationship. They found that a positive and significant risk-return relationship occurred during periods of low volatility, while a relationship could not be observed during periods of high volatility. Though past studies have tried to combine the multi-factor model with a regime switching framework, none of them have systematically investigated the time-variations in both, risk factors excess returns and risk loadings (known as betas). It is also vital to make comparisons among major asset-pricing models. Last, but not least, few researchers have investigated the time-varying risk-return relationship in the Chinese stock market. As the Chinese economy has had tremendous growth during the last 30 years, it has become one of the biggest markets in the world. Along with its economic growth, the Chinese financial market is also growing fast and has played an important role in the Asian-Pacific area. However, unlike other important stock markets, the Chinese stock market still faces serious problems that have been caused by the dual economic system, restrictions on foreign currency conversion, government interventions, and information asymmetries. Since the Chinese stock market is immature, highly restricted, and less inclusive, it is driven more by liquidity and speculative sentiment, rather than by economic fundamentals. That is why China has spent more than half of its time in a bear market, which has also been confirmed by our results. On the other hand, a bull state occurs from time to time, along with interventions by the government. Every time the government tries to “stabilize” a market or issue new policies, it brings a short period of prosperity dominated by irrational retail investors and speculators. As Brunnermeier et al.(2017) show in their theoretical model, although government intervention helps stabilize such an unregulated market, it also amplifies the effects of policy errors and worsens informational efficiency. Based on their predictions, we expect that the validity of equilibrium models may be distorted during some periods when pricing errors are increased by noise factors. Moreover, since Lin et al.(2012) and Guo et al.(2017) point that Int. J. Financial Stud. 2018, 6, 54 3 of 19

Fama-French factors can proxy the latent risk factors in the Chinese stock market, we focus on the two typical asset-pricing models (i.e., CAPM and Fama–French three-factor model) and put them under Markov regime switches. It was found that the Chinese stock market can be depicted with two regimes in which the multi-factor asset-pricing model deviates from one to the other. Hence, investigations on the features of the Chinese stock market, and the portraits of time-varying risk factors and betas, are of great signiﬁcance to investors. In this paper, we study two typical multi-factor asset-pricing models under the Markov regime switches for the Chinese stock market. We ﬁrst distinguish different market states and examine the time-varying risk factors. Then, we allow risk loadings (i.e., betas) to switch across regimes. The results may shed light on the state-dependent risk-return relationship.

2. Data and Methodology

2.1. Data

2.1.1. Benchmark Portfolios Following the conventions of Fama and French(1996, 2006), we constructed 25 portfolios (denoted as FF25) based on quintile intersections of size and B/M, and take them as the benchmark portfolios. We studied all firms that were listed on the Shanghai Stock Exchange and the Shenzhen Stock Exchange, during the period from July 1995 to March 2015. However, since stock returns from July of year t to June of year t + 1 are matched with the accounting data in December of year t – 1, firms that did not have December fiscal year-end data or 36 months of stock return data were excluded from the dataset. Every year, at the end of June, all stocks were ranked and allocated by their size quintile cutoffs and B/M quintile cutoffs. The size breakpoints of year t were the market capitalizations of each stock by the end of June of year t. The B/M for June of year t is the book value of year-end t – 1 divided by the market value in December of year t – 1. Then, the 25 portfolios were held for a year and the monthly average raw returns for each portfolio were calculated.

2.1.2. Constructing Risk Factors Based on the methodology of Fama and French(1993), at the end of June every year, all the sample stocks were divided into six groups based on their size and B/M value cutoffs. The size breakpoint was the median market capitalization of all stocks at the end of June of year t. The B/M breakpoint for June of year t was the book value of last fiscal year-end in December year t – 1 divided by market equity for December year t – 1. Based on the cutoffs, growth portfolio was composed of the lowest 30% B/M firms, while value group included the highest 30% B/M firms. The remaining 40% of stocks comprised the medium group. Following Equations (1) and (2), value-weighted return difference between the value and growth portfolios within each of the size groups was calculated and averaged. The return difference was a value common risk factor, denoted by HML. We adopted the same approach to calculate SMB. Table1 presents the formation of the common risk factor. The market risk factor was also the excess return on the market portfolio.

Table 1. Formation of common risk factors.

Book-to-Market (B/M) Size Growth Medium Value (30%) (40%) (30%) Small (50%) S/L S/M S/H Big (50%) B/L B/M B/H Note: Size refers to the market capitalization of the stocks in June year t. Book-to-Market (B/M) is the book equity of last ﬁscal year-end in December divided by market value for December, year t – 1. “L”, “M”, and “H” represent the low, medium, and high B/M levels, respectively. All of the stocks were sorted based on size and B/M cutoffs. Six groups were formed, denoted as “S/L”, “S/M”, “S/H”, “B/L”, “B/M”, and “B/H”. Int. J. Financial Stud. 2018, 6, 54 4 of 19

HML = (RS/H + RB/H − RS/L − RB/L) ÷ 2 (1)

SMB = (RS/L + RS/M + RS/H − RB/L − RB/M − RB/H) ÷ 3 (2)

2.2. Methodology and Models

2.2.1. The Framework of a Markov Regime-Switching Model The MRS is a ﬂexible framework that is proﬁcient in handling the variations caused by heterogeneous states of the world. In this paper, we focused on the regime-dependent risk factors and risk loadings in multi-factor models. Following the conventions of Hamilton(1989, 1994), we modeled the regimes as follows. For a regime-switching model, the transition of states is stochastic. However, the switching process followed a Markov chain and was driven by a transition matrix, which controls the probabilities of making a switch from one state to another. Considering that the Chinese stock market has manifested bear and bull markets, which along with business cycles, vary between expansions and recessions, we allowed two states, namely, a bear state and a bull state, in this model. The transition matrix is represented as: " # P 1 − P Π = 11 22 (3) 1 − P11 P22 where Pij is the probability of switching from state i to state j. Denoting ψt−1 as the matrix of available information at time t – 1, the probability of State 1 or 2 is calculated following Equations (4) and (5):

1 − P22 π1 = Pr(S0 = 1|ψ0) = (4) 2 − P11 − P22

1 − P11 π2 = Pr(S0 = 2|ψ0) = (5) 2 − P22 − P11

To estimate a regime-switching model where the states are unknown, we consider f (yt|St = j, Θ) as the likelihood function for state j on a set of parameters (Θ). Then the full log likelihood function of the model is given by: T 2 lnL = ∑ ln ∑( f (yt|St = j, Θ)Pr(St = j)) (6) t=1 j=1 which is a weighted average of the likelihood function in each state, and the weights are the states’ probabilities. Applying Hamilton’s filter, the estimates of probabilities can be acquired by doing an iterative algorithm. Finally, the estimates in the model were obtained by finding the set of parameters that maximized the log likelihood equation. To investigate a time-varying risk-return relationship, we put the multi-factor asset-pricing models under Markov regime switches. We first analyzed time-series variations in risk premiums for each risk factor by using a multivariate MRS model. Then we allowed beta in the multi-factor asset-pricing model to switch under a univariate MRS setting. Two models were adopted in this research. The first model was the CAPM, including the market factor (MKT). The second model was the Fama–French three-factor model, which incorporates the size factor (SMB) and value factor (HML).

2.2.2. CAPM with Markov Switching (MR–CAPM) In the case of a CAPM, the market risk factor was ﬁrst studied to identify two regimes.

λMKT = µm,St + em,t (7) Int. J. Financial Stud. 2018, 6, 54 5 of 19 with: ∼ N 2 em,t 0, σm,St (8)

St = 1, 2 (9) where λMKT is the market risk premium, St is an indicator variable that denotes the possible two states, e is the residual vector that follows the normal distribution, and σ2 is the variance vector at state S . t St t In the second step, the market beta and residual in CAPM were assumed to be regime-dependent.

Ri,t − R f ,t = αi, St + βi, St ·MKTi,t + et (10) ∼ N 2 et 0, σSt (11)

St = 1, 2 (12) where Ri,t − R f ,t is the excess return for portfolio i, αi, St is the unexplained return, βi, St is the risk loading, and MKTi,t is excess return on market portfolio. St is an indicator variable that denotes the possible two states. e is the residual vector that follows the normal distribution and σ2 is the variance t St vector at state St. The model was applied independently for the 25 benchmark portfolios. The matrix of estimates for parameters in the model is reported in section three.

2.2.3. Fama–French Three-Factor Model with Markov Switching (MR-FF3 Model) In a Fama–French three-factor model, three factors (i.e., the market factor (MKT), the size factor (SMB), and the value factor (HML)) are proposed to explain the size anomaly and value anomaly. According to the formation of common risk factors, SMB and HML are history returns on the hedging portfolios (small minus big, high B/M minus low B/M), known as RSMB and RHML. So, if these factors (i.e., SMB and HML) originate from the portfolios/stocks in the market, and the return series emerge from the market, it is sagacious to guess that SMB and HML factors in the Fama–French three-factor model may vary over time and could be non-linear dynamic. Therefore, the risk premiums for the common risk factors can be represented as a matrix λ:

λ = [λMKT, λSMB, λHML] (13)

Since the common risk factors are mimicking portfolios, excess returns on risk factors have no autoregressive terms, following a simple mean-variance MRS model:

λMKT = µm,St + em,t (14)

λSMB = µs,St + es,t (15)

λHML = µh,St + eh,t (16) with: ∼ N 2 em,t 0, σm,St (17) ∼ N 2 es,t 0, σs,St (18) e ∼ N 0, σ2 (19) h,t h,St

St = 1, 2 (20)

Cov(em,t, es,t, eh,t,) = 0 (21) Int. J. Financial Stud. 2018, 6, 54 6 of 19

where St is an indicator variable that denotes the possible two states. em,t, es,t, and eh,t refer to the residual vectors for MKT, SMB, and HML, and follow the normal distribution. σ2 , σ2 , and σ2 are m,St s,St h,St the variance vectors for the three factors at state St. In the second step, betas for three factors and the residual in the Fama–French three-factor model are assumed to be regime-dependent.

Ri,t − R f ,t = αi, St + βi, St ·MKTt + si, St ·SMBt + hi, St ·HMLt + et (22) ∼ N 2 Int. J. Financial Stud. 2018, 6, x FOR PEER REVIEW et 0, σSt 6 of 19 (23)

St = 1, 2 (24) 푅푖,푡 − 푅푓,푡 = 훼푖,푆푡 + 훽푖,푆푡 ∙ 푀퐾푇푡 + 푠푖,푆푡 ∙ 푆푀퐵푡 + ℎ푖,푆푡 ∙ 퐻푀퐿푡 + 휖푡 (22) R − R i α β s h where i,t f ,t is the excess return for portfolio 2 , i, St is the unexplained return, i, St(23, )i , St , i, St 휖푡~푁(0, 휎푆푡) are the risk loadings on three factors, and MKTi, SMBt, HMLt are the three factors. St is an indicator 푆푡 = 1, 2 (24) variable that denotes the possible two states. et is the residual vector that follows the normal where 푅2 − 푅 is the excess return for portfolio i, 훼 is the unexplained return, 훽 , 푠 , ℎ are distribution. σ푖,푡 is the푓,푡 variance vector at state S . 푖,푆푡 푖,푆푡 푖,푆푡 푖,푆푡 St t Similarly,the risk the loadings MR-FF3 on three model factors, will andbe estimated 푀퐾푇푖, 푆푀퐵푡 for, 퐻푀퐿 each푡 are of the the three 25 portfolios. factors. St is an indicator variable that denotes the possible two states. 휖푡 is the residual vector that follows the normal 2 distribution. 휎 is the variance vector at state St. 3. Empirical Results푆푡 Similarly, the MR-FF3 model will be estimated for each of the 25 portfolios. In this section, we present the estimates of multi-factor asset-pricing models under Markov regime3. switches. Empirical First, Results in the market process, we conducted the multivariate MRS model to analyze the regime-dependentIn this section, variations we present among the estimates common of risk multi factors.-factor asset Then,-pricing in the models beta process,under Markov a univariate MRS modelregime is switches. applied First, to investigate in the market variations process, we in conduct risk loadings.ed the multivariate MRS model to analyze the regime-dependent variations among common risk factors. Then, in the beta process, a 3.1. Estimationunivariate of MRS MR-CAPM model is applied to investigate variations in risk loadings.

3.1.1. Risk3.1. FactorEstimation Variations of MR-CAPM In3.1.1. the market Risk Factor process, Variations we ﬁrst analyzed the risk factor variations under Markov switching to determine regimes.In the market Following process, we the first model analyze developedd the risk factor in Section variations 2.2.2 under, we Markov applied switching the Perlin to (2014) Matlab Packagedetermine and regimes. estimated Following the theparameters, model developed as shown in S inection Table 2.2.2,2 and we Figure applied1 , the determining Perlin (2014) the two regimesMatlab as a bear Package and and a bull estimate state,d respectively.the parameters, as shown in Table 2 and Figure 1, determining the two regimes as a bear and a bull state, respectively.

Figure 1.FigureMarket 1. Market excess excess returns return ands and smoothed smoothed regimeregime p probabilitiesrobabilities in a in Markov a Markov Regime Regime Switching Switching CapitalC Assetapital PricingAsset P Modelricing M (MR-CAPM),odel (MR-CAPM and), GDP and Growth. GDP Growth Notes:. Notes: The ﬁgure The figure shows shows the conditional the mean ofconditional market return mean (blueof market line) return andthe (blue smoothed line) and probabilitythe smoothed of probability being either of being a bear either market a bear (red area) or a bullmarket market (red (green area) area),or a bull along market with (green the variationsarea), alongin with GDP thegrowth variations rate in andGDP macroeconomicgrowth rate and events macroeconomic events in the period between July 1995 and March 2015. in the period between July 1995 and March 2015. Figure 1 plots the conditional mean of market return and the smoothed probablilities of regimes 1 and 2 in the sample period. The red area refers to the smoothed probability of regime 1, while the green area implies that of regime 2. At any time point, the sum of probablilities of regimes 1 and 2 should be equal to one. It is shown that regime 1 dominated most of the sample period, inferring that the Chinese stock market has been a bear market for most of the time. Further, if we

Int. J. Financial Stud. 2018, 6, 54 7 of 19

Figure1 plots the conditional mean of market return and the smoothed probablilities of regimes 1 and 2 in the sample period. The red area refers to the smoothed probability of regime 1, while the green area implies that of regime 2. At any time point, the sum of probablilities of regimes 1 and 2 should be equal to one. It is shown that regime 1 dominated most of the sample period, inferring that the Chinese stock market has been a bear market for most of the time. Further, if we compare the period of regime 2 with real world events that occurred in the time horizon, we can find that regime 2 has captured most of the astonishing booms and crises, and ups and downs. The market was inspired before Hong Kong returned to China in 1997. However, as the Asian financial crisis hit the Hong Kong market, the Chinese stock market also came to a bear state. In 2000, the international dot-com bubble occured, and the Chinese stock market experienced a short rise before coming to a long bear state from 2001 to 2006, because the state-owned shares (previously illiquid shares) were reduced and dumped into the market. Then, in 2006, as the government released several policies related to the split-share reform, the market stepped into a bull regime. However, speculations were driven by irrationalities, as it was more or less like a bubble. Along with the snow disaster in southern China and the Wenchuan earthquake, following the overwhelming subprime crisis along with the global economic slowdown, the Chinese stock market was dragged into another bear regime. Table2 shows that the conditional mean of market return is relatively lower in regime 1, but higher in regime 2, aligning with the definitions of regimes 1 and 2. The transition matrix reveals that the probability of switching from bear to bull is 0.03, which is smaller than the probability of switching from bull to bear (0.06). If the current state is regime 1, it is less likely to switch to the bull market, because the probability of keeping the current state is 0.97. Hence, it is natural to find that the expected duration of staying at a bear market is 34.05 months, which is much longer than that of a bull market. Thus, the Chinese market has remained a bear market for most of the time.

Table 2. Parameter estimates of the MR-CAPM.

Regime St = 1 (Bear) St = 2 (Bull)

Transition matrix Π St = 1 (Bear) 0.97 (0.00) 0.06 (0.00) St = 2 (Bull) 0.03 (0.00) 0.94 (0.00) Expected Duration 34.05 18.01 MKT µm −0.005 (0.38) 0.048 (0.00) σm 0.059 (0.00) 0.122 (0.00) Note: This table reports the estimates of parameters in the two-state MRS model developed in Section 2.2.2. The p-values are reported in the parentheses under the corresponding mean. Π is the state transition matrix, which reports the probability of switching from one state to another. The sample period is from July 1995 to March 2015, on a monthly basis.

3.1.2. Risk Loading Variations In the beta process, we allow risk loading for each risk factor to switch across regimes. Figure2 plots the market betas in the MR-CAPM model for the 25 characterized portfolios. The left subﬁgure plots the estimates in regime 1, and the right subﬁgure plots the estimates in regime 2. In both regimes, the market betas are non-zeros. If we compare two typical characterized portfolios, P5 (the 5th portfolio characterized by the highest B/M and the smallest size) and P21 (the 21st portfolio characterized by the lowest B/M and the biggest size), it will help reveal the return dispersions among portfolios. The risk loading of P5 is higher than the loading of P21 in regime 1, but the relation is reversed in regime 2. Thus, we can say that the return dispersion between P5 and P21 is explained by the risk dispersion between P5 and P21 in regime 1. However, the risk explanation no longer holds in regime 2. Thus, in a bear market, a positive risk-return relationship holds, while in a bull market, the trade-off between risk and return follows other patterns. Int. J. Financial Stud. 2018, 6, 54 8 of 19 Int. J. Financial Stud. 2018, 6, x FOR PEER REVIEW 8 of 19

Market Betas in MR‐CAPM for 25 Por olios

1.171

1.066 1.105 1.027 1.201 1.2291.073 0.954 0.953 1.400 1.058 1.064 1.262 1.164 1.082 1.207 0.987 0.813 1.200 1.152 1.017 1.140 1.052 1.000 0.791 1.132 1.072 1.1970.976 1.113 0.952 0.950 1.178 Beta 0.800 1.203 0.959 0.983 1.124 1.070 0.973 Small 0.600 1.120 1.205 1.051 0.400 0.876 1.029 1.000 1.316 1.200 0.200 0.986 0.984 0.000 1.008 Big Low 0.959

Big Regime 1 0.670 High 0.757

Size Quin le Low

Regime 2 Small High

B/M Quin le

FigureFigure 2. 2.Market Market Betas Betas in in MR-CAPM MR‐CAPM for for 25 25 Size-B/M Size‐B/M Portfolios. Portfolios. Notes: Notes: The The three three-dimensional‐dimensional space space showsshows the the market market betas betas in thein the two-regime two‐regime MR-CAPM MR‐CAPM for for 25 Size-B/M25 Size‐B/M portfolios. portfolios. The The left left group group plots estimatesplots estimates of regime of regime 1 and the1 and right the subﬁgureright subfigure plots plots those those of regime of regime 2. The 2. The vertical vertical axis axis denotes denotes risk betas.risk betas. The horizontal The horizontal space space denotes denotes the the 25 25 portfolios. portfolios. The The horizontal horizontal axis denotes the the B/M B/M value value magnitude.magnitude. From From left left to to right, right, the the B/M B/M value of portfolios increases. increases. The The depth depth axis axis refers refers to tosize size magnitude.magnitude.As As depth depth increases, increases, the the size size of portfolios of portfolios increases. increases. The portfolios The portfolios are marked are marked with numbers with fromnumbers 1 to 25. from 1 to 25.

3.2.3.2. Estimation Estimation of of MR-FF3 MR‐FF3 Model Model

3.2.1.3.2.1. Risk Risk Factor Factor Variations Variations ToTo better better understandunderstand how how the the risk risk-return‐return relationship relationship deviates, deviates, we put we the puttraditional the traditional Fama– Fama–FrenchFrench three three-factor‐factor model model in the in framework the framework of Markov of Markov regime regime switches. switches. We first We analyzed ﬁrst analyzed how risk how riskfactors factors vary vary as regime as regime switches. switches. Figure Figure 3 plots3 plots the time the‐series time-series variations variations of risk ofpremiums risk premiums on MKT, on SMB, and HML. The red and green areas denote the smoothed probabilities of a bear market MKT, SMB, and HML. The red and green areas denote the smoothed probabilities of a bear market (regime 1) and a bull market (regime 2), respectively. It was shown that a bear market dominated (regime 1) and a bull market (regime 2), respectively. It was shown that a bear market dominated most of the observation period. According to the estimates in Table 3, if the current state is a bear most of the observation period. According to the estimates in Table3, if the current state is a bear market, the probability of staying at the current state is as high as 0.95. Further, since the probability market, the probability of staying at the current state is as high as 0.95. Further, since the probability of of transmitting from bear to bull is 5%, lower than the probability of transmitting from bull to bear transmitting(11%), it is from more bear likely to bullto be is a 5%, bear lower market, than whose the probability expected ofduration transmitting is as long from as bull 18.89, to bear almost (11%), it istwice more that likely of a to bull be market. a bear market, whose expected duration is as long as 18.89, almost twice that of a bull market.

Int. J. Financial Stud. 2018, 6, 54 9 of 19 Int. J. Financial Stud. 2018, 6, x FOR PEER REVIEW 9 of 19

Figure 3. Risk premium series and smoothed regime probabilities in MR-CAPM. Notes: The figure Figure 3. Risk premium series and smoothed regime probabilities in MR-CAPM. Notes: The ﬁgure shows the conditional means of risk premiums and the smoothed probability of being either a bear shows the conditional means of risk premiums and the smoothed probability of being either a bear market (red area) or a bull market (green area). The observation period is from July 1995 to March market (red area) or a bull market (green area). The observation period is from July 1995 to March 2015. 2015.

TableTable3 provides 3 provides the the estimates estimates of of parameters parameters in in the the MR-FF3 MR-FF3 model. model. Aligning Aligning with with the the ﬁndingsfindings in Chenin ( Chen2017), (2017), in a bear in market a bear market (regime (regime 1), the risk 1), premiumsthe risk premiums of SMB and of SMB HML and was HML slightly was higher. slightly It is becausehigher in. It a bear is because market, in when a bear the market, market when return th ise lowmarket and return the business is low is and under the downturn, business is investors under askdownturn, for higher investors compensation ask for on size-related higher compensation risk and value-related on size-related risk. risk According and value to Cochrane-related ( risk.2005 ), duringAccording bad times, to Cochrane when investors (2005), during value abad little times, bit of when extra investors wealth, “good”value a stockslittle bit that of payextra off wealth, well are wanted“good” by stocks investors that andpay getoff well a higher are wanted price. Otherby investors stocks thatand cannotget a higher provide price. good Other payoffs stocks in that these timescannot are provide“bad” stocks good and payoffs will in have these a lower times areprice “bad” with stocks a higher and expected will have return. a lower Then, price the with return a dispersionshigher expected between return. “bad” Then, and the “good” return stocks dispersions are expected between to “bad” expand. and Therefore,“good” stocks return are dispersionsexpected to expand. Therefore, return dispersions between “small” and “big” stocks and “high B/M” and between “small” and “big” stocks and “high B/M” and “low B/M” stocks are expected to expand, “low B/M” stocks are expected to expand, and thus, risk premiums on SMB and HML are higher in and thus, risk premiums on SMB and HML are higher in a bear market. a bear market. However, in a bull market, when the market return is high, investors ask for lesser compensation. However, in a bull market, when the market return is high, investors ask for lesser This is consistent with the expectation that during the bull market, risk-loving investors that ask for a compensation. This is consistent with the expectation that during the bull market, risk-loving lowerinvestors price that of risk ask for for aa unitlower of price risk areof risk increasing. for a unit Thus, of risk in are regime increasing. 1, as theThus, risk in premiums regime 1, as in the SMB andrisk HML premiums are higher, in SMBthe risk-aversion and HML are levels higher, in the the market risk-aversion are also increasing. levels in the However, market in are regime also 2, followingincreasing. the However, same rule, in the regime risk-aversion 2, following levels the insame the rule, market the arerisk decreasing.-aversion levels in the market are decreasing. Table 3. Parameter estimates of the MR-FF3 model. Table 3. Parameter estimates of the MR-FF3 model. Regime St = 1 (Bear) St = 2 (Bull) Regime St = 1 (Bear) St = 2 (Bull) Transition matrix Π St = 1 (Bear) 0.95 (0.00) 0.11 (0.00) Transition matrix Π St = 1 (Bear) 0.95 (0.00) 0.11 (0.00) St = 2 (Bull) 0.05 (0.00) 0.89 (0.00) St = 2 (Bull) 0.05 (0.00) 0.89 (0.00) Expected Duration 18.89 9.36 MKTExpected Durationµ m −0.00118.89 (0.85) 0.0379.36 (0.02) MKT σmμm −0.0600.001 (0.00)(0.85) 0.037 0.123 (0.02) (0.00) SMB µσs m 0.0600.009 (0.00) (0.01) 0.123 0.008 (0.00) (0.28) σs 0.031 (0.00) 0.057 (0.00) SMB μs 0.009 (0.01) 0.008 (0.28) HML µh 0.004 (0.28) 0.003 (0.65) σs 0.031 (0.00) 0.057 (0.00) σh 0.027 (0.00) 0.003 (0.00) HML μh 0.004 (0.28) 0.003 (0.65) Note: This table reports the estimates of parameters in the two-state MRS model developed in Section 2.2.3. σh 0.027 (0.00) 0.003 (0.00) The p-values are reported in the parentheses under the corresponding mean. Π is the state transition matrix, whichNote: reports This the table probability reports of the switching estimates from of one parameters state to the other. in the The two sample-state period MRS is model from July developed 1995 toMarch in 2015Section on a monthly 2.2.3. The basis. p-values are reported in the parentheses under the corresponding mean. Π is the state transition matrix, which reports the probability of switching from one state to the other. The sample period is from July 1995 to March 2015 on a monthly basis.

Int. J. Financial Stud. 2018, 6, 54 10 of 19

Int. J. Financial Stud. 2018, 6, x FOR PEER REVIEW 10 of 19 3.2.2. Risk Loading Variations 3.2.2.Adopting Risk Loading the same Variations approach as in Section 3.1.2, we allow risk loadings of the MR-FF3 model to vary acrossAdopting regimes. the same Figures approach4–6 plotings as in Section on MKT, 3.1.2, SMB, we andallow HML risk inloadings the MR-FF3 of the modelMR‐FF3 for model the 25 characterizedto vary across portfolios, regimes. respectivelyFigures 4–6 plot the riskthe risk load. loadings on MKT, SMB, and HML in the MR‐FF3 modelIn either for the ﬁgure, 25 characterized the subﬁgures portfolios, plot estimates respectively. for regime 1 and regime 2, respectively. It is shown that inIn regime either 1, figure, risk loadings the subfigures on SMB plot and estimates HML have for typicalregime patterns.1 and regime In regime 2, respectively. 1, betas of It SMBis factorshown increased that in regime as the size1, risk of loadings each portfolio on SMB decreased and HML from have bigtypical to small. patterns. Thus, In regime return 1, dispersions betas of betweenSMB factor big and increased small stocksas the weresize capturedof each portfolio by the SMBdecreased factor. from Meanwhile, big to small. betas Thus, of HML return factor increaseddispersions as B/M between value big of and each small portfolio stocks increased, were captured implying by the that SMB return factor. dispersions Meanwhile, between betas of low andHML high factor B/M increased stocks were as B/M captured value by of HMLeach portfolio factors. Therefore,increased, implying it is in regime that return 1 that dispersions a three-factor asset-pricingbetween low model and high can B/M explain stocks the were expected captured returns by HML on stocks. factors. However Therefore, in it regime is in regime 2, beta 1 loadings that a hadthree a reversed‐factor asset pattern,‐pricing in thatmodel big can size explain and low the B/M expected portfolios returns had on higherstocks. riskHowever loadings. in regime Thus, 2, in a beta loadings had a reversed pattern, in that big size and low B/M portfolios had higher risk bull market, investing on such big and low B/M stocks may undertake higher risks, and a positive loadings. Thus, in a bull market, investing on such big and low B/M stocks may undertake higher risk-return relationship no longer holds. risks, and a positive risk‐return relationship no longer holds.

MKT Betas in MR‐FF3 for 25 Por olios

1.149 0.895 1.600 0.950 0.996 0.878 0.999 1.090 0.996 1.400 1.050 1.016 1.058 1.002 1.025 1.146 1.200 0.991 1.019 0.992 1.066 0.966 1.000 1.027 1.210 0.926 Beta 0.931 0.997 1.534 0.800 0.987 0.942 1.041 1.107 1.189 1.013 0.600 1.068 0.971 Small 0.773 1.042 1.033 0.400 1.075 0.931 1.066 0.200 1.279 1.056 0.990 1.024 0.000 1.027 1.133 0.952 0.981 Low Big 0.965 Regime 1 0.765 0.952 Big High

Low Size Quin le

Regime 2 Small

High

B/M Quin le

FigureFigure 4. 4.Market Market betas betas in in MR-FF3 MR‐FF3 for for 25 25 size-B/M size‐B/M portfolios. Notes: Notes: The The three three-dimensional‐dimensional space space showsshows the the market market betas betas in in the the two-regime two‐regime MR-FF3MR‐FF3 forfor 25 size-B/Msize‐B/M portfolios. portfolios. The The left left group group plots plots estimatesestimates of of regime regime 1 1 and and the the right right subﬁgure subfigure plots those of of regime regime 2. 2. The The vertical vertical axis axis denotes denotes risk risk betas.betas. The The horizontal horizontal space space denotes denotes thethe 2525 portfolios.portfolios. The The horizontal horizontal axis axis denotes denotes the the B/M B/M value value magnitude.magnitude. From From left left to to right, right, the the B/MB/M value of of portfolios portfolios increases. increases. The The depth depth axis axis refers refers to tosize size magnitude.magnitude. As As depth depth increases, increases, the the size size ofof portfoliosportfolios increases.

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SMB Betas in MR‐FF3 for 25 Por olios SMB Betas in MR‐FF3 for 25 Por1.341olios 1.341 1.240 1.019 1.240 1.019 1.119 1.131 0.789 1.400 1.119 0.846 1.131 0.789 1.400 0.919 0.808 0.846 1.200 1.002 0.649 1.054 1.200 1.233 0.919 0.808 1.054 1.002 0.789 0.649 1.000

1.233 0.562 1.000 1.021 0.789 0.800 0.562 0.512 1.080 0.947 1.021 0.672 0.507 1.127 0.512 0.800 0.659 0.463 0.600 1.127 1.080 0.947 0.672 0.498 0.507 0.857 0.463 0.600Beta 0.659 0.696 0.532 0.400 0.498 0.857 0.064 Beta 0.968 0.777 0.696 0.415 0.532 0.2000.400 0.555 0.760 0.064 0.968 0.7770.818 0.415 0.200 Small 0.760 0.000 0.555 0.818 0.443 ‐0.159 Small ‐0.0000.200 0.443 ‐0.159 0.431 ‐0.295 ‐‐0.4000.200 0.449 0.431 ‐0.295 ‐0.302 ‐‐0.6000.400 ‐0.249 0.449 ‐0.302 ‐0.600 Big Small 0.229 0.053 ‐0.249 Low Big Small 0.229 0.053 ‐0.147 ‐0.245 Low ‐0.147 ‐0.245 Regime ‐0.173 Regime1 ‐0.173 High 1 ‐0.318 ‐0.287 High Big‐ 0.318 ‐0.287 Low Size Quin le Big ‐0.439 Low Size Quin le ‐0.439 Regime Regime2 2 High High B/M Quin le B/M Quin le

FigureFigure 5. SMB 5. SMB betas betas in MR-FF3in MR‐FF3 for for 25 25 size-B/M size‐B/M portfolios.portfolios. Notes: Notes: The The three three-dimensional‐dimensional space space Figure 5. SMB betas in MR‐FF3 for 25 size‐B/M portfolios. Notes: The three‐dimensional space showsshows the SMBthe SMB betas betas in thein the two-regime two‐regime MR-FF3 MR‐FF3 forfor 2525 size-B/Msize‐B/M portfolios. portfolios. The The right right group group plots plots shows the SMB betas in the two‐regime MR‐FF3 for 25 size‐B/M portfolios. The right group plots estimates of regime 1 and the left subfigure plots those of regime 2. The vertical axis denotes risk estimatesestimates of regime of regime 1 and 1 and the leftthe subﬁgureleft subfigure plots plots those those of regimeof regime 2. 2. The The vertical vertical axis axis denotes denotes risk risk betas. betas. The horizontal space denotes the 25 portfolios. The horizontal axis denotes the B/M value The horizontalbetas. The spacehorizontal denotes space the denotes 25 portfolios. the 25 portfolios. The horizontal The horizontal axis denotes axis denotes the B/M the value B/M magnitude. value magnitude. From right to left, the B/M value of portfolios increases. The depth axis refers to size Frommagnitude. right to left, From the right B/M to valueleft, the of B/M portfolios value of increases. portfolios Theincreases. depth The axis depth refers axis to refers size magnitude.to size magnitude. As depth increases, the size of portfolios decreases. As depthmagnitude. increases, As depth the size increases, of portfolios the size decreases.of portfolios decreases.

HML Betas in MR‐FF3 for 25 Por olios HML Betas in MR‐FF3 for 25 Por olios

1.735 1.735 1.795 1.795 2.000 ‐0.036 2.000 ‐0.005 ‐0.036 1.500 ‐0.120 ‐0.005 ‐0.698 1.500 ‐0.120 ‐0.220‐0.698 1.000 ‐0.156 ‐0.222 ‐0.534 ‐0.083 0.054‐0.220 1.000 0.426‐ 0.156 ‐0.222 ‐0.534 ‐0.145 0.054 ‐0.468 ‐0.562 0.500 ‐0.083 0.426 Beta Small 0.329 ‐0.468‐0.190 ‐ 0.562 ‐0.164 0.500 0.253 0.417‐0.145 Beta ‐0.173 0.114 ‐0.668 0.000 Small 0.329 ‐0.384 0.671 ‐0.2720.186 ‐0.190 ‐0.164 0.253 0.417 0.1140.336 ‐0.241 0.498 ‐0.173 ‐0.668 0.000 0.671 0.091 ‐0.384 ‐0.389 ‐0.325 0.671 ‐0.2720.186 0.498 ‐0.269 0.336 ‐0.241 ‐0.500 Small 0.091 0.671 ‐1.019‐0.389 ‐0.041 ‐0.325 1.022 ‐ 0.269 0.010 ‐0.500 Small ‐0.013 ‐0.041 1.022 ‐1.019 ‐1.000 ‐0.013 0.135 0.010 Big ‐0.362 ‐1.000 0.233 0.135 ‐1.500 Big 0.233 ‐1.002 ‐0.362 0.624 0.272 ‐1.500 Low 0.624 ‐1.002‐ 0.117 ‐0.468 0.272 ‐0.941 Low ‐0.117 ‐0.468 ‐0.941 Regime 1 Big Regime 1 High Big Size Quin le High Size Quin le Low Low Regime 2 Regime 2 High

High B/M Quin le B/M Quin le

Figure 6. HML betas in MR‐FF3 for 25 size‐B/M portfolios. Notes: The three‐dimensional space Figure 6. HML betas in MR‐FF3 for 25 size‐B/M portfolios. Notes: The three‐dimensional space Figureshows 6. HML the HML betas betas in MR-FF3in the two for‐regime 25 size-B/M MR‐FF3 for portfolios. 25 size‐B/M Notes: portfolios. The three-dimensionalThe right group plots space shows the HML betas in the two‐regime MR‐FF3 for 25 size‐B/M portfolios. The right group plots showsestimates the HML of regime betas in 1 and the two-regimethe left subfigure MR-FF3 plots for those 25 of size-B/M regime 2. portfolios. The vertical The axis right denotes group risk plots estimates of regime 1 and the left subfigure plots those of regime 2. The vertical axis denotes risk estimatesbetas. of The regime horizontal 1 and space the left denotes subﬁgure the 25 plots portfolios. those of The regime horizontal 2. The axis vertical denotes axis the denotes B/M value risk betas. betas. The horizontal space denotes the 25 portfolios. The horizontal axis denotes the B/M value The horizontal space denotes the 25 portfolios. The horizontal axis denotes the B/M value magnitude. From right to left, the B/M value of portfolios increases. The depth axis refers to size magnitude. As depth increases, the size of portfolios decreases. Int. J. Financial Stud. 2018, 6, 54 12 of 19

3.2.3. Robustness Test Using a Hedging Portfolio To compare the performance of unconditional factor models and regime-dependent factor models, we further conducted a time-series regression of excess returns for a hedging portfolio on risk factors. The hedging portfolio is a zero-cost portfolio that has a long position on the 5th portfolio of FF25 portfolios, and a short position on the 21st portfolio of the FF25 portfolios, denoted as a “5–21” portfolio. Because the 5th portfolio was the one that had the smallest size and the highest B/M value, while the 21st portfolio was the one that had the biggest size and the lowest B/M value, return spreads between the two portfolios should be the largest and related with firm characteristics of size and B/M. Table4 reports the estimates of exposures to risk factors for each of the four models: unconditional CAPM, MR-CAPM, unconditional FF3 model, and MR-FF3 model. As is shown in Panel A of Table4, the unconditional CAPM fails to explain the abnormal return of the hedging portfolio because there is a significant intercept and an insignificant market beta. However, when the model is conditioned on regimes, we found that the CAPM had an asymmetric pattern under two regimes, in which regime 1 had a significant market beta. The Fama–French three-factor model was also improved by adjusting under Markov regime switches. The MR-FF3 model explained more unexplained returns and depicted a regime-dependent risk exposure pattern. Statistics in Panel B implied that am MR-FF3 model had the least pricing errors and best performance under different criterions.

Table 4. Time-series regressions of “5–21” portfolio returns on risk factors.

Panel A: Estimates for the Four Models Model Regime Intercept MKT SMB HML Unconditional CAPM 0.011 0.061 (0.038) (0.303) MR-CAPM St = 1 0.010 0.161 (0.006) (0.000) St = 2 0.017 −0.052 (0.223) (0.698) Unconditional FF3 −0.002 −0.004 1.436 1.437 (0.301) (0.868) (0.000) (0.000) MR-FF3 St = 1 −0.004 0.031 1.439 1.235 (0.010) (0.091) (0.000) (0.000) St = 2 0.018 −0.183 1.042 2.105 (0.076) (0.067) (0.000) (0.000) Panel B: Statistics for the Four Models Log Model SSR AIC BIC HQC Likelihood Unconditional 1.470 266.017 −2.228 −2.199 −2.216 CAPM MR-CAPM 1.472 313.237 −2.576 −2.459 −2.529 Unconditional 0.222 489.909 −4.100 −4.042 −4.077 FF3 MR-FF3 0.179 544.315 −4.492 −4.317 −4.421 Note: This table reports the parameter estimates and statistics for the unconditional CAPM, MR-CAPM, unconditional FF3 model, and MR-FF3 model. Excess returns of a “5–21” hedging portfolio are regressed on the risk factor premiums. The sample period is from July 1995 to March 2015 on a monthly basis. p-values are reported in the parentheses under the corresponding estimates. SSR is the sum of squared residuals. AIC refers to Akaike information criterion, while BIC refers to Schwarz criterion, and HQC denotes Hannan–Quinn information criterion.

3.3. Out-of-Sample Analysis on MR-FF3 Model Based on the MR-FF3 model developed in Section 3.2, we conducted an out-of-sample analysis to examine its predictability. For each of the 25 portfolios, the estimation window ranged from July 1995 to March 2015, while the forecasting period ranged from April 2015 to December 2017. The MR-FF3 model was ﬁrst estimated in the expanding window regressions, and then the Markov regime transition Int. J. Financial Stud. 2018, 6, 54 13 of 19 matrix and conditional mean parameters were estimated under a root mean squared prediction error (RMSPE) criterion. !0.5 1 T−1 RMSPE = (R − (R ))2 (25) − ∑ t+1 E t+1 T 1 t=1

E(Rt+1) is the one-step-ahead expected return under regime switching, which is a weighted average of the returns in Regime 1 and Regime 2, where the weights were given by the transition probabilities conditional on the prevailing state at time t, as shown in the following equation: " # h i P(St+1 = 1|St = i) E(Rt+1) = P(St+1 = 1|St = i)·µ1 + P(St+1 = 2|St = i)·µ2 = µ1 µ2 · (26) P(St+1 = 2|St = i)

h i where µ1 µ2 is the mean forecast for each state:

µ1 = Coef ficient Estimates|St = 1·Independent Datat+1 (27)

µ2 = Coef ficient Estimates|St = 2·Independent Datat+1 (28) Here in model MR-FF3:

µ1 = α1 + β1·MKTt+1 + s1·SMBt+1 + h1·HMLt+1 (29)

µ2 = α2 + β2·MKTt+1 + s2·SMBt+1 + h2·HMLt+1 (30) h i where βSt sSt hSt is the coefﬁcient vector that depends on state (St = 1, 2). h i MKTt+1 SMBt+1 HMLt+1 is the realized factor return vector at t + 1, which is the independent data used to predict the expected return. " # P(S = 1|S = i) t+1 t is calculated based on: P(St+1 = 2|St = i)

" # " # " # P 1 − P π P ·π + (1 − P )·(1 − π ) 11 22 · 1 = 11 1 22 1 (31) 1 − P11 P22 1 − π1 (1 − P11)·π1 + P22·(1 − π1)

" # P11 1 − P22 h i0 where is the transition probability matrix and π1 1 − π1 is the ﬁltered 1 − P11 P22 probability vector at t. Therefore, P(St+1 = 1|St = i) = P11·π1 + (1 − P22)·(1 − π1) (32)

P(St+1 = 2|St = i) = (1 − P11)·π1 + P22·(1 − π1) (33) Based on the calculations above, we can estimate the one-step-ahead forecasts. Since the estimation window incorporates all the previous information, it is an expanding window estimation. The forecasting horizon is one step, from t to t + 1, stepping over one month. Figure7 illustrates the methodology adopted in this section. The out-of-sample analysis was conducted for each of the 25 Fama–French portfolios. Figure8 plots the forecasting returns and actual returns for portfolios 1, 5, 21, and 25, respectively. Table5 further reports the performance of the MR-FF3 model in in-sample and out-of-sample ﬁtting. It was noticed that the in-sample RMSE was unusually larger than out-of-sample RMSE for the reason that idiosyncratic variance declined over time. Furthermore, we calculated the arithmetic average of the 25 portfolio estimated returns at each time point during the forecasting window. Figure9 shows the Int. J. Financial Stud. 2018, 6, x FOR PEER REVIEW 14 of 19 Int. J. Financial Stud. 2018, 6, 54 14 of 19

average forecasting returns versus true values. It was shown that the differences between forecasting andInt. J. trueFinancial values Stud. were2018, 6 close, x FOR to PEER zero. REVIEW 14 of 19

Figure 7. One-Step-Ahead forecasting of an MR-FF3 model. Notes: The figure illustrates the expanding estimation window and the forecasting window of an out-of-sample analysis for an MR - FF3 model. FigureFigure 7.7.One-Step-Ahead One-Step-Ahead forecasting forecasting of anof MR-FF3 an MR model.-FF3 m Notes:odel. TheNotes: ﬁgure The illustrates figure illustratesthe expanding the estimationexpanding windowestimation and window the forecasting and the windowforecasting of anwindow out-of-sample of an out analysis-of-sample for ananalysis MR-FF3 for model. an MR- FF3 model.

Figure 8. Out-of-sample forecasting return vs. actual return for each portfolio. Notes: The ﬁgure FiguredepictsFigure 8. Out 8 the. -Outof one-step-ahead,-sample-of-sample forecasting forecasting out-of-sample return return forecastingvs. vs. actualactual returns return (red for for each line) each portfolio and portfolio actual. Notes:. monthly Notes: The returnsThe figure figure depicts(bluedepicts the line), onethe during-onestep-step-ahead the-ahead period, out, out- fromof--ofsample- Aprilsample 2015forecasting forecasting to December returns 2017 (red(red for line theline) 1st, and) and 5th, actual actual 21st, monthly and monthly 25th returns of returns the (blueFama–French( blueline) line, during), during 25 the portfolios theperiod period (P01,from fromP05, April April P21, 2015 2015 P25). to to December December 20172017 for for the the 1 st1,st 5,th 5,th 21, st21, stand, and 25th 25 ofth the of the FamaFama–French–French 25 portfolios 25 portfolios (P01, (P01, P05, P05, P21, P21, P25) P25). .

Int.Int. J. Financial Stud.Stud. 2018,, 6,, 54x FOR PEER REVIEW 1515 ofof 1919

Table 5. Performance of the MR-FF3 in in-sample and out-of-sample fitting. Table 5. Performance of the MR-FF3 in in-sample and out-of-sample ﬁtting. MR-FF3 In-Sample RMSE Out-of-Sample RMSE 1995.07–2015.03 2015.04–2017.12 MR-FF3 In-Sample RMSE Out-of-Sample RMSE Portfolio 1 0.045 0.039 1995.07–2015.03 2015.04–2017.12 Portfolio 5 0.034 0.019 Portfolio 1 0.045 0.039 Portfolio 21 0.027 0.020 Portfolio 5 0.034 0.019 PortfolioPortfolio 2125 0.027 0.020 HedgingPortfolio Portfolio 25 of “5–21” 0.027 0.0200.016 Note: ThisHedging table reports Portfolio the ofperformance “5–21” of the 0.027MR-FF3 model in in-sample 0.016 and out-of-sample Note:fitting This. Results table reportsof portfolio the performance 1, 5, 21, and of the 25 MR-FF3of the FF25 model portfolios in in-sample and and a “ out-of-sample5–21” hedging ﬁtting. portfolio Results are of portfolio 1, 5, 21, and 25 of the FF25 portfolios and a “5–21” hedging portfolio are reported. The in-sample period spansreported from. The July in 1995-sample to March period 2015, spans while from the out-of-sample July 1995 to periodMarch is 2015, from Aprilwhile 2015 the toout December-of-sample 2017, period on a monthlyis from April basis. RMSE2015 to is December the Root Mean 2017, Squared on a monthly Error. basis. RMSE is the Root Mean Squared Error.

0.3

0.2

0.1

-0.1 Average Return Average -0.2

-0.3

-0.4

AvgForecast AvgTrue Diff

FigureFigure 9.9. AverageAverage out-of-sampleout-of-sample forecastingforecasting returnreturn vs.vs. actualactual return.return. Notes: The ﬁgurefigure depictsdepicts thethe one-step-ahead,one-step-ahead, out-of-sampleout-of-sample forecasting forecasting returns returns (denoted (denoted by Avg by Forecast), Avg Forecast) actual, monthly actual monthly returns (denotedreturns (denoted by AvgTrue), by AvgTrue) and their, and differences their differences (denoted (denoted by Diff), by during Diff), theduring period the from period April from 2015 April to December2015 to December 2017 for 2017 theFama–French for the Fama– 25French portfolios. 25 portfolios.

4. Discussion InIn thisthis study,study, wewe foundfound thatthat therethere werewere twotwo significantsignificant regimesregimes (i.e.,(i.e., bearbear andand bull)bull) existingexisting inin thethe ChineseChinese stock market.market. It was shown that the bear market dominateddominated most of the sample period, aligning with the fact that the Chinese stock market is still an emergingemerging marketmarket andand hashas beenbeen facingfacing problemsproblems causedcaused byby dualdual economiceconomic characteristicscharacteristics andand restrictionsrestrictions byby governmentgovernment policies.policies. On the other hand,hand, the so-calledso-called bullbull marketmarket waswas characterizedcharacterized by increasing risk-takersrisk-takers (risk-lovers)(risk-lovers) in thethe market, generating lowerlower riskrisk premiumspremiums onon commoncommon riskrisk factorsfactors (i.e.,(i.e., SMBSMB andand HML).HML). To understand understand how how asset-pricing asset-pricing models models perform perform under under different different regimes, regimes, we we first first adjusted adjust theed CAPMthe CA byPM introducing by introducing the two the regimes. two regimes. It was found It wa ins found regime in 1 that regime a positive 1 that risk-return a positive relationship risk-return wasrelationship persistent wa whens persistent market when beta wasmarket significantly beta was significantly positive. However, positive. in However, a bull market, in a bull when market, there werewhen negative there were betas, negative a trade-off betas, between a trade- riskoff between and return risk had and other return patterns. had other patterns. Furthermore, we proposedproposed a MR-FF3MR-FF3 model to investigate the risk-returnrisk-return trade trade-off-off deviations.deviations. WeWe observedobserved riskrisk factorfactor variationsvariations andand riskrisk loadingloading variationsvariations acrossacross twotwo regimes.regimes. It waswas foundfound in regimeregime 11 thatthat the the factor factor excess excess returns returns were were relatively relatively higher, higher, consistent consistent with with the hypothesisthe hypothesis that athat bear a marketbear market has a higherhas a higher risk-aversion risk-aversion level. Moreover, level. Moreover, investigations investigations of the beta of process the beta implied process that impl returnied dispersionsthat return among dispersions characterized among portfolios characterized came from portfolios risk loading came patterns.from risk Specifically, loading portfolios patterns. thatSpecifically, have higher portfolios returns that endure have greater higher exposure returns to end commonure greater risk factors exposure (i.e., toSMB common and HML). risk Thus, factors a (i.e., SMB and HML). Thus, a multi-factor asset-pricing model worked well in regime 1. However,

Int. J. Financial Stud. 2018, 6, 54 16 of 19 multi-factor asset-pricing model worked well in regime 1. However, in State 2, beta loadings had a reversed pattern, where portfolios characterized with a big size and low B/M had higher risk loadings. Meanwhile, since the risk price in regime 1 was higher than risk price in regime 2, the risk-return relationship did not hold any more in a bull market1. Moreover, we conducted an out-of-sample analysis on the Fama–French three-factor model under Markov regime switches. It was shown that an MR-FF3 model performed well in one-step-ahead forecasting. To sum up, for an investor in the Chinese stock market, variations in risk factors dominated the changes from regime 1 to 2. Though investors do not know the exact current state, they could infer the market state based on information available from newspapers and government policies, especially from the estimates of excess returns on common risk factors. If it is in regime 1, investment strategies based on a three-factor model may be helpful. However, if it is in a bull market, when the multi-factor asset-pricing models deviate, investing on big stocks and low B/M stocks, which were originally regarded as “good” stocks, may be much riskier, because risk loadings on them are higher during this period.

Author Contributions: J.C. and Y.K. conceived and designed the experiments; J.C. performed the experiments, analyzed the data, and wrote the paper. Acknowledgments: We thank anonymous referees for their insightful comments. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A. Proof of Statement 1 According to Campbell et al.(1997), all the exact factor-pricing models allow one to estimate the expected return on a given asset. In case the factors are the excess returns on traded portfolios, risk premiums can be estimated directly from the sample means of the excess returns on the portfolios. The relation between expected return and risk can be written as:

R = β·λ (A1)

R: A vector of asset excess returns. β: A vector of risk loadings. λ: A vector of risk premiums on risk factors. Assume in either State 1 (S1) or State 2 (S2), there is R = β·λ. For certain portfolio, RS1 = RS2 (A2)

There is, βS1 ·λS1 = βS2 ·λS2 (A3)

S1: State 1. S2: State 2. It is known: > RP5 RP21 (A4)

P5: The ﬁfth portfolio of 25 Fama–French size-B/M characterized portfolios, which was characterized with the smallest size and the highest B/M. P21: The 21st portfolio of 25 Fama–French size-B/M characterized portfolios, which was characterized with the biggest size and the lowest B/M. Therefore, there are: S S β 1 − β 1 ·λS1 > 0 (A5) P5 P21 βS2 − βS2 ·λS2 > 0 (A6) P5 P21

1 Proof of this statement is shown in AppendixA. Int. J. Financial Stud. 2018, 6, 54 17 of 19

S S β 1 − β 1 ·λS1 = βS2 − βS2 ·λS2 (A7) P5 P21 P5 P21 Empirical results in the risk premium process showed that:

λS1 > λS2 > 0 (A8)

Based on Equations (A5)–(A8), we have:

βS1 − βS1 > 0 (A9) P5 P21

βS2 − βS2 > 0 (A10) P5 P21 Based on Equations (A7)–(A10), there must be:

βS1 − βS1 < βS2 − βS2 (A11) P5 P21 P5 P21

However, as the empirical results in the beta process in Section 3.2.2 showed, there are:

βS1 > βS2 (A12) P5 P5

βS1 < βS2 (A13) P21 P21 Equations (A12) and (A13) imply that:

βS1 − βS1 > βS2 − βS2 (A14) P5 P21 P5 P21

We get a theoretical inference (Equation (A11)) contradicting our empirical result (Equation (A14)). Therefore, the assumption that the risk-return relationship holds in either state is wrong. As we ﬁnd in regime 1, the risk-return relationship holds well, it is in regime 2 when the market is full of speculations and irrationality that the risk-return relationship no longer holds.

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