CAPM and the Efficacy of Higher Moment CAPM:
Empirical Evidence from Taiwan Stock Market
Pei-Yu Lee S2299364 Supervisor: Dr. L. Dam
EBM866B20: Master Thesis Finance (2014-2015-1) (Repair) February, 7th 2015
Abstract
The thesis examines whether co-skewness and co-kurtosis are priced in the Taiwan stock market over the period from July 2005 to June 2014, using the two-step approach by Fama and MacBeth (1973). All 819 companies listed at the Taiwanese Stock Exchange (TWSE) are sorted into 42 portfolios using standard industrial classification (SIC2). The empirical findings indicate that the inclusion of higher moments into the traditional CAPM does not help to explain the behavior of portfolio returns in the Taiwanese stock market. As a robustness check, the thesis further tests the period before the financial crisis in June 2008. Still, the results also show that the higher moment CAPM is inadequate for Taiwan’s equity market.
1 1. Introduction
The thesis examines the efficacy of the CAPM and the higher moment CAPM in the Taiwanese stock market for a 10-year period from July 2005 to June 2014. The Capital Asset Pricing Model (CAPM) was first developed by Sharpe (1964) and Lintner (1965). In the traditional CAPM, assets returns are assumed to be normally distributed and investors have quadratic preferences, and the model suggests that the systematic risk is the only relevant risk measure when examining stock returns, and there should be a positive trade-off between the systematic risk (covariance) and expected returns. However, many empirical results indicate that the traditional CAPM, which uses only one risk factor to explain the risk-return relationship, is not capable of fully explaining the behavior of asset returns. Furthermore, the assumption that stock returns are normally distributed is flawed. Many empirical findings suggest that financial asset returns are not normally distributed and have a fat-tailed distribution. Richardson and Smith (1993) find that other than the non-normality, the distribution of stock returns also exhibits skewness and excess kurtosis. While some researchers propose to add more factors into the model to remedy the insufficiency of the traditional CAPM, for instance, Fama and French (1993) include firm size and book to market ratio, there are researchers who propound the non-linear form of the CAPM, id est, higher moments. The concept of the higher moment CAPM was initially proposed by Rubinstein (1973). Kraus and Litzenberger (1976) incorporate the effect of skewness into the traditional CAPM, and they find that investors have preference for positive skewness. A negatively skewed distribution would mean a greater chance of extremely bad outcomes, and a risk-averse investor would prefer a positively skewed distribution since it means frequent small negative outcomes and a few extreme gains. Fang and Lai (1997) and Dittmar (2002) provide empirical evidence that kurtosis also needs to be taken into account when examining the risk premium of an asset, besides skewness, and they find investors dislike kurtosis. A risk-averse resent kurtosis since it has a tendency to yield extremes outcomes that is either far above or low below the mean. Investors would expect positive corresponding premium when there is high covariance, high kurtosis, and negative skewness.
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moments, co-skewness1 and co-kurtosis are priced in Taiwan stock market.
Economic growth in emerging markets performs relatively well than developed markets. However, emerging markets are different from developed markets with respect to features such as market efficiency, market structure, and institutional infrastructure. There are plenty of reasons to believe that emerging and developed markets share different sources of risk. Emerging markets exposes to risks such as the incompetence of the exchange rate system, unstable industrial structure, and corruption of the government. Bekaert (1995) states that emerging markets are different from mature markets like the U.S. market because of having a more volatile financial environment, lacking of high-quality regulatory, the limited market size, as well as barriers to international investments. It is not surprising that risk factors will differ across emerging and developed markets. MSCI also documents that the volatility of stock returns in the ten largest emerging markets (based on market capitalization) is higher than the ten largest developed markets for the period 1994-2014. Harvey (1995) points out that distribution of stock returns in emerging markets are commonly observed to be leptokurtic2 and skewed. Harvey and Siddique (1995, 2001, and 2002) propose a series of research regarding the skewness effect on asset pricing and they conclude that the higher moment CAPM are better explained than the traditional mean-variance CAPM in emerging markets. In sum, it is worth investigating whether higher moment CAPM helps explaining the behavior of asset returns in emerging markets.
There are studies examining the efficacy of the higher moment CAPM in emerging markets, for instance, Hasan, Kamil, Mustafa, and Baten (2013) study Bangladesh stock market, Messis, Iatridis, Blanas (2007) examine Athens equity market, and Javid (2009) looks into Pakistani equity market, however, there is no research focusing on the Taiwanese stock market. In addition, compared to the other emerging countries, Taiwan is characterized as a high-tech, high liquidity, and well regulated market. Especially, industrial structure in Taiwan is considerably different from other countries. In the last decade, high-tech industry has become the dominant
1 Skewness is a measure of the asymmetry of the probability distribution. Kurtosis is the degree of peakedness of the probability distribution. Harvey (2002) defines co-skewness as an asset’s contribution to a well-diversified portfolio’s total skewness. In the higher moment CAPM model, co-skewness (co-kurtosis) is the prediction of an asset’s excess return on the market excess return squared (cubed). Literature commonly focuses on co-moments instead of skewness and kurtosis to reflect the statement that investors do not receive compensations for diversifiable skewness and keurtosis in equilibrium.
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role in Taiwan’s economy. According to Taiwan National Statistics, the high-tech product accounts for more than 60 percent of the export during the past ten years. Furthermore, many empirical studies find that the stock returns are not normally distributed in the Taiwan stock market. Thus, it becomes interesting to examine whether higher moments are priced in the Taiwan equity market.
The moments of stock returns are recognized as time-varying in nature, which implies that the relationship between co-moments and asset returns is time-varying as well. The thesis adopts a two-step approach introduced by Fama and Macbeth (1973) to capture the effect of higher moments. In the first stage, the time-series regressions are performed to measure covariance, co-skewness, and co-kurtosis. In the second stage, the covariance, co-skewness, and co-kurtosis obtained from the first stage are used as explanatory variables in the cross-sectional regressions to see whether they are priced. In addition, the thesis study the distribution of asset returns based on sorted portfolios instead of individual firms. When examining whether higher moments are priced in a market, some studies test the model using stock returns from individual companies, yet one of the shortcomings is that they are not able to cover all individuals in the market, hence, the result becomes unrepresentative. Hung, Shackleton, and Xu (2003) propose that it has two advantages to test the model’s predictability using portfolios. One is the factor of interest will be concentrated by grouping stocks into portfolios. Second, portfolio returns are less noisy than individual stock returns and will reduce the unexplained systematic risk in cross-sectional prediction. Hence, in the thesis, the daily stock returns of 819 companies listed at TWSE are sorted into 42 industry portfolios using standard industrial classification (SIC2). The thesis compares validity of the CAPM and the higher moment CAPM in the Taiwan equity market. TWSE index is used as a proxy of market portfolio. The Taiwan government 90-day money market rate is used as a proxy of risk-free assets.
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either co-skewness or co-kurtosis is priced in the Taiwanese stock market.
The paper examines whether the inclusion of skewness and kurtosis into the CAPM is a decent approach when predicting the expected stock returns in emerging markets. Section 2 discusses previous literature. Section 3 provides the methodology. Section 4 describes the data. Section 5 presents the empirical results and robustness check. Section 6 concludes.
2. Literature Review
The capital asset pricing model is one of the most influential theories on asset pricing. The CAPM is developed by Sharpe (1964) and Lintner (1965). The model describes the relationship between risk and expected return of an asset and suggests that the beta coefficient is the only relevant risk measure and there should be a positive relationship between beta and average return. The model predicts that the intercept equals to the risk free rate and the beta coefficient is the excess market return.
Fama and MacBeth (1973) are one of the firsts to examine the validity of the CAPM with data, which are all the stocks listed on New York Stock Exchange during 1935 to 1968. They use a two-step approach to estimate the CAPM. In the first stage, the systematic risk, beta, is estimated using time-series regressions. In the second stage, the cross-sectional regressions are performed by allowing the beta coefficients obtained from the first stage to vary over time to see whether the risks are priced in the market. They find that on average there is a positive trade-off between risk and expected returns. However, there is growing empirical evidence suggesting that the beta of the Sharper - Lintner CAPM does not measure the systematic risk completely. Black, Jensen, and Scholes (1972), Blume and Friend (1973), Fama and MacBeth (1973) consistently find that the intercept of the model is larger than the average risk free rate and the beta coefficient is smaller than the average excess market return compared to that in the Sharper- Lintner model which predicts that the intercept equals to the risk free rate and the beta coefficient is the excess market return.
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ratio minus low). The corresponding coefficients of these two factors can take positive and negative values, which makes the three-factor model a better approach than the CAPM in explaining returns. Carhart (1997) extend the Fama-French three-factor model by including the momentum factor, which can be constructed by subtracting the lagged one month returns of the highest return portfolio from the lowest one. The size (SMB), value (HML), and momentum factor help to capture the pattern of returns in the cross-sectional regressions.
Besides adding more factors into the CAPM model to capture the returns pattern, the empirical findings also suggest to examine the non-linear form of the model. The concept of higher moment CAPM model is initially proposed by Rubinstein (1973). Kraus and Litzenberger (1976) are the first to test the three-moment model using coskewness as a supplement to the systematic risk to explain the relationship between risk and return. They find that the three-moment model did explain the insufficiency of the risk-return relationship which is not explained by the traditional CAPM. Fang and Lai (1997) and Dittmar (2002) extend the analysis by adding the fourth moment, cokurtosis into the model and provide empirical evidence, yet, they fail to find significant relation that cokurtosis has impact on asset pricing.
Hung, Shackleton, and Xu (2003) suggest that when examining the cross-sectional asset returns, it is common practice to test the model’s predictive ability on portfolios instead of individual stocks. They suggest that test the model’s predictive ability using portfolios has two advantages. First, it reduces the estimation error of the stocks’ factor itself. Second, portfolio returns are less noisy than individual stock returns and will reduce the unexplained systematic risk in cross-sectional prediction.
Hwang and Satchell (1999) state that emerging markets are not likely to yield appreciable results in a mean-variance world, hence, adding the coskweness and cokurtosis into the CAPM could help to explain the relationship between risk and return in emerging markets. Hung, Shackleton, and Xu (2003) state that investors concern about portfolio skewness and kurtosis when the market returns are skewed or leptokurtic. They suggest that if investors have preference for portfolio skewness and kurtosis, each stock’s distribution to coskewness and cokurtosis may affect its required return.
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CAPM efficacy in the Athens stock market. They find that investors have a preference for the portfolio when stock prices exhibit positive skewness sign. Regarding the systematic variance and systematic kurtosis, they fail to identify significant relationship between risk and return.
Hasan, Kamil, Mustafa, and Baten (2013) use monthly stock returns from 80 non-financial companies listed in DSE to test the higher moment model. They find insignificant relationships between beta and excess returns both in the CAPM and the higher moment CAPM. However, they find that when including higher moments, adjusted R-square increases. For the period studied, they find co-skewness risk is compensated by the market. They state that the higher moment CAPM performs relatively well when examining the risk-return relationship in emerging market like Bangladesh stock market.
3. Methodology
3.1 Normality Test of Stock Returns
It is well recognized that stock returns do not follow a normal distribution. Many empirical findings indicate that stock returns often have a fat-tail distribution and are more peaked at the mean. Empirical evidence shows that the majority of emerging countries’ stock returns is as well not normally distributed. One prerequisite to examine the higher moment CAPM is that stock returns need to be non-normally distributed.
The thesis checks the normality of stock returns using SKTEST in Stata. SKTEST is a test for normality using skewness and kurtosis. The null hypothesis of normality is rejected only when both statistics are jointly rejected, i.e. joint probability is larger than the critical value from χ 2 distribution.
H0: Stock returns follow the normal distribution
H1: Stock returns do not follow normal distribution
3.2 The Traditional CAPM
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risk, and there should be a positive trade-off between expected returns and systematic risk.
The traditional CAPM model specifies the relationship between risk and expected returns as follows:
𝐸 𝑅𝑖 − 𝑅𝑓 = 𝛽𝑖(𝐸(𝑅𝑚) − 𝑅𝑓) (1)
𝛽𝑖 =𝐶𝑜𝑣(𝑅𝑖,𝑅𝑚)
𝑉𝑎𝑟 (𝑅𝑚 ) (2)
where 𝐸 𝑅𝑖 − 𝑅𝑓 is the risk premium, which stands for the differences between the
expected return of an asset i and a risk free rate, 𝛽𝑖 is the systematic variance of an asset i, 𝐸(𝑅𝑚) is the expected return on the market index
The traditional CAPM has three testable implications:
1. The relationship between the expected return on a security and its risk is linear, i.e. the intercept is zero.
2. 𝛽𝑖 is a complete measure of the risk of security i. No other measure of risk of i appears in equation (1).
3. In a market of risk-averse investors, higher risk should be associated with higher expected return, i.e. the market risk premium is positive.
3.3 Fama and MacBeth CAPM test
Fama and MacBeth (1973) are one of the firsts to examine the validity of the CAPM with data. They adopt a two-step approach to test the CAPM.
First step, the beta coefficient is estimated from the time-series regression of historical excess returns on the market portfolios.
(Time-Series Regression) 𝑅𝑖𝑡 − 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 𝑅𝑚𝑡 − 𝑅𝑓 + 𝜀𝑡 (3)
where 𝑅𝑖𝑡 represents the return on portfolio i at time t, 𝑅𝑓 is the risk-free rate (The
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In the second step, I create a cross-section of average returns of assets and regress average returns on betas. 𝛽𝑖 obtained from the first stage is the regressor now,
and we estimate the risk premium 𝛾𝛽 for the factors.Equation (4) is estimated using
ordinary least squares (OLS).
(Cross-Section Regression) 𝑅 − 𝑅𝑖 = 𝛾𝑓 0+ 𝛾𝛽𝛽𝑖 + 𝜀𝑖 (4) where 𝑅𝑖 is the return on portfolio i, 𝛽𝑖 is the systematic risk of an portfolio i obtained from the first stage, and 𝜀𝑖 is the regression residual, 𝛾0 is the intercept, 𝛾𝛽 is the premium associated with the beta risk.
3.3 The Higher Moment Models
To examine the non-linear form of the capital asset pricing model, Kraus and Litzenberger (1976) extend the CAPM by incorporate the third moment, coskewness into the model. Fang and Lai (1997) and Dittmar (2002) further include the fourth moment, cokurtosis into the model.
Following the inclusion of coskewness and cokurtosis into the traditional CAPM, the higher moment CAPM can be written as:
𝑅𝑖𝑡 − 𝑅𝑓𝑡 = 𝛼𝑖 + 𝛽𝑖 𝑅𝑚𝑡 − 𝑅𝑓𝑡 + 𝛿𝑖 𝑅𝑚𝑡 − 𝑅𝑓𝑡 2+ 𝜅𝑖 𝑅𝑚𝑡 − 𝑅𝑓𝑡 3+ 𝜀𝑖𝑡 (5)
Following the two-step approach by Fama and MacBeth, the first step would be running the time-series regressions on the above equation (5). In the time-series regression, the risk factors, 𝛽𝑖, 𝛿𝑖, and 𝜅𝑖 are assumed to be constant over time.
𝛿𝑖 = 𝐶𝑜𝑣(𝑅𝑖,𝑅𝑚2)
𝐸 𝑅𝑚−𝐸(𝑅𝑚) 3 (6)
𝜅𝑖 = 𝐶𝑜𝑣(𝑅𝑖,𝑅𝑚3)
𝐸 𝑅𝑚−𝐸(𝑅𝑚) 4 (7)
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obtained from the first stage are used as explanatory variables in the cross-sectional regressions on the above equation (8). Equation (8) is estimated using ordinary least squares (OLS).
where 𝛽𝑖, 𝛿𝑖, 𝜅𝑖 stand for covariance, coskewness, and cokurtosis, respectively, 𝛾1,
𝛾2, and 𝛾3 represent the coefficients associated with the covariance, coskewness, and cokurtosis, respectively, and 𝑅 − 𝑅𝑖 is the average excess return of the market 𝑓 portfolio
One way of analyzing the effect is by sorting portfolios based on skewness into HISKEW and LOWSKEW portfolios and checks whether this sort generates significant alpha in a benchmark model, such as the CAPM. The approach I take here, however, is running Fama-Macbeth regressions to check whether the factors are priced.
Hung, Shackleton, and Xu (2003) suggest that when examining the cross-sectional asset returns, it is common practice to test the model’s predictive ability on portfolios instead of individual stocks. Hence, in the thesis, the daily stock returns of 819 companies listed at TWSE are sorted into portfolios using standard industrial classification.
We know that there might be volatility clustering in returns, as is clear from the GARCH literature. However, since OLS will still give us unbiased and consistent estimate of the factor betas, we can still measure the betas using OLS to check whether the factors are priced in the second stage regression - the only caveat is that we have to keep in mind that the standard errors may be biased in judging the significance of the factor price.
3.4 Hypotheses
Following Fang and Lai (1997), the thesis tests the following hypotheses. To test the hypotheses, firstly, the beta coefficients have to be obtained from the time-series regressions. Secondly is the regressing of the estimated beta coefficients against the average excess returns.
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Hypothesis 1 deals with the assumption of the Sharpe-Linter CAPM that there is unrestricted riskless lending and borrowing. However, the joint hypothesis of the EMH and validity of the asset pricing model by Campbell, Lo, and, Mackinlay suggest that the intercept might also indicate a problem with the specification of the model.
Hypothesis 2 : The risk premium is positive and equals to the average excess return of the portfolio
(Second step) 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜+ 𝛾1 𝛽𝑖+ 𝜀𝑖 (9) Hypothesis 3 : The asset returns is symmetrically distributed, while co-skewness is priced in equation (10):
(Second step) 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜 + 𝛾1 𝛽𝑖 + 𝛾2𝛿𝑖+ 𝜀𝑖 (10) Hypothesis 4 : The asset returns is symmetrically distributed, while co- kurtosis is priced in equation (11):
(Second step) 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜 + 𝛾1 𝛽𝑖 + 𝛾3𝜅𝑖 + 𝜀𝑖 (11) Equation (8), (9), (10), and (11) are estimated using ordinary least Squares (OLS).
4. Data
The data are daily stock returns of all 819 companies listed at the Taiwanese Stock Exchange (TWSE) from 2005 to 2014. TWSE index is used as a proxy of market portfolio. The Taiwan government 90-day money market rate is used as a proxy of risk-free assets. The sample period is for the 2346 days in a 10-year period from July 2005 to June 2014. All 819 companies’ daily stock returns are sorted into 42 portfolios using standard industrial classification (SIC2). All the data used in this study are from Datastream. There are plenty of reasons why a company has incomplete stock return history, for instance, bankruptcy, liquidation, privatization, or no dealing after quotation cancellation. In the study, all the missing values of daily stock returns are dropped.
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Table 1. Descriptive Statistics of Industry Portfolio Returns for Taiwan, 2005-2014
Observations Mean S.D. Min Max Skewness Excess Kurtosis P(SKEW) P(KURT) Market Index TWSE index 98532 (0.34) 1.27 (7.38) 6.31 (0.54) 3.64
Portfolios (by sic2)
15 2346 0.04 1.73 (6.66) 5.98 (0.36) 2.02 0.00 0.00 16 2346 0.02 1.75 (7.78) 6.60 (0.30) 2.36 0.00 0.00 17 2346 0.01 1.58 (8.73) 7.72 0.22 5.39 0.00 0.00 20 2346 0.03 1.42 (7.06) 5.84 (0.32) 2.65 0.00 0.00 22 2346 0.00 2.41 (18.42) 16.74 (0.16) 13.49 0.00 0.00 23 2346 0.01 1.53 (7.07) 6.25 (0.36) 2.50 0.00 0.00 25 2346 (0.01) 0.81 (7.21) 6.77 0.84 27.30 0.00 0.00 26 2346 0.02 1.28 (6.22) 6.11 (0.34) 4.39 0.00 0.00 28 2346 0.02 0.91 (4.01) 4.36 (0.20) 2.26 0.00 0.00 29 2346 0.03 1.58 (7.19) 6.74 0.04 3.29 0.48 0.00 30 2346 0.05 1.03 (5.97) 4.99 (0.45) 3.30 0.00 0.00 31 2346 0.01 2.93 (7.26) 7.94 0.25 0.84 0.00 0.00 32 2346 0.02 1.18 (6.25) 5.22 (0.55) 2.91 0.00 0.00 33 2346 0.02 1.24 (8.93) 5.83 (0.67) 4.46 0.00 0.00 34 2346 0.01 0.78 (5.18) 3.11 (0.73) 3.50 0.00 0.00 35 2346 0.03 1.61 (6.56) 5.95 (0.51) 1.92 0.00 0.00 36 2346 0.01 1.24 (6.49) 5.58 (0.68) 3.03 0.00 0.00 37 2346 0.02 0.69 (6.88) 5.55 (0.06) 10.12 0.21 0.00 38 2346 0.02 1.23 (6.39) 5.12 (0.72) 3.31 0.00 0.00 39 2346 0.05 1.29 (7.64) 14.83 1.56 18.04 0.00 0.00 42 2346 0.00 2.11 (6.98) 6.48 (0.10) 1.32 0.05 0.00 44 2346 0.02 1.74 (6.78) 6.63 (0.07) 2.22 0.19 0.00 45 2346 (0.03) 1.95 (6.93) 6.63 0.06 2.48 0.22 0.00 47 2346 0.04 1.70 (11.74) 6.16 0.12 5.34 0.01 0.00 48 2346 0.02 0.83 (4.35) 4.48 0.00 3.33 1.00 0.00 49 2346 0.02 1.57 (20.51) 24.81 0.39 73.75 0.00 0.00 50 2346 0.05 1.30 (6.87) 6.04 (0.35) 3.06 0.00 0.00 51 2346 0.02 1.86 (7.80) 6.59 (0.19) 2.21 0.00 0.00 53 2346 0.04 1.51 (5.84) 5.58 (0.03) 1.89 0.55 0.00 54 2346 0.07 1.84 (7.25) 6.76 0.37 2.51 0.00 0.00 55 2346 0.02 1.37 (6.98) 6.67 (0.16) 5.35 0.00 0.00 57 2346 0.04 2.12 (7.74) 7.24 0.13 2.36 0.01 0.00 58 2346 0.01 2.20 (8.58) 6.67 (0.15) 1.61 0.00 0.00 60 2346 (0.01) 1.55 (6.57) 6.53 (0.19) 2.43 0.00 0.00 61 2346 (0.02) 1.85 (7.26) 6.76 0.06 3.54 0.21 0.00 62 2346 0.02 1.57 (7.20) 6.69 (0.26) 3.34 0.00 0.00 63 2346 0.03 1.36 (7.03) 6.51 (0.19) 5.04 0.00 0.00 65 2346 0.05 1.39 (6.41) 5.89 (0.05) 2.72 0.35 0.00 70 2346 0.02 1.61 (6.66) 5.96 0.10 3.13 0.04 0.00 73 2346 0.06 1.34 (8.78) 6.33 (0.14) 3.95 0.01 0.00 79 2346 0.03 1.96 (7.28) 9.40 0.36 3.04 0.00 0.00 87 2346 0.01 1.56 (9.35) 7.62 (0.11) 2.73 0.03 0.00
The table presents the first four moments of daily stock returns of 42 portfolios and the market return of TWSE market. The sample period is for the 2346 days in the 10-year period from July 2005 to June 2014. The Taiwan government 90-day money market rate is used as a proxy of risk-free assets. 819 companies’ daily stock returns are sorted into 42 portfolios using standard industrial classification (SIC2). From Table1, it can be seen that the minimum portfolio returns ranges from -20.51% to -4.01%, while the maximum portfolio returns ranges from 3.11% to 24.81%. Excess kurtosis can be as high as 73.75, ranging from 0.8438 to 73.75. The skewness ranges from -0.03 to 1.56. In column 7, 14 out of 42 portfolios exhibit positive skewness. Excess kurtosis in column 8 shows that the distribution of daily stock returns of all 42 portfolios is more peaked at the mean. In the last two columns, the SKTEST shows that 34 out of 42 portfolio returns are not normally distributed
Source: Datastream.
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from 3.11% to 24.81%. Excess kurtosis ranges from 0.8438 to 73.75. The skewness ranges from -0.03 to 1.56. In order to examine the higher moment CAPM, stock returns need to be non-normally distributed. The normality test shows that 34 out of 42 portfolio returns are not normally distributed.
5. Empirical Results and Robustness Check
5.1 CAPM and the higher moment CAPM
To compare the single factor CAPM with the higher moment CAPM, first the beta coefficients have to be obtained. The CAPM has econometric restrictions that the intercept term should be zero and there should be a positive relationship between market risk and return.
Table 2 presents the time-series regression results of equation (3) and describes the relationship between beta and return. The beta coefficients of all 42 industry portfolios exhibit significant positive signs. All the beta coefficients are statistically significant at 1% significance level. The betas range from 0.090 to 1.098. The adjusted R2 ranges from to 0.014 to 0.691. The result is consistent with the CAPM that systematic risk is positively related to returns, which indicates that investors in Taiwan receive premiums for taking the extra risk.
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Table 2. Beta Coefficients of Industry Portfolios of Taiwan Stock Market, 2005-2014
SIC2 SIC description α t
β
t Adj. R
2
15 General Building Contractors 0.00380 (14.84)***
0.987
(50.56)*** 0.521 16 Heavy Construction, Except Building 0.00342
(12.73)***
0.957
(46.72)*** 0.482 17 Special Trade Contractors 0.00192
(6.34)***
0.548
(23.68)*** 0.193 20 Food & Kindred Products 0.00283
(12.46)***
0.746
(43.13)*** 0.442 22 Textile Mill Products 0.00123
(2.44)*
0.362
(9.40)*** 0.036 23 Apparel & Other Textile Products 0.00268
(10.51)***
0.759
(38.97)*** 0.393 25 Furniture & Fixtures 0.000238
(1.39)
0.0901
(6.91)*** 0.020 26 Paper & Allied Products 0.00245
(11.79)***
0.655
(41.24)*** 0.420 28 Chemical & Allied Products 0.00216
(18.00)***
0.565
(61.81)*** 0.620 29 Petroleum & Coal Products 0.00267
(9.64)***
0.710
(33.62)*** 0.325 30 rubber & Miscellaneous Plastics Products 0.00242
(15.47)***
0.577
(48.45)*** 0.500 31 Leather & Leather Products 0.00291
(4.98)***
0.840
(18.87)*** 0.132 32 Stone, Clay, & Glass Products 0.00271
(17.30)***
0.727
(60.83)*** 0.612 33 Primary Metal Industries 0.00283
(17.76)***
0.786
(64.69)*** 0.641 34 Fabricated Metal Products 0.00153
(12.37)***
0.412
(43.66)*** 0.448 35 Industrial Machinery & Equipment 0.00344
(14.60)***
0.930
(51.79)*** 0.533 36 Electronic & Other Electric Equipment 0.00284
(19.27)*** 0.814 (72.49)*** 0.691 37 Transportation Equipment 0.00121 (9.88)*** 0.298 (31.78)*** 0.301 38 Instruments & Related Products 0.00271
(15.54)***
0.729
(54.73)*** 0.561 39 Miscellaneous Manufacturing Industries 0.00114
(4.22)***
0.198
(9.59)*** 0.037 42 Trucking & Warehousing 0.00330
(8.99)*** 0.971 (34.69)*** 0.339 44 Water Transportation 0.00347 (12.90)*** 0.951 (46.37)*** 0.478 45 Transportation by Air 0.00259 (7.44)*** 0.853 (32.16)*** 0.306 47 Transportation Services 0.00206 (6.09)*** 0.495 (19.18)*** 0.135 48 Communications 0.00137 (9.26)*** 0.359 (31.80)*** 0.301 49 Electric, Gas, & Sanitary Services 0.000743
(2.23)*
0.147
(5.81)*** 0.014 50 Wholesale Trade - Durable Goods 0.00300
(15.67)***
0.746
(51.19)*** 0.528 51 Wholesale Trade - Nondurable Goods 0.00308
(9.46)***
0.839
(33.85)*** 0.328 53 General Merchandise Stores 0.00293
(11.73)*** 0.756 (39.68)*** 0.402 54 Food Stores 0.00296 (8.43)*** 0.659 (24.62)*** 0.205 55 Automative Dealers & Service Stations 0.00261
(11.72)***
0.699
14
57 Furniture & Homefurnishings Stores 0.00368 (9.91)***
0.960
(33.95)*** 0.329 58 Eating & Drinking Places 0.00379
(10.36)*** 1.098 (39.41)*** 0.398 60 Depository Institutions 0.00331 (17.35)*** 0.997 (68.59)*** 0.667 61 Nondepository Institutions 0.00323 (11.34)*** 1.006 (46.28)*** 0.477 62 Security & Commodity Brokers 0.00365
(19.42)*** 1.029 (71.79)*** 0.687 63 Insurance Carriers 0.00295 (14.91)*** 0.789 (52.28)*** 0.538 65 Real Estate 0.00285 (12.24)*** 0.687 (38.73)*** 0.390 70 Hotels & Other Lodging Places 0.00266
(9.46)*** 0.736 (34.30)*** 0.334 73 Business Services 0.00276 (12.13)*** 0.641 (36.89)*** 0.367 79 Amusement & Recreation Services 0.00220
(5.65)***
0.565
(19.05)*** 0.134 87 Engineering & Management Services 0.00318
(14.37)***
0.917
(54.39)*** 0.558 Note: t statistics in parenthesis.
*, **, *** indicate significance level at 1%, 5%, 10%, respectively
The table presents the time-series regression results of equation (3) 𝐸 𝑅𝑖 − 𝑅𝑓= 𝛽𝑖(𝐸(𝑅𝑚) − 𝑅𝑓). The beta
coefficients of all 42 portfolios exhibit significantly positive signs. The Adjusted R-squared ranges from 0.02 to 0.69. The intercepts of 42 portfolios are significantly not different from zero
Graph 1 presents the actual and predicted average excess returns for 42 industry portfolios for the traditional CAPM.
GRAPH 1. Actual and Predicted Returns of 42 Industry Portfolios
15
Table 3. Beta, Coskewness, and Cokurtosis of Taiwan stock market, 2005-2014
SIC2 SIC description α β Skewness Kurtosis Adj. R2 15 General Building Contractors 0.00381 (13.79)*** 0.971 (35.28)*** -0.228 (-0.31) 11.10 (0.60) 0.521 16 16 Heavy Construction, Except Building 0.00346 (11.93)*** 0.890 (30.89)*** -0.744 (-0.96) 48.46 (2.51)* 0.484 17 Special Trade Contractors 0.00176 (5.37)*** 0.489 (15.02)*** 0.754 (0.86) 58.92 (2.70)** 0.195 20 Food & Kindred
Products 0.00286 (11.68)*** 0.687 (28.25)*** -0.645 (-0.98) 42.70 (2.62)** 0.445 22 Textile Mill Products 0.00147
(2.69)** 0.446 (8.22)*** -1.093 (-0.75) -84.33 (-2.32)* 0.037 23 Apparel & Other
Textile Products 0.00278 (10.10)*** 0.681 (24.92)*** -1.225 (-1.66) 51.70 (2.82)** 0.397 25 Furniture & Fixtures 0.000231
(1.25) 0.143 (7.80)*** 0.434 (0.88) -40.08 (-3.27)** 0.026 26 Paper & Allied
Products 0.00238 (10.71)*** 0.540 (24.41)*** -0.328 (-0.55) 93.79 (6.33)*** 0.433 28 Chemical & Allied
Products 0.00212 (16.33)*** 0.579 (44.95)*** 0.423 (1.22) -6.87 (-0.80) 0.620 29 Petroleum & Coal
Products 0.00228 (7.62)*** 0.711 (23.91)*** 2.835 (3.54)*** 33.12 (1.66) 0.328 30 rubber & Miscellaneous Plastics Products 0.00236 (14.02)*** 0.540 (32.23)*** 0.112 (0.25) 32.76 (2.92)** 0.502 31 Leather & Leather
Products 0.00303 (4.82)*** 0.681 (10.88)*** -2.063 (-1.22) 111.70 (2.67)** 0.136 32 Stone, Clay, & Glass
Products 0.00275 (16.28)*** 0.684 (40.70)*** -0.587 (-1.29) 30.09 (2.67)** 0.614 33 Primary Metal Industries 0.00283 (16.48)*** 0.731 (42.82)*** -0.37 (-0.80) 43.38 (3.80)*** 0.644 34 Fabricated Metal Products 0.00167 (12.53)*** 0.409 (30.82)*** -1.036 (-2.89)** -9.93 (-1.12) 0.450 35 Industrial Machinery & Equipment 0.00365 (14.36)*** 0.885 (35.07)*** -1.809 (-2.65)** 16.85 (1.00) 0.536 36 Electronic & Other
Electric Equipment 0.00291 (18.32)*** 0.805 (50.87)*** -0.615 (-1.44) 0.54 (0.05) 0.691 37 Transportation Equipment 0.00126 (9.53)*** 0.327 (24.81)*** -0.142 (-0.40) -26.64 (-3.02)** 0.303 38 Instruments & Related
Products 0.00282 (14.95)*** 0.695 (37.08)*** -0.976 (-1.93) 17.72 (1.41) 0.562 39 Miscellaneous Manufacturing Industries 0.00137 (4.69)*** 0.263 (9.06)*** -1.143 (-1.46) -68.83 (-3.54)*** 0.042 42 Trucking & Warehousing 0.00353 (8.91)*** 0.923 (23.41)*** -2.000 (-1.88) 18.12 (0.69) 0.340 44 Water Transportation 0.00354 (12.18)*** 0.921 (31.88)*** -0.705 (-0.90) 17.09 (0.88) 0.478 45 Transportation by Air 0.00248 (6.59)*** 0.845 (22.59)*** 0.739 (0.73) 15.54 (0.62) 0.305 47 Transportation Services 0.00172 (4.72)*** 0.439 (12.10)*** 2.009 (2.05)* 71.15 (2.93)** 0.138 48 Communications 0.00123 (7.72)*** 0.321 (20.24)*** 0.725 (1.69) 40.78 (3.84)*** 0.305 49 Electric, Gas, &
16 Service Stations (9.60)*** (22.91)*** (2.07)* (9.74)*** 57 Furniture & Homefurnishings Stores 0.00363 (9.06)*** 0.895 (22.47)*** -0.117 (-0.11) 54.06 (2.03)* 0.330 58 Eating & Drinking
Places 0.00390 (9.88)*** 1.027 (26.19)*** -1.316 (-1.24) 45.11 (1.72) 0.400 60 Depository Institutions 0.00328 (15.91)*** 1.031 (50.36)*** 0.479 (0.87) -23.28 (-1.70) 0.668 61 Nondepository Institutions 0.00291 (9.47)*** 0.966 (31.57)*** 2.008 (2.43)* 58.02 (2.83)** 0.479 62 Security & Commodity Brokers 0.00344 (17.01)*** 0.987 (48.98)*** 1.162 (2.14)* 49.49 (3.67)*** 0.689 63 Insurance Carriers 0.00264 (12.43)*** 0.700 (33.21)*** 1.624 (2.85)** 95.19 (6.74)*** 0.547 65 Real Estate 0.00293 (11.66)*** 0.654 (26.18)*** -0.829 (-1.23) 18.40 (1.10) 0.391 70 Hotels & Other
Lodging Places 0.00254 (8.38)*** 0.660 (21.87)*** 0.313 (0.38) 68.65 (3.40)*** 0.337 73 Business Services 0.00279 (11.34)*** 0.589 (24.07)*** -0.553 (-0.84) 38.32 (2.34)* 0.369 79 Amusement & Recreation Services 0.00233 (5.53)*** 0.557 (13.30)*** -0.959 (-0.85) -3.81 (-0.14) 0.133 87 Engineering & Management Services 0.00303 (12.69)*** 0.950 (40.01)*** 1.313 (2.05)* -12.43 (-0.78) 0.559 Note: t statistics in parenthesis.
*, **, *** indicate significance level at 10%, 5%, 1%, respectively Table 3 presents the time series regression results using equation (6).
𝑅𝑝𝑡− 𝑅𝑓𝑡 = 𝛼𝑖+ 𝛽𝑖 𝑅𝑚𝑡− 𝑅𝑓𝑡 + 𝛿𝑖 𝑅𝑚𝑡− 𝑅𝑓𝑡 2+ 𝜅𝑖 𝑅𝑚𝑡− 𝑅𝑓𝑡 3+ 𝜀𝑖𝑡
The daily excess portfolio returns are the dependent variables. The excess market returns, the square, and the cube of the excess market returns are the independent variables. All the beta coefficients are statistically significant at 1% significance level. The coefficients on the square of the excess market returns range from -2.06 to 3.35. The coefficients on the cube of the excess market returns range from -84.33 to 153.10.
Table 4 presents the cross-sectional regressions of the traditional CAPM, the third moment CAPM (inclusion of co-skewness), the fourth moment CAPM (inclusion of co-kurtosis), and the higher moment CAPM (inclusion of both co-skewness and co-kurtosis) for 42 industry portfolios. In the cross-sectional regression result, the adjusted R2 decreases from 0.013 to -0.010 when introducing co-skewness as an additional explanatory variable to the regression. The adjusted R2 decreases from 0.013 to 0.009 when incorporates higher moments. The adjusted R2 increases from 0.013 to 0.034 when incorporating co-kurtosis into the regression, which might be an indication that the inclusion of the effect of the kurtosis into the traditional CAPM is a better approach than the inclusion of skewness.
17
Table 4. Cross-Sectional Regression for 42 Industry Portfolios, July 2005- June 2014
Model 𝛾𝑜(%) t-value 𝛾1𝜷(%) t-value 𝛾2𝑺𝑲𝑬𝑾(%) t-value 𝛾3𝑲𝑼𝑹𝑻(%) t-value Adj. R2
Traditional CAPM 0.03 3.39*** (0.02) (1.25) 0.013 Third Moment 0.03 3.33*** (0.02) (1.20) 0.00 0.29 -0.010 Fourth Moment 0.03 3.48*** (0.02) (1.55) 0.00 1.37 0.034 Higher Moment CAPM 0.03 3.43*** (0.02) (1.54) 0.00 (0.13) 0.00 1.33 0.009 Note: t statistics in parenthesis.
*, **, *** indicate significance level at 10%, 5%, 1%, respectively
The table presents the cross-sectional regression for 42 industry portfolios from July 2005 to June 2014. The time-series regressions result using equation (5) from Table 3 are used as explanatory variables in the cross-sectional regressions. Row 2, 3, 4, and 5 are the cross-sectional regression result of equation (9): 𝑅 −𝑖
𝑅𝑓
= 𝛾𝑜+ 𝛾1 𝛽𝑖+ 𝜀𝑖, (10): 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜+ 𝛾1 𝛽𝑖+ 𝛾2𝛿𝑖+ 𝜀𝑖, (11): 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜+ 𝛾1 𝛽𝑖+ 𝛾3𝜅𝑖+ 𝜀𝑖, and (8): 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜+ 𝛾1 𝛽𝑖+ 𝛾2𝛿𝑖+ 𝛾3𝜅𝑖+ 𝜀𝑖, respectively. Constants of these four models are all significant at 1% significance level. The coefficients (𝛾1) associated with the systematic risk all exhibits a negative sign. In column 6, the corresponding premium related to the skewness all exhibits an insignificant positive relationship with respect to the average excess returns. In column 7, the corresponding premium related to the kurtosis also exhibits an insignificant positive relationship with respect to the average excess returns.
help to explain the behavior of the stock returns in Taiwan stock market for the 10-year period from July 2005 to June 2014.
5.2 Robustness check
The financial environment is chaotic during the financial crisis period in 2008, we argue that the inclusion of the crisis period might be the reason that we cannot find significant result that co-moments are priced in Taiwan stock market. Hence, as a robustness check, the thesis exclude the period after the financial crisis in 2008.
Table 5 presents the descriptive statistics of daily stock returns of 42 industry portfolios (by SIC2) for sub-sample period of 781 days from July 2005 to June 2008 before the financial crisis. In column 7, 28 out of 42 portfolios exhibit negative skewness. Excess kurtosis in column 8 shows that the distribution of daily stock returns of all 42 portfolios is more peaked at the mean. The minimum portfolio returns ranges from -0.09% to 0.00%, while the maximum portfolio returns ranges from 0.00% to 0.08%. In the last column, the normality test shows that 26 out of 42 portfolios returns have non-normal distributions. Compared to the 10-year period, there is less portfolios that returns are non-normally distributed.
18
Table 5. Descriptive Statistics of Industry Portfolio Returns for Taiwan, July 2005- June 2008
Portfolios Observations Mean S.D. Min Max Skewness Excess Kurtosis P(SKEW) P(KURT) 15 781 0.09 1.72 (6.07) 5.88 (0.35) 1.17 0.00 0.00 16 781 0.02 1.92 (7.78) 6.18 (0.25) 1.21 0.01 0.00 17 781 (0.04) 1.08 (8.73) 7.72 (0.66) 21.08 0.00 0.00 20 781 0.05 1.39 (7.06) 5.30 (0.44) 2.82 0.00 0.00 22 781 (0.00) 0.32 (1.48) 1.06 (0.46) 2.22 0.00 0.00 23 781 0.02 1.22 (5.44) 4.86 (0.20) 2.48 0.03 0.00 25 781 (0.01) 0.00 (0.01) (0.00) (0.55) (1.08) 0.00 0.00 26 781 0.02 1.38 (6.22) 5.94 (0.50) 3.11 0.00 0.00 28 781 0.04 0.81 (3.92) 3.31 (0.39) 2.54 0.00 0.00 29 781 0.07 1.51 (7.17) 6.66 (0.14) 4.00 0.12 0.00 30 781 0.02 0.94 (4.92) 3.96 (0.54) 3.26 0.00 0.00 31 781 0.06 3.40 (7.24) 7.94 0.21 0.01 0.02 0.84 32 781 0.04 1.06 (5.83) 4.78 (0.67) 3.34 0.00 0.00 33 781 0.05 1.16 (5.95) 5.46 (0.51) 2.93 0.00 0.00 34 781 0.01 0.57 (2.69) 1.73 (0.54) 2.13 0.00 0.00 35 781 (0.05) 1.71 (6.56) 5.18 (0.39) 1.20 0.00 0.00 36 781 0.01 0.99 (4.91) 4.00 (0.90) 3.50 0.00 0.00 37 781 (0.00) 0.37 (1.58) 1.60 0.10 2.86 0.26 0.00 38 781 0.01 0.90 (4.81) 3.47 (0.82) 4.42 0.00 0.00 39 781 (0.01) 0.04 (0.20) 0.15 (0.68) 3.76 0.00 0.00 42 781 0.04 2.09 (6.93) 6.09 (0.09) 0.76 0.31 0.00 44 781 0.10 1.99 (6.55) 5.98 (0.06) 0.73 0.48 0.00 45 781 (0.02) 1.54 (6.62) 6.24 0.03 4.57 0.70 0.00 47 781 0.06 1.33 (6.67) 6.16 0.93 12.63 0.00 0.00 48 781 0.03 0.77 (3.98) 4.48 0.01 3.38 0.89 0.00 49 781 (0.00) 0.07 (0.31) 0.35 0.36 6.51 0.00 0.00 50 781 0.02 1.07 (5.65) 4.28 (0.45) 3.61 0.00 0.00 51 781 0.05 2.07 (7.80) 6.53 0.02 1.20 0.85 0.00 53 781 0.06 1.65 (5.63) 5.37 (0.04) 1.54 0.66 0.00 54 781 0.07 1.99 (7.24) 6.75 0.42 2.07 0.00 0.00 55 781 0.02 1.33 (6.93) 6.67 (0.01) 4.91 0.93 0.00 57 781 0.07 2.54 (7.74) 7.24 0.11 1.15 0.19 0.00 58 781 (0.06) 2.16 (8.58) 6.47 (0.36) 1.29 0.00 0.00 60 781 (0.03) 1.39 (6.57) 5.31 (0.24) 2.08 0.01 0.00 61 781 (0.04) 1.75 (7.21) 6.70 0.14 3.07 0.11 0.00 62 781 0.05 1.43 (6.75) 5.61 (0.09) 1.66 0.28 0.00 63 781 0.04 1.27 (7.03) 5.87 (0.03) 4.32 0.73 0.00 65 781 0.07 1.37 (5.22) 4.73 0.08 2.23 0.33 0.00 70 781 0.07 1.79 (6.66) 5.85 0.26 1.91 0.00 0.00 73 781 0.05 1.46 (6.57) 6.16 0.12 2.29 0.15 0.00 79 781 (0.02) 1.81 (6.99) 6.75 0.33 2.67 0.00 0.00 87 781 (0.01) 1.41 (9.35) 7.62 (0.24) 3.79 0.01 0.00
The table presents the first four moments of daily stock returns of 42 portfolios. The sub-sample period is for the 781 days from July 2005 to June 2008 before the financial crisis. 819 companies’ daily stock returns are sorted into 42 portfolios using standard industrial classification (SIC2). From Table8, it can be seen that the minimum portfolio returns ranges from -0.09% to 0.00%, while the maximum portfolio returns ranges from 0.00% to 0.08%. Excess kurtosis can be as high as 21.08, ranging from -1.08 to 21.08. The skewness ranges from -0.90 to 0.93. In column 7, 14 out of 42 portfolios exhibit positive skewness. Excess kurtosis in column 8 shows that the distribution of daily stock returns of all 42 portfolios is more peaked at the mean.
Source: Datastream.
which suggest that the beta is positively related to returns. 18 out of 42 industry portfolios exhibit negative relationship between skewness and return. 6 out of 42 industry portfolios exhibit negative relationship between kurtosis and return.
19
Table 6. Beta, Coskewness, and Cokurtosis of Taiwan stock market, July 2005 to June 2008
SIC2 α β Skewness Kurtosis Adj. R2 15 0.005 0.859 -0.051 -3.118 0.360 16 0.005 0.894 -1.212 55.787 0.387 17 0.001 0.239 -1.523 -28.321 0.068 20 0.004 0.612 0.514 99.903 0.397 22 0.001 0.163 0.072 13.195 0.458 23 0.003 0.480 0.536 51.800 0.275 25 0.000 0.000 -0.012 -0.126 0.052 26 0.003 0.521 -1.779 105.265 0.374 28 0.003 0.509 0.499 24.889 0.620 29 0.004 0.683 3.806 84.436 0.320 30 0.002 0.396 1.289 96.570 0.402 31 0.004 0.630 3.137 214.489 0.086 32 0.003 0.572 -0.045 70.810 0.568 33 0.004 0.683 1.969 81.920 0.592 34 0.001 0.231 -0.526 30.878 0.351 35 0.004 0.832 -0.084 47.825 0.397 36 0.003 0.619 0.699 48.487 0.649 37 0.001 0.188 0.746 13.733 0.382 38 0.002 0.460 0.936 66.857 0.489 39 0.000 0.013 0.004 2.546 0.197 42 0.005 0.815 -3.385 -32.356 0.238 44 0.006 0.967 -3.001 -35.764 0.359 45 0.003 0.579 -1.529 65.185 0.297 47 0.001 0.187 0.983 35.409 0.033 48 0.001 0.251 1.160 60.527 0.226 49 0.000 0.014 0.040 7.627 0.188 50 0.003 0.507 0.787 72.217 0.429 51 0.004 0.788 3.354 145.507 0.272 53 0.004 0.696 -0.906 27.609 0.300 54 0.004 0.781 4.838 77.359 0.222 55 0.003 0.370 -0.424 137.479 0.264 57 0.006 0.909 -4.421 11.900 0.232 58 0.005 0.993 -2.262 26.816 0.358 60 0.004 0.864 1.309 35.132 0.586 61 0.003 0.757 2.494 54.326 0.284 62 0.004 0.864 1.796 46.774 0.557 63 0.003 0.652 2.691 113.915 0.491 65 0.004 0.555 -2.721 13.662 0.308 70 0.004 0.602 -0.304 58.834 0.210 73 0.003 0.542 3.184 124.220 0.273 79 0.002 0.443 -0.954 18.914 0.102 87 0.003 0.811 0.747 -46.271 0.409 Note: t statistics in parenthesis.
*, **, *** indicate significance level at 10%, 5%, 1%, respectively The table presents the time series regression results using equation (5).
𝑅𝑝𝑡− 𝑅𝑓𝑡 = 𝛼𝑖+ 𝛽𝑖 𝑅𝑚𝑡− 𝑅𝑓𝑡 + 𝛿𝑖 𝑅𝑚𝑡− 𝑅𝑓𝑡 2+ 𝜅𝑖 𝑅𝑚𝑡− 𝑅𝑓𝑡 3+ 𝜀𝑖𝑡
The daily excess portfolio returns are the dependent variables. The excess market returns, the square, and the cube of the excess market returns are the independent variables. All the beta coefficients are positively related to the excess returns. The coefficients on the square of the excess market returns range from -4.421 to 4.838. The coefficients on the cube of the excess market returns range from -46.271 to 214.489. Adjusted R-squared ranges from 0.033 to 0.649.
20
Table 7. Cross-Sectional Regression for 42 Industry Portfolios, July 2005- June 2008
Model 𝛾𝑜(%) t-value 𝛾1𝜷(%) t-value 𝛾2𝑺𝑲𝑬𝑾(%) t-value 𝛾3𝑲𝑼𝑹𝑻(%) t-value Adj. R2
Traditional CAPM 0.00 0.29 0.00 1.56 0.034 Third Moment 0.00 0.20 0.03 1.59 0.00 0.79 0.025 Fourth Moment 0.00 (0.13) 0.03 1.47 0.00 1.23 0.460 Higher Moment CAPM 0.00 (0.12) 0.03 1.46 0.00 0.10 0.00 0.92 0.021 Note: t statistics in parenthesis.
*, **, *** indicate significance level at 10%, 5%, 1%, respectively
The table presents the cross-sectional regression for 42 industry portfolios from July 2005 to June 2008. The time-series regressions result using equation (5) from Table 6 are used as explanatory variables in the cross-sectional regressions. Row 2, 3, 4, and 5 are the cross-sectional regression result of equation (9): 𝑅 −𝑖
𝑅𝑓
= 𝛾𝑜+ 𝛾1 𝛽𝑖+ 𝜀𝑖, (10): 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜+ 𝛾1 𝛽𝑖+ 𝛾2𝛿𝑖+ 𝜀𝑖, (11): 𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜+ 𝛾1 𝛽𝑖+ 𝛾3𝜅𝑖+ 𝜀𝑖, and (8):
𝑅 − 𝑅𝑖 = 𝛾𝑓 𝑜+ 𝛾1 𝛽𝑖+ 𝛾2𝛿𝑖+ 𝛾3𝜅𝑖+ 𝜀𝑖, respectively. Constants of these four models are all significant at 1% significance level. The coefficients (𝛾1) associated with the systematic risk all exhibits a negative sign. In column 6,
the corresponding premium related to the skewness all exhibits an insignificant positive relationship with respect to the average excess returns. In column 7, the corresponding premium related to the kurtosis also exhibits an insignificant positive relationship with respect to the average excess returns. The adjusted R2 ranges from 0.021 to 0.460.
premium related to kurtosis also exhibits an insignificant relationship with respect to the average access returns. The robustness check shows that the inclusion of the effect of skewness and kurtosis does not provide a better explanatory power than the traditional CAPM even exclude the period after the financial crisis in 2008.
However, the adjusted R-squares are higher in the subsample for all four models than in the 10-year period, which might be an indication that excluding the financial crisis period improves the validity of the models. It is also worth noted that the adjusted R2 of the fourth moment CAPM increases from 0.034 in the 10-year studied period to 0.460 in the subsample, which indicates that the inclusion of kurtosis effect is more suitable when excluding the chaotic financial crisis period.
6. Conclusion
21
The thesis examines the efficacy of the higher moment CAPM in Taiwan stock market using daily stock returns for a 10-year period from July 2005 to June 2014. Taiwan is different from other emerging countries because of having an intensive high-tech industry structure, a high liquidity and well regulated market. All 819 companies listed at the Taiwan Stock Exchange are sorted into 42 industry portfolios using standard industrial classification (sic2).
The efficacy of the higher moment CAPM is tested using a two-step approach by Fama and MacBeth (1973). In the first stage, the covariance, co-skewness, and co-kurtosis are estimated using rolling time-series regressions. In the second stage, I create a cross-section of average returns of the assets and regress average returns on these three factors to see whether they are priced in the Taiwan stock market.
The empirical finding of the thesis is consistent with the traditional CAPM that systematic risk is positively related to returns, however, the inclusion of the higher moments (skewness and kurtosis) into the traditional CAPM does not help to explain the behavior of stock returns in Taiwan’s equity market from July 2005 to June 2014. In the cross-section result, we can see that neither co-skewness nor co-kurtosis is priced. As a robustness check, the thesis further tests the period before the financial crisis. Yet, the result shows no difference than the 10-year studied period. Although there are studies documented that the higher moment CAPM has better explaining power than the traditional CAPM in emerging markets, the empirical finding of the thesis suggest otherwise.
Limitations and Suggestions
22
23 Reference
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