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financial economics / jel cio, c50, gi0
Ibrahim A. Onour
Modeling and assessing systematic risk in stock markets in major oil exporting countries*
keywords
abstract
beta;
CAPM;
GARCH;
volatility;
asymmetry
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*An earlier version of this paper is posted at Researchgate by the same author as a working paper: DOI: 10.13140/RG.2.2.29977.44644.
Introduction. This paper aims to assess time variability of beta coefficients (systematic risk) of Capital Asset Pricing Model (CAPM) using data from five key sectors in Saudi Arabia and Kuwait stock markets.
Material and methods. To assess time - varying systematic risk we employed symmetric as well as asymmetric conditional volatility specifications to account for skewness and leptkurtosis of high frequency financial time series to better specify conditional higher moments.
Results & discussions. The results of the paper support significant evidence of time-varying beta coefficients for all sectors included in the study, in particular the banking sector, and relatively with a lesser degree, the food, and the service sectors in both countries. For the banking sector in Saudi Arabia, the beta coefficients variability during the sample period estimated between (0.18 to 22.1), and also for Kuwait stock market the beta coefficient of the banking sector variability estimated between (0.16 to 22.1). This result invalidates, at least in the context of the sample country's banking sectors, the standard application of (CAPM) that assumes constant beta coefficients. Also indicated in the paper, time-varying beta estimates are consistent with a modified version of CAPM prediction that is portfolios with wider range of beta variations expected to yield higher return values and those with lower range of beta variations yield lower returns.
Conclusion. In this new context, risk is no longer is a point estimate as implied by the standard CAPM model, but it is a range of values. Our findings also show the size and the range of beta variations are sensitive to skewness and fat tailedness that characterize asset returns distribution.
Onour, I. A. (2021). Modeling and assessing systematic risk in stock markets in major oil exporting countries. Economic consultant, 35 (3), 18-29. doi: 10.46224/ecoc.2021.3.3
introduction
How should a rational investor measure the risk of stock market investments? The search for an answer to this question became the major task in financial economics and that led to the development of Capital Asset Pricing Model (CAPM) which become the centre piece in modern finance textbooks. The CAPM decompose risk valuation into risk size (risk premium) and risk price (beta*). According to CAPM the required rate of return on a company's stock (or the cost of equity capital) depends on three components among which the stock's equity beta which measures the risk of company's stock relative to the market risk; or putting it differently, the risk each dollar invested in equity i contributes to the market portfolio. CAPM predict low beta stocks should offer low stock returns and higher beta stocks should offer higher stock returns. This imply stocks with higher risks should yield higher returns to compensate for the additional risk borne.
An appropriate specification of time-varying volatility depends on what empirical regularities the model should capture. Some important regularities for asset returns volatility include the so-called "volatility clustering" phenomena which refers to the situation of large change of asset returns followed by large change of either sign, and small changes followed by small changes. Also another related phenomena is the "leverage effect" which refers to the different response of volatility to bad news as compared to good news. To account for these type of asymmetric effect of news on traded asset returns' volatility in this paper Glosten, Jagannathan, and Runkle [11] specification of GARCH model is adopted. GJR-GARCH specification separates the effect of negative news on volatility from that of positive news.
While there is a considerable amount of research in this area for industries in developed nd in some emerging stock markets, Mckenzie et al [15] for U.S., banks; Brooks, Faff, and Aritt [3] for Australian financial sector; Faff, Hillier, and Hillier [13] for U.K., industry portfolio; Yu [17] for New Zealand, Moonis and Shah [16] for Indian Market, and Kanwer [14] for Karachi Market.
The relevance of the current research stems from the observation that portfolio managers need constantly update and re-estimate the relationship between risk factors and returns, contrary to traditional application of CAPM that assumes constant beta coefficients.
As a result, Adrian and Franzoni [1], Mumtaz and Theodoridis [20]; Casarin et al. [4]; and Pacifico [21], suggest that models without time-varying beta coefficients fail to capture investors' risk aversion characteristics and may lead to inaccurate estimates of risk dynamics. Similar work on GCC and less developed stock markets is lacking. The main purpose of the current research is to assess time-variability of beta coefficients in Saudi and Kuwait stock markets using data on six key sectors in both countries. This paper contributes to the existing literature by taking into account leverage effect, and skewed-fat-tailed aspects of volatility when estimating beta coefficients.
The reminder of the paper is structured as follows. The next sections include methods, results, and conclusions.
* Systematic or systemic risk is the overall economy wide risk that cannot be avoided via diversification strategy.
materials and methods
Although the simple GARCH specification is widely used in the empirical research of finance, there are substantial evidences that volatility of asset returns characterized by time varying asymmetry (Glosten, Jagannathan and Runkle [11]). As a result, to avoid misspecification of the conditional variance equation, a leverage term in the GARCH specification is included. The GARCH-type specification introduced by Glosten, et al, [11] allows a quadratic response of volatility to news with different coefficients for good and bad news, but maintains the assertion that the minimum volatility will result when there is no news*. Given the capital market model specified as follows:
i,(i)
where et = <jtz
i.id
zt ~ /(^0,1)
q p
and c~t - a0 + ^aqe2t-q + ^Spcr2t-P + £t
q=1 p= 1
where Rmt is the return on market portfolio, and R.t is the return on sector i, and rj. and ft. are the associated portfolio mean, and beta respectively.
Beta coefficient in (1) reflect the sensitivity of industry return to change in market return. f(.) is the density function of the standardized residuals, zt, where E(zt) =0, v(zt) =1, and a> is a vector of the parameters reflecting skewness and kurtosis parameters. In GARCH-type models the variance covariance matrix of the different portfolios and the market index returns are not constant over time. In this case Beta defined as:
p garch = cow(Rtt,Rmt) ^
var(/?J
so that equation (1) becomes,
R„ =V:+0„a"'CHR„ +e„ (3)1
One approach to estimating ^iGARCH is to estimate conditional covariance, cov(R.tRmt) and conditional market variance v(R ). Adopting asymmetric GARCH- type model the problem can be reduced to estimating the following specifications of variance and covariance equations:
where I denotes indicator function taking on the values of 1 when e > 0, and 0 otherwise.
* Any selection of an appropriate ARCH/GARCH model requires having a good idea of what empirical regularities the model should capture. Among documented other regularities in the literature are thick tails that characterize asset returns, and volatility clustering, which refers to the phenomena that large changes in volatility tend to be followed by large changes of either sign, and small changes to be followed by small changes.
The threshold ARCH (TARCH) model of Zakoian [18] corresponds to equation (4) with X = 1, whereas GJR-GARCH - type specification treats equation (4) with X = 2, to allow for quadratic response of volatility to news with different coefficients for good and bad news, while maintaining the possibility that minimum volatility occur when there is no news. Similarly, the variance of market portfolio from equation (4) hold with the change of the subscript from i to m.
The situation that a+ > 0 captures the asymmetric relationship between news (et) and volatility. For example, when e then I=1 and the conditional variance becomes,
,<x2,,-
(J2it = CO + Yj V; +
Ui UP
and when then I=0 and the conditional variance becomes
(j2u =0) + ^ a~e2ij-j+YjSj(j2i,t
Therefore, the negative news result in a variance level different from that associated with positive news. This type of investors behavior imply risk aversion attitudes depend on the magnitude of risk investors expecting to face.
Given that the error terms, in (4) are conditional means; namely Rit=eit, Rmt=e . Then the conditional covariance of industry and market portfolio can be computed by:
co \(Rit, Rmt) = pim 4o1ua1mt
(5)
where is p. the correlation coefficient between Rt and R t.
' im it mt
It is well documented that even asymmetric GARCH models fail to fully account for skewness and leptkurtosis of high frequency financial time series when they are assumed to follow normal or symmetric student's t-distributions. This has led to the use of asymmetric nonnormal distributions to better specify conditional higher moments. An important candidate in this respect is Hansen's [19] distribution. Despite other distributions that allow for skewness and excess kurtosis we choose Hansen's distribution due to its simplicity and its superiority in empirical performance.
Given the standardized errors
i
CT2,
= z, with mean zero and variance one, then Hansen's [19]
autoregressive conditional density model with skewed error terms specified as:
Specification of conditional distribution of the standardized residuals, Zt, in equation (6) is determined by two parameters, Kurtosis (0) and the skewness parameter (0).The two parameters are restricted to 0 > 2 and -1 < 0 < 1. When 0 = 0 the skewed t-distribution tend to symmetric t-distribution, and when 0 ^ x>, tend to standardized normal distribution.
Hansen's skewed t-distribution is fat tailed, and skewed to the left (right) when 0 is less (greater) than zero. Similar to the case of Student's t-distribution, when 0 > 2. Hansen's skewed t-distribution is well defined and its second moment exist, while skewness exist if 0 > 3 and kurtosis is defined if 0 > 4.
results
Data employed in this study are daily stock price indices related to five key sectors beside the aggregate stock indices for each of Saudi Arabia and Kuwait stock markets. The sample period covers from December-6-2011 to January-9-2020, including 2000 observations. The sectors included in this study are, banks; food; industrial; real estate; and the service sectors. Results in tables (1&2) indicate that all sectors in the two markets yield positive mean returns. The high values kurtosis statistics indicate the stock price returns distribution is characterized by high peakness (fat tailedness). The negative skewness results indicate that Saudi and Kuwait portfolio industry exhibit a higher probability of a negative returns, which is similar to the case in some developed and emerging markets as indicated by Harvey and Siddique (1999). The Jarque-Bera (JB) test statistics for both markets provides clear evidence of rejecting the null-hypothesis of normality for the unconditional distribution of the daily stock price changes for all sectors. The sample autocorrelation statistic indicated by Ljung-Box, Q statistic, show the Q(5) test statistic reject the null hypothesis of uncorrelated price changes for five lags for all sectors in both markets. The high values for Q2(5) test statistic in both markets suggest that conditional homoskedasticity can be rejected for all sectors. To test the presence of hetroskidasticity more formally the LM test is employed. Results of LM statistics for ARCH (1) and ARCH (5) error terms confirm the significance of ARCH effects in the data.
Table 1
Summary statistics of stock returns of Saudi stock market (log differenced)
Banks Sector Food Sector Industrial Sector Real Estate Sector Service Sector Market Index
Mean 0.9E-1 0.7E-3 0.8E-3 0.5E-3 0.2E-1 0.1E-2
Skewness -16.2 -23.1 -10.3 -11.6 -16.1 -18.1
Excess kurtosis 74 54 25 87 68 60
JB test (p-value) 8851 (0.00) 7480 (0.00) 1349 (0.00) 8364 (0.00) 9212 (0.00) 6523 (0.00)
Q(5) (p-value) 265 (0.00) 134 (0.00) 657 (0.00) 289 (0.00) 567 (0.00) 123 (0.00)
Q2(5) (p-value) 235 (0.00) 541 (0.00) 871 (0.00) 456 (0.00) 790 (0.00) 267 (0.00)
LM ARCH(1) (p-value) 89 (0.00) 0.60 (0.47) 234 (0.00) 12.1 (0.01) 45.4 (0.00) 45.0 (0.00)
LM ARCH(5) (p-value) 154 (0.00) 432 (0.00) 678 (0,00) 567 (0.00) 321 (0.00) 356 (0.00)
Table 2
Summary statistics of stock returns of Kuwait stock market (log differenced)
Banks Sector Food Sector Industrial Sector Real Estate Sector Service Sector Market Index
Mean 0.12E-1 0.5E-2 0.1E-2 0.3E-4 0.9E-1 0.8E-2
Skewness -11.2 -25.1 -16.3 -10.6 -14.1 -12.1
Excess kurtosis 84 59 28 77 58 65
JB test (p-value) 1885 (0.00) 2748 (0.00) 5134 (0.00) 1836 (0.00) 2921 (0.00) 5652 (0.00)
Q(5) (p-value) 262 (0.00) 130 (0.00) 659 (0.00) 285 (0.00) 562 (0.00) 120 (0.00)
Q2(5) (p-value) 835 (0.00) 741 (0.00) 971 (0.00) 156 (0.00) 290 (0.00) 967 (0.00)
LM ARCH(1) (p-value) 56 (0.00) 19 (0.01) 123 (0.00) 15 (0.02) 38 (0.03) 51 (0.00)
LM ARCH(5) (p-value) 154 (0.00) 432 (0.00) 678 (0,00) 567 (0.00) 321 (0.00) 356 (0.00)
Estimation of beta coefficients based on conditional volatility of stock returns, assuming asymmetric GARCH specification under Normal distribution (GJR-N); and Skewed t-distribution (GJR-skt) of error terms for each of the two markets reported in tables (3&4). Tables (A1-A4) in the appendix include estimation results of the parameters of equations (4), (6), and (7) for each of the two markets. The significance and positive sign of the volatility parameter (¿) for all sectors in the two markets indicate evidence of volatility persistence in both markets. The significance of the asymmetry coefficient (a+) for the Normal distribution for all five sectors in the two markets is an indication that positive shocks (or good news) have more significant effect on volatility than bad news. However, the response of each sector in the two markets to bad news differ from one sector to another in each of the two markets. That is to say, negative shocks in industrial, real estate, and service sectors in the two markets better captured by skewed-t-distribution, as skeweness and kurtosis parameters are significant factors in analyzing the impact of negative news on volatility of these sectors. This result imply that while there is evidence of leverage effect, it seems conditional on the size of the expected effects on portfolio returns. For example, since investors in equity seek short term profit gains they always care about the good news, but the bad news as it is not frequent, it is assessed on its expected size effect before considering hedging aspect, because it incur additional cost.
Results of the skewed t-distribution also indicate significance of the Kurtosis coefficient (0) for all sectors in the two markets, implying that fat-tailed student t-density is needed to fully model the distribution of return. Despite the significance of the Kurtosis coefficient (0) for all sectors, the log-likelihood values strongly suggest that the Normal distribution (GJR-N) outperform, the skewed-t distribution GARCH/ARCH model.
When looking at the range of variation of beta coefficients it become apparent that both models support evidence of time-varying beta values for all sectors. Based on the Normal distribution results, Beta values for Banks, Food, and Service sectors in both markets exhibit wider range of variation compared to the other two sectors. The other two sectors, Industry and Real estate, show relatively stable variation of betas. When comparing return values in tables (1& 2) with the range statistics in table (3 & 4) (under the Normal distribution) we
conclude a modified version of CAPM prediction that is portfolios with wider range of beta variations expected to yield higher return values.
Table 3
Beta Coefficients of Saudi stock indices
Sectors GJR-GARCH I Normal dist. GJR-GARCH I Sk-t dist.
Banks 1.21 4.56
(low/high) (0.18/22.1) (0.14/5.26)
Range 21.8 5.12
P 0.88 0.91
Food 0.10 3.12
(low/high) (0.02/8.1) (0.5/4.87)
Range 8.0 4.37
P 0.78 0.88
Industry 2.79 5.4
(low/high) (0.60/3.90) (0.20/3.8)
Range 3.30 3.6
P 0.71 0.73
Real Estate 0.14 2.12
(low/high) (0.06/0.74) (1.10/5.17)
Range 0.6 4.1
P 0.77 0.98
Service 0.79 3.04
(low/high) (0.28/6.1) (0.13/3.77)
Range 5.9 3.6
P 0.88 0.76
Note: The first row entries are mean values of Betas. Range statistics refer to the difference between high and low values. p denotes correlation coefficient between volatilities of market index and sector portfolio.
Table 4
Beta Coefficients of Kuwait stock indices
Sectors GJR-GARCH I Normal dist. GJR-GARCH I Sk-t dist.
Banks 2.50 3.76
(low/high) (0.16/22.1) (0.41/4.47)
Range 21.9 4.1
P 0.89 0.78
Food 0.57 3.08
(low/high) (0.01/7.1) (0.7/4.97)
Range 7.1 4.2
P 0.88 0.98
Industry 0.68 2.5
(low/high) (0.51/1.81) (0.12/4.8)
Range 1.30 4.7
P 0.70 0.82
Real Estate 0.71 1.72
(low/high) (0.16/1.76) (1.17/4.17)
Range 1.6 3.0
P 0.81 0.75
Service 0.76 1.50
(low/high) (0.40/8.1) (0.13/3.73)
Range 7.6 3.6
P 0.90 0.89
Note: The first row entries are mean values of Betas. Range statistics refer to the difference between high and low values. p denotes correlation coefficient between volatilities of market index and sector portfolio.
To verify which model better fit the data set, the predictive ability performance and the log-likelihood criterias employed. Root Mean Square Error (RMSE) and Diebold-Mariano (1995) test results included in table (5 & 6), indicate that the GJR-Normal distribution model yield the lowest values of the RMSE loss functions for all sectors compared to GJR-sk-t distribution specification of volatility. DM test statistic results confirm that the predictive power of the two models are significantly different for all sectors; implying that GJR-Normal distribution model yield superior forecast performance for forward-looking beta values.
Table 5
RMSE Loss functions and Diebold & Mariano test Saudi market
RMSE Loss Functions 1 GJR-N GJR-sk(t) D&M statistic
Banks 1 p-value 0.001 0.005 3.68 (0.00)
Food 1 p-value 0.003 0.007 3.21 ( 0.00)
Industry 1 p-value 0.001 0.10 5.37 (0.00)
Real estate 1 p-value 0.004 0.009 8.8 (0.00)
Service 1 p-value 0.006 0.011 3.05 ( 0.00)
Note: The loss functions are based on five days ahead forecast errors.
Table 6
RMSE Loss functions and Diebold & Mariano test Kuwait market
RMSE Loss Functions 1 GJR-N GJR-sk(t) D&M statistic
Banks 1 p-value 0.004 0.025 3.12 (0.00)
Food 1 p-value 0.002 0.008 4.01 ( 0.00)
Industry 1 p-value 0.009 0.01 4.3 (0.00)
Real estate 1 p-value 0.007 0.09 6.9 (0.00)
Service 1 p-value 0.006 0.009 3.55 ( 0.00)
Note: The loss functions are based on five days ahead forecast errors.
conclusion
In this paper time-varying beta estimates for major key sectors in Saudi Arabia and Kuwait capital markets are explored. To account for the asymmetric effect of news on volatility of asset returns Glosten, Jagannathan, and Runkle (1993) specification is adopted under two alternative assumptions about stock returns distribution, the Normal distribution (GJR-N) and skewed t-distribution (GJR-sk) specification. Based on forecasting performance and on log-likelihood values, the evidence generated in this paper overwhelmingly supports GJR-Normal distribution specification over GJR-skewed t-distribution specification. The result in the paper support evidence of time-varying beta coefficients for all sectors included in the study. From the results of the two specifications of conditional volatility it is evident that the size and the
range of beta variations are sensitive to skewness and tail distribution of asset returns. An important policy implication of time-varying beta evidence is that portfolios with wider range of beta variations expected to yield higher return values and those with lower range of beta variations yield lower returns. In this new context risk is no longer is a point estimate as in the case of the standard CAPM model, but it is a range of beta variability.
references
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appendix
Table 1
GARCH(1.1)/ARCH(q) Saudi stock market*
Banks I GARCH(1,1) Food I ARCH(1) Industry I ARCH(1)
GJR-t GJR- GJR-t GJR- GJR-t GJR-
skew normal skew normal skew normal
m (p-value) 0.31 (0.00) 0.13 (0.00) 0.05 (0.00) 0.01 (0.00) 0.09 (0.00) 0.01 (0.00)
ô (p-value) 0.86 (0.00) 0.12 (0.00) 0.09 (0.02) 0.51 (0.00) 0.48 (0.00) 0.41 (0.00)
а+ (p-value) 0.01 (0.64) 0.06 (0.00) 0.22 (0.30) 0.01 (0.00) 0.08 (0.60) 0.09 (0.00)
аr (p-value) 0.07 (0.55) -0.10 (0.69) -1.5 (0.79) 0.71 (0.18) 0.10 (0.00) -0.40 (0.91)
0 (p-value) 0.56 (0.25) -- 0.78 (0.66) -- 0.87 (0.76) --
в (p-value) 1.97 (0.00) -- 0.92 (0.00) -- 1.91 (0.00) --
LnL 1204 3131 2956 5388 1161 1864
*The lag parameters (p, q) determined based on stationarity restrictions. An examination of the coefficients in GARCH specification in table 1 and 2 reveals that hit for Banks, Industrial, Real Estate, and Service sectors follow stationary ARCH(1), whereas Food sector follows ARCH(3); that is, the condition \a\ < 1 is satisfied for all sectors. Results in table 1 also reveals market portfolio index follows GARCH(1,1) process, and the stationarity conditions, [a2 > 0, ô > 0, (a + ô) < 1], are satisfied.
Table 2
GARCH(1.1)/ARCH(q): Saudi stock market
Real estate 1 ARCH(1) Service I GARCH(1,1)
GJR-t GJR- GJR-t GJR-
skew normal skew normal
m (p-value) 0.01 (0.00) 0.08 (0.00) 0.01 (0.00) 0.08 (0.00)
ô (p-value) 0.41 (0.00) 0.39 (0.00) 0.13 (0.00) 0.10 (0.00)
а+ (p-value) 0.72 (0.30) 0.43 (0.00) 0.01 (0.60) 0.60 (0.00)
а- (p-value) 0.60 (0.00) -0.30 (0.99) 2.3 (0.00) -1.20 (0.99)
0 (p-value) 0.91 (0.15) -- 0.89 (0.16) --
в (p-value) 2.7 (0.00) -- 1.21 (0.00) --
LnL 2753 4233 2172 2812
Table 3
GARCH(1.1)/ARCH(q) Kuwait stock market*
Banks I GARCH(1,1) Food I ARCH(1) Industry I ARCH(1)
GJR-t GJR- GJR-t GJR- GJR-t GJR-
skew normal skew normal skew normal
m (p-value) 0.01 (0.00) 0.10 (0.00) 0.01 (0.00) 0.11 (0.00) 0.01 (0.00) 0.02 (0.00)
ô (p-value) 0.16 (0.00) 0.02 (0.00) 0.19 (0.02) 0.21 (0.00) 0.08 (0.00) 0.11 (0.00)
a+ (p-value) 0.11 (0.64) 0.26 (0.00) 0.12 (0.30) 0.08 (0.00) 0.01 (0.60) 0.01 (0.00)
a- (p-value) 0.01 (0.15) -0.13 (0.19) -1.1 (0.29) 0.91 (0.38) 0.20 (0.00) -0.10 (0.21)
0 (p-value) 0.16 (0.21) -- 0.38 (0.61) -- 0.57 (0.26) --
0 (p-value) 1.37 (0.00) -- 0.32 (0.00) -- 1.01 (0.00) --
LnL 9209 5130 1957 4385 2160 3861
*The lag parameters (p, q) determined based on stationarity restrictions. An examination of the coefficients in GARCH specification in table 3 and 4 reveals that h for Banks, Industrial, Real Estate, and Service sectors follow stationary ARCH(1), whereas Food sector follows ARCH(3); that is, the condition \a\ < 1 is satisfied for all sectors. Results in table 1 also reveals market portfolio index follows GARCH(1,1) process, and the stationarity conditions, [a2 > 0, ô > 0, (a + ô) < 1], are satisfied.
Table 4
GARCH(1.1)/ARCH(q): Kuwait stock market
Real estate I ARCH(1) Service I GARCH(1,1)
GJR-t GJR- GJR-t GJR-
skew normal skew normal
m (p-value) 0.03 (0.00) 0.01 (0.00) 0.08 (0.00) 0.09 (0.00)
ô (p-value) 0.21 (0.00) 0.79 (0.00) 0.43 (0.00) 0.12 (0.00)
a+ (p-value) 0.12 (0.20) 0.33 (0.00) 0.51 (0.20) 0.10 (0.00)
a- (p-value) 0.50 (0.00) -0.10 (0.91) 0.3 (0.00) -2.20 (0.90)
0 (p-value) 0.51 (0.65) -- 0.49 (0.19) --
0 (p-value) 1.7 (0.00) -- 0.21 (0.00) --
LnL 9753 9233 8172 7812
information about the author
Ibrahim A. Onour (Sudan, Khartoum) - Professor of Financial Econometrics. School of Management Studies. University of Khartoum. ORCID ID: 0000-0002-9967-6873. Scopus Author ID: 6506520131. ResearcherID: AAG-9902-2021. E-mail: onour@uofk.edu
Available: https://statecounsellor.wordpress.com/2021/08/26/onour-2/
Received: Feb 12, 2021 I Accepted: Jul 7, 2021 I Published: Sep 1, 2021
Editor: Mohamed R. Abonazel, PhD in Statistics and Econometrics. Cairo University, EGYPT
Copyright: © 2021 Onour, I. A. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Competing interests: The authors have declared that no competing interests exist.
