Mathematical finance with applications

Wong, Wing Keung (Ed.); Guo, Xu (Ed.); Ortobelli Lozza, Sergio (Ed.) Book — Published Version Mathematical finance with applications Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Wong, Wing Keung (Ed.); Guo, Xu (Ed.); Ortobelli Lozza, Sergio (Ed.) (2020) : Mathematical finance with applications, ISBN 978-3-03943-574-6, MDPI, Basel, https://doi.org/10.3390/books978-3-03943-574-6 This Version is available at: https://hdl.handle.net/10419/237807 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Mathematical Finance with Applications • Wing-Keung Wong, Xu Guo and Sergio Ortobelli Lozza Mathematical Finance with Applications Printed Edition of the Special Issue Published in Journal of Risk and Financial Management www.mdpi.com/journal/jrfm Wing-Keung Wong, Xu Guo and Sergio Ortobelli Lozza Edited by Mathematical Finance with Applications Mathematical Finance with Applications Editors Wing-Keung Wong Xu Guo Sergio Ortobelli Lozza MDPI •Basel •Beijing •Wuhan •Barcelona •Belgrade •Manchester •Tokyo •Cluj •Tianjin Editors Wing-Keung Wong Asia University Taiwan Xu Guo Beijing Normal University China Sergio Ortobelli Lozza University of Bergamo Italy Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Journal of Risk and Financial Management (ISSN 1911-8074) (available at: https://www.mdpi.com/ journal/jrfm/special issues/mathematical finance applications). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year,Article Number, Page Range. ISBN 978-3-03943-573-9 (Hbk) ISBN 978-3-03943-574-6 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editors .............................................. vii Wing-Keung Wong Editorial Statement for Mathematical Finance Reprinted from: J. Risk Financial Manag. 2020,13, 18, doi:10.3390/jrfm13020018 .......... 1 Imran Yousaf, Shoaib Ali and Wing-Keung Wong An Empirical Analysis of the Volatility Spillover Effect between World-Leading and the Asian Stock Markets: Implications for Portfolio Management Reprinted from: J. Risk Financial Manag. 2020,13, 226, doi:10.3390/jrfm13100226 .......... 5 Mohammad Rafiqul Islam and Nguyet Nguyen Comparison of Financial Models for Stock Price Prediction Reprinted from: J. Risk Financial Manag. 2020,13, 181, doi:10.3390/jrfm13080181 .......... 33 Imran Yousaf, Shoaib Ali and Wing-Keung Wong Return and Volatility Transmission between World-Leading and Latin American Stock Markets: Portfolio Implications Reprinted from: J. Risk Financial Manag. 2020,13, 148, doi:10.3390/jrfm13070148 .......... 53 Haim Levy The Investment Home Bias with Peer Effect Reprinted from: J. Risk Financial Manag. 2020,13, 94, doi:10.3390/jrfm13050094 .......... 73 Zhe Li Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks Reprinted from: J. Risk Financial Manag. 2020,13, 16, doi:10.3390/jrfm13010016 .......... 93 Alex Golodnikov, Viktor Kuzmenko and Stan Uryasev CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles Reprinted from: J. Risk Financial Manag. 2019,12, 107, doi:10.3390/jrfm12030107 ..........111 Sel Ly, Kim-Hung Pho, Sal Ly and Wing-Keung Wong Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas Reprinted from: J. Risk Financial Manag. 2019,12, 42, doi:10.3390/jrfm12010042 ..........133 L´aszl´o Nagy and Mih´aly Ormos Friendship of Stock Market Indices: A Cluster-Based Investigation of Stock Markets Reprinted from: J. Risk Financial Manag. 2018,11, 88, doi:10.3390/jrfm11040088 ..........161 Rafiuddin Ahmed and Rafiqul Bhuyan Capital Structure and Firm Performance in Australian Service Sector Firms: A Panel Data Analysis Reprinted from: J. Risk Financial Manag. 2020,13, 214, doi:10.3390/jrfm13090214 ..........177 Jian Huang and Huazhang Liu Examination and Modification of Multi-Factor Model in Explaining Stock Excess Return with Hybrid Approach in Empirical Study of Chinese Stock Market Reprinted from: J. Risk Financial Manag. 2019,12, 91, doi:10.3390/jrfm12020091 ..........193 v About the Editors Wing-Keung Wong obtained his Ph.D. from the University of Wisconsin-Madison, the USA with a major in Business Statistics (Statistics and Finance) and obtained his Bachelor degree from the Chinese University of Hong Kong, Hong Kong, with a major in Mathematics and a double minor in Economics and Statistics. Currently, he is a Chair Professor at the Department of Finance, Asia University. He was a Full Professor at the Department of Economics, Hong Kong Baptist University, and Deputy Director at Risk Management Institute, National University of Singapore. Professor WONG appears in “Who’s Who in the World” and gets Albert Nelson Marquis Lifetime Achievement Award. 2017, Marquis Who’s Who. His Erdos number is 3. He is ranked top 1% by Social Science Research Network and in the list of top Taiwan economists and Asian economists and top economists by RePEc. He has published more than three hundred papers including papers published in some top journals. He has more than 9400 citations in Google Scholar, more than 6900 citations in Researchgate, and more than 3200 citations in Mendeley. His h-index is 54, (40 since 2015) and i10-index is 205, (182 since 2015) by Google Scholar citation. He has been serving international academies, Government, society, and universities, providing consultancy to several Government departments and corporations, and giving lectures and seminars to several universities. For example, he has been serving as editor, guest leading editor, advisor, associate editor for some international journals, appointed as an advisor/member of various international associations/institutes, serving as a referee for many journals/conferences, supervising solely or jointly several overseas graduate students, appointed as an external reviewer and external examiner by other universities, and invited by many universities/institutions to present papers or conduct seminars. Xu Guo is an Associate Professor in School of Statistics, Beijing Normal University. He was an Assistant Research Professor in the Department of Statistics, Pennsylvania State University from Sep. 2018 to Feb. 2020. He got his Ph.D. degree from the Department of Mathematics, Hong Kong Baptist University in 2014. His research interests are model specification testing, missing data, semiparametric regression analysis, risk management and decision making under risk and uncertainty. Up to now, he has published more than 30 papers in numerous international journals, such as Journal of the Royal Statistical Society: Series B, Biometrika, Journal of Multivariate Analysis, Computational Statistics & Data Analysis, Insurance: Mathematics and Economics, Economics Letters, Economic Modelling, North American Journal of Economics and Finance, etc. Sergio Ortobelli Lozza is a full professor in Mathematical Finance at the University of Bergamo. He holds a Ph.D. in Computational Methods for Financial and Economic Forecasting and Decisions from the University of Bergamo. His research, published in various academic journals in mathematics and finance, focuses on the application of probability theory and operational research to portfolio theory, risk management, and option theory. He taught numerous courses at the Universities of Bergamo, Calabria and Milan, including basic and advanced calculus, measure theory, stochastic processes, portfolio theory, and advanced mathematical finance. From 1999 till 2020 he wrote more than 170 refereed scientific works together with several well known Italian and international professors. Some of these papers have been well recognized by the scientific community. vii JRFM 2020,13, 226 Severalstudies have examinedlinkagesbetween the equitymarkets during the1997 Asian financial crisis (In Francis et al. 2001;Wan and Wong 2001;Yang et al. 2003), and the last 2008 global financial crisis (Yilmaz 2010;Cheung et al. 2007;Kim et al. 2015;Li and Giles 2015; Lean et al. 2015 ; Vieito et al. 2015 ; Zhu et al. 2019) and some studies, see, for example, Fung et al. (2011) and Guo et al. (2017) , develop theories to explain that crisis. However, the linkages between equity markets during the Chinese stock market crash of 2015have been rarely examined. The Chinese stock market experienced a major crash in 2015 (Zhu et al. 2017;Yousaf and Hassan 2019;Yousaf et al. 2020;Yousaf and Ali 2020). The CSI 300 index increased before reaching 5178 points in mid-June of 2015. Then, it took a roller-coaster ride and dropped by up to 34% in just 20 days; Chinese stock market also lost 1000 points within just one week. Around 50% of Chinese stocks lost more than half of their pre-crash market value. The Chinese stock market crash affected many other commodities and financial markets, including Asian (Allen 2015) and the US stock markets (The causes and consequences of China’s market crash 2015). Despite the importance of the Chinese crash for international portfolio managers, few studies have examined how it was transmitted to other national financial markets. Xiong et al. (2018) investigate the time-varying correlation between economic policy uncertainty and Chinese stock market returns during the Chinese crash of 2015, while Yousaf and Hassan (2019) examine the linkages between crude oil and emerging Asian stock markets during this crisis. However, research on the linkages between stock markets has not been investigated yet for the 2015 Chinese crash. Therefore, this study focuses on providing useful insights about this issue for the Asian region, which has attracted considerable attention from finance practitioners and academics due to its position as the center of global economic activity in the 21st century 1 . While using the US and Chinese equity markets as the indicators of global markets, we explore whether global investors can get the maximum benefit of diversification by adding emerging Asian market stocks in their portfolios. In literature, several studies have examined the linkages between the global (US and China) and emerging Asian equity markets during the Asian financial crisis, and the US financial crisis (Yang et al. 2003;Beirne et al. 2013;Jin 2015; Li and Giles 2015), but not in the Chinese stock market crash. We address the above-mentioned literature gap by examining the return and volatility spillover from the US and China to the emerging Asian equity markets during the Chinese stock market crash by using the VAR-AGARCH model that was developed by Ling and McAleer (2003). Moreover, we examine the ability of spillovers during the full sample period and the 2008 US financial crisis to provide comparative insights to investors about whether the impact of the Chinese crash on equity market spillovers was different from those in the other sample periods. Our findings show that return spillover was observed from the US and China to the Asian stock market during the US financial crisis and the Chinese stock market crash. Volatility was also transmitted from the US to the majority of the Asian stock markets during the Chinese stock market crash. However, volatility was transmitted from China to the majority of the Asian stock markets during the US financial crisis. Overall, as the return and volatility transmission vary across pairs of stock markets and financial crises, investors have to adjust their asset allocations from time to time to improve their profits. Therefore, we also estimate the optimal weights and hedge ratios during the full sample period, the US financial crisis, and the Chinese stock market crash. Our findings imply that fewer US stocks were required to minimize the risk for Asian stock investors during the US financial crisis compared to during the Chinese crash. In contrast, fewer Chinese stocks were needed to minimize the risk for Asian stock investors during the Chinese stock market crash as compared to during the US crisis. Overall, our findings draw several important implications for risk management and portfolio diversification that could be useful for investors and policymakers related to the US and Asian stock markets. 1Source: https://www.ft.com/content/520cb6f6-2958-11e9-a5ab-ff8ef2b976c7. 6 JRFM 2020,13, 226 The rest of the paper is organized as follows: Section 2provides the literature review. Section 3 describes the data and methodology. Section 4reports the findings, and Section 5concludes the whole discussion. 2. Literature Review The analysis of both return and volatility spillover between stock markets is crucial for investors in designing optimal portfolios. According to modern portfolio theory, the gains of international portfolio diversification decrease when the correlation of security returns increases and vice versa. Michaud et al. (1996) discuss the advantages of a low correlation between the developed and emerging markets for international portfolio diversification. Due to this trend, investors can benefit by investing in emerging markets that are weakly interconnected with developed markets. However, this correlation becomes higher during an economic crisis, suggesting low diversification benefits when diversification is most required. 2.1. Linkages between US, China, and Asian Stock Markets Many studies have been conducted to investigate the link between different stock markets during the last three decades. Liu and Pan (1997) examine the mean and the volatility spillover from the US and Japan to Singapore, Hong Kong, Thailand, and Taiwan. The results show that the US market is more dominant than the Japanese stock market in transmitting return and volatility effects to four Asian stock markets. Huang et al. (2000) investigate the link between the US, Japan, and South China growth triangle. The US stock market significantly and dominantly affects the south Chinese growth triangle compared to the impact of Japan on China’s stock market. The return spillover has been also found to be significant from the US to Hong Kong and Taiwan, and from Hong Kong to the Taiwanese stock market. Miyakoshi (2003) estimates the return and volatility spillover between the US, Japan, and seven Asian stock markets (South Korea, Taiwan, Singapore, Thailand, Indonesia, and Hong Kong). It finds a significant return spillover from the US to Asian markets, whereas no return spillover is found from Japan to Asian stock markets. Moreover, the volatility spillover from Japan to other Asian stock markets is observed to be dominant as compared to the volatility spillover from the US to Asian stock markets. Johansson and Ljungwall (2009) examine the association between stock markets of China, Hong Kong, and Thailand. It reports a significant return spillover from Taiwan to China and the Hong Kong stock market. In contrast, volatility spillover runs from Hong Kong to Taiwan and from Taiwan to the Chinese stock market. Zhou et al. (2012) estimate the spillover between Chinese and international (the US, the UK, France, Germany, Japan, India, Hong Kong, Taiwan, South Korea, and Singapore) stock markets from 1996 to 2009. Before 2005, the Chinese stock market was affected by spillover from other international markets. After 2005, volatility spillover was significantly transmitted from China to most of the other international stock markets. Chien et al. (2015) report on the significant financial integration between China and the ASEAN-5 (Indonesia, Malaysia, the Philippines, Singapore, and Thailand) stock markets. Huo and Ahmed (2017) provide significant evidence of both return and volatility effects from China to the Hong Kong equity market. 2.2. Linkages between US, China, and Asian Stock Markets during Crisis Manystudies haveexaminedthelinkagesbetween marketsduring crisisperiods. Yang et al. (2003) investigate the short and long-run relationship between the US, Japan, and ten Asian stock markets, mainly focusing on the Asian financial crisis of 1997–1998. This study reports a strengthened long-run co-integration among these stock markets during the Asian financial crises. The degree of integration is found to change during crises and non-crisis periods. Beirne et al. (2013) look at the volatility spillover from developed to emerging stock markets during periods of turbulence in mature stock markets. It finds that volatility in mature markets affects the conditional variances in emerging stock markets. 7 JRFM 2020,13, 226 Moreover, the spillover effect from developed to emerging markets is also changed during times of turbulence in mature markets. Jin (2015) examines the mean and volatility spillover between China, Taiwan, and Hong Kong. It reveals that financial crises have a substantial and positive effect on expected conditional variances, but also that the size and dynamics of impacts vary from market to market. Li and Giles (2015) investigate the volatility spillover across the US, Japan, and four Asian developing economies during the Asian financial crisis of 1997 and the US financial crises of 2008. The results revealed that there is a presence of a volatility spillover effect from the USA to Asian developing economies and Japan. This study also finds a bidirectional volatility spillover between US and Asian markets that occurred during the Asian financial crisis. Gkillas et al. (2019) explore integration and co-movement between 68 international stock markets (including in the Asian region) during the US financial crisis. Overall, several studies have examined the return and volatility spillover from the US to Asian markets during the Asian financial crisis of 1997 and the US financial crisis of 2008. However less has been done on both return and volatility transmission from China to the emerging Asian stock markets during the US financial crisis and the Chinese stock market crash. Moreover, no study has examined return and volatility spillovers from the US to the emerging Asian stock markets during the Chinese crash. Therefore, this study addresses these above-mentioned literature gaps. 3. Data and Methodology 3.1. Data We based our empirical investigation on daily data of accepted benchmark stock indices of nine Asian countries and the US. The Emerging Asian stock markets include China, India, South Korea, Indonesia, Pakistan, Malaysia, the Philippines, Thailand, and Taiwan. The emerging Asian economies were selected from the list of countries, including the MSCI (Morgan Stanley Capital International) emerging market index. The data of stock indices were taken from the Data Stream database. The index is assumed to be the same on non-trading days (holidays except weekends) as on the previous trading day, as suggested by Malik and Hammoudeh (2007) and many others.2 This study used the full sample period from 1 January 2000 to 30 June 2018 and studies the following two sub-samples: the first sub-period from 1 August 2007 to 31 July 2010 presenting the period with the US financial crisis; and the second sub-period from 1 June 2015 to 30 May 2018 presenting the period with the Chinese Stock market crash. We note that Yousaf and Hassan (2019) also use similar time frames for the US financial crisis and the Chinese stock market crash. This study followed He (2001) and many others to use three-year data for each crisis for short-run analysis. Changes in market correlations take place continuously not only as a result of crises but also due to the consequences of many financial, economic, and political events. Moreover, Arouri et al. (2015) have also used the daily data covering periods shorter than three years to estimate the return and volatility spillover between gold and Chinese stock markets in US financial crisis by applying the VAR-GARCH model. The difference in the opening time of US and Asian stock markets was adjusted by using lags where necessary. 2 In time-series data, if there are missing values, there are two ways to deal with the incomplete data: (a) omit the entire record that contains information, (b) Impute the missing information. We used 10 series in this paper and if we wanted to omit the missing data for one series then the data of all other nine series needed to be removed as well for that specific day. So, if we omitted the data for days where values are missing at specific days, then we lost the data for many days, which is not good for getting realistic results. Therefore, we followed many studies, for example, Malik and Hammoudeh (2007), and imputed the missing data by using previous day data. Indeed, there are many methods used to impute the missing data and every method has pros and cons, but we used this imputation method following past literature. Moreover, our missing observations were less than one percent of overall data, therefore the imputation method should not create a larger effect than that on results. 8 JRFM 2020,13, 226 3.2. Methodology This study estimated the return and volatility transmissions using the Vector AutoregressiveGeneralized Autoregressive Conditional Heteroskedasticity (VAR-AGARCH) model proposed by McAleer et al. (2009). Several studies have previously used the VAR-GARCH and VAR-AGARCH model to estimate spillover between different asset classes (Arouri et al. 2011;Arouri et al. 2012; Jouini 2013;Yousaf and Hassan 2019). This model includes the Constant Conditional Correlation (CCC-GARCH) model of Bollerslev (1990) as a special case. The selection of the model was based on three reasons. First, the most commonly used multivariate models are the BEKK (Baba, Engle, Kraft, and Kroner) model and the DCC (dynamic conditional correlation) model. These models often suffer from unreasonable parameter estimates and data convergence problems (Bouri 2015). The VAR-AGARCH model overcomes these problems regarding parameters and data convergence. Second, it incorporates asymmetry into the model. Third, this model can be used to calculate the optimal weights and hedge ratios. Ling and McAleer (2003) propose the multivariate VAR-GARCH Model to estimate the return and volatility transmission between the different series. For two series, the VAR-GARCH model has the following specifications for the conditional mean equation3: Rt=μ+FRt−1+etwith et=D1/2 tηt, (1) in which Rt represents a 2 × 1 vector of daily returns 4 on the stocks xand yat time t, μ denotes a 2 × 1 vector of constants, F is a 2 × 2 matrix of parameters measuring the impacts of own lagged and cross mean transmissions between two series, et is the residual of the mean equation for the two stocks returns series at time t, ηt indicates a 2 × 1 vector of independently and identically distributed random vectors, and D1/2 t =diag ( hx t , hy t ), where hx t and hy t representing the conditional variances of the returns for stocks x and y, respectively, are given as hx t=Cx+a11ex t−12+a21ey t−12+b11hx t−1+b21hy t−1, (2) hy t=Cy+a12ex t−12+a22ey t−12+b12hx t−1+b22hy t−1. (3) Equations (2) and (3) reveal how shock and volatility are transmitted over time and across the markets under investigation. Furthermore, the conditional covariance between returns from two different stock markets can be estimated as follows: hx,y t=p×hx t×hy t. (4) In the above equation, hx,y t refers to the conditional covariance between the returns of two stock markets ( x , y ) at time t. Moreover, p indicates the constant conditional correlation between the returns of two stock markets (x,y). The VAR − GARCH model assumes that positive or negative shocks have the same impact on the conditional variance. To estimate the spillover between different markets, we estimated spillover between two stock markets by using the VAR − AGARCH Model proposed by the McAleer et al. (2009). 3 Several studies, for example, Hammoudeh et al. (2009), Arouri et al. (2011), and Dutta et al. (2018) have applied the VAR for the conditional mean equation. 4Stock Returnst=ln Stock Indext Stock Indext−1. 9 JRFM 2020,13, 226 The VAR AGARCH model incorporates asymmetry in the model as well. Specifically, instead of using Equations (2) and (3), the conditional variance of the VAR AGARCH model was defined as follows: hx t=Cx+a11Aex t−12+a21Aey t−12+b11hx t−1+b21hy t−1+a11Bex t−1(ex t−1<0)), (5) hy t=Cy+a12Aex t−12+a22Aey t−12+b12hx t−1+b22hy t−1+a22Bey t−1(ey t−1<0)). (6) In the above equations, A ex t−12 and Bex t−1(ex t−1<0)) as well as Aey t−12 and Bey t−1(ey t−1<0)) reveal the relationships between a market’s volatility and both positive and negative own lagged returns, respectively (Lin et al. 2014). Equations (5) and (6) show that the conditional variance of each market depends upon its past shock and past volatility, as well as the past shock and past volatility of other markets. In Equation (5), ex t−12 and ey t−12 explain how the past shocks of both x and y affect the current conditional volatility of x . Moreover, hx t−1 and hy t−1 measure how the past volatilities of both x and y affect the current conditional volatility of x . The parameters of the VAR-AGARCH model can be estimated by using the Quasi-Maximum Likelihood estimation (QMLE) and using the BFGS algorithm.5 The estimates of the VAR-AGARCH model can be used to calculate optimal portfolio weights. This study followed Kroner and Ng (1998) to calculate the optimal portfolio weights for the pairs of the stock market (x,y) as: wxy,t=hy,t−hxy,t hx,y−2hxy,t+hy,t(7) wxy,t=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0if Wxy,t<0 wxy,tif 0≤wxy,t≤1 1if wxy,t>1 , where wxy,t is the weight of stock( x ) in a $1 stock(x)-stock( y ) portfolio at time t, hxy,t is the conditional covariance between the two stock markets, hx,t and hy,t are the conditional variance of stock( x ) and stock(y), respectively, and 1-wxy,tis the weight of stock(y) in a $1 stock(x)-stock(y) portfolio. It is also essential to estimate the risk-minimizing optimal hedge ratios for the portfolio of different stocks. The estimates of the VAR-AGARCH model can also be used to calculate optimal hedge ratios. This study followed Kroner and Sultan (1993) to calculate the optimal hedge ratios as: βxy,t=hxy,t hy,t, (8) where βxy,t represents the hedge ratio. This shows that a short position in the stock ( y ) market can hedge a long position in the stock (x). Lastly, RATS 10.0 software is used for estimations. 4. Empirical Results 4.1. Descriptive Analysis Table 1reports the summary statistics of the daily returns for the US, China, and eight emerging Asian stock markets, namely India, Korea, Indonesia, Pakistan, Malaysia, the Philippines, Thailand, and Taiwan. The average returns of the Pakistani stock market are the highest out of these markets, whereas the lowest returns are found in the US stock market during the full sample period. The unconditional volatility is lower in Malaysia and the US market and is highest in the 5 Arouri et al. (2011), Sadorsky (2012), and Allen et al. (2013) use the Quasi-Maximum Likelihood estimation (QMLE) and use the BFGS algorithm to estimate the parameters in the VAR-GARCH model. 10 JRFM 2020,13, 226 Chinese stock market. The skewness is negative in all cases, kurtosis is higher than 3 for all stocks, and Jarque–Bera statistics do not accept the hypothesis of the normality for all stocks. Moreover, we applied the Ljung–Box Q test for autocorrelation to the standardized residuals and squared standardized residuals. The coefficients both Q(12) and Q 2 (12) were found to be signifcant for all series. ARCH effects were also statistically significant for all series.6 Table 1. Summary Statistics. Mean Median Max Min Std. Dev. Skewness Kurtosis Jarque-Bera Q-Stat ARCH Test USA 0.00016 0.00055 0.10958 −0.09470 0.01200 −0.20353 11.57202 14802.7 a37.24 a206.42 a CHN 0.00045 0.00096 0.09401 −0.09256 0.01570 −0.31725 8.21506 5547.4 a54.64 a180.10 a IND 0.00050 0.00094 0.15990 −0.11809 0.01472 −0.22234 10.54239 11474.1 a84.62 a283.89 a INDO 0.00046 0.00113 0.07623 −0.10954 0.01357 −0.85402 10.92376 13206.3 a154.0 a457.66 a KOR 0.00028 0.00080 0.11284 −0.12805 0.01509 −0.57337 9.64860 9149.3 a24.06 a210.02 a MYS 0.00020 0.00041 0.04503 −0.09979 0.00816 −0.85496 13.33067 22038.9 a226.8 a267.36 a PAK 0.00081 0.00109 0.08507 −0.07741 0.01359 −0.34875 6.83764 3058.01 a165.5 a594.62 a PHL 0.00038 0.00055 0.16178 −0.13089 0.01309 0.23024 19.78304 56658.3 a96.40 a161.15 a TAIW 0.00018 0.00070 0.06525 −0.09936 0.01356 −0.27454 6.59593 2659.6 a77.68 a201.54 a THA 0.00044 0.00064 0.10577 −0.16063 0.01316 −0.70520 12.86191 19948.5 a70.19 a656.27 a Notes: aindicates the statistical significance at 1% level. 4.2. Return, Shock and Volatility Spillover Analysis 4.2.1. Stock Market Linkages between the USA and Asia from the Full Sample Period Table 2represents the return and volatility spillover between US and Asian stock markets during the full sample period. The lagged stock returns were found to significantly affect the current stock returns in all studied Asian stock markets except for Korea. This highlights the possibility of short-term predictions of current returns through past returns in the Asian stock markets. Moreover, the autoregressive term of the USA stock market was found to be significant as well. This depicts that past returns help to predict current returns in the American stock market. The estimate of return spillover from one market to another market can be estimated by using the coefficient of lagged return of one market (i.e., the US) onto another market (i.e., India) and vice versa. The return spillover from the USA to all Asian stock markets is significant. This implies that US stock market prices play an important role in predicting the prices of all Asian stock markets during the full sample period. These results are in line with the findings of Huyghebaert and Wang (2010), which find a significant return spillover from the USA to Asian markets. This shows that the effect of the returns of the American stock market are significantly transmitted to the Asian stock markets. However, the return spillover from all Asian stock markets to the USA was found to be insignificant. This implies that Asian stock market prices are not helpful in predicting the prices of the US stock market during the full sample period. 6 We applied both Augmented Dickey–Fuller (ADF), and Phillip–Perron (PP) tests to examine the stationarity of all returns series and found that all returns series are stationary, but we do not report these results in Table form for the sake of brevity. 11 JRFM 2020,13, 226 Table 2. Estimates of bivariate Vector Autoregressive-Generalized Autoregressive Conditional Heteroskedasticity (VAR-AGARCH) for the USA and Asian stock markets during a full sample period. IND USA INDO USA KOR USA MYS USA PAK USA PHL USA TAIW USA THA USA Panel A: Mean Equation Constant 4.90 × 10−4a (0.001) 2.48 × 10−4c (0.023) 5.13 × 10−4a (0.001) 2.40 × 10−4b (0.022) 1.55 × 10−4 (0.252) 2.08 × 10−4c (0.059) 1.08 × 10−4 (0.195) 2.46 × 10−4b (0.026) 8.79 × 10−4a (0.000) 2.07 × 10−4b (0.048) 3.53 × 10−4b (0.030) 2.20 × 10−4a (0.001) 1.28 × 10−4 (0.296) 2.64 × 10−4b (0.016) 6.11 × 10−4a (0.000) 2.71 × 10−4c (0.010) ro t−1 0.100 a (0.000) −7.65 × 10−4 (0.933) 0.136 a (0.000) −0.010 (0.260) 0.017 (0.241) 0.016 (0.126) 0.168 a (0.000) −0.022 (0.232) 0.168 a (0.000) −4.52 × 10−3 (0.648) 0.118 a (0.000) −1.87 × 10−3 (0.132) 0.054 a (0.000) 0.014 (0.232) 0.093 a (0.000) −8.99 × 10−3 (0.423) ru t−1 0.231 a (0.000) −0.041 a (0.009) 0.301 a (0.000) −0.032 b (0.031) 0.420 a (0.000) −0.037 b (0.017) 0.201 a (0.000) −0.034 b (0.035) 6.40 × 10−3b (0.017) −0.030 b (0.040) 0.421 a (0.000) −0.030 b (0.014) 0.386 a (0.000) − 0.037 b (0.021) 0.224 a (0.000) − 0.037 b (0.021) Panel B: Variance Equation Constant 2.68 × 10−6a (0.000) 1.72 × 10−6a (0.000) 1.46 × 10−5a (0.000) 1.47 × 10−6a (0.000) 1.40 × 10−6a (0.000) 1.71 × 10−6a (0.000) 6.67 × 10−7a (0.000) 1.70 × 10−6a (0.000) 7.61 × 10−6a (0.000) 2.30 × 10−6a (0.000) 3.82 × 10−5a (0.000) −1.46 × 10−6a (0.000) 1.11 × 10−6a (0.000) 1.64 × 10−6a (0.000) 4.90 × 10−6a (0.000) 1.72 × 10−6a (0.000) eo t−120.056 a (0.000) 0.038 a (0.000) 0.118 a (0.000) 0.057 a (0.000) 0.053 a (0.000) 0.020 a (0.001) 0.090 a (0.000) 7.04 × 10−3a (0.002) 0.100 a (0.000) −1.84 × 10−3a (0.000) 0.125 a (0.000) 0.041 a (0.000) 0.037 a (0.000) 0.035 a (0.000) 0.080 a (0.000) 8.93 × 10−3 (0.315) eu t−128.17 × 10−3a (0.001) −5.68 × 10−3 (0.401) 2.30 × 10−3 (0.538) −0.011 c (0.052) 6.81 × 10−3b (0.032) −0.011 (0.108) 4.97 × 10−3 (0.524) −0.011 c (0.073) 5.98 × 10−3a (0.000) −6.98 × 10−3c (0.083) −4.96 × 10−3a (0.000) −0.014 a (0.000) 6.71 × 10−3 (0.110) −8.59 × 10−3 (0.228) 7.89 × 10−3b (0.012) −9.45 × 10−3 (0.119) ho t−1 0.877 a (0.000) −0.030 a (0.000) 0.579 a (0.000) 0.085 (0.130) 0.906 a (0.000) −0.018 (0.290) 0.853 a (0.000) 2.77 × 10−3 (0.378) 0.791 a (0.000) 1.71 × 10−4 (0.369) 0.506 a (0.000) −0.029 (0.142) 0.925 a (0.000) − 0.030 a (0.000) 0.813 a (0.000) 5.47 × 10−3 (0.599) hu t−1 −4.98 × 10−3b (0.031) 0.896 a (0.000) 6.82 × 10−4 (0.915) 0.908 a (0.000) −2.59 × 10−3 (0.407) 0.900 a (0.000) 0.010 (0.355) 0.894 a (0.000) −7.80 × 10−3a (0.000) 0.892 a (0.000) 0.036 a (0.000) 0.903 a (0.000) 2.08 × 10−4 (0.968) 0.897 a (0.000) −4.67 × 10−3c (0.076) 0.900 a (0.000) Asymmetry 0.098 a (0.000) 0.172 a (0.000) 0.199 a (0.000) 0.164 a (0.000) 0.068 a (0.000) 0.173 a (0.000) 0.060 a (0.000) 0.183 a (0.000) 0.143 a (0.000) 0.193 a (0.000) 0.143 a (0.000) 0.167 a (0.000) 0.051 a (0.000) 0.167 a (0.000) 0.157 a (0.000) 0.175 a (0.000) Panel C: Constant Conditional Correlation p0,u0.198 a (0.000) 0.104 a (0.000) 0.185 a (0.000) 0.082a (0.000) 0.031b (0.030) 0.054 a (0.000) 0.142 a (0.000) 0.134 a (0.000) Panel D: Diagnostic Tests LogL 30500.4 30678.873 30692.048 33342.351 30597.4 30633.0 30931.2 30683.3 AIC −9.520 −9.912 −10.060 −11.074 −10.044 −10.537 −10.030 −10.174 SIC −9.126 −9.518 −9.671 −10.680 −9.040 −10.143 −9.780 −9.779 12 JRFM 2020,13, 226 Table 2. Cont. IND USA INDO USA KOR USA MYS USA PAK USA PHL USA TAIW USA THA USA Panel D: Diagnostic Tests JB 533.870 a (0.000) 381.111 a (0.000) 688.360 a (0.000) 528.060 a (0.000) 870.230 a (0.000) 391.040 a (0.000) 1921.200 a (0.000) 500.410 a (0.000) 3442.14 a (0.000) 508.010 a (0.000) 10842.7 a (0.000) 438.390 a (0.000) 295.380 a (0.000) 462.680 a (0.000) 20331.6 a (0.000) 560.990 a (0.000) Q(12) 16.143 (0.185) 22.788 (0.303) 9.012 (0.702) 15.021 (0.240) 5.829 (0.924) 18.049 (0.114) 13.179 (0.356) 14.149 (0.291) 55.870 c (0.097) 15.244 (0.228) 21.845 b (0.039) 15.008 (0.241) 18.533 c (0.100) 19.566 c (0.076) 33.450 c (0.087) 16.279 (0.179) Q2(12) 13.026 (0.367) 9.413 (0.667) 19.566 c (0.076) 8.416 (0.752) 5.242 (0.949) 9.919 (0.623) 14.799 (0.253) 10.027 (0.614) 5.970 (0.918) 0.000 a (0.000) 1.823 (0.923) 0.000 a (0.000) 16.456 (0.171) 10.976 (0.531) 0.928 (0.965) 11.617 (0.477) Notes: The number of lags for VAR was decided using SIC (Schwartz information criterion) and AIC (Akaike information criterion) criteria. JB, Q(12), and Q 2 (12) indicate the empirical statistics of the Jarque–Bera test for normality, while Ljung–Box Q statistics with order 12 for autocorrelation were applied to the standardized residuals and squared standardized residuals, respectively. USA, United States of America; IND, India; INDO, Indonesia; KOR, South Korea; MYS, Malaysia; PAK, Pakistan; PHL, the Philippines; TAIW, Taiwan; THA, Thailand. Values in parentheses are the p-values. a,b,cindicate the statistical significance at 1%, 5%, and 10%, respectively. 13 JRFM 2020,13, 226 ARCH coefficient captures the shock dependence, while the GARCH coefficient captures the persistence of volatility in conditional variance equations. The findings reveal that the sensitivity of past own shocks (ARCH term) is significantly positive for all Asian Stock Markets in the short run. In addition, the sensitivity of past own volatility (the GARCH term) was found to be significant for all stock markets (including the Asian and American Markets), thus the ARCH (1) volatility model was determined to be more appropriate in this case. The coefficient of past own volatility was than the coefficients of past own shocks in all Asian stock markets, implying that past own volatilities are more critical for prediction of future volatility as compared to past own shocks. The conditional volatility of India’s, South Korea’s, the Philippines’, Pakistan’s, and Thailand’s stock markets was found to be significantly affected by shocks in the American stock market. These results are similar to the findings of Syriopoulos et al. (2015), which show that past shocks in the American market significantly affect the market volatility of India, Brazil, and Russia. Therefore, this implies that shock in the American stock market leads to an increase in the volatility of the majority of Asian markets. The past volatility of the American stock market significantly influenced the conditional volatility of India’s, The Philippines’, Pakistan’s, and Thailand’s stock markets. These results confirm the previous findings of Li and Giles (2015), which finds a significant volatility spillover from the USA to emerging Asian stock markets. Further, Syriopoulos et al. 2015 found a significant volatility spillover from the USA to India. In addition, the past volatility of the majority of Asian Markets (Except for India and Taiwan) has not been significantly transmitted to the American stock market. The asymmetric coefficients of all Asian stock markets are significant and positive, showing that negative news (or unexpected shocks) for the American stock market has more ability to increase the volatility of all Asian Stock markets as compared to positive news. Besides, the asymmetric coefficient of the American stock market is positively significant, demonstrating that negative unexpected shocks in Asian Stock markets will increase the volatility more in the American Stock market as compared to a positive shock. Constant conditional correlation (CCC) is positively significant for all pairs of stock markets. However, cross-market correlation is weak in almost all pairs, indicating that investors can get substantial gains by having these pairs in the same portfolio. 4.2.2. Stock Market Linkages between China and Asia from the Full Sample Period Table 3reports the return and volatility spillover between the Chinese and other Asian stock markets during the full sample period. The current stock returns of Asian stock markets are significantly affected by their own lagged stock returns. This highlights the possibility of short-term predictions of current returns through past returns in the Asian stock markets. Moreover, Chinese stock returns are also significantly influenced by their own single period lagged returns. These findings depict that stock prices can be predicted in the short term in the Chinese stock market. The return spillover is not significant from China to the majority of other Asian markets except for the Indian, Philippines, and Thai stock markets. Besides, the return transmission from Asian markets to the Chinese market is insignificant except for in the case of the Indian Stock market. Moreover, there is a presence of bi-directional return transmission between the Indian and Chinese stock markets. This implies that Chinese (Indian) stock market prices play an important role in predicting the prices of Indian (Chinese) stock markets during the full sample period. The coefficient of past own shock of all Asian markets (including China) was found to be significant; thus, past shocks affect current conditional volatility in Asian stock markets. Besides, the sensitivity of past own volatility for all Asian markets was found to be significant as well. 14 JRFM 2020,13, 226 Table 3. Estimates of bivariate VAR−AGARCH for China’s and other Asian stock markets during the full sample period. IND CHN INDO CHN KOR CHN MYS CHN PAK CHN PHL CHN TAIW CHN THA CHN Panel A: Mean Equation Constant 5.06 × 10−4a (0.000) 3.25 × 10−4c (0.053) 5.37 × 10−4a (0.000) 3.50 × 10−4b (0.022) 2.71 × 10−4c (0.065) 3.26 × 10−4b (0.043) 1.69 × 10−4b (0.036) 3.74 × 10−4b (0.023) 8.64 × 10−4a (0.000) 3.39 × 10−4b (0.029) 3.32 × 10−4b (0.031) 3.50 × 10−4b (0.030) 2.18 × 10−4c (0.089) 3.51 × 10−4b (0.027) 5.05 × 10−4a (0.000) 2.12 × 10−4 (0.139) ro t−1 0.137 a (0.000) 0.036 a (0.004) 0.152 a (0.000) 0.017 (0.237) 0.083 a (0.000) 0.011 (0.345) 0.191 a (0.000) 0.018 (0.456) 0.167 a (0.000) 6.27 × 10−3 (0.576) 0.144 a (0.000) −0.028 b (0.048) 0.100 a (0.000) 0.017 (0.241) 0.125 a (0.000) 0.018 (0.183) rc t−1 −0.018 c (0.096) 0.049 a (0.001) −6.05 × 10−3 (0.603) 0.050 a (0.000) −0.013 (0.269) 0.055 a (0.000) 7.11 × 10−3 (0.295) 0.053 a (0.000) 0.010 (0.272) 0.055 a (0.000) 0.025 b (0.043) 0.055 a (0.000) 2.26 × 10−3 (0.841) 0.051 a (0.000) −0.020 (0.103) 0.049 a (0.001) Panel B: Variance Equation Constant 2.14 × 10−6a (0.000) 1.14 × 10−6a (0.001) 4.30 × 10−6a (0.000) 1.19 × 10−6a (0.002) 1.00 × 10−6a (0.000) 1.37 × 10−6a (0.000) 5.84 × 10−7a (0.000) 1.26 × 10−6a (0.000) 7.29 × 10−6a (0.000) 1.15 × 10−6a (0.002) 1.36 × 10−5a (0.000) 1.19 × 10−6b (0.048) 9.17 × 10−7a (0.000) 1.21 × 10−6a (0.000) 1.01 × 10−6a (0.001) 1.19 × 10−6a (0.001) eo t−120.051 a (0.000) −1.95 × 10−3 (0.413) 0.074 a (0.000) −1.77 × 10−4 (0.950) 0.031a (0.000) 7.03 × 10−3 (0.241) 0.074 a (0.000) 3.80 × 10−4 (0.668) 0.100 a (0.000) −9.47 × 10−4 (0.509) 0.053 a (0.000) 8.20 × 10−3b (0.016) 0.033 a (0.000) 3.74 × 10−3b (0.034) 0.070 a (0.000) 0.023 a (0.000) ec t−12−5.03 × 10−3c (0.081) 0.069 a (0.000) 9.29 × 10−3c (0.099) 0.066 a (0.000) −1.32 × 10−3 (0.551) 0.070 a (0.000) −6.92 × 10−3 (0.422) 0.069 a (0.000) 3.56 × 10−3 (0.453) 0.064 a (0.000) 8.77 × 10−4 (0.754) 0.068 a (0.000) 9.94 × 10−3a (0.000) 0.069 a (0.000) 7.51 × 10−3b (0.025) 0.062 a (0.000) ho t−1 0.876 a (0.000) 9.48 × 10−3a (0.001) 0.840 a (0.000) 0.011 b (0.041) 0.929 a (0.000) −6.06 × 10−3a (0.000) 0.871 a (0.000) 2.91 × 10−3b (0.022) 0.792 a (0.000) 1.39 × 10−3 (0.458) 0.797 a (0.000) 2.22 × 10−3 (0.656) 0.922 a (0.000) −1.34 × 10−3 (0.508) 0.878 a (0.000) −8.97 × 10−3a (0.001) hc t−1 7.61 × 10−3b (0.033) 0.920 a (0.000) −4.86 × 10−3 (0.482) 0.923 a (0.000) 2.24 × 10−3 (0.314) 0.918 a (0.000) 0.010 (0.278) 0.921 a (0.000) −1.29 × 10−3 (0.816) 0.923 a (0.000) 1.53 × 10−3 (0.816) 0.921 a (0.000) −8.00 × 10−3a (0.002) 0.921 a (0.000) −3.06 × 10−3 (0.497) 0.919 a (0.000) Asymmetry 0.105 a (0.000) 0.016 b (0.048) 0.096 a (0.000) 0.012 (0.138) 0.067 a (0.000) 0.019 b (0.023) 0.073 a (0.000) 0.016 c (0.055) 0.139 a (0.000) 0.019 b (0.012) 0.096 a (0.000) 0.018 b (0.035) 0.072 a (0.000) 0.016 b (0.050) 0.073 a (0.000) 0.030 a (0.001) Panel C: Constant Conditional Correlation p0,c0.158 a (0.000) 0.164 a (0.000) 0.211 a (0.000) a (0.000) 0.047 a (0.001) 0.131 a (0.000) 0.218 a (0.000) 0.146 a (0.000) Panel D: Diagnostic Tests LogL 28559.5 28690.5 28584.3 31339.5 28776.2 28564.8 28882.9 28821.945 AIC 9.191 −9.543 −9.621 −10.762 −9.824 −9.920 −9.766 −9.770 SIC −8.797 −9.149 −9.227 −10.368 −9.430 −9.520 −9.372 −9.375 15 JRFM 2020,13, 226 Table 5. Cont. IND CHN INDO CHN KOR CHN MYS CHN PAK CHN PHL CHN TAIW CHN THA CHN Panel D: Diagnostic Tests JB 501.810 a (0.000) 693.810 a (0.000) 541.320 a (0.000) 686.040 a (0.000) 374.450 a (0.000) 673.900 a (0.000) 541.990 a (0.000) 694.160 (0.000) 590.770 a (0.000) 756.610 a (0.000) 1903.28 a (0.000) 678.760 a (0.000) 356.750 a (0.000) 671.310 a (0.000) 358.350 a (0.000) 686.450 a (0.000) Q(12) 14.715 (0.257) 7.454 (0.826) 11.152 (0.516) 7.664 (0.811) 12.457 (0.410) 8.501 (0.745) 11.203 (0.512) 6.933 (0.862) 18.800 c (0.093) 9.775 (0.636) 11.248 (0.508) 7.940 (0.790) 13.853 (0.310) 8.794 (0.720) 12.923 (0.375) 7.812 (0.800) Q2(12) 8.980 (0.705) 13.615 (0.326) 5.438 (0.942) 14.090 (0.295) 6.297 (0.900) 14.103 (0.294) 6.502 (0.889) 14.401 (0.276) 6.331 (0.899) 20.886 b (0.052) 27.233 a (0.007) 13.701 (0.320) 5.748 (0.928) 15.255 (0.228) 6.937 (0.862) 14.507 (0.270) Notes: The number of lags for VAR was decided using SIC (Schwartz information criterion) and AIC (Akaike information criterion) criteria. JB, Q(12) and Q 2 (12) indicate the empirical statistics of the Jarque–Bera test for normality, while Ljung–Box Q statistics of order 12 for autocorrelation were applied to the standardized residuals and squared standardized residuals, respectively. USA, United States of America; IND, India; INDO, Indonesia; KOR, South Korea; MYS, Malaysia; PAK, Pakistan; PHL, the Philippines; TAIW, Taiwan; THA, Thailand. Values in parentheses are the p-values. a,b,cindicate the statistical significance at 1%, 5%, and 10%, respectively. Table 6. Estimates of bivariate VAR−AGARCH for American and Asian stock markets during the Chinese Stock Market Crash. IND USA INDO USA KOR USA MYS USA PAK USA PHL USA TAIW USA THA USA Panel A: Mean Equation Constant 3.17 × 10−4 (0.227) 5.16 × 10−4b (0.020) −8.21 × 10−5a (0.743) 4.52 × 10−4b (0.047) 2.08 × 10−4 (0.395) 5.26 × 10−4b (0.013) −5.44 × 10−5 (0.721) 4.01 × 10−4b (0.055) 3.33 × 10−4 (0.225) 4.55 × 10−4b (0.030) −3.09 × 10−4 (0.341) 4.77 × 10−4b (0.029) −3.28 × 10−6 (0.988) 5.05 × 10−4b (0.011) 1.19 × 10−4 (0.557) 5.36 × 10−4a (0.005) ro t−10.077 b (0.032) −0.044 (0.166) 0.107 a (0.001) −0.028 (0.359) −0.020 (0.560) −0.018 (0.588) 0.089 b (0.013) 0.022 (0.685) 0.284 a (0.000) −0.020 (0.367) 0.032 (0.342) −0.032 (0.188) 0.031 (0.359) 0.076 (0.120) 0.147 a (0.000) −0.024 (0.519) ru t−1 0.233 a (0.000) −0.082 b (0.024) 0.241 a (0.000) −0.069 c (0.065) 0.351 a (0.000) −0.079 b (0.016) 0.234 a (0.000) −0.070 c (0.080) 0.088 b (0.016) −0.077 b (0.044) 0.416 a (0.000) −0.077 b (0.046) 0.405 a (0.000) − 0.081 b (0.030) 0.148 a (0.000) −0.054 (0.173) Panel B: Variance Equation Constant 3.75 × 10−6b (0.020) 4.42 × 10−6a (0.000) 2.05 × 10−6a (0.005) 4.16 × 10−6a (0.000) 1.11 × 10−5a (0.001) −1.41 × 10−6 (0.670) 3.61 × 10−7c (0.092) 3.28 × 10−6a (0.004) 4.09 × 10−6a (0.000) 4.13 × 10−6a (0.000) 4.62 × 10−6b (0.025) 1.00 × 10−6 (0.714) 2.60 × 10−6a (0.008) 3.85 × 10−6a (0.000) 1.12 × 10−6a (0.001) 7.03 × 10−7 (0.518) eo t−12−0.034 b (0.037) 0.079 a (0.000) 2.40 × 10−3 (0.911) 0.086 a (0.003) 0.149 a (0.001) 0.073 (0.270) 0.088 a (0.000) 0.018 (0.110) −0.018 (0.402) 0.011 (0.425) 0.032 (0.277) 0.063 (0.130) −0.020 (0.271) 0.071 (0.160) −0.023 (0.146) 0.020 (0.311) eu t−121.73 × 10−3 (0.915) 0.051 (0.102) −6.47 × 10−4 (0.950) 0.065 c (0.064) 9.22 × 10−3 (0.738) 0.020 (0.534) −0.077 a (0.000) 0.046 (0.216) 4.99 × 10−3 (0.296) 0.015 (0.640) −0.014 (0.145) 0.037 (0.343) 0.030 (0.162) 0.068 c (0.059) 0.061 b (0.046) 0.014 (0.650) ho t−1 0.896 a (0.000) −0.066 a (0.000) 0.891 a (0.000) −0.058 (0.202) 0.511 a (0.000) 0.049 (0.243) 0.865 a (0.000) −7.32 × 10−3 (0.556) 0.856 a (0.000) −0.020 (0.155) 0.870 a (0.000) −0.051 a (0.004) 0.890 a (0.000) − 0.054 a (0.001) 0.913 a (0.000) −0.017 (0.219) 22 JRFM 2020,13, 226 Table 6. Cont. IND USA INDO USA KOR USA MYS USA PAK USA PHL USA TAIW USA THA USA Panel B: Variance Equation hu t−1 −0.012 (0.644) 0.754 a (0.000) 0.021 (0.306) 0.696 a (0.000) 0.204 (0.149) 0.672 a (0.000) 0.303 a (0.002) 0.642 a (0.000) −0.012 a (0.002) 0.797 a (0.000) 0.070 (0.166) 0.711 (0.290) −0.018 (0.515) 0.726 a (0.000) 0.157 c (0.072) 0.645 a (0.000) Asymmetry 0.126 a (0.002) 0.270 a (0.000) 0.123 a (0.001) 0.314 a (0.000) −0.079 (0.183) 0.346 a (0.000) 0.040 (0.231) 0.317 a (0.000) 0.253 a (0.000) 0.258 a (0.000) 0.081 b (0.035) 0.338 a (0.000) 0.131 a (0.001) 0.296 a (0.000) 0.164 a (0.000) 0.343 a (0.000) Panel C: Constant Conditional Correlation p0,u0.324 a (0.000) 0.123 a (0.000) 0.281 a (0.000) 0.146 a (0.000) 0.059 a (0.078) 0.103 a (0.001) 0.214 a (0.000) 0.224 a (0.000) Panel D: Diagnostic Tests LogL 5580.558 5500.464 5640.723 5895.388 5471.88 5390.25 5605.08 5699.52 AIC −13.712 −13.614 −14.028 −14.525 −13.330 −13.434 −13.877 −13.881 SIC −13.418 −13.321 −13.734 −14.232 −13.037 −13.140 −13.583 −13.587 JB 72.640 a (0.000) 204.890 a (0.000) 58.520 a (0.000) 412.790 a (0.000) 134.540 a (0.000) 168.440 a (0.000) 22.680 a (0.000) 442.380 a (0.000) 30.880 a (0.000) 428.120 a (0.000) 27.070 a (0.000) 422.700 a (0.000) 84.480 a (0.000) 230.350 a (0.000) 68.060 a (0.000) 154.390 a (0.000) Q(12) 16.833 (0.156) 8.760 (0.723) 12.383 (0.415) 9.384 (0.670) 4.583 (0.970) 10.350 (0.585) 7.511 (0.822) 9.705 (0.642) 11.341 (0.500) 10.270 (0.592) 9.383 (0.670) 12.352 (0.418) 20.039 c (0.066) 13.680 (0.322) 6.178 (0.907) 7.190 (0.845) Q2(12) 7.383 (0.831) 3.713 (0.988) 18.894 c (0.091) 8.117 (0.776) 8.144 (0.774) 5.426 (0.942) 8.099 (0.777) 5.282 (0.948) 7.532 (0.821) 5.161 (0.952) 12.838 (0.381) 7.290 (0.838) 7.271 (0.839) 8.802 (0.720) 24.298 b (0.019) 5.675 (0.932) Notes: The number of lags for VAR was decided using SIC (Schwartz information criterion) and AIC (Akaike information criterion) criteria. JB, Q(12) and Q 2 (12) indicate the empirical statistics of the Jarque–Bera test for normality, while Ljung–Box Q statistics of order 12 for autocorrelation were applied to the standardized residuals and squared standardized residuals, respectively. USA, United States of America; IND, India; INDO, Indonesia; KOR, South Korea; MYS, Malaysia; PAK, Pakistan; PHL, the Philippines; TAIW, Taiwan; THA, Thailand. Values in parentheses are the p-values. a,b,cindicate the statistical significance at 1%, 5%, and 10%, respectively. 23 JRFM 2020,13, 226 The return spillover from the USA to all Asian Stock markets was significant during the Chinese crisis. This implies that US stock market prices played an important role in predicting the prices of all Asian stock markets during the Chinese stock market crash. Moreover, the return spillover from Asia to the US stock market was insignificant. The coefficient of past shock was insignificant in the majority of Asian markets except for in India, Korea, and Malaysia. The sensitivity of past own shocks of the USA was insignificant in most of the cases. In addition, the coefficient of past own volatility significantly affected the conditional volatility of all Asian markets. The conditional volatility of the majority of Asian stock markets (except Malaysia and Thailand) was not significantly affected by the shocks in the US stock market. In addition, past shocks in most of the Asian stock markets (Except in India and Indonesia) did not influence the conditional volatility of the US stock market. The volatility transmission from the USA to most of the Asian stock markets (except Malaysia, Pakistan, and Thailand) was found to be insignificant during the Chinese Crisis. On the other hand, volatility spillover from most of the Asian stock markets to the USA stock market was evidently insignificant. The asymmetric coefficients of all Asian stock markets (except Korea and Malaysia) were significant and positive, showing that negative news from the US stock market has a greater ability to increase the volatility of all Asian Stock markets as compared to positive news. However, the asymmetric coefficient of the US stock market is significant and positive. Constant conditional correlation was positively significant for all pairs of stock markets. However, CCC was weak in the majority of pairs. 4.2.6. Stock Market Linkages between China and Asia from the Chinese Stock Market Crash Table 7reports the return and volatility spillover between Chinese and Asian stock markets during the Chinese stock market crash. There is significant evidence that lagged returns influence the current stock returns of Asian Stock markets (Except in Korea, the Philippines, and Taiwan). This shows the short-term predictability in stock price changes in the Asian stock markets. Moreover, Chinese stock market returns were not affected by their lags during the Chinese stock market crash. The return spillover was found to be insignificant from China to all Asian markets. However, the return spillover was found to be insignificant from the majority of Asian markets to the Chinese market, except for India and Taiwan, during the Chinese stock market crash. The coefficient of past own shock did not significantly influence the conditional variance of the most of the Asian stock markets except in India, Malaysia, and Thailand. Moreover, the sensitivity to past own shock of the Chinese stock market was insignificant during the Chinese crash. However, the sensitivity of past own volatility was found to be significant for all Asian stock markets. The conditional volatility of India, Indonesia, Taiwan, and Thailand was significantly affected by the shocks in the Chinese stock market. However, the shocks in the majority of Asian stock markets (except India and the Philippines) did not influence the Chinese stock market. The past volatility of China significantly impacted the conditional volatility of the stock markets of India, Indonesia, Taiwan, and Thailand. However, volatility spillover was not found from most of the Asian stock markets (except India, Taiwan, and Thailand) to the Chinese stock market during the Chinese stock market crash. The asymmetric coefficients of all Asian stock markets (except Malaysia and the Philippines) were significant and positive, showing that negative news of the US stock market has a greater ability to increase the volatility of Asian stock markets as compared to positive news. Asymmetric coefficients of China were significant and positive in all pairs, demonstrating that negative news for any Asian markets except for India had a greater ability to increase the volatility of Chinese stock markets as compared to positive news during the Chinese crash. Constant conditional correlation was positively significant for all pairs of stock markets, but CCC was weak in the majority of pairs. 24 JRFM 2020,13, 226 Table 7. Estimates of bivariate VAR−AGARCH for Chinese and Asian stock markets during the Chinese Stock Market Crash. IND CHN INDO CHN KOR CHN MYS CHN PAK CHN PHL CHN TAIW CHN THA CHN Panel A: Mean Equation Constant 3.32 × 10−4 (0.168) 5.74 × 10−5 (0.835) 1.10 × 10−5 (0.964) 1.39 × 10−4 (0.615) 3.28 × 10−4 (0.147) 8.49 × 10−5 (0.715) −1.35 × 10−5 (0.933) 9.62 × 10−5 (0.715) 4.13 × 10−4 (0.145) 1.16 × 10−4 (0.663) 5.57 × 10−5 (0.875) 1.07 × 10−4 (0.699) 2.52 × 10−4 (0.310) 9.50 × 10−5 (0.714) 1.73 × 10−4 (0.359) 1.56 × 10−4 (0.561) ro t−1 0.167 a (0.000) 0.106 b (0.022) 0.144 a (0.000) −0.034 (0.500) 0.084 (0.350) −0.014 (0.715) 0.125 a (0.000) 0.062 (0.322) 0.288 a (0.000) −0.012 (0.660) 0.056 (0.141) −1.15 × 10−4 (0.997) 0.086 (0.280) 0.106 b (0.014) 0.155 a (0.000) 0.021 c (0.620) rc t−1 −0.029 (0.185) 0.037 (0.356) −0.012 (0.550) 0.069 (0.330) −0.019 (0.361) 0.062 c (0.077) 0.021 (0.222) 0.053 (0.181) 0.014 (0.452) 0.062 (0.120) 0.033 (0.274) 0.062 (0.113) 5.37 × 10−3 (0.836) 0.021 (0.580) −8.57 × 10−4 (0.965) 0.070 b (0.052) Panel B: Variance Equation Constant 3.77 × 10−6a (0.000) 1.79 × 10−6b (0.012) 4.23 × 10−6c (0.010) −4.44 × 10−8 (0.930) 3.43 × 10−6 (0.139) 9.89 × 10−7 (0.425) 1.42 × 10−6 (0.126) −2.42 × 10−7 (0.768) 4.07 × 10−6a (0.000) 8.60 × 10−7a (0.005) 6.97 × 10−6b (0.032) 1.31 × 10−6 (0.259) 9.22 × 10−6a (0.000) 3.22 × 10−6b (0.017) 2.64 × 10−6a (0.001) −8.98 × 10−7b (0.031) eo t−12−0.063 a (0.000) −5.50 × 10−3a (0.002) 0.052 (0.119) 5.85 × 10−4 (0.901) 0.024 (0.413) 5.44 × 10−3 (0.107) 0.102 a (0.003) −8.63 × 10−4 (0.677) −0.013 (0.560) −2.66 × 10−5 (0.986) 0.058 (0.144) 0.018 b (0.022) −0.028 (0.209) 5.81 × 10−3 (0.347) −0.039 b (0.021) −2.55 × 10−3 (0.364) ec t−120.070 a (0.000) 0.035 b (0.031) −0.018 a (0.007) 0.012 (0.567) 8.96 × 10−3 (0.618) 0.023 (0.290) −0.013 (0.780) 9.71 × 10−3 (0.611) 5.15 × 10−3 (0.180) 8.77 × 10−3 (0.656) 0.021 (0.137) 0.020 (0.342) 0.062 a (0.008) 0.027 (0.176) −0.033 a (0.000) 0.017 (0.355) ho t−1 0.880 a (0.000) 0.016 a (0.000) 0.789 a (0.000) 8.85 × 10−3 (0.176) 0.879 a (0.000) −2.40 × 10−3 (0.665) 0.768 a (0.000) 9.26 × 10−3 (0.283) 0.841 a (0.000) −1.18 × 10−3 (0.517) 0.824 a (0.000) −9.03 × 10−3 (0.137) 0.659 a (0.000) 0.021 b (0.038) 0.818 a (0.000) 0.015 b (0.028) hc t−1 −0.096 a (0.000) 0.945 a (0.000) 0.039 b (0.025) 0.940 a (0.000) −0.014 (0.742) 0.944 a (0.000) 0.100 (0.343) 0.934 a (0.000) −6.00 × 10−3 (0.109) 0.949 a (0.000) −0.026 (0.341) 0.943 a (0.000) −0.118 b (0.017) 0.944 a (0.000) 0.112 a (0.000) 0.933 a (0.000) Asymmetry 0.171 a (0.000) 0.029 (0.145) 0.163 a (0.000) 0.059 b (0.021) 0.044 c (0.077) 0.045 c (0.058) 0.087 (0.112) 0.065 b (0.015) 0.263 a (0.000) 0.058 c (0.010) 0.061 (0.210) 0.048 b (0.045) 0.276 a (0.000) 0.047 b (0.040) 0.229 a (0.000) 0.049 b (0.029) Panel C: Constant Conditional Correlation p0,c0.204 a (0.000) 0.151 a (0.000) 0.282 a (0.000) 0.166 a (0.000) 0.109 a (0.003) 0.179 a (0.000) 0.319 a (0.000) 0.202 a (0.000) Panel D: Diagnostic Tests LogL 5216.702 5157.83 5258.48 5520.00 5143.34 5026.77 5246.59 5354.32 AIC −12.108 −12.093 −12.437 −12.929 −11.856 −11.866 −12.268 −12.364 SIC −11.814 −11.799 −12.143 −12.635 −11.562 −11.573 −11.975 −12.070 25 JRFM 2020,13, 226 Table 7. Cont. IND CHN INDO CHN KOR CHN MYS CHN PAK CHN PHL CHN TAIW CHN THA CHN JB 245.340 a (0.000) 365.250 a (0.000) 212.590 a (0.000) 369.930 a (0.000) 325.490 a (0.000) 305.710 a (0.000) 214.270 a (0.000) 342.420 a (0.000) 156.230 a (0.000) 422.300 a (0.000) 216.090 a (0.000) 338.660 a (0.000) 301.330 a (0.000) 269.370 a (0.000) 227.030 a (0.000) 329.200 a (0.000) Q(12) 31.405 a (0.002) 16.802 (0.157) 10.948 (0.533) 15.522 (0.214) 27.305 a (0.007) 16.590 (0.166) 15.069 (0.238) 16.940 (0.152) 46.439 a (0.000) 20.284 c (0.062) 26.154 c (0.010) 15.115 (0.235) 13.912 (0.306) 11.123 (0.518) 24.373 b (0.018) 15.957 (0.193) Q2(12) 55.871 a (0.000) 17.976 (0.116) 47.212 a (0.000) 21.484 b (0.044) 48.551 a (0.000) 17.416 (0.135) 70.480 a (0.000) 21.463 b (0.044) 70.632 a (0.000) 30.028 a (0.003) 50.180 a (0.000) 23.783 b (0.022) 44.260 a (0.000) 16.310 (0.177) 88.957 a (0.000) 31.358 a (0.002) Notes: The number of lags for VAR was decided using SIC (Schwartz information criterion) and AIC (Akaike information criterion) criteria. JB, Q(12) and Q 2 (12) indicate the empirical statistics of the Jarque–Bera test for normality, while Ljung–Box Q statistics of order 12 for autocorrelation were applied to the standardized residuals and squared standardized residuals, respectively. USA, United States of America; IND, India; INDO, Indonesia; KOR, South Korea; MYS, Malaysia; PAK, Pakistan; PHL, the Philippines; TAIW, Taiwan; THA, Thailand. Values in parentheses are the p-values. a,b,cindicate the statistical significance at 1%, 5%, and 10%, respectively. 26 JRFM 2020,13, 226 4.3. Optimal Weights and Hedge Ratio Portfolio Implications Table 8indicates the optimal weights and hedge ratios for the pairs of Asia-US stock portfolios during the full sample period, US financial crisis, and the Chinese stock market crash. 7 The range of optimal weights is 0.37 for IND/USA to 0.68 for MYS/USA during the period of the full sample, indicating that for a $1 India-USA portfolio, 37 cents should be invested in Indian stocks and the remaining 63 cents in the US stock market. The average optimal portfolio weights vary from 0.38 for IND/USA to 0.80 for MYS/USA during the US financial crisis and range from 0.37 for PHL/USA to 0.69 for MYS/USA during the Chinese stock market crash. Overall, the optimal weights of US stock in Asia-USA portfolios are higher during the Chinese stock market crash compared to the US financial crisis. This implies that investors should have maintained more US stocks in their portfolio of Asia-USA during the Chinese stock market crash compared to the Asian stocks during the US financial crisis. Table 9presents the optimal weights and hedge ratios for the pairs of Asia-China stock portfolio during the full sample period, US financial crisis, and the Chinese stock market crash. 8 The range of optimal weights is from 0.56 for IND/CHN, KOR/CHN, PHL/CHN to 0.81 for MYS/CHN during the full sample period. The average optimal portfolio weights vary from 0.53 for IND/CHN to 0.90 for MYS/CHN during the US financial crisis and range from 0.52 for PHL/CHN to 0.82 for MYS/CHN during the Chinese stock market crash. Overall, for the majority of Asia-China portfolios, the optimal weights of Chinese stocks were almost equal or higher during the Chinese stock market crash and the US financial crisis. This suggests that portfolio managers and investors should have maintained almost the same investment in Chinese stock in their majority of the portfolio of Asia-China during both the Chinese crash and the US financial crisis. Table 8presents the optimal hedge ratios for the pairs of Asia-USA stock portfolio during the full sample period, US financial crisis, and the Chinese stock market crash. Regarding the hedge ratio, the range of average hedge ratio is 0.04 for PAK/USA to 0.27 for IND/USA during the period of the full sample, showing that a long position of $1 in Pakistani stocks can be hedged for a short position of 4 cents in US stocks. During the US financial crisis, the average optimal hedge ratios varied from 0.08 for PAK/USA to 0.36 for IND/USA. The average optimal hedge ratio ranged from 0.09 for PAK/USA to 0.36 for IND/USA during the Chinese stock market crash. For the majority of pairs of Asia-USA, the hedge ratios were lower in the US financial crisis compared with the Chinese stock market crash. This suggests that few US stocks were required to minimize the risk of Asian stock investors during the US financial crisis as compared to during the Chinese crash. Table 9provides the optimal hedge ratios for the pairs of a Asia-China stock portfolio during the full sample period, US financial crisis, and the Chinese stock market crash. The range of average hedge ratio is 0.04 for PAK/CHN to 0.21 for KOR/CHN during the period of the full sample. During the US financial crisis, the average optimal hedge ratios varied from 0.03 for PAK/CHN to 0.32 for KOR/CHN. The average optimal hedge ratio ranged from 0.09 for MYS/CHN to 0.26 for TAIW/CHN during the Chinese stock market crash. Overall, for the Asia-China pairs, the hedge ratio was lower during the Chinese stock market crash compared to the hedge ratios in the US financial crisis. This implies that fewer Chinese stocks were needed to minimize the risk for Asian stock investors during the Chinese stock market crash as compared to during the US crisis. 7 We calculated the optimal weights by using both VAR-GARCH and VAR-AGARCH models, but we reported the optimal weights only from the VAR-AGARCH model for the purpose of brevity. 8 We calculated the optimal weights by using both VAR-GARCH and VAR-AGARCH models, but we reported the optimal weights only from the VAR-AGARCH model for the purpose of brevity. 27 JRFM 2020,13, 226 Table 8. Optimal Weights and Hedge Ratios for Asia/USA pairs. IND/USA INDO/USA KOR/USA MYS/USA PAK/USA PHL/USA TAIW/USA THA/USA Full Sample Period wSU t0.37 0.41 0.40 0.68 0.41 0.42 0.43 0.41 βSU t0.27 0.13 0.24 0.06 0.04 0.06 0.17 0.17 US Financial Crisis wSU t0.38 0.51 0.54 0.80 0.52 0.57 0.52 0.52 βSU t0.36 0.18 0.21 0.11 0.08 0.09 0.15 0.22 Chinese Stock Market Crash wSU t0.46 0.44 0.51 0.69 0.44 0.37 0.49 0.53 βSO t0.36 0.15 0.29 0.11 0.09 0.14 0.22 0.22 Note: wSU tand βSU trefer to the optimal weights and hedge ratios, respectively. Table 9. Optimal Weights and Hedge Ratios for Asia/China pairs. IND/CHN INDO/CHN KOR/CHN MYS/CHN PAK/CHN PHL/CHN TAIW/CHN THA/CHN Full Sample Period wSC t0.56 0.57 0.56 0.81 0.57 0.56 0.59 0.59 βSC t0.15 0.15 0.21 0.09 0.04 0.12 0.20 0.13 US Financial Crisis wSC t0.53 0.63 0.68 0.90 0.64 0.64 0.66 0.67 βSC t0.31 0.24 0.32 0.13 0.03 0.22 0.30 0.19 Chinese Stock Market Crash wSC t0.65 0.61 0.66 0.82 0.58 0.52 0.66 0.73 βSC t0.17 0.13 0.23 0.09 0.11 0.18 0.26 0.14 Note: wSC tand βSC trefer to the optimal weights and hedge ratios, respectively. 28 JRFM 2020,13, 226 5. Conclusions In this paper, we extend the previous work by examining the return and volatility transmissions from the US and China to the eight emerging Asian stock markets including India, Indonesia, Korea, Malaysia, Pakistan, the Philippines, Taiwan, and Thailand during the Chinese stock market crash by using the VAR-AGARCH model. Moreover, we also examine the spillovers during the full sample period and the 2008 US financial crisis to provide comparative insights to investors about whether the impact of the Chinese crash on equity market spillovers is different from the crashes in other sample periods. Lastly, we also estimate the optimal weights and hedge ratios during the full period and all sub-periods. Our comprehensive analysis reveals that both return and volatility spillover vary across different pairs of stock markets and during financial crises. The findings of return spillover indicate a significant spillover from the USA to Asian stock markets during the full sample period, the US financial crisis, and the Chinese stock market crash. This implies that US stock market prices play an important role in predicting the prices of the majority of Asian stock markets during the full period and all the sub-periods. However, the return spillover is not significant from China to emerging Asian stock markets during the US financial crisis and the Chinese stock market crash, implying that Chinese stock prices cannot be used for predicting the prices of the majority of Asian stock markets during any of the crisis periods in our study. Our volatility spillover analysis reveals that the volatility was transmitted from the US to the majority of Asian markets during the full sample period and the Chinese stock market crash, but such a conclusion cannot be drawn for during the US financial crisis. This implies that portfolio investors of Asian stock markets could have gotten the maximum benefits of diversification by holding US stocks in their portfolio during the US financial crisis. However, the volatility spillover was transmitted from China to a majority of Asian markets during the full sample period and US financial crisis, but such a conclusion cannot be reached for during the Chinese crash, implying that portfolio investors of Asian stock markets could have gotten the maximum benefits of diversification by holding Chinese stocks in their portfolio during the Chinese stock market crash. Based on optimal weights results, the weights of the US stocks in the Asia-USA portfolios are higher during the Chinese crash compared to the US financial crisis, implying that investors should keep more US stocks in their portfolio of the Asia-USA stocks during the Chinese stock market crash, compared to the US financial crisis. For the majority of Asia-China portfolios, the optimal weights of Chinese stocks were almost equal or higher during the Chinese stock market crash and the US financial crisis. This suggests that portfolio managers and investors should have maintained almost the same investment in the Chinese stocks in their portfolio of the Asia-China majority pairs during both the Chinese crash and the US financial crisis. Regarding the hedge ratios, for most of the Asia-USA pairs, the hedge ratios were smaller in the US financial crisis than in the Chinese stock market crash. This suggests that few US stocks were required to minimize the risk for Asian stock investors during the US financial crisis as compared to during the Chinese crash. In contrast, for the Asia-China pairs, the hedge ratio was smaller during the Chinese stock market crash compared to that in the US financial crisis. This implies that fewer Chinese stocks were needed to minimize the risk for Asian stock investors during the Chinese stock market crash as compared to the US crisis. Overall, our findings provide several important implications for risk management and portfolio diversification that could be useful for investors and for policymakers related to the US and Asian stock markets. Author Contributions: Conceptualization, estimations, formal analysis, original draft preparation I.Y.; Data collection, methodology writing, and review of draft S.A.; review, editing, and funding W.-K.W. All authors have read and agreed to the published version of the manuscript. Funding: This research has been supported by Air University, Asia University, China Medical University Hospital, The Hang Seng University of Hong Kong, Research Grants Council (RGC) of Hong Kong (project 29 JRFM 2020,13, 226 number 12500915), and the Ministry of Science and Technology (MOST, Project Numbers 106-2410-H-468-002 and 107-2410-H-468-002-MY3), Taiwan. Acknowledgments: The first author gratefully acknowledge Arshad Hassan (department of Management and Social Sciences, Capital University of Science and Technology, Islamabad) for their valuable suggestions. The third author would like to thank Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement. Conflicts of Interest: The authors declare no conflict of interest. References Allen, Katie. 2015. Why is China’s stock market in crisis? The Guardian, July 8. Allen, David E., Ron Amram, and Michael McAleer. 2013. Volatility spillovers from the Chinese stock market to economic neighbours. Mathematics and Computers in Simulation 94: 238–57. [CrossRef] Arouri, Mohamed El Hedi, Amine Lahiani, and Duc Khuong Nguyen. 2011. Return and volatility transmission between world oil prices and stock markets of the GCC countries. Economic Modelling 28: 1815–25. [CrossRef] Arouri, Mohamed El Hedi, Jamel Jouini, and Duc Khuong Nguyen. 2012. 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Available online: https: //www.economist.com/news/2015/08/24/the-causes-and-consequences-of-chinas-market-crash (accessed on 24 September 2020). 31 JRFM 2020,13, 181 Table 1. ARIMA (p,d,q) model comparison. Model AIC BIC AICc ARIMA(0,2,0) −7144.11 −7139.03 −7144.11 ARIMA(0,2,1) −7981.48 −7971.31 −7981.47 ARIMA(0,2,2) −7979.84 −7964.59 −7979.82 ARIMA(0,2,3) −7981.77 −7961.43 −7981.73 ARIMA(0,2,4) −7980.02 −7954.61 −7979.97 ARIMA(1,2,0) −7455.29 −7445.12 −7455.28 ARIMA(1,2,1) −7979.81 −7964.56 −7979.79 ARIMA(1,2,2) −7982.16 −7961.83 −7982.13 ARIMA(1,2,3) −7983.04 −7957.62 −7982.98 ARIMA(1,2,4) −7981.53 −7951.03 −7981.45 ARIMA(2,2,0) −7631.73 −7616.48 −7631.71 ARIMA(2,2,1) −7981.50 −7961.16 −7981.46 ARIMA(2,2,2) −7982.88 −7957.46 −7982.83 ARIMA(2,2,3) −7982.62 −7952.12 −7982.55 ARIMA(2,2,4) −7979.91 −7944.32 −7979.81 ARIMA(3,2,0) −7692.63 −7672.30 −7692.60 ARIMA(3,2,1) −7979.84 −7954.42 −7979.79 ARIMA(3,2,2) −7981.50 −7951.00 −7981.43 ARIMA(3,2,3) −7977.84 −7942.25 −7977.74 ARIMA(3,2,4) −7978.07 −7937.40 −7977.95 ARIMA(4,2,0) −7738.80 −7713.39 −7738.75 ARIMA(4,2,1) −7980.13 −7949.63 −7980.06 ARIMA(4,2,2) −7978.69 −7943.11 −7978.60 ARIMA(4,2,3) −7978.28 −7937.61 −7978.15 ARIMA(4,2,4) −7984.84 −7939.09 −7984.69 Table 2. ARIMA(0,2,1) model summary. Model Arima(x = tr.stock, order = c(0, 2, 1)) MA(1) Coefficient −1.00 Standard Error 0.0027 Sigma-squared estimated as 0.0000718 Log likelihood 3992.74 AIC −7981.48 AICc −7981.47 BIC −7971.31 Training set error measures ME RMSE MAE MPE MAPE MASE ACF1 0.00013 0.00846 0.00573 0.00220 0.10561 0.99516 −0.01682 The model in difference equation is given as Wt=et+θet−1 ∇2Yt=et+θet−1 Yt−2Yt−1+Yt−2=et+θet−1 Yt=2Yt−1−Yt−2+et+θet−1. (2) Finally, substituting the MA parameter θ=− 1 in Equation (2), the model for Yt=log Xt is given as Yt=2Yt−1−Yt−2+et−et−1. (3) A fixed window of 1194 past observed stock prices have been used to predict each of the next-day prices using the model in Equation (3). Hence, the training dataset moved and the end price of the 38 JRFM 2020,13, 181 window was updated with the actual price. The results and diagnostics of this model are discussed in Section 4.1. 3.3. Stochastic Model Geometric Brownian Motion A process that generates some outcomes which are time-dependent but can not be said ahead of time is known as a stochastic process. A stochastic process {W(t): 0 ≤t≤T} is a standard Brownian motion on [0, T]if 1. W(0)=0 2. It has independent increments. That is, for any t1 , t2 , ... , tn , W(t2)−W(t1) , W(t3)− W(t4)...,W(tn)−W(tn−)are independent random variables. 3. For every 0 ≤s<t≤T,W(t)−W(s)∼N(0, t−s). A stochastic process {X(t): 0 <t<T} is said to be a general Brownian motion with a drift parameter μ and diffusion coefficient σ2 if X(t)−μt σ is a standard Brownian motion, written as X(t)∼BM(μ,σ2). The general Brownian motion still follows first two properties of the standard Brownian motion. However, the third property is modified as X(t)−X(s)∼N(μ(t−s) , σ2(t−s)) for any 0 ≤s<t<T . 3.3.1. Geometric Brownian Motion (GBM) Model If X(t)∼BM(μ,σ2)then X(t)satisfies the stochastic differential equation (Yang and Aldous 2015) dX(t)=μt+σdW(t), (4) where, W(t) is the standard Brownian motion or Wiener process. If the stochastic process is defined as X(t)=log S(t) then dS(t)=μS(t)dt +σS(t)dW(t) is the stochastic differential equation for the stock price random process. For a given time t> 0, the standard model for stock prediction can be given from the stochastic differential equation by integration S(t)=S(0)+μt 0S(r)dr +σt 0S(r)dW(r). (5) A more explicit formula can be derived using Ito’s formula (Ševcovic et al. 2011) to the function F(log S(t),t) dF =∂F ∂t+μ∂F ∂S(t)+1 2σ2∂2F ∂2S(t)dt +σ∂F ∂S(t)dW(t), which results dlog S(t)= 1 S(t)dS(t)+1 2−1 S2(t)(dS(t))2 =μdt +σdW(t)+1 2−1 S2(t)(μS(t)dt +σS(t)dW(t))2 =(μ−1 2σ2)dt +σdW(t). For any time t>0, the differential can be written as log S(t)=log S(0)+(μ−1 2σ2)t+σW(t) Or, S(t)=S(0)e(μ−1 2σ2)t+σW(t). (6) 39 JRFM 2020,13, 181 3.3.2. Geometric Brownian Motion Model forS&P500Index: GBM(μ,σ2) Simulation For a given time set, t0= 0 <t1<t2<...<tn , the stock price S(t) at time t0 , t1 , ... , tn can be generated by S(ti+1)=S(ti)e(μ−1 2σ2)(ti+1−ti)+σ√(ti+1−ti)Zi+1, (7) where Z1 , Z2 , ...Zn are independent and identically distributed standard normals and i=0, (n−1) . In our case, the time interval ti+1−ti= 1 for all i=0, (n−1) , since we are predicting the next-day price. Hence, the model becomes S(ti+1)=S(ti)e(μ−1 2σ2)+σZi+1. (8) Using the model in Equation (8), we simulate a large number of prices, and from that we take the average to predict the next-day price. For our data, this large number is 100,000. A total of 63 predictions have been made using this model. A fixed window of 1194 past observed stock prices have been used to predict each of the next-day prices. Hence, the training dataset moved, and the end price of the window was updated with the actual price. The results and diagnostics of this model are discussed in Section 4.2. 3.4. Artificial Neural Network This section describes how an artificial neural network can be used to predict the stock price and how to build a model based on the stock data forS&P500index. 3.4.1. Model Descriptions Artificial neural network is one of the most popular machine learning techniques for nonlinear approximations because of its ability to deal with a large number of functions with a high degree of accuracy (Chen et al. 2003). The idea of ANN came from the structure of the animal brain, more specifically, from the human neural system. It is based on the idea of how brain works, how the neurons in the brain receive information from the input neurons, analyse it, and finally identify the object or pattern. Fundamentally, the mechanism has three layers—input layer, hidden layers, and output layer. Each layer consists of neurons or nodes. The hidden part may consist of many layers, however, for the time series analysis and forecasting, the single hidden layer feed forward network is the most widely used model structure (Zhang et al. 1998). A simple three layer neural network has the following mathematical form Yt=W0+ q ∑ j=1 Wj.g(W0,j+ p ∑ i=1 Wi,j.Yt−i)+t, (9) where, Wi,j and Wj for i= 1,2, ... , p , j= 1,2, ... , q are known as connection weights. The parameter p and q are the number of input and output nodes respectively. The network involves an activation function which plays a very important role because it converts the input signals to be used for the neurons or nodes in the next layer, eventually the output neuron. The most widely used activation functions are the logistic and hyperbolic functions (Khashei and Bijari 2010), which are shown in Equations (10) and (11) sig(x)= 1 1−e−x(10) tan−1(x)=1−e−2x 1+e−2x. (11) 40 JRFM 2020,13, 181 Most of the modelers prefer the hyperbolic tangent function as the activation functions because of its faster convergence, and it makes the optimization easier. Hence, we used this activation function in our model. There is no systematic rule of choosing the number of neurons or nodes, q in the hidden layer (Khashei and Bijari 2010). In most of the cases it is data-dependent and chosen on the basis of trial and error. 3.4.2. Artificial Neural Network for S& P 500 Index The model proposed for S & P 500 in this section is a three layer model—input, hidden, and the output layer. The input layer consists of a total of seven nodes which are daily Open, Close, High, Low, Average, Volume, and Return. The variable Average is the average of daily Open, Close, High, and Low. The Volume was converted to million units. Daily return was calculated by this formula rt=log St St−1 , where St is the adjusted closed price and day one return, r0 was considered zero. The output layer has only one node that corresponds to the predicting variable Adjusted Close price. The number of the nodes in the hidden layer was chosen based on the error measures in Equation (1) for different combinations of the hidden nodes, which are displayed in Table 3. From this Table 3, we see that model ANN(7-15-1) has the lowest APE, AAE, ARPE, and RMSE and the highest adjusted R2value. Table 3. Error measures for different network structures. MODEL R2APE AAE ARPE RMSE 7-2-1 0.3137 0.0231 7.0435 0.1118 0.3344 7-3-1 0.4158 0.0211 6.412 0.1018 0.319 7-4-1 0.0422 0.028 8.5128 0.1351 0.3676 7-5-1 0.0368 0.0292 8.8796 0.1409 0.3754 7-6-1 0.3716 0.0215 6.5496 0.104 0.3224 7-7-1 0.3431 0.0226 6.8687 0.109 0.3302 7-8-1 0.3907 0.0219 6.6781 0.106 0.3256 7-9-1 0.4036 0.0212 6.4713 0.1027 0.3205 7-10-1 0.4108 0.0214 6.5237 0.1036 0.3218 7-11-1 0.5155 0.0195 5.9284 0.0941 0.3068 7-12-1 0.5392 0.0187 5.6802 0.0902 0.3003 7-13-1 0.4777 0.0194 5.8945 0.0936 0.3059 7-14-1 0.4676 0.0196 5.9702 0.0948 0.3078 7-15-1 0.6216 0.0167 5.0928 0.0808 0.2843 7-16-1 0.5701 0.0182 5.5382 0.0879 0.2965 The original dataset had 1257 observations, but the dataset used in this method was modified in this way—all the predictor and predicting variables have the same length of 1256, however, the predictor variables started from day 1 to the 1256th day and the predicting variable day 2 to the 1257th day. Then, the dataset was divided into two parts to run the model. The test dataset contained the last 63 actual stock prices (adjusted close) which were compared to the predicted prices. The best model was selected on the basis of the adjusted R2and four error measures (Table 3). The model architecture is shown in Figure 5and the result of this model is discussed in Section 4.3. 41 JRFM 2020,13, 181 Figure 5. Artificial neural network architecture. 4. Results In this section, the result of from the above three models is discussed and a window of the predicted and actual price is shown together with a graphical presentation. Finally, we assess how our model is performing by model diagnostics. 4.1. Autoregressive Integrated Moving Average 4.1.1. ARIMA Model Result The model ARIMA( 0,2,1 ) in Equation (3) produces the prediction in a logarithmic scale, which is then converted back to the original scale by the formula Predicted Price =eprediction . Total 63 trading days have been predicted by the model and compared with the actual prices which has been shown in Table 4with individual prediction error calculated by the formula, error =actual −predicted actual ×100. (12) From Table 4, we see that the forecast errors are less than one dollar for the daily period from 12 November, 2019 to 30 December 2019, with the relative errors within the range of 0.00003 to 0.00292. Figure 6shows the graphical representation of the actual and predicted stock price by the model. The black line represents the actual stock price and the red line represents the predicted stock price for S & P 500. Figure 6also shows that the ARIMA (0,2,1) predicted prices follow closely to the trend of the actual prices. 42 JRFM 2020,13, 181 Table 4. Prediction by ARIMA(0,2,1) model. Date Actual Predicted Error 11-12-2019 305.69 305.17 0.17 11-13-2019 305.79 305.81 0.01 11-14-2019 306.23 305.91 0.10 11-15-2019 308.45 306.35 0.68 11-18-2019 308.68 308.57 0.04 11-19-2019 308.59 308.80 0.07 11-20-2019 307.44 308.71 0.41 11-21-2019 306.95 307.56 0.20 11-22-2019 307.63 307.07 0.18 11-25-2019 310.01 307.75 0.73 11-26-2019 310.72 310.14 0.19 11-27-2019 312.10 310.85 0.40 11-29-2019 310.94 312.23 0.41 12-2-2019 308.30 311.07 0.90 12-3-2019 306.23 308.43 0.72 12-4-2019 308.12 306.36 0.57 12-5-2019 308.68 308.25 0.14 12-6-2019 311.50 308.81 0.86 12-9-2019 310.52 311.64 0.36 12-10-2019 310.17 310.65 0.15 12-11-2019 311.05 310.30 0.24 12-12-2019 313.73 311.18 0.81 12-13-2019 313.92 313.86 0.02 12-16-2019 316.08 314.05 0.64 12-17-2019 316.15 316.21 0.02 12-18-2019 316.17 316.29 0.04 12-19-2019 317.46 316.31 0.36 12-20-2019 318.86 317.60 0.40 12-23-2019 319.34 319.00 0.11 12-24-2019 319.35 319.48 0.04 12-26-2019 321.05 319.49 0.49 12-27-2019 320.97 321.20 0.07 12-30-2019 319.20 321.12 0.60 4.1.2. ARIMA Model Diagnostics The performance of the ARIMA(0,2,1) model was assessed by the analysis of the four error measures state in Equation (1) and the residuals plot, which is depicted in Figure 7and those four error measures are tabulated in Table 5. Table 5. Prediction error by ARIMA(0,2,1) model. MODEL APE AAE ARPE RMSE ARIMA(0,2,1) 0.0044 1.3651 0.0217 0.1472 From the results in Table 5, we see that all the error measures are comparatively very low to the actual prices, this indicates that the model is performing better in its prediction. From Figure 7it is clear that the residuals do not follow any special pattern, they are a randomized plot. Correlations in the few lags are significant. Overall, the model fits very well to predict the stock price. 43 JRFM 2020,13, 181 Figure 6. ARIMA (0,2,1) model prediction. Figure 7. ARIMA(0,2,1) model residual analysis. 4.2. Stochastic Model 4.2.1. Stochastic Model Result The model proposed in Equation (8) requires the calculation of 63 distinct values of the means μ and standard deviations σ of the daily returns. Both of the parameters were calculated on the basis of the same number of returns each time. Predicted values, actual values, and individual errors are shown in Table 6, and the errors were calculated by the same formula (12) used in the previous model. The results in Tables 5and 6are almost the same except at some points. Thus, Figures 7and 8are almost identical. Figure 8displays the graphical representation of the actual and predicted stock prices from the stochastic model. The black line represents the actual stock price and the green line represents the predicted stock price forS&P500index. 44 JRFM 2020,13, 181 Table 6. Prediction by geometric Brownian motion. Date Actual Predicted Error 11-12-2019 305.69 305.17 0.17 11-13-2019 305.79 305.81 0.01 11-14-2019 306.23 305.91 0.10 11-15-2019 308.45 306.35 0.68 11-18-2019 308.68 308.59 0.03 11-19-2019 308.59 308.80 0.07 11-20-2019 307.44 308.71 0.41 11-21-2019 306.95 307.58 0.21 11-22-2019 307.63 307.06 0.19 11-25-2019 310.01 307.74 0.73 11-26-2019 310.72 310.15 0.18 11-27-2019 312.10 310.86 0.40 11-29-2019 310.94 312.23 0.41 12-2-2019 308.30 311.09 0.90 12-3-2019 306.23 308.43 0.72 12-4-2019 308.12 306.36 0.57 12-5-2019 308.68 308.25 0.14 12-6-2019 311.50 308.82 0.86 12-9-2019 310.52 311.63 0.36 12-10-2019 310.17 310.64 0.15 12-11-2019 311.05 310.30 0.24 12-12-2019 313.73 311.21 0.80 12-13-2019 313.92 313.88 0.01 12-16-2019 316.08 314.06 0.64 12-17-2019 316.15 316.21 0.02 12-18-2019 316.17 316.27 0.03 12-19-2019 317.46 316.31 0.36 12-20-2019 318.86 317.59 0.40 12-23-2019 319.34 319.00 0.11 12-24-2019 319.35 319.47 0.04 12-26-2019 321.05 319.51 0.48 12-27-2019 320.97 321.20 0.07 12-30-2019 319.20 321.11 0.60 Figure 8. Stochastic model geometric Brownian motion prediction. 4.2.2. Stochastic Model Diagnostics The performance of the stochastic model was assessed by the analysis of the four error measures stated in Equation (1) and the residual plot which is depicted in Figure 9and the calculation of the four different error measures, as shown in Table 7. 45 JRFM 2020,13, 181 Table 7. Prediction error by geometric Brownian motion. MODEL APE AAE ARPE RMSE GBM 0.0044 1.3341 0.0212 0.1455 Figure 9. GBM model diagnostics. The standardized residual plot is random and the mean passes through the zero line. A few of the residuals at the lower end are outside of the band in the Q-Q plot of the residuals. Still, both of the plots depict the approximate normal behavior of the residuals. 4.3. Artificial Neural Network 4.3.1. ANN(7-15-1) Results Both of the training and test datasets were converted to normal and the prediction was converted back to the original scale by inverse transformation. The model required approximately 7000 steps with an error of 0.1972. Actual prices, predicted prices, and the corresponding errors are displayed in Table 8. Table 8. Prediction by ANN (7-15-1) model. Date Actual Predicted Error 11-12-2019 305.69 302.13 1.17 11-13-2019 305.79 302.52 1.07 11-14-2019 306.23 302.26 1.30 11-15-2019 308.45 302.39 1.97 11-18-2019 308.68 303.56 1.66 11-19-2019 308.59 304.19 1.43 11-20-2019 307.44 303.54 1.27 11-21-2019 306.95 302.52 1.44 11-22-2019 307.63 302.86 1.55 11-25-2019 310.72 305.06 1.82 11-27-2019 312.10 305.81 2.02 11-29-2019 310.94 306.38 1.47 12-2-2019 308.30 306.02 0.74 12-3-2019 306.23 303.59 0.86 12-4-2019 308.12 301.63 2.11 12-5-2019 308.68 304.03 1.51 46 JRFM 2020,13, 181 Table 8. Cont. Date Actual Predicted Error 12-6-2019 311.50 304.28 2.32 12-9-2019 310.52 306.05 1.44 12-10-2019 310.17 306.07 1.33 12-11-2019 311.05 305.23 1.87 12-12-2019 313.73 305.55 2.61 12-13-2019 313.92 306.48 2.37 12-16-2019 316.08 306.73 2.96 12-17-2019 316.15 307.56 2.72 12-18-2019 316.17 308.05 2.57 12-19-2019 317.46 308.40 2.85 12-20-2019 318.86 308.04 3.39 12-23-2019 319.34 306.20 4.11 12-24-2019 319.35 308.94 3.26 12-26-2019 321.05 309.71 3.53 12-27-2019 320.97 310.31 3.32 12-30-2019 319.20 309.93 2.90 The predicted errors in Table 8are much higher than those in Tables 5and 6. Precise comparisons of the three models are given in the next section. The graph associated with this result is displayed in Figure 10. The black line represents the actual stock price and the blue line represents the predicted stock price for the S & P 500 index. From the graph, it is clear that the model is working better at the beginning of the prediction interval. Figure 10. ANN(7-15-1) prediction. 4.3.2. ANN(7-15-1) Model Diagnostics The performance of the neural network ANN(7-15-1) was assessed by the analysis of the four error measures stated in Equation (1) and the standardized residuals plot, which is depicted in Figure 11, and the calculation of the four different error measures are shown in Table 9. Table 9. Prediction error by ANN(7-5-1) model. MODEL APE AAE ARPE RMSE ANN(7-15-1) 0.0167 5.09279 0.08084 0.28432 47 JRFM 2020,13, 148 Li and Giles 2014 ;Gulzar et al. 2019) and the global financial crisis (Ta¸sdemir and Yalama 2014; Bekiros 2014 ;Mensi et al. 2016;Gamba-Santamaria et al. 2017). However, the linkages between equity markets are rarely examined during the crash of the Chinese stock market in 2015. The Chinese stock market crashed in 2015 (Han and Liang 2016;Ahmed and Huo 2019;Yousaf and Hassan 2019). The CSI 300 index had reached up to 5178 points until mid-June in 2015. Then, it took roller-coaster ride and dropped up to 34% in just 20 days, also losing 1000 points within just one week. Around 50% of the Chinese stocks lost more than half of their pre-crash market value. This crash adversely affected the many other financial markets around the globe (Fang and Bessler 2017). Despite the importance of the Chinese crash to international portfolio managers, only Ahmed and Huo (2019) examined the volatility transmission between the Chinese and Asian stock markets during the crash of the Chinese stock market in 2015. The empirical research remains surprisingly limited on the area of linkages between equity markets during the crash of the Chinese stock market. The US and China are the most significant trading partners of the emerging Latin American economies. From 2000 to 2017, the trade volume of China (US) is increased by 21 (2.5)-fold with emerging Latin American economies. The trade volume of leading economies grew at a different rate with the emerging Latin American (LA) economies; thus, spillover can also be changed between the China-LA and US-LA pairs during the last two decades. Johnson and Soenen (2003) also suggest that trade increases the financial integration between countries’ stock markets. Previously, several studies have examined the spillovers between the US and Latin American stock markets ( Meric et al. 2001 ; Arouri et al. 2015;Ben Rejeb and Arfaoui 2016;Cardona et al. 2017;Gamba-Santamaria et al. 2017; Ramirez-Hassan and Pantoja 2018;Yousaf and Ahmed 2018;Fortunato et al. 2019; Coleman et al. 2018 ). However, the linkages between the China and Latin American stock markets have not yet been explored, especially during the global financial crisis and the crash of the Chinese stock market. Based on the above-mentioned literature gaps, this study aims to examine the return and volatility spillover between the world-leading (the US and China) and emerging Latin American stock markets during the full sample period, the global financial crisis, and the crash of the Chinese stock market. Additionally, this study estimates the optimal weights and hedge ratios during all the sample periods. Our study makes the following contributions to the literature. First, regarding return spillover, the findings reveal a unidirectional return transmission from Mexico to the US stock market during the global financial crisis. During the crash of the Chinese stock market, the return spillover is found to be unidirectional from the US to the Brazil, Chile, Mexico, and Peru stock markets. Moreover, the results indicate a unidirectional return transmission from China to the Brazil, Chile, Mexico, and Peru stock markets during the global financial crisis and the crash of the Chinese stock market. Regarding volatility spillover, the results show the bidirectional volatility transmission between the US and the stock markets of Chile and Mexico during the global financial crisis. During the Chinese crash, a bidirectional volatility transmission is observed between the US and Mexican stock markets. Furthermore, the volatility spillover is unidirectional from China to the Brazil stock market during the global financial crisis. During the Chinese crash, the volatility spillover is bidirectional between the China and Brazil stock markets. The contributions of this study are four-fold. First, this study provides a comprehensive analysis of spillover between the world-leading and emerging LA stock markets during the crash of the Chinese stock market. Second, it contributes to the literature of the China-LA stock markets by examining the spillovers during the global financial crisis. Lastly, the BEKK-GARCH model is applied to estimate the spillovers, optimal weights, and hedge ratios, which provide better statistical properties compared to the many other GARCH models. The rest of the paper is organized as follows: Section 2provides a review of the literature. The empirical method is described in Section 3. Section 4consists of the data and the preliminary analysis. The empirical results are reported in Section 5. Finally, Section 5 concludes the discussion. 54 JRFM 2020,13, 148 2. Literature Review Markowitz’s modern portfolio theory can describe the relationship between different stock markets in order to build an optimum portfolio. The rationale behind this concept is to combine risky assets with less risky or risk-free assets in the portfolio (Markovitz 1959). For example, the leading stock market shows a higher volatility during the financial crisis, and as a result the portfolio investors need to diversify their portfolios by investing in weakly integrated emerging stock markets. Therefore, an analysis of risk transmission between different equity markets is essential for portfolio managers to identify opportunities for portfolio diversification across markets and over time. Over the past decade, there has been a growing body of literature examining the information transmissions (return and volatility) between the US and LA stock markets during the crisis and non-crisis periods. Meric et al. (2001) report significant co-movements between the US and LA (Brazil, Argentina, Chile, and Mexico) stock markets during the period 1984–1995. Fernández-Serrano and Sosvilla-Rivero (2003) report the cointegration across the US and LA equity markets. Sharkasi et al. (2005) investigate the spillover across the US and Brazil stock markets. They provide evidence of co-movements between the US and Brazil stock markets. Diamandis (2009) investigates the linkages and common trends between the US and four Latin American (Argentina, Brazil, Chile, and Mexico) stock markets. Because the four Latin American countries initiated a phase of financial liberalization in the late 1980s and early 1990s, this study also explores whether the removal of foreign-exchange controls had any effect on the potential linkages. Firstly, this study finds that the US stock market is partially integrated with four LA stock markets. Secondly, the five stock markets have four significant common permanent components/trends which influence their system in the long run. Thirdly, the results indicate significant short-term deviations from standard stochastic patterns during the 1994–1996 Mexican crisis and the 2001 financial crisis. Beirne et al. (2013) use the tri-variate GARCH-BEKK model to estimate the volatility transmission from mature markets to 41 emerging (including 8 Latin American) stock markets. The volatility transmission is observed to be significant from many mature markets to the emerging stock markets. Additionally, there is evidence of changes in the parameters of volatility spillovers during turbulent or crisis periods. Graham et al. (2012) estimate the integration between the US and 22 emerging equity markets and find evidence of strong co-movements across the US, Brazil, and Mexico equity markets. Hwang (2014) examined the spillover between the US and LA equity markets during the global financial crisis. The study found that the integration between the US and LA equity markets became stronger during the global financial crisis. Using the VAR-GARCH model, Arouri et al. (2015) estimate the return and volatility transmissions between the US and LA (Brazil, Argentina, Mexico, Chile, and Columbia) stock markets from 1993 through to 2012. The return spillover is seen to be significant from the US to the Argentina, Mexico, and Colombia stock markets. It also provides evidence of a volatility transmission from the US to a few LA stock markets. Syriopoulos et al. (2015) use the VAR-GARCH model and find that the return and volatility spillover is significant between the US and BRICS (Brazil, Russia, India, China, and South Africa) equity markets (at the sectoral level). Mensi et al. (2016) reveal the strong dynamic correlation between US and BRICS equity markets during the global financial crisis of 2008. Ben Rejeb and Arfaoui (2016) examine the volatility transmission between developed (US and Japan) and emerging (Latin American and Asian) stock markets using standard GARCH models and a quantile regression approach. This study reveals a significant presence of volatility transmission in these markets. The volatility transmission is seen to be closely associated with the crisis period and geographical proximity. A lower and upper quantiles analysis shows that interdependence between markets decreases during a bearish trend, while it increases during bullish markets. Using the GARCH model, Bhuyan et al. (2016) observes return and volatility transmissions from the US to BRICS stock markets. Al Nasser and Hajilee (2016) provide evidence of short-run integration between developed (US, UK, and Germany) and emerging stock markets (Brazil, Mexico, Russia, China, and Turkey). However, 55 JRFM 2020,13, 148 in the long run, the cointegration is only found to be significant between Germany and emerging Asian stock markets. Gamba-Santamaria et al. (2017) examine the directional volatility transmission between the US and the four LA stock markets (Brazil, Chile, Mexico, and Columbia) using the framework of Diebold and Yilmaz (2012). Brazil is found to be the net volatility transmitter for most of the sample period, whereas Columbia, Chile, and Mexico are the net receivers of volatility. Moreover, the US stock market is observed to be the net transmitter of volatility to the four LA stock markets. Besides this, the magnitude of volatility transmission is increased from the US to LA stock markets during the global financial crisis of 2008.1 Yousaf and Ahmed (2018) study the influence of the US and Brazil on the Mexico, Argentina, Chile, and Peru stock markets by using GARCH in a mean approach. The study concludes that the return effects are dominantly transmitted from the US to the Mexico, Argentina, Chile, and Peru stock markets. Moreover, the volatility transmission is found to be dominant from Brazil to the Mexico, Argentina, Chile, and Peru stock markets. Cardona et al. (2017) use the MGARCH-BEKK model to estimate the volatility transmission between the US and the six LA stock markets (Brazil, Argentina, Mexico, Chile, Peru, and Colombia). They report the significant volatility transmission from the US to all LA stock markets. Moreover, only Brazil transmits volatility effects to the US stock market. Ramirez-Hassan and Pantoja (2018) provide evidence of co-movements between the returns of the US and six LA stock markets after the global financial crisis of 2008. Fortunato et al. (2019) provide evidence of return transmission from the US to the Brazil, Chile, Columbia, Mexico, and Peru equity markets. Coleman et al. (2018) find the co-movements between the US and LA (Brazil, Chile, Mexico, Peru, Venezuela, and Argentina) stock markets. Su (2020) reports the dominant risk transmission from the G7 (US, Japan, UK, Germany, France, Italy, and Canada) countries to the BRICS (Brazil, Russia, India, China, and South Africa) stock markets. However, fewer studies have examined the spillovers between the China and Latin American stock markets during the crisis and non-crisis periods. Garza-Garc í a and Vera-Ju á rez (2010) study the impact of US and Chinese macroeconomic variables on the stock markets of Brazil, Mexico, and Chile. The macroeconomic variables (the US and Chinese) are observed to be integrated with the LA stock markets. Additionally, the US macroeconomic variables Granger affect the Brazilian and Mexican stock markets. On the other hand, the Chinese macroeconomic variables Granger affect the stock markets of Mexico and Chile. Horvath and Poldauf (2012) find that the Chinese stock market is weakly correlated with the Brazil, Australia, Canada, Germany, Japan, Hong Kong, South Africa, Russia, US, and UK stock markets. Sharma et al. (2013) apply the VAR model to examine the linkages between the BRICS (Brazil, Russia, India, China, and South Africa) stock markets. This study finds a return transmission from Brazil (India) to the Russia, India (Brazil), China, and South Africa equity markets. Moreover, the return transmission is only observed from China to the Russian stock market. Bekiros (2014) looks at the contagion effect between Brazil, Russia, India, and China by using several multivariate GARCH models. 1 Our study is different from the study of Gamba-Santamaria et al. (2017) in the following aspects. Gamba-Santamaria et al. (2017) examine the volatility spillover between the US and four Latin American markets (Brazil, Chile, Mexico, and Columbia) during the US financial crisis, whereas our study is examining the volatility as well as return spillover between the leading (US and China) markets and four Latin American markets (Brazil, Chile, Mexico, and Peru) during the US financial crisis and the crash of the Chinese stock market. More specifically, firstly our study examines the return as well as volatility spillovers, whereas Gamba-Santamaria et al. (2017) examine the directional volatility spillovers. Second, our study is examining the spillovers between two world-leading (the US and China) markets and four LA markets, whereas Gamba-Santamaria et al. (2017) examine the spillovers between US and four LA markets. Third, our study is focusing on the spillovers during the global financial crisis and the crash of the Chinese stock market in 2015, whereas Gamba-Santamaria et al. (2017) examine the spillovers during the US financial crisis. Fourth, our study is using the BEKK-GARCH model, whereas Gamba-Santamaria et al. (2017) employ the approach of Diebold and Yilmaz (2012). Lastly, our full data sample is from January 2001 to May 2020, whereas Gamba-Santamaria et al. (2017) use the sample period from January 2003 to January 2016. Apart from the differences, the study of Gamba-Santamaria et al. (2017) is very beneficial for understanding the linkages among the US and LA stock markets. 56 JRFM 2020,13, 148 This study concludes that there exists a higher integration between Brazil, Russia, India, and China after the global financial crisis. Ahmad and Sehgal (2015) estimate the volatility of the BRIICKS (Brazil, Russia, India, Indonesia, China, South Korea, and South Africa) stock markets by using the Markov regime-switching (MS) in the mean-variance model. It suggests that investors should allocate investment in the China, Russia, and India emerging stock markets. While investigating the relationship between the Chinese and foreign stock markets (US, Brazil, India, and Germany), Cao et al. (2017) reported a bi-directional causality between the China and foreign stock markets. Previous work does not provide evidence of return and volatility spillover between leading (US and China) and Latin American stock markets during the global financial crisis and the crash of the Chinese stock market. Therefore, this study addresses the above-mentioned literature gaps. 3. Data and Methodology In this section, we will discuss the data and methodology used in our paper. We first discuss the data. 3.1. Data This study uses the daily data of benchmark stock indices of the US (S&P 500); China (SSE Composite Index); and four emerging LA stock markets—namely, Brazil (IBOVESPA index), Chile (IPSA index), Mexico (S&P/BMV IPC Index), and Peru (S&P/BVL Peru General TR PEN Index). The data of stock indices are taken from the Data Stream database. The index is assumed to be the same on non-trading days (holidays except weekends) as on the previous trading day, as suggested by Malik and Hammoudeh (2007), and Ali et al. (2020). This study uses the full sample period from 1 January 2000, to 29 May 2020, and studies the following two sub-samples: the first sub-period from 1 August 2007, to 30 July 2010, presenting the period with the US financial crisis; and the second sub-period from 1 June 2015, to 31 May 2018, presenting the period with the Chinese stock market crash. We note that Yousaf and Hassan (2019) also use similar timeframes for the global financial crisis and the crash of the Chinese stock market. This study follows He (2001) to use three-year data for each crisis for a short-run analysis. Changes in the market correlations take place continuously, not only as a result of the crises but also due to the consequences of many financial, economic, and political events. This study uses the same time for both the crisis periods to make the coefficient comparable. The difference in the opening time of the China and LA stock markets has been adjusted in the estimations. 3.2. Methodology The econometric specification used in this study has two components. First, a vector autoregression (VAR) with one lag is used to model the returns. 2 This allows for autocorrelations and cross-autocorrelations in the returns. Second, a multivariate BEKK-GARCH model is used to model the time-varying variances and covariances developed by Engle and Kroner (1995). 3 BEKK-GARCH has the attractivecharacteristicsthattheconditionalcovariancematricesarepositivedefinite (Chang et al. 2011) . Several studies have used the BEKK-GARCH model to estimate the spillover between different asset classes; see, for example, Chang et al. (2011), Sadorsky (2012), Beirne et al. (2013), Chang et al. (2017), Cardona et al. (2017), and Sarwar et al. (2020). Moreover, we will estimate the optimal weights and hedge ratios using the BEKK-GARCH model. 2The number of lags is selected on the basis of the AIC and SIC criteria. 3We apply the BEKK-GARCH model on the valuable suggestion of a respected reviewer. 57 JRFM 2020,13, 148 This study aims to examine the return and volatility spillover between the stock markets, and thus we firstly focus on return spillover. For any pair of two series, the following are the specifications for the conditional mean equation: Rt=μ+∅Rt−1+etwith et=H1/2 tηt. (1) Rt=(Rx t , Ry t) is the vector of returns on the stock market indices x and y at time t, respectively; ∅ is the 2 × 2 matrix of parameters, measuring the impacts of own lagged and cross mean transmissions between two series; et=ex t,ey t is the vector of error terms of the conditional mean equations for the two series at time t; ηt=ηx t,ηy t indicates a sequence of independently and identically distributed random errors; and Ht=Hx tHxy t Hxy tHy tdenotes the conditional variance-covariance matrix of return series of xand y. In addition, H1/2 tis the 2 ×2 symmetric positive definite matrix. The full BEKK–GARCH, which imposes positive definiteness restrictions for Ht, is given by: Ht=CC+Aet−1e t−1A+BHt−1B, (2) where Aand Bare (n × n) coefficient matrices and CC is the decomposition of the intercept matrix. Each element (i,j)th in Ht depends on the corresponding (i,j)th element in ( et−1e t−1) and Ht−1 . Accordingly, past shocks and volatility are allowed to directly spill over from a market to another, and they are captured by the coefficients of the Aand Bmatrices. More specifically, the BEKK-GARCH matrices can be expanded as follows: hx t=Cx+α2 xex t−12+2αxαyxex t−1ey t−1+α2 yxey t−12+β2 xhx t−1+2βxβyxhxy t−1+β2 yxhy t−1(3) hxy t=Cxy +αxαxyex t−12+αyxαxy +αxαyex t−1ey t−1+αyxayey t−12+βsβxyhx t−1+βyxβxy +βxβyhxy t−1+βyxβyhy t−1(4) hy t=Cy+α2 xyex t−12+2αxyαyex t−1ey t−1+α2 yey t−12+β2 xyhx t−1+2βxyβyhxy t−1+β2 yhy t−1(5) The BEKK-GARCH parameters are estimated by the maximum likelihood method using the BFGS algorithm. In addition to the return and volatility spillover, we also compute the optimal weights and hedge ratios for each pair of stocks. The conditional variance and covariances are used for calculating the optimal portfolio weights and hedge ratios. This study follows Kroner and Ng (1998) in calculating the optimal portfolio weights of different pairs of stock markets: wxy t=hy t−hxy t hx t−2hxy t+hy t , (6) wxy t=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0,If w xy t<0 wxy t,If 0≤wxy t≤1 1,If w xy t>1 where wxy t is the weight of stock( x ) in a $1 stock(x)-stock( y ) portfolio at time t; hxy t is the conditional covariancebetween thetwo stock markets; hx t and hy t aretheconditional varianceof stock( x )and stock( y ), respectively; and 1 −wxy t is the weight of stock( y ) in a $1 stock( x )-stock( y ) portfolio. As suggested by Kroner and Sultan (1993): βxy t=hxy t hy t (7) where βxy t represents the hedge ratio. This shows that a short position in the stock ( y ) market can hedge a long position in stock (x). 58 JRFM 2020,13, 148 4. Empirical Results and Implications In this section, we will discuss our empirical results and implications. We first discuss our preliminary analysis. 4.1. Descriptive Statistics Table 1reports the summary statistics of the daily returns for the US; China; and four emerging LA stock markets—namely, Brazil, Chile, Mexico, and Peru. Among them, Brazil and Peru have the highest mean return, and the US has the smallest mean return during the full sample period. On the other hand, Chile has the smallest standard deviation, while Brazil has the largest standard deviation. Thus, Peru provides the highest mean return, with a relatively smaller risk in the LA stock markets. Overall, the skewness is significantly negative, the kurtosis is significantly higher than three for all stocks, and the Jarque–Bera statistics reject normality hypothesis for all series, inferring that all the returns are negatively skewed and fat-tailed. Moreover, Table 1also confirms that there are 1% significant autocorrelation and ARCH (autoregressive conditional heteroskedasticity) effects for all returns. We also apply both Augmented Dickey–Fuller (ADF) and Phillip–Perron (PP) tests to examine the stationarity of all the returns and exhibit the results in Table 2. The table indicates that all the series are 1% significant, inferring that all the returns are stationary. Table 1. Summary statistics. Markets Mean Std. Dev. Skewness Kurtosis J-B Stat Q-Stat ARCH US 0.00016 0.0124 −0.364 *** 14.045 *** 27181.3 *** 56.584 *** 548.40 *** CHN 0.00040 0.0155 −0.330 *** 8.2116 *** 6121.9 *** 60.119 *** 189.01 *** BRAZ 0.00047 0.0183 −0.403 *** 9.6439 *** 9937.1 *** 24.957 *** 686.82 *** CHIL 0.00030 0.0105 −0.878 *** 19.883 *** 37,432.8 *** 148.49 *** 180.34 *** MEXI 0.00024 0.0128 −0.086 * 8.3698 *** 6403.18 *** 108.33 *** 173.49 *** PERU 0.00047 0.0133 −0.549 *** 15.441 *** 34605.3 *** 290.64 *** 796.97 *** Notes: US—United States of America; CHN—China; BRAZ—Brazil; MEXI—Mexico; CHIL—Chile. Q-stat denotes the Ljung–Box Q-statistics. ARCH test refers to the LM-ARCH test. ***, * indicate the statistical significance at 1% and 10%, respectively. Table 2. Unit root tests ADF (t-Test) Phillips-Perron Test Markets None Constant Constant and Trend None Constant Constant and Trend US −79.73 *** −79.94 *** −79.96 *** −80.00 *** −80.01 *** −80.05 *** CHN −32.44 *** −32.48 *** −32.49 *** −69.97 *** −69.89 *** −69.88 *** BRAZ −72.04 *** −72.08 *** −72.08 *** −72.03 *** −72.08 *** −72.08 *** CHIL −33.59 *** −33.64 *** −33.67 *** −63.11 *** −63.11 *** −63.10 *** MEXI −50.70 *** −50.73 *** −50.73 *** −63.67 *** −63.68 *** −63.68 *** PERU −31.06 *** −31.12 *** −31.15 *** −61.82 *** −61.75 *** −61.64 *** Notes: US—United States of America; CHN—China; BRAZ—Brazil; MEXI—Mexico; CHIL—Chile; ADF—Augmented Dickey Fuller. *** indicate the statistical significance at 10%, respectively. 4.2. Return and Volatility Spillover between the US and LA Stock Markets We turn to apply the BEKK-GARCH model to examine the return and volatility spillovers between the US and LA stock markets in the full sample period, the global financial crisis, and the crash of the Chinesestockmarketandexhibit theresultsin Tables3–5. Wenotethatthe1%significantautocorrelation and ARCH effects for all returns, as shown in Table 1, justify the use of the BEKK-GARCH model in our analysis. 59 JRFM 2020,13, 148 Table 3. Estimates of BEKK-GARCH for the US and Latin American stock markets during the full sample period Brazil and US Chile and US Mexico and US Peru and US Coefficient p-Value Coefficient p-Value Coefficient p-Value Coefficient p-Value Panel A. Mean Equation μ10.075 *** 0.000 0.098 * 0.062 0.058 *** 0.000 0.047 *** 0.000 ∅11 −0.031 ** 0.036 0.007 0.722 0.038 ** 0.027 0.137 *** 0.000 ∅12 0.023 *** 0.005 0.000 0.938 0.021 0.109 −0.019 0.185 μ20.056 *** 0.000 0.062 *** 0.000 0.051 *** 0.000 0.058 *** 0.000 ∅21 0.055 ** 0.028 0.111 *** 0.000 0.050 *** 0.005 0.081 *** 0.000 ∅22 −0.077 *** 0.000 −0.057 *** 0.001 −0.071 *** 0.000 −0.042 *** 0.006 Panel B. Variance Equation c11 0.219 *** 0.000 2.496 *** 0.000 0.117 *** 0.000 0.161 *** 0.000 c21 0.069 *** 0.004 0.124 *** 0.009 0.050 *** 0.006 0.056 *** 0.003 c22 0.118 *** 0.000 0.042 0.571 0.124 *** 0.000 0.122 *** 0.000 α11 0.225 *** 0.000 0.003 0.562 0.264 *** 0.000 0.309 *** 0.000 α12 −0.014 0.319 0.001 0.611 0.008 0.609 0.004 0.836 α21 0.036 0.421 0.151 ** 0.026 −0.026 0.390 0.010 0.541 α22 0.338 *** 0.000 0.341 *** 0.000 0.314 *** 0.000 0.300 *** 0.000 β11 0.966 *** 0.000 0.701 *** 0.000 0.959 *** 0.000 0.942 *** 0.000 β12 0.005 0.158 −0.003 0.400 0.005 0.349 −0.002 0.789 β21 −0.101 ** 0.050 0.095 0.203 0.090 ** 0.043 −0.004 0.538 β22 0.933 *** 0.000 0.944 *** 0.000 0.938 *** 0.000 0.947 *** 0.000 Panel C. Diagnostic Tests LogL −16,174.1 −21,397.4 −13,816.1 −14,378.1 AIC 6.789 8.588 5.970 6.370 SIC 6.799 8.635 6.017 6.417 Q1[20] 30.320 * 0.065 1.791 0.720 19.075 0.517 328.759 * 0.031 Q2[20] 18.920 0.527 19.184 0.510 19.493 0.490 17.728 0.605 Q2 1[20] 29.942 0.182 0.004 0.659 34.776 ** 0.021 30.071 0.198 Q2 2[20] 27.782 0.115 22.616 0.308 25.161 0.185 33.181 0.146 Notes: US, United States of America; CHN, China; BRAZ, Brazil; CHIL, Chile; MEXI, Mexico. Variable order is the Latin American stock market (1) and China (2). In the mean equations, μ denotes the constant terms, whereas ∅12 denotes the return spillover from the Latin American stock market to the US stock market. In the variance equation, c denotes the constant terms, α denotes the ARCH terms, and β denotes the GARCH terms. In the variance equation, α12 indicates the shock spillover from the Latin American stock market to the US stock market, whereas β12 denotes the long − term volatility spillover from the Latin American stock market to the US stock market. Number of lags for VAR is decided using the SIC and AIC criteria. JB, Q(20), and Q 2 (20) indicate the empirical statistics of the Jarque–Bera test for normality, Ljung–Box Q statistics of order 20 for autocorrelation applied to the standardized residuals, and squared standardized residuals, respectively. Values in parentheses are the p-Value. ***, **, * indicate the statistical significance at 1%, 5%, and 10%, respectively. Tables 3–5report the return and volatility spillovers between the US and LA stock markets during the full sample period, the global financial crisis, and the crash of the Chinese stock market, respectively. Referring to coefficients ∅11 and ∅22 in Panel A, the results show that the lagged returns significantly influence the current returns in the US and the majority of LA stock markets during the full sample period, the global financial crisis, and the crash of the Chinese stock market. It highlights the possibility of the short-term prediction of current returns through past returns in the US and the majority of the LA stock markets. Our results are consistent with the findings of Syriopoulos et al. (2015) and Arouri et al. (2015) , which observe a significant impact of past returns on current returns in the US and LA stock markets. 60 JRFM 2020,13, 148 Table 4. Estimates of BEKK-GARCH for US and Latin American stock markets during the global financial crisis. Brazil and US Chile and US Mexico and US Peru and US Coefficient p-Value Coefficient p-Value Coefficient p-Value Coefficient p-Value Panel A. Mean Equation μ10.101 * 0.068 0.113 *** 0.000 0.055 0.246 −0.014 0.744 ∅11 −0.044 0.450 0.118 *** 0.001 0.023 0.680 0.138 *** 0.000 ∅12 0.021 0.581 0.019 0.440 0.096 ** 0.043 −0.009 0.770 μ20.019 0.665 0.046 0.361 0.064 0.144 0.029 0.518 ∅21 0.040 0.578 0.020 0.308 0.020 0.707 0.063 0.107 ∅22 −0.128 *** 0.007 −0.164 *** 0.000 −0.188 *** 0.000 −0.100 *** 0.003 Panel B. Variance Equation c11 0.271 ** 0.044 0.287 *** 0.000 0.218 ** 0.017 0.291 *** 0.000 c21 0.098 0.546 0.040 0.417 −0.035 0.730 0.173 *** 0.000 c22 0.129 ** 0.039 0.153 *** 0.000 0.000 0.799 0.109 *** 0.007 α11 0.421 * 0.069 0.483 *** 0.000 0.117 0.220 0.453 *** 0.000 α12 0.139 ** 0.022 −0.055 0.333 −0.104 * 0.057 0.087 0.255 α21 −0.237 0.128 −0.020 0.550 0.249 *** 0.000 −0.088 0.581 α22 0.138 0.145 0.292 *** 0.000 0.295 *** 0.000 0.226 *** 0.002 β11 0.902 *** 0.000 0.841 *** 0.000 1.063 *** 0.000 0.896 *** 0.000 β12 -0.041 * 0.082 0.051 ** 0.034 0.218 *** 0.001 −0.034 0.197 β21 0.071 0.482 0.024 * 0.083 −0.183 *** 0.000 0.014 0.687 β22 0.990 *** 0.000 0.937 *** 0.000 0.797 *** 0.000 0.969 *** 0.000 Panel C. Diagnostic Tests LogL −2766 −2438.432 −2514.181 −2804.767 AIC 7.792 7.026 7.132 8.025 SIC 8.018 7.253 7.359 8.251 Q1[20] 15.405 0.753 15.396 0.753 15.749 0.732 18.138 0.578 Q2[20] 19.980 0.459 22.469 0.316 25.023 0.201 19.998 0.458 Q2 1[20] 23.671 0.257 17.388 0.628 15.064 0.773 13.734 0.844 Q2 2[20] 37.237 ** 0.011 45.203 *** 0.001 33.878 ** 0.027 37.570 *** 0.010 Notes: US, United States of America; CHN, China; BRAZ, Brazil; CHIL, Chile; MEXI, Mexico. Variable order is the Latin American stock market (1) and China (2). In the mean equations, μ denotes the constant terms, whereas ∅12 denotes the return spillover from the Latin American stock market to the US stock market. In the variance equation, c denotes the constant terms, α denotes the ARCH terms, and β denotes the GARCH terms. In the variance equation, α12 indicates the shock spillover from the Latin American stock market to the US stock market, whereas β12 denotes the long-term volatility spillover from the Latin American stock market to the US stock market. Number of lags for VAR is decided using the SIC and AIC criteria. JB, Q(20), and Q 2 (20) indicate the empirical statistics of the Jarque–Bera test for normality, Ljung–Box Q statistics of order 20 for autocorrelation applied to the standardized residuals, and squared standardized residuals, respectively. Values in parentheses are the p-Value. ***, **, * indicate the statistical significance at 1%, 5%, and 10%, respectively. Regarding the interdependence of returns in the mean equation (see coefficients ∅12 and ∅21 in Panel A), the results indicate the unidirectional return spillover from the US to the majority of LA stock markets during the full sample period and the crash of the Chinese Stock Market. They imply that the past US returns can be used to predict the current returns of the LA markets during the full sample period and the crash of the Chinese Stock Market. These results are consistent with the previous findings of Arouri et al. (2015), who find the unidirectional return spillover from the US to the LA stock markets. Moreover, the return transmission is also significant from the Brazil to the US stock market during the full sample period. In contrast, the return transmissions are not found to be significant between the US and the majority of the LA stock (except Mexico) markets during the global financial crisis. These results suggest that the US (LA) stock returns are not useful in predicting the returns in the majority of the LA (US) stock markets during the global financial crisis. The results also reveal a unidirectional volatility spillover from Mexico to the US stock market during the global financial crisis. Based on the variance equation (see coefficients of α11 in Panel B), the results show that the conditional volatility of the majority of LA stock markets depends on their past shocks during all the sample periods. In addition, the coefficients of the past own shocks ( α22) are highly significant for the US in all the sample periods. Besides this, the sensitivity of past own volatility ( β11 and β22) is significant for the US and LA stock markets during all the sample periods. These results are consistent 61 JRFM 2020,13, 148 with the findings of Syriopoulos et al. (2015), which find that the past own volatility is a significant determinant of the future volatility of BRICS countries (including Brazil). Further, the coefficients of past own volatility are higher compared to the coefficients of the past own shocks in the US and LA stock markets, suggesting that the past own volatilities are more critical for the prediction of future volatility than the past own shocks during all the sample periods. Referring to the coefficient α12 and α21 in Panel B, the past shocks of the US stock market significantly influence the conditional volatility of just the Chile stock market during the full sample period. During the global financial crisis, the shock transmission is unidirectional from Brazil to the US and bidirectional between the US and Mexican stock markets. Moreover, the conditional volatility of the Mexican stock market is significantly affected by the US during the crash of the Chinese stock market. Table 5. Estimates of BEKK-GARCH for the US and Latin American stock markets during the crash of the Chinese stock market. Brazil and US Chile and US Mexico and US Peru and US Coefficient p-Value Coefficient p-Value Coefficient p-Value Coefficient p-Value Panel A. Mean Equation μ10.090 0.105 0.030 0.210 0.013 0.585 0.088 ** 0.029 ∅11 −0.047 0.172 0.077 * 0.082 −0.032 0.469 0.076 * 0.089 ∅12 0.016 0.302 −0.035 0.134 0.002 0.937 −0.025 0.550 μ20.064 *** 0.004 0.075 *** 0.001 0.072 *** 0.001 0.061 0.158 ∅21 0.129 * 0.064 0.120 *** 0.001 0.137 *** 0.000 0.086 * 0.093 ∅22 −0.066 ** 0.040 −0.052 0.112 −0.059 * 0.083 −0.050 * 0.085 Panel B. Variance Equation c11 0.268 *** 0.005 0.361 *** 0.004 0.599 *** 0.000 0.159 * 0.066 c21 0.151 ** 0.017 0.011 0.720 0.164 *** 0.000 0.117 0.122 c22 0.124 * 0.076 0.186 *** 0.000 0.124 0.150 0.089 0.790 α11 0.196 *** 0.001 0.522 *** 0.001 0.434 *** 0.000 0.278 ** 0.011 α12 0.008 0.821 −0.024 0.326 0.033 0.298 0.142 0.331 α21 0.023 0.744 −0.028 0.648 −0.115 * 0.052 0.019 0.850 α22 0.430 *** 0.000 0.421 *** 0.000 0.381 *** 0.000 0.313 0.416 β11 0.958 *** 0.000 0.686 *** 0.001 −0.359* 0.077 0.949 *** 0.000 β12 −0.008 0.571 0.018 0.411 −0.068 *** 0.000 −0.042 0.464 β21 0.013 0.697 0.076 0.227 0.528 *** 0.000 −0.014 0.766 β22 0.880 *** 0.000 0.879 *** 0.000 0.915 *** 0.000 0.917 *** 0.000 Panel C. Diagnostic Tests LogL −2078 −1582.556 −1545.033 −1733.842 AIC 5.759 4.585 4.429 4.891 SIC 5.986 4.812 4.655 5.118 Q1[20] 21.413 0.373 33.001 ** 0.034 21.955 0.343 31.804 ** 0.045 Q2[20] 24.907 0.205 24.713 0.213 24.601 0.217 25.783 0.173 Q2 1[20] 6.942 0.897 85.117 *** 0.000 29.827 * 0.073 16.276 0.699 Q2 2[20] 8.249 0.890 9.945 0.969 10.383 0.961 8.909 0.984 Notes: US, United States of America; CHN, China; BRAZ, Brazil; CHIL, Chile; MEXI, Mexico. Variable order is the Latin American stock market (1) and China (2). In the mean equations, μ denotes the constant terms, whereas ∅12 denotes the return spillover from the Latin American stock market to the US stock market. In the variance equation, c denotes the constant terms, α denotes the ARCH terms, and β denotes the GARCH terms. In the variance equation, α12 indicates the shock spillover from the Latin American stock market to the US stock market, whereas β12 denotes the long-term volatility spillover from the Latin American stock market to the US stock market. Number of lags for VAR is decided using the SIC and AIC criteria. JB, Q(20), and Q 2 (20) indicate the empirical statistics of the Jarque–Bera test for normality, Ljung–Box Q statistics of order 20 for autocorrelation applied to the standardized residuals, and squared standardized residuals, respectively. Values in parentheses are the p-Value. ***, **, * indicate the statistical significance at 1%, 5%, and 10%, respectively. Regarding the cross-market volatility spillover (see coefficients β12 and β21 in Panel B), the results indicate that the volatility transmission is unidirectional from the US to the Brazil and Mexican stock markets during the full sample period. In contrast, the results reveal the bidirectional volatility transmission between the US and two LA stock markets (Chile and Mexico), whereas there was unidirectionalvolatilitytransmission fromBrazilto theUS stockmarket during theglobal financialcrisis. 62 JRFM 2020,13, 148 TheseresultsareincontrastwiththefindingsofWangetal. (2017), whichreportaninsignificantvolatility spillover between the US and Brazil stock markets during the global financial crisis. The considerable trade volumes between the US and two LA stock markets (Brazil and Mexico) explain the volatility linkages between the stock markets of the concerned countries. Johnson and Soenen (2003) also suggest that trade increases the financial contagion effects between the stock markets of concerned countries. From the Latin American region, Mexico is the biggest trading partner of the US; therefore, volatility linkages are also observed between Mexico and the US stock market during the global financial crisis. These findings suggest that portfolio investors can get the maximum benefit of diversification by making a portfolio of US and Peru stocks during the global financial crisis. Lastly, a bidirectional volatility transmission is observed between the US and Mexican stock markets during the crash of the Chinese stock market. It implies that portfolio investors can diversify risk by making a portfolio of the US and LA stock markets (except Mexico) during the crash of the Chinese stock market. 4.3. Return and Volatility Spillover between China and the LA Stock Markets Tables 6–8represent the return and volatility transmissions between China and the LA stock markets during the full sample period, the global financial crisis, and the crash of the Chinese stock market. The difference in the opening time of the China and LA stock markets has been adjusted where necessary in the estimations. Referring to the coefficient ∅11 in Panel A, the results indicate that the lagged returns of the majority of LA stock markets (except Brazil) largely determine their current returns during the full sample period and the crash of the Chinese stock market. During the global financial crisis, the past returns significantly affect the current returns of the Chile and Peru stock markets. This implies that the past returns can be used for the short-term prediction of the current LA stock returns. These results confirm the previous findings of Arouri et al. (2015). Referring to the coefficient ∅22 in Panel A, the lagged returns significantly influence the current returns in the Chinese stock market during the full sample period. In contrast, the current returns of the Chinese stock market are not influenced by their past returns during the global financial crisis and the crash of the Chinese stock market. This implies that the past returns cannot be used for the short-term prediction of the current Chinese stock returns during the crisis period. Based on the cross-market return spillover (see the coefficients ∅12 and ∅21 in Panel A), the results reveal the unidirectional return transmissions from China to the majority of LA stock markets during all the sample periods. These results contradict the previous findings of Aktan et al. (2009) and Sharma et al. (2013) , who report the insignificant impact of the Chinese stock returns on the Brazilian stock returns. In addition, the return transmission is also significant from Brazil to China during the crash of the Chinese stock market. From the variance equation (see coefficients α11 and α22 Panel B), the findings show that the lagged shocks significantly influence the conditional volatility of the China and LA stock markets during all the sample periods. Referring to the coefficients β11 and β22 , the results show that the current conditional volatility depends on their past volatility in the China and LA stock markets during the all sample periods. The critical finding is that the coefficients of past own volatility are seen to be higher compared to the past own shocks. This difference suggests that past own volatilities rather than past shocks are more important for the prediction of the current volatility in the China and LA stock markets. Refer to the coefficients α12 and α21 in panel B, the shock transmission is unidirectional from Brazil and Peru to the Chinese stock market, whereas bidirectional shock transmission is observed between the China and Mexican stock markets during the full sample period. The results reveal that the past shocks in the Brazil and Mexican stock markets significantly affect the conditional volatility of the Chinese stock market during the global financial crisis. 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For example, it is argued that wishing to have a large correlation with the Standard and Poor’s 500 stock index (S&P 500 stock index) may induce an increase in the domestic investment weight by American investors and, hence, rationalize the IHB. While this argument isvalid in the mean-variance framework, employing bivariate first-degree stochastic dominance (BFSD), we prove that this intuition is generally invalid. Counter intuitively, employing “keeping up with the Joneses” (KUJ) preference with actual international data even enhances the IHB phenomenon. Keywords: investment home bias (IHB); bivariate first-degree stochastic dominance (BFSD); keeping up with the Joneses (KUJ); correlation loving (CL) JEL Classification: D81; C91 1. Introduction The investment home bias (IHB) is well documented. For example, the US equity market accounts for about 35% of the world equity market, yet about 75% of Americans’ equity investment is allocated to the US market. Hence, the US equity IHB is in the magnitude of 40%. For most commonly employed utility functions, the univariate expected utility maximization also does not support the relatively large domestic investment weight; hence, the IHB puzzle emerges. It is advocated that the bivariate expected utility maximization rationalizes partially or fully the IHB. In this study, we employ the “keeping up with the Joneses” (KUJ) preference, where the investor’s wealth and the peer group’s wealth are the two attributes of this utility function, analyzing the peer effect on the empirically observed IHB phenomenon. The basic idea and the intuition of the KUJ argument for rationalizing the IHB phenomenon is as follows: suppose that you know your univariate utility function and that, for a given joint distribution of returns corresponding to the various international markets, you derive with this utility function the optimal investment weights in the domestic market as well as in the foreign markets under consideration. Furthermore, suppose that the optimal domestic investment weight is, say, p%. Now, suppose that you decide to consider, in addition to the joint distribution of returns, one more factor: you also want the performance of your portfolio to be as close as possible to the performance of a certain local stock index. For example, the American investor wants the return on her portfolio to be as close as possible to the return on the S&P 500 index, which, for simplicity of the discussion, is assumed to be the peer’s portfolio (the same analysis applies to any other local stock index). Thus, if the investor benefits from having a relatively large correlation with the S&P index, she may have an incentive to increase the domestic investment weight (which generally increases the correlation with the S&P stock index, and if the domestic investment weight is 100%, this correlation is +1) beyond JRFM 2020,13, 94; doi:10.3390/jrfm13050094 www.mdpi.com/journal/jrfm 73 JRFM 2020,13,94 what is obtained by a maximization of a univariate expected utility function. Therefore, employing KUJ preferences may rationalize the IHB.1 Indeed, Lauterbachand Reisman (2004) usethe KUJ preferenceprove that theIHB is rationalizedby incorporating the peer effect. However, they use the mean-variance model with some approximations to achieve this result. We analyze in this paper the impact of incorporating the peer effect on the IHB in the most general bivariate expected utility case, not relying on the mean-variance framework and where no approximations of the various mathematical formulas are employed. We define the precise conditions, which guarantee the IHB rationalization, by adding the peer effect. We find that, in this unrestricted analysis, the appealing intuitive explanation of the IHB rationalization by the peer effect is generally wrong. Thus, we conclude that one should seek other economic explanations for the observed IHB phenomenon. For some interesting economic suggestions, see Coeurdacier and Rey (2013) and Berriel and Bhattarai (2013)2or other behavioral explanations. We employ in this study distribution-free bivariate first-degree stochastic dominance (BFSD), with no assumptions on the shape of the bivariate preferences and no approximations. We prove that, despite the above appealing intuition of the peer effect on the optimal domestic investment weight, using the bivariate preferences, the IHB may increase or decrease relative to the univariate optimal domestic investment weight. Moreover, we demonstrate with actual international data that adding the peer effect, counter intuitively, even intensifies the IHB from the American investor’s point of view. Hence, the IHB still exists. The structure of the rest of this paper is as follows. Section 2provides a brief literature review. Section 3presents bivariate first-degree stochastic dominance (BFSD) rule and the implied theoretical results. We analyze the various factors affecting the IHB and show that bivariate preferences rationalize the IHB phenomenon only in a limited and unrealistic case. Section 4is devoted to the commonly employed KUJ preferences, which is a specific set of all the bivariate preferences. We show empirically that the peer effect with KUJ preferences even enhances the IHB puzzle. Section 5concludes. 2. Literature Review Vanp é e and DeMoore (2012) show that the IHB exists in virtually all countries. The magnitude of the IHB phenomenon is relatively large, characterizing various periods, assets, and countries. While about three decades ago the American investment in the local market was more than 90%, implying a very large IHB, in recent years the IHB phenomenon has been mitigated, yet it is still about 40%. When it comes to fixed-income assets, the home bias is even larger. This phenomenon is not unique to the US and characterizes many capital markets (for a report on the IHB in various countries, regarding equity and fixed-income assets, see (Philips et al. 2012)). Actually, there is evidence that the home bias is even worse than reported (see Baxter and Jermann 1997). Researchers have analyzed various possible key explanations for the IHB. It is agreed that some portion ofthedomesticoverinvestment maybeinduced byinternationaltrade barriers, foreign exchange risk, and regulation, as well as by a domestic peer group effect. However, with the increase in the rapid flow of information and market efficiency observed over the last few decades, the trade barriers, including possible asymmetrical information, have drastically declined. This may account for the observed slight decrease in the domestic overinvestment phenomenon. However, since 1998, the equity IHB of American investors has stabilized at about 40% (see Levy and Levy 2014). 1 Note, we analyze whether the peer effect increases the optimal domestic weight, which partially or fully rationalizes the IHB. The reason is that it is possible that the peer effect increases the optimal domestic weight by, say, 1%, but the IHB is, say, 40%, a case where other factors are needed to explain the observed IHB. In our study, we find empirically that the peer effect even enhanced the IHB; hence, the distinction between partial and full IHB rationalization is irrelevant. 2 They consider portfolio diversification when macroeconomic factors are incorporated into a two-country general equilibrium model, called the “Open Economy Financial Macroeconomics” model. They conclude that, with this equilibrium model, the home bias is less of a puzzle. Berriel and Bhattarai (2013) also suggest a macroeconomic model (related to the positive association between government spending and return on local stocks) to explain the home bias. 74 JRFM 2020,13,94 While most empirical studies analyze the IHB at the country level (see French and Poterba 1991; Tesar and Werner 1995), Kang and Stultz (1997), who study the IHB puzzle in Japan, analyze it at the individual firm level, showing that foreign investors hold disproportionally more Japanese shares of firms in the manufacturing industries, large firms, and firms with good accounting performance. Similarly, Dahlquist and Robertsson (2001) identify the characteristics of Swedish firms that attract foreign investors. Lewis (1999), who analyzes the effect of each economic factor that is considered as a barrier for efficient international diversification on the IHB, concludes that the trade barriers cannot explain the magnitude of the existing IHB. Therefore, the IHB puzzle is still an interesting research topic.3 Obviously, if the IHB does not incur economic loss, it does not constitute an economic puzzle. Indeed, the intensity of the IHB economic cost changes over time. Levy (2016) analyzes the trend in the IHB phenomenon over time. Moreover, he distinguishes between the economic home bias (EHB), which measures the economic loss in terms of the differences in the certainty equivalent of two alternative international diversification strategies (with and without a home bias) and the IHB, which simply measures the deviations between the optimal international investment weights and the actual investment weights. He reports that, while the EHB was very large in the past, in the last 15 years, the EHB from the American investment point of view has become negligible, despite the existence of about 40% IHB. This reduction in the EHB is induced by the increasing trend in the international correlations. Thus, it seems that for the American investors the IHB is not a major economic puzzle. However, he also reports that for other countries, e.g., France, the EHB is still very large, and the economic puzzle exists. Moreover, in recent years, we have trend reversal in correlations, and a decrease in the average correlation between various markets has been recorded. As a result of this trend reversal, the EHB has recently increased, even for American investors. Thus, for most countries and with the recent trend reversal in correlation also for the US, the IHB still constitutes an economic puzzle that needs an explanation. The employment of the KUJ preference, namely incorporation of the peer effect, is considered as one of the promising paths in explaining the IHB puzzle. We employ in this paper a bivariate preference. Generally, with bivariate preference, the two variables can take many forms, e.g., wealth and health, climate and income, etc. Our study deals with investment choices. Hence, the two variables are the individual’s wealth and the peer group’s wealth. The peer group’s wealth can be the return on a certain domestic portfolio, and in our case, as mentioned above, we assume, for the simplicity of the discussion and without loss of generality, that it is the return on S&P 500 stock index. The common view is that the relevant bivariate utility function has a positive cross derivative (we will elaborate on this issue below) and that investors want, among other things, the performance of their portfolio to be as close as possible to the performance of the peer’s portfolio, i.e., a large correlation with the S&P stock index is desired. 4 Therefore, we focus our analysis on the positive cross-derivative case. Obviously, despite the desire for having a relatively large correlation with the S&P index, the investor will shift from a portfolio with a small correlation to a portfolio with a large correlation, only if the bivariate expected utility increases by such a shift. We turn to analyze the conditions under which indeed such shift takes place, namely that the IHB can be rationalized with the peer effect. 3 It is interesting to note that, even in a case in which there are no transparent trade barriers, there is a tendency to invest in firms that are geographically located close to the investor’s location. This phenomenon is well documented within the US (see Coval and Moskowitz 1999,2001;Huberman 2001). This indicates that the home bias is a complex phenomenon that is not easy to explain with conventional economic factors. 4 Tsetlin and Winkler (2009) advocate that correlation aversion prevails. However, in their model, the two attributes of the bivariate preference directly affect the utility of the decision maker, for example, income and quality of life. In our model, the two attributes are different: the individual’s wealth and the peer group’s wealth. As relative wealth may affect the individual’s utility, it is advocated in the literature that, when some conditions hold, correlation loving prevails. 75 JRFM 2020,13,94 3. Bivariate First-Degree Stochastic Dominance (BFSD) and the IHB We would like to stress at the outset that most of the mathematical formulas given in the first part of this section are not new and exist in the literature, albeit in different forms and in different connotations. However, we use these mathematical results, to the best of our knowledge for the first time to analyze the peer effect with KUJ preferences on the IHB phenomenon. 3.1. The Sufficient Conditions for BFSD Implying the IHB Rationalization Consider an individual with a bivariate preference U(w , wP) , where w denotes the return on the selected international portfolio by the investor under consideration, and wp denotes the return on the peer’s portfolio. We compare two bivariate investment portfolios, Fand G, where the domestic investment weight in portfolio Fis larger than the domestic investment weight in portfolio G(we will elaborate later on the selected portfolios, Fand G). Our aim is to examine the conditions under which F dominates Gby BFSD with the above bivariate utility function, where we first assume two assumptions on the preferences: ∂Uw,wp/∂w≡U1≥ 0 (monotonicity) and ∂2Uw,wp/∂w∂wp≡U12 ≥ 0 (later on we consider also U12 ≤ 0, a case usually not considered in KUJ economic research but emerges as important to our analysis). There is no constraint on the derivative ∂Uw,wp/∂wp≡U2 , which can be negative, zero, or positive. 5 If such dominance exists, then all investors, regardless of the precise shape of the bivariate preference, will switch from Gto F. Hence, the optimal domestic investment weight increases, and therefore the peer effect rationalizes the IHB phenomenon. Note that the main ingredient of the KUJ preference is that the cross derivative (U12) is positive, implying that the individual’s marginal utility increases with an increase in the peer group wealth (see Ljungqvist and Uhlig 2000). 6 Therefore, as explained before, it seems that the investor with a positive cross derivative would incline to overinvest domestically, as she prefers her wealth to be positively correlated with the peer’s wealth. While the above intuitive explanation is appealing, in the following proposition, it is formally shown that generally only under some specific conditions, indeed a positive cross derivative is tantamount to correlation loving, where correlation loving implies that, by increasing the domestic investment weight, the bivariate expected utility increases. Namely, if the conditions required in the proposition are intact, the investor increases her bivariate expected utility by overinvesting domestically (relative to the optimal univariate expected utility maximization optimal domestic investment weight), and by doing so, the correlation increases. Thus, if the proposition required conditions hold in practice, we have by the KUJ preferences a rationalization of the IHB, and the IHB puzzle may vanish. As we explain below, in practice, the required conditions for IHB rationalization are not intact. Before stating the proposition, we need the following definition: Definition 1. Definition of correlation loving (CL): The investor is CL if and only if, by increasing the correlation between her portfolio and the peer’s portfolio, the expected bivariate utility increases. Hence, CR investors who maximize the bivariate expected utility would increase the domestic investment weight relative to the optimal univariate expected utility weight. 5Note that a negative sign implies jealousy, and a positive sign implies altruism (see Dupor and Liu 2003). 6 Numerous studies suggest replacing the univariate expected utility analysis with the expected bivariate utility analysis with various definitions of the two variables: past and present consumption, consumption of the individual, and consumption of the peer group, the wealth obtained by the individual and the opponent in an ultimatum game, and so forth. For studies that assume that the utility is derived not from the absolute wealth (or consumption) of the individual but from the relative wealth (or consumption), in which the wealth’s position relative to the peer group plays an important role, as well as for other factors that do not affect the classic univariate expected utility but affect the bivariate expected utility, see, for example, Abel (1990), Constantinides (1990), Bolton (1991), Rabin (1993,1998), Gal í (1994), Campbell and Cochrane (1999), Bolton and Ockenfels (2000), Dupor and Liu (2003), Zizzo (2003), and Demarzo et al. (2008). 76 JRFM 2020,13,94 Proposition 1. Suppose that the investor faces two alternate bivariate prospects, F(w , wp) and G(w , wp) , where w as well as wp can take only two different outcomes. As there are only two outcomes, they can be rearranged to have either a correlation of +1 or a correlation of − 1. Diversification between w and wp is not allowed, implying that the marginal distributions are identical (namely, Fw=Gw and FwP=GwP , regardless of the outcomes arrangement; see, for example, Table 1). Under these specific conditions, the investor with a bivariate preference is CL if and only if the cross derivative is positive, namely U12 ≥ 0. Specifically, under the conditions of the proposition, with CL, the prospect with a correlation of +1 yields a higher bivariate expected utility than any other possible prospect. (For proof, with some other notation, see (Eeckhoudt et al. 2007)). Thus, if the conditions of the proposition were intact, the American investor who likes her investment performance to be as close as possible to the S&P index would have a higher expected utility by increasing the domestic investment weight. Actually, under the conditions of the proposition, having a correlation of +1 with the S&P index is optimal, implying that investing 100% domestically is optimal, which creates a negative IHB puzzle (because in practice less than 100% is invested domestically). In short, if the conditions of Proposition 1 are intact, we have: CL ⇔U12 >0 (1) Note that investing more intensively domestically, hence increasing the correlation between the investor’s portfolio and the peer’s portfolio, generally does not imply CR as defined above. The reason is that, with investment in practice, by increasing the domestic investment weight, although the correlation increases, generally, other parameters of the portfolio may also change, the marginal distributions may change (hence, the conditions of the proposition are violated), and the bivariate expected utility may decrease. Therefore, the American investor may decide not to decrease the domestic investment weight, despite the desire to have large correlation with the S&P index. However, by the above definition, the investor is CL only if, after considering all effects, the bivariate expected utility increases. However, note that, by Proposition 1, the marginal distributions are kept unchanged, and the correlation can take only the extreme values of either +1or − 1. This is because in Eeckhoudt et al. (2007) original proposition, each variable can get only two possible values. Hence, by reordering these values, the marginal distributions are kept unchanged. Also, diversification between wand wp is not allowed, because if it is allowed, the marginal distribution of the individual’s wealth, w, generally will not be kept constant. Thus, the statement given in Proposition 1 is suitable to some choices, where the variables are, for example, wealth and health, with only two outcomes (say, bad and good health, high and low income, etc.). As we shall see below, with international diversification, we have more than two outcomes corresponding to each prospect, and diversification is allowed. Hence, the marginal distributions generally change when the selected diversification changes. Therefore, a positive cross derivative in our analysis does not necessarily imply CL. As a result, we may even obtain an IHB phenomenon enhanced with bivariate preferences relative to the univariate IHB, despite the fact that a positive cross derivative is assumed. Let us turn now to the conditions for BFSD of the distribution of returns of the portfolio with the IHB over the distribution of returns with no IHB. The two portfolios that we compare, Fand G, have bivariate density functions, denoted by f(w , wP) and g(w , wP) , respectively. As we focus on the possible IHB rationalization, it is assumed, as explained before, that Fstands for a portfolio with an IHB, that is, the domestic weight in this portfolio is larger than the corresponding weight in G. Thus, if the domestic investment weight in Gis equal to the optimal theoretical univariate expected utility maximization domestic weight (say, the international market portfolio), the BFSD of Fover G implies that the peer effect rationalizes the IHB phenomenon, as all investors would prefer Fover 77 JRFM 2020,13,94 G. 7 Assuming that ∂2w,wp/∂w∂wp)≡U12 > 0 with KUJ preference 8 to explain various observed economic phenomena is very common. As seen in Proposition 1, this assumption is an important ingredient also needed to rationalize the IHB phenomenon, so long as the conditions of Proposition 1 hold. Therefore, we examine the role of the cross derivative on the BFSD relation. To examine possible rationalization of the observed IHB with KUJ preferences, we extend the expected utility univariate analysis to the bivariate expected utility analysis by adding the peer effect. The expected bivariate utility of portfolios Fand Gis given by: EFU=w wwp wpU(w,wp)fw,wpdwdwp EGU=w wwp wpU(w,wp)gw,wpdwdwp (2) where w and w denote the minimal and maximal values of w (which can be −∞ and ∞ ); similarly, wP and wPdenote the minimal and maximal values of wP. Thus, Δi≡EFU−EGU=w wwp wp U(w,wp)fw,wp−gw,wpdwdwp Integrating by parts the above equation with respect to both variables yields: Δi≡EFU−EGU=wp wpw wU12[F(w,wp)−G(w,wp)]dwdwp+w wU1[G(w)− F(w)]dw +wp wpU2[G(wp)−F(wp)]dwp ≡A+B+C (3) where Δi denotes the expected utility difference corresponding to the ith investor, F(w , wP) and G(w , wP) are the two bivariate cumulative distributions, F(w)=Fw,wp is the marginal cumulative distribution function of w , Fwp=Fw,wp is the marginal cumulative distribution function of wP , and U1 , U2 , and U12 denote the partial derivatives: U1≡∂U/∂w , U2≡∂U/∂wP , and U12 ≡∂2U/∂w∂wP , respectively. For the derivation of Equation (3) with slightly different notations, see Levy and Paroush (1974, p. 131) and Atkinson and Bourguignon (1982, pp. 185–86). 9 Note that, as the marginal utility of the peer’s portfolio is identical under the various investment strategies (with and without intensive domestic investment). Namely, we have Gwp=Fwp , therefore term Cin Equation (3) is equal to zero. Thus, the rest of the paper relate only to terms Aand B. Let us first analyze the relation between Equation (3) (with C=0) and the conditions given in Proposition 1. If each of the two random variables, w and wp , has only two possible different outcomes, the correlation is either +1or − 1. Also, when diversification between wand wp is not allowed, the marginal distributions are equal (namely, G(w)=F(w) , see also the example given in 7 Obviously, we have a different optimum portfolio for each utility function, but, as we shall see below, the analysis is intact, independent of the assumed preference. 8 The KUJ and CUJ literature is very extensive; hence, we mention here only a few of these studies. Abel (1990) and Gal í (1994) use this bivariate framework to explain optimal choices. Ljungqvist and Uhlig (2000) examine the role of tax policies in economics with CUJ utility functions. Campbell and Cochrane (1999) assume that the preference is a function of the relative consumption, when the individual’s consumption is measured relative to the weighted average of the past consumption of all individuals. In these models, when the peer group’s variable (e.g., consumption) is a lagged variable, the model is commonly called the CUJ model, and when the individual’s variable and the peer group variable relate to the same time period (e.g., return on investment), it is commonly called the KUJ model. In this paper, we analyze the optimal portfolio investment decision in the KUJ set-up. 9 Note that Equation (2) is reduced to the well-known univariate formula employed to derive the FSD rule, where U12 =U2= 0. For more details, see Hadar and Russell (1969) and Hanoch and Levy (1969). Although we focus in this paper on FSD, one can assume risk aversion and employ stronger investment rules; for example, see Rothschild and Stiglitz (1970) and Levy (2015). 78 JRFM 2020,13,94 Table 1. Hence, in this specific case also term Bis equal to zero, and we are left with term A.IfF represents the +1 correlation and Gthe − 1 correlation, we must have with the two outcomes case that Fw,wp≥Gw,wp (see example 1 in Table 1. Hence, in this case by Equation (3), with B=C=0, the condition U12 ≥ 0isasufficient condition for dominance of the joint distribution with the +1 correlation over the joint distribution with the − 1 correlation (see Equation (3)). 10 It is easy to verify that, in this specific case, U12 ≥ 0 is a necessary and sufficient condition for dominance. 11 Thus, Equation (3) is perfectly consistent with Proposition 1, so long as the conditions given in the proposition are intact. However, Equation (3) corresponds to the general case, as it covers the more realistic scenarios where more than two outcomes are possible. Diversification is allowed, and the marginal distributions are not necessarily equal, hence term Bis not necessarily equal to zero. As we shall see in this general and realistic case, U12 ≥ 0 is neither a necessary nor a sufficient condition for dominance. We turn now to analyze the possible dominance of the portfolio with the IHB over a portfolio with no IHB in the most general case. To examine possible rationalization of the IHB phenomenon with bivariate preferences, let us first take a deeper look at the marginal distributions corresponding to the international diversification issue analyzed in this paper. Returning to Equation (3), note that, as mentioned above, it is reasonable to assume that the third term on the right-hand side of Equation (3), term C, is equal to zero, as the investor in the capital market generally cannot affect the peer group investment decision; hence, the peer’s group marginal distribution is identical under Fand G. This condition conforms to the requirement in Proposition 1, even in the case where more than two outcomes exist. This is a reasonable assumption with the investment choices that we analyze in this study but not with ultimatum games in which the individual decision affects the opponent’s outcome. Moreover, in the portfolio investment case, this term is equal to zero, regardless of whether the peer group portfolio is domestic or international. Thus, regarding the issue that we investigate in this paper (investment with a particular stock index as the peer group’s portfolio), as advocated above, the sign of the derivative U2 is irrelevant. Namely, term C=0 and there is no need to assume jealousy (U2<0) or altruism (U2> 0 ) to obtain our results corresponding to the portfolio investment case. Thus, as for the analysis of the IHB, term Cis equal to zero, and Equation (3) is reduced to: Δi=A+B. (4) However, note that generally we cannot assume that also term Bis equal to zero, as by changing the diversification strategy, we change the marginal distribution of the individual’s wealth. Thus, with international portfolio diversification, the condition of equal marginal distributions (see term Bof Equation (3)) required by Proposition 1 does not hold. To be able to determine whether the peer effect induces an increase in the optimal domestic investment relative to the univariate expected utility optimal domestic investment weight, we need to be more specific regarding the definitions of portfolios Fand Gunder consideration. We examine here the possible existence of BFSD by considering the two specific portfolios with direct implication to the IHB issue analyzed in this paper. These two portfolios are denoted by FAand GM, as defined below. Definition 2. GM is the portfolio with the international market weights. If the American investor holds this market portfolio, she would invest 35% (which is the weight of the American market in the world market) domestically; hence, the IHB does not exist. Distribution FA stands for the actual aggregate portfolio held by the American investors. Namely, the actual domestic weight held by the American investor is 75%; hence, holding this portfolio implies an IHB of 40%. 10 Actually, it is required to have at least one strict inequality with the distribution functions as well as with the cross derivative to avoid the trivial case of having Δi= 0. In the rest of the paper, when we write such inequalities, we always mean that there is at least one strict inequality, but to avoid a complex writing, we will not write it down everywhere. 11 If U12 < 0, in some range, one can always find a bivariate preference, such that outside this range the cross derivative is close to zero; hence, Δiis negative. Therefore, to guarantee that Δiis non-negative, the cross derivative cannot be negative. 79 JRFM 2020,13,94 Specifically, with the 11 countries for α= 2, we find with no peer group effect (a univariate utility function) that the optimal investment weight in the US is 0.49. Employing Equation (13) (Abel’s preference), we find that the domestic investment in the US decreases rather than increases. For example, for α= 2 and γ= 1, the US domestic weight decreases from 0.49 to 0.29. The same is true for the other KUJ preference suggested in this study. Employing Equation (14), we find that with the KUJ domestic peer group effect, for α= 2 and β= 0.5, the optimal investment weight in the US decreases from 0.49 with no peer group effect to 0.39 with the peer group effect (this figure is not reported in a table). Thus, with the KUJ preference with a positive cross derivative, adding the domestic peer group effect decreases rather than increases the optimal US investment weight. Hence, with these specific KUJ preferences, we cannot rationalize the home bias empirically, which confirms the assertion that a positive cross derivative and CL are generally not equivalent. With a positive cross derivative, the investor wishes to increase the correlation by increasing the domestic investment weight but may not do it because other parameters (e.g., mean return) may induce a decrease in the bivariate expected utility by such action. Thus, if all investors have CRRA preference with various risk aversion parameters, the IHB is even intensified with the peer effect, despite the fact that a positive cross derivative implies that, other things being held unchanged, the American investor wants her portfolio to be as close as possible to the S&P stock index. We turn now to examine whether incorporating the peer effect with a negative cross derivative may increase the optimal domestic investment weight. We employ the function [w1−α 1−α]/[w1−β p 1−β] ( with α<1 and β<1 ). It is easy to verify that the derivative with respect to w is positive (monotonicity) and that the cross derivative is negative. Once again, counter intuitively, we find that with this bivariate utility function with a negative cross derivative, the optimal domestic investment weight of the American investor in the US increases rather than decreases, due to the peer effect. For example, for α= 2 and γ= 0.5, we obtain with this function that the domestic investment weight increases from 0.49 to 0.54. Therefore, we conclude that, with historical data, a positive cross derivative is neither sufficient (see Proposition 1) nor necessary (as demonstrated empirically with a preference with a negative cross derivative) for IHB rationalization. 5. Concluding Remarks It iswelldocumented inthe literature that, inthe univariateexpectedutility framework, theoptimal international diversified portfolio generally reveals an investment home bias (IHB), which constitutes a major economic puzzle. In another research strand, it is suggested that investor’s welfare is generally determined by relative wealth, relative consumption, relative success in investment, and so on, leading to the development of the bivariate expected utility paradigm, in which keeping up with the Joneses (KUJ) and catching up with the Joneses (CUJ) preferences are probably the most widely employed preferences in the bivariate framework. In this study, we combine these two research strands by investigating whether switching from a univariate preferences framework to multivariate preferences framework enables the rationalization of the empirically observed IHB phenomenon. As with the bivariate preference, with a positive cross derivative, other things being held constant, the investor wishes the performance of her portfolio to be as close as possible to the performance of a certain local stock index (the peer effect), it is suspected that with a peer effect the investor tends to overinvest domestically (relative to the univariate expected utility domestic optimal investment weight). Hence, the employment of the bivariate preference with a positive cross derivative may rationalize the IHB. We find, theoretically and empirically, that this intuitive explanation is misleading. While it is proven in the literature that, under some approximation, employing the mean-variance framework with a peer effect indeed rationalizes the IHB, we show in this paper that for unrestricted preferences not confining the analysis to the mean-variance model, counter intuitively, the IHB cannot be rationalized by the peer effect, even when the cross derivative of the bivariate preference is assumed to be positive. 86 JRFM 2020,13,94 We employ the bivariate first-degree stochastic dominance (BFSD) rule and prove theoretically that bivariate preferences with a positive cross derivative rationalizes the observed IHB, only in the unrealistic case in which the marginal distributions of all possible portfolio under consideration are identical. Of course, this does not hold in practice, as not all international markets are identical, and therefore also the marginal distributions of various selected diversified portfolios are not identical. Thus, even with peer effect, overinvesting domestically may be an inferior investment strategy, hence the IHB cannot be explained by the peer effect. With actual empirical international stock market data (obviously, with unequal empirical marginal distributions), we find that the commonly employed KUJ preference with a positive cross derivative, which intuitively implies a desire to increase the correlation by overinvesting domestically, decreases rather than increases the domestic investment weight, hence the peer effect even enhances the IHB puzzle. Moreover, once again counter intuitively, we find that, with a bivariate preference with a negative cross derivative, the optimal domestic investment increases. Thus, a positive cross derivative is neither necessary nor sufficient for IHB rationalization. In sum, employing a general bivariate utility function with peer effect, with no constraints on the preference employed, generally cannot rationalize the empirically observed IHB. Employing the commonly employed specific KUJ bivariate preferences also does not rationalize the IHB. As the IHB is an empirical fact, to rationalize this phenomenon, one needs to seek other explanations and other research strands, as the intuitive explanation of the peer effect for rationalizing the IHB phenomenon is misleading. Funding: This research received no external funding. Acknowledgments: I would like to thank three anonymous referees of this Journal for their helpful comments which greatly improved the paper. Conflicts of Interest: The authors declare no conflict of interest. 87 JRFM 2020,13,94 Appendix A Table A1. The annual rates of return 1988–2012. Year USA Canada Germany France The Netherlands Norway Sweden UK Australia Japan Emerging Markets 1988 0.16 0.18 0.21 0.39 0.16 0.43 0.49 0.06 0.38 0.36 0.40 1989 0.31 0.25 0.47 0.37 0.37 0.46 0.33 0.22 0.11 0.02 0.65 1990 −0.02 −0.12 −0.09 −0.13 −0.02 0.01 −0.20 0.10 −0.16 −0.36 −0.11 1991 0.31 0.12 0.09 0.19 0.19 −0.15 0.15 0.16 0.36 0.09 0.60 1992 0.07 −0.11 −0.10 0.03 0.03 −0.22 −0.14 −0.04 −0.10 −0.21 0.11 1993 0.10 0.18 0.36 0.22 0.37 0.43 0.38 0.24 0.37 0.26 0.75 1994 0.02 −0.02 0.05 −0.05 0.13 0.24 0.19 −0.02 0.06 0.22 −0.07 1995 0.38 0.19 0.17 0.15 0.29 0.07 0.34 0.21 0.12 0.01 −0.05 1996 0.24 0.29 0.14 0.22 0.29 0.29 0.38 0.27 0.18 −0.15 0.06 1997 0.34 0.13 0.25 0.12 0.25 0.07 0.13 0.23 −0.10 −0.24 −0.12 1998 0.31 −0.06 0.30 0.42 0.24 −0.30 0.15 0.18 0.07 0.05 −0.25 1999 0.22 0.54 0.21 0.30 0.07 0.32 0.81 0.12 0.19 0.62 0.66 2000 −0.13 0.06 −0.15 −0.04 −0.04 0.00 −0.21 −0.12 −0.09 −0.28 −0.31 2001 −0.12 −0.20 −0.22 −0.22 −0.22 −0.12 −0.27 −0.14 0.03 −0.29 −0.02 2002 −0.23 −0.13 −0.33 −0.21 −0.20 −0.07 −0.30 −0.15 0.00 −0.10 −0.06 2003 0.29 0.55 0.65 0.41 0.29 0.50 0.66 0.32 0.51 0.36 0.56 2004 0.11 0.23 0.17 0.19 0.13 0.54 0.37 0.20 0.32 0.16 0.26 2005 0.06 0.29 0.11 0.11 0.15 0.26 0.11 0.07 0.18 0.26 0.35 2006 0.15 0.18 0.37 0.35 0.32 0.46 0.45 0.31 0.33 0.06 0.33 2007 0.06 0.30 0.36 0.14 0.21 0.32 0.01 0.08 0.30 −0.04 0.40 2008 −0.37 −0.45 −0.45 −0.43 −0.48 −0.64 −0.49 −0.48 −0.50 −0.29 −0.53 2009 0.27 0.57 0.27 0.33 0.43 0.89 0.66 0.43 0.77 0.06 0.79 2010 0.15 0.21 0.09 −0.03 0.02 0.12 0.35 0.09 0.15 0.16 0.19 2011 0.02 −0.12 −0.17 −0.16 −0.12 −0.09 −0.15 −0.03 −0.11 −0.14 −0.18 2012 0.16 0.10 0.32 0.23 0.21 0.20 0.23 0.15 0.22 0.08 0.19 88 JRFM 2020,13,94 Appendix B Table A2. The correlation matrix for the period 1988–2012. USA Canada Germany France The Netherlands Norway Sweden UK Australia Japan Emerging Markets USA 1 0.68 0.79 0.83 0.85 0.48 0.76 0.86 0.58 0.44 0.52 Canada 0.68 1 0.77 0.76 0.75 0.84 0.88 0.79 0.81 0.67 0.78 Germany 0.79 0.77 1 0.90 0.89 0.70 0.80 0.84 0.72 0.59 0.68 France 0.83 0.76 0.90 1 0.88 0.66 0.84 0.83 0.75 0.62 0.67 The Netherlands 0.85 0.75 0.89 0.88 1 0.73 0.77 0.93 0.75 0.45 0.66 Norway 0.48 0.84 0.70 0.66 0.73 1 0.79 0.75 0.83 0.55 0.76 Sweden 0.76 0.88 0.80 0.84 0.77 0.79 1 0.80 0.79 0.80 0.74 UK 0.86 0.79 0.84 0.83 0.93 0.75 0.80 1 0.78 0.42 0.65 Australia 0.58 0.81 0.72 0.75 0.75 0.83 0.79 0.78 1 0.63 0.83 Japan 0.44 0.67 0.59 0.62 0.45 0.55 0.80 0.42 0.63 1 0.67 Emerging Markets 0.52 0.78 0.68 0.67 0.66 0.76 0.74 0.65 0.83 0.67 1 89 JRFM 2020,13,94 References Abel, Andrew B. 1990. Asset prices under habit formation and catching up with the Joneses. American Economic Review 80: 38–42. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 91 Journal of Risk and Financial Management Article Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks Zhe Li Business School, Nanjing Normal University, Nanjing 210023, China; [email protected] Received: 21 November 2019; Accepted: 12 January 2020; Published: 17 January 2020 Abstract: Recently, a large number of empirical studies indicated that individual equity options exhibit a strong factor structure. In this paper, the importance of systematic and idiosyncratic volatility and jump risks on individual equity option pricing is analyzed. First, we propose a new factor structure model for pricing the individual equity options with stochastic volatility and jumps, which takes into account four types of risks, i.e., the systematic diffusion, the idiosyncratic diffusion, the systematic jump, and the idiosyncratic jump. Second, we derive the closed-form solutions for the prices of both the market index and individual equity options by utilizing the Fourier inversion. Finally, empirical studies are carried out to show the superiority of our model based on the S&P 500 index and the stock of Apple Inc. on options. The out-of-sample pricing performance of our proposed model outperforms the other three benchmark models especially for short term and deep out-of-the-money options. Keywords: equity option pricing; factor models; stochastic volatility; jumps JEL Classification: G13 1. Introduction Most of the existing literature studies on option pricing are for index options, and there are very few about equity options. One approach to modeling equity options is to employ the state-of-the-art model in the index option literature, a stochastic volatility model with jumps (see, for example, Bates 1996,2000 ;Bakshi et al. 1997;Duffie et al. 2000;Eraker et al. 2003; Broadie et al. 2007 ; Christoffersen et al. 2012;Andersen et al. 2015;Bardgett et al. 2019), but to ignore any underlying factor structure. In Bakshi and Kapadia (2003a), the research results indicated that the volatility risk premium is negative in index options by examining the statistical properties of delta hedged option portfolios, i.e., a portfolio of a long call option position hedged by a short position in the stock. On the one hand, stock returns have a significant market component; the emergence of market volatility risk premiums is bound to have an impact on individual equity option pricing. On the other hand, from the economic point of view, the risk neutral distributions of individual equities are systematically different from the market index. Thus, it is necessary to explore how volatility risk is priced in individual equity options, which also can produce additional insights into the pricing structure of individual equity options (see Bakshi et al. 2003). As is well known, the beta of a stock represents the sensitivity of the risk of the individual equity with respect to the systematic risk of the market and is very useful for portfolio construction in the capital asset pricing model. Therefore, under the assumption that stock returns include a market component and an idiosyncratic component, Bakshi and Kapadia (2003b) developed a factor model for equity option valuation and investigated the pricing of market volatility risk in individual equity options. Their empirical results showed that volatility risk premiums in equity options are smaller than in index options. JRFM 2020,13, 16; doi:10.3390/jrfm13010016 www.mdpi.com/journal/jrfm 93 JRFM 2020,13,16 Afterwards, Fouque and Kollman (2011) proposed a continuous-time capital asset pricing model (CAPM) where the dynamics of the market index have a stochastic volatility driven by a fast mean reverting process. Moreover, they derived the analytical approximation pricing formulas for both the market index and individual equity call options using a singular perturbation method. Meanwhile, a calibration method for the beta parameter was also presented based on the estimated model parameters of both the market index and individual equity option prices. Subsequently, Fouque and Tashman (2012) extended the constant beta-parameter factor model of Fouque and Kollman (2011) by considering a piecewise-linear relationship between the individual asset and the market index and proposed a regime switching factor model for the pricing market index and individual equity options. Supposing that stock return is linearly related to market index return in terms of the beta parameter, Carr and Madan (2012) developed a factor model for individual equity option pricing under a purely discontinuous Lévy process via fast Fourier transform, in which the variance gamma process for the dynamics of both the market index and stock was taken as an example for illustration. By supposing a continuous-time CAPM with Lévy processes, Wong et al. (2012) also derived analytical solutions to the index and equity options and explored the corresponding static hedging with index futures. Christoffersen et al. (2018) empirically studied the equity volatility levels, skews, and term structures by using equity option prices and principal component analysis. The results indicated that the equity options had a strong factor structure, and then, they developed an equity option pricing model with a CAPM factor structure and stochastic volatility, which allowed for mean reverting stochastic volatility for the dynamics of both the market factor and individual equity. Recently, Xiao and Zhou (2018) proposed a GARCH-jump model for individual stock returns that took into account four types of risks: the systematic and idiosyncratic jumps and the systematic and idiosyncratic diffusive volatility. By using a dataset consisting of the S&P 500 index and 15 individual stock prices, their empirical results indicated that idiosyncratic jumps were a key determinant of expected stock return. 1 Instead of using only stock returns, Kapadia and Zekhnini (2019) used both stock and option data to decompose the four risk premiums associated with systematic and idiosyncratic diffusive and jump risks and also documented that idiosyncratic jumps are important determinants of the mean returns of a stock from both an ex post and ex ante perspective. Motivated by the above mentioned insights, we propose to price individual equity options in stochastic volatility jump-diffusion models with a market factor structure, which can be seen as a generalized version of Christoffersen et al. (2018). Specifically, in our proposed model, the individual equity prices are driven by the market factor, as well as an idiosyncratic component that also has stochastic volatility and jump. Due to the model belonging to the affine class, we derive the closed-form solutions for the prices of both the market index and individual equity options by utilizing the Fourier inversion. In addition, we provide the empirical results to test the pricing performance of the proposed factor model based on the S&P 500 index and the stock of Apple Inc. (AAPL) on options. Toward this end, we empirically compare the pricing performance of the proposed model with those of the other three classical two factor stochastic volatility models being taken as benchmark models. Empirical results presented here confirm that the equity option pricing model considering systematic and idiosyncratic volatility and jump risks may offer a good competitor to the models of Bates (2000) , Christoffersen et al. (2009), or Christoffersen et al. (2018) for some other option markets. The remainder of the paper proceeds as follows. In Section 2, we present a novel factor model for equity option valuation and derive the corresponding closed-form solutions. In Section 3, 1 In fact, the work of Xiao and Zhou (2018) is a complement to the recent studies that disentangle the four types of risks in equity premiums, such as Bégin et al. (2020), who developed a GARCH-jump model in which an individual firm’s systematic and idiosyncratic risk have both a Gaussian diffusive and a jump component. Their empirical results showed that normal diffusive and jump risks have drastically different effects on the expected return of individual stocks by using 20 years of returns and options on the S&P 500 and 260 stocks. 94 JRFM 2020,13,16 empirical studies are carried out to show the pricing performance of our proposed model. Finally, some conclusions are stated in Section 4. 2. Equity Option Valuation In this section, we introduce a general class of stochastic volatility models with jumps for the dynamics of both the market factor and individual equity prices and derive closed form solutions to the prices of the European equity call and put options. 2.1. Model Description Consider a filtered probability space (Ω , F , Q) with information filtration {Ft}0≤t≤T satisfying the usual conditions (increase, right-continuous, and augmented), where Q is a risk neutral measure. We model an equity market consisting of N firms with a single market factor, It (usually approximated by a market index in practice). The individual stock prices are denoted by Si t , for i= 1,2, ... , N . For the sake of convenience, we ignore the superscript i , and denote (St)t≥0 the pricing process of an individual stock. Investors also have access to a risk free bond that pays a return rate of r . To start, the market factor Itevolves under a risk neutral measure Qas: dIt It−=rdt +VI,tdWI 1,t+R(ey−1)˜ Ny(dt,dy), (1) dVI,t=κI(θI−VI,t)dt +σIVI,tdWI 2,t, (2) where It− stands for the value of It before a possible jump occurs, y∈R=R\{ 0 } , VI,t is the variance of market factor, θI denotes the long run variance, κI captures the mean reversion speed of VI,t to θI , σI measures the volatility of volatility, 2 κIθI≥σ2 I to ensure that the process VI,t remains strictly positive2,WI 1,tand WI 2,tare correlated standard Brownian motions, i.e., the innovations to the market return and volatility are correlated with correlation coefficient ρI , Cov dWI 1,t,dWI 2,t=ρIdt , and ˜ Ny(dt , dy)=Ny(dt , dy)−νy(dy)dt is a compensated jump measure, where Ny(dt , dy) is the jump measure and the Lévy kernel (or density) νy(dy)satisfies Rmin(1, y2)νy(dy)<∞. Furthermore, we separate the effects of the market factor on individual equities’ returns into two types of risks: the systematic diffusive volatility and jump. More specifically, the diffusive random variation of individual equities’ returns is dependent on the Brownian motion that drives market returns through the coefficient βdif f . In addition, the discontinuous movements in the market return can also trigger jumps in individual equities’ returns through the coefficient βjump . Therefore, the individual equity prices are driven by the market factor, as well as an idiosyncratic component that also has stochastic volatility and jump, whose process under a risk neutral measure Qfollows:3 dSt St−=rdt +βdi f f VI,tdWI 1,t   Systematic diffusive +R(eβjumpy−1)˜ Ny(dt,dy)   Systematic jump +VS,tdWS 1,t   Idiosyncratic diffusive +R(eξ−1)˜ Nξ(dt,dξ)   Idiosyncratic jump , (3) dVS,t=κS(θS−VS,t)dt +σSVS,tdWS 2,t, (4) 2One can refer to Assumption 2.1 of Cheang et al. (2013) and Cheang and Garces (2019) for a more detailed explanation. 3 Obviously, our proposed model for the dynamics of the market factor and individual equity prices is an extension of Christoffersen et al. (2018). In fact, our model also can be regarded as a further generalization of Cheang et al. (2013) and Cheang and Garces (2019) by taking into account the factor structure. 95 JRFM 2020,13,16 where Cj,market(St , T , Kj) is the market price of the equity call option contract from the sample and CΘS j,model(St , T , Kj) represents the model price calculated using Equation (13) and the vector of model input parameters ΘS. On the basis of the above calibration method, Table 1presents the risk neutral parameter estimates across various model specifications. Note that the values of the diffusive beta βdif f and jump beta βjump for our proposed model were 0.3891 and 0.8429, respectively. The corresponding value of βdif f for the 2-FSV model was 0.2457. Obviously, both our proposed model and the 2-FSV model showed that AAPL tended to have a relatively low exposure to diffusive market movements. However, the jump exposure coefficient βjump = 0.8429 indicated that the AAPL had a strong exposure to market jumps, which meant that the factor structure of the jumps was much stronger than the one of the diffusive movements. The reason for this result may be related to the sample data we selected. If we can get more sample data in the future, we will do an in-depth analysis. Moreover, we also can see that the values of correlation ρ were strongly negative for four models, capturing the so-called leverage effect both in the index and individual equity. Table 1. Estimated parameters. Note: This table shows the average of the estimated parameters obtained by minimizing the root mean squared pricing errors between the market price and the model price for each option on 8 May 2019. Standard errors are reported in parentheses . Parameters Our 2-FSV 2-SV 2-SVJ SPX AAPL SPX AAPL AAPL AAPL VI,0/V1,0 0.0133 0.0119 0.0239 0.0181 (0.0000) (0.0000) (0.0002) (0.0001) VS,0/V2,0 0.0470 0.0514 0.0197 0.0176 (0.0000) (0.0000) (0.0002) (0.0002) κI/κ10.2496 0.2929 0.3489 0.4064 (0.0212) (0.0148) (0.0118) (0.0311) κS/κ20.2454 0.1504 0.4131 0.4108 (0.0288) (0.0797) (0.0729) (0.0171) θI/θ10.2820 0.3066 0.3314 0.2817 (0.0181) (0.0317) (0.0534) (0.0348) θS/θ20.2303 0.3683 0.2447 0.3415 (0.0190) (0.0590) (0.0365) (0.0423) σI/σ10.3472 0.3932 0.1615 0.1898 (0.0127) (0.0137) (0.0081) (0.0106) σS/σ20.1496 0.1640 0.2206 0.1970 (0.0056) (0.0135) (0.0386) (0.0059) λI0.0450 (0.0017) λS0.3413 0.3065 (0.2463) (0.1194) μI0.1657 (0.0599) μS0.0889 0.0333 (0.0391) (0.0042) δI0.0850 (0.0113) δS0.0679 0.0534 (0.0078) (0.0013) βdif f 0.3891 0.2457 (0.0381) (0.0983) βjump 0.8429 (0.8091) ρI/ρ1−0.9290 −0.8498 −0.9222 −0.7445 (0.0063) (0.0080) (0.0096) (0.0297) ρS/ρ2−0.9926 −0.8938 −0.7673 −0.7817 (0.0001) (0.0469) (0.1632) (0.0549) 102 JRFM 2020,13,16 3.3. Pricing Performance In this subsection, we present the empirical results for the calibrated models. In order to investigate the impacts of the systematic and idiosyncratic volatility and jump risks on equity option pricing, we took the 2-FSV, 2-SV, and 2-SVJ models as benchmark models to evaluate the pricing performance of our proposed model. Figures 1–10 exhibit the predicted prices of the four model specifications and market prices listed on 9 May 2019, with 11, 16, 21, 26, 31, 51, 71, 96, 116, and 181 trading days to expiry, respectively. Here, the predicted prices (out-of-sample pricing) were calculated by the in-sample calibration parameters reported in Table 1. One can clearly observe from the left panels of Figures 1–10 that the option prices obtained by theoretical models were generally closer to the market prices for different strike prices. To further investigate the pricing performance of the four models, the right panels of Figures 1–10 show the relative price differences (relative errors) between the theoretical model prices and market prices. 5 For simplicity, we refer to a call option as deep out-of-the-money (DOTM) if S/K≤ 0.90; out-of-the-money (OTM) if 0.90 <S/K≤ 0.97; at-the-money (ATM) if 0.97 <S/K≤ 1.03; in-the-money (ITM) if 0.97 <S/K≤ 1.10; and deep in-the-money (ITM) if 1.10 <S/K . Moreover, we considered options less than 60 days to expiration as short term; options with 60–120 days to expiration as medium term; and options larger than 120 days to expiration as long term. For the options with 11, 16, 21, 26, 31, and 51 trading days to expiry, the relative pricing errors produced by our proposed model were all significantly lower than those of 2-FSV, 2-SV, and 2-SVJ models in the case of DOTM options, while the relative errors of all models were slightly higher. It is also worth noting that the pricing performance of the stochastic model with jump behavior was much better than that of the model without jump in the case of deep out-of-money. For the options with 71, 96, 116, and 181 trading days to expiry, we did not find the same conclusions as the above short term options. In conclusion, the pricing performance of equity option valuation model considering market and idiosyncratic volatility and jump risks was significantly improved for short term and DOTM options. 180 200 220 240 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 Relative Error In-sample, T=May 24, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 140 160 180 200 220 240 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Relative Error Out-of-sample, T=May 24, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 1. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 24 May 2019. 5 The relative error is defined by |Cmodel−Cmarket| Cmarket × 100%, where Cmodel and Cmarket denote the theoretical model option prices and the real market prices, respectively. 103 JRFM 2020,13,16 180 200 220 240 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative Error In-sample, T=May 31, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 140 160 180 200 220 240 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Relative Error Out-of-sample, T=May 31, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 2. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 31 May 2019. 160 180 200 220 240 Strike Price 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Relative Error In-sample, T=June 7, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 160 180 200 220 240 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative Error Out-of-sample, T=June 7, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 3. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 7 June 2019. 104 JRFM 2020,13,16 160 180 200 220 240 260 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 Relative Error In-sample, T=June 14, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 160 180 200 220 240 260 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative Error Out-of-sample, T=June 14, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 4. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 14 June 2019. 150 200 250 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Relative Error In-sample, T=June 21, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 150 200 250 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Relative Error Out-of-sample, T=June 21, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 5. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 21 June 2019. 105 JRFM 2020,13,16 140 160 180 200 220 240 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative Error In-sample, T=July 19, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 160 180 200 220 240 260 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Relative Error Out-of-sample, T=July 19, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 6. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 19 July 2019. 150 200 250 Strike Price 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Relative Error In-sample, T=August 16, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 150 200 250 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative Error Out-of-sample, T=August 16, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 7. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 16 August 2019. 106 JRFM 2020,13,16 140 160 180 200 220 240 260 Strike Price 0 0.05 0.1 0.15 0.2 0.25 Relative Error In-sample, T=September 20, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 160 180 200 220 240 260 Strike Price 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Relative Error Out-of-sample, T=September 20, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 8. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 20 September 2019. 150 200 250 300 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative Error In-sample, T=October 18, 2019 Our model 2-FSV model 2-SV model 2-SVJ model 150 200 250 300 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Relative Error Out-of-sample, T=October 18, 2019 Our model 2-FSV model 2-SV model 2-SVJ model Figure 9. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 18 October 2019. 107 JRFM 2020,13,16 100 150 200 250 300 350 Strike Price 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Relative Error In-sample, T=January 17, 2020 Our model 2-FSV model 2-SV model 2-SVJ model 100 150 200 250 300 350 Strike Price 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Relative Error Out-of-sample, T=January 17, 2020 Our model 2-FSV model 2-SV model 2-SVJ model Figure 10. The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity T= 17 January 2019. To summarize the model calibration results, we also adopted the RMSE as a measure of the goodness of fit. Table 2reports the out-of-sample pricing errors for the four models across different maturities. Note from Table 2that our proposed model generally outperformed the other three models in terms of out-of-sample pricing errors. In fact, the same was true for in-sample, whose pricing errors were generally lower than those of the out-of-sample. We will not repeat them here. To measure the extent to which a model was better or worse than another, we defined the improvement rate as the relative differences between the pricing errors from the benchmark model and our proposed model, i.e., Improvement rate =RMSEbenchmark −RMSEour RMSEbenchmark ×100% where RMSEour and RMSEbenchmark denote the RMSE implied by our model and benchmark model, respectively. A positive (or negative) value of improvement rate meant that our model yielded lower (or higher) pricing errors than benchmark model, implying that the pricing performance of the former was better (or worse) than that of the latter by a percentage of that value. From the last column of Table 2, we can see that our model was superior to the 2-SVJ model across different maturities, which meant that it was necessary to consider the market factor structure in equity option pricing. From the third last column of Table 2, the improvement rate indicated that our model slightly outperformed the 2-FSV model in terms of short term options, but was worse than that of both medium and long term. In spite of this, our empirical study presented here could at least illustrate that the equity option pricing model considering systematic and idiosyncratic volatility and jump risks may offer a good competitor of the models of Bates (2000), Christoffersen et al. (2009), or Christoffersen et al. (2018) for some other equity option markets. 108 JRFM 2020,13,16 Table 2. Out-of-sample pricing errors. Note: This table shows the out-of-sample pricing errors across different maturities. Pricing errors are reported as the root mean squared errors (RMSE) of option prices for four models. RMSE Our 2-FSV 2-SV 2-SVJ Improvement Rate Maturity Our vs. 2-FSV Our vs. 2-SV Our vs. 2-SVJ 24 May 2019 0.2573 0.2574 0.2596 0.2707 0.0373% 0.8803% 4.9568% 31 May 2019 0.2507 0.2508 0.2564 0.2652 0.0392% 2.2499% 5.4846% 7 June 2019 0.2343 0.2347 0.2527 0.2474 0.1764% 7.2947% 5.3044% 14 June 2019 0.1992 0.2041 0.2261 0.2099 2.4278% 11.9155% 5.0858% 21 June 2019 0.1824 0.1827 0.1873 0.1916 0.1399% 2.5963% 4.7934% 19 July 2019 0.3256 0.3301 0.3326 0.3383 1.3434% 2.0948% 3.7368% 16 August 2019 0.2856 0.2835 0.2879 0.2922 −0.7573% 0.7946% 2.2384% 20 September 2019 0.3177 0.3159 0.3162 0.3222 −0.5932% -0.4851% 1.4002% 18 October 2019 0.1185 0.1180 0.1215 0.1272 −0.4458% 2.4886% 6.8593% 17 January 2020 0.4882 0.4882 0.4893 0.4943 −0.0071% 0.2182% 1.2201% 4. Conclusions In Christoffersen et al. (2018), the issues of the equity volatility levels, skews, and term structures were investigated by using equity option prices and the principal component analysis method. Their empirical results indicated that the equity options had a strong factor structure, and then, they developed an equity option pricing model with a CAPM factor structure and stochastic volatility. In addition, jumps in stock returns of individual firms were triggered by either systematic events or idiosyncratic shocks. Some recent studies indicated that idiosyncratic jumps were a key important determinant of expected stock; see, for example, Xiao and Zhou (2018), Kapadia and Zekhnini (2019) and Bégin et al. (2020). Motivated by these insights, we developed a novel model for pricing individual equity options that incorporated a market factor structure, which could be seen as a generalized version of the work by Christoffersen et al. (2018) . Specifically, in our model, the individual equity prices were driven by the market factor, as well as an idiosyncratic component that also had stochastic volatility and jump. Due to our model belonging to the affine class, we derived the closed-form solutions for the prices of both the market index and individual equity options by utilizing the Fourier inversion. In addition, we provided the empirical results to test the pricing performance of our proposed factor model based on the S&P 500 index and the AAPL stock on options. Toward this end, we empirically compared the pricing performance of our proposed model with those of the other three classical two factor stochastic volatility models being taken as benchmark models. The out-of-sample pricing performance of equity option valuation model considering market and idiosyncratic volatility and jump risks was significantly improved for short term and DOTM options. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 110 Journal of Risk and Financial Management Article CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles Alex Golodnikov 1, Viktor Kuzmenko 1and Stan Uryasev 2,* 1V.M. Glushkov Institute of Cybernetics, 40, pr. Akademika Glushkova, 03187 Kyiv, Ukraine 2 Applied Mathematics & Statistics, Stony Brook University, B-148 Math Tower, Stony Brook, NY 11794, USA *Correspondence: Stanislav[email protected] Received: 16 May 2019; Accepted: 20 June 2019; Published: 26 June 2019 Abstract: A popular risk measure, conditional value-at-risk (CVaR), is called expected shortfall (ES) in financial applications. The research presented involved developing algorithms for the implementation of linear regression for estimating CVaR as a function of some factors. Such regression is called CVaR (superquantile) regression. The main statement of this paper is: CVaR linear regression can be reduced to minimizing the Rockafellar error function with linear programming. The theoretical basis for the analysis is established with the quadrangle theory of risk functions. We derived relationships between elements of CVaR quadrangle and mixed-quantile quadrangle for discrete distributions with equally probable atoms. The deviation in the CVaR quadrangle is an integral. We present two equivalent variants of discretization of this integral, which resulted in two sets of parameters for the mixed-quantile quadrangle. For the first set of parameters, the minimization of error from the CVaR quadrangle is equivalent to the minimization of the Rockafellar error from the mixed-quantile quadrangle. Alternatively, a two-stage procedure based on the decomposition theorem can be used for CVaR linear regression with both sets of parameters. This procedure is valid because the deviation in the mixed-quantile quadrangle (called mixed CVaR deviation) coincides with the deviation in the CVaR quadrangle for both sets of parameters. We illustrated theoretical results with a case study demonstrating the numerical efficiency of the suggested approach. The case study codes, data, and results are posted on the website. The case study was done with the Portfolio Safeguard (PSG) optimization package, which has precoded risk, deviation, and error functions for the considered quadrangles. Keywords: quantile; VaR; quadrangle; CVaR; conditional value-at-risk; expected shortfall; ES; superquantile; deviation; risk; error; regret; minimization; CVaR estimation; regression; linear regression; linear programming; portfolio safeguard; PSG 1. Introduction We start the introduction with a quick outline of the main result of this paper. The conditional value-at-risk(CVaR)is apopularrisk measure. Itiscalledexpectedshortfall (ES)infinancialapplications and it is included in financial regulations. This paper provides algorithms for the estimation of CVaR with linear regression as a function of factors. This task is of critical importance in practical applications involving low probability events. By definition, CVaR is an integral of the value-at-risk (VaR) in the tail of a distribution. VaR can be estimated with the quantile regression by minimizing the Koenker–Bassett error function. This paper shows that CVaR can be estimated by minimizing a mixture of the Koenker–Bassett errors with an additional constraint. This mixture is called the Rockafellar error and it has been earlier used for CVaR estimation without a rigorous mathematical justification. One more equivalent variant of CVaR regression can be done by minimizing a mixture of CVaR deviations for finding all coefficients, except JRFM 2019,12, 107; doi:10.3390/jrfm12030107 www.mdpi.com/journal/jrfm 111 JRFM 2019,12, 107 Figure 2. Five equally probable atoms. Risk in the CVaR quadrangle and mixed-quantile quadrangle with Set 1 of parameters for α=0.5. 4. Set 2 of Parameters for the Mixed-Quantile Quadrangle This section gives an alternative expression for the risk Rα(X) and deviation Dα(X) in the CVaR quadrangle for a discrete uniformly distributed random variable. This expression is based on the following Set 2 of parameters. This set of parameters has the same number of parameters as Set 1 but different values of weights and confidence levels. Similar to Section 3, let X be a discrete random value with support, xi , i= 1,2, ... , ν , and ProbX=xi= 1 /ν for i= 1,2, ... , ν . Denote xmax =max i=1,...,νxi . For this random value CVaR1(X)=VaR− 1(X)=xmax. Set 2 of parameters: • partition of the interval [βνα−1,1] : δ= 1 /ν , βi=iδ , for i=να− 1, να , ... , ν , where να=να+ 1, with zbeing the largest integer less than or equal to z;δα=βνα−α. •confidence levels: βi,i=να−1, να,...,ν. •weights: qν=0; qν−1=δ 1−α×[2ln(2)] ≈δ 1−α×1.386294361, (if ν−1>ν α) qν−2=δ 1−α×23ln3 2+ln1 2≈δ 1−α×1.046496288, (if ν−2>ν α) qν−j=δ 1−α×j(j+1)lnj+1 j+(j−1)lnj−1 j,(if j>2,ν−j>ν α) qνα=δ 1−α×jδ−δα+(j+1)ln1−α δj+(j−1)lnj−1 j, (if να<ν−1, j=ν−να) qνα−1=δ 1−α×jδα+(j−1)lnδ(j−1) 1−α,(if να−1<ν−1, j=ν−να+1) if να=ν−1, then qνα=δ 1−α×21+ln1−α δ−1, qνα−1=δ 1−α×2lnδ 1−α−1+2 if να=ν, then qνα−1=1. Lemma 2. Let X be a discrete random value with ν equally probable atoms. Then, risk and deviation of the CVaR quadrangle for X are given by the following expressions with parameters from Set 2. 1. CVaR2 Risk: Rα(X)=1 1−α1 α CVaRβ(X)dβ=ν−1 i=να−1qiCVaRβi(X)(4) 118 JRFM 2019,12, 107 2. CVaR2 Deviation: Dα(X)=1 1−α1 α CVaRβ(X)dβ−E[X]=ν−1 i=να−1qiCVaRβi(X)−E[X](5) Proof. Appendix Dcontains proof of the lemma.  Note. Equations (4) and (5) are valid if CVaRβν−1is replaced by CVaRγwith an arbitrary γ∈[βν−1,1]. Corollary 2. For the random value X defined in Lemma 2, risk and deviation of the CVaR quadrangle coincide with risk and deviation of the mixed-quantile quadrangle with r=ν−να+ 1, λk=qνα−2+k , αk=βνα−2+k , k=1,...,r. Proof. Right hand sides in Equations (4) and (5) define risk and deviation of the mixed-quantile quadrangle because qi>0, i=να−1,...,ν−1, and ν−1 i=να−1qi=1.  Lemma 3. Let X be a discrete random value with equally probable atoms xi , i= 1,2, ... , ν , ProbX=xi= 1 /ν . Then, statistic of the mixed-quantile quadrangle defined by the Set 2 of parameters is a range containing statistic of the CVaR quadrangle. Proof. Appendix Econtains proof of the lemma.  5. On the Estimation of CVaR with Mixed-Quantile Linear Regression This section formulates regression problems using the CVaR quadrangle and mixed-quantile quadrangle. For discrete final distributions with equally probable atoms, we prove some equivalence statements for the CVaR and mixed-quantile quadrangles. Further, we demonstrate how to estimate CVaR by using the linear regression with error and deviation from the mixed-quantile quadrangle. We want to estimate variable V using a linear function f(Y)=C0+CTY of the explanatory factors Y=(Y1,...,Yn) . Let  E be an error from some quadrangle (further we consider, mixed-quantile and CVaR quadrangles), and  D and  S be a deviation and a statistic, respectively, corresponding to this quadrangle. Below we consider optimization statements for solving regression problems. General Optimization Problem 1 Minimize error  Eand find optimal C∗ 0,C∗ min C0∈R,C∈Rm E(Z(C0,C)) where Z(C0,C)=V−C0−CTY. General Optimization Problem 2 •Step 1. Find an optimal vector C∗by minimizing deviation: min C∈Rm D(Z0(C)) where Z0(C)=V−CTY. •Step 2. Assign C∗ 0: C∗ 0∈ S(Z0(C∗)) 119 JRFM 2019,12, 107 Error  E(X)is called nondegenerate if: inf X:EX=D E(X)>0 for constants D0. Rockafellar et al. (2008), p. 722, proved the following decomposition theorem. Theorem 1. (Error-Shaping Decomposition of Regression). Let  E be a nondegenerate error and  D=min C  E(X−C) be the corresponding deviation, and let  S be the associated statistic. Point ( C∗ 0 , C∗ ) is a solution of the General Optimization Problem 1 if and only if C∗ is a solution of the General Optimization Problem 2, Step 1 and C∗ 0∈ S(Z0(C∗)) with Step 2. According to the decomposition theorem, when  E ,  D , and  S are elements of the CVaR quadrangle, the following Optimization Problems 1 and 2 are equivalent. Optimization Problem 1 Minimize error from the CVaR quadrangle: min C0∈R,C∈RmEα(Z(C0,C)) where Z(C0,C)=V−C0−CTY. Optimization Problem 2 •Step 1. Find an optimal vector C∗by minimizing deviation from the CVaR quadrangle: min C∈RmDα(Z0(C)) where Z0(C)=V−CTY. •Step 2. Calculate: C∗ 0=CVaRα(Z0(C∗)) In Optimization Problem 2 in Step 2 the statistic equals S(Z0(C∗)) =CVaRα(Z0(C∗)) , which is the specification of the inclusion operation in the Optimization Problem General 2 in Step 2. Optimization Problem 2 is used in Rockafellar et al. (2014) for the construction of the linear regression algorithms for estimating CVaR. According to the decomposition theorem, when  E ,  D , and  S are elements of a mixed-quantile quadrangle, the following Optimization Problems 3 and 4 are equivalent. Optimization Problem 3 Minimize error from the mixed-quantile quadrangle: min C0∈R,C∈RmE(Z(C0,C)) Optimization Problem 4 •Step 1. Find an optimal vector C∗by minimizing deviation from the mixed-quantile quadrangle: min C∈RmD(Z0(C)) 120 JRFM 2019,12, 107 •Step 2. Assign: C∗ 0∈S (Z0(C∗)) Corollaries 1 and 2 can be used for constructing the linear regression for estimating CVaR. Let Y be a random vector of factors for estimating the random value V . We consider that the linear regression function f(Y)=C0+CTY approximates CVaR of V , where C0∈R , C∈Rm are variables in the linear regression. The residual is denoted by Z(C0,C)=V−C0+CTYand Z0(C)=V−CTY. Further we provide a lemma about linear regression problems based on Corollary 1 with the Set 1 of parameters. The main statement here is that the Optimization Problems 3 and 4 for the mixed-quantile quadrangle can be used to solve linear regression problems for estimating CVaR. This is the case because CVaR and mixed-quantile quadrangles have the same Statistic and Deviation. Lemma 4. Let the residual random value Z(C0,C)=V−C0+CTY be discretely distributed with ν equally probable atoms. Let us consider the CVaR quadrangle with error Eα(Z(C0,C)) , deviation Dα(Z0(C)) , and statistic Sα(Z0(C∗)) . Let us also consider the mixed-quantile quadrangle with the error E(Z(C0,C)) , deviation D(Z0(C)) , and statistic S(Z0(C∗)) with parameters defined by Set 1 and r=ν−να+ 1, λk=pνα−1+k , αk=γνα−1+k , k= 1, ... , r . Then, the Optimization Problems 1–4 are equivalent, i.e., the sets of optimal vectors of these optimization problems coincide. Moreover, let C∗ 0,C∗ be a solution vector of the equivalent Optimization Problems 1–4. Then: C∗ 0=CVaRα(Z0(C∗)) EαC∗ 0,C∗=EC∗ 0,C∗=D(Z0(C∗))=Dα(Z0(C∗)) Proof. This lemma is a direct corollary of the decomposition Theorem 1 and Corollary 1 of Lemma 1. Indeed, Corollary 1 implies that the Optimization Problems 2 and 4 are equivalent. Further, the decomposition theorem implies that Optimization Problems 1 and 2 and the Optimization Problems 3 and 4 are equivalent.  Further we provide a lemma about linear regression problems based on Corollary 2 with the Set 2 of parameters. The main statement is that the Optimization Problem 4, Step 1 for the Mixed-Quantile Quadrangle can be used to solve linear regression problem for estimating CVaR. This is the case because CVaR and Mixed-Quantile Quadrangles have the same Deviation. After obtaining vector of coefficients C∗, intercept is calculated, C∗ 0=CVaRα(Z0(C∗)). Lemma 5. Let the residual random value Z(C0,C)=V−C0+CTY be discretely distributed with ν equally probable atoms. Let the mixed-quantile quadrangle with deviation D(Z0(C)) be defined by parameters of Set 2 and r=ν−να+ 1, λk=qνα−2+k , αk=βνα−2+k , k= 1, ... , r . Then, C∗ 0,C∗ is a solution of Optimization Problem 1, if and only if, C∗ 0,C∗is a solution of the following two-step procedure: •Step 1. Find an optimal vector C∗by minimizing deviation from the mixed-quantile quadrangle: min C∈RmD(Z0(C)) •Step 2. Calculate C∗ 0=CVaRα(Z0(C∗)). Proof. This lemma is a direct corollary of the decomposition Theorem 1 and Corollary 2 or Lemma 2. Indeed, the Corollary 2 implies that the Optimization Problems 2 and 4 are equivalent. Further, since deviations of the CVaR and mixed-quantile quadrangles coincide, we can use Step 1 to calculate optimal coefficients C∗ . Further, the intercept is calculated with C∗ 0=CVaRα(Z0(C∗)) because CVaR is the statistic in the CVaR quadrangle.  121 JRFM 2019,12, 107 For the Set 2 of parameters, the deviations in the CVaR and mixed-quintile quadrangles coincide. The two-step procedure in Optimization Problem 4 can be used to solve linear regression problems with Set 2 parameters for the mixed-quantile deviation. Also, the minimization of the Rockafellar error with the Set 2 of parameters may result in a correct C∗ . The statistic of CVaR belongs to the statistic of the mixed-quantile quadrangle. Therefore, the optimization of the Rockafellar error with the Set 2 of parameters may lead to a wrong value of intercept C∗ 0 . This potential incorrectness can be fixed by assigning C∗ 0=CVaRα(Z0(C∗)). 6. Case Study: Estimation of CVaR with Linear Regression and Style Classification of Funds The case study described in this section is posted online (see Case Study (2016)). The codes and data are available for downloading and verification. Every optimization problem is presented in three formats: Text, MATLAB, and R. Calculations were done with a PC with a 3.14 GHz processor. We have applied CVaR regression to the return-based style classification of a mutual fund. We regress a fund return by several indices as explanatory factors. The estimated coefficients represent the fund’s style with respect to each of the indices. A similar problem with a standard regression based on the mean squared error was considered by Carhart (1997) and Sharpe (1992). They estimated the conditional expectation of a fund return distribution (under the condition that a realization of explanatory factors is observed). Basset and Chen (2001) extended this approach and conducted the style analyses of quantiles of the return distribution. This extension is based on the quantile regression suggested by Koenker and Bassett (1978). The Case Study (2014), “Style Classification with Quantile Regression” implemented this approach and applied quantile regression to the return-based style classification of a mutual fund. For thenumerical implementationofCVaR linearregression, we usedthe PortfolioSafeguard(2018) package. Portfolio Safeguard (PSG) can solve nonlinear and mixed-integer nonlinear optimization problems. A special feature of PSG is that it includes precoded nonlinear functions: CVaR2 error ( cvar2_err ) and CVaR2 deviation ( cvar2_dev ) from the CVaR quadrangle, Rockafellar error ( ro_err ) from the mixed-quantile quadrangle, and CVaR deviation (cvar_dev) from the quantile quadrangle. We implemented the following equivalent variants of CVaR regression: •Minimization of the CVaR2 error (PSG function cvar2_err). •Two-step procedure with CVaR2 deviation (PSG function cvar2_dev). •Minimization of the Rockafellar error (PSG function ro_err) with the Set 1 of parameters. • The two-step procedure using mixed CVaR deviation with the Set 1 and Set 2 of parameters. This is calculated as a weighted sum of CVaR deviations (PSG function cvar_dev )fromthe quantile quadrangle. PSG automatically converts the analytic problem formulations to the mathematical programming codes and solves them. We included in Appendix Aconvex and linear programming problems for the minimization of the Rockafellar error with the Set 1 of parameters. These formulations are provided for verification purposes. They can be implemented with standard commercial software. For instance, the linear programming formulation can be implemented with the Gurobi optimization package. If Gurobi is installed in the computer, PSG can use Gurobi code as a subsolver. With the CARGRB solver in PSG, by setting the linearize option to 1, it is possible to solve the linear programming problem with Gurobi. However, this conversion will deteriorate the performance, compared to the default PSG solver VAN. For small problems it will not be noticeable. However, for problems with a large number of scenarios (e.g., with 10 8 observations), the standard PSG solver VAN will dramatically outperform the Gurobi linear programming implementation. In this case, Gurobi may not even start on a small PC because of a shortage of memory. Nevertheless, if the number of observations is small (e.g., 10 3 ) and the number of factors is very large (e.g., 10 7 ), it is recommended that the linear programming formulation is used. 122 JRFM 2019,12, 107 We regressed the CVaR of the return distribution of the Fidelity Magellan Fund on the explanatory variables: Russell 1000 Growth Index (RLG), Russell 1000 Value Index (RLV), Russell Value Index (RUJ), and Russell 2000 Growth Index (RUO). The dataset includes 1264 historical daily returns of the Magellan Fund and the indices, which were downloaded from the Yahoo Finance website. The data (design matrix for the regression) is posted on the Case Study (2016) website. The CVaR regression was done with the confidence levels α= 0.75 and α= 0.9. Calculation results are in Tables 1and 2, respectively. Here is the description of the columns of the tables: • Optimization Problem #: Optimization Problem number, as denoted in Section 5; also it is the problem number in the case study posted online, see, Case Study (2016). •Set #: Set of parameters for the mixed-quantile quadrangle. •Objective: Optimal value of the objective function. •RLG: coefficient for the Russell Value Index. •RLV: coefficient for the Russell 1000 Value Index. •RUJ: coefficient for the Russell Value Index. •RUO: coefficient for the Russell 2000 Growth Index. •Intercept: regression intercept. •Solving Time: solver optimization time. Table 1. Optimization outputs: estimating CVaR with the linear regression, α=0.75. Optimization Problem # Set # Objective RLG RLV RUJ RUO Intercept Solving Time (s) 1N/A 0.01248 0.486 0.581 − 0.0753 −6.22 ×10−36.98 ×10−30.02 2N/A 0.01248 0.486 0.582 − 0.0753 −6.22 ×10−36.98 ×10−30.02 3 Set 1 0.01247 0.486 0.582 − 0.0753 −6.22 ×10−36.96 ×10−30.03 4 Set 1 0.01251 0.486 0.582 − 0.0752 −6.22 ×10−36.98 ×10−30.11 4 Set 2 0.01248 0.486 0.582 − 0.0753 −6.23 ×10−36.98 ×10−30.03 Table 2. Optimization outputs: estimating CVaR with the linear regression, α=0.9. Optimization Problem # Set # Objective RLG RLV RUJ RUO Intercept Solving Time (s) 1N/A 0.016656 0.472 0.606 −0.078 −7.052 ×10−31.05 ×10−20.02 2N/A 0.016656 0.472 0.606 −0.078 −7.052 ×10−31.05 ×10−20.02 3 Set 1 0.016656 0.472 0.606 −0.078 −7.052 ×10−31.05 ×10−20.02 4 Set 1 0.016656 0.472 0.606 −0.078 −7.052 ×10−31.05 ×10−20.11 4 Set 2 0.016656 0.472 0.606 −0.078 −7.052 ×10−31.05 ×10−20.04 Tables 1and 2show calculation results for the considered equivalent problems. We observe that regression coefficients coincide for all problems in Tables 1and 2. This confirms the correctness of theoretical results and the numerical implementation. Also, we want to point out that the regression coefficients are quite similar for α=0.75 (Table 1) and α=0.9 (Table 2). The calculation time in majority of cases was around 0.02–0.04 s, except for the case with the mixed CVaR deviation for Set 1, which took 0.11 s. The PSG calculation times were quite low because the solver “knows” analytical expressions for the functions and can take advantage of this knowledge. 7. Conclusions The quadrangle risk theory Rockafellar and Uryasev (2013) and the decomposition theorem Rockafellar et al. (2008) provided a framework for building a regression with relevant deviations. Solution of a regression problem is split in two steps: (1) minimization of deviation from the corresponding quadrangle, and (2) determining of intercept by using statistic from this quadrangle. For CVaR regression, Rockafellar et al. (2014) reduced the optimization problem at Step 1 to 123 JRFM 2019,12, 107 a high-dimension linear programming problem. We suggested two sets of parameters for the mixed-quantile quadrangle and investigated its relationship with the CVaR quadrangle. The Set 1 of parameters corresponds to CVaR regression in Rockafellar et al. (2014), where the Set 2 is a new set of parameters. For the Set 1 of parameters, the minimization of error from CVaR Quadrangle was reduced to the minimization of the Rockafellar error from the mixed-quantile quadrangle. For both sets of parameters, the minimization of deviation in CVaR quadrangle is equivalent to the minimization of deviation in mixed-quantile quadrangle. We presented optimization problem statements for CVaR regression problems using CVaR and mixed-quantile quadrangles. Linear regression problem for estimating CVaR were efficiently implemented in Portfolio Safeguard (2018) with convex and linear programming. We have done a case study for the return-based style classification of a mutual fund with CVaR regression. We regressed the fund return by several indices as explanatory factors. Numerical results validating the theoretical statements are placed to the web (see Case Study (2016)). Supplementary Materials: Data and codes used in the case study can be downloaded: 1. Case Study (2016): Estimation of CVaR through Explanatory Factors with CVaR (Superquantile) Regression. http://www.ise.ufl. edu/uryasev/research/testproblems/financial_engineering/on-implementation-of-cvar-regression/.2.Case Study (2014): Style Classification with Quantile Regression. http://www.ise.ufl.edu/uryasev/research/testproblems/ financial_engineering/style-classification-with-quantile-regression/. Author Contributions: Conceptualization, S.U.; Formal analysis, V.K.; Investigation, A.G.; Methodology, S.U.; Software, V.K.; Supervision, S.U.; Writing—original draft, A.G. Funding: Research of Stan Uryasev was partially funded by the AFOSR grant FA9550-18-1-0391 on Massively Parallel Approaches for Buffered Probability Optimization and Applications. Conflicts of Interest: The authors declare no conflict of interest. Appendix A CVaR Regression with Rockafellar Error: Convex and Linear Programming The value of the Rockafellar error with given set of parameters λk , αk ( αk∈(0,1) , k= 1, ...r , r k=1λk= 1) for a random value X is a minimum w.r.t. a set of variables B1 , ... , Br of a mixture of Koenker–Bassett Error functions with one linear constraint on these variables: Rockafellar_Error (X)λ1,α1,...,λr,αr=min B1,...,Br⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r  k=1 λkEαk(X−Bk) r  k=1 λkBk=0⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ where Eαk(X−Bk)=Eαk 1−αk[X−Bk]++[X−Bk]−is the normalized Koenker–Bassett error. By using regret from the mixed-quantile quadrangle, we express the Rockafellar error as follows: Rockafellar_Error (X)λ1,α1,...,λr,αr=min B1,...,Br⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r  k=1 λk 1−αk E[X−Bk]+ r  k=1 λkBk=0⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭−E[X]. For the linear regression problem, the random variable X is defined by a set of differences between observed values Vi and linear functions C0+CTYi , where Yi is a vector of explanatory factors, i= 1, 2, ... , ν . Vectors C and Y have m components, C=(C1,...,Cm) , Y=(Y1,...,Ym) , and C0 is a scalar. Residuals Xi=Vi−C0−CTYi are values (scenarios) of atoms of the random value X . We consider that all atoms have equal probabilities. The estimation of V with factors Y is done by minimizing the error w.r.t. variables C,C0. Further we use the Set 1 of parameters. Let us denote: E[V]=1 v v  i=1 Vi,E[Y]=1 v v  i=1 Yi. 124 JRFM 2019,12, 107 Appendix A.1 Convex Programming Formulation for CVaR Regression Minimize the Rockafellar error: min B1,...,Br,C0,C1,...,Cm,⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r  k=1 λk (1−αk)v v  i=1Vi−C0−CTYi−Bk+−E[V]+C0+CTE[Y]⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭(A1) subject to the constraint: r  k=1 λkBk=0. (A2) This optimization problem has a convex objective and one linear constraint. Appendix A.2 Linear Programming Formulation for CVaR Regression Equations (A1) and (A2) are reduced to linear programming with additional variables and constraints: min A11,...,Arv B1,...,Br,C0,C1,...,Cm, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r  k=1 λk (1−αk)v v  i=1 Aki −E[V]+C0+CTE[Y]⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭(A3) subject to constraints: r  k=1 λkBk=0 (A4) Aki ≥Vi−C0−CTYi−Bk,k=1,...,r,i=1,...,v(A5) Aki ≥0, k=1,...,r,i=1,...,v(A6) The linear function C∗ 0+C∗TY estimates CVaRα(V) as a function of explanatory factors Y , where C∗ 0and C∗are optimal values of variables for Equations (A1) and (A2) or (A3)–(A6). Appendix B Codes Implementing Regression Optimization Problems This appendix contains codes implementing Optimization Problems 1–4 described in Section 5. Codes and solution results are posted at internet link: Case Study (2016). Codes are written in Portfolio Safeguard (PSG) Text, MATLAB, and R environments. Here are codes in the Text environment. Optimization Problem 1 Code in PSG Text format: minimize cvar2_err(0.75,matrix_s) The keyword “ minimize ” indicates that the objective function is minimized. The objective function cvar2_err(0.75,matrix_s) calculates error Eα(Z(C0,C)) in CVaR quadrangle with confidence level α=0.75. The matrix_s contains scenarios of residual of the regression Z(C0,C)=V−C0−CTY. Optimization Problem 2 Code in PSG Text format: minimize cvar2_dev(0.75,matrix_s) value: cvar_risk(0.75,matrix_s) 125 JRFM 2019,12, 107 The code includes two parts. The first part begins with the keyword “ minimize ” indicating that the objective function is minimized. It implements Step 1 of the Optimization Problem 2, which minimizes deviation from the CVaR quadrangle (for determining the optimal vector C∗ of regression coefficients without an intercept). The PSG function cvar2_dev(0.75,matrix_s) calculates deviation Dα(Z0(C)) with α =0.75. The matrix_s contains scenarios of the residual of the regression without an intercept Z0(C)=V−CTY. The second part of the code begins with the keyword “ value .” This part implements Step 2 of the Optimization Problem 2 for calculating the optimal value of intercept C∗ 0 . The PSG function cvar_risk(0.75,matrix_s) calculates CVaRα(Z0(C∗)) with α=0.75 at the optimal point C∗. Optimization Problem 3 Code in PSG Text format with parameters from Set 1. minimize ro_err(matrix_s, matrix_coeff) The keyword “ minimize ” indicates that the objective function is minimized. The objective function ro_err(matrix_s, matrix_coeff) calculates the Rockafellar error E(Z(C0,C)) in the mixed-quantile quadrangle. The matrix_s contains scenarios of the residual of the regression Z(C0,C)=V−C0−CTY . The matrix_coeff includes vectors of weights and confidence levels for Set 1 with α=0.75. Optimization Problem 4 Code in PSG Text format with parameters from Set 1 for α=0.75. minimize vector_c*cvar_dev(vector_a, matrix_s) value: cvar_risk(0.75,matrix_s) The code includes two parts. The first part begins with the keyword “ minimize ” indicating that the objective function is minimized. It implements Step 1 of the Optimization Problem 4, which minimizes deviation from the mixed-quantile quadrangle (for determining optimal vector C∗ of regression coefficients without intercept). The inner product vector_c*cvar_dev(vector_a, matrix_s) calculates the mixed CVaR deviation D(Z0(C)) . The function cvar_dev corresponds to the CVaR deviation from the quantile quadrangle. Vector vector_c contains weights for CVaR Deviation mix corresponding to Set 1. Vector vector_a contains confidence levels defined by Set 1. The matrix_s contains scenarios of the residual of the regression Z(C0,C)=V−C0−CTY. The second part of the code begins with the keyword “ value .” This part implements Step 2 of the Optimization Problem 4, calculating the optimal value of intercept C∗ 0 . The PSG function cvar_risk(0.75,matrix_s) calculates CVaRα(Z0(C∗)) with α=0.75 at the optimal point C∗. Appendix C Proof of the Lemma 1 Proof. According to the definition, βv= 1. However, while proving this lemma, we consider βv a bit smaller than 1, i.e., βv= 1 −ε>β v−1 , ε> 0, to avoid division by 0. Then, we will consider limit ε→0 to finish the proof. Note that for α<1 and for the considered partition, βi−1<β ifor all i=να,...,ν. First, let us prove that γi∈(βi−1,βi)for βi<1. Consider three functions of σ,δfor σ>0, δ≥0: f1(δ)=δ σ+δ,f2(δ)=ln1+δ σ,f3(δ)=δ σ When δ= 0, all functions equal to 0 and have equal derivatives. When δ> 0, it is true for derivatives that: f1(δ)<f2(δ)<f3(δ) 126 JRFM 2019,12, 107 Hence, when >0, similar inequalities are valid for functions: δ σ+δ<ln1+δ σ<δ σ. There exist some δγ such that 0 <δ γ<δ and δ σ+δγ=ln1+δ σ .If δ=βi−βi−1 , σ= 1 −βi , γ=1−σ−δγ, then γ=γiand βi−1<γ<β i. Therefore, γi∈(βi−1,βi). Further, we show how to calculate the integral 1−ε αCVaRβ(X)dβ as a sum of integrals over the partition: Rα(X)=lim ε→0 1 1−α1−ε α CVaRβ(X)dβ=lim βv→1 1 1−α ν  i=ναβi βi−1 CVaRβ(X)dβ. Let us denote Cβi=CVaRβi(X) and Vi=VaRγi(X) . Note that VaRγ(X) is a singleton for every γ∈(βi−1,βi)and it equals Vi, because γi∈(βi−1,βi). Below, we use value Vi for the closed interval [βi−1,βi] while calculating the integral over this interval because the value of the integral does not depend on the finite values of VaRγ(X) at the boundary points βi−1,βi. Using the definition of CVaR (Rockafellar and Uryasev (2002), Proposition 8 CVaR for scenario models) we write: βi βi−1CVaRβ(X)dβ=βi βi−1 1 1−βCβi(1−βi)+Vi(βi−β)dβ =Cβi(1−βi)βi βi−1 1 1−βdβ+Viβi βi−1 βi−β 1−βdβ =Cβi(1−βi)ln1−βi−1 1−βi+Vi(βi−βi−1)−(1−βi)ln1−βi−1 1−βi (A7) Let us make the transformation of Equation (A7) using the expression for γi in the Set 1 definition: βi βi−1CVaRβ(X)dβ=(βi−βi−1)Vi+1 βi−βi−1ln1−βi−1 1−βiCβi−Vi(1−βi) =(βi−βi−1)Vi+1 1−γiCβi−Vi(1−βi)=(βi−βi−1)1 1−γiCβi(1−βi)+Vi(βi−γi) =(βi−βi−1)CVaRγi(X) The last equality is valid because γi∈(βi−1,βi). Taking into account that lim ε→0γν= 1, CVaR1−ε(X)=CVaR1(X) , and VaR1−ε(X)=VaR1(X) , we obtain: Rα(X)=lim ε→0 ν  i=να βi−βi−1 1−αCVaRγi(X)= ν  i=να piCVaRγi(X) Deviation is calculated as follows: Dα(X)= ν  i=να piCVaRγi(X)−E[X] By the definition of CVaR (Rockafellar and Uryasev (2002)): CVaRα(X)= ν  i=να piVaRγi(X)=Sα(X) Lemma 1 is proved.  127 JRFM 2019,12,42 1. Introduction Determining distributions of the functions of random variables is a very crucial task and this problem has been attracted a number of researchers because there are numerous applications in Risk Management, Finance, Economics, Science, and, many other areas, see, for example, (Donahue 1964; Ly et al. 2016;Nadarajah and Espejo 2006;Springer 1979). Basically, the distributions of an algebraic combination of random variables including the sum, product, and quotient are focused on some common distributions along with the assumptions of independence or correlated through Pearson’s coefficient or dependence via multivariate normal joint distributions (Arnold and Brockett 1992; Bithas et al. 2007;Cedilnik et al. 2004;Hinkley 1969;Macalos and Arcede 2015;Marsaglia 1965; Matovi´c et al. 2013;Meki´cet al. 2012;Nadarajah and Espejo 2006;Nadarajah and Kotz 2006a, 2006b;Pham-Gia et al. 2006;Pham-Gia 2000;Rathie et al. 2016;Sakamoto 1943). Regarding ratio, it often appears in the problems of constructing statistics used in hypothesis testing and estimating issues. Some well-known distributions are results of such quotients. For example, the quotient of a Gaussian random variable divided by a square root of an independent chi-distributed random variable follows the t -distribution while the F-distribution is derived via the ratio of two independent chi-squared distributed random variables. To relax independence assumption, it is necessary to develop a framework for modeling dependence structures of random vectors in more general sense. To do so, Dolati et al. (2017) develop the distribution for X/Yin which both Xand Yare positive. In our paper, we first extend the theory developed by Dolati et al. (2017) to relax the positive assumption for the variables by developing the theory on both density and distribution function (CDF) for the quotient Y=X1 X2 of two dependent or independent continuous random variables X1 and X2 in which X1 and X2 could be any real number. Thereafter, we develop a theory on both density and distribution function for the ratio of one variable over the sum of two variables Z=X1 X1+X2 of two dependent or independent continuous random variables X1 and X2 by using copulas to capture the structures between X1and X2. Since the density and the CDF formula of the ratios of both Y and Z are in terms of integrals and are very complicated, we cannot obtain the exact forms of the densities and the CDFs. To circumvent the difficulty, in this paper, we propose to use a Monte Carlo algorithm, numerical analysis, and graphical approach to study behavior of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of standard normal random variables on several different copulas, including Gaussian, Studentt , Clayton, Gumbel, Frank, and Joe Copulas and we find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, skewness and scale effects. For instance, when X1 and X2 tend to be more co-monotonic indicated by increasing the parameters of copulas, then the median of Y is shifted to be higher and its shape tends to be more symmetric. In the meantime, the median of Z is equally unchanged one-half and the shape always has symmetry. We note that the approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewedt distribution and skewedt Copulas. 1 Thus, our findings are useful for academics, practitioners, and policy makers. The rest of the paper is organized as follows. In Sections 2and 3, we will briefly discuss the background theory and copula theory related to the theory developed in our paper. In Section 4, we provide main results on the quotients of dependent and independent random variables. Section 5 proposes using the Monte Carlo to deal with complex integrals and estimate some percentiles by using some special copulas, and investigate their effects on the behavior of ratios of two standard normal random variables. The last section provides the conclusions. 1We would like to thank the anonymous reviewer for giving us helpful comments so that we could draw this conclusion. 134 JRFM 2019,12,42 2. Background Theory We first review some previous work on the weighted sum, for example, in constructing portfolio Y1that is composed of two dependent assets defined by Y1=w1X1+w2X2, (1) in which the random variables Xi,t ( i= 1,2) denotes the rate of return at time t for the asset defined in terms of the following random quotient: Xi,t=Pi,t−Pi,t−1 Pi,t−1,, where Pi,t denotes the price of the i th asset at time t . Note that Xi is assumed to be absolutely continuous with the cumulative distribution functions (CDF) Fi . Suppose that (X1 , X2) follows copula C , then the CDF, FY1(y),ofY1defined in (1) satisfies: FY1(y)=1{w2<0}+sgn(w2)1 0 ∂ ∂uC*u,F2*y−w1F−1 1(u) w2++du, (2) where sgn(·)denotes the sign function such that sgn(x)=%1, if x>0, −1, if x<0. Then, the CDF, FY1 , can be used to estimate the distortion risk measure of the portfolio defined by Rg[Y1]=∞ 0g(FY1(y))dy +0 −∞[g(FY1(y)) −1]dy, where g is a distortion function and FY1(y)= 1 −FY1(y) is a survival function of Y1 . Readers may refer to Ly et al. (2016) for more detailed information. In the credit model, the total loss is defined as the aggregation of the product of risk factors. Thus, it is necessary to find the distribution for the product case, for instance, Y2given by Y2=X1X2. (3) Ly et al. (2019) show that the CDF of Y2can be determined by FY2(y)=F1(0)+1 0sgn F−1 1(u)∂ ∂uC*u,F2*y F−1 1(u)++du. (4) 3. Copulas In this section, we will briefly discuss the copula theory related to the theory developed in our paper. Readers may refer to (Cherubini et al. 2004;Joe 1997;Nelsen 2007;Tran et al. 2015,2017) for more information. Let I=[ 0,1 ] be the closed unit interval and I2=[ 0,1 ]×[ 0,1 ] be the closed unit square interval. We first state the most basic definition of copula in two dimensions in the following: Definition 1. (Copula) A 2-copula (two-dimensional copula) is a function C: I2→I satisfying the following conditions: (i) C(u,0)=C(0, v)=0for any u,v∈I; (ii) C(u,1)=u and C(1, v)=v for any u,v∈I; and 135 JRFM 2019,12,42 (iii) for any u1,u2,v1,v2∈Iwith u1≤u2and v1≤v2, C(u2,v2)+C(u1,v1)−C(u2,v1)−C(u1,v2)≥0. In copula theory, Sklar proposed a very important theorem in 1959 called Sklar’s Theorem (Cherubini et al. (2004); Joe (1997); Nelsen (2007)), which plays the most important role in this theory. It tells us that given a random vector (X1 , X2) with absolutely continuous marginal distribution functions FX1 and FX2 , respectively, and its joint distribution function denoted by H , and then there exists a unique copula Csuch that H(x1,x2)=C,FX1(x1),FX2(x2)-, h(x1,x2)= ∂2 ∂x1∂x2H(x1,x2)=c,FX1(x1),FX2(x2)fX1(x1)fX2(x2)-, (5) where c(u , v):=∂2 ∂u∂vC(u , v) denotes density of copula C , fXi is probability density function (PDF) of Xi , i= 1,2, and h(x1 , x2) is the joint density function of X1 and X2 . Copula is used to combine several univariate distributions together into bivariate [multivariate] settings so as the copula C can capture the dependence structure of (X1,X2)[(X1,···,Xn)]. For any copula C, we have the bounds W(u,v)≤C(u,v)≤M(u,v), where the copula W(u , v):=max(u+v− 1,0 ) captures counter-monotonicity structure; that is, X2=f(X1) a.s., where f is strictly decreasing, while the copula M(u , v):=min(u , v) is used to capture comonotonicity; that is, X2=f(X1) a.s., where f is strictly increasing. In case X1 and X2 are independent, they follow copula denoted by Π(u , v):=uv . Copulas can be used not only to model the dependence structure of the variables, but also capture the correlation between the variables. The Kendall’s coefficient τcan be expressed in terms of copulas as shown in the following: τ(X1,X2)=τ(C)=4 I2C(u,v)dC(u,v)−1. (6) In the next section, we will derive the two main propositions regarding formulas that can be used to determine the probability density and probability distribution of the quotient of dependent random variables by using copulas. In addition, we will apply the results to derive some corollaries on PDFs, CDFs, and median of the ratios in case X1 and X2 are normal distributed and they follow the Gaussian copulas. 4. Theory We now develop two propositions on both density and distribution functions for the quotient Y:=X1 X2 and the ratio of one variable over the sum of two variables Z:=X1 X1+X2 of two dependent random variables X1 and X2 by using copulas. We first develop the proposition on the density and distribution functions for the quotient Y=X1 X2as stated in the following: Proposition 1. Supposing that (X1 , X2) is a vector of two absolutely continuous random variables X1 and X2 with the marginal distributions F1 and F2 , respectively, let C be an absolutely continuous copula modeling dependence structure of the random vector (X1,X2), and define Y as Y:=X1 X2. (7) 136 JRFM 2019,12,42 Then, the density fY(y)and distribution functions FY(y)of Y are fY(y)=1 0F−1 2(v)cF1yF−1 2(v),vf1yF−1 2(v)dv, (8) FY(y)=F2(0)+1 0sgn F−1 2(v)∂ ∂vCF1yF−1 2(v),vdv, (9) respectively, where F−1 2denotes the inverse function of F2, c is the density of copula C, and sgn(·)stands for a sign function such that sgn(x)=%1, if x >0, −1, if x <0. Proof. Letting ⎧ ⎨ ⎩ Y1:=X1 X2, Y2:=X2. We note that since X2 is absolutely continuous, P(X2= 0 )= 0; that is, X2= 0 almost surely. Hence, the transformation Y1=X1 X2 always exists with probability 1 and we can obtain its inverse transformation by using %X1=Y1Y2, X2=Y2, and their corresponding Jacobian J= Y2Y1 01 =Y2. Then, we obtain the joint density of Y1and Y2such that h(y1,y2)=f(y1y2,y2)|y2| =|y2|c(F1(y1y2),F2(y2)) f1(y1y2)f2(y2). This yields the density of Y1: fY1(y1)= ∞  −∞|y2|c(F1(y1y2),F2(y2)) f1(y1y2)f2(y2)dy2(10) =1 0F−1 2(v)cF1y1F−1 2(v),vf1y1F−1 2(v)dv. (11) As a result, the CDF of Y1is determined by FY1(t)=1 0t −∞F−1 2(v)cF1y1F−1 2(v),vf1y1F−1 2(v)dy1dv. (12) By changing variable, u=F1y1F−1 2(v) , we get du =F−1 2(v)f1y1F−1 2(v)dy1 and we note that F−1 2(v)≥0⇐⇒ v∈[0, F2(0)], and F−1 2(v)≤0⇐⇒ v∈[F2(0),1]. 137 JRFM 2019,12,42 This yields FY1(t)=−F2(0) 0F1(tF−1 2(v)) 1 ∂2 ∂u∂vC(u,v)dudv +1 F2(0)F2(tF−1 1(v)) 0 ∂2 ∂u∂vC(u,v)dvdu =−F2(0) 0∂ ∂vCF1tF−1 2(v),v−∂ ∂vC(1, v) dv +1 F2(0) ∂ ∂vCF1tF−1 2(v),vdv (13) =F2(0)+1 0sgn F−1 2(v)∂ ∂vCF1tF−1 2(v),vdv. Thus, the assertions of Proposition 1hold. From Proposition 1and applying Equation (8) , we obtain the following corollary on both density and distribution functions for the quotient Y=X1 X2 of two independent random variables X1 and X2 by using copulas: Corollary 1. When X1 and X2 are independent, then its copula C(u , v)=uv has the density c(u , v)= 1, ∀u,v∈Iand the density fY(y)of the ratio Y :=X1 X2of two independent random variables becomes fY(y)=∞ −∞|x|f1(xy)f2(x)dx. This result is well known in the literature. Next, we apply Equation (9) to derive both density and distribution function for Y=X1 X2 in case X1 and X2 are normal random variables and their dependence structure is captured by Gaussian Copulas. We first obtain the following corollary: Corollary 2. Assume that X1∼N(μ1 , σ2 1) , X2∼N(μ2 , σ2 2) and (X1 , X2) follows Gaussian Copulas Cr(u , v) , |r|< 1, given in (46) . Then, the density fY(y) and distribution function FY(y) of Y=X1 X2 have the forms fY(y)=1 0sgnσ2Φ−1(v)+μ2ϕ(yσ2−rσ1)Φ−1(v)+yμ2−μ1 σ1√1−r2σ2Φ−1(v)+μ2 σ1√1−r2dv, (14) FY(y)=Φ−μ2 σ2+1 0sgnσ2Φ−1(v)+μ2Φ(yσ2−rσ1)Φ−1(v)+yμ2−μ1 σ1√1−r2dv, (15) respectively, where ϕ(x) and Φ(x) are PDF and CDF of the standard normal distribution, respectively, and Φ−1(x)denotes for the inverse function of Φ(x). Proof. Let X1∼N(μ1 , σ2 1) , and X2∼N(μ2 , σ2 2) ; then, their CDFs and inverse functions can be expressed in the following form: Fi(x)=Φx−μi σi,F−1 i(v)=σiΦ−1(v)+μi,i=1,2. Given Gaussian Copulas Cr(u , v) with |r|< 1, one can obtain its derivative ∂Cr ∂v(u , v) , see Meyer (2013), as shown in the following: ∂Cr ∂v(u,v)=ΦΦ−1(u)−rΦ−1(v) √1−r2. 138 JRFM 2019,12,42 Now, applying Equation (9), we can simplify it to be FY(y)=Φ−μ2 σ2+1 0sgnσ2Φ−1(v)+μ2ΦΦ−1Φyσ2Φ−1(v)+yμ2−μ1 σ1−rΦ−1(v) √1−r2 =Φ−μ2 σ2+1 0sgnσ2Φ−1(v)+μ2Φ(yσ2−rσ1)Φ−1(v)+yμ2−μ1 σ1√1−r2dv. Taking derivative of Fy(y) with respect to y , one gets the density fY(y) defined as in (14) . The assertions of Corollary 2hold. We note that the probability exhibited in (15) can be easily computed by using the following Monte Carlo algorithm: For each y∈R , we generate V from the uniform distribution on the unit interval [0,1]with sample size N, say N= 10,000, and then the estimated probability is given by . Fy(y)≈Φ−μ2 σ2+1 N N ∑ i=1 sgnσ2Φ−1(vi)+μ2Φ(yσ2−rσ1)Φ−1(vi)+yμ2−μ1 σ1√1−r2. (16) Using the result, we obtain the following corollary: Corollary 3. If X1∼N(μ1 , σ2 1) , X2∼N( 0, σ2 2) , and (X1 , X2) follows Gaussian Copulas Cr(u , v) , |r|< 1, given in (46), then the median of Y =X1 X2satisfies median(Y)=rσ1 σ2,for all μ1∈R. (17) Proof. Since X2∼N( 0, σ2 2) , Φ(−μ2 σ2)=Φ( 0 )= 0.5. Hence, it is sufficient to prove that the integral term given in (15) is equal to zero. In fact, we find that FYrσ1 σ2=0.5 +1 0sgn(σ2Φ−1(v))Φ−μ1 σ1√1−r2dv =0.5 +Φ−μ1 σ1√1−r2−1/2 0dv +1 1/2 dv =0.5. Hence, the quantity rσ1 σ2is the median of Y. The proof is complete. We turn to develop the proposition on density and distribution functions for the ratio of one variable over the sum of two variables Z:=X1 X1+X2 of two dependent random variables X1 and X2 by using copulas as stated in the following: Proposition 2. Suppose that (X1 , X2) is a vector of two absolutely continuous random variables X1 and X2 with the marginal distributions F1 and F2 , respectively, and let C be an absolutely continuous copula modeling dependence structure of the random vector (X1,X2), and define Z as Z:=X1 X1+X2. (18) Then, the density fZ(z)and distribution function FZ(z)of Z are fZ(z)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩1 0|F−1 1(u)| z2cu,F21−z zF−1 1(u)f21−z zF−1 1(u)du,if z =0, f1(0)1 0F−1 2(v)cF1(0),vdv,if z =0, (19) 139 JRFM 2019,12,42 FZ(z)=1{z≥0}+1 0sgn F−1 1(u)∂ ∂uCu,F2−F−1 1(u)−∂ ∂uCu,F21−z zF−1 1(u)du,(20) respectively, where 1{·} denotes an indicator function, F−1 i denotes the inverse function of Fi for i= 1,2, c is the density of copula C, and sgn(·)is the sign function such that 1{z≥0}=%1, if z ≥0, 0, if z <0, and sgn(x)=%1, if x >0, −1, if x <0. Proof. By defining ⎧ ⎨ ⎩ Z1=X1 X1+X2, Z2=X1+X2. Here, we note that, since X1 and X2 are absolutely continuous, P(X1+X2= 0 )= 0; that is, X1+X2= 0 almost surely. Hence, the transformation Z1=X1 X1+X2 always exists with probability 1 and we obtain the following inverse transformation: %X1=Z1Z2, X2=Z2−Z1Z2, and the Jacobian J= Z2Z1 −Z21−Z1=Z2. Thus, the joint density of Z1and Z2becomes h(z1,z2)=f(z1z2,z2−z1z2)|z2| =|z2|c(F1(z1z2),F2(z2−z1z2)) f1(z1z2)f2(z2−z1z2), which leads us to get the density of Z1such that fZ1(z1)= ∞  −∞|z2|c(F1(z1z2),F2(z2−z1z2)) f1(z1z2)f2(z2−z1z2)dz2. (21) If z1=0, then by taking v:=F2(z2),weget fZ1(0)=f1(0)1 0F−1 2(v)cF1(0),vdv. If z1>0, then by taking u:=F1(z1z2), we obtain fZ1(z1)=1 0F−1 1(u) z2 1 cu,F21−z1 z1F−1 1(u)f21−z1 z1F−1 1(u)du. If z1<0, then also by taking u:=F1(z1z2), we yield fZ1(z1)=0 1F−1 1(u) −z2 1 cu,F21−z1 z1F−1 1(u)f21−z1 z1F−1 1(u)du. (22) 140 JRFM 2019,12,42 Hence, for z1=0, we obtain fZ1(z1)=1 0F−1 1(u) z2 1 cu,F21−z1 z1F−1 1(u)f21−z1 z1F−1 1(u)du for z1=0. As a consequence, the distribution of Z1becomes FY1(t)=1 0t −∞F−1 1(u) z2 1 cu,F21−z1 z1F−1 1(u)f21−z1 z1F−1 1(u)dz1du. (23) Setting v=F21−z1 z1F−1 1(u)=⇒dv =−F−1 1(u) z2 1f21−z1 z1F−1 1(u)dz1, and note that F−1 1(u)≥0⇐⇒ u≥F1(0), and F−1 1(u)≤0⇐⇒ u≤F1(0), we consider two cases as follows: (i) Case 1: For t<0, we have FZ1(t)=F1(0) 0F2(1−t tF−1 1(u)) F2(−F−1 1(u)) ∂2 ∂u∂vC(u,v)dvdu −1 F1(0)F2(1−t tF−1 1(u)) F2(−F−1 1(u)) ∂2 ∂u∂vC(u,v)dvdu =F1(0) 0∂ ∂uCu,F21−t tF−1 1(u)−∂ ∂uCu,F2−F−1 1(u)du −1 F1(0)∂ ∂uCu,F21−t tF−1 1(u)−∂ ∂uCu,F2−F−1 1(u)du =1 0sgn F−1 1(u)∂ ∂uCu,F2−F−1 1(u)−∂ ∂uCu,F21−t tF−1 1(u)du. (24) (ii) Case 2: For t≥0, we first split the integrals FZ1(t)=1 00 −∞F−1 1(u) z2 1 cu,F21−z1 z1F−1 1(u)f21−z1 z1F−1 1(u)dz1du +1 0t 0F−1 1(u) z2 1 cu,F21−z1 z1F−1 1(u)f21−z1 z1F−1 1(u)dz1du (25) =:I1+I2. We then apply (24) to obtain the following expression for the integral I1: I1=FZ1(0)=1 0sgn F−1 1(u)∂ ∂uCu,F2−F−1 1(u)−∂ ∂uCu, lim t→0−F21−t tF−1 1(u) du =1 0sgn F−1 1(u)∂ ∂uCu,F2−F−1 1(u)du +F1(0) 0∂ ∂uC(u,1)du −1 F1(0)∂ ∂uC(u,0)du =1 0sgn F−1 1(u)∂ ∂uCu,F2−F−1 1(u)du +F1(0), (26) and obtain the following expression for the integral I2: I2=F1(0) 0F2(1−t tF−1 1(u)) 0 ∂2 ∂u∂vC(u,v)dvdu −1 F1(0)F2(1−t tF−1 1(u)) 1 ∂2 ∂u∂vC(u,v)dvdu =F1(0) 0∂ ∂uCu,F21−t tF−1 1(u)−∂ ∂uCu,0) du −1 F1(0)∂ ∂uCu,F21−t tF−1 1(u)−∂ ∂uCu,1 du =1−F1(0)−1 0sgn F−1 1(u)∂ ∂uCu,F2−F−1 1(u)du. (27) 141 JRFM 2019,12,42 From (26) and (27), we get the following for t≥0, FZ1(t)=1+1 0sgn F−1 1(u)∂ ∂uCu,F2−F−1 1(u)−∂ ∂uCu,F21−t tF−1 1(u) du. (28) Combining (24) and (28) imply (20), we complete the proof. In the situation X1 and X2 are independent, applying Proposition 2, we obtain the following corollary: Corollary 4. When X1 and X2 are independent, then its copula C(u , v)=uv has the density c(u , v)= 1, ∀u , v∈I and the density fZ(z) and distribution function FZ(z) for the ratio of one variable over the sum of two variables Z :=X1 X1+X2of two independent random variables X1and X2become fZ(z)= ∞  −∞|x|f1(xz)f2((1−x)z)dx and FZ(z)=1{z≥0}+1 0sgn F−1 1(u)F2−F−1 1(u)−F21−z zF−1 1(u) du, =1{z≥0}+∞ −∞sgn(x)F2(−x)−F21−z zx f1(x)dx, respectively. Next, we apply Equation (20) to derive the distribution function of Z:=X1 X1+X2 in the situation that both X1 and X2 are normal distributed such that their dependence structure can be captured by Gaussian Copulas as shown in the following corollary: Corollary 5. Assume that X1∼N(μ1 , σ2 1) , X2∼N(μ2 , σ2 2) , and (X1 , X2) follows Gaussian Copulas Cr(u,v),|r|<1, given in (46). Then, distribution function FZ(z)of Z :=X1 X1+X2has the form FZ(z)= = 1{z≥0}+2Φμ1 σ1−1−1 0sgnσ1Φ−1(u)+μ1Φ(σ1+rσ2)Φ−1(u)+μ1−μ2 σ2√1−r2du −1 0sgnσ1Φ−1(u)+μ1Φ[(1−z)σ1−zrσ2]Φ−1(u)−z(μ1+μ2)+μ1 zσ2√1−r2du, (29) where Φ(x)and Φ−1(x)are CDF and its inverse of the standard normal random variable, respectively. Proof. Let X1∼N(μ1 , σ2 1) and X2∼N(μ2 , σ2 2) , the CDFs and their inverse functions can be written in the form Fi(x)=Φx−μi σi,F−1 i(v)=σiΦ−1(v)+μi,i=1,2. Given Gaussian Copulas Cr(u , v) , |r|< 1, we apply the results from Meyer (2013) to obtain its derivative ∂Cr ∂u(u,v)as shown in the following: ∂Cr ∂u(u,v)=ΦΦ−1(v)−rΦ−1(u) √1−r2. 142 JRFM 2019,12,42 Applying Equation (20), one can simplify it to be FZ(z)=1{z≥0}+1 0sgnσ1Φ−1(u)+μ1ΦΦ−1Φ−σ1Φ−1(u)+μ1−μ2 σ2−rΦ−1(u) √1−r2du −1 0sgnσ1Φ−1(u)+μ1ΦΦ−1Φ1−z z!σ1Φ−1(u)+μ1"−μ2 σ2−rΦ−1(u) √1−r2du =1{z≥0}+1 0sgnσ1Φ−1(u)+μ1Φ−(σ1+rσ2)Φ−1(u)+μ1−μ2 σ2√1−r2du −1 0sgnσ1Φ−1(u)+μ1Φ[(1−z)σ1−zrσ2]Φ−1(u)−z(μ1+μ2)+μ1 zσ2√1−r2du =1{z≥0}+2Φμ1 σ1−1−1 0sgnσ1Φ−1(u)+μ1Φ(σ1+rσ2)Φ−1(u)+μ1−μ2 σ2√1−r2du −1 0sgnσ1Φ−1(u)+μ1Φ[(1−z)σ1−zrσ2]Φ−1(u)−z(μ1+μ2)+μ1 zσ2√1−r2du. In the last step of the above, we use the property Φ(−x)=1−Φ(x)and 1 0sgnσ1Φ−1(u)+μ1du =−Φ−μ1 σ1 0du +1 Φ−μ1 σ1du =1−2Φ−μ1 σ1=2Φμ1 σ1−1. The proof is complete. We note that the probability given in (29) can also be easily computed by using the following Monte Carlo algorithm: For each z∈R , we first generate U from the uniform distribution on the unit interval [ 0,1 ] with sample size N , say N= 10,000. Then, we obtain the following estimated probability: . FZ(z)≈1{z≥0}+2Φμ1 σ1−1−1 N N ∑ i=1 sgnσ1Φ−1(ui)+μ1Φ(σ1+rσ2)Φ−1(ui)+μ1−μ2 σ2√1−r2 −1 N N ∑ i=1 sgnσ1Φ−1(ui)+μ1Φ[(1−z)σ1−zrσ2]Φ−1(ui)−z(μ1+μ2)+μ1 zσ2√1−r2. (30) Using the above results, we obtain the following corollary: Corollary 6. Assume that X1∼N( 0, σ2) , X2∼N( 0, σ2) and (X1 , X2) follows Gaussian Copulas Cr(u , v) , |r|<1, given in (46). Then, the median of Z :=X1 X1+X2is equal to 1 2. Proof. Because X1∼N( 0, σ2) , X2∼N( 0, σ2) and sgnσΦ−1(u)=sgnΦ−1(u) , we obtain CDF of the ratio Z:=X1 X1+X2from (29) as shown in the following: FZz=1{z≥0}−1 0sgnΦ−1(u)Φ(1+r)Φ−1(u) √1−r2+Φ[1−z−zr]Φ−1(u) z√1−r2du. (31) We get FZ1 2=1−1 0sgnΦ−1(u)Φ(1+r)Φ−1(u) √1−r2+Φ[1−r]Φ−1(u) √1−r2du. (32) 143