risky asset returns, since the stock markets have the characteristics of heavy-tailness, non-zero skewness, positive excess kurtosis and there exists tail dependence among the international stock markets. Then, we provide two different forms of portfolio. For the first portfolio, we consider that our assets include the risky assets and risk-free asset. For the second one, we consider the risky asset as our assets and use the
leverage of risky assets to invest. Our risky assets include the developed countries and developing countries. The assets of developed countries include the S&P500, DAX, CAC40 and FTSE100 stock indices. The assets of developing countries include the BSE SENSEX, IBOVESPA, RTS and SSE Composite Index stock indices. Our risk-free asset is the U.S. three-month LIBOR rates.
For optimal asset allocation strategy, we want to make our asset allocation more dynamic and efficient. We change our asset allocation every three months and
re-estimate the parameters based on the past five years data, select the optimal
weights of portfolio and re-assess the optimal asset allocation at each decision date.
For the empirical analysis, we divide our asset allocation into three parts. For the first part, we use the assets of developed countries including the S&P500, DAX, CAC40 and FTSE100. For the second part, we use the assets of developing countries
including the BSE SENSEX, IBOVESPA, RTS and SSE Composite Index. As for the third part, we use the mixing assets of developed countries and developing countries including the S&P500, FTSE100, RTS and SSE Composite Index. Empirically, when the performances of stock markets are good, we suggest that our asset allocation uses the skewness as the objective function. When the performances of stock markets are bad, we suggest that our asset allocation uses the variance as the objective function.
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MAJD distributions. From Equation (2-3), (2-4), (2-5), (3-4), (3-5) and (3-6), we have the characteristic function ofR t ; that is
P( )‧
the MANIG distributions. From Equation (2-12), (2-13), (2-14), (3-4), (3-5) and (3-6), we have the characteristic function ofR t ; that is
P( )‧
MAJD distributions. From Equation (2-3), (2-4), (2-5), (3-4), (3-5) and (3-13), we have the characteristic function ofR t ; that is
P( )‧
the MANIG distributions. From Equation (2-12), (2-13), (2-14), (3-4), (3-5) and (3-13), we have the characteristic function ofR t ; that is
P( )‧ 國
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Appendix C
Table C-1-Table C-19 show the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 from other periods.
Table C-20-Table C-38 show the summary statistics on the returns of BSE SENSEX, IBOVESPA, RTS and SSE Composite Index from other periods.
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-1 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is the period is from April 2, 2001 to March 27, 2006. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010 Table C-2.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-2 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from July 2, 2001 to June 26, 2006. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-3 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period isfrom October 1, 2001 to September 25, 2006. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010 Table C-4.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-4 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from January 7, 2002 to December 25, 2006. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-5 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period isfrom April 1, 2002 to March 26, 2007. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010
Table C-6.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-6 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from July 1, 2002 to June 25, 2007. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance.
All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-7 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from October 7, 2002 to September 24, 2007. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010
Table C-8.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-8 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from January 6, 2003 to December 31, 2007. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010
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Descriptive statistics of developed countries of weekly returns of stock indices
Table C-9 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is April 7, 2003 to March 31, 2008. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance.
All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Jarque-Bera test P-value 0.0091 0.0010 0.0010 0.0010
Table C-10.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-10 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period isfrom July 7, 2003 to June 30, 2008. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-11 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from October 6, 2003 to September 29, 2008. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Jarque-Bera test P-value 0.0366 0.0010 0.0010 0.0010
Table C-12.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-12 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from January 5, 2004 to December 29, 2008. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-13 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from April 5, 2004 to March 30, 2009. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010 Table C-14.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-14 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period isfrom July 5, 2004 to June 29, 2009. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-15 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period isfrom October 4, 2004 to September 28, 2009. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010 Table C-16.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-16 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from January 6, 2005 to December 31, 2009. We can see S&P500 index has the lowest average weekly return and variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-17 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from April 4, 2005 to March 29, 2010. We can see FTSE100 index has the lowest average weekly returns, S&P500 index has the lowest weekly variance and DAX index has the highest average weekly returns. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010 Table C-18.
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-18 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from July 3, 2005 to June 28, 20010. We can see FTSE100 index has the lowest average weekly return, S&P500 index has the lowest weekly variance and DAX index has the highest average weekly return and variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
‧
Descriptive statistics of developed countries of weekly returns of stock indices
Table C-19 provides the summary statistics on the returns of S&P500, DAX, CAC40 and FTSE100 and its period is from the period is from October 2, 2005 to September 27, 20010. We can see FTSE100 index has the lowest average weekly return, S&P500 index has the lowest weekly variance, DAX index has the highest average weekly return and CAC40 has the highest weekly variance. All stock indices have
significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics S&P500 DAX CAC40 FTSE100
Jarque-Bera test P-value 0.0010 0.0010 0.0010 0.0010 Table C-20.
Descriptive statistics of developing countries of weekly returns of stock indices
Table C-20 provides the summary statistics on the returns of BSE SENSEX, IBOVESPA, RTS and SSE Composite Index and its period is from April 2, 2001 to March 27, 2006. We can see SSE Composite Index has the lowest average weekly return and variance, RTS index has the highest average weekly return and IBOVESPA has the highest weekly variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics BSE SENSEX IBOVESPA RTS SSE
Mean 0.0037 0.0028 0.0073 -0.0025
Jarque-Bera test P-value 0.0010 0.0099 0.0039 0.0010
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Descriptive statistics of developing countries of weekly returns of stock indices
Table C-21 provides the summary statistics on the returns of BSE SENSEX, IBOVESPA, RTS and SSE Composite Index and its period is from July 2, 2001 to June 26, 2006. We can see SSE Composite Index has the lowest average weekly return and variance and RTS index has the highest average weekly return and IBOVESPA has the highest weekly variance. All stock indices have significant non-zero skewness and positive excess kurtosis. According to the Jarque-Bera test, all P-value of stock indices are low under 0.05 level of alpha that implies the assumption of normality is rejected.
Statistics BSE SENSEX IBOVESPA RTS SSE
Mean 0.0032 0.0024 0.0058 -0.0022
Jarque-Bera test P-value 0.0010 0.0034 0.0010 0.0010 Table C-22.
Descriptive statistics of developing countries of weekly returns of stock indices
Table C-22 provides the summary statistics on the returns of BSE SENSEX, IBOVESPA, RTS and SSE Composite Index and its period is from October 1, 2001 to September 25, 2006. We can see SSE Composite
Table C-22 provides the summary statistics on the returns of BSE SENSEX, IBOVESPA, RTS and SSE Composite Index and its period is from October 1, 2001 to September 25, 2006. We can see SSE Composite