4. Empirical analysis
4.3 The funding value for our objective functions under MAJD, MAVG and
4.3.3 The funding values on the assets of mixing of developed countries and
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4.3.3 The funding values on the assets of mixing of developed countries and developing countries
a) For the performance of funding value on the first portfolio:
Table 11 shows the returns of portfolio for (P1), (P2), (P3), (P4) and (P5) on the mixing of developed countries and developing countries under MAJD, MAVG and MANIG distributions. Table 12 and Figure 5 show the funding values for (P1), (P2), (P3), (P4) and (P5) on the mixing of developed countries and developing countries under MAJD, MAVG and MANIG distributions.
From Table 11 and Table 12, we can see that the returns of portfolio for (P3) are the highest during 2007, but are the lowest during the first half year of 2008. However, the returns of portfolio for (P2), (P3), (P4) and (P5) are higher than the return of
portfolio for (P1) during the last half year of 2008, especially the returns of portfolio for (P2) and (P3) under MAJD distributions. From Table 12 and Figure 5, we can see that the performance of funding values for (P3) are the best at the December 27, 2010, especially the performance of funding value for (P3) under MAJD distributions.
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The returns of portfolio for (P1), (P2), (P3), (P4) and (P5) on the mixing of
developed countries and developing countries under MAJD, MAVG, and MANIG distributions
Table 11 shows the returns of portfolio for (P1), (P2), (P3), (P4) and (P5) on the mixing of developed countries and developing countriesunder MAJD, MAVG and MANIG distributions. (P1) uses the mean of portfolio as objective function. (P2) uses the variance of portfolio as objective function. (P3) uses the skewness of portfolio as objective function. (P4) uses the kurtosis of portfolio as objective function. (P5) uses the skewness-kurtosis of portfolio as objective function.
Obj. Fun. Model 2006/3/27 2006/6/26 2006/9/25 2006/12/26 2007/3/26 2007/6/25 2007/9/24 2007/12/31 2008/3/31 2008/6/30 (P1) 0.2534 -0.1370 -0.0096 0.1886 0.0313 -0.0767 0.0667 0.0099 -0.0959 0.1037 Obj. Fun. Model 2008/9/29 2008/12/29 2009/3/30 2009/6/29 2009/9/29 2009/12/29 2010/3/29 2010/6/28 2010/9/27 2010/12/27 (P1) -0.6656 -0.2485 0.3352 0.1814 -0.0049 0.1227 0.0458 -0.2819 0.0376 0.0565
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The funding values for (P1), (P2), (P3), (P4) and (P5) on the mixing of developed countries and developing countries under MAJD, MAVG, and MANIG
distributions
Table 12 shows the funding values for (P1), (P2), (P3), (P4) and (P5) on the mixing of developed countries and developing countries under MAJD, MAVG and MANIG distributions. (P1) uses the mean of portfolio as objective function. (P2) uses the variance of portfolio as objective function. (P3) uses the skewness of portfolio as objective function. (P4) uses the kurtosis of portfolio as objective function. (P5) uses the skewness-kurtosis of portfolio as objective function. According to the equation (3-4), the funding values for (P1) and (P2), are the same under MAJD, MAVG and MANIG
distributions.
Obj. Fun. Model 2006/3/27 2006/6/26 2006/9/25 2006/12/26 2007/3/26 2007/6/25 2007/9/24 2007/12/31 2008/3/31 2008/6/30 (P1) 125.344 108.175 107.141 127.347 131.338 121.259 129.342 130.627 118.100 130.343 Obj. Fun. Model 2008/9/29 2008/12/29 2009/3/30 2009/6/29 2009/9/29 2009/12/29 2010/3/29 2010/6/28 2010/9/27 2010/12/27 (P1) 43.586 32.754 43.733 51.667 51.411 57.718 60.361 43.348 44.977 47.518
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Figure 5.
The funding values for (P1), (P2), (P3), (P4) and (P5) on the mixing of developed countries and developing countries under MAJD, MAVG, and MANIG distributions
Figure 5 shows the funding values from table 12.
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b) For the performance of funding value on the second portfolio:
Table 13 shows the returns of portfolio for (P1)’, (P2)’, (P3)’, (P4)’ and (P5)’, using the leverage (k) equal to 1, on the mixing of developed countries and
developing countries under MAJD, MAVG and MANIG distributions. Table 14 and Figure 6 show the funding values for (P1)’, (P2)’, (P3)’, (P4)’ and (P5)’, using the leverage (k) equal to 1, on the mixing of developed countries and developing countries under MAJD, MAVG and MANIG distributions. About the returns and funding values of portfolio of leverage (k) equal to 2 and 3 are put in Appendix G.
From Table 13 and Table 14, as leverage (k) equal to 1, we can see that the returns of portfolio for (P3)’ are the highest during 2007, but are the lowest during the first half year of 2008. From Table 14 and figure 6, we can see that the performance of funding value for (P3)’ is the best at the December 27, 2010, especially the
performance of funding value for (P3)’ under MAJD distributions. From Appendix G, we can see that when the leverage (k) is getting larger, it causes that the returns of portfolio for (P3)’, (P4)’ and (P5)’ are higher than the leverage (k) equal to 1 during 2007 and the returns of portfolio for (P3)’, (P4)’ and (P5)’ are lower than the leverage (k) equal to 1 during 2008. Therefore, when the leverage (k) is getting larger, the funding values have more significant ascension during 2007 and the funding values have more significant decline during the 2008 financial crisis.
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The returns of portfolio for (P1)’, (P2)’, (P3)’, (P4)’ and (P5)’, using the leverage (k) equal to 1, on the mixing of developed countries and developing countries under MAJD, MAVG, and MANIG distributions
Table 13 shows the returns of portfolio for (P1)’, (P2)’, (P3)’, (P4)’ and (P5)’, using the leverage (k) equal to 1, on the mixing of developed countries and developing countriesunder MAJD, MAVG and MANIG distributions. (P1)’ uses the mean of portfolio as objective function. (P2)’ uses the variance of portfolio as objective function. (P3)’ uses the skewness of portfolio as objective function. (P4)’
uses the kurtosis of portfolio as objective function. (P5)’ uses the skewness-kurtosis of portfolio as objective function.
Obj. Fun. Model 2006/3/27 2006/6/26 2006/9/25 2006/12/26 2007/3/26 2007/6/25 2007/9/24 2007/12/31 2008/3/31 2008/6/30 (P1)’ 0.2534 -0.1370 -0.0096 0.1886 0.0313 -0.0767 0.0667 0.0099 -0.0959 0.1037 Obj. Fun. Model 2008/9/29 2008/12/29 2009/3/30 2009/6/29 2009/9/29 2009/12/29 2010/3/29 2010/6/28 2010/9/27 2010/12/27
(P1)’ -0.6656 -0.2307 0.0501 0.1379 -0.0049 0.1233 0.0533 -0.2819 0.0376 0.0565
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The funding values for (P1)’, (P2)’, (P3)’, (P4)’ and (P5)’, using the leverage (k) equal to 1, on the mixing of developed countries and developing countries
Table 14 shows the funding values for (P1)’, (P2)’, (P3)’, (P4)’ and (P5)’, using the leverage (k) equal to 1, on the mixing of developed countries and developing countriesunder MAJD, MAVG and MANIG distributions. (P1)’ uses the mean of portfolio as objective function. (P2)’ uses the variance of portfolio as objective function. (P3)’ uses the skewness of portfolio as objective function. (P4)’
uses the kurtosis of portfolio as objective function. (P5)’ uses the skewness-kurtosis of portfolio as objective function. According to the equation (3-4), the funding values for (P1)’ and (P2)’ are the same under MAJD, MAVG and MANIG distributions.
Obj. Fun. Model 2006/3/27 2006/6/26 2006/9/25 2006/12/26 2007/3/26 2007/6/25 2007/9/24 2007/12/31 2008/3/31 2008/6/30 (P1)’ 125.344 108.175 107.141 127.347 131.338 121.259 129.342 130.627 118.100 130.343 Obj. Fun. Model 2008/9/29 2008/12/29 2009/3/30 2009/6/29 2009/9/29 2009/12/29 2010/3/29 2010/6/28 2010/9/27 2010/12/27
(P1)’ 43.586 33.531 35.209 40.067 39.868 44.784 47.170 33.875 35.148 37.133
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Figure 6.
The funding values for (P1)’, (P2)’, (P3)’, (P4)’ and (P5)’, using the leverage (k) equal to 1, on the mixing of developed countries and developing countries
Figure 6 shows the funding values from table 14.
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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 theleverage 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.