In this section, we give the descriptions of empirical datasets used in this study and present the results of the empirical study
5.1. Data Description
Three empirical datasets are considered in the empirical evaluation of the MVSEM. The first considered dataset consists of monthly returns on 20 industry portfolios in the United States and they are taken from Kenneth French’s web site [
51]. The 20 industries considered are Games, Books, Apparel, Chemicals, Construction, Steel, Fabricated Products, Electrical Equipment, Automotive, Carry, Telecommunications, Services, Business Equipment, Paper, Transportation, Wholesale, Retail, Meals, Finance and others. The period of dataset is from January 1993 to December 2007 (L = 180 monthly observations).
The second dataset consists of monthly seven international equity indexes, which are taken from Morgan Stanley web site [
52], US, UK, Japan, Germany, France, Italy, and Canada (G-7 countries) for the period from January 1970 to September 2010 (L = 489 monthly observations).
Last dataset includes monthly returns of 15 assets, which are traded on the Istanbul Stock Exchange (ISE) in Turkey, from different sectors: Financial Institutions, Manufacturing Industry and Technology sectors. The dataset are taken from the ISE web site [
53]. The dataset period is from January 1994 to December 2007 (L = 168 monthly observations).
It should be emphasized that all these datasets are adjusted for capital splits and stock dividends. The summary statistics for these datasets are presented in
Table 1,
Table 2 and
Table 3, respectively.
Table 1.
Descriptive statistics and normality test results for industry dataset.
Table 1.
Descriptive statistics and normality test results for industry dataset.
Portfolio | Mean | Variance | Skewness | Kurtosis | JB Test |
---|
X1 | 0.0102 | 0.0034 | −0.5559 | 1.3831 | 23.6184 |
X2 | 0.0075 | 0.0017 | 0.1444 | 0.5086 | 2.5656 |
X3 | 0.0071 | 0.0033 | −0.3311 | 2.4903 | 49.8034 |
X4 | 0.0096 | 0.0023 | 0.2047 | 1.6211 | 20.9661 |
X5 | 0.0096 | 0.0025 | −0.5567 | 1.1828 | 19.7904 |
X6 | 0.0121 | 0.0062 | 0.1268 | 1.7839 | 24.3485 |
X7 | 0.0127 | 0.0037 | −0.3756 | 1.0758 | 12.9134 |
X8 | 0.0148 | 0.0033 | −0.2002 | 0.2437 | 1.6479 |
X9 | 0.0075 | 0.0042 | −0.2532 | 0.7203 | 5.8153 |
X10 | 0.0140 | 0.0032 | −0.7880 | 2.3812 | 61.1530 |
X11 | 0.0064 | 0.0029 | −0.0366 | 1.5864 | 18.9161 |
X12 | 0.0109 | 0.0050 | −0.1061 | 0.8156 | 5.3264 |
X13 | 0.0133 | 0.0076 | −0.4505 | 1.3098 | 18.9553 |
X14 | 0.0082 | 0.0021 | 0.0259 | 1.6327 | 20.0137 |
X15 | 0.0084 | 0.0023 | −0.4984 | 1.3597 | 21.3170 |
X16 | 0.0071 | 0.0019 | −0.5083 | 1.4661 | 23.8741 |
X17 | 0.0082 | 0.0025 | −0.0993 | 0.3521 | 1.2258 |
X18 | 0.0089 | 0.0023 | −0.4481 | 0.7969 | 10.7884 |
X19 | 0.0113 | 0.0022 | −0.3932 | 2.8659 | 66.2398 |
X20 | 0.0040 | 0.0026 | −0.3992 | 2.1389 | 39.0915 |
Table 2.
Descriptive statistics and normality test results for international dataset.
Table 2.
Descriptive statistics and normality test results for international dataset.
Portfolio | Mean | Variance | Skewness | Kurtosis | JB Test |
---|
X1 | 0.007 | 0.002 | −0.664 | 2.424 | 155.670 |
X2 | 0.008 | 0.003 | −0.891 | 3.494 | 313.363 |
X3 | 0.008 | 0.004 | −0.446 | 1.571 | 66.500 |
X4 | 0.007 | 0.004 | −0.635 | 1.932 | 108.918 |
X5 | 0.004 | 0.005 | −0.116 | 0.800 | 14.146 |
X6 | 0.007 | 0.004 | −0.012 | 0.573 | 6.698 |
X7 | 0.008 | 0.004 | 0.333 | 5.537 | 633.630 |
Table 3.
Descriptive statistics and normality test results of ISE dataset.
Table 3.
Descriptive statistics and normality test results of ISE dataset.
Portfolio | Mean | Variance | Skewness | Kurtosis | JB Test |
---|
X1 | 0.043 | 0.027 | 0.262 | 1.143 | 11.059 |
X2 | 0.031 | 0.060 | 0.227 | 1.365 | 14.482 |
X3 | 0.037 | 0.035 | −0.455 | 2.855 | 62.845 |
X4 | 0.041 | 0.042 | −0.077 | 0.928 | 6.191 |
X5 | 0.040 | 0.043 | 0.354 | 2.233 | 38.410 |
X6 | 0.035 | 0.041 | 0.261 | 1.307 | 13.875 |
X7 | 0.036 | 0.035 | 0.081 | 1.253 | 11.181 |
X8 | 0.038 | 0.044 | −0.258 | 3.476 | 86.450 |
X9 | 0.040 | 0.046 | −0.367 | 1.403 | 17.551 |
X10 | 0.033 | 0.045 | −0.055 | 1.979 | 27.492 |
X11 | 0.038 | 0.034 | 0.152 | 2.094 | 31.333 |
X12 | 0.023 | 0.039 | 0.263 | 1.475 | 17.156 |
X13 | 0.036 | 0.041 | 0.690 | 2.042 | 42.549 |
X14 | 0.025 | 0.045 | 0.794 | 3.120 | 85.785 |
X15 | 0.029 | 0.036 | 0.562 | 1.852 | 32.857 |
The statistics in
Table 1,
Table 2 and
Table 3 give some insight into the characteristics of the return data. As can be seen from these tables, the Jack-Bera test for the most of return distribution of three empirical datasets reject the null hypothesis for normality at the 5% significance level.
5.2. Results of the Empirical Study
In the empirical study, we choose different values of
in MVSEM, which can be interpreted as risk preference of investors such as
,
,
as parallel in the studies [
2,
4,
54]. It is known that the results of MVSEM with
are equal to that with
when used with the weighted sum method [
46,
47]. Therefore, in MVSEM, when (
) are taken as
, the MVSM is obtained. In other words, while equal weights are assigned to variance and skewness, entropy is weighted at zero. Likewise, the choice of
indicates that variance, skewness and entropy are of equal importance to investors.
We evaluate empirically the performances of the MVSEM with the chosen relative to the EWM, MinVM, MVM and MVSM using 20 industry portfolios, 7 international portfolios, 15 ISE portfolios. However, since the qualitative results regarding the MVSEM with (2/4,1/4,1/4), (1/4,2/4,1/4), (1/4,1/4,2/4) are quite similar to the MVSEM with (1/3,1/3,1/3), we report the results only for the MVSEM with (1/3,1/3,1/3) in this study. The results for the other values of are available from authors.
All computations needed in the empirical study are conducted using the MATLAB program. It is also emphasized that the average CPU time to obtain portfolios via the MVSEM increases rapidly as the sample size and the number of assets increase due to the MVSEM’s computational complexity and the long computing time needed.
For the industry dataset, we present the results for window length W = 120 in
Table 4. As seen in this table, the all considered the MVSEMs provide best results in terms of all performance measures except GRRs, which favor the EWM. Moreover, it should be noted about
Table 4 that the MVM, MinVM and MVSM show the worst performance with respect to all considered performance measures. On the other hand, it is seen that the PT values of portfolios obtained from the MVSEMs are less than that of the MVM, MinVM and MVSM. This is a natural result since the resulting portfolios from MVSEMs shrink towards the equally-weighted portfolio due to entropy term. Moreover, the
-values for the differences in Sharpe ratios show that while the difference for the MVSEMs are not statistically significant at 5% level, that for the MVM and MVSM is statistically significant.
Table 5 presents the results of industry dataset for window length W = 150. It can be observed that the MVSEMs performs better than MVM, MinV and MVSM according to all considered performance measures except GRR (0.5,2). Taking into consideration the performance of the EWM, the EWM works better than MVSEMs according to only the SSR and GRRs. In terms of the PT, it is seen from
Table 5 that the values of PT of all MVSEMs are smaller than the MVM, MinVM and MVSM.
In
Table 6 and
Table 7, we present the results of international dataset for W = 120 and 150, respectively. Taking into account the results of
Table 6 and
Table 7 together, it is seen that the MVSEMs outperform the considered other models in terms of the most of the performance measures. Comparing the PT for the models, we observe that the values of PT for the MVSEMs are substantially less than that for the others. On the other hand, the p-values for the differences in Sharpe ratios show that none of the models yield significantly different the SR with respect to the EWM.
The results for the ISE dataset are showed in
Table 8 and
Table 9 for window length W = 120 and 150, respectively. The obtained results for W = 120 show that the MVSEMs are able to provide a good performance relative to the EWM, MVM, MinVM and MVSM in terms of most of performance measures. Besides, for W = 150, the MVSEMs significantly outperform the other portfolio models according to all considered performance measures. Furthermore, the MVSEMs yield the lowest values of the PT for W = 120 and 150.
Overall, we can say that portfolios obtained from the MVSEMs perform better in terms of variety portfolio performance measures than the EWM, MinVM, MVM and MVSM. Besides, the MVSEMs are able to provide smaller PT when compared to the other models.
Table 4.
The results of portfolio performance measures for industry dataset and W = 120.
Table 4.
The results of portfolio performance measures for industry dataset and W = 120.
Models | SR | ASR | MADR | SSR | FTR(0.5,2) | FTR(1.5,2) | FTR(1,1) | GRR(0.5,2) | GRR(1.5,2) | GRR(1,1) | PT | p-value |
---|
EWM | 0.3776 | 0.3809 | 0.4672 | 1.0923 | 0.2451 | 1.4576 | 2.5817 | 114.71 | 9.6396 | 1.6717 | – | – |
MinVM | 0.3157 | 0.3160 | 0.3954 | 0.8287 | 0.2015 | 1.2453 | 2.2020 | 108.64 | 7.5617 | 1.4303 | 0.1106 | 0.0524 |
MVM | 0.3007 | 0.3005 | 0.3808 | 0.7745 | 0.1936 | 1.2026 | 2.1344 | 100.06 | 6.9410 | 1.3627 | 0.1798 | 0.0380 |
MVSM | 0.2885 | 0.2883 | 0.3641 | 0.7348 | 0.1887 | 1.1768 | 2.0653 | 99.89 | 6.8775 | 1.3541 | 0.1693 | 0.0227 |
MVSEM(1/2,0,1/2) | 0.3820 | 0.3854 | 0.4763 | 1.1032 | 0.2486 | 1.4581 | 2.6123 | 110.97 | 9.3111 | 1.6435 | 0.0539 | 0.6877 |
MVSEM(0,1/2,1/2) | 0.3824 | 0.3858 | 0.4768 | 1.1049 | 0.2489 | 1.4590 | 2.6148 | 110.73 | 9.3038 | 1.6436 | 0.0541 | 0.7046 |
MVSEM(1/3,1/3,1/3) | 0.3822 | 0.3855 | 0.4764 | 1.1031 | 0.2486 | 1.4582 | 2.6122 | 110.98 | 9.3115 | 1.6435 | 0.0539 | 0.6873 |
Table 5.
The results of portfolio performance measures for industry dataset and W = 150.
Table 5.
The results of portfolio performance measures for industry dataset and W = 150.
Models | SR | ASR | MADR | SSR | FTR(0.5,2) | FTR(1.5,2) | FTR(1,1) | GRR(0.5,2) | GRR(1.5,2) | GRR(1,1) | PT | p-value |
---|
EWM | 0.3441 | 0.3381 | 0.4234 | 1.4145 | 0.0543 | 1.0790 | 2.3529 | 69.8352 | 3.7987 | 0.9413 | – | – |
MinVM | 0.2447 | 0.2416 | 0.2975 | 0.9225 | 0.0418 | 0.9083 | 1.8224 | 67.8758 | 3.5597 | 0.9029 | 0.2370 | 0.0299 |
MVM | 0.1819 | 0.1798 | 0.2258 | 0.6328 | 0.0373 | 0.7791 | 1.5633 | 67.7187 | 3.3612 | 0.8654 | 0.2448 | 0.0008 |
MVSM | 0.1715 | 0.1697 | 0.2117 | 0.5948 | 0.0352 | 0.7713 | 1.5259 | 69.5960 | 3.4701 | 0.8816 | 0.1636 | 0.0005 |
MVSEM(1/2,0,1/2) | 0.3448 | 0.3395 | 0.4248 | 1.3693 | 0.0546 | 1.0792 | 2.3610 | 66.1523 | 3.5817 | 0.9237 | 0.0373 | 0.2721 |
MVSEM(0,1/2,1/2) | 0.3452 | 0.3404 | 0.4258 | 1.3737 | 0.0546 | 1.0793 | 2.3618 | 65.9892 | 3.5753 | 0.9231 | 0.0374 | 0.3005 |
MVSEM(1/3,1/3,1/3) | 0.3450 | 0.3397 | 0.4251 | 1.3692 | 0.0545 | 1.0790 | 2.3609 | 66.1638 | 3.5822 | 0.9238 | 0.0374 | 0.2718 |
Table 6.
The results of portfolio performance measures for international dataset and W = 120.
Table 6.
The results of portfolio performance measures for international dataset and W = 120.
Models | SR | ASR | MADR | SSR | FTR(0.5,2) | FTR(1.5,2) | FTR(1,1) | GRR(0.5,2) | GRR(1.5,2) | GRR(1,1) | PT | p-value |
---|
EWM | 0.1543 | 0.1505 | 0.2082 | 0.1248 | 1.8625 | 1.0392 | 1.5038 | 18.4701 | 1.7883 | 0.7751 | – | – |
MinVM | 0.1691 | 0.1637 | 0.2266 | 0.1357 | 1.9361 | 1.0398 | 1.5588 | 19.4430 | 1.6470 | 0.7289 | 0.1199 | 0.6527 |
MVM | 0.1495 | 0.1456 | 0.1997 | 0.1204 | 1.8745 | 1.0318 | 1.4785 | 19.4162 | 1.8105 | 0.7912 | 0.1405 | 0.3722 |
MVSM | 0.1491 | 0.1452 | 0.1995 | 0.1200 | 1.8652 | 1.0296 | 1.4781 | 19.3950 | 1.7964 | 0.7864 | 0.1371 | 0.3692 |
MVSEM(1/2,0,1/2) | 0.1692 | 0.1657 | 0.2278 | 0.1211 | 1.8278 | 1.0407 | 1.4823 | 19.5161 | 1.8915 | 0.8122 | 0.0694 | 0.6601 |
MVSEM(0,1/2,1/2) | 0.1692 | 0.1656 | 0.2275 | 0.1209 | 1.8274 | 1.0407 | 1.4821 | 19.5121 | 1.8919 | 0.8123 | 0.0694 | 0.6577 |
MVSEM(1/3,1/3,1/3) | 0.1695 | 0.1659 | 0.2287 | 0.1213 | 1.8279 | 1.0414 | 1.4825 | 19.5166 | 1.8925 | 0.8123 | 0.0691 | 0.6600 |
Table 7.
The results of portfolio performance measures for international dataset and W = 150.
Table 7.
The results of portfolio performance measures for international dataset and W = 150.
Models | SR | ASR | MADR | SSR | FTR(0.5,2) | FTR(1.5,2) | FTR(1,1) | GRR(0.5,2) | GRR(1.5,2) | GRR(1,1) | PT | p-value |
---|
EWM | 0.1738 | 0.1688 | 0.2363 | 0.1655 | 1.2311 | 1.0132 | 1.5858 | 19.1082 | 1.8555 | 0.7980 | – | – |
MinVM | 0.1786 | 0.1724 | 0.2422 | 0.1677 | 1.2665 | 0.9970 | 1.6000 | 19.8151 | 1.7237 | 0.7517 | 0.1063 | 0.6392 |
MVM | 0.1696 | 0.1648 | 0.2269 | 0.1611 | 1.2429 | 1.0123 | 1.5570 | 20.1864 | 1.8797 | 0.7968 | 0.1032 | 0.3820 |
MVSM | 0.1693 | 0.1648 | 0.2263 | 0.1615 | 1.2388 | 1.0180 | 1.5559 | 20.9188 | 1.9436 | 0.8084 | 0.1461 | 0.3729 |
MVSEM(1/2,0,1/2) | 0.1787 | 0.1725 | 0.2428 | 0.1698 | 1.2484 | 1.0368 | 1.5870 | 20.9453 | 1.9941 | 0.8213 | 0.0652 | 0.6456 |
MVSEM(0,1/2,1/2) | 0.1786 | 0.1724 | 0.2428 | 0.1697 | 1.2483 | 1.0369 | 1.5873 | 20.9437 | 1.9945 | 0.8214 | 0.0652 | 0.6449 |
MVSEM(1/3,1/3,1/3) | 0.1789 | 0.1726 | 0.2432 | 0.1701 | 1.2486 | 1.0373 | 1.5875 | 20.9460 | 1.9952 | 0.8216 | 0.0650 | 0.6455 |
Table 8.
The results of portfolio performance measures for ISE dataset and W = 120.
Table 8.
The results of portfolio performance measures for ISE dataset and W = 120.
Models | SR | ASR | MADR | SSR | FTR(0.5,2) | FTR(1.5,2) | FTR(1,1) | GRR(0.5,2) | GRR(1.5,2) | GRR(1,1) | PT | p-value |
---|
EWM | 0.3701 | 0.3605 | 0.4461 | 1.0448 | 0.1781 | 1.2098 | 2.3632 | 26.8589 | 3.6586 | 1.1685 | – | – |
MinVM | 0.3563 | 0.3472 | 0.4286 | 0.9790 | 0.1779 | 1.1624 | 2.2576 | 31.3438 | 4.4686 | 1.2872 | 0.1704 | 0.3994 |
MVM | 0.3892 | 0.3782 | 0.4577 | 1.1198 | 0.1797 | 1.2606 | 2.4595 | 31.9823 | 4.3659 | 1.2691 | 0.1954 | 0.6552 |
MVSM | 0.3759 | 0.3652 | 0.4517 | 1.0613 | 0.1759 | 1.2183 | 2.3907 | 29.6981 | 3.8523 | 1.1747 | 0.2108 | 0.5471 |
MVSEM(1/2,0,1/2) | 0.3910 | 0.3803 | 0.4721 | 1.1310 | 0.1848 | 1.2621 | 2.5055 | 28.5031 | 3.7766 | 1.2962 | 0.0993 | 0.8928 |
MVSEM(0,1/2,1/2) | 0.3897 | 0.3798 | 0.4711 | 1.1293 | 0.1846 | 1.2624 | 2.5019 | 28.4221 | 3.7844 | 1.2993 | 0.0982 | 0.8762 |
MVSEM(1/3,1/3,1/3) | 0.3915 | 0.3808 | 0.4727 | 1.1321 | 0.1857 | 1.2625 | 2.5060 | 28.5150 | 3.7781 | 1.2967 | 0.0990 | 0.8829 |
Table 9.
The results of portfolio performance measures for ISE dataset and W = 150.
Table 9.
The results of portfolio performance measures for ISE dataset and W = 150.
Models | SR | ASR | MADR | SSR | FTR(0.5,2) | FTR(1.5,2) | FTR(1,1) | GRR(0.5,2) | GRR(1.5,2) | GRR(1,1) | PT | p-value |
---|
EWM | 0.3962 | 0.3940 | 0.5038 | 2.3084 | 0.0317 | 1.1254 | 2.7379 | 77.8862 | 8.0211 | 1.6044 | – | – |
MinVM | 0.3867 | 0.3710 | 0.5191 | 1.9718 | 0.0300 | 0.9582 | 2.5791 | 24.7371 | 2.7496 | 0.9575 | 0.1761 | 0.4645 |
MVM | 0.4740 | 0.4645 | 0.6001 | 2.9144 | 0.0343 | 1.3085 | 3.2330 | 53.0313 | 4.9134 | 1.2229 | 0.1986 | 0.8267 |
MVSM | 0.4183 | 0.4129 | 0.5300 | 2.4419 | 0.0336 | 1.1570 | 2.7795 | 60.0778 | 5.4989 | 1.2898 | 0.1961 | 0.6347 |
MVSEM(1/2,0,1/2) | 0.4807 | 0.4673 | 0.6046 | 2.9134 | 0.0359 | 1.3220 | 3.2437 | 88.0680 | 9.5432 | 1.7751 | 0.1275 | 0.8477 |
MVSEM(0,1/2,1/2) | 0.4785 | 0.4649 | 0.6045 | 2.9239 | 0.0350 | 1.3140 | 3.2396 | 85.0866 | 9.2271 | 1.7431 | 0.1256 | 0.8308 |
MVSEM(1/3,1/3,1/3) | 0.4837 | 0.4683 | 0.6054 | 2.9421 | 0.0356 | 1.3316 | 3.2492 | 89.1761 | 9.6398 | 1.7799 | 0.1268 | 0.8466 |