Parameter’s Sensitivity Test

Order 3 Parameter

4.2 Parameter’s Sensitivity Test

Sensitivity testing on the distributions' parameters was carried out to determine the most significant parameter contributing to generating the maximum wealth.

For each portfolio, there was a different parameter that influenced the terminal wealth. In this section, an order 1 universal portfolio with 1500 trading days was generated to observe changes in the parameters on the wealth generated by the

1.0000 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000

0 94 188 282 376 470 564 658 752 846 940 1034 1128 1222 1316 1410

Wealth Achieved

Trading Day

Portfolio C (Order 2)

Pareto Loglogistic Paralogistic Burr

Transformmed Gamma

1.0000 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000

0 94 188 282 376 470 564 658 752 846 940 1034 1128 1222 1316 1410

Wealth Achieved

Trading Day

Portfolio C

Pareto Loglogistic Paralogistic Burr

Transformmed Gamma

portfolio. Each distribution's parameter value was tested by selecting a value from 1 to 3000. Tables 4.68, 4.69, and 4.70 show the highest wealth achieved by each distribution with the selected parameter value.

4.2.1 Pareto Distribution (𝜶_𝟏, 𝜶_𝟐, 𝜶_𝟑, 𝜽_𝟏, 𝜽_𝟐, 𝜽_𝟑)

Three parameters in Pareto distribution affect the wealth of Portfolio A. As the value of 𝛼₂, 𝛼₃ and 𝜃₁ increases, the wealth of Portfolio A generated will increase. While for 𝛼₁ , 𝜃₂ and 𝜃₃, they are set to be 50 as they are either need to be as small as possible or have not much effect on the wealth generated by the portfolio. From Figure 4.10, we observe that when 𝛼₂, 𝛼₃ and 𝜃₁increases together, and there will be a drastic change in the terminal wealth. The highest wealth achieved by Pareto distribution by changing these parameters is 2.3398, which we set the parameter to be (50, 100, 100, 100, 50, 50). From Table 4.15, when 𝜃₁ increases to 100, generating a wealth of 2.0308, the highest wealth compared to the other two parameters. Based on the empirical result, we found that 𝜃₂ and 𝜃₃ need to be small and 𝛼₂ has not much effect on the wealth generated by Portfolio A. Therefore, to obtain the highest wealth of the portfolio, we set 𝜃₂ and 𝜃₃ to be 1, and 𝛼₂ to be 5 which following the parameter condition of the Pareto distribution, 𝛼 needs to be more than 4 to generate order 3 universal portfolio. The highest wealth generated by Pareto distribution is 2.7708 within order 2 and order 3 with parameters (5, 170, 170, 170, 1, 1).

Figure 4.10: The Performance of the universal portfolio generated by the Pareto distribution by changing i) 𝛼₂, ii) 𝛼₃, iii) 𝜃₁, and iv) 𝛼₂, 𝛼₃, 𝜃₁ for Portfolio A.

For Portfolio B, two parameters will affect the wealth of Portfolio B. As the value of 𝛼₂ and 𝜃₃ increases, the wealth of Portfolio B generated will increase. While for 𝛼₁ , 𝛼₃, 𝜃₁ and 𝜃₂, they are set to be 50 as they are either need to be as small as possible or have not much effect on the wealth generated by the portfolio. From Figure 4.11, we observe that when 𝛼₂ and 𝜃₃ increases simultaneously; there will be a drastic change in the terminal wealth. The highest wealth achieved by Pareto distribution by changing these parameters is 5.1782, which we set the parameter to be (50, 100, 50, 50, 50, 100). Both 𝛼₂ and 𝜃₃ have almost the same influence on the wealth generated. Based on the result, we found that 𝛼₃, 𝜃₁ and 𝜃₂ need to be small and 𝛼₁ has not much effect on the wealth generated by Portfolio B. Therefore, to obtain the highest wealth of the portfolio, we set 𝛼₁ and 𝛼₃ to be 100 and 5 respectively, 𝜃₁ and 𝜃₂ to be 1. The highest wealth achieved by Pareto distribution for Portfolio B is 6.7857 with parameters (100, 100, 5, 1, 1, 100) for order 3 universal portfolio.

Figure 4.11: The Performance of the universal portfolio generated by the Pareto distribution by changing i) 𝛼₂, ii) 𝜃₃And iii) 𝛼₂, 𝜃₃ for Portfolio B.

For Portfolio C, two parameters will affect the wealth of Portfolio C. As the value of 𝛼₂ and 𝜃₃ increase, the wealth of Portfolio C generated will increase. While for 𝛼₁ , 𝛼₃, 𝜃₁ and 𝜃₂, they are set to be 50 as they are either need to be as small as possible or have not much effect on the wealth generated by the portfolio. From Figure 4.12, we observe that when 𝛼₂ and 𝜃₃ increases

simultaneously; there will be a drastic change in the terminal wealth. The highest wealth achieved by Pareto distribution by changing these parameters is 3.0705, which we set the parameter to be (50, 100, 50, 50, 50, 100). Both 𝛼₂ and 𝜃₃ have almost the same influence on the wealth generated. According to the analysis, we found that 𝛼₁, 𝜃₁ and 𝜃₂ need to be small and 𝛼₃ has not much effect on the wealth generated by Portfolio C. Therefore, to obtain the highest wealth of the portfolio, we set 𝛼₁ and 𝛼₃ to be 5 and 170 respectively, 𝜃₁ and 𝜃₂ to be 1. The highest wealth that we can generate for Portfolio C is within order 3 with (5, 170, 170, 1, 1, 100) as the parameter of Pareto distribution, which is 4.5595.

Figure 4.12: The Performance of the universal portfolio generated by the Pareto distribution by changing i) 𝛼₂, ii) 𝜃₃, and iii) 𝛼₂, 𝜃₃ for Portfolio C.

Table 4.13: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝛼₂ value for Portfolio A.

Portfolio 𝜶_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3345 0.3320 0.3334 1.7284 60 0.3594 0.2824 0.3582 1.7406 70 0.3783 0.2447 0.3770 1.7496 80 0.3930 0.2153 0.3916 1.7564 90 0.4048 0.1919 0.4033 1.7617 100 0.4143 0.1729 0.4128 1.7659

Table 4.14: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝛼₃ value for Portfolio A.

Portfolio 𝜶_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3345 0.3320 0.3334 1.7284 60 0.3596 0.3568 0.2836 1.7834 70 0.3787 0.3756 0.2457 1.8260 80 0.3935 0.3903 0.2162 1.8597 90 0.4054 0.4019 0.1927 1.8868 100 0.4150 0.4114 0.1736 1.9091

Table 4.15: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝜃₁ value for Portfolio A.

Portfolio 𝜽_𝟏 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3345 0.3320 0.3334 1.7284

60 0.3876 0.3056 0.3068 1.8006

70 0.4350 0.2819 0.2830 1.8667

80 0.4773 0.2609 0.2619 1.9268

90 0.5148 0.2422 0.2430 1.9814

100 0.5482 0.2255 0.2263 2.0308

Table 4.16: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝛼₂, 𝛼₃ and 𝜃₁ value for Portfolio A.

Portfolio 𝜶_𝟐 𝜶_𝟑 𝜽_𝟏 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 50 50 0.3345 0.3320 0.3334 1.7284 60 60 60 0.4453 0.2768 0.2779 1.8812 70 70 70 0.5443 0.2275 0.2282 2.0250 80 80 80 0.6271 0.1862 0.1868 2.1504 90 90 90 0.6938 0.1529 0.1533 2.2549 100 100 100 0.7466 0.1265 0.1268 2.3398 Table 4.17: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝛼₂ value for Portfolio B.

Portfolio 𝜶_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3313 0.3338 0.3348 3.5574

60 0.3561 0.2839 0.3600 3.7674

70 0.3749 0.2460 0.3791 3.9330

80 0.3895 0.2165 0.3940 4.0657

90 0.4012 0.1929 0.4059 4.1738

100 0.4107 0.1737 0.4156 4.2634

Table 4.18: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝜃₃ value for Portfolio B.

Porforlio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3313 0.3338 0.3348 3.5574

60 0.3049 0.3071 0.3879 3.7704

70 0.2814 0.2833 0.4354 3.9676

80 0.2604 0.2621 0.4776 4.1486

90 0.2417 0.2432 0.5151 4.3140

100 0.2250 0.2264 0.5486 4.4646

Table 4.19: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝛼₂ and 𝜃₃ value for Portfolio B.

Portfolio 𝜶_𝟐 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 50 0.3313 0.3338 0.3348 3.5574 60 60 0.3253 0.2599 0.4148 3.9897 70 70 0.3106 0.2055 0.4839 4.3660 80 80 0.2924 0.1652 0.5424 4.6857 90 90 0.2733 0.1349 0.5918 4.9541 100 100 0.2547 0.1119 0.6334 5.1782 Table 4.20: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝛼₂ value for Portfolio C.

Portfolio 𝜶_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3359 0.3333 0.3308 2.6276

60 0.3611 0.2835 0.3554 2.7211

70 0.3802 0.2456 0.3741 2.7929

80 0.3952 0.2162 0.3887 2.8492

90 0.4071 0.1926 0.4003 2.8944

100 0.4168 0.1735 0.4097 2.9313

Table 4.21: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝜃₃ value for Portfolio C.

Portfolio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3359 0.3333 0.3308 2.6276

60 0.3094 0.3071 0.3835 2.6953

70 0.2856 0.2837 0.4307 2.7515

80 0.2645 0.2627 0.4728 2.7980

90 0.2456 0.2441 0.5103 2.8364

100 0.2288 0.2274 0.5438 2.8681

Table 4.22: The performance of Order 1 universal portfolio, which generated by the Pareto distribution with selected 𝛼₂ and 𝜃₃ value for Portfolio C.

Portfolio 𝜶_𝟐 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 50 0.3359 0.3333 0.3308 2.6276

60 60 0.3302 0.2599 0.4099 2.7844

70 70 0.3157 0.2059 0.4784 2.8977

80 80 0.2975 0.1657 0.5368 2.9775

90 90 0.2783 0.1355 0.5862 3.0327

100 100 0.2596 0.1124 0.6280 3.0705 4.2.2 Loglogistic Distribution (𝜸_𝟏, 𝜸_𝟐, 𝜸_𝟑, 𝜽_𝟏, 𝜽_𝟐, 𝜽_𝟑)

Four parameters will affect the wealth of Portfolio A. As the value of 𝜃₁ increases and the value of 𝛾₁, 𝜃₂ and 𝜃₃ decreases, the wealth of Portfolio A generated will increase. While for 𝛾₂ and 𝛾₃, they are set to be 50 as they are either need to be as small as possible or have not much effect on the wealth generated by the portfolio. From Figure 4.13, we observe that when 𝛾₁, 𝜃₂ and, 𝜃₃ decreases simultaneously; there will be a drastic change in the terminal wealth. The highest wealth achieved by Loglogistic distribution by changing these parameters is 2.4939, which we set the parameter to be (10, 50, 50, 100, 10, 10). From Table 4.23, when 𝜃₁ increases to 100; it can generate a wealth of 1.9617, the highest wealth compared to changing the other three parameters. Therefore, it indicates that 𝜃₁ has more influence on the wealth generated by Portfolio A. Based on the result obtained, we found that 𝛾₂ and 𝛾₃ has not much effect on the wealth generated by Portfolio A. Therefore, to obtain the highest wealth of the portfolio, we set 𝛾₂ and 𝛾₃ to be 100. The highest wealth generated by Loglogistic distribution is 2.7544 within order 3 with parameters (5, 100, 100, 100, 1, 1).

Figure 4.13: The Performance of the universal portfolio generated by the Loglogistic distribution by changing i) 𝜃₃, ii) 𝛾₁, iii) 𝜃₂And iv) 𝛾₁, 𝜃₂, 𝜃₃ for Portfolio A.

Those are four parameters that will affect the wealth of Portfolio B. As the value of 𝛾₂ and 𝜃₃ increases and the value of 𝛾₃ and 𝜃₂ decreases, the wealth of Portfolio B generated will increase. While for 𝛾₁ and 𝜃₁, they are set to be 50 as they are either need to be as small as possible or have not much effect on the wealth generated by the portfolio. From Figure 4.14, we observe that when 𝛾₃ and 𝜃₂ decreases together; there will be a drastic change in the terminal wealth. The highest wealth achieved by Loglogistic distribution by changing these parameters is 5.3998, which we set the parameter to be (50, 100, 10, 50, 10, 100). From Table 4.26, when 𝜃₃ decreases to 10; it can generate a wealth of 2.0070, the highest wealth generated by changing the other three parameters. Therefore, it indicates that 𝜃₃ has more influence on the wealth generated by Portfolio B. According to the analysis, we found that 𝛾₁ and 𝜃₁ has not much effect on the wealth generated by Portfolio B. Therefore, to obtain the highest wealth of the portfolio, we set 𝛾₁ and 𝜃₁ to be 5 and 1, respectively. The highest wealth generated by Loglogistic distribution is 6.7328 within order 3 with parameters (5, 100, 5, 1, 1, 100).

Figure 4.14: The Performance of the universal portfolio generated by the Loglogistic distribution by changing i) 𝛾₃, ii) 𝜃₂And iii) 𝛾₃, 𝜃₂ for Portfolio B.

Four parameters will affect the wealth of Portfolio C. As the value of 𝜃₁ and 𝜃₃ increases and the value of 𝛾₃ and 𝜃₂ decreases, the wealth of Portfolio C generated will increase. While for 𝛾₁ and 𝛾₂, they are set to be 50 as they have not much effect on the wealth generated by the portfolio. From Figure 4.15, we observe that when 𝛾₃ and 𝜃₂ decreases simultaneously; there will be a drastic change in the terminal wealth. The highest wealth achieved by Loglogistic distribution by changing these parameters is 3.1713, which we set the parameter to be (50, 50, 10, 100, 10, 100). From Table 4.31, when 𝜃₂ decrease to 10; it can generate a wealth of 3.0831, the highest wealth generated by changing the other three parameters. Therefore, it indicates that 𝜃₂ has more influence on the wealth generated by the Portfolio C. Hence, the highest wealth can be achieved by setting 𝛾₁ and 𝛾₂ to be 100 respectively for portfolio C, which is 3.2619 within order 3 with parameters (100, 100, 5, 100, 1, 100).

Figure 4.15: The Performance of the universal portfolio generated by the Loglogistic distribution by changing i) 𝛾₃, ii) 𝜃₂, and iii) 𝛾₃, 𝜃₂ for Portfolio C.

Table 4.23: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₁ value for Portfolio A.

Portfolio 𝜽_𝟏 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3333 0.3333 0.3333 1.7281 60 0.3750 0.3125 0.3125 1.7847 70 0.4118 0.2941 0.2941 1.8357 80 0.4445 0.2777 0.2778 1.8817 90 0.4738 0.2631 0.2631 1.9236 100 0.5001 0.2500 0.2500 1.9617

Table 4.24: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝛾₁ value for Portfolio A.

Portfolio 𝜸_𝟏 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3333 0.3333 0.3333 1.7281

40 0.3335 0.3333 0.3333 1.7283

30 0.3338 0.3331 0.3331 1.7287

20 0.3346 0.3327 0.3327 1.7299

10 0.3394 0.3303 0.3303 1.7362

Table 4.25: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₂ value for Portfolio A.

Portfolio 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3333 0.3333 0.3333 1.7281

40 0.3572 0.2857 0.3572 1.7398

30 0.3846 0.2307 0.3846 1.7528

20 0.4167 0.1666 0.4167 1.7671

10 0.4546 0.0909 0.4546 1.7828

Table 4.26: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₃ value for Portfolio A.

Portfolio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3333 0.3333 0.3333 1.7281

40 0.3572 0.3571 0.2857 1.7807

30 0.3846 0.3846 0.2307 1.8428

20 0.4167 0.4167 0.1666 1.9169

10 0.4546 0.4546 0.0909 2.0070

Table 4.27: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝛾₁, 𝜃₂ and 𝜃₃ value for Portfolio A.

Portfolio 𝜸_𝟏 𝜽_𝟐 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 50 50 0.3333 1.7281 1.7281 1.7281 40 40 40 0.5559 0.2221 0.2221 2.0440 30 30 30 0.6258 0.1871 0.1871 2.1501 20 20 20 0.7162 0.1419 0.1419 2.2921 10 10 10 0.8394 0.0803 0.0803 2.4939 Table 4.28: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝛾₂ value for Portfolio B.

Portfolio 𝜸_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3333 0.3333 0.3333 3.5515

60 0.3334 0.3333 0.3334 3.5518

70 0.3334 0.3332 0.3334 3.5520

80 0.3334 0.3332 0.3334 3.5521

90 0.3334 0.3332 0.3334 3.5522

100 0.3334 0.3332 0.3334 3.5522

Table 4.29: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₃ value for Portfolio B.

Portfolio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3333 0.3333 0.3333 3.5515 60 0.3125 0.3125 0.3750 3.7187 70 0.2941 0.2941 0.4118 3.8705 80 0.2777 0.2778 0.4445 4.0086 90 0.2631 0.2631 0.4738 4.1348 100 0.2500 0.2500 0.5001 4.2505

Table 4.30: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝛾₃ value for Portfolio B.

Portfolio 𝜸_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3333 0.3333 0.3333 3.5515 40 0.3333 0.3333 0.3335 3.5520 30 0.3331 0.3331 0.3338 3.5532 20 0.3327 0.3327 0.3346 3.5566 10 0.3303 0.3303 0.3394 3.5752

Table 4.31 The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₂ value for Portfolio B.

Portfolio 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3333 0.3333 0.3333 3.5515 40 0.3571 0.2857 0.3572 3.7514 30 0.3846 0.2307 0.3846 3.9923 20 0.4167 0.1666 0.4167 4.2873 10 0.4546 0.0909 0.4546 4.6558

Table 4.32: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝛾₃ and 𝜃₂ value for Portfolio B.

Portfolio 𝜸_𝟑 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 50 0.3333 0.3333 0.3333 3.5515 40 40 0.2630 0.2103 0.5267 4.4691 30 30 0.2774 0.1663 0.5563 4.7227 20 20 0.2928 0.1170 0.5902 5.0230 10 10 0.3053 0.0610 0.6337 5.3998

Table 4.33: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₁ value for Portfolio C.

Portfolio 𝜽_𝟏 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3333 0.3333 0.3333 2.6217

60 0.3750 0.3125 0.3125 2.6439

70 0.4118 0.2941 0.2941 2.6621

80 0.4445 0.2777 0.2777 2.6772

90 0.4738 0.2631 0.2631 2.6899

100 0.5001 0.2500 0.2500 2.7007

Table 4.34: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₃ value for Portfolio C.

Portfolio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3333 0.3333 0.3333 2.6217

60 0.3125 0.3125 0.3750 2.6748

70 0.2941 0.2941 0.4118 2.7192

80 0.2778 0.2778 0.4445 2.7567

90 0.2631 0.2631 0.4737 2.7885

100 0.2500 0.2500 0.5001 2.8157

Table 4.35: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝛾₃ value for Portfolio C.

Portfolio 𝜸_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3333 0.3333 0.3333 2.6217

40 0.3333 0.3333 0.3335 2.6219

30 0.3331 0.3331 0.3338 2.6223

20 0.3327 0.3327 0.3346 2.6235

10 0.3304 0.3304 0.3393 2.6297

Table 4.36: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝜃₂ value for Portfolio C.

Portfolio 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3333 0.3333 0.3333 2.6217

40 0.3572 0.2857 0.3571 2.7106

30 0.3846 0.2307 0.3846 2.8144

20 0.4167 0.1666 0.4167 2.9368

10 0.4546 0.0909 0.4546 3.0831

Table 4.37: The performance of Order 1 universal portfolio, which generated by the Loglogistic distribution with selected 𝛾₃ and 𝜃₂ value for Portfolio C.

Portfolio 𝜸_𝟑 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 50 0.3333 0.3333 0.3333 2.6217 40 40 0.4166 0.1666 0.4168 2.9370 30 30 0.4344 0.1303 0.4353 3.0070 20 20 0.4532 0.0906 0.4562 3.0842 10 10 0.4692 0.0469 0.4840 3.1713 4.2.3 Paralogistic distribution (𝜶_𝟏, 𝜶_𝟐, 𝜶_𝟑, 𝜽_𝟏, 𝜽_𝟐, 𝜽_𝟑)

Three parameters will affect the wealth of Portfolio A. As the value of 𝜃₁ increases and the value of 𝜃₂ and 𝜃₃ decreases, the wealth of Portfolio A generated will increase. While for 𝛼₁, 𝛼₂ and 𝛼₃, they are set to be 50 as they have not much effect on the wealth generated by the portfolio. From Figure 4.16, we observe that when 𝜃₂ and, 𝜃₃ decreases together; there will be a drastic change in the terminal wealth. The highest wealth achieved by Paralogistic distribution by changing these parameters is 2.4839, which we set the parameter to be (50, 50, 50, 100, 10, 10). From Table 4.40, when 𝜃₃ increases to 10; it can generate a wealth of 2.2361, which is the highest wealth compared to changing the other two parameters. Therefore, it indicates that 𝜃₃ has more influence on the wealth generated by Portfolio A. Therefore, to obtain the highest wealth of the portfolio, we set 𝛼₁, 𝛼₂ and 𝛼₃ to be 4. The highest wealth that can generate by Paralogistic distribution is 2.7702 within order 2 with parameters (4, 4, 4, 3500, 1, 1).

Figure 4.16: The Performance of the universal portfolio generated by the Paralogistic distribution by changing i) 𝜃₂, ii) 𝜃₃ and iii) 𝜃₂, 𝜃₃ for Portfolio A.

Three parameters will affect the wealth of Portfolio B. As the value of 𝜃₃ increases and the value of 𝜃₁ and 𝜃₂ decreases, the wealth of Portfolio B generated will increase. While for 𝛼₁, 𝛼₂ and 𝛼₃, they are set to be 50 as they have not much effect on the wealth generated by the portfolio. From Figure 4.17, we observe that when 𝜃₁ and 𝜃₂ decreases simultaneously; there will be a drastic change in the terminal wealth. The highest wealth achieved by Paralogistic distribution by changing these parameters is 2.4839, which we set the parameter to be (50, 50, 50, 10, 10, 100). From Table 4.43, we noticed that when 𝜃₁ decreases to 10 by itself; the terminal wealth will decrease. Therefore, it indicates that 𝜃₁ and 𝜃₂ need to decrease together to increase the wealth generated by the Portfolio B. Therefore, to obtain the highest wealth of the portfolio, we set 𝛼₁, 𝛼₂ and 𝛼₃ to be 4. The highest wealth generated by Paralogistic distribution is 6.7710 within order 2 with parameters (4, 4, 4, 1, 1, 500).

Figure 4.17: The Performance of the universal portfolio generated by the Paralogistic distribution by changing i) 𝜃₁ and ii) 𝜃₁, 𝜃₂ for Portfolio B.

Three parameters will affect the wealth of Portfolio C. As the value of 𝜃₃ increases and the value of 𝛼₂ and 𝜃₂ decreases, the wealth of Portfolio C generated will increase. While for 𝛼₁, 𝛼₃ and 𝜃₁, they are set to be 50 as they have not much effect on the wealth generated by the portfolio. From Figure 4.18, we observe that when 𝛼₂ and 𝜃₂ decreases simultaneously; there will be a

drastic change in the terminal wealth. The highest wealth achieved by Loglogistic distribution by changing these parameters is 3.1737, which we set the parameter to be (50, 10, 50, 50, 10, 100). From Table 4.47, when 𝜃₂ decreases to 10, it can generate a wealth of 3.1559, which is comparable to the wealth generated by decreasing both 𝛼₂ and 𝜃₂ together. Therefore, it indicates that 𝜃₂ has more influence on the wealth generated by the Portfolio C. To obtain the highest wealth of the portfolio, we set 𝛼₁, 𝛼₃ and 𝜃₁ to be 100 respectively.

The highest wealth generated by Paralogistic distribution is 3.2598 within order 2 with parameters (5, 5, 150, 100, 1, 100).

Figure 4.18: The Performance of the universal portfolio generated by the Paralogistic distribution by changing i) 𝛼₂, ii) 𝜃₂and iii) 𝛼₂, 𝜃₂ for Portfolio C.

Table 4.38: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₁ value for Portfolio A.

Portfolio 𝜽_𝟏 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3333 0.3333 0.3333 1.7281

60 0.3750 0.3125 0.3125 1.7847

70 0.4118 0.2941 0.2941 1.8357

80 0.4445 0.2778 0.2778 1.8817

90 0.4737 0.2631 0.2631 1.9235

100 0.5000 0.2500 0.2500 1.9616

Table 4.39: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₂ value for Portfolio A.

Portfolio 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3333 0.3333 0.3333 1.7281

40 0.5264 0.2105 0.2631 1.9830

30 0.5556 0.1666 0.2778 2.0065

20 0.5883 0.1176 0.2941 2.0323

10 0.6251 0.0625 0.3125 2.0608

Table 4.40: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₃ value for Portfolio A.

Portfolio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 0.3333 0.3333 0.3333 1.7281

40 0.5264 0.2631 0.2105 2.0174

30 0.5556 0.2778 0.1666 2.0806

20 0.5883 0.2941 0.1176 2.1528

10 0.6251 0.3125 0.0625 2.2361

Table 4.41: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₂ and 𝜃₃ value for Portfolio A.

Portfolio 𝜽_𝟐 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

A

50 50 0.3333 0.3333 1.7281 1.7281 40 40 0.5556 0.2222 0.2222 2.0437 30 30 0.6251 0.1875 0.1875 2.1491 20 20 0.7144 0.1428 0.1428 2.2892 10 10 0.8334 0.0833 0.0833 2.4839 Table 4.42: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₃ value for Portfolio B.

Portfolio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3333 0.3333 0.3333 3.5515

60 0.3125 0.3125 0.3750 3.7187

70 0.2941 0.2941 0.4118 3.8704

80 0.2778 0.2778 0.4445 4.0085

90 0.2631 0.2631 0.4737 4.1347

100 0.2500 0.2500 0.5000 4.2503

Table 4.43: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₁ value for Portfolio B.

Portfolio 𝜽_𝟏 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3333 0.3333 0.3333 3.5515

40 0.2857 0.3571 0.3571 3.5405

30 0.2308 0.3846 0.3846 3.5194

20 0.1666 0.4167 0.4167 3.4836

10 0.0909 0.4546 0.4546 3.4262

Table 4.44: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₂ value for Portfolio B.

Portfolio 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 0.3333 0.3333 0.3333 3.5515

40 0.3571 0.2857 0.3571 3.7514

30 0.3846 0.2308 0.3846 3.9922

20 0.4167 0.1666 0.4167 4.2872

10 0.4546 0.0909 0.4546 4.6557

Table 4.45: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₁ and 𝜃₂ value for Portfolio B.

Portfolio 𝜽_𝟏 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

B

50 50 0.3333 0.3333 0.3333 3.5515 40 40 0.2222 0.2222 0.5556 4.5007 30 30 0.1875 0.1875 0.6251 4.8257 20 20 0.1428 0.1428 0.7144 5.2623 10 10 0.0833 0.0833 0.8334 5.8753

Table 4.46: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₃ value for Portfolio C.

Portfolio 𝜽_𝟑 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3333 0.3333 0.3333 2.6217

60 0.3125 0.3125 0.3750 2.6748

70 0.2941 0.2941 0.4118 2.7192

80 0.2778 0.2778 0.4445 2.7566

90 0.2631 0.2631 0.4737 2.7884

100 0.2500 0.2500 0.5000 2.8157

Table 4.47: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝜃₂ value for Portfolio C.

Portfolio 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3333 0.3333 0.3333 2.6217

40 0.2631 0.2105 0.5264 2.8879

30 0.2778 0.1666 0.5556 2.9679

20 0.2941 0.1176 0.5883 3.0568

10 0.3125 0.0625 0.6251 3.1559

Table 4.48: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝛼₂ value for Portfolio C.

Portfolio 𝜶_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 0.3333 0.3333 0.3333 2.6217

40 0.2510 0.2469 0.5021 2.8213

30 0.2526 0.2423 0.5052 2.8298

20 0.2552 0.2343 0.5105 2.8443

10 0.2608 0.2174 0.5217 2.8753

Table 4.49: The performance of Order 1 universal portfolio, which generated by the Paralogistic distribution with selected 𝛼₂ and 𝜃₂ value for Portfolio C.

Portfolio 𝜶_𝟐 𝜽_𝟐 b1500, 1 b1500, 2 b1500, 3 Terminal Wealth, S1500

C

50 50 0.3333 0.3333 0.3333 2.6217

40 40 0.2640 0.2078 0.5282 2.8929

30 30 0.2797 0.1609 0.5594 2.9783

20 20 0.2970 0.1090 0.5940 3.0723

10 10 0.3158 0.0525 0.6317 3.1737

4.2.4 Burr Distribution (𝜶_𝟏, 𝜶_𝟐, 𝜶_𝟑, 𝜽_𝟏, 𝜽_𝟐, 𝜽_𝟑, 𝜸_𝟏, 𝜸_𝟐, 𝜸_𝟑)

Three parameters will affect the wealth of Portfolio A. As the value of 𝜃₁ increases and the value of 𝜃₂ and 𝜃₃ decreases, the wealth of Portfolio A generated will increase. While for 𝛼₁, 𝛼₂, 𝛼₃, 𝛾₁, 𝛾₂ and 𝛾₃ , they are set to be 50 as they have not much effect on the wealth generated by the portfolio. The highest wealth achieved by Burr distribution by decreasing 𝜃₂ and 𝜃₃ together is 2.4839, which we set the parameter to be (50, 50, 50, 100, 10, 10, 50, 50, 50).

We also observed that when 𝜃₃ decreases to 10, generating a wealth of 2.2361, which is the highest wealth compared to changing the other two parameters.

Therefore, it indicates that 𝜃₃ has more influence on the wealth generated by Portfolio A. Therefore, to obtain the highest wealth of the portfolio, we set

𝛼₁, 𝛼₂, 𝛼₃ and 𝛾₁ to be 1 while 𝛾₂ and 𝛾₃ to be 100. The highest wealth generated by Burr distribution is 2.7475 within order 2 with parameters (1, 1, 1, 100, 1, 1, 5, 100, 100).

Figure 4.19: The Performance of the universal portfolio generated by the Burr distribution by changing i) 𝜃₂, ii) 𝜃₃ and iii) 𝜃₂, 𝜃₃ for Portfolio A.

Four parameters will affect the wealth of Portfolio B. As the value of 𝜃₃ and 𝛾₃ increases and the value of 𝜃₁ and 𝜃₂ decreases, the wealth of Portfolio B generated will increase. While for 𝛼₁, 𝛼₂, 𝛼₃, 𝛾₁ and 𝛾₂ they are set to be 50 as they do not affect the wealth generated by the portfolio. From Figure 4.20, we observe that when 𝜃₃ and 𝛾₃ increases together, the highest wealth achieved by Burr distribution is 4.2995, which we set the parameter to be (50, 50, 50, 50, 50, 100, 50, 50, 100). From Figure 4.21, we observe that when 𝜃₁ and 𝜃₂ decrease together, the highest wealth achieved by Burr distribution

is 5.9073, which we set the parameter to

be (50, 50, 50, 10, 10, 100, 50, 50, 100). From Table 4.58, we noticed that when 𝜃₂ decreases to 10 by itself; it will generate a wealth of 5.4067, which is the highest compare to by changing the other parameters. Therefore, it indicates that 𝜃₂ is sensitive in affecting the wealth of Portfolio B. Therefore, to obtain the highest wealth of the portfolio, we set 𝛼₁, 𝛼₂ and 𝛼₃ to be 1 while 𝛾₁ and 𝛾₂ to be 5. The highest wealth that able to generate by Burr distribution is 6.7621 within order 1 with parameters (1, 1, 1, 1, 1, 1, 500, 5, 5, 100).

Figure 4.20: The Performance of the universal portfolio generated by the Burr distribution by changing i) 𝜃₃, ii) 𝛾₃ and iii) 𝜃₃, 𝛾₃ for Portfolio B.

Figure 4.21: The Performance of the universal portfolio generated by the Burr distribution by changing i) 𝜃₁, ii) 𝜃₂ and iii) 𝜃₁, 𝜃₂ for Portfolio B.

Four parameters will affect the wealth of Portfolio C. As the value of 𝜃₁ and 𝜃₃ increases and the value of 𝜃₂ and 𝛾₂ decreases, the wealth of Portfolio C generated will increase. While for 𝛼₁, 𝛼₂, 𝛼₃, 𝛾₁ and 𝛾₃ , they are set to be 50 as they have not much effect on the wealth generated by the portfolio. From Table 4.62, we observe that when increasing both 𝜃₁ and 𝜃₃, the highest wealth achieved by Burr distribution is 2.8729, which we set the parameter to be (50,

50, 50, 100, 50, 100, 50,50, 50). From Table 4.56, we observe that when 𝜃₂ and 𝛾₂ decreases together, the highest wealth achieved by Burr distribution is 3.1937, which we set the parameter to be (50, 50, 50, 100, 10, 100, 50, 10, 50).

From Table 4.63, when 𝜃₂ decreases to 10; it can generate a wealth of 3.1672, which is higher than changing the other parameters. Therefore, it indicates that 𝜃₂ has more influence on the wealth generated by the Portfolio C. To obtain the highest wealth of the portfolio, we set 𝛼₁, 𝛼₂ and 𝛼₃ to be 150 while 𝛾₁ and 𝛾₃ to be 100. The highest wealth generated by Burr distribution is 3.2584 within order 1 with parameters (150, 150, 150, 200, 1, 200, 100, 5, 100).

Figure 4.22: The Performance of the universal portfolio generated by the Burr distribution by changing i) 𝜃₁, ii) 𝜃₃ and iii) 𝜃₁, 𝜃₃ for Portfolio C.