A Novel Framework for Portfolio Selection Model Using Modified ANFIS and Fuzzy Sets
Abstract
:1. Introduction
1.1. Related Work
1.2. Concept of the Proposed Framework
1.2.1. Phase 1. Optimization of Newly Derived Parameters
1.2.2. Phase 2. Design of New Six-Layered Structure of ANFIS
1.2.3. Economic and Statistical Significance of Using the ANFIS Structure
- Statistical Significance: ANFIS structure is utilized for providing a prediction capability in portfolio selection. Since forecast of expected returns may be a desirable feature in view of the investor’s selections. This prediction scheme should be capable of accurately modeling the desired parameters based on existing data points. This can be achieved by means of an ANFIS structure that has a fuzzy inference system. Furthermore, situations where ANFIS with different data points are desirable for selective time duration, an adaptive structure needs to be employed. This adaptive nature is required for a different set of parameters.
- Economic Significance: Designing of efficient models for portfolio selection can be suitably crafted using the computational index represented by the output of the last node in the ANFIS structure. While obtaining the expected returns in case of multi-assets data, this index can be employed as a decision parameter. The expected returns would correspondingly alter in the view of the selected value of this index. Even though a nominal value of this index is employed while designing, this scheme could substantially alter the values of expected returns. These designs may also be employed for finding an optimal solution in the areas beyond financial applications. The investor would like to select an option that yields lower values of risk.
1.2.4. Advantageous of Using the Proposed Methodology
2. Overview of the Basic Mean-Variance Model
Combined_desired_value | Desired value of (Input). |
Parameters used while correlating with CVaR. | |
Parameters used while correlating with CVaR. | |
, | Nodes of Layer 2 (Modified ANFIS). |
Nodes of Layer 3 (Modified ANFIS). | |
Nodes of Layer 5 (Modified ANFIS). | |
, | Parameters used while correlating with CVaR. |
Weight associated with and . | |
The values of parameter of or . | |
Cost associated with or . | |
This term is representing loss incurred in the investment process. | |
, | Weight associated with and respectively. |
This is a control parameter that represents maximal weighted cost associated with . | |
Maximum value of risk. | |
Minimum value of risk. | |
, | Scaling parameters. |
,, | Cost coefficient used in calculating cost, . |
, , | Parameters used while correlating with CVaR. |
Lagrangian Function. | |
Sub parameters generated from using fuzzy sets. | |
Membership values from fuzzy sets associated with sub parameters . | |
Membership values of fuzzy sets = . | |
This parameter represents the total cost associated with parameters | |
Values used in minimization equation = . | |
, | Cost coefficients used in minimization problem. |
Input parameter used in ANFIS. | |
, | Minimum and maximum value of . |
Nodes of Layer 1 (Modified ANFIS). |
3. Proposed Novel Portfolio Selection Model Based on Costs Associated with and
3.1. Correlate Parameter with Parameter That Is Being Computed from the Basic Mean-Variance Model
3.2. Correlate Parameter with Conditional-Value-at-Risk (CVaR)
- (c)
- Combined_desired_value:
4. Generating Novel Sub-Parameters from Parameters βnew and Finding Optimal Values of Sub-Parameters Using Fuzzy Sets
- Fuzzy Set A is used when the value of parameter β_new lies in Category ACategory A: A0 ≤ βnew ≤ A25,
- Fuzzy Set B1 is used when the value of parameter lies in Category BCategory B: B20 ≤ βnew ≤ B25,
- Fuzzy Set B2 is used when the value of parameter lies in Category C
4.1. Description of Various Equations Used in Fuzzy Sets
4.1.1. Category A: Fuzzy Set-A
4.1.2. Category B: Fuzzy Set-B1
4.1.3. Category C: Fuzzy Set-
- (a)
- The values of cost-coefficients as used in the following equation:
- (b)
- The values of these coefficients are given in Table 4.
- (c)
- The specified value of .
4.2. Mathematical Modeling of Module for Computing Sub Parameters
4.3. Equations for Parameters (i = 1 to 8)
4.4. Description of Different Outputs of the Module for Sub-Parameters
5. Need for Design of a Model Using Fuzzy Logic
- , , ,
- Rule 1:
- Rule 2:
- Rule 3:
- Rule 4:
- Rule 5:
- Rule 6:
- Rule 7:
- Rule 8:
6. New Model Using Modifications in ANFIS: Six-Layered Structure
- Rule 1:
- Rule 2:
Proposed Modifications in Fifth Layer (Layer 5) for Modified ANFIS Using the Cuckoo Intelligence Algorithm
General Framework of the Modification
- Rule 1:
- Rule 2: .
7. Performance Analysis and Experimental Results
7.1. Discussion on Modifications of the Range Used within a Cuckoo Intelligence Algorithm (Layer 5 of Modified 6 Layered ANFIS)
- = 0.15 = 0.6,
- = 0.1 = 0.4,
- = 0.05 = 0.2.
7.2. Computational Results with Analysis of ANFIS
7.2.1. Comparison of Various Outputs Obtained by Changing Different Layers of ANFIS
7.2.2. Comparison of Values of Output Nodes of the Last Layer in Existing and Modified ANFIS
7.2.3. Economic Significance of the ANFIS Methodology for Additional Constraints
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Existing ANFIS (Number of Layers = 5) | Modified ANFIS (Number of Layers = 6) | ||
---|---|---|---|
Inputs = , | |||
S. No | Layer | Nodes of Existing ANFIS | Nodes of Modified ANFIS |
1. | First (Layer 1) | ||
2. | Second (Layer 2) | ||
3. | Third (Layer 3) | ||
4. | Fourth (Layer 4) | ||
5. | Fifth (Layer 5) | ||
6. | Last layer | None |
S. No | Portfolio Return | Portfolio Weights | |||
---|---|---|---|---|---|
Risk | Col.5 | Col.8 | Col.10 | ||
1 | 0.2572 | 0.1622 | 0.5630 | 0 | 0.4370 |
2 | 0.2775 | 0.1641 | 0.5005 | 0 | 0.4995 |
3 | 0.2979 | 0.1697 | 0.4379 | 0 | 0.5621 |
4 | 0.3183 | 0.1787 | 0.3754 | 0 | 0.6246 |
5 | 0.3387 | 0.1905 | 0.3128 | 0 | 0.6872 |
6 | 0.3590 | 0.2047 | 0.2502 | 0 | 0.7498 |
7 | 0.3794 | 0.2207 | 0.1554 | 0.0580 | 0.7866 |
8 | 0.3998 | 0.2376 | 0.0485 | 0.1376 | 0.8139 |
9 | 0.4202 | 0.2557 | 0 | 0.1124 | 0.8876 |
10 | 0.4405 | 0.2773 | 0 | 0 | 1.000 |
S. No | ||||
---|---|---|---|---|
1. | 1.0 | 0.0001 | 95.8799 | 70.5697 |
2. | 0.9 | 0.1 | 95.8799 | 70.5697 |
3. | 0.7 | 0.3 | 95.8800 | 70.5667 |
4. | 0.51 | 0.49 | 97.8193 | 68.4395 |
5. | 0.5 | 0.5 | 101.0353 | 64.9947 |
6. | 0.49 | 0.51 | 104.3293 | 61.5669 |
7. | 0.48 | 0.52 | 107.7027 | 58.1558 |
8. | 0.47 | 0.53 | 111.1578 | 54.7610 |
9. | 0.46 | 0.54 | 114.6992 | 51.3820 |
10. | 0.45 | 0.55 | 118.3202 | 48.0185 |
11. | 0.4 | 0.6 | 119.1606 | 47.2520 |
12. | 0.1 | 0.9 | 119.1606 | 47.2518 |
S. No | ||
---|---|---|
1 | 0.01 | 0.8 |
2 | 0.015 | 0.9 |
3 | 0.02 | 0.96 |
4 | 0.011 | 0.85 |
5 | 0.009 | 0.88 |
6 | 0.008 | 0.80 |
7 | 0.007 | 0.81 |
8 | 0.006 | 0.87 |
S. No | Category | Range of Values of βnew | Parameter BCi |
---|---|---|---|
1 | A | 70.56–70.56 | |
2 | A | 68.43–70.56 | |
3 | A | 61.56–64.99 | |
4 | B | 55.00–58.15 | |
5 | B | 51.52–54.76 | |
6 | B | 49.00–51.38 | |
7 | C | 47.25–48.01 | |
8 | C | 46.50–47.25 |
S. No. | Parameter | Si min | Si max |
---|---|---|---|
1 | 0.05 | 0.8 | |
2 | 0.21 | 0.21 | |
3 | 0.01 | 0.75 | |
4 | 0.15 | 0.4 | |
5 | 0.31 | 0.31 | |
6 | 0.1 | 0.4 | |
7 | 0.05 | 0.1 | |
8 | 0.05 | 0.15 |
S. No | |||||||||
---|---|---|---|---|---|---|---|---|---|
1 | 2.9 | 0.579 | 0.210 | 0.750 | 0.400 | 0.310 | 0.400 | 0.100 | 0.150 |
2 | 2.7 | 0.414 | 0.210 | 0.750 | 0.400 | 0.310 | 0.365 | 0.100 | 0.150 |
3 | 2.6 | 0.355 | 0.210 | 0.750 | 0.400 | 0.310 | 0.324 | 0.100 | 0.150 |
4 | 2.3 | 0.221 | 0.210 | 0.676 | 0.400 | 0.310 | 0.232 | 0.100 | 0.150 |
5 | 1.9 | 0.155 | 0.210 | 0.393 | 0.400 | 0.310 | 0.188 | 0.92 | 0.150 |
6 | 1.45 | 0.99 | 0.210 | 0.153 | 0.336 | 0.310 | 0.149 | 0.70 | 0.120 |
7 | 1.35 | 0.88 | 0.210 | 0.104 | 0.314 | 0.310 | 0.142 | 0.66 | 0.113 |
S. No | |||||||||
---|---|---|---|---|---|---|---|---|---|
1 | 2.9 | 62.51 | 69.5 | 70.57 | 49.0 | 54.56 | 58.15 | 46.5 | 48.01 |
2 | 2.7 | 63.22 | 69.5 | 70.57 | 49.0 | 54.56 | 57.87 | 46.5 | 48.01 |
3 | 2.6 | 63.47 | 69.5 | 70.57 | 49.0 | 54.56 | 57.55 | 46.5 | 48.01 |
4 | 2.3 | 64.05 | 69.5 | 70.46 | 49.0 | 54.56 | 56.82 | 46.5 | 48.01 |
5 | 1.9 | 64.33 | 69.5 | 70.06 | 49.0 | 54.56 | 56.48 | 46.5 | 48.01 |
6 | 1.45 | 64.57 | 69.5 | 69.71 | 49.38 | 54.56 | 56.17 | 46.72 | 47.82 |
7 | 1.35 | 64.61 | 69.5 | 69.64 | 49.51 | 54.56 | 56.12 | 46.75 | 47.82 |
S. No | Parameters | Lower Limit | Upper Limit | Coefficients | Optimized Values by Executing Cuckoo Intelligence | Scaled Optimized Values | Optimized Value of Fifth Layer (Layer 5) | |
---|---|---|---|---|---|---|---|---|
1. | 150 | 600 | 0.001562 | 338.8589 | 8.1945 | |||
7.92 | 0.3388589 | |||||||
300 | ||||||||
2. | 100 | 400 | 0.00194 | 333.7502 | ||||
7.85 | 0.3337502 | |||||||
320 | ||||||||
3. | 50 | 200 | 0.00482 | 127.3909 | ||||
7.97 | 0.1273909 | |||||||
329 |
S. No | Parameters | Lower Limit | Upper Limit | Coefficients | Optimized Values by Executing Cuckoo Intelligence | Scaled Optimized Values | Optimized Value of Fifth Layer (Layer 5) | |
---|---|---|---|---|---|---|---|---|
1. | 500 | 800 | 0.001562 | 529.7067 | 0.5297067 | 10.3241 | ||
7.92 | ||||||||
300 | ||||||||
2. | 200 | 600 | 0.00194 | 377.013 | 0.377013 | |||
7.85 | ||||||||
320 | ||||||||
3. | 50 | 300 | 0.00482 | 171.2219 | 0.1712219 | |||
7.97 | ||||||||
329 |
S. No | Coefficients Used in Modified Fifth Layer | Values |
---|---|---|
1. | 0.45335985 | |
2. | 0.27411745 | |
3. | 0.49140574 | |
4. | 0.34975271 | |
5. | 0.01405009 | |
6. | 0.02201897 |
Selected Layer in Existing and Modified ANFIS | Nodes Used in the Selected Layer | Chosen Values of the Selected Nodes |
---|---|---|
First layer (layer 1) | 0.3477 | |
0.1944 | ||
0.6811 | ||
0.8250 |
Selected Layer in Existing ANFIS | Nodes Used in the Selected Layer | Output |
---|---|---|
Second layer (Layer 2) | 0.2368 | |
0.1604 | ||
Third layer (Layer 3) | 0.5963 | |
0.4037 | ||
Fourth layer (layer 4) | 1154.7 | |
451.52 | ||
Last output layer (layer 5) | Output node of Existing ANFIS | 1606.3 |
Selected Layer in Modified ANFIS | Nodes Used in the Selected Layer | Output |
---|---|---|
Second layer (layer 2) | * | 0.4040 |
* | 0.4050 | |
Fifth layer (layer 5) | 967.1194 | |
559.8678 | ||
Last output layer (layer 6) | Output node of Modified ANFIS | 1527.0 |
Selected Layer in Modified ANFIS | Nodes Used in the Selected Layer | Output |
---|---|---|
Second layer (layer 2) | 0.2368 | |
0.1604 | ||
Fifth layer (layer 5) | 644.3709 | |
281.5149 | ||
Last output layer (layer 6) | Output node of Modified ANFIS | 925.8858 |
Selected Layer in Modified ANFIS | Nodes Used in the Selected layer | Output |
---|---|---|
Second layer (layer 2) | * | 0.4040 |
* | 0.4050 | |
Fifth layer (layer 5) | 539.6790 | |
349.0620 | ||
Last output layer (layer 6) | Output node of Modified ANFIS | 888.7410 |
S. No. | Particular Case | Selected Layers in Which Changes Are Incorporated | Value of Output Node of the Last Layer | ANFIS Structure Used |
---|---|---|---|---|
1. | Case 1 | No layer changed (Existing ANFIS) | 1606.3 | Existing ANFIS |
2. | Case 2 | Second layer (layer 2) | 1527.0 | Modified ANFIS |
3. | Case 3 | Fifth layer (layer 5) | 925.8858 | Modified ANFIS |
4. | Case 4 | Second and Fifth layer | 888.7410 | Modified ANFIS |
Portfolio Returns (r0) | Allocation | Portfolio Risk | |||||
---|---|---|---|---|---|---|---|
B1 | B2 | B3 | B4 | B5 | |||
PP1 | 0.2560 | - | - | - | - | 0.5628 | 0.1622 |
PP2 | 0.2786 | - | - | - | - | 0.5002 | 0.1641 |
PP3 | 0.3012 | - | - | - | - | 0.4377 | 0.1698 |
PP4 | 0.3238 | - | - | - | - | 0.3752 | 0.1788 |
PP5 | 0.3464 | - | - | - | - | 0.3126 | 0.1907 |
PP6 | 0.3690 | - | - | - | - | 0.2501 | 0.2050 |
PP7 | 0.3916 | - | - | - | - | 0.1876 | 0.2212 |
PP8 | 0.4142 | - | - | - | - | 0.1251 | 0.2390 |
PP9 | 0.4368 | - | - | - | - | 0.0438 | 0.2579 |
PP10 | 0.4594 | - | - | - | - | 0.0 | 0.2779 |
B6 | B7 | B8 | B9 | B10 | |||
- | - | - | - | 0.4372 | |||
- | - | - | - | 0.4998 | |||
- | - | - | - | 0.5623 | |||
- | - | - | - | 0.6248 | |||
- | - | - | - | 0.6874 | |||
- | - | - | - | 0.7499 | |||
- | - | - | - | 0.8124 | |||
- | - | - | - | 0.8749 | |||
- | - | - | - | 0.9259 | |||
- | - | - | - | 1.0 |
Portfolio Returns (r0) | Allocation | Portfolio Risk | |||||
---|---|---|---|---|---|---|---|
B1 | B2 | B3 | B4 | B5 | |||
PP1 | 0.2606 | - | - | - | - | 0.5639 | 0.1622 |
PP2 | 0.2833 | - | - | - | - | 0.5012 | 0.1642 |
PP3 | 0.3059 | - | - | - | - | 0.4386 | 0.1698 |
PP4 | 0.3286 | - | - | - | - | 0.3759 | 0.1789 |
PP5 | 0.3512 | - | - | - | - | 0.3133 | 0.1908 |
PP6 | 0.3739 | - | - | - | - | 0.2506 | 0.2052 |
PP7 | 0.3965 | - | - | - | - | 0.1880 | 0.2215 |
PP8 | 0.4192 | - | - | - | - | 0.1253 | 0.2393 |
PP9 | 0.4418 | - | - | - | - | 0.0411 | 0.2583 |
PP10 | 0.4645 | - | - | - | - | 0.0 | 0.2783 |
B6 | B7 | B8 | B9 | B10 | |||
- | - | - | - | 0.4361 | |||
- | - | - | - | 0.4988 | |||
- | - | - | - | 0.5614 | |||
- | - | - | - | 0.6241 | |||
- | - | - | - | 0.6887 | |||
- | - | - | - | 0.7494 | |||
- | - | - | - | 0.8120 | |||
- | - | - | - | 0.8747 | |||
- | - | - | - | 0.9240 | |||
- | - | - | - | 1.0 |
Portfolio Returns (r0) | Allocation | Portfolio Risk | |||||
---|---|---|---|---|---|---|---|
B1 | B2 | B3 | B4 | B5 | |||
PP1 | 0.2555 | - | - | - | - | 0.5626 | 0.1622 |
PP2 | 0.2781 | - | - | - | - | 0.5001 | 0.1641 |
PP3 | 0.3007 | - | - | - | - | 0.4376 | 0.1698 |
PP4 | 0.3233 | - | - | - | - | 0.3751 | 0.1788 |
PP5 | 0.3459 | - | - | - | - | 0.3126 | 0.1907 |
PP6 | 0.3685 | - | - | - | - | 0.2501 | 0.2050 |
PP7 | 0.3911 | - | - | - | - | 0.1875 | 0.2212 |
PP8 | 0.4137 | - | - | - | - | 0.1250 | 0.2390 |
PP9 | 0.4363 | - | - | - | - | 0.0441 | 0.2579 |
PP10 | 0.4589 | - | - | - | - | 0.0 | 0.2779 |
B6 | B7 | B8 | B9 | B10 | |||
- | - | - | - | 0.4374 | |||
- | - | - | - | 0.4999 | |||
- | - | - | - | 0.5624 | |||
- | - | - | - | 0.6249 | |||
- | - | - | - | 0.6874 | |||
- | - | - | - | 0.7499 | |||
- | - | - | - | 0.8125 | |||
- | - | - | - | 0.8750 | |||
- | - | - | - | 0.9261 | |||
- | - | - | - | 1.0 |
Portfolio | Expected Return with Existing ANFIS | Expected Return with Modified ANFIS |
---|---|---|
PP1 | 0.2555 | 0.2606 |
PP2 | 0.2781 | 0.2833 |
PP3 | 0.3007 | 0.3059 |
PP4 | 0.3233 | 0.3286 |
PP5 | 0.3459 | 0.3512 |
PP6 | 0.3685 | 0.3739 |
PP7 | 0.3911 | 0.3965 |
PP8 | 0.4137 | 0.4192 |
PP9 | 0.4363 | 0.4418 |
PP10 | 0.4589 | 0.4645 |
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Kumar, C.; Doja, M.N. A Novel Framework for Portfolio Selection Model Using Modified ANFIS and Fuzzy Sets. Computers 2018, 7, 57. https://doi.org/10.3390/computers7040057
Kumar C, Doja MN. A Novel Framework for Portfolio Selection Model Using Modified ANFIS and Fuzzy Sets. Computers. 2018; 7(4):57. https://doi.org/10.3390/computers7040057
Chicago/Turabian StyleKumar, Chanchal, and Mohammad Najmud Doja. 2018. "A Novel Framework for Portfolio Selection Model Using Modified ANFIS and Fuzzy Sets" Computers 7, no. 4: 57. https://doi.org/10.3390/computers7040057