Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty
Abstract
1. Introduction
2. Methods
2.1. Asymmetric Fractality of the ETF Price Series
2.2. Recurrent Neural Network Group
2.3. Black–Litterman Portfolio with Asymmetric Fractality
Algorithm 1 Black–Litterman portfolio with asymmetric fractality |
Input: price series of ith ETF Output: portfolio weight with 10 ETFs
|
3. Experiments and Data
3.1. Experiments
3.2. Data
4. Results
4.1. Prediction Results
4.2. Portfolio Evaluation
5. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
BL | Black–Litterman (portfolio) |
ETF | exchange-traded fund |
EMH | efficient market hypothesis |
MFDFA | multifractal detrended fluctuation analysis |
A-MFDFA | asymmetric multifractal detrended fluctuation analysis |
RNN | recurrent neural network |
LSTM | long short-term memory |
GRU | gated recurrent unit |
BiLSTM | bidirectional long short-term memory |
BiGRU | bidirectional gated recurrent unit |
MAE | mean absolute error |
RMSE | root mean squared error |
MDD | maximum drawdown |
Appendix A
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Hyperparameter | Search Space |
---|---|
Number of layer | [1, 2, 3] |
Number of neurons | [32, 64, 128, 256, 512] |
Dropout rate | [0, 0.1, 0.2, 0.3, 0.4, 0.5] |
Activation function | [tanh, ReLU] |
Optimizer | Adam |
Batch size | [32, 64] |
Epochs | 100 |
Ticker | MSCI Country ETFs |
---|---|
EWA | iShares MSCI Australia |
EWC | iShares MSCI Canada |
EWG | iShares MSCI Germany |
EWJ | iShares MSCI Japan |
EWT | iShares MSCI Taiwan |
EWU | iShares MSCI United Kingdom |
EWW | iShares MSCI Mexico |
EWY | iShares MSCI South Korea |
EWZ | iShares MSCI Brazil |
EZA | iShares MSCI South Africa |
ETF | Mean | Max | Min | Standard Deviation | Skewness | Kurtosis | Jarque– Bera Test | ADF Test 1 |
---|---|---|---|---|---|---|---|---|
EWA | 0.0003 | 0.2075 | −0.1611 | 0.0171 | −0.03 | 13.82 | 42,664.9 * | −18.4 * |
EWC | 0.0003 | 0.1286 | −0.1332 | 0.0138 | −0.45 | 11.30 | 28,715.1 * | −14.2 * |
EWG | 0.0004 | 0.1979 | −0.1269 | 0.0159 | 0.05 | 11.73 | 30,765.2 * | −16.2 * |
EWJ | 0.0003 | 0.1582 | −0.1041 | 0.0129 | 0.18 | 10.37 | 24,072.8 * | −17.8 * |
EWT | 0.0003 | 0.1416 | −0.1563 | 0.0164 | −0.17 | 8.86 | 17,574.6 * | −14.9 * |
EWU | 0.0002 | 0.1706 | −0.1202 | 0.0142 | −0.25 | 14.61 | 47,735.4 * | −16.6 * |
EWW | 0.0005 | 0.2149 | −0.1525 | 0.0171 | 0.06 | 10.51 | 24,689.1 * | −16.2 * |
EWY | 0.0004 | 0.2242 | −0.1581 | 0.0192 | 0.59 | 15.77 | 55,843.7 * | −16.3 * |
EWZ | 0.0005 | 0.2558 | −0.2309 | 0.0237 | −0.24 | 9.87 | 21,836.5 * | −17.1 * |
EZA | 0.0004 | 0.2292 | −0.2008 | 0.0213 | −0.07 | 9.14 | 18,669.6 * | −17.0 * |
Ticker | LSTM | BiLSTM | GRU | BiGRU |
---|---|---|---|---|
EWA | 0.4446 | 0.4056 | 0.3024 | 0.4830 |
EWC | 0.7288 | 0.6832 | 0.7486 | 1.0484 |
EWG | 0.5057 | 0.5707 | 0.3995 | 0.4526 |
EWJ | 0.8806 | 1.0497 | 0.8528 | 0.9151 |
EWT | 1.9431 | 1.5749 | 2.2252 | 1.8637 |
EWU | 0.5175 | 0.6803 | 0.5683 | 0.5359 |
EWW | 1.0188 | 1.2924 | 0.9032 | 0.9651 |
EWY | 1.8421 | 1.4657 | 1.2411 | 1.4163 |
EWZ | 1.1970 | 1.2120 | 1.1880 | 0.9775 |
EZA | 1.0658 | 1.1811 | 1.0804 | 1.1062 |
Mean | 1.0144 | 1.0116 | 0.9510 | 0.9764 |
Ticker | LSTM | BiLSTM | GRU | BiGRU |
---|---|---|---|---|
EWA | 0.5470 | 0.5039 | 0.3991 | 0.5825 |
EWC | 0.8831 | 0.8524 | 0.9563 | 1.2546 |
EWG | 0.6391 | 0.6966 | 0.5102 | 0.5821 |
EWJ | 1.1018 | 1.2694 | 1.0577 | 1.1252 |
EWT | 2.2230 | 1.7876 | 2.5269 | 2.1602 |
EWU | 0.6629 | 0.8258 | 0.7214 | 0.6911 |
EWW | 1.2755 | 1.5547 | 1.1270 | 1.2422 |
EWY | 2.1501 | 1.8247 | 1.5557 | 1.7773 |
EWZ | 1.5004 | 1.4797 | 1.4282 | 1.2561 |
EZA | 1.3482 | 1.4823 | 1.3516 | 1.4474 |
Mean | 1.2331 | 1.2277 | 1.1634 | 1.2119 |
Method | BL | BL- | BL- | BL- | BL-- |
---|---|---|---|---|---|
LSTM | 1.1021 | 1.2455 | 1.2185 | 1.2216 | 1.3496 |
GRU | 1.3016 | 1.4483 | 1.4073 | 1.4862 | 1.5393 |
BiLSTM | 1.1223 | 1.1826 | 1.1940 | 1.2008 | 1.1844 |
BiGRU | 1.1251 | 1.2569 | 1.2092 | 1.1998 | 1.3567 |
Metrics | Base | MW | BL | BL- | BL- | BL- | BL-- |
---|---|---|---|---|---|---|---|
Mean Return | 0.0358 | 0.0374 | 0.3376 | 0.3886 | 0.3752 | 0.3960 | 0.4136 |
Standard Dev. | 0.2082 | 0.1766 | 0.2440 | 0.2545 | 0.2524 | 0.2530 | 0.2557 |
Sharpe Ratio | 0.0759 | 0.0985 | 1.3016 | 1.4483 | 1.4073 | 1.4862 | 1.5393 |
MDD | 0.4811 | 0.3764 | 0.3699 | 0.3844 | 0.3805 | 0.3752 | 0.3759 |
Year | Metrics | Base | MW | BL | BL- | BL- | BL- | BL-- |
---|---|---|---|---|---|---|---|---|
2016 | Mean Return | 0.091 | 0.0939 | 0.7079 | 0.7683 | 0.7602 | 0.7645 | 0.7526 |
Standard Dev. | 0.22 | 0.173 | 0.3362 | 0.3389 | 0.3388 | 0.3389 | 0.3392 | |
Sharpe Ratio | 0.4129 | 0.5418 | 2.1015 | 2.2625 | 2.2392 | 2.2514 | 2.2143 | |
MDD | 0.1042 | 0.1022 | 0.1393 | 0.1393 | 0.1393 | 0.1393 | 0.1393 | |
2017 | Mean Return | 0.2061 | 0.2027 | 0.351 | 0.4051 | 0.4547 | 0.3764 | 0.4168 |
Standard Dev. | 0.1128 | 0.0766 | 0.1013 | 0.1272 | 0.132 | 0.1262 | 0.1462 | |
Sharpe Ratio | 1.824 | 2.6417 | 3.4597 | 3.178 | 3.4373 | 2.9765 | 2.8453 | |
MDD | 0.0353 | 0.0252 | 0.0293 | 0.0393 | 0.0393 | 0.0365 | 0.0484 | |
2018 | Mean Return | −0.2154 | −0.1613 | 0.1223 | 0.2153 | 0.2158 | 0.3744 | 0.3936 |
Standard Dev. | 0.1877 | 0.1413 | 0.2199 | 0.2436 | 0.2434 | 0.2607 | 0.2615 | |
Sharpe Ratio | −1.1453 | −1.1394 | 0.5549 | 0.8821 | 0.8847 | 1.4331 | 1.5021 | |
MDD | 0.2664 | 0.2125 | 0.1392 | 0.1425 | 0.1408 | 0.1156 | 0.1156 | |
2019 | Mean Return | 0.1227 | 0.1856 | 0.2043 | 0.2599 | 0.2419 | 0.2725 | 0.2834 |
Standard Dev. | 0.1347 | 0.1055 | 0.1301 | 0.137 | 0.1349 | 0.136 | 0.1367 | |
Sharpe Ratio | 0.9093 | 1.7568 | 1.5668 | 1.8933 | 1.7902 | 2.0002 | 2.0687 | |
MDD | 0.1167 | 0.0665 | 0.1248 | 0.091 | 0.1082 | 0.0963 | 0.0932 | |
2020 | Mean Return | 0.0498 | 0.1307 | 0.4523 | 0.5095 | 0.4529 | 0.4883 | 0.4881 |
Standard Dev. | 0.3663 | 0.2951 | 0.4005 | 0.4142 | 0.4062 | 0.4066 | 0.4077 | |
Sharpe Ratio | 0.1357 | 0.4419 | 1.1272 | 1.2277 | 1.1129 | 1.1985 | 1.1949 | |
MDD | 0.4063 | 0.3038 | 0.3699 | 0.3844 | 0.3805 | 0.3752 | 0.3759 | |
2021 | Mean Return | 0.035 | 0.0813 | 0.2396 | 0.2589 | 0.248 | 0.2458 | 0.2489 |
Standard Dev. | 0.1583 | 0.1385 | 0.2014 | 0.2115 | 0.2115 | 0.2008 | 0.2081 | |
Sharpe Ratio | 0.2206 | 0.5859 | 1.1875 | 1.2216 | 1.17 | 1.2215 | 1.1937 | |
MDD | 0.1166 | 0.0901 | 0.1417 | 0.1603 | 0.1483 | 0.1483 | 0.1483 | |
2022 | Mean Return | -0.203 | -0.1965 | 0.3503 | 0.3778 | 0.3568 | 0.3448 | 0.4107 |
Standard Dev. | 0.2186 | 0.2017 | 0.2383 | 0.2435 | 0.2411 | 0.2378 | 0.2385 | |
Sharpe Ratio | −0.9267 | −0.9719 | 1.4669 | 1.5484 | 1.4768 | 1.4473 | 1.7184 | |
MDD | 0.2781 | 0.2673 | 0.2257 | 0.2062 | 0.2201 | 0.2047 | 0.1973 | |
2023 | Mean Return | 0.1541 | 0.1501 | 0.2807 | 0.3232 | 0.2805 | 0.311 | 0.325 |
Standard Dev. | 0.1575 | 0.1453 | 0.1731 | 0.1826 | 0.1792 | 0.1814 | 0.1833 | |
Sharpe Ratio | 0.9761 | 1.0311 | 1.6179 | 1.7671 | 1.5619 | 1.7105 | 1.7698 | |
MDD | 0.1367 | 0.1147 | 0.1396 | 0.1439 | 0.1539 | 0.1503 | 0.157 |
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Cho, P.; Lee, M. Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty. Fractal Fract. 2024, 8, 642. https://doi.org/10.3390/fractalfract8110642
Cho P, Lee M. Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty. Fractal and Fractional. 2024; 8(11):642. https://doi.org/10.3390/fractalfract8110642
Chicago/Turabian StyleCho, Poongjin, and Minhyuk Lee. 2024. "Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty" Fractal and Fractional 8, no. 11: 642. https://doi.org/10.3390/fractalfract8110642
APA StyleCho, P., & Lee, M. (2024). Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty. Fractal and Fractional, 8(11), 642. https://doi.org/10.3390/fractalfract8110642