Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty

Abstract

This study investigates the profitability of portfolios that integrate asymmetric fractality within the Black–Litterman (BL) framework. It predicts 10-day-ahead exchange-traded fund (ETF) prices using recurrent neural networks (RNNs) based on historical price information and technical indicators; these predictions are utilized as BL views. While constructing the BL portfolio, the Hurst exponent obtained from the asymmetric multifractal detrended fluctuation analysis is employed to determine the uncertainty associated with the views. The Hurst exponent describes the long-range persistence in time-series data, which can also be interpreted as the uncertainty in time-series predictions. Additionally, uncertainty is measured using asymmetric fractality to account for the financial time series’ asymmetric characteristics. Then, backtesting is conducted on portfolios comprising 10 countries’ ETFs, rebalanced on a 10-day basis. While benchmarking to a Markowitz portfolio and the MSCI world index, profitability is assessed using the Sharpe ratio, maximum drawdown, and sub-period analysis. The results reveal that the proposed model enhances the overall portfolio return and demonstrates particularly strong performance during negative trends. Moreover, it identifies ongoing investment opportunities, even in recent periods. These findings underscore the potential of fractality in adjusting uncertainty for diverse portfolio optimization applications.

1. Introduction

According to the efficient market hypothesis (EMH) [1], the financial market is weak-form efficient because all existing information is reflected in market prices. If the market contains all information, the market prices should follow a random walk. However, many empirical studies suggest that the financial market is inefficient [2,3,4,5]. If the market is inefficient, it has certain patterns that make market predictions possible [6]. One method of measuring market efficiency is fractality. Fractality refers to self-similarity, which explains long memory. Hurst [7] measured the long-term storage capacity of reservoirs to present long-range dependence in hydrology. Since then, the Hurst exponent has been widely used in various fields such as finance, physics, and cardiology. One of the most widely used methodologies to measure fractality is the multifractal detrended fluctuation analysis (MFDFA) [8]. This method allows us to obtain the generalized Hurst exponent, which measures long-memory dependence and multifractality. When a major event occurs in the market, there is a tendency for long-memory dependence to emerge, and numerous studies have explored this subject. Li demonstrated that multifractal characteristics appear during market slumps [9], while Ameer et al. investigated the multifractality of BRICS and emerging markets before and after COVID-19 [10]. Choi compared market efficiency during the COVID-19 pandemic and the global financial crisis [11], and Wang et al. found that anti-persistence in the gold market increases when a major event occurs [12]. In the stock market, characteristics that change according to market conditions have been identified, leading to the proposal of the Asymmetric-MFDFA (A-MFDFA) [13] to measure long-memory dependence based on market conditions; thus, an asymmetric market efficiency measure also exists [14]. Studies on asymmetric multifractality have been conducted in various assets, including ESG markets [15], the technological and renewable energy sectors [16], and the natural gas market [17]. Yang and Feng performed an asymmetric multifractal analysis across different regimes, constructing an inefficiency spillover network [18], while Lee et al. demonstrated the asymmetric multifractal behavior in the cryptocurrency market by combining the CAPM filter with A-MFDFA [19].

These efficiencies have been employed to improve the forecasting performance of the financial time series. A multifractal least squares support vector machine detrended fluctuation analysis (MF-LSSVM-DFA), constructed by converting polynomial fitting to LSSVM, has demonstrated its predictive capability on its own [20], whereas a multifractal asymmetric detrended cross-correlation analysis (MF-ADCCA) has shown effectiveness in stock return forecasting [21]. Cabezas-Rivas et al. detected price persistence through fractional geometry, demonstrating low efficiency and high predictability [22], while Kojić et al. captured asset predictability by combining multifractal detrended cross-correlation analysis with Granger causality [23]. Additionally, two stock prediction models have been developed: one that combines Lie group methods with long short-term memory (LSTM) [24] and another that integrates fractional Brownian motion with particle swarm optimization [25]. Similarly, efforts have been made to improve the utilization of deep learning predictions. Yang et al. developed a model that determines early warning levels using the MFDFA and predicts tunnel settlement using particle swarm optimization long short-term memory (PSO-LSTM), and they integrated both results to produce a comprehensive warning outcome [26]. Efficiency has also been applied to investment analysis. Ameer et al. measured market efficiency using MFDFA with the coronavirus disease 2019 (COVID-19) outbreak as the reference point, arguing that pre-COVID-19 (post-COVID-19), China (Brazil) had the highest inefficiency and investment opportunities [10]. According to the aforementioned study, high market inefficiency can yield investment opportunities.

The Black–Litterman (BL) portfolio incorporates an investor’s subjective views into the optimization process [27]. Therefore, the most crucial step in this model is to effectively represent the investor’s views. Studies have used predictions generated by machine learning techniques for this. Barua and Sharma predicted price series using convolutional neural network–bidirectional long short-term memory (CNN-BiLSTM), which has high predictive accuracy, and formulated their views accordingly [28]. Meanwhile, Zhu and Yen advanced price series prediction by integrating a Transformer and a Generative Adversarial Network (GAN), and developed their views based on this approach [29]. These studies aimed to enhance portfolio profitability by improving the price time series’ predictive performance. Other studies have focused on enhancing portfolio performance by considering market characteristics. To consider macro information from the market, Li and Chen divided the economic cycle based on the economic growth gap and inflation and constructed their views accordingly [30]. Zheng et al. utilized predictive views that incorporate weather information to construct a commodity BL portfolio [31]. Barua and Sharma incorporated investor sentiment into their views using a fear/greed indicator [32]. Furthermore, the BL model requires determining the uncertainty of views. The results can vary depending on this uncertainty. Therefore, attempts have been made to adjust for this uncertainty. Teplova et al. estimated the confidence of BL views using the copula function [33]. Meanwhile, Fuhrer and Hock derived an equilibrium model estimation procedure for uncertainty [34]. At this point, we can hypothesize that fractality is profitable in determining the uncertainty of the BL portfolio. Specifically, to capture the differing market behavior in bull and bear markets, it is possible to propose the hypothesis that asymmetric fractality enhances the profitability of the BL portfolio.

The proposed model employs machine learning predictions as views while determining their uncertainty using the Hurst exponent. When the Hurst exponent is $0.5$, the market exhibits random walk characteristics, indicating the highest uncertainty. Conversely, if the Hurst exponent deviates from $0.5$, indicating that the market displays persistent or anti-persistent characteristics, uncertainty is reduced. The proposed model calculates the asymmetric Hurst exponent to capture the asymmetric market fractality. Financial time series exhibit different characteristics during uptrends and downtrends; therefore, asymmetric fractality is employed to consider both features. This approach calculates two asymmetric Hurst exponents, one for uptrends and the other for downtrends, and the average is then used to adjust the uncertainty of the views. Using recurrent neural networks (RNNs), past price data and technical indicators are employed to forecast the prices of each exchange-traded fund (ETF) 10 days ahead, and views based on these predictions are assigned using asymmetric fractality. The RNN models are trained on data from the past ten years to predict data for the upcoming year, and a new model is trained each year. The portfolio is rebalanced every 10 days with benchmarks including the buy-and-hold strategy, the Markowitz portfolio [35], and the BL portfolio based on MFDFA. To evaluate profitability performance, the Sharpe ratio, Maximum Drawdown (MDD), and sub-period analyses are conducted.

This study’s novelty is that it introduces the use of fractality to quantify the uncertainty of the BL portfolio for the first time. In addition, it enhances portfolio performance by incorporating the asymmetric characteristics of financial time series. This approach maximizes the integration of deep learning and econophysics, thereby broadening the scope of portfolio selection.

The remainder of this paper consists of the Methods section, which explains the foundations of the generalized Hurst exponent, RNN, and portfolio theory; the Experiments and Data section, which details the backtesting procedures; the Results section, which provides an in-depth analysis of rebalancing the ETF portfolio based on the obtained results; and the Discussion and Conclusions section, which interprets the results and draws conclusions.

2. Methods

2.1. Asymmetric Fractality of the ETF Price Series

The asymmetric fractality of the ETF price series is calculated to adjust the subjective views of investors used to optimize the BL portfolio. As fractality is related to the volatility of returns, it can be considered an alternative that reflects investors’ subjective views or uncertainties. Particularly, asymmetric fractality has the advantage of distinguishing between periods when market prices increase and decrease, thereby reflecting investors’ changing outlooks based on market conditions. A-MFDFA is employed to measure the asymmetric fractality [13,14]. The generalized Hurst exponent, which is a feature of asymmetric fractality, is then calculated using this methodology. A-MFDFA involves several steps, which are summarized below. Consider a price time series $\{P_t:t=1,2,\dots ,N\}$ and return time series $\{r_t:t=1,2,\dots ,N\}$; then, $P_t=P_{t-1}exp\left(r_t\right)$ for $t=1,2,\dots ,N$.

Step 1: Defining the profile of the return time series, $y_t$, as follows:

$$\begin{array}{c}\hfill y_t=\sum _{j=1}^t(r_j-\overline{r}),t=1,2,\dots ,N\end{array}$$

(1)

where $\overline{r}={\sum }_{j=1}^N{r}_j/N$.

Step 2: Dividing the price time series $P_t$ and profile $y_t$ into $N_n$ (where $N_n$ is the largest integer less than or equal to $N/n$) non-overlapping sub-time-series of equal length n (scale). This procedure is repeated from the other end of $\left\{P_t\right\}$ and $\left\{y_t\right\}$ to obtain 2$N_n$ sub-time-series. Subsequently, let $A_j=\{a_{j,k},k=1,2,\dots ,n\}$ be the jth n-length sub-time-series of $\left\{P_t\right\}$ and $B_j=\{b_{j,k},k=1,2,\dots ,n\}$ be the jth n-length sub-time-series of $\left\{r_t\right\}$ for $j=1,2,\dots ,2N_n$. Notably, $5\le n\le N/4$ [8].

Step 3: Fitting the local linear regression model using the least-squares method for each sub-time-series $A_j$ and $B_j$ to calculate the local trend and fluctuation function. $L_{A_j}\left(k\right)={\beta }_{1A_j}+{\beta }_{2A_j}k$, and $L_{B_j}\left(k\right)={\beta }_{1B_j}+{\beta }_{2B_j}k$, where k is the horizontal coordinate. The sign of ${\beta }_{2A_j}$ indicates the local trend in the price index. In addition, the fluctuation function, $F_j\left(n\right)$, is defined as follows:

$$\begin{array}{c}\hfill F_j\left(n\right)=\sum _{k=1}^n{\left(b_{j,k}-L_{B_j}\left(k\right)\right)}^2/n,j=1,2,\dots ,2N_n\end{array}$$

(2)

Step 4: Classifying the sub-time-series into uptrend and downtrend based on the sign of ${\beta }_{2A_j}$. The directional q-order average fluctuation functions, $F_q^{+}\left(n\right)$ and $F_q^{-}\left(n\right)$, can be defined as follows:

$$F_q^{+}\left(n\right)={\left(\sum _{j=1}^{2N_n}(1+sign\left({\beta }_{2A_j}\right)){\left[F_j\left(n\right)\right]}^{q/2}/M^{+}\right)}^{1/q}$$

(3)

$$F_q^{-}\left(n\right)={\left(\sum _{j=1}^{2N_n}(1-sign\left({\beta }_{2A_j}\right)){\left[F_j\left(n\right)\right]}^{q/2}/M^{-}\right)}^{1/q}$$

(4)

where $M^{+}={\sum }_{j=1}^{2N_n}(1+sign\left({\beta }_{2A_j}\right))$, $M^{-}={\sum }_{j=1}^{2N_n}(1-sign\left({\beta }_{2A_j}\right))$ and $sign\left(x\right)$ is the sign of the value x. The MFDFA model’s average fluctuation function, $F_q\left(n\right)$, is:

$$\begin{array}{c}\hfill F_q\left(n\right)={\left(\sum _{j=1}^{2N_n}{\left[F_j\left(n\right)\right]}^{q/2}/\left(2N_n\right)\right)}^{1/q}\end{array}$$

(5)

Step 5: Calculating the generalized Hurst exponents to capture the time series’ asymmetric fractality. Let $H_q,H_q^{+}$, and $H_q^{-}$ be the overall, uptrend, and downtrend generalized Hurst exponents, respectively. These values follow the power-law scaling relationship $F_q\left(n\right)\propto n^{H_q}, F_q^{+}\left(n\right)\propto n^{H_q^{+}}$, and $F_q^{-}\left(n\right)\propto n^{H_q^{-}}$, respectively.

The time series is persistent when $0.5<H_q<1$ and anti-persistent when $0<H_q<0.5$. If $H_q=0.5$, the time series follows a random walk process. We use the Hurst exponent with $q=2$. After calculating the Hurst exponent for the uptrend, downtrend, and overall trend, the degree to which these values deviate from 0.5 indicates the extent to which they diverge from a normal random walk. Consequently, this deviation is reflected in the BL portfolio as a view on uncertainty.

2.2. Recurrent Neural Network Group

While constructing the views for the BL portfolio, price time series forecasts are used, with RNNs predicting 10-day-ahead prices. RNNs are one of the most commonly used methods for processing and predicting time series data owing to their sequential data processing capabilities. Indeed, several updated models are available [36]. Specifically, we utilize four RNN models: long short-term memory (LSTM), bidirectional LSTM (BiLSTM), gated recurrent unit (GRU), and bidirectional GRU (BiGRU). First, LSTM addresses the long-term dependency problem using gate mechanisms, allowing it to retain important information over extended sequences in time-series data [37]. Along with the input and output gates, the forget gate plays a crucial role in determining the amount of previous memory to be discarded. The following equations describe each gate:

$$i_t=\sigma (W_i{x}_t+V_i{h}_{t-1}+b_i)$$

(6)

$$o_t=\sigma (W_o{x}_t+V_o{h}_{t-1}+b_o)$$

(7)

$$f_t=\sigma (W_f{x}_t+V_f{h}_{t-1}+b_f)$$

(8)

$$c_t=f_t\odot c_{t-1}+i_t\odot tanh(W_c{x}_t+V_c{h}_{t-1}+b_c)$$

(9)

$$h_t=o_t\odot tanh\left(c_t\right)$$

(10)

where $i_t$, $o_t$, $f_t$, $c_t$, $h_t$, $x_t$, $\sigma$, and $\odot$ correspond to the input gate, output gate, forget gate, memory cell, hidden state, input vector, logistic sigmoid function, and element-wise multiplication operator, respectively. W, V, and b are the model parameters to be learned at each time step t.

Second, the GRU simplifies the LSTM architecture by combining forget and input gates into a single update gate. This makes it computationally more efficient while still effectively capturing dependencies in sequential data [38]. The reset gate effectively clears past information, whereas the update gate controls the rate at which past and present information is updated. The equations for these two gates are as follows:

$$\begin{array}{cc}\hfill u_t& =\sigma (W_u{x}_t+V_u{h}_{t-1}+b_u)\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill r_t& =\sigma (W_r{x}_t+V_r{h}_{t-1}+b_r)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill h_t& =u_t\odot h_{t-1}+(1-u_t)\odot tanh(W_h{x}_t+V_h(r_t\odot h_{t-1})W_h)\hfill \end{array}$$

(13)

where $u_t$ and $r_t$ correspond to the update and reset gates, respectively.

Finally, BiLSTM and BiGRU are both types of bidirectional RNNs that process data in both the forward and backward directions. This allows them to capture contextual information from both the past and future within a sequence. BiLSTM uses the LSTM architecture, which is designed to effectively handle long-term dependencies. Meanwhile, BiGRU employs the GRU architecture, which is more streamlined and computationally less demanding [39,40]. The hidden state of the bidirectional networks is expressed as follows:

$$\begin{array}{c}\hfill h_t=[\overrightarrow{h_t},\overleftarrow{h_t}]\end{array}$$

(14)

where $\overrightarrow{h_t}$ and $\overleftarrow{h_t}$ represent the hidden states in the forward and backward directions, respectively. The final hidden state combines both hidden states to simultaneously process past and future information.

2.3. Black–Litterman Portfolio with Asymmetric Fractality

The proposed model integrates asymmetric fractality into the BL framework to construct an optimal ETF portfolio. The BL portfolio reflects the investor’s views. The return ${\mu }_{BL}$ and covariance matrix ${\mathrm{\Sigma }}_{BL}$ are calculated as follows [41,42,43]:

$${\mu }_{BL}={[{\left(\tau \mathrm{\Sigma }\right)}^{-1}+P^T{\mathrm{\Omega }}^{-1}P]}^{-1}[{\left(\tau \mathrm{\Sigma }\right)}^{-1}\pi +P^T{\mathrm{\Omega }}^{-1}Q]$$

(15)

$${\mathrm{\Sigma }}_{BL}=\mathrm{\Sigma }+{[{\left(\tau \mathrm{\Sigma }\right)}^{-1}+P^T{\mathrm{\Omega }}^{-1}P]}^{-1}$$

(16)

where $\tau$, $\mathrm{\Sigma }$, P, $\mathrm{\Omega }$, $\pi$, and Q correspond to the scale number, covariance matrix of the assets, identifying matrix of the views, uncertainty matrix of the views, implied return vector, and view vector, respectively. The uncertainty of the views follows a normal distribution $N(0,\mathrm{\Omega })$. First, the view vector Q is assigned each asset’s predicted returns. For instance, if a specific asset’s price is expected to increase by 3% in 10 days, the corresponding value in Q is set to 3%. Similarly, the predicted returns for all assets are incorporated into the view vector. Next, the identifying matrix of the views, P, records the assets to which the views apply. Since this study uses absolute views for predictions across all assets, P is an identity matrix. The covariance matrix of the assets, $\mathrm{\Sigma }$, is calculated from the past 60 days of price data to determine the covariance between assets. The scale number $\tau$, used as a weight-on-views scalar, is set to $0.05$. The implied return vector $\pi$ incorporates the prior estimates of the returns observed in the market. Finally, the uncertainty matrix of the views, $\mathrm{\Omega }$, is specified based on asymmetric fractality.

When the Hurst exponent is $0.5$, the time series exhibits the highest randomness, thus, the uncertainty should be the highest. Conversely, when the Hurst exponent is 0 or 1, the time series is persistent or anti-persistent, respectively, resulting in the lowest uncertainty. Therefore, the uncertainty $u_i$ of an asset i can be measured as follows:

$$\begin{array}{cc}\hfill u_i& =0.5-|0.5-H_2\left(i\right)|\hfill \end{array}$$

(17)

where $H_2\left(i\right)$ represents the Hurst exponent of asset i from the MFDFA. Thus, the maximum value of $u_i$ is $0.5$, and the Hurst exponents above $0.5$ exhibit a symmetrical pattern around 0.5 when compared to the Hurst exponents below $0.5$. Similarly, the uncertainties derived from the A-MFDFA, $u_i^{+}$ and $u_i^{-}$, can be constructed as follows:

$$\begin{array}{cc}\hfill u_i^{+}& =0.5-|0.5 - H_2^{+}\left(i\right)|\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill u_i^{-}& =0.5-|0.5 - H_2^{-}\left(i\right)|\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill u_i^{+-}& =(u_i^{+}+u_i^{-})/2\hfill \end{array}$$

(20)

where $H_2^{+}\left(i\right)$ and $H_2^{-}\left(i\right)$ represent the uptrend and downtrend Hurst exponents of asset i by A-MFDFA, respectively. To account for both trends, the final uncertainty, $u_i^{+-}$, is determined by averaging each uncertainty, as given by Equation (20). Since $\mathrm{\Omega }$ is a diagonal matrix, each asset i’s uncertainty $u_i^{+-}$ is placed on the diagonal of $\mathrm{\Omega }$, yielding an uncertainty matrix that reflects asymmetric fractality. Similarly, we can construct the uncertainty matrix using $u_i^{+}$, $u_i^{-}$, and $u_i$, with these models serving as benchmarks. Several studies have been conducted to determine $\mathrm{\Omega }$ [44,45,46], and this study aims to demonstrate whether determining the uncertainty of views through asymmetric fractality is effective. All uncertainty matrices are scaled, and the final portfolio weights, w, are determined as follows:

$$\begin{array}{cc}\hfill \delta & ={\displaystyle \frac{r_{mkt}-r_f}{{\sigma }_{mkt}^2}}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill w& ={\left(\delta {\mathrm{\Sigma }}_{BL}\right)}^{-1}{\mu }_{BL}\hfill \end{array}$$

(22)

where $\delta$, $r_{mkt}$, $r_f$, and ${\sigma }_{mkt}$ are the risk aversion coefficient, expected market return, risk-free rate, and variance of the market portfolio, respectively. The pseudocode of the Black–Litterman portfolio, which adjusts the uncertainty of views through asymmetric fractality, is shown in Algorithm 1.

Algorithm 1 Black–Litterman portfolio with asymmetric fractality

Input: price series $P_i=\{p_{i1},\dots ,p_{i(t-10)}\}$ of ith ETF   Output: portfolio weight $W=\{W_1,\dots ,W_{10}\}$ with 10 ETFs       -  Predict the 10-day-ahead price of each ETF  1: for $i=1:10$ do  2:     Calculate 13 technical indicators  3:     Prepare the stock prediction models with training data  4:     Predict $p_{it}$ with the trained models  5:     Construct P, Q, and Σ of the BL portfolio using $p_{it}$  6: end for       -  Calculate the asymmetric fractality  7: for $i=1:10$ do  8:     Compute the uptrend and downtrend Hurst exponents $H_2^{+}\left(i\right)$ and $H_2^{-}\left(i\right)$  9:     $u_i^{+}=0.5-|0.5-H_2^{+}\left(i\right)|$ 10:    $u_i^{-}=0.5-|0.5-H_2^{-}\left(i\right)|$ 11:    $u_i^{+-}=(u_i^{+}+u_i^{-})/2$ 12:    Construct $\mathrm{\Omega }$ of the BL portfolio using $u_i^{+-}$ 13: end for 14: W = BlackLitterman(P, Q, $\mathrm{\Sigma }$, $\mathrm{\Omega }$)

3. Experiments and Data

3.1. Experiments

To demonstrate the asymmetric Hurst exponent-based BL portfolio’s profitability, the backtesting process is structured into three stages, as depicted in Figure 1. Initially, 10 global ETFs are selected to comprise the portfolio, and their daily open, high, low, and close prices are extracted. Using these price data, technical indicators are calculated, and a predictive model is developed to forecast the 10-day-ahead close prices based on historical price data and technical indicators. To normalize the data, a min–max scaler is used to transform each variable between 0 and 1. The model is trained on a 10-year training set to predict the subsequent year’s test set, with a new model being trained and tested annually. Then, the prediction models, including LSTM, GRU, BiLSTM, and BiGRU, are trained. Hyperparameter tuning is performed through a grid search over combinations of hyperparameters as shown in

Table 1. Hyperparameter tuning for RNNs.

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

. The final model is determined through a 5-fold cross-validation. The prediction performance is assessed using the mean absolute error (MAE) and root mean squared error (RMSE) metrics. The predictions serve as views in the BL portfolio, with the view’s uncertainty determined by the Hurst exponent. The Hurst exponent is calculated daily from the ETF prices using the A-MFDFA. A Hurst exponent of $0.5$ indicates that the time series exhibits a random walk, representing the most uncertain state. Therefore, based on a Hurst exponent of $0.5$, we determine an uncertainty measure wherein uncertainty decreases as the Hurst exponent deviates from 0.5. This helps in determining asset allocation in the BL portfolio, with backtesting performed by rebalancing every 10 days. For comparison, we construct a buy-and-hold strategy on the MSCI World Index and Markowitz portfolio. Profitability is assessed using the Sharpe ratio, MDD, and sub-period analysis. All computations are performed using Python 3.10, with the TensorFlow 2.10 and PyPortfolioOpt 1.5.5 libraries [47]. The hardware setup consists of an AMD Ryzen 9 4.5 GHz processor, 32 GB of RAM, and an NVIDIA GTX 4090 GPU with 24 GB of memory.

The models are categorized into four types based on how the Hurst exponent is calculated, as illustrated in Figure 2. First, using the MFDFA, $H_2$ is calculated. A model utilizing the uncertainty $\mathrm{\Omega }\left(H_2\right)$, which folds the upper part of the values below 0.5, is denoted as BL-${\mathrm{H}}_2$. Second, using the A-MFDFA, $H_2^{+}$ and $H_2^{-}$ can be calculated. The models that use $\mathrm{\Omega }\left(H_2^{+}\right)$ and $\mathrm{\Omega }\left(H_2^{-}\right)$ are similarly referred to as BL-${\mathrm{H}}_2^{+}$ and BL-${\mathrm{H}}_2^{-}$, respectively. Finally, the proposed model combines $H_2^{+}$ and $H_2^{-}$ to construct the uncertainty, and is referred to as BL-${\mathrm{H}}_2^{+}$-${\mathrm{H}}_2^{-}$. Figure 2 illustrates the process of converting EZA’s price series into the view’s uncertainty. The same uncertainty calculation is performed for the remaining nine ETFs to construct the BL portfolio.

3.2. Data

The portfolio comprises 10 MSCI ETFs representing different countries. The information regarding these ETFs is provided in

Table 2. ETFs to be used for the portfolio.

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

. The open, high, low, and close price series of these ETFs are from Yahoo Finance. The following 13 technical indicators are calculated using these data: moving average, exponential moving average, moving average convergence divergence, relative strength index, accumulation/distribution oscillator, stochastics fast, stochastics slow, Bollinger band, fear/greed index, Williams R%, commodity channel index, on-balance volume, and Ichimoku cloud. These technical indicators are widely used to forecast future price information [28]. A moving window of 500 days is used for calculating the generalized Hurst exponent using the A-MFDFA. The first prediction model is trained on data from 2006 to 2015 and used to forecast the close prices for 2016. Similarly, the second model is trained on data from 2007 to 2016 and forecasts data for 2017, resulting in eight models. To measure portfolio performance, the risk-free rate is the three-month Treasury Bill Rate.

Multifractal concepts such as MFDFA and A-MFDFA, which are used to calculate the Hurst exponent, can be applied to stationary data [13]. However, because the price series predicted by the proposed model is non-stationary, additional steps are required. Typically, the Hurst exponent is estimated using the log return series. The descriptive statistics for each ETF’s return series are presented in

Table 3. Descriptive statistics of return series.

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 *

Note that ¹ denotes the Augmented Dickey–Fuller Test, and * denotes the 1% level of significance.

. At the 1% significance level, the Jarque–Bera (JB) test results indicate that all 10 ETFs’ return series do not follow a normal distribution. Additionally, because the kurtosis is significantly higher than 3, this confirms that the return series follows the fat-tail distribution. This is a characteristic feature of financial time series. Next, at the 1% significance level, the Augmented Dickey–Fuller (ADF) test indicates that the return series of all 10 ETFs are stationary, thereby validating the data for computing the Hurst exponent.

4. Results

4.1. Prediction Results

As previously stated, we predict the 10-day-ahead close price of each ETF using the past 60 days of price data and technical indicators. Predictions are made over eight years using four RNNs: LSTM, BiLSTM, GRU, and BiGRU. The error metrics, MAE and RMSE, are calculated as follows [48]:

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n|a_i-p_i|$$

(23)

$$\mathrm{RMSE}={\left[{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(a_i-p_i)}^2\right]}^{{\displaystyle \frac{1}{2}}}$$

(24)

where $a_i$ and $p_i$ are the actual and predicted values, respectively. Lower values of both MAE and RMSE indicate better predictive performance of the models. The results are shown in

Table 4. MAE of prediction results; best results are highlighted in bold.

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

and

Table 5. RMSE of prediction results; best results are highlighted in bold.

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

. The metrics for each ETF’s predictions are calculated across the four RNNs and the average performance is computed. According to Table 4 and Table 5, the best-performing model varied by ETF. However, predictions using GRU generally showed superior average performance. Therefore, the BL view utilizes the GRU model predictions.

4.2. Portfolio Evaluation

Figure 3 illustrates the price and asymmetric Hurst exponent series for the EZA ETF. Other ETFs’ corresponding series are shown in Figure A1. Most Hurst exponents are close to 0.5, indicating a general tendency toward random walk behavior. Some exhibit persistent (anti-persistent) characteristics when below (above) 0.5. Notably, $H_2^{+}$ and $H_2^{-}$ exhibit more significant variations than $H_2$. The Hurst exponent-based prediction’s uncertainty can be quantified and used as the uncertainty view in the BL portfolio.

The profitability metrics, Sharpe ratio, and MDD, are calculated as follows [48]:

$$\begin{array}{cc}\hfill \mathrm{Sharpe}\mathrm{ratio}& ={\displaystyle \frac{r-r_f}{\sigma }}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill MDD& ={\displaystyle \frac{pv-lp}{pv}}\hfill \end{array}$$

(26)

where r, $r_f$, $pv$, and $lp$ are the return, risk-free rate, peak value, and the lowest return after the peak, respectively. A higher Sharpe ratio and lower MDD indicate better profitability performance of the models. The profitability performances are shown in

Table 6. Sharpe ratio by Hurst exponent; best results are highlighted in bold.

Method	BL	BL-${\mathbf{H}}_2$	BL-${\mathbf{H}}_2^{+}$	BL-${\mathbf{H}}_2^{-}$	BL-${\mathbf{H}}_2^{+}$-${\mathbf{H}}_2^{-}$
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

. The BL model, which applies the same uncertainty to all predictions, exhibits a lower Sharpe ratio than the BL-${\mathrm{H}}_2$ model, which uses Hurst exponent-adjusted uncertainty. Thus, adjusting the uncertainty based on the Hurst exponent improves the model’s effectiveness. Similarly, the BL-${\mathrm{H}}_2^{+}$ and BL-${\mathrm{H}}_2^{-}$ models are constructed based on the asymmetric Hurst exponents $H_2^{+}$ and $H_2^{-}$ using the A-MFDFA. For BL-${\mathrm{H}}_2^{+}$, only the BiLSTM prediction shows slightly better performance than BL-${\mathrm{H}}_2$; meanwhile, BL-${\mathrm{H}}_2^{-}$ demonstrates significantly higher performance than BL-${\mathrm{H}}_2$ in the GRU prediction. By combining $H_2^{+}$ and $H_2^{-}$ to account for both positive and negative trends, the BL-${\mathrm{H}}_2^{+}$-${\mathrm{H}}_2^{-}$ model achieves the highest Sharpe ratio across all prediction models, except BiLSTM. Additionally, the BL-${\mathrm{H}}_2^{+}$-${\mathrm{H}}_2^{-}$ model outperforms the BL-$H_2$ model across all four prediction models. Thus, incorporating the asymmetric Hurst exponent based on trends effectively quantifies the prediction uncertainty.

Table 7. Profitability metrics; best results are highlighted in bold.

Metrics	Base	MW	BL	BL-${\mathbf{H}}_2$	BL-${\mathbf{H}}_2^{+}$	BL-${\mathbf{H}}_2^{-}$	BL-${\mathbf{H}}_2^{+}$-${\mathbf{H}}_2^{-}$
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

compares the profitability metrics with the benchmarks. First, the base model (Base), which employs a buy-and-hold strategy on the MSCI World Index, has the lowest Sharpe ratio and the highest MDD. Subsequently, the Markowitz portfolio (MW), which exhibits the lowest standard deviation, exhibits the potential for a lower-risk portfolio. However, it shows a significantly lower Sharpe ratio than that of all BL models. All BL models use GRU predictions, which demonstrate the highest performance. The BL model without the Hurst exponent demonstrates the lowest Sharpe ratio, but the lowest MDD. The two A-MFDFA-based models exhibit a lower standard deviation and MDD than those based on the BL-${\mathrm{H}}_2$. The proposed BL-${\mathrm{H}}_2^{+}$-${\mathrm{H}}_2^{-}$ model, which has the highest standard deviation, achieves the highest mean return, and thus, highest Sharpe ratio. Additionally, the proposed BL-${\mathrm{H}}_2^{+}$-${\mathrm{H}}_2^{-}$ model’s MDD is comparable to that of the Markowitz portfolio, indicating its ability to maintain a low level of risk.

The proposed model is updated annually by training a new model to predict close prices, which may lead to variations in the predictive performance and portfolio returns. Accordingly,

Table 8. Sub-period analysis; best results are highlighted in bold.

Year	Metrics	Base	MW	BL	BL-${\mathbf{H}}_2$	BL-${\mathbf{H}}_2^{+}$	BL-${\mathbf{H}}_2^{-}$	BL-${\mathbf{H}}_2^{+}$-${\mathbf{H}}_2^{-}$
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

examines the temporal changes in profitability. The BL model has the highest Sharpe ratio and the second lowest MDD in 2017, which is the highest Sharpe ratio observed over the entire period. The MFDFA-based BL-${\mathrm{H}}_2$ model achieves the highest Sharpe ratios in 2016, 2020, and 2021. While it demonstrates the best performance over these three years, the differences compared with other Hurst exponent-based models are not substantial. The proposed BL-${\mathrm{H}}_2^{+}$-${\mathrm{H}}_2^{-}$ model records the highest Sharpe ratios in 2018, 2019, 2022, and 2023. Notably, in 2018 and 2022, it shows exceptional performance compared with the other models, coinciding with periods of negative trends in the global financial markets, such as the 2018 financial crisis and 2022 stock market decline. Financial markets are known to exhibit persistent return series during bear markets. Thus, the proposed model effectively captures these market characteristics in its uncertainty estimation. Additionally, the model’s highest performance in 2022 and 2023 shows that investment opportunities remain viable.

5. Discussion and Conclusions

This study presents a portfolio that adjusts for the uncertainty of the BL prediction view through asymmetric fractality. The portfolio demonstrates higher profitability than other benchmarks based on the Sharpe ratio and in-depth sub-period analysis. The portfolio is constructed using ETFs from 10 countries, and the Hurst exponent is calculated using both MFDFA and A-MFDFA. The results show that the Hurst exponent, which characterizes a random walk when it equals $0.5$, can effectively represent the uncertainty. Furthermore, the proposed asymmetric fractality-based method has superior performance. This demonstrates that a portfolio that accounts for the market’s asymmetric characteristics can be successfully constructed. The proposed model also shows stronger returns during negative trends and delivers superior performance, suggesting that investment opportunities remain available. This highlights various possibilities for advancing investment strategies using deep learning predictions. Further, the overall returns can be improved by calculating the prediction uncertainty based on the Hurst exponent. Overall, the asymmetric fractality allows for the quantification of the market’s asymmetric characteristics.

Barua and Sharma optimize their BL portfolio using the fear/greed indicator [32], making it possible to compare profitability with our study due to the identical asset composition. We can compare the annual Sharpe ratios from 2017 to 2022, with distinct characteristics evident in each investment strategy. Their portfolio outperforms in bull markets (2017, 2019, and 2021), while ours shows a higher Sharpe ratio in bear markets (2018, 2020, and 2022). Thus, our approach, which accounts for the market’s asymmetric efficiency, yields higher profitability during periods of low market returns. This demonstrates the hypothesis that a portfolio considering asymmetric market efficiency can be more profitable, especially during bear markets.

A limitation of this study is that while the portfolio designed by considering asymmetric fractality achieved higher returns during bear markets, it slightly sacrifices returns in bull markets. Future research can focus on developing portfolios that can maximize returns in bull markets while simultaneously accounting for asymmetric behavior. While the portfolio is constructed using 10 countries’ ETFs, the validity of this investment strategy can also be tested using other assets. Moreover, research integrating various deep learning prediction models and fractality measurement methods can allow for a more in-depth analysis of the causal relationship between fractality and uncertainty.