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<reference>
Enhancing Black-Litterman Portfolio via Hybrid Forecasting
Model Combining Multivariate Decomposition and Noise
Reduction
Ziye Yang1 , Ke Lu2 , Yang Wang3 , Jerome Yen2*

arXiv:2505.01781v2 [cs.CE] 25 May 2025

1

Institute of Collaborative Innovation, University of Macau, Macau, 999078, China.
2
Faculty of Technology, University of Macau, Macau, 999078, China.
3
Shenzhen Institutes of Advanced Technology, Shenzhen, 518055, China.

*Corresponding author(s). E-mail(s): jeromeyen@um.edu.mo;
Contributing authors: mc46468@um.edu.mo; yc37949@um.edu.mo;
yang.wang1@siat.ac.cn;
Abstract
Modern portfolio construction demands robust methods for integrating data-driven insights into asset
allocation. The Black-Litterman model offers a powerful Bayesian approach to adjust equilibrium
returns using investor views to form a posterior expectation along with market priors. Mainstream
research mainly generates subjective views through statistical models or machine learning methods,
among which hybrid models combined with decomposition algorithms perform well. However, most
hybrid models do not pay enough attention to noise, and time series decomposition methods based
on single variables make it difficult to fully utilize information between multiple variables. Multivariate decomposition also has problems of low efficiency and poor component quality. In this study, we
propose a novel hybrid forecasting model—SSA-MAEMD-TCN— to automate and improve the view
generation process. The proposed model combines Singular Spectrum Analysis (SSA) for denoising,
Multivariate Aligned Empirical Mode Decomposition (MA-EMD) for frequency-aligned decomposition, and Temporal Convolutional Networks (TCNs) for deep sequence learning to capture complex
temporal patterns across multiple financial indicators. Empirical tests on the Nasdaq 100 Index stocks
show a significant improvement in forecasting performance compared to baseline models based on
MAEMD and MEMD. The optimized portfolio performs well, with annualized returns and Sharpe
ratios far exceeding those of the traditional portfolio over a short holding period, even after accounting
for transaction costs.
Keywords: Black-Litterman model, Portfolio optimization, Asset price prediction, Decomposition ensemble
model, Temporal convolutional networks, Singular spectrum analysis

1 Introduction

the Mean-Variance (MV) framework (Markowitz,
1952). However, MV the framework faces many
challenges in practical applications. Its extreme
sensitivity to input parameters often leads to significant changes in asset weights (Best & Grauer,

In 1952, Markowitz introduced Modern Portfolio Theory (MPT) to revolutionize portfolio
management by optimizing risk return through

1

1991). In addition, traditional MV optimizers
amplify the estimation errors and ignore the input
uncertainty (Michaud, 1989).
Faced with such limitations, Fisher Black and
Robert Litterman proposed the Black-Litterman
(BL) model (Black & Litterman, 1992). The model
is based on Bayesian statistics that combine the
implied return from market equilibrium with subjective investor views to generate a new posterior
distribution of returns, which can be used to build
a new portfolio to increase its logic (He & Litterman, 2002; Idzorek, 2007; Satchell & Scowcroft,
2007). The most important thing is that investors
or fund managers could use such BL framework
to justify their extra earning by providing such
investor views.
Despite the advantages of the BL model, how
to more accurately predict asset movements to
form the investor view is still a significant challenge for researchers. In our review of the relevant
literature, we found that most studies use different forecasting models to analyze historical data.
These approaches include traditional econometric
models for time series that are linearly dominant
(Beach & Orlov, 2007; Duqi, Franci, & Torluccio,
2014; Palomba, 2008), as well as hybrid models
that incorporate traditional econometric analyzes
and machine learning (Kara, Ulucan, & Atici,
2019; Pyo & Lee, 2018).
Artificial intelligence has flourished in recent
years. Deep learning methods show significant
advantages over statistical methods in capturing
complex nonlinear relationships in data through
automatic feature engineering and complex neural
network structures (Gao, Zhang, & Yang, 2020;
Y. Tang et al., 2022). Decomposition Ensemble
models (DEMs), consisting of the combination
of deep learning model and the Empirical mode
decomposition (EMD) algorithm (Huang et al.,
1998) or its modifications (Torres, Colominas,
Schlotthauer, & Flandrin, 2011; Wu & Huang,
2009; Yeh, Shieh, & Huang, 2010), produced
breakthroughs to the financial market in supporting asset movement forecasting with mixed
relationships. Decomposition techniques can effectively capture the different patterns in time series,
which can help improve the model’s ability to
learn linear and nonlinear relationships in the
data (C. Zhang, Sjarif, & Ibrahim, 2024). DEMs
provide better predictive power (Lin, Yan, Xu,
Liao, & Ma, 2021; Shu & Gao, 2020; Y. Zhang,

Yan, & Aasma, 2020). Recently, new approaches
based on DEMs have been developed and widely
used to support the generation of investor views
with better performance of portfolio return (Barua
& Sharma, 2023; Rezaei, Faaljou, & Mansourfar,
2021).
The above decomposition-ensemble methods
are based on univariate prediction, while multivariate analyses can better explain stock price
movements with more information to include
(Jiang, 2021; Kumbure, Lohrmann, Luukka, &
Porras, 2022). The decomposition algorithm cannot guarantee that the number of IMFs is the
same for each variable due to differences in the
frequency of data for different variables, preventing multivariate forecasting. Therefore, scholars
focused on Multivariate Empirical Mode Decomposition (MEMD) (Rehman & Mandic, 2010).
This method can jointly decompose multidimensional signals and ensure that the IMFs obtained
from the decomposition in each dimension are
matched in both the number and frequency spectrum. In recent years, Memd-based multivariate
decomposition ensemble models (M-DEMs) have
been widely used in financial markets and outperformed univariate DEM in terms of prediction
accuracy (J. Bai et al., 2023; Deng, Huang, Hasan,
& Bao, 2022; Yao, yang Zhang, & Zhao, 2023; Zou
& He, 2022). Among them, Yao et al. (2023) proposed the MEMD-TCN hybrid model for stock
index prediction, and emphasized that the TCN
took less memory in training, provided higher
accuracy, and avoided the problem of gradient
explosion or gradient vanishing, which is common
in RNNs.
Research into multivariate decomposition algorithms became popular, with a subset of articles
pointing out that MEMD requires longer computation times when dealing with high-dimensional
data (Lang et al., 2018; Mourad, 2023). In addition to the inefficiency, Cai, Li, Zhang, and
Wu (2025) experimentally demonstrated that the
decomposition quality of MEMD is far inferior to
that of EMD by using periodicity and symmetry metrics. Such limitation affects the subsequent
prediction accuracy. Therefore, they proposed a
new algorithm - Multivariate Aligned Empirical Modal Decomposition (MA-EMD). Based
on EMD, the algorithm measures the difference
between IMFs based on the frequency representation of the extreme points that support the
2

alignment with IMFs. So far, no research that
applied the MA-EMD algorithm to forecast the
movement of financial assets has been found, and
this is one of our motivations in exploring its
effectiveness and usefulness.
In addition, one issue that cannot be ignored
is how to deal with noise in financial data. These
noises may originate from the fluctuations of market transactions, interference from external factors, and so on (Black, 1986). Many studies have
shown that the accuracy of financial time series
forecasting can be effectively enhanced by appropriate noise reduction (Alrumaih & Al-Fawzan,
2002; Bao, Yue, & Rao, 2017; Hassani, Dionisio, &
Ghodsi, 2010; Yu, Ming, Sumei, & Shuping, 2020).
But the majority of the current decompositionensemble models do not fully consider the noises
that exists in the original data (J. Wang & Liu,
2024). As a result, models might overfit the noisy
data during the training process, and the predictions with such models might be biased or
underfitting when it is difficult to capture the special patterns contained in the data (Akşehir &
Kılıç, 2024).
To address these problems, this paper proposes a new SSA-MAEMD-TCN model based
on the ”M-DEM” forecasting framework, aiming at improving the forecasting accuracy and
removing the bias of the investor view for the
Black-Litterman model. The difference between
this research and previous research include: First,
we replace MEMD with the MA-EMD algorithm
for more efficient and accurate sequence decomposition. Secondly, Singular Spectrum Analysis
(Broomhead & King, 1986) is very effective in
noise reduction of financial data(Lahmiri, 2018;
Q. Tang, Fan, Shi, Huang, & Ma, 2021; C.H. Wang, Yuan, Zeng, & Lin, 2024), so SSA was
used to reduce its impact.
The US stock data from January 2020 to
September 2023 was selected to test the proposed model’s predictive performance and the
performance of the portfolio constructed. We document the differences in the predictive metrics
between SSA-MAEMD-TCN and the benchmark
model, as well as compare the performance of the
Black-Litterman based on the generated predictive model with the other benchmark portfolio
models in terms of return, volatility, and other
performance measures over different holding periods.

The contributions of this paper include the
following three aspects:

• This paper explores the application of the MAEMD algorithm in financial time series forecasting and proves its superiority in dealing with
financial data through empirical research, which
provides a reference for subsequent related
research.
• The SSA-MAEMD-TCN model is proposed to
improve the approach based on M-DEM: the
introduction of MA-EMD supports a more
efficient and better-quality decomposition; the
introduction of SSA effectively reduces the
impacts of noise in financial data on the subsequent forecasting steps.
• The proposed hybrid model generates more
accurate and less biased investor views for the
Black-Litterman model. The empirical results
show that the methodology generates much
higher returns in the U.S. stock market.
The article is organized as follows: Section 2
reviews past approaches to generating subjective
views and the application of M-DEM and SSA
in financial markets. Section 3 delves into the
core technique’s theoretical foundations. Section
4 introduces the proposed model and details the
research process. Section 5 evaluates the experimental results, focusing on forecast accuracy and
portfolio performance. Finally, Section 6 summarizes the main findings and discusses future
research directions.

2 Related Work
This research focuses on the academic areas of
generating investor perspectives to enhance BL
portfolio and applying M-DEMs and SSA method
in financial market.
Early studies mainly generated subjective
views through traditional econometric models. Beach and Orlov (2007) first applied the
EGARCH-M model to global asset allocation,
dynamically estimated the expected returns and
conditional variances of assets in 20 countries,
and generated dynamic opinions for the BL
model, with returns exceeding the market equilibrium weight and Markowitz optimal allocation. Palomba (2008) combined the multivariate
GARCH model with the BL model and used the
FDCC model to estimate and predict asset returns
3

and covariance matrices, achieving better riskreturn balance and higher information ratio in
tactical asset allocation (TAA). Duqi et al. (2014)
used the EGARCH-M model to capture volatility clustering, leverage effects, and asymmetry,
construct stock portfolios at different risk levels, and calculate expected excess returns through
the BL model, performing well in risk-stratified
allocation.
As machine learning advances, blending econometric and machine learning models effectively
generates investor views in BL models. Pyo and
Lee (2018) combined three machine learning models (Gaussian Process Regression, Support Vector
Regression, and Artificial Neural Networks) with
the GARCH model to forecast asset volatility
in the KS200. They divided assets into high-risk
and low-risk groups, constructing a BL portfolio
reflecting investor views. Results showed this portfolio had higher Sharpe ratios and positive alpha.
Kara et al. (2019) used the GARCH model to predict stock indices and Support Vector Regression
to map volatility forecasts to return forecasts, generating investor-view vectors for the BL model.
This approach outperformed market indices in the
BIST-30 and DJI, showing better portfolio returns
and Sharpe ratios across holding periods. Lei
(2019) proposed a PCA-AHP framework to tackle
overfitting in high-dimensional asset allocation,
reducing prediction errors in Chinese stock-market
sector indices via dimensionality reduction and
weight optimization in the BL model.
Recently, deep learning and decomposition
ensemble models (DEMs) have also been gradually
applied to generate subjective views and improve
the performance of asset portfolios. Rezaei et al.
(2021) developed a CEEMD-CNN-LSTM hybrid
model for stock price prediction. Investor views
based on its predictions significantly enhance
asset allocation effectiveness. Barua and Sharma
(2022) used a CNN-BiLSTM model to predict future stock prices and a dynamic EWMA
covariance matrix to emphasize recent data.
Their follow-up study (Barua & Sharma, 2023)
built a CEEMDAN-GRU-XGBoost framework,
using GRU to predict sentiment indicators and
XGBoost to generate ETF views. The improved
BL model outperformed others over six investment
periods.
Most DEMs are univariate forecasting models that only decompose and forecast closing

prices. Multivariate versions of EMD have been
developed, including B-EMD (Rilling, Flandrin,
Gonçalves, & Lilly, 2007), T-EMD (ur Rehman &
Mandic, 2009), and MEMD (Rehman & Mandic,
2010). MEMD is widely used in financial forecasting and forms Multivariate Decomposition
Ensemble models (M-DEMs). Deng et al. (2022)
proposed the MEMD-LSTM model, combining
MEMD for multidimensional time-frequency analysis and feature extraction with OATM-optimized
LSTM hyperparameters. Experiments show that
its multistep prediction surpasses EMD-LSTM
and traditional LSTM. Zou and He (2022)
developed a MEMD-CNN multiscale model for
risk prediction, outperforming single-scale models like ARMA-GARCH and VMD-CNN. J. Bai
et al. (2023) built a hybrid prediction model
blending ”fuzzy computing” and ”decompositionensemble”. Using MPE for feature selection, MI
and NRS for filtering, and MEMD for decomposition, features entered an LSTM network. Though
capturing market complexity, it struggles with
unexpected events and has prediction delays. Yao
et al. (2023) proposed a hybrid MEMD-TCN
model for stock index prediction, decomposing
multidimensional data like opening, high, low,
closing prices and trading volumes. The model
outperforms univariate DEM and M-DEM combined with recurrent neural networks (RNNs) in
stock index forecasting in several countries. It also
emphasizes the superiority of TCNs compared to
rnns.
However, MEMD suffers from computational
inefficiency when dealing with multivariate data.
Lang et al. (2018) proposed the FMEMD algorithm, which is far more computationally efficient
than MEMD when dealing with large-scale multidimensional signals. Thirumalaisamy and Ansell
(2018) proposed the FAMVEMD algorithm, which
is much more practical when dealing with onedimensional and three-dimensional synthetic signals. In addition to being computationally slow,
Cai et al. (2025), standing for the prediction
task, also pointed out that MEMD suffers from
relaxing the quality constraints on subsequences,
generating unnecessary frequency components,
which negatively affects the final prediction performance. For this reason, they proposed Multivariate Aligned Empirical Mode Decomposition
(MA-EMD). The algorithm first applies the standard EMD to each variable individually and then
4

takes the IMFs of the target variable as a reference, uses the Kullback-Leibler dispersion (KLD)
to measure the frequency similarity between the
IMFs of different variables and aligns them to the
corresponding sub-prediction models. It is experimentally demonstrated that M-DEM combined
with MA-EMD outperforms the MEMD-based
model regarding prediction accuracy.
In addition, reducing noise in financial data
is another effective strategy to improve prediction results. However, existing DEM and M-DEM
frameworks seldom consider the effects of noise
when predicting financial data (J. Wang & Liu,
2024). Singular Spectrum Analysis (SSA) (Broomhead & King, 1986) is a powerful noise reduction algorithm that is very effective in financial
forecasting tasks. Lahmiri (2018) proposed an
intraday stock price prediction method based on
SSA and Support Vector Regression (SVR) combined with Particle Swarm Optimization (PSO).
Experimental results show that the method outperforms traditional prediction models in terms of
accuracy. Q. Tang et al. (2021) used wavelet transform (WT) and SSA to reduce the raw data and
then input the smoothed sequences into LSTM
for prediction. This approach resulted in better
prediction performance than feeding the original
sequences directly into LSTM. (C.-H. Wang et al.,
2024) proposed a hybrid deep learning framework
SSA-PSO-LSTM for predicting stock index prices
and their fluctuations. Experimental results show
that this method has higher prediction accuracy
during high volatility than the remaining models,
such as PSO-LSTM.
In summary, existing research has achieved
some results in generating investor views and
applying M-DEM, but there is still room for
improvement. This paper proposes a hybrid SSAMAEMD-TCN model which provides a new
method for generating investor opinions and fills
the research gap of M-DEMs in this field.

Fig. 1 illustrates the key parameters and process
of the model.

3.1.1 Calculating Priori Returns:
Starting from market equilibrium conditions, prior
estimates of asset expected returns are derived
through reverse optimization. Specifically, assuming that the market portfolio weights are based
on market capitalization, the implied equilibrium
returns are obtained using the reverse process of
mean-variance optimization, as shown below:
Π = λΣwmkt

(1)

where, Π is the implied excess return vector (N ×
1), λ is the risk aversion parameter. In this paper,
we set λ = 2.5 to represent the world average
risk aversion index. Σ is the covariance matrix of
excess returns (N ×N ), wmkt is the vector of asset
weights based on market capitalization (N × 1).

3.1.2 Setting Investor Views:
Investor views are treated as new information,
which can be either absolute or relative. These
views are then converted into three parameters.
View Matrix P : The view matrix P is a k×n
matrix, where k is the number of views and n is
the number of assets. It maps the investor’s views
onto specific assets, indicating which assets correspond to each view. Since we are only considering
absolute views, each row of the P matrix will have
exactly one non-zero element, which is located in
the column corresponding to the asset and equals
1. Specifically, suppose you have k absolute views,
corresponding to assets i1 , i2 , . . . , ik (where ij is
the index of the asset, and 1 ≤ ij ≤ n), then the
matrix P can be represented as:




0 0 ··· 1 ··· 0
0 0 · · · 1 · · · 0


P = . . . . . .
 .. .. . . .. . . .. 

3 Technical Background

(2)

0 0 ··· 1 ··· 0

3.1 Black-Litterman Model

View Return Vector Q: This is a k × 1 vector representing the specific expected returns for
each view. It captures the expected return for each
individual view from the investor’s perspective. If
we have forecasted the returns of k stocks to be
R1 , R2 , . . . , Rk , the Q matrix can be represented

To solve the problems of input sensitivity and estimation error maximization in traditional meanvariance optimization, Fischer Black and Robert
Litterman developed the Black-Litterman model.

5

Fig. 1 The process of Black-Litterman model (Idzorek, 2007).

as:

3.1.3 Calculating Posterior Returns:




R1
R2 
 
Q= . 
 .. 

The priori return distribution and the subjective view distribution are combined using bayesian
methods to calculate the posterior estimate of
asset expected returns, as shown below:

(3)

Rk
View Error Covariance Matrix Ω: The
view error covariance matrix Ω is a k × k diagonal matrix representing each view’s uncertainty or
confidence level. The elements along the diagonal
represent the variance for each view, reflecting the
investor’s confidence in that view. When setting
up the Ω matrix, we need to determine a scalar τ .
In this example, we use 500 days of historical data
for calculating the assets’ priori returns and priori
covariances and therefore choose τ = 1/500.
(p1 Σp′1 )τ
0
···
′

0
(
p
Σ
p
)
τ
···
2
2

Ω=
..
..
..

.
.
.



0

0

0
0
..
.

· · · (pk Σp′k )τ

(τ Σ)−1 Π + P T Ω−1 Q
(5)
And the formula for the posterior covariance
(Σpost ) is:
Σpost = Σ + (τ Σ)−1 + P T Ω−1 P



−1

(6)

3.1.4 Optimizing Asset Allocation:
The posterior returns and the posterior covariance
matrix are input into the mean-variance model for
optimization to obtain the specific asset allocation
weights wbl , as shown below:







−1

µpost = (τ Σ)−1 + P T Ω−1 P

(4)

−1

wbl = (λΣpost )

where, pi is the asset weight vector for the i-th
view, reflecting the investor’s assessment of that
asset’s importance or weight in the view. Σ is the
covariance matrix of asset returns, indicating the
correlations and volatilities of the asset returns.

µpost

(7)

3.2 Multivariate Aligned Empirical
Mode Decomposition
(MA-EMD)
MA-EMD is an improved version of MEMD, aiming to solve the problems in multivariate decomposition, such as degradation of decomposition

6

3.2.2 Frequency Alignment Module

quality and computational inefficiency. In MAEMD, each variable is independently decomposed
by the standard EMD to produce a series of
IMFs. However, EMD leads to different IMFs
for eachvariable, which the KLD-based frequency
alignment module can solve. The main task of
this module is to use the extreme value intervals
of IMFs to characterize their frequency properties
and then measure the similarity of the two IMFs
using the Kullback-Leibler Divergence (KLD).
Finally, frequency alignment is achieved by assigning each IMF of the related variable (OHLV) to
the IMF of the target variable (Close) with the
smallest KLD.

The algorithm uses the spacing between extreme
values as a frequency representation method.
The extreme value spacing distribution is sufficient for high-frequency IMFs. However, as the
extreme values become more and more sparse, the
applicability of this indicator for low-frequency
IMFs gradually decreases. Cai and Li (2024)
concluded that partial decomposition does not
degrade the performance of the decomposition
ensemble model. Therefore, the MA-EMD algorithm stops the decomposition process by setting
a threshold ω of the number of extreme values. In
this paper, we take ω = 20.
The module uses Kullback-Leibler divergence
(KLD) to measure the similarity between the
distributions of the extreme value intervals of
two IMFs. Before calculating the KLD value, the
probability distribution of each IMF needs to be
expanded to include all possible extreme value
intervals for each IMF. Then, Laplace smoothing
is applied to ensure that the probabilities are not
zero. The calculation formula is as follows:
(
Nx
+ ϵ, if x ∈ X
′
P (X = x) = N −1
(11)
ϵ,
if x ∈ X ′ \ X

3.2.1 EMD Algorithm
Empirical Mode Decomposition (EMD) is an
adaptive signal processing technique suitable for
nonlinear and non-stationary signals. Its core is
decomposing complex signals into a set of IMFs
with different time scales. The first step is to
determine the local extreme values of the signal
x(t) and use spline interpolation to generate the
upper envelope u(t) and the lower envelope l(t).
Then calculate the average value m(t) of the two
envelopes:
u(t) + l(t)
m(t) =
(8)
2
Following this, the mean envelope is subtracted
from the original signal to obtain the intrinsic
mode function (IMF):

h(t) = x(t) − m(t)

where: Nx refers to the number of occurrences of
the mechanism interval x in the IMF; N is the
total number of extrema points; X is the set of
extrema intervals that exist in this particular IMF;
X ′ represents the set of extended extrema intervals
across all IMFs; ϵ is a very small constant, ϵ =
0.000001.
The final probability distribution P ′′ (X = x)
and the formula for calculating the similarity
between distributions using KLD are given by Eqs.
12 and 13, respectively.

(9)

A qualified IMF must verify that h(t) satisfies two conditions. First, the meaning of the
upper and lower envelopes must be zero. Second, the difference between the number of extreme
values and the number of zero crossings of h(t)
must not exceed 1. If these two conditions are not
met, the iterative decomposition process continues
until the conditions are met. After the decomposition is completed, the original signal x(t) can
be expressed as the sum of multiple IMFs and a
residual term.

x(t) =

n
X

hi (t) + rn (t)

P ′ (X = x)
′
x∈X ′ P (X = x)

(12)

P ′′ (x)
Q′′ (x)

(13)

P ′′ (X = x) = P
DKL (P ′′ ∥ Q′′ ) =

X
x∈X

P ′′ (x) log

where P and Q represent the probability distributions for the related variable and the target
variable, respectively, and X is the extended range
of the two distributions.

(10)

i=1

7

3.3 Temporal Convolutional
Network (TCN)

3.3.2 Dilated Convolution
Dilated convolution introduces a skip interval in
the convolutional kernel, allowing the receptive
field of TCN to grow exponentially. For example, when the dilation factor is 2, the convolution
kernel skips one data point during the convolution operation. As the network depth increases,
the dilation factor gradually increases, enabling
the network to capture longer temporal dependencies, as shown in Fig. 2. For a dilation factor d,
the dilated convolution operation Fd at the s-th
element of the sequence is defined as:

TCN is a deep learning model specialized in processing time series data. Its design is inspired
by Convolutional Neural Networks (CNNs) structure and optimized and improved for time series
prediction tasks. The core innovation of TCN is
the combination of dilated convolution and causal
convolution techniques.

3.3.1 Causal Convolution
Causal convolution is one of the core components of TCN, ensuring that the network uses
only the current and past information for predictions, without leaking any future information. In
causal convolution, the convolution operation is
performed only on the current time step and the
data before it, achieved by adding zero-padding
to the input data. For a one-dimensional sequence
input x ∈ Rn and a convolutional filter f :
{0, . . . , k − 1} → R, the causal convolution operation F at the s-th element of the sequence is
defined as:

F (s) = (x ∗ f )(s) =

k−1
X

f (i) · xs−i

Fd (s) = (x ∗d f )(s) =

k−1
X

f (i) · xs−d·i

(15)

i=0

where d is the dilation factor, and k is the size of
the convolutional kernel.

3.3.3 Residual Connections
To alleviate the issues of vanishing and exploding gradients during the training of deep networks,
TCN introduces residual connections. Residual
connections make it easier for the network to learn
identity mappings by directly adding the input
to the output. In TCN, each residual block typically consists of two dilated convolution layers,
weight normalization, ReLU activation functions,
and a dropout layer. If the dimensions of the input
and output are not consistent, a 1x1 convolution
can be used to adjust the dimensions to allow for
element-wise addition. As shown in Fig. 3. For an
input x and the output of a convolutional layer
F (x), the output of a residual block o is:

(14)

i=0

where, k is the size of the convolutional kernel, and s is the index of the current time step.
This convolution ensures that the output F (s)
depends only on x0 , . . . , xs , and not on future
inputs xs+1 , . . . , xn .

o = Activation(x + F (x))

(16)

where Activation is a nonlinear activation function, such as ReLU.

3.4 Singular Spectral Analysis
(SSA)
SSA is a classic method for processing time series
data. It embeds data into a high-dimensional
space and uses a sliding window combined with
singular value decomposition (SVD) to extract the
main patterns.
The first step of SSA involves reconstructing
the time series into a matrix containing lagged

Fig. 2 A dilated causal convolution with dilation factors
d = 1, 2, 4 and filter size k = 3 (S. Bai, Kolter, & Koltun,
2018).

8

Fig. 3 The residual block in TCNs. Left plot: an 1 × 1 convolution is added when the residual input and output have
different dimensions. Right plot: an example of the residual connection in TCN (S. Bai et al., 2018).

{σ1 , σ2 , . . . , σr } are partitioned into m groups
{I1 , I2 , . . . , Im }, with each group corresponding to
a submatrix. For the i-th group, the submatrix Xi
is formed by multiplying the corresponding singular values, left singular vectors, and right singular
vectors:
ki
X
σij uij viTj
(19)
Xi =

vectors. Starting with a time series of length N ,
denoted as {x1 , x2 , ..., xN }, a window size L is
selected, typically much smaller than N (usually
L < N/2). This window is used to construct
a trajectory matrix X , which has L rows and
K = N − L + 1 columns. The matrix X is built
by sliding the window across the time series, with
each row representing a ”snapshot” of the series
at a given time point. The structure of the time
series over time is thus captured in this matrix:



x1
 x2

X= .
 ..

x2
x3
..
.


. . . xK
. . . xK+1 

.. 
..
.
. 

xL xL+1 . . .

j=1

where uij and vij are the ij -th columns of U and
V , respectively.
The final step involves reconstructing the
time series from the grouped submatrices. Antidiagonal averaging is performed on each submatrix to obtain the reconstructed series. For each
group {I1 , I2 , . . . , Im }, the corresponding submatrix Xi is averaged along the anti-diagonals to
produce a sequence ŷi of length N = L + K − 1.
The formula for anti-diagonal averaging is:

(17)

xN

With the trajectory matrix in hand, the next
step is to decompose it further using SVD. SVD
is a technique that breaks down the matrix into
three parts: two orthogonal matrices and a diagonal matrix containing the singular values. The
SVD of matrix X is expressed as:

X = U ΣV T

n

ŷi (k ) =

k
1 X
Xi (j, k − j + 1)
nk j=1

(20)

where nk is the number of elements on the k th anti-diagonal. The reconstructed original time
series ŶN is obtained by summing these sequences:

(18)

where U ∈ RL×L contains the left singular vectors, Σ ∈ RL×K is the singular value matrix with
diagonal values representing the strength of each
component, and V T ∈ RK×K contains the right
singular vectors.
Following the decomposition, the time series
is divided into multiple linearly independent
subseries through grouping. The singular values

ŶN =

m
X

ŷi

(21)

i=1

This reconstruction process combines the decomposed subseries to produce a denoised version of
the original time series.
9

Table 1 Stock Pool for Portfolio Construction
Stock

Sector

Stock

Table 2 The statistical analysis results of selected
stocks

Sector

AAPL
Technology
COST Consumer Discretionary
MSFT
Technology
AZN
Health Care
GOOG
Technology
AMGN
Health Care
GOOGL
Technology
ISRG
Health Care
NVDA
Technology
PEP
Consumer Staples
INTC
Technology
HON
Industrials
AMD
Technology
LIN
Industrials
AMZN Consumer Discretionary TMUS
Telecommunications
TSLA Consumer Discretionary CMCSA Telecommunications
NFLX Consumer Discretionary CSCO
Telecommunications

4 Experiment design

Stock

Max

Min

AAPL

194.7577

54.4499

Mean

Std

MSFT

354.6355 129.6211 248.1334 50.0254

134.6913 32.1692

GOOGL 149.1255

52.4557

NVDA

49.3326

4.8924

104.9878 24.7227
18.8433

TSLA

409.9700

24.0813

206.1424 88.8008

9.4758

AMGN

270.2472 156.2849 213.2018 18.7573

PEP

185.9797

90.6220

144.5176 21.3784

HON

217.2645

93.6586

179.9061 25.2358

Before decomposition, normalizing the data is not
negligible, as shown in the following equations:

4.1 Data description
zt,i =

To construct an investment portfolio, we carefully selected 20 representative stocks from various
industries within the NASDAQ 100 index as the
targets for our portfolio. These stocks cover key
areas such as technology, healthcare, and consumer sectors and are known for their high growth
potential and innovative capabilities, as shown in
Table 1. We collected the daily trading data of
these stocks from January 1, 2020, to August 31,
2023, from Yahoo Finance.
Among the 20 stocks, we focus on eight
key stocks, including Apple (AAPL), Microsoft
(MSFT), Alphabet Class A (GOOGL), Nvidia
(NVDA), Tesla (TSLA), Amgen (AMGN), PepsiCo (PEP), and Honeywell International (HON).
These stocks hold significant market positions in
their respective sectors, and detailed statistical
analyses are presented in Table 2.

xt,i − µi
σi

(22)

Where, xt,i is the raw value of the i-th feature at
time t. µi is the mean of the i-th feature, σi is the
standard deviation of the i-th feature.
The final step of the prediction framework is to
predict the IMF for each alignment using a TCN.
We use the sliding window method to divide the
data into many short time intervals using a fixedlength seven-step input, with the eighth data point
serving as a label. The window gradually moved
forward until the last window reached. Regarding the division of the dataset, 70% of the data is
assigned to the training set, 15% to the validation
set, and the remaining 15% as the test set (out-ofsample data). Finally, the predicted values of each
IMF are summed up to give the final prediction.
See Table 3 for specific parameters of the model.

4.2 Proposed Methodology

Table 3 Hyperparameter tuning for TCNs.

In this paper, we propose a novel stock price prediction model, SSA-MAEMD-TCN, and consider
the model as a way to enhance the subjective view
of the investor and thus optimize the performance
of Black-Litterman’s portfolio. See Fig. 4 for the
overall framework.
(1) Predict stock prices:
This stage aims to predict stock prices using
a hybrid model. First, the original time series
data are subjected to SSA denoising to reduce
the noise interference. The denoised data are then
decomposed into multiple IMFs using MA-EMD,
as shown in Fig. 5, where it can be visualized that
the number of IMFs for each variable is equivalent.

Hyperparameter

Chosen Value

Kernel sizes
Hidden size
Learning rate
Layer num
Dropout rate
Epochs
Batch size
Activation function
Optimizer

2
[64, 128]
0.001
2
[0, 0.3]
[50, 100]
[16, 32]
ReLU
Adam

(2) Construct the portfolio:
The next step is to calculate the market equilibrium return based on inputs such as market
10

Fig. 4 Procedure of the complete Black-Litterman strategy.

Close

Open

High

Low

Volume

Fig. 5 COHLV of MSFT was decomposed using MA-EMD.

capitalization, daily return rate, and risk-free rate.
Inverse optimization is used to obtain the equilibrium return after determining the risk aversion
coefficient and constructing the covariance matrix.
The asset prices predicted in the first stage are

converted into returns to generate subjective opinions. Finally, the posterior excess return and posterior covariance are obtained according to Eqs. 5
and 6, and the optimized portfolio weights are
obtained using Eq. 7.

11

4.3 Assessment metric

weights of each asset in the portfolio. The lower
the HHI value, the more diversified the portfolio
is.
n
X
HHI =
(wi )2
(27)

4.3.1 Predictive model evaluation
metrics
To evaluate the performance of the proposed
hybrid model, we employ three metrics for assessment: RMSE, MAPE, and the R2 , based on
Eqs. (23, 24, 25). RMSE measures the difference between the predicted and actual values; the
smaller the value, the closer the model’s prediction is to the exact value. MAPE expresses the
error as a percentage, mainly reflecting the difference between the predicted and actual values. R²
measures the model’s goodness of fit, and its value
ranges from 0 to 1. The closer the value of R² to
1, the better the model’s fit to the data.
v
u
u1 N
X
(yt − ŷt )2
(23)
RMSE = t
N t=1

i=1

where wi represents the weight of the i-th stock,
and n is the total number of stocks in the market.

5 Result Discussion
5.1 Prediction results
In this section, we perform a comprehensive comparative analysis of the proposed SSA-MAEMDTCN hybrid model with the benchmark models
(MAEMD-TCN, MEMD-TCN, MAEMD-LSTM)
to evaluate their performance in stock price prediction tasks. We used three evaluation metrics:
RMSE, MAPE and R2 . These metrics can reflect
the predictive accuracy and goodness-of-fit of the
models from different perspectives.
From the data in Table 4, SSA-MAEMD-TCN
is significantly better than MAEMD-TCN for all
the stocks. For example, the RMSE of the former
is 1.1355 in GooGL, while that of the latter is
1.7628, and the predicted values of SSA-MAEMDTCN are closer to the actual values. The former
also shows better performance in terms of MAPE
and R². In addition, the prediction performance of
SSA-MAEMD-TCN is better than that of all the
benchmark models mentioned above. The advantage of the proposed model is the introduction
of SSA for data preprocessing and noise reduction. This step dramatically reduces the impact of
noise in financial data on subsequent predictions,
enabling the model to accurately capture valid
information in the data and improve the overall
prediction results. Thus, the use of SSA for data
preprocessing is necessary.
MAEMD-TCN outperforms MEMD-TCN in
the prediction of most of the stocks. In the case of
AAPL, for example, the RMSE of MAEMD-TCN
is 1.9308, which is lower than that of MEMDTCN (2.3602). Regarding MAPE, the value of
MAEMD-TCN is 0.0089, which is lower than that
of MEMD-TCN (0.0109), indicating that its prediction error is minor. R² also shows that the
fit of the MAEMD-TCN model (0.9796) to the
data is better than that of MEMD-TCN (0.9674).
MA-EMD solves the problem of the deficiency of
MEMD in multivariate decomposition. Secondly,

N

MAPE =

2

1 X yt − ŷt
× 100%
N t=1
yt

PN

R = 1 − Pt=1
N

(yt − ŷt )2

2
t=1 (yt − ȳ )

(24)

(25)

In the aforementioned formula, yt denotes the
actual value, ŷt denotes the predicted value, ȳ is
the mean of the actual values, and N represents
the total number of samples.

4.3.2 Portfolio evaluation metrics
The Sharpe Ratio is a commonly used metric in finance for measuring the performance of
investment portfolios. It assesses the risk-adjusted
return of a portfolio, that is, the excess return per
unit of total risk.
Sharpe Ratio =

Rp − Rf
σp

(26)

where, Rp is the expected return of the portfolio.
Rf is the risk-free rate. σp is the standard deviation of the portfolio’s returns, representing the
total risk.
The Herfindahl-Hirschman Index (HHI) is
used to quantify the concentration and diversification of a portfolio. The HHI reflects the concentration of a portfolio by adding the squares of the
12

Table 5 Learning ability of different models for
high-frequency imfs.

Table 4 Prediction accuracy of different models.
Stock
AAPL

metric Model-11 Model-22 Model-33 Model-44

RMSE
MAPE
R2
MSFT
RMSE
MAPE
R2
GOOGL RMSE
MAPE
R2
NVDA RMSE
MAPE
R2
TSLA
RMSE
MAPE
R2
AMGN RMSE
MAPE
R2
PEP
RMSE
MAPE
R2
HON
RMSE
MAPE
R2

1.5629
0.0076
0.9866
2.9177
0.0078
0.9899
1.1355
0.0080
0.9919
0.8591
0.0215
0.9899
4.4226
0.0138
0.9874
1.4308
0.0050
0.9866
1.0452
0.0047
0.9652
1.2415
0.0050
0.9604

1.9308
0.0089
0.9796
3.3560
0.0083
0.9866
1.7628
0.0119
0.9805
0.9658
0.0218
0.9876
5.6694
0.0206
0.9793
2.5348
0.0088
0.9579
2.1527
0.0104
0.8533
1.3809
0.0058
0.9508

2.3602
0.0109
0.9674
4.4554
0.0122
0.9765
2.1211
0.0154
0.9716
1.3608
0.0252
0.9753
5.4443
0.0220
0.9809
2.9539
0.0120
0.9429
1.6807
0.0080
0.9106
1.4816
0.0063
0.9435

Stock

2.9410
0.0125
0.9491
5.4004
0.0139
0.9652
1.9092
0.0137
0.9772
1.5789
0.0322
0.9624
5.9504
0.0221
0.9771
3.6569
0.0129
0.9124
2.2738
0.0105
0.8226
1.8478
0.0075
0.9122

SSA-MAEMD-TCN
IMF

IMF 1
IMF 2
IMF 3
IMF 1
MSFT
IMF 2
IMF 3
IMF 1
GOOGL IMF 2
IMF 3
IMF 1
NVDA
IMF 2
IMF 3
IMF 1
TSLA
IMF 2
IMF 3
IMF 1
AMGN IMF 2
IMF 3
IMF 1
PEP
IMF 2
IMF 3
IMF 1
HON
IMF 2
IMF 3
AAPL

1:SSA-MAEMD-TCN, 2:MAEMD-TCN, 3:MEMD-TCN,
4:MAEMD-LSTM

it uses KLD to evaluate the frequency similarity
of different variable IMFs to ensure the consistency and matching of the decomposed IMFs in
frequency. This improved decomposition method
provides TCN with higher-quality input features,
enabling MAEMD-TCN to capture patterns and
regularities in time series accurately.
In comparing MAEMD-LSTM and MAEMDTCN, the later shows advantages in all three
metrics. Compared with LSTM, the advantage of
TCN is that it can effectively extend the sensory field to capture long-distance dependencies in
time series, is less prone to the problem of gradient vanishing or gradient explosion during the
training process, and has a more stable prediction performance. Therefore, MAEMD-TCN can
more effectively extract the time series’ shortrange and long-range dependency features, thus
realizing more accurate prediction results.
The following section will analyze how the
noise reduction step improves the model performance. The EMD-based decomposition algorithm
decomposes the sequence into multiple IMFs, and
the forecasting model outputs the forecast values of these components and integrates them.
The high-frequency components (such as IMF 1)
contain a lot of noise, which will interfere with

MAEMD-TCN

RMSE

2

R

IMF

RMSE

R2

0.0155
0.0052
0.0046
0.0130
0.0072
0.0088
0.0181
0.0081
0.0081
0.0157
0.0062
0.0093
0.0218
0.0066
0.0105
0.0333
0.0121
0.0153
0.0173
0.0048
0.0060
0.0235
0.0088
0.0109

0.8530
0.9921
0.9978
0.9263
0.9863
0.9933
0.8900
0.9865
0.9954
0.7815
0.9851
0.9909
0.8254
0.9944
0.9931
0.8620
0.9927
0.9934
0.7539
0.9882
0.9968
0.8523
0.9917
0.9956

IMF 1
IMF 2
IMF 3
IMF 1
IMF 2
IMF 3
IMF 1
IMF 2
IMF 3
IMF 1
IMF 2
IMF 3
IMF 1
IMF 2
IMF 3
IMF 1
IMF 2
IMF 3
IMF 1
IMF 2
IMF 3
IMF 1
IMF 2
IMF 3

0.0369
0.0122
0.0045
0.0547
0.0184
0.0104
0.0397
0.0130
0.0075
0.0400
0.0102
0.0084
0.0310
0.0125
0.0070
0.0615
0.0276
0.0141
0.0341
0.0120
0.0067
0.0457
0.0129
0.0067

-0.4201
0.9424
0.9960
-2.9550
0.8912
0.9820
-0.3396
0.9437
0.9906
-0.6051
0.9434
0.9874
0.5355
0.9467
0.9940
0.5971
0.9450
0.9917
0.1095
0.9430
0.9907
0.3082
0.9606
0.9957

the model’s learning of effective trends. Through
SSA denoising, SSA-MAEMD-TCN significantly
outperforms MAEMD-TCN in predicting highfrequency components. As shown in Table 5,
RMSE and R² indicators represent the model’s
learning ability.
In IMF 1 of AAPL, RMSE decreased from
0.036864 to 0.015491 (58.0%), and MSFT from
0.054744 to 0.012974 (76.3%). The R² indicator
also improved significantly, from -0.339 to 0.890 in
the GOOGL data. The denoising effect still exists
in the sub-high frequency component (IMF 2),
but the improvement is weakened. For example,
the RMSE of NVDA is reduced by 39.8%, and
the R² of GOOGLE is improved by 3.23%. The
noise impact is already low in the intermediate
frequency component (IMF 3), and the denoising
optimization effect is limited. In addition, combined with Fig. 6, after denoising, the prediction
model has better fitting effects on the training
set, validation set, and test set. These results
show that SSA can effectively separate noise from
trend information, allowing the model to more
accurately capture the trend characteristics in the
series.

13

Fig. 6 Effectiveness of different models in fitting IMF 1 of AAPL (The data has not been denormalized).

5.2 Portfolio Optimization

The empirical results in Fig. 7 confirm the
significant advantages of the BL model for short
holding periods. When looking at the overall data,
all strategies exhibit high Sharpe ratios and decent
annualized returns under different holding periods, indicating that the market is in a state where
the trend is more pronounced or the volatility is
relatively orderly, giving all investment strategies
a chance to achieve profitability.
With a holding period of 1 day, the BL model
achieves an annualized return of more than 80%
and a Sharpe ratio of about 4.5, which significantly
outperforms the other models. This is mainly due
to the proposed hybrid model, which improves
the accuracy of the subjective view. By keenly
capturing market fluctuations, the BL model adequately adjusts the portfolio, resulting in high
returns and better risk control during the short
holding period. As the holding period lengthens,
the advantages of BL’s annualized returns and
Sharpe ratios diminish. Longer periods increase
market uncertainty, and subjective views are no
longer adapted to market conditions. Based on
these outdated views, portfolio adjustments are
no longer effective, and yield and volatility control advantages are no longer as pronounced in the
short term.
To further verify the above experimental findings, we count the times the four models obtain
the highest return and the times they get the
lowest return under different holding periods, as
shown in Table 6. For the 1-day and 3-day holding
periods, the BL model obtains the highest return
53 times and 55 times, accounting for 40.15%
and 42.31%, respectively. In contrast, the lowest
returns are only 7 times, 5.30% and 5.38%, highlighting its stability and superiority in short-term

In this section, we construct investor views based
on the prediction results of the proposed model
for eight stocks and construct an investment portfolio through the BL model. We select 132 trading days from February 23, 2023, to August 31,
2023 (out-of-sample data in the previous section)
as the backtesting period. We select the meanvariance model (MV), the equal-weight model
(EW), and the market-weighted portfolio model
(MW) as benchmark models. This paper adopts
two backtesting methods: ”rolling data scheme”
and ”rebalancing strategy.” In addition, we also
compare the weight distribution of the portfolios
constructed by the BL model and the MV model
to evaluate whether the BL model can make up
for the shortcomings of the MV model.

5.2.1 Rolling data scheme
This study uses a rolling data scheme to assess the
effectiveness of portfolios under different holding
periods. Starting from the initial date of the specified test period, the portfolio is constructed based
on the forecast data of that day. It is assumed
to remain unchanged for a predetermined holding period (e.g., 10 days). After the first holding
period, the portfolio is reconstructed using forecast data from the second day of the test period,
and the process is repeated until the end of the
test period. The portfolio’s overall performance is
assessed by calculating and annualizing the average return over the holding period, which is taken
to be 1, 3, 5, 10, and 20 days. The focus is
on observing the relationship between annualized
returns and Sharpe ratios and the length of the
holding period.

14

Fig. 7 Evaluate portfolio performance using annualized returns and Sharpe ratios.

Table 6 The number of times with the highest and lowest return at rolling runs.
Period

Number of Runs

1
3
5
10
20
Overall
Percentage

132
130
128
123
113
626

Highest Return
Lowest Return
BL-Highest EW-Highest MW-Highest MV-Highest BL-Lowest EW-Lowest MW-Lowest MV-Lowest
53 (40.15%)
24
1
54
7 (5.30%)
22
49
54
55 (42.31%)
20
10
45
7 (5.38%)
40
32
51
40 (31.25%)
19
14
55
14 (10.94%)
32
31
51
28 (22.76%)
25
22
48
8 (6.50%)
46
24
45
26 (23.01%)
25
24
38
14 (12.39%)
29
26
44
202
113
71
240
50
169
162
245
32.27%
18.05%
11.34%
38.34%
7.99%
27.00%
25.88%
39.14%

Table 7 Performance of portfolio management with
rebalancing

investment. When the holding period is gradually extended, the number of times that the BL
model obtains the highest rate of return gradually
decreases, and the number of times that it obtains
the lowest rate of return gradually increases, especially when the holding period is 20 days, it
reaches 14 times, accounting for 12.39%, which
indicates that its volatility and riskiness increase
in a more extended holding period.

Period
1

3

5

5.2.2 Rebalancing Strategy
10

The rebalancing strategy dynamically manages an
investment portfolio by adjusting asset weights
to meet the investor’s initial goals. When market
fluctuations cause asset weights to deviate from
the preset ratio, the target weights are restored
by buying and selling assets. This adjustment can
be a regular adjustment based on time intervals
or an irregular adjustment based on the degree
of deviation of asset weights. This paper uses the
”periodic rebalancing strategy” for experiments
and assumes that the transaction cost is 0.2%.
The holding period is consistent with the previous
article. Table 7 shows the performance of four different portfolio strategies under different holding
periods.

20

Metrics

BL

MV

EW

MW

Annual Return
Annual Volatility
Sharpe Ratio
Annual Return
Annual Volatility
Sharpe Ratio
Annual Return
Annual Volatility
Sharpe Ratio
Annual Return
Annual Volatility
Sharpe Ratio
Annual Return
Annual Volatility
Sharpe Ratio

1.0335
0.1889
4.2154
0.7963
0.1748
3.6301
0.6689
0.1660
3.1850
0.6056
0.1317
3.4953
0.6264
0.1424
3.3877

0.3399
0.2038
1.3967
0.5411
0.1841
2.4209
0.5801
0.1466
3.1359
0.5750
0.1332
3.2902
0.5322
0.1604
2.5836

0.5247
0.1508
2.8288
0.5240
0.1437
2.9633
0.5236
0.1370
3.0292
0.5261
0.1174
3.4048
0.5368
0.1095
3.7542

0.5951
0.1826
2.6629
0.5973
0.1731
2.8157
0.6010
0.1647
2.9027
0.5996
0.1390
3.2879
0.5946
0.1386
3.3066

The BL model shows a significant holding
period effect: when the holding period is extended
from 1 day to 20 days, the expected annualized
return decreases from 1.0335 to 0.6264, and the
Sharpe ratio decreases from 4.2154 to 3.3877. It
is the only strategy whose Sharpe ratio decreases
with the increase of the holding period. This
shows that the BL model achieves higher returns

15

Fig. 8 Cumulative return of 1-day rebalancing.

Table 8 Performance of portfolio management with
1-day rebalancing

in a short holding period by combining market
equilibrium returns and subjective views through
a Bayesian framework. Although the transaction
costs are high, its excellent performance makes
the strategy returns less affected by transaction
costs. However, as the holding period increases,
the adaptability of investors’ views decreases,
and the performance of the BL model gradually
approaches that of the MW model. The annualized return of the MV model shows a trend of
first rising and then falling with the increase of
the holding period: from 0.3399 in a 1-day holding
period to 0.5801 in a 5-day holding period, and
then falls back to 0.5322 in 20 days (a decrease of
7.4%), while the volatility shows a trend of first
falling and then rising. In the short holding period,
the Sharpe ratio of the MV model is significantly
lower than that of the BL model, exposing the
problems of high transaction costs and parameter
misalignment under high-frequency adjustments.
In contrast, the EW and MW portfolios maintain relatively stable annualized returns under different holding periods, with gradually decreasing
volatility and gradually increasing Sharpe ratios,
verifying the characteristics of passive allocation
to achieve long-term stability by diversifying risks.
However, the passive investment strategy lacks
flexibility, and its returns and Sharpe ratios are
lower than the BL model’s in the short holding
period (1-3 days).

EW

MW

MV

BL

daily average return (%) 0.1886 0.2124 0.1323 0.3354
daily volatility(%)
0.9498 1.1505 1.2840 1.1900
Cumulative Return
0.2748 0.3117 0.1780 0.5414
Sharpe Ratio
2.8288 2.6629 1.3967 4.2154

We focus on analyzing a special case of the
rebalancing strategy: 1-day rebalancing, which is
essentially a portfolio model that generates daily
portfolio weights, as shown in Table 8 and Fig. 8.
In the framework of this strategy, the BL model
has a significantly higher average daily and cumulative return than the other strategies, with a
Sharpe ratio of 4.2154, much higher than the MV
model’s 1.3967. Although the daily volatility of
1.190% is slightly higher than that of the equalweighted portfolio’s 0.9498%, the high Sharpe
ratio confirms the effectiveness of this strategy
dynamically adjusting to the short holding period.
Combined with the accurate prediction results of
the hybrid model, the BL model achieves a balance between maximizing return and controlling
risk in a very short holding period.
The EW portfolio reflects the defensive nature
of the passive strategy with the lowest volatility and stable returns. Still, its cumulative return

16

Fig. 9 Comparison of different portfolio weights and returns in 2023.07.26

Table 9 Portfolio Weight Distribution

is only 50% of the BL model’s. The MV model
has the lowest daily return and the highest daily
volatility. These data show that its reliance on
historical data for optimization has the risk of
overfitting, and the transaction costs of highfrequency position adjustments further weaken
the model performance. In short, the BL model
is thoroughly dominant in high-frequency active
investment strategies.

Mean of stock number
Std of stock number
Mean of HHI
Std of HHI

MV

BL

11.190
8.183
0.245
0.157

18.030
0.937
0.131
0.009

Further deconstructing the concentration difference by HHI, the mean HHI of the BL model is
0.131, which is much lower than that of the MV
model (0.245). Its HHI standard deviation is only
0.009, which is equivalent to 5.73% of the volatility of the MV model. The MV model presents high
concentration and allocation instability compared
to the BL model. Regarding portfolio diversification, the BL model is superior across the board in
the diversification and robustness dimensions.
Fig. 9 shows the weights and returns of different investment portfolios on a particular day. The
BL model combines the views of investors, so its
investment portfolio can be regarded as an optimization of the MW portfolio. Therefore, ideally,
the single-day return of the BL model is higher.
The portfolio distribution and return of the MV
model in this day are not good enough.

5.2.3 Portfolio Diversification
In this section, we discuss the degree of portfolio
diversification for the Black-Litterman and MeanVariance models. We select the daily weights
under 1-day rebalance to test whether the BL
model has an advantage in terms of the degree
of diversification of the underlying. As seen from
the two-dimensional empirical results of weight
distribution and Herfindahl Index (Table 9), the
Black-Litterman and the Mean-Variance models
present a significant difference in underlying diversification. In terms of weight distribution, the BL
model covers 18.03 stocks with a standard deviation of 1.039, which is significantly better than
the MV model’s 11.19 stocks. Meanwhile, the former’s standard deviation is even lower than that
of the latter, suggesting that the BL model has a
more stable underlying distribution.

17

6 Conclusions

learning and online learning can be explored to
develop adaptive portfolio optimization strategies
that allow investment portfolios to automatically
adjust and optimize in response to changing market conditions.

The traditional Black-Litterman model faces challenges in the generation of investor views. This
study innovatively constructs an SSA-MAEMDTCN hybrid deep learning framework, which
integrates Singular Spectrum Analysis (SSA),
Multivariate Aligned Empirical Mode Decomposition (MA-EMD), and Temporal Convolutional
Networks (TCN) to achieve high-precision stock
price forecasting and provide reliable inputs for
the Black-Litterman model. Empirical results
demonstrate that, in terms of asset price prediction, the hybrid model significantly outperforms three benchmark models—MAEMD-TCN,
MEMD-TCN, and MAEMD-LSTM—in key metrics such as RMSE, MAPE, and R². In addition,
we also find that the prediction performance of
medium and high frequency sequences can be significantly improved by using the SSA algorithm
to process the noise reduction of the original
sequences
In portfolio management, we conducted a comprehensive performance evaluation of the investment portfolios constructed based on this model
using two backtesting methods: the rolling data
scheme and the rebalancing strategy. Comparative analyses with benchmark models such as
the Mean-Variance, Equal-Weighted, and MarketWeighted models further illustrate the superiority
of the investment portfolios generated by combining the hybrid and Black-Litterman models.
Additionally, we examined the diversification of
the investment portfolios. Compared with the
mean-variance model, the Black-Litterman model
can generate more diversified portfolios, effectively reducing concentration risk and enhancing
portfolio robustness.
Future research can focus on two main directions. First, the current model primarily relies
on daily data for single-step predictions, which
is suitable for short-term trading but has limitations in medium- and long-term investments. By
developing multi-step prediction models, we can
forecast stock price trends over longer horizons,
such as one week, one month, or even longer.
This enables investors to make informed decisions
for medium- and long-term investments. Second,
further research is needed to improve portfolio optimization methods, especially in dynamic
environments. Techniques such as reinforcement

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</reference>

<statements>
1. A hybrid is feasible and already exists in several published forms: Black-Litterman or inverse-optimization backbones combined with machine-learning or LLM views
2. It combines Singular Spectrum Analysis for denoising, Multivariate Aligned Empirical Mode Decomposition for frequency-aligned decomposition, and Temporal Convolutional Networks for deep sequence learning
3. Empirical tests on Nasdaq 100 stocks show significant forecasting improvement over MAEMD and MEMD baselines, and the optimized portfolio achieves annualized returns and Sharpe ratios far exceeding those of the traditional portfolio over a short holding period, even after transaction costs
4. The same source warns that as the holding period lengthens, the advantages of Black-Litterman returns and Sharpe ratios diminish because subjective views become less adapted to market conditions
5. The evidence supports hybrid frameworks, but not a symmetric merger of all three families: LLM and decomposition-ensemble models are used as view-generation layers inside Black-Litterman rather than as standalone allocators
6. The hybrids that appear strongest assign each component a specific job: a data-driven view layer supplied by LLMs or decomposition-ensemble forecasts
7. SSA-MAEMD-TCN Black-Litterman | SSA denoising, multivariate aligned EMD, and TCN forecasting as automated view generation for Black-Litterman | View quality and noise in forecasting | Short-holding advantage; benefits diminish as holding period lengthens
8. The SSA-MAEMD-TCN architecture addresses a different bottleneck: forecast quality and noise
9. Mainstream hybrid view-generation models often neglect noise and use univariate decompositions that fail to exploit multivariate information
10. SSA-MAEMD-TCN uses singular spectrum analysis to extract main patterns from embedded high-dimensional data, multivariate aligned empirical mode decomposition with Kullback-Leibler divergence to align intrinsic mode functions, and temporal convolutional networks whose dilated and causal convolutions expand the receptive field exponentially
11. On Nasdaq 100 stocks, it outperforms MAEMD and MEMD baselines, and the resulting Black-Litterman portfolio shows strong short-holding performance after transaction costs
12. LLM and decomposition-ensemble hybrids report strong results, but over limited test periods and short holding horizons
13. The most defensible framework combines a stable equilibrium or regularized mean-variance backbone, machine-learning or LLM-generated views with explicit uncertainty calibration
14. What would change the judgment is not another backtest with a higher Sharpe ratio, but long-sample out-of-sample results after realistic costs, robustness across market regimes and holding horizons, transparent attribution that survives auditability, and deployment evidence showing that learned or hybrid allocators displace regularized mean-variance and Black-Litterman workflows in live robo-advisor or institutional asset-allocation settings.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.