Some authors propose to shortlist the assets before the PO process, i.e., only the shortlisted assets are allowed to take part in the financial decision-making process through optimization. This preliminary selection of assets may depend on various crucial factors that govern the asset performance, which the PO models may fail to capture. An attempt to integrate ML with PO is made by [19] by employing support vector machines (SVM) for asset pre-selection in the PO model by selecting only those assets that have the capacity to reach a certain returns target. [20] proposes to integrate the long short-term memory networks (LSTM) with the mean-variance model for portfolio selection. The primary purpose of LSTM in this work is to capture the long-term dependencies of asset returns and pre-select the assets based on these criteria before PO. A step approach is followed by [21] for portfolio selection, namely (i) asset price prediction through an artificial neural network and fundamental analysis, (ii) asset shortlisting by differential evolution and fundamental analysis, and (iii) PO through a genetic algorithm.

There are other ways in literature by which ML techniques are adopted to facilitate the process of PO. [22] develops the prediction-based PO models by using deep neural networks to forecast asset returns, maximize the portfolio predicted returns, and minimize the downside absolute predicted deviation errors. Monthly excess returns and volatility are forecasted via a random forest model, and optimal weights are computed through a standard linear model. Some clustering techniques are also incorporated in the PO models as a means to reduce the number of assets taken into consideration for the optimization process. [23] apply some commonly used clustering techniques like K-means, fuzzy C-means, and self-organizing maps, and stocks are selected from each cluster for the mean-variance model. Agglomerative hierarchical clustering is used by [24] to arrange resources into a hierarchical structure; the clustering procedure is based on the correlation between the assets. Through extensive analysis, they show that alpha values for different factor models are significantly improved. [25] uses three clustering techniques, namely agglomerative hierarchy, K-means, and partitioning around medoids for dimensionality reduction, and observes the portfolio returns after selecting assets from the clustering process. An investment decision-making strategy is put forward by [26] by applying SVM to classify assets into different classes, and an appropriate set of assets from these classes is selected according to the preference of the investor. [19] develops a portfolio selection strategy with a mean-variance model where assets are shortlisted through SVM. A comparison of the proposed model with other related techniques illustrates the superiority of the developed approach. Stock return prediction through machine learning techniques, including support vector regression integrated with the mean-variance model for optimal portfolio selection by [27].

Moreover, recent work by Kaur et al. [28, 29, 30, 31] gives new directions to the ongoing research on the new field of integration of machine learning techniques, especially SVM with the Markowitz mean-variance portfolio optimization model [28] later on extended to sentiment analysis in a book chapter [29] showcasing investor’s sentiment to strategy: machine learning in emotion-based asset allocation. Further revised the mean-variance model to a conditional value-at-risk model based on machine learning techniques, specifically by modifying the more computationally efficient method, Least Squares Support Vector Machines (LSSVM) [30]. Afterwards, extend binary class classification to novel multi-class classification scenarios [31]. Based on getting motivation from recent advancements in the new ongoing motivational field of integrating machine learning with PO, this research gives new avenues to existing literature, starting from Markowitz and moving forward till SSD. Hence, we attempted to integrate SVM with SSD to enhance superior asset classification and obtain optimal portfolios from the shortlisted preselected assets by the SVM model.

Riding on recent developments by Kaur et al. [28, 29, 30, 31], the coupling of machine learning methods—especially Support Vector Machines (SVM)—with traditional portfolio optimization (PO) models is a promising avenue of research in financial modeling. Previous efforts showed how the legacy Markowitz mean–variance model [28] could be improved via machine learning-based asset selection and subsequently progressed towards sentiment-based portfolio building [29]. Later, the Conditional Value-at-Risk (CVaR) model was further improved by the addition of Least Squares SVM (LSSVM) and its fuzzy extension (FLSSVM) [30] for improved representation of tail-risk and effectiveness under uncertainty in markets. Recent developments towards multi-class classification [31] have only extended the application of SVM-based portfolio models further. Taken together, these advances underscore the increasing significance of machine learning-based risk management in portfolio design and inspire research toward still more responsive and dominance-homologous methods.

Nonetheless, there are some unresolved challenges. Most prior research is based solely on optimization-oriented models for investment choice, ignoring the preselection of assets, which is a key component of enhancing portfolio efficiency. Additionally, classical SVM models—albeit robust in classification—place equal weight on all data points and therefore cannot emphasize temporal significance, uncertainty, and noise associated with finance data. It is with this deficiency that Fuzzy Support Vector Machines (Fuzzy SVM) are used as an efficient asset shortlisting process before optimization.

Fuzzy SVM brings with it an important improvement over standard SVM by the inclusion of a fuzzy membership function that attributes different weights to observations according to how reliable or up-to-date they are. This allows the model to give greater weight to more informative and up-to-date data points, thus lowering the impact of outliers and market noise. As illustrated in recent research studies by Kaur et al. [30, 31], fuzzy and hybrid SVM methodologies have exhibited better generalization, stability, and predictive performance in sophisticated financial settings, and hence are especially relevant to turbulent market situations.

After asset shortlisting via fuzzy SVM, the next step is the optimization of the identified assets via Second-Order Stochastic Dominance (SSD) constraints because the Markowitz model cannot provide maximum investor satisfaction or utility. SSD ensures that the resulting portfolio dominates a benchmark in regard to expected utility for any risk-averse investor in accordance with non-decreasing and concave utility functions. In contrast to conventional mean–risk models (e.g., mean–variance), SSD optimization is consistent with investor rationality and describes a more realistic risk–return trade-off. Mathematically, it can be presented as an LP problem, which ensures computational tractability.

Motivation Summary

• Preselection of Assets: All PO models optimally make a balance between larger asset weights, which usually result in inefficiency and overfitting. It is beneficial to include a fuzzy SVM-based preselection step to enhance portfolio quality and computational efficiency.

• Relevance and Effectiveness of Data: Fuzzy SVM is computationally more efficient and reduces the impact of noise and uncertainty due to the presence of a membership function, thereby enhancing adaptability to dynamic financial environments.

• Dominance-Consistent Optimization: SSD constraints are helpful in logical decision-making according to the needs of risk-averse investor behavior by getting rid of the deficiencies of classical mean–risk models.

• Computational Efficiency: Fuzzy SVM (convex quadratic formulation) and SSD (linear program) both achieve global optimality, whereas the cutting-plane algorithm ensures scalability to large datasets.

In conclusion, this study proposes a two-stage smart framework—Fuzzy SVM for asset categorization and SSD-constrained optimization for portfolio allocation—promoting a more adaptive, dominance-consistent, and risk-sensitive approach to contemporary portfolio management.

The structure of the paper is as follows: Section 2 covers the mathematical notations and preliminaries of the PO model with SSD constraints, and also explains the SVM and FSVM models in more detail. Section 3 describes the methodology carried out in this study, along with the data set. It further includes the performance measures and out-of-sample comparison analysis of the optimal portfolios obtained from the proposed model. Additionally, the integration of both SVM and FSVM scenarios provides a comparative analysis of out-of-sample optimal portfolios generated from the SSD model. Section 4 includes the observations, and Section 5 concludes the article.

Assume n number of assets considered for investment, and we take u as n×1 vector P n where each component ui=1 and ui≥0, since we ui, i=1,...,n of vector u is the weight assigned to the respective asset. Thus i=1 assume that short-selling is not allowed. Take U to be a polyhedral set of admissible portfolios. As we note the price of a stock at a discrete finite-time point called scenarios, let us take the total number of T scenarios

with probability vector p=(p1,p2,...,pT ), pj=1,pj≥0,∀j=1,2,...,T. Let Rj=(r1j,r2j,...,rnj) represents

th

the return of assets at j time point j=1,2,...,T. So the return of the portfolio u at any time point j denoted by Rj(u)= rijui. Thus the portfolio return is a random variable Ru finitely distributed over

(R1(u),R2(u),...,RT (u)) with the probability pj,j=1,2,...,T . Hence, the expected value of portfolio return is

P T P n P n P T P T P n

E[Ru]= ( rijui)pj= ( rijpj)ui. Let ( rijpj)=ˆ ri. Therefore, E[Ru]= r ˆiui.

j=1 i=1 i=1 j=1 j=1 i=1

Now consider the case when all the T scenarios are equally probable i.e., pj= , j=1,2,...,T. Let (R1(u),R2(u),...,RT (u)) and (R1(u ˆ),R2(u ˆ),...,RT (u ˆ)) be the portfolio returns of portfolio u and u ˆ

respectively.

A portfolio u is preferred to portfolio u ˆ concerning FSD iff the corresponding portfolio returns

Let FR be the cumulative distribution function for portfolio u. Mathematically, FR (η)=P (Ru≤η). Then the portfolio u dominates another portfolio u ¯ w.r.t. first order stochastic dominance if FR (η)≤FR (η) ∀η∈R. Equivalently, under the

u u ¯

discrete distribution of returns, if q1,...,qT are portfolio returns for u and q ¯1,...,q ¯T are portfolio returns for u ¯, arranged in ascending order, then qt≤q ¯t, For each j in the range 1 through T , inclusively, there is a presence of at least a single strict disparity.

The portfolio u dominates portfolio u ¯ w.r.t. SSD if E((η−Ru) )≤E((η−Ru ¯) ) ∀ η∈R where E(.) is the expectation. Equivalently,

under the discrete distribution of returns, if R1(u),R2(u)....,R P r (u) and P r R1(u ¯),R2(u ¯)....,R (u ¯) are the portfolio returns for u and

Rt(u)≥ Rt(¯ u), r=1,...,T with atleast one strict inequality. returns for u and u ¯, arranged in ascending order, then

t=1 t=1

The PO model that determines the optimal portfolio dominating benchmark/market index, say M w.r.t. SSD, is given by [13]:

max E(Ru) subject to: E((η − Ru) ) ≤ E((η − M) ) ∀η ∈ R, u ∈ U, Here, M represents the stochastic variable characterizing benchmark returns. Let m1,m2,...,mT be the benchmark returns in respective T scenarios.

max E(Ru) subject to: E((η − Ru) ) ≤ E((η − M) ), ∀η = m1, m2, . . . , mT , u ∈ U,

In the scenario of discrete return distributions, we derive the subsequent model:

max r ˆiui

T n +

subject to: mk − rijui pj ≤ βk, k = 1, 2, . . . , T

j=1 i=1

u ∈ U, where βk=E[(ηk−M) ]. Introducing the auxiliary variable

djk = mk − rijui

The foregoing quandary lends itself to the formulation of the subsequent Linear Programming (LP) model:

(SSD) max r ˆiui

subject to : djkpj ≤ βk, k = 1, ..., T,

djk ≥ mk − rijui , k = 1, ..., T, j = 1, ..., T, (1)

djk ≥ 0, k = 1, ..., T, j = 1, ..., T, u ∈ U

Although the above SSD model is an LP, it has a large number of constraints due to (1). Therefore, in instances where the number of scenarios is sufficiently large, [13] recommends the utilization of the cutting plane algorithm to effectively expedite the computational process for resolving the (SSD) model.

The concepts of SVM for two-class classification problems are summarized in this subsection. A mathematical description of hyperplanes serves as the foundation for SVM. The SVM algorithm aims to find a hyperplane in an m-dimensional space, where m is the number of features that distinctly classify the data points.

Let L be the number of training points partitioned into two classes. Let (xl,yl)∈R ×{−1,1} denote a point, where m represents the number of observed features for each xl;l=1,2,...,L and yl indicates to which of the two classes the point2belongs. m m If the ,x ,...,x )∈R , we

m 1 l l l

training points are linearly separable, there exists w=(w1,w2,...,wm)∈R and b∈R such that for each xl=(x have:

1 2 m

b + w1xl + w2xl + . . . + wmxl ≥ +1, for which yl = +1 and

1 2 m

b + w1xl + w2xl + . . . + wmxl ≤ −1, for which yl = −1.

The SVM Problem determines the hyperplane f(xl)=w x+b, which optimally separates the training points into two classes. Here, optimality is two-fold. The goal is to minimize the cumulative classification errors and increase the margin between two parallel hyperplanes supporting vectors from different classes. In the traditional hard margin SVM model put forward by [33], reducing the empirical risk—a trade-off between the previously listed goals—and taking care of the structural risk are the main priorities.

The SVM approach solves the following optimization problem to determine the ideal hyperplane of separation between two classes of data points:

The datasets sourced from the Thomson Reuters Eikon data stream are utilized for comparative analysis among the aforementioned models. We use the daily closing price, opening price, high, low, and volume traded of each asset from 1 January 2015 to 8 August 2022 for designing the feature vectors used in SVM/FSVM.

• Dataset 1: Bovespa 90 (Brazil) • Dataset 2: DAX 40 (Germany) • Dataset 3: Dow Jones Industrial Average (U.S.A.) • Dataset 4: EURO 50 (Europe) • Dataset 5: FTSE 100 (U.K.) • Dataset 6: Hang Seng (Hong Kong) • Dataset 7: Nifty 100 (India) • Dataset 8: Nikkei 400 (Japan) Further, daily closing prices of the market index and its constituents are employed for PO through SSD. The number of assets for Datasets 1 to 8 is 71, 35, 29, 47, 94, 55, 80, and 379, respectively. Note that to prove the viability of the suggested strategy, we have made an effort to include significant marketplaces worldwide.

The out-of-sample returns from all rolling windows are concatenated to perform the comparative analysis on the basis of the following measures. With t=1,...T1, which represents the total number of out-of-sample data points, let rt be the out-of-sample returns.

• Average: Out-of-sample mean returns calculated as r ˆ= rt/T1.

• Median: Out-of-sample median returns.

• Min: Minimum out-of-sample return, i.e., min= min rt

t=1,...,T1

• Max: Maximum out-of-sample returns, i.e., max= max rt

t=1,...,T1 (r −r ˆ)2

• SD: Out-of-sample standard deviation given by SD =

(T1−1)

• VaR0.95: Out-of-sample value at risk (VaR) at confidence level 0.95.

• CVaR0.95: Out-of-sample conditional value at risk ([4]) at confidence level 0.95.

• SR: Sharpe ratio (SR) ([38]) is defined as the ratio of excess average returns over the risk-free returns and the

r ˆ−rf

standard deviation, SR= . This ratio is defined only when r>rf ˆ . For convenience, we take rf =0.

SD

• STARR: Stable Tail-Adjusted Return Ratio (STARR) is defined in a similar manner as SR, except the risk

r ˆ−rf

measure used to compute STARR is taken as CVaRα, i.e., STARR= . Again the ratio is defined only when

CV aRα

r>r ˆ f and CV aRα>0. We take α=0.95

E(Rp)−rf

• Sortino: Sortino is defined as follows: Sortino = where E(Rp) is the expected portfolio return and σd is

σd

the downside risk standard deviation.

For convenience and comparison purposes, we take rf =0.

Tables 3,4, 5 and 6 report the performance results of the out-of-sample returns on all eight data sets. We compare the performance of FSVM_SSD with SVM_SSD and SSD in the following subsections.