In the initial phase of this review, we conducted a systematic examination of the Hierarchical Risk Parity (HRP) algorithm, along with various scholarly works aimed at refining or expanding its core methodology. Since its original introduction by De Prado (2016), HRP has quickly captured the interest of the portfolio optimization community, prompting numerous researchers to apply and adapt the approach across a wide spectrum of financial markets and distinct asset classes.

Early studies immediately recognized HRP’s potential. Alipour et al. (2016), for instance, benchmarked HRP against Minimum Variance, Inverse Variance Parity (IVP), and Quantum-Inspired HRP (QHRP) across a global set of 38 futures contracts, which included equity, bond, and commodities like oil, wheat, and gold. Their comparative analysis clearly established QHRP as the superior performer among all models tested. Focusing on the Indian market, Jothimani (2017) pioneered an HRP model based on Gerber statistics (HRP-GS), employing machine-learning-driven optimization techniques. This model’s efficacy was evaluated against both the standard HRP and Global Minimum Variance portfolios derived from historical correlations and Gerber statistics. Subsequently, Raffinot (2018) put forward the Hierarchical Equal Risk Contribution (HERC) model, which integrates HRP with the HCAA method. By testing this model on European, US, and 365 S&P 500 stocks, Raffinot demonstrated that HERC portfolios structured using Conditional Value at Risk (CVaR) provided statistically superior risk-adjusted performance, especially when incorporating Conditional Drawdown at Risk (CDaR) structures.

Further validating the model’s attractiveness, Liutov (2018) analysed closing prices for 30 DJIA stocks between 1990 and 2017 and concluded that HRP offered a more appealing option for investors compared to the traditional Mean-Variance Optimization (MVO) model. Building on this, Jothimani and Bener (2019) introduced the HRP-HC (Historical Correlation–Based HRP) alongside the HRP-GS models, evaluating them against 250 firms on the Toronto Stock Exchange over a decade (2007–2016). Their findings consistently pointed to the HRP-GS model’s superiority. Contrasting results were noted by Uyar (2019), who assessed HRP, Minimum Variance, and IVP portfolios using BIST, FTSE, and DAX index returns. Uyar’s work indicated that HRP’s performance lagged on the BIST and FTSE indices but proved positive on the DAX. In 2020, Bechis constructed various portfolios using 30 DJIA stocks and 15 ETFs, prioritizing profitability, diversification, and risk minimization. Compared to models like Minimum Variance, IVP, CLA, equally weighted, and Randomized Weighted Portfolios (RWP), HRP consistently delivered well-diversified portfolios. Supporting this, Barziy and Chlebus (2020) conducted an evaluation of 32 ETFs spanning US, Asian, and European markets using high-frequency, 30-minute data, which confirmed that HRP successfully outperformed MVO, IVP, and CLA.

The application of HRP has also rapidly expanded to the volatile cryptocurrency space. For instance, Papenbrock et al. (2021) developed a tree-based HRP model leveraging Adaptive Serial Risk Parity (ASRP), showing that it performed better than the classical HRP using data from the 14 largest cryptocurrencies. Similarly, Jaeger et al. (2021) compared HRP with the Equal Risk Contribution (ERC) model using a comprehensive 2000–2020 dataset, ultimately concluding that HRP generated more favourable outcomes. Additionally, Burggraf (2021) confirmed HRP’s risk-reduction advantage using a large dataset of 61 cryptocurrencies. Outside of crypto, Nourahmadi and Sadeqi (2021) applied HRP to the 50 largest stocks on the Tehran Stock Exchange, identifying performance that was superior to Minimum Variance, equal weighting, and standard Risk Parity portfolios.

Subsequently, several researchers focused on introducing enhanced HRP variations. The upHRP algorithm, proposed by Amor et al. (2022), successfully demonstrated improvements over Minimum Variance, IVP, and the baseline HRP. Likewise, Shahbazi and Byun (2022), analysing 61 cryptocurrencies, identified HRP as the most effective strategy in terms of risk-adjusted returns. Nourahmadi and Sadeqi (2022) revisited the Tehran Stock Exchange, testing HRP against Minimum Variance on 30 stocks, and once again reported stronger performance from the HRP method.

In 2023, Sen and Dutta applied HRP and Eigen Portfolios across seven sectors of the Indian National Stock Exchange, concluding that the HRP approach generated higher performance. A related study by Sen (2023) compared HRP with Reinforcement Learning (RL)–based portfolios, finding that RL portfolios achieved better results based on metrics like returns, risk, and Sharpe ratios. Reis et al. (2023) conducted a critical out-of-sample comparison, evaluating HRP against various traditional allocation methods using pre- and post-pandemic data for 25 Brazilian stocks, where HRP was shown to generally outperform its alternatives. Further methodological improvements came from Uyar and Yavuz (2023), who introduced the FRAKTAL-HRP method, which demonstrated a stronger Sharpe ratio performance than Minimum Variance portfolios when tested on the 30 largest Frankfurt Stock Exchange stocks.

Contemporary contributions continue to evaluate and build upon the HRP framework. Palit and Prybutok (2024) confirmed HRP’s superior performance over MVO using sector-level S,P 500 data. Deković and Šimović (2025) presented methodological enhancements to the HRP algorithm that resulted in significantly increased computational efficiency for large-scale, real-time portfolio management. Through extensive Monte Carlo simulations, Antonov et al. (2025) compared HRP with the classical Markowitz model and successfully validated HRP’s inherent advantage. Arian et al. (2025) addressed HRP’s limitation to a single feature by introducing QHRP and Kernel-based HRP (KHRP), with both variations proving to outperform the original HRP. Finally, Ramirez-Carrillo et al. (2025) applied HRP to the NUAM region (Chile, Colombia, and Peru). While their comparative study showed that the maximum Sharpe ratio portfolio generated the highest returns, HRP was observed to offer a more balanced and favourable risk–return trade-off. A summary of the literature review on the HRP method is presented in Table 1.

The second stage of the literature review involved synthesizing relevant studies on the MEREC (Method based on the Removal Effects of Criteria) technique, which has demonstrated wide-ranging applicability across a multitude of domains. MEREC has established itself as a versatile tool, successfully addressing complex decision problems in sectors from technology and strategy to sustainability and regional economics.

In the realm of strategic decision-making and technology, MEREC has been instrumental in areas such as cloud service provider selection (Popović et al., 2021), determining effective e-commerce development strategies (Popović et al., 2022), and aiding in the selection of smart buildings (Elsayed, 2024).

The method’s utility extends significantly to energy, materials, and logistics: it has been used for the selection of renewable energy sources (Goswami et al., 2022), in the choice of phase change materials (Nicolalde et al., 2022), for electric car selection (Puška et al., 2023), developing low carbon tourism strategies (Mishra et al., 2022), and in practical logistics applications like transport planning (Simić et al., 2022) and the selection of pallet trucks (Ulutaş et al., 2022) and even a spray-painting robot (Shanmugasundar et al., 2022).

Furthermore, MEREC is frequently employed in performance evaluation and financial analysis. This includes portfolio construction for investment alternatives (Fidan, 2022), hospital location selection (Hadi and Abdullah, 2022), assessing the entrepreneurship and innovation performance of universities (Satıcı, 2022; Yaşar and Ünlü, 2023) and evaluating the vulnerability performance of countries (Altıntaş, 2023).

More recent applications highlight its growing role in macro- and micro-economic analysis: evaluating the macroeconomic performance of OECD countries (Ersoy, 2023), analysing the performance of the Serbian economy (Lukić, 2023), assessing the vulnerability performance of nations (Altıntaş, 2023), and evaluating countries’ innovation performance (Ecer and Ayçin, 2023). Sectoral and financial studies also leverage MEREC, including analyses of Türkiye’s carbon emission profile (Pelit and Avşar, 2025), the sustainable competitive position of Türkiye (Kara et al., 2024a), sustainable industrial development potential (Komasi et al., 2025), and the financial performance of banks in Bosnia and Herzegovina (Mastilo et al., 2024), along with general financial performance evaluation (Oğuz and Satir, 2024) (Table 2).

The final stage of the literature review involved summarizing the body of work related to the WEDBA (Weighted Euclidean Distance Based Approach) method. This technique has found extensive application across a diverse set of fields. Notable uses include the evaluation of e-learning websites (Garg, 2017), selection of software reliability growth models (Gupta et al., 2018), and the assessment of production system flexibility (Jain and Ajmera, 2019).

WEDBA has also been applied to various machinery and technological selection problems, such as stacker selection (Ulutaş, 2020) and the determination of appropriate blockchain technology (Kaska, 2020), as well as technology selection for vertical farm alternatives (Tolga and Başar, 2022).

Furthermore, the method is a popular tool in financial and performance analysis: it has been used for the evaluation of financial performance (Işık, 2021; Erzurumlu and Gençtürk, 2025), assessing the performance of banks (Keskin and Türkoğlu, 2022; Şimşek, 2022), and the evaluation of companies operating in the private pension system (Tuş and Aytaç Adalı, 2025). Other significant applications include the evaluation of foundation universities (Demir, 2021), analysing the performance of regions concerning information technology use (Ecemiş and Coşkun, 2022), and cloud service provider selection (Arman and Kundakcı, 2023).

On a broader scale, WEDBA has been instrumental in evaluating economic freedom sub-criteria (Karaköy et al., 2023), assessing supply chain performance of countries (Kara et al., 2024b), airport performance (Işıldak et al., 2023), and the academic achievements of Turkish universities (Kara et al., 2024c). Studies related to the WEDBA method were summarized and presented in Table 3.

The primary justification for selecting the HRP algorithm in this study rests on its capacity to minimize the risk of excessive concentration within portfolios and its ability to generate stable outcomes even when the covariance estimation is prone to errors. Nevertheless, the inherent limitations of HRP, namely the concatenation problem and its inability to accurately define the optimal number of clusters, necessitate a methodological adjustment. To directly address these issues, the Elbow method has been seamlessly integrated into the HRP framework.

The essential reason for incorporating the MEREC and WEDBA methods into the evaluation of HRP-derived portfolios is that these techniques offer objective, multi-criteria, and purely data-driven assessment capabilities. While MEREC provides an objective mechanism for calculating criterion weights, WEDBA ensures a more precise discernment of the performance differences among various investment alternatives. This synergistic combination significantly strengthens both the methodological originality and the practical applicability of the overall HRP-MCDM approach for professional investors.

Research in the field of finance consistently shows that investors fundamentally rely on two core strategies when constructing their portfolios. The first, rooted in traditional financial thought, emphasizes simple diversification. This traditional view operates on the principle that risk can be mitigated merely by ensuring assets are not excessively concentrated within a single group. The second, and more profound, approach originates from Harry Markowitz’s seminal 1952 work, which laid the groundwork for what is commonly known as Modern Portfolio Theory (MPT), the mean-variance framework, or Markowitz portfolio optimization. MPT fundamentally asserts that portfolio risk is best minimized not just by diversifying holdings, but by simultaneously considering the precise interrelationships (correlations) among those assets. Following Markowitz’s pioneering contribution, portfolio optimization problems have conventionally been formulated as quadratic programming models that map the risk–return profiles of various financial instruments (Pfitzinger and Katzke, 2019). Given that this framework became the cornerstone of contemporary portfolio optimization, exhaustive research has been dedicated to mapping the portfolios located on the “efficient frontier”. This frontier represents the optimal set of portfolios that either maximize expected return for a predefined level of risk or minimize risk for a target return level (Nourahmadi and Sadeqi, 2021; Reis et al., 2023). Portfolios situated on this boundary are, by definition, deemed optimal.

Despite its enduring presence and theoretical importance, the practical application of MPT has introduced significant challenges. The concentration problem remains a persistent issue, often manifesting as optimal portfolios that allocate disproportionately large weights to a limited subset of assets. This drawback has prompted numerous studies to propose alternative mitigation strategies. Furthermore, as practitioners integrate an increasing number of assets into portfolios, the escalating complexity of inter-asset correlations necessitates deeper diversification, a process that ironically can lead to unstable solutions. De Prado (2016) famously dubbed this instability and pressure for excessive diversification the “Markowitz Curse” (De Prado, 2016; Uyar, 2019; Kaae et al., 2022).

To bypass the deficiencies linked to quadratic optimization – such as instability, susceptibility to estimation errors, and concentration – De Prado (2016) introduced the Hierarchical Risk Parity (HRP) algorithm. HRP is a heuristic methodology explicitly designed to circumvent the practical limitations of MPT (De Prado, 2016; De Lio Pérego, 2021). It draws on concepts from mathematical modelling, graph theory, and machine learning, relying on hierarchical clustering to mirror the hypothesized layered structure inherent in financial markets. Crucially, HRP departs from the conventional assumption of direct pairwise relationships, acknowledging that many assets are only indirectly connected through a chain of intermediate relationships (De Lio Pérego, 2021).

A substantial benefit of the HRP framework is its ability to eliminate the requirement to invert the covariance matrix. This matrix inversion is one of the primary drivers of estimation error in optimization, making HRP a more stable alternative (Sjöstrand et al., 2020; Reis et al., 2023). By utilizing a hierarchical clustering structure derived from machine learning, HRP successfully addresses the inherent weaknesses of conventional risk-based optimization methods (Jain and Jain, 2019). The algorithm functions by iteratively assigning inverse-volatility weights to clusters of similar assets, recursively subdividing these clusters until each individual asset is isolated. This hierarchical process typically yields more stable portfolios that frequently demonstrate superior performance compared to those produced under MPT (Lagowski, 2022).

HRP offers several clear advantages over the Markowitz model. While Markowitz-based portfolios are prone to assigning heavy weights to a limited number of assets, HRP achieves a more uniform distribution of risk through hierarchical diversification (Sen, 2023). Furthermore, HRP applies inverse-variance weighting only within groups of correlated assets, and by narrowing these groups through recursive subdivision, it assigns weights solely among assets within the same cluster, rather than across the entire portfolio (Pfitzinger and Katzke, 2019; Jain and Jain, 2019; Bechis, 2020). HRP also exhibits greater temporal stability than the Markowitz model, which is highly susceptible to abrupt allocation shifts triggered by minor changes in market conditions. HRP’s foundation in hierarchical structure makes it notably less sensitive to short-term volatility, resulting in smoother and more reliable allocation patterns over time (Sen, 2023).

The Elbow method is a commonly recognized heuristic for determining the optimal number of clusters (k) within a given dataset. The fundamental principle underpinning this technique is that the ideal cluster count corresponds to the point where introducing further clusters results in only a marginal, negligible improvement in the model’s performance or the variance explained by the clustering structure.

This methodology relies heavily on a graphical interpretation of the results. To apply it, the proportion of variance accounted for by the clustering is plotted against different values for k. Initial clusters naturally capture a significant amount of the dataset’s inherent information. However, as k continues to increase, the incremental benefit—the addition to the explained variance—experiences a sharp decline, creating a distinct “elbow” shape on the graph. This inflection point is interpreted as the most appropriate number of clusters, signifying the optimal value for k.

Implementation of the Elbow method is systematic: the value of k is typically initiated at two and is increased sequentially. At each step, clusters are computed, and a corresponding cost metric (often a measure of within-cluster variance) is calculated. While this cost metric decreases substantially for lower k values, it inevitably reaches a plateau as k continues to rise. The point at which this levelling-off begins effectively signals the optimal cluster size; beyond this threshold, additional clusters offer diminishing returns, as they become increasingly similar to existing ones, thus contributing little to the overall model’s explanatory power.

The Elbow method is a widely used approach for identifying the optimal number of clusters or groups within a dataset (Kodinariya and Makwana, 2013; Bholowalia and Kumar, 2014; Abrar et al., 2023). The fundamental principle of this method is that the ideal number of clusters corresponds to the point beyond which adding an additional cluster yields only marginal improvement in model performance (Bholowalia and Kumar, 2014).

The MEREC method is one of the objective methods for criteria weighting developed by Keshavarz-Ghorabaee et al. (2021). In contrast to other criteria weighting methods, the MEREC method focuses on the changes in the overall weighting of the criteria by excluding the relevant criterion when calculating the importance of each criterion (Keshavarz-Ghorabaee et al., 2021).

The implementation steps of the MEREC method are shown in equations (10)–(18) (Keshavarz-Ghorabaee et al., 2021; Keshavarz-Ghorabaee, 2021).

Step 1: The first step of the MEREC method is to create the decision matrix. The decision matrix, which consists of m alternatives and n criteria, is created using equation (10). (10) X = x 11 x 12 … x 1 n x 21 x 22 … x 2 n ⋮ ⋮ ⋮ ⋮ x m 1 x m 2 … x m n .

In the decision matrix in equation (10), x i j shows the value of alternative i in relation to criterion j. This value should be greater than zero. If there is a negative value x i j in the decision matrix, it should be converted to a positive value using appropriate methods. In this study, negative values were converted to positive values using the Z-score standardization transformation. Equation (11) and equation (12) were used for the Z-score standardization transformation (Zhang et al., 2014; Ayçin and Güçlü, 2020; Kundakcı and Arman, 2023). (11) z i j = x i j − x ‾ j σ i ( i = 1 , 2 , … , m ; j = 1 , 2 , … n ) , (12) z i j ′ = z i j + A ( i = 1 , 2 , … , m ; j = 1 , 2 , … n ) .

In equation (12), the value of A stands for > | min z i j |.

Step 2: The decision matrix is normalized using equation (13) for benefit criteria and equation (14) for cost criteria. (13) x i j ∗ = min ( x i j ) x i j , (14) x i j ∗ = x i j max ( x i j ) .

The normalization process is similar to the process used in methods such as WASPAS but differs in that it switches between formulas for benefit and cost criteria. In contrast to many other studies, the MEREC method converts all criteria into minimization-type criteria (Keshavarz-Ghorabaee et al., 2021).

Step 3: A logarithmic measure with equal criteria weighting is used to determine the overall performance values of the alternatives. This logarithmic measure is based on a non-linear function shown in Fig. 5.

Using the values obtained with the normalization matrix, it can be ensured that smaller values of x i j ∗ result in a larger overall performance value. This value is calculated using equation (15). (15) S i = ln ( 1 + ( 1 n ∑ j | ln ( x i j ∗ ) | ) ) .

Step 4: As with the determination of the overall performance value of the alternatives, a logarithmic criterion is also used in this step. In contrast to the previous step, in this step the performance values of the alternatives are calculated by subtracting them individually for each criterion. These values are calculated using equation (16). (16) S i ′ = ln ( 1 + ( 1 n ∑ k , k ≠ j | ln ( x i j ∗ ) | ) ) .

Step 5: The E j value, which is determined with the help of equation (15) and equation (16) and indicates the removal effect of the j. criterion, is calculated for each criterion with the help of equation (17). (17) E j = ∑ i | S i ′ − S i | .

Step 6: Using the E j values calculated by equation (17), the importance weights of each criterion are calculated by equation (18). (18) w j = E j ∑ k E k .

The WEDBA method, introduced by Rao and Singh (2012), is an MCDM technique grounded in the principle that the most favourable alternative is the one with the shortest Euclidean distance to the ideal solution, whereas the least desirable alternative is the farthest from the non-ideal solution (Rao and Singh, 2012). In this framework, the overall performance index of each alternative is computed based on its Euclidean distance to both the ideal and all non-ideal reference points. This structure ensures that alternatives are evaluated relative to ideal and non-ideal benchmarks rather than being compared directly with one another, thereby necessitating the use of Euclidean distance as the primary measure (Rao and Singh, 2012).

The procedural steps of the WEDBA method are presented in equations (19)–(28) (Rao and Singh, 2012; Işık, 2021).

Step 1: The first step of the WEDBA method is to construct the decision matrix. The decision matrix is created using equation (10).

Step 2: The decision matrix is normalized using equation (19) for benefit criteria and equation (20) for cost criteria. (19) x i j ∗ = x i j max ( x i j ) , (20) x i j ∗ = min ( x i j ) x i j .

Step 3: The normalized values obtained by equation (19) and equation (20) are standardized by equation (21). (21) y i j = x i j ∗ − μ j σ j .

The expressions μ j and σ j in equation (21) indicate the mean and standard deviation of criterion j. respectively. The value of μ j is calculated using equation (22) and σ j is calculated using equation (23). (22) μ j = ∑ i = 1 m x i j ∗ m , (23) σ j = ∑ i = 1 m ( x i j ∗ − μ j ) 2 m .

Step 4: Ideal values are calculated using equation (24) and anti-ideal values using equation (25). (24) y i j + = ( y i j ) , (25) y i j − = ( y i j ) .

Step 5: The weighted Euclidean distances of the alternatives to the ideal points are calculated with the help of equation (26) and the weighted Euclidean distances to the non-ideal points are calculated by using equation (27). (26) WED i + = ∑ j = 1 n { w j x ( y i j − y i j + ) } 2 , (27) WED i − = ∑ j = 1 n { w j x ( y i j − y i j − ) } 2 .

Step 6: Equation (28) is used to calculate the index score, which is the last step of the WEDBA method. (28) IS i = WED i − WED i − + WED i + .

An increase in an alternative’s index value indicates that it is closer to the ideal solution. Therefore, the alternative with the highest index value is the best in terms of performance, while the one with the lowest index value is the weakest alternative.

To compare the portfolios created using the HRP algorithm and the MCDM methods, evaluation criteria are needed. Although there are many alternatives in the literature for comparing portfolio performance, the Sharpe ratio, also known as the reward to volatility ratio based on standard deviation, is preferred in this study. The Sharpe ratio, which helps investors understand how well their investments are compensated for the risk they bear, is calculated using equation (29) (Samarakoon and Hasan, 2006; Bechis, 2020). (29) S p = r p − r j σ p .

S p : Sharpe ratio,

r p : Portfolio return,

r f : Risk-free interest rate,

σ p : Represents the standard deviation of the portfolio.

Diversifying the portfolio with securities with low or negative correlation increases the Sharpe ratio as it reduces the risk of the portfolio in general (Srivastava and Mazhar, 2018). In other words, a high portfolio return, and a low standard deviation increase the Sharpe ratio, while a low portfolio return and a high standard deviation decrease the Sharpe ratio. Therefore, among alternative investment portfolio strategies, investors should favour the portfolio with a high Sharpe ratio.

In this study, portfolios were systematically constructed based on the Ward, single, complete, average, weighted, centroid, and median linkage methods within the HRP algorithm. The analysis utilized the daily closing prices of stocks listed in the BIST 100 Index for the period spanning January 2018 to December 2022. The dataset used in this study consists of the daily closing prices of BIST 100 Index stocks and is available for open access at https://zenodo.org/records/18140067.

The data acquisition and initial processing were conducted using the Python programming language. Yahoo Finance was selected as the primary data source due to its extensive coverage of major global indices (such as the Dow Jones, NASDAQ, and S&P 500) and its proven reliability for empirical finance research. However, because comprehensive historical data for all 100 BIST constituents were not fully available throughout the 2018–2022 period, the final, clean dataset was limited to 72 stocks for which complete information could be retrieved.

In strict adherence to the HRP framework, portfolio weights were precisely calculated using the following equations for each linkage method: equation (3) for the single linkage method, equation (4) for the complete linkage method, equation (5) for the average linkage method, equation (6) for the weighted linkage method, equation (7) for the centroid linkage method, equation (8) for the Ward linkage method, and equation (9) for the median linkage method. The resulting final portfolio allocations are comprehensively detailed in Table 5.

Table 5 presents a comprehensive overview of the portfolio outcomes generated by applying the standalone HRP algorithm to the BIST 100 Index data.

An initial review of the results reveals that portfolios constructed using the single and centroid linkage methods exhibited the most robust performance, delivering the strongest results in terms of both average returns and Sharpe ratios (a key risk-adjusted performance metric).

When the portfolios are assessed solely based on risk, as measured by standard deviation, the Ward linkage method resulted in the highest recorded level of volatility. Conversely, the centroid linkage method was the most effective in reducing risk, producing the lowest volatility across all tested criteria.

Further examination of the return distribution statistics indicates that the skewness values for all linkage approaches are left-skewed relative to a standard normal distribution. This finding suggests a financial reality where there is a greater concentration and frequency of negative returns than positive ones. Regarding kurtosis (a measure of the”tailedness” of the distribution), the highest values were observed in portfolios formed using the average and median linkage methods. The lowest kurtosis values, however, were associated with the weighted and Ward linkage approaches. These results imply that portfolios constructed with the weighted and Ward linkage methods exhibit a higher probability of experiencing extreme price fluctuations (both large gains and large losses) compared to those created using the other clustering criteria.

Since the introduction of the HRP algorithm by De Prado (2016), various researchers have successfully identified several inherent shortcomings and weaknesses. The most significant of these practical issues are the concatenation problem and the algorithm’s inability to determine the optimal number of clusters.

The chaining problem stems from the sensitivity of HRP’s linking methods to specific data values. This sensitivity can cause clusters to become unduly long, distributed, and heterogeneous, resulting in groupings that contain assets which, based on their risk profile, should not logically be clustered together. This situation is highly undesirable for both researchers and investors, as the addition of a seemingly minor asset can disproportionately alter a cluster’s variance. Such structural instability leads to distorted asset weighting and potentially erroneous investment decisions.

The second major issue arises when the optimal number of clusters cannot be determined accurately. In this scenario, the algorithm may treat every individual entity as a separate cluster, effectively failing to recognize the relevant underlying structure of the data. This failure can be characterized as a form of overfitting, which carries potentially detrimental consequences. Overfitting occurs when a model is so closely tailored to specific, potentially noisy, historical observations that it fails to capture the true, general structure of the market. Attempting to fit the model too tightly to these slightly erroneous data points can introduce significant biases and substantially diminish the model’s predictive power.

To effectively eliminate the structural problems caused by the HRP algorithm and to establish the necessary framework for the Multi-Criteria Decision-Making (MCDM) methods, the popular Elbow method has been integrated into the HRP algorithm.

This integration was performed specifically to determine the optimal number of clusters. The first practical step of this enhanced HRP algorithm was to partition the assets—based on the daily closing data of companies traded in the BIST 100 Index—into distinct clusters according to the predefined linkage methods.

Based on the single linkage method, companies listed on the BIST 100 Index are grouped into six clusters, as shown in Fig. 6.

Based on the complete linkage method, companies listed on the BIST 100 Index are grouped into six clusters, as shown in Fig. 7.

Based on the average linkage method, companies listed on the BIST 100 Index are grouped into six clusters, as shown in Fig. 8.

Based on the centroid linkage method, companies listed on the BIST 100 Index are grouped into six clusters, as shown in Fig. 9.

Based on the median linkage method, companies listed on the BIST 100 Index are grouped into seven clusters, as shown in Fig. 10.

Based on the weighted linkage method, companies listed on the BIST 100 Index are grouped into seven clusters, as shown in Fig. 11.

Based on the Ward linkage method, companies listed on the BIST 100 Index are grouped into eight clusters, as shown in Fig. 12.

The distinct cluster structures derived from the enhanced HRP algorithm—integrated with the Elbow method—are presented across the various linkage methods as follows: Fig. 5 illustrates the clusters generated using single linkage, Fig. 6 depicts those obtained via complete linkage, Fig. 7 shows the results for average linkage, Fig. 8 for centroid linkage, Fig. 9 for median linkage, Fig. 11 for weighted linkage, and Fig. 12 for Ward linkage.

Following this robust clustering phase, a separate MCDM procedure was applied to the asset sets produced under each linkage method. Within this hybrid framework, the assignment of criterion weights is a critical step, as it has a direct and substantial influence on the final portfolio outcomes. For this reason, the necessary criterion weights corresponding to each specific cluster were computed independently using the MEREC method, an objective weighting technique.

The MEREC method was selected for several strategic reasons: it is a relatively recent addition to the literature, has only been utilized in a limited number of financial studies, and its characteristics align well with the BIST 100 dataset used in this research. Methodologically, MEREC offers notable advantages: it possesses a solid mathematical foundation, avoids unnecessary computational complexity, operates entirely without subjective input from decision makers, and has been applied in conjunction with the WEDBA method in a small number of prior studies. Crucially, its integration with the HRP algorithm is novel, further validating its use in this research.

Once the criterion weights for every cluster were objectively determined, the financial performance of the firms within each cluster was rigourously assessed using the WEDBA method. WEDBA was chosen because, despite being one of the earlier techniques in the MCDM literature, it remains underutilized in practical financial decision-making applications. It has appeared in only a few finance-related studies—often in combination with MEREC—and, like MEREC, has not been previously employed alongside the HRP algorithm. Additionally, its established computational structure is highly suitable for analysing the characteristics of the specific dataset used in this study.

The final portfolio outcomes produced through the integrated HRP-MCDM framework—using the Ward, single, complete, average, weighted, centroid, and median linkage methods—are concisely summarized in Table 6. In addition, the application steps for the MEREC and WEDBA methods are available for open access at https://zenodo.org/records/18140067 for the clusters obtained under each linkage criterion.

Table 6 provides the detailed performance outcomes for the portfolios constructed under the hybrid HRP-MCDM framework using the stocks listed in the BIST 100 Index.

A meticulous inspection of these results reveals that the portfolios generated using the single and Ward linkage methods delivered the most robust overall performance, particularly in relation to both mean return and the crucial Sharpe ratio.

When the portfolios are comparatively assessed based on risk (standard deviation), the average-linkage portfolios exhibited the highest recorded volatility. Conversely, the weighted-linkage portfolios successfully displayed the lowest risk levels.

In terms of distribution characteristics, the skewness values consistently suggest that the return distributions across all linkage methods are negatively skewed relative to the standard normal distribution, implying a tendency toward larger negative return events. Concerning kurtosis, the single and Ward linkage portfolios demonstrated the most pronounced “heavy-tail” behaviour (leptokurtosis), while the complete and average linkage portfolios showed the lowest values. These findings suggest that the complete and average linkage methods are associated with a greater likelihood of extreme return fluctuations compared to other methods.

Finally, following the implementation of portfolio optimization using the traditional MEREC and WEDBA MCDM methods alone, a key finding emerged: a comparative analysis of the results in Table 6 indicates that the hybrid HRP-MCDM portfolios generally outperform those created exclusively through traditional, standalone MCDM procedures.

Beyond this general performance comparison, a more nuanced evaluation yields several additional insights into the structural behaviour of the proposed hybrid model. The consistent increase in Sharpe ratios—most prominently observed for the single and Ward linkage strategies—demonstrates that augmenting HRP with objective MCDM-based evaluation effectively reduces the algorithm’s sensitivity to clustering noise and substantially enhances the overall risk–return alignment.

The empirical evidence also suggests that linkage techniques which inherently produce more homogeneous or economically meaningful clusters benefit disproportionately from the MCDM integration. This indicates that cluster quality directly mediates the effectiveness of the hybrid model. Although the HRP-MCDM approach might marginally increase volatility for some specific linkage configurations, this is typically offset by a corresponding increase in average returns, which ultimately strengthens the risk-adjusted performance. This observed pattern confirms that the combination of MEREC-based weighting and WEDBA-based ranking systematically shifts allocations toward financially stronger firms within each HRP-generated cluster.

Overall, the empirical results decisively demonstrate that the hybrid model not only optimizes weight reallocation but also systematically mitigates structural weaknesses inherent in the original HRP algorithm—particularly its tendency to over-allocate to underperforming assets in elongated cluster structures—thereby establishing a more resilient, balanced, and superior portfolio optimization procedure.