Keywords

1 Introduction

The construction of the optimal portfolio plays a critical role behind any investment decision as it is premised on the concept of maximization of the return on investment. It is imperative to contemplate on the selection of appropriate stocks in the right proportion for constructing a portfolio with an objective to increase return while minimizing risk. The essence of effective portfolio management lies in optimum utilization of the capital in terms of allocation of the stocks constituting the portfolio and balancing the same taking risk-return trade-off into account, since risk tolerance level and expectation of return vary from investor to investor. Stated in this field, Markowitz’s mean–variance approach [1] is considered to be the first systematic attempt to formulate a diversified investment decision with an objective of ensuring high return at low-risk level [2, 3]. In tune with this work, Tobin [4] postulated the concept of efficient frontier and capital market line, wherein the author argued for the persistence of the portfolio structure irrespective of risk tolerance level at a consistent expectation level; only relative proportions of stocks change otherwise. Extending the work, Markowitz [5] introduced expected return–semi-variance based analysis which considers both the extremes while determining the efficiency of the portfolio. Sharpe [6, 7] further worked on Markowitz’s analysis and noted that optimal portfolio (among all efficient portfolios) depends on expected return and risk preference of the investors. These classical approaches paved the way to the modern portfolio theory (MPT). MPT aims to maximize the expected return of the portfolio at a given risk level or minimizing portfolio risk at a given return by optimally allocating the total available fund to different assets [8]. However, while forming the portfolio, it is essential to consider the influence of the assets on each other, i.e., only on the basis of individual stock performance one cannot form a portfolio [9]. In effect, the distribution of the returns decides effective optimization of the portfolio. In the context of portfolio management, a risk is perceived as the total risk of a portfolio which has two components; systematic risk or market risk and unsystematic risk or diversifiable risk. By prudent stock selection and distribution of appropriate weightage for them within a portfolio, unsystematic risk can be reduced to a considerable extent [10].

In line with the seminal work of Markowitz [1], over the years, several studies have been made on portfolio selection and optimization. There has been a growing stream of alternative methods suggested by several researchers and practitioners in the stated field. Examples include artificial neural network [11], genetic algorithm [12], particle swarm optimization [13], simulated annealing [14], ensemble [15], decision tree [16], clustering [17], multi-criteria decision-making approach [18] to name a few. It is evident from these researches that in order to optimize the portfolio and correctly predict its return, selection of the right number of right stocks in the right proportion is of paramount importance. There has been a plethora of research conducted on portfolio selection using unsupervised learning methods like clustering. Clustering finds its importance in pattern identification, classification, and detection of an anomaly while selecting stocks to form a portfolio. Further, it is useful in finding interrelationship or co-movements of the stocks [19]. Although hierarchical clustering is a dominant method in this regard, k-means clustering and C-means clustering have also drawn significant attraction of the researchers [20].

In this study, the authors have used k-means clustering for segregating 53 heterogeneous companies with different sectors enlisted with the NSE during the year 2012–2017 on the basis of monthly stock return (60 months) into three distinct clusters; low stock price (LSP), mid stock price (MSP), and high stock price (HSP) in order to construct a portfolio. Further, GRG method has been used to optimize the constructed portfolio of the stocks belonging to the MSP cluster through dynamic allocation of the weights to different stocks with an objective to minimize risk while maximizing return. The rest of this paper is organized as follows. In Sect. 2, data and methodology are discussed. Section 3 presents the findings. Finally, Sect. 4 concludes the paper while highlighting some of the implications and future scope.

2 Data and Methodology

The broad objective of this study is to find out the distribution of the stocks under the portfolio ensuring maximum return at minimum risk. It requires appropriate weight assignment to individual stocks forming the portfolio.

2.1 Sample

The sample for this study consists of 53 numbers of heterogeneous companies with different sectors enlisted with the National Stock Exchange (NSE), India through convenience sampling (refer Table 1). The study period is the year 2012–2017. The data were collected from the published secondary database. Monthly returns of those stocks for 60 months have been considered for analysis. In order to calculate monthly returns in case of the Index or Stock (since the data collected is month wise), we have used the formula Ln(P1/P0), Ln(P2/P1) and so on since the behavior of data is continuous. After clustering, we have worked on MSP cluster.

Table 1 Stocks under the study (primary level)
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2.2 Methods

In this study, a three-stage approach has been followed. First, k-means clustering, an unsupervised learning technique has been applied in order to classify the stocks based on the monthly returns for constructing the portfolio; second, GRG, a nonlinear optimization technique has been selected for deciding weights for the stocks with an objective to minimize risk while maximizing return. Finally, we have applied perception mapping using mean and standard deviation of the monthly returns of the stocks belong to MSP.

The expected return on a portfolio is computed as follows:

$$ E \left( {R_{p} } \right) = \mathop \sum \limits_{i = 1}^{N} w_{i} E \left( {R_{i} } \right) $$
(1)

where

E(Rp):

the expected return on the portfolio

N:

the number of Index or stocks in the portfolio

wi:

the proportion of the portfolio invested in Index or ith Stock

E(Ri):

the expected return on ith Stock.

The risk is calculated as follows:

$$ {\upsigma }^{2} _{{\text{p}}} = \left( {{\text{w}}_{{\text{A}}} } \right)^{2} {\upsigma }^{2} _{{\text{A}}} + \left( {{\text{w}}_{{\text{B}}} } \right)^{2} {\upsigma }^{2} _{{\text{B}}} + 2\,{\text{w}}_{{\text{A}}} {\text{w}}_{{\text{B}}} ~{\upsigma }_{{{\text{A,B}}}} $$
(2)

where σp: Standard deviation of the portfolio; σA and σB: Standard deviation of the stocks A and B; wA and wB: Weights assigned to the stocks A and B.

K-means Clustering. It is a type of unsupervised learning, which distribute the unlabeled data into a specific number of groups or clusters (represented by the variable K) on the basis of feature similarity. Each such cluster is represented by its centroid which is a collection of feature values pertaining to that cluster. This algorithm takes a set of “m” number of data points in “n” dimensions into “K” number of clusters through an iterative process. The objective is to minimize within-cluster sum of squares [21]. It starts with an initial estimate of clusters wherein a particular data point \( {\text{x}}_{\text{j}} \left( {{\text{where}},\,{\text{j}} = 1,2, \ldots \,{\text{m}}} \right) \) is allocated to a particular cluster \( {\text{c}}_{\text{i}} \left( {{\text{where}},\,{\text{i}} = 1,2, \ldots {\text{k}}} \right) \) depending on the squared Euclidean distance between the data point and the centroid of the respective cluster. The points which find their distances with respect to the centroid of a particular cluster minimum are assigned to that particular cluster. Then, the process recomputes all the centroids and again finds the distances of all the data points with respect to each newly constructed centroids for assigning the data points to newly formed clusters. This process repeats itself until limiting condition is reached, i.e., either there is no change of clusters by the data points (i.e., within a particular cluster all data points are homogeneous) or within group sum of squares are minimized or the prefixed maximum number of iterations is conducted.

Generalized Reduced Gradient Method (GRG). This is a general version of the reduced gradient method, which solves optimization problems with nonlinear constraints and arbitrary bounds. The method is described as follows as explained by Lasdon et al. [22].

The structure of general nonlinear problem is given by

$$ {\text{Minimize}}\,{\text{f}}_{0} \left( {\text{X}} \right) $$
(3)
$$ {\text{subject}}\,{\text{to}}\,{\text{the}}\,{\text{constraints}}\,{\text{g}}_{\text{i}} \left( {\text{X}} \right) = 0;{\text{where}},\,{\text{i}} = 1,2, \ldots {\text{m}} $$
(4)
$$ {\text{Where}},\,{\text{and}}\,{\text{l}}_{\text{j}} \le {\text{X}}_{\text{j}} \le {\text{u}}_{\text{j}} ;{\text{where}}, {\text{j}} = 1,2, \ldots {\text{n}} $$
(5)

Here, \( u_{j} \) and \( l_{j} \) indicate the upper boundary and lower boundary, respectively, where, \( u_{j} > l_{j} \) assuming m < n in order to avoid infeasibility of the solution or unique solution. The above forms are general since inequality constraints get transformed to equalities by adding slack variables. Following this, for solving the problem, basic variables (m) get expressed in terms of remaining nonbasic (n-m) variables. If \( \bar{X} \) denotes a feasible solution point and Y be the vector representing basic variables and Z be the vector of nonbasic variables on X, then after partitioning, Eqs. (4) and (5) can be written as

$$ {\text{X}} = \left( {{\text{Y}},{\text{Z}}} \right),{\bar{\text{X}}} = \left( {{\bar{\text{Y}}},{\bar{\text{Z}}}} \right) $$
(6)
$$ {\text{g}}_{\text{i}} \left( {{\text{Y}},{\text{Z}}} \right) = 0 $$
(7)

Here, it is assumed that both the objective and constraint functions are differentiable. The transformed objective function is given as

$$ {\text{F}}\left( {\text{Z}} \right) = {\text{f}}\left( {{\text{Y}}\left( {\text{Z}} \right),{\text{Z}}} \right) $$
(8)

Accordingly, the nonlinear problem is transformed at least for Z close to \( \bar{Z} \), to a reduced problem given by

$$ {\text{Minimize}}\,{\text{F}}\left( {\text{Z}} \right) $$
(9)
$$ {\text{Subject}}\,{\text{to}}\,{\text{l}}^{{\prime }} \le {\text{Z}} \le {\text{u}}^{{\prime }} $$
(10)

The GRG method actually solves the original problem as stated above by solving Eqs. (9) and (10). This study addresses a nonlinear problem since in our study, it has been observed that the monthly rate of return is having quadratic and cubic nature. Hence, the data suits the applicability of GRG method in our case.

3 Results and Discussions

Table 2 describes the distribution of the stocks (Table 1) into three defined clusters such as LSP, MSP, and HSP.

Table 2 Number of companies in each cluster
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In our study, we have selected MSP for further analysis since in LSP cluster we have 44 companies which make the portfolio too stretched for a common investor. Also, HSP cluster shows opposite nature and therefore, it has not been considered. Also, LSP signifies comparatively prematured or poor performance at the market and HSP indicates a bit saturated performance and comparatively less growth. We have considered to form a portfolio based on the stocks belonging to the MSP cluster. Table 3 lists out the stocks under study along with their monthly expected rate of return and standard deviations.

Table 3 Stocks under MSP cluster
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Further, we have performed a normality test (refer Table 4) in order to comply with the conditions of Markowitz [1]. It is seen from the result that the rate of returns of the stocks satisfies normality condition.

Table 4 Normality test
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In order to understand whether diversification is possible among the stocks, we have conducted a correlation analysis (refer Table 5). The determinant value (0.16548) obtained from the correlation matrix suggests that the stocks under the portfolio can be diversified. Hence, it is a problem of dynamic allocation of weights among the stocks.

Table 5 Correlation matrix
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Table 6 shows the variance (diagonal values) and covariance (off-diagonal values) analysis for determining risk.

Table 6 Variance and covariance table
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Table 7 shows the weights of the stocks calculated dynamically using GRG method. The optimum risk of the portfolio under study is 0.11% as calculated through the GRG method.

Table 7 Distribution of the weights to the stocks under the portfolio
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For validation purpose, we have performed perception mapping (refer Fig. 1). Table 8 shows the combined values of AVROR and SD. These two values are calculated as follows:

Fig. 1
figure 1

Perceptual map

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Table 8 Combined AVROR and SD
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$$ {\text{Combined}}\,{\text{AVROR}}\left( {\text{CAR}} \right) = {\text{Average}}\left( {\text{AVROR}} \right) $$
(11)
$$ \begin{aligned} {\text{Combined}}\,{\text{SD}}\left( {\text{CSD}} \right) & = \frac{{\sqrt {\sum \left( {d_{i}^{2} + SD_{i}^{2} } \right)} }}{n}; \end{aligned} $$
(12)

where

di:

Average return for ith Stock-CAR; i = 1, 2, …n.

Further, we have used the values of CAR and CSD to construct a perceptual map, wherein the axes are shifted from the origin to the point having CAR and CSD values. This results into generation of four new quadrants representing (High Return, High SD), (High SD, Low Return), (Low SD, High Return), and (Low SD, Low Return), respectively.

It is seen from the above figure that only P&G falls in the third quadrant, which characterizes high growth and low risk. It suggests that out of the stocks considered for the study, P&G stands alone, matched with the stated requirements. Therefore, it is logical to state that the earlier findings are being validated by the perceptual map.

4 Conclusion

In this study, we have attempted to assign optimal weights to the individual stocks constituting a portfolio in a dynamic environment. In order to select a portfolio, we have applied k-means clustering on a heterogeneous set of stocks listed in the NSE which are selected based on convenient sampling. Next, we have used a nonlinear optimization technique such as GRG for allocating weights with an objective to minimize the risk while maximizing the return. Allocation of the weights to the individual stocks forming the portfolio is having a significant impact on return on investment from the same. Further, in order to validate the results, a perception mapping of stocks under study has been performed which commensurate the earlier findings. This study is unique in the sense that initially, it started with unsupervised learning, but in the process, it came up with the bridging effect of unsupervised learning to a supervised learning validating all the way. This study may further be extended to analyze the clusters in the same way other than MSP for investigating any relation with the present one. Further, sector-wise portfolio performance may also be assessed for a comparative analysis in a dynamic and uncertain environment.