Sectoral portfolio optimization by judicious selection of ...

arX

iv:2

106.

1148

4v1

[q-

fin.

PM]

22

Jun

2021

Sectoral portfolio optimization by judicious selection of

financial ratios via PCA

Vrinda Dhingra1, Amita Sharma2, Shiv K. Gupta1*

Abstract

Embedding value investment in portfolio optimization models has always been achallenge. In this paper, we attempt to incorporate it by first employing principal com-ponent analysis sector wise to filter out dominant financial ratios from each sector andthereafter, use the portfolio optimization model incorporating second order stochasticdominance criteria to derive the final optimal investment. We consider a total of 11well known financial ratios corresponding to each sector representing four categoriesof ratios, namely Liquidity, Solvency, Profitability, and Valuation. PCA is then ap-plied sector wise over a period of 10 years from April 2004 to March 2014 to extractdominant ratios from each sector in two ways, one from the component solution andother from each category on the basis of their communalities. The two step SectoralPortfolio Optimization (SPO) model integrating the SSD criteria in constraints is thenutilized to build an optimal portfolio. The strategy formed using the former extractedratios is termed as PCA-SPO(A) and the latter one as PCA-SPO(B).

The results obtained from the proposed strategies are compared with the SPO modeland one step nominal SSD model, with and without financial ratios for computationalstudy. Empirical performance of proposed strategies is assessed over the period of 6years from April 2014 to March 2020 using a rolling window scheme with varying out-of-sample time line of 3, 6, 9, 12 and 24 months for S&P BSE 500 market. We observethat the proposed strategy PCA-SPO(B) outperforms all other models in terms ofdownside deviation, CVaR, VaR, Sortino ratio, Rachev ratio, and STARR ratios overalmost all out-of-sample periods. This highlights the importance of value investmentwhere ratios are carefully selected and embedded quantitatively in portfolio selectionprocess.

Keywords Principal component analysis, Financial Ratios, Sectoral portfolio op-timization, Second order stochastic dominance, In-sample and out-of-sample analysis

1 Department of Mathematics, Indian Institute of Technology Roorkee-247667, India.e-mail: Vrinda Dhingra: [email protected]

e-mail: Shiv K. Gupta: [email protected] Institute of Management Technology, Ghaziabad-201001, India.e-mail: Amita Sharma: [email protected]*Corresponding author.

1

http://arxiv.org/abs/2106.11484v1

1 Introduction

Benjamin Graham (Graham and Dodd 1934; Graham 1949) and Warren Buffet are amongthe famous personalities who emphasized on the importance of using fundamental analysisinto investment, also known as value investment. The aim of fundamental analysis is to pickfundamentally strong assets to produce robust performance of a portfolio over the investmentperiod. Financial analysis (FA) which is a subset of fundamental analysis, focuses on financialratios (FRs) to identify the strengths and weaknesses of a company in terms of its financialperformance e.g. sales, earnings, debt, liquidity, and liabilities among others. Financial ratio(FR) is the ratio of two appropriately chosen numerical values of the financial entities of acompany to figure out the relative performance. For example, current ratio collates currentassets to current liabilities, gauging a firm’s ability to meet its short term obligations whereasnet profit margin is the net profit earned by a company as a percentage of its total revenue,highlighting overall growth of the firm. The broad categories into which the FRs can beclassified are: profitability, liquidity, solvency, and valuation.

The importance and usefulness of FRs has been repeatedly demonstrated in various stud-ies (Beaver 1966; Chen and Shimerda 1981; Delen et al. 2013; Arkan 2016). Researchers,accountants, and financial analysts have been using financial ratios for the assessment of risks,forecasting failures, predicting future profits, and liquidity positions among others. The im-portance of FRs is not limited to forecasting and predictive analysis, it has also been adoptedin portfolio selection. Several researchers (Edirisinghe and Zhang 2007; Samaras et al. 2008)have incorporated the FRs by employing MCDM techniques to filter out elite stocks andthen construct an optimal portfolio. Yu et al. (2009) and Silva et al. (2014) used severalfundamental indicators like ROE (return on equity), Debt-ratio, and P/E ratio jointly withtechnical indicators to generate optimal portfolios using evolutionary algorithms. On theother hand, few others (Sharma and Mehra 2017; Tarczynski and Tarczynska- Luniewska2017) emphasized on the importance of sector-wise comparison of the assets when FRs isone of the selection criteria.

Considering the movement in share price alone is a naive way to evaluate the performanceof an asset. It is therefore advisable to invest in a company which is doing good in termsof its financial entries or FRs. However, when and how to include the FRs quantitativelyin portfolio selection process impacts its performance significantly. For instance, it is well-known that the sectors behave differently owing to their varied business functionalities andoperational structures and therefore, comparison between two companies belonging to differ-ent sectors 1 on the basis of same FRs is not wise. A very few studies focus on this importantstep. The authors (Sharma and Mehra 2017) attempted to resolve this issue by proposinga two-step investment scheme, termed as SPO in their paper, by first solving the secondorder stochastic dominance (SSD) portfolio optimization model for each sector separately onthe basis of their FR values and then a final optimal portfolio is obtained by combining theweights derived from step 1 along with their mean return performance. They used the SSDcriteria from the benchmark index in constraints to generate portfolios in both the steps.

SPO, though having several advantages over the classical SSD model, incorporates value

1In financial markets, sectors are the groups of stocks classified according to their similar business func-tionality and operations

2

investment in the form of four fixed FRs common for all sectors. However, it is well knownthat ratios that are found to be critical for one sector may not be so for another sectorbecause companies in different sectors have different market and capital structures. Forinstance, the banking and finance sector rely on credit calculations, therefore, liquidity andsolvency ratios are more important to this sector whereas the construction industry beingcapital intensive relies hugely on equipment to build projects and thus profitability ratios aremost valuable among construction based firms. Hence, the choice of ratio(s) in each sectoris crucial in the decision making process and thus considering fixed set of FRs for all sectorsunable to reflect the importance of value investment.

To address this issue, we propose to apply Principal Component Analysis (PCA) sectorwise to extract most dominant financial ratios for each sector and thereafter use the SPOtwo step scheme to generate final optimal portfolio. PCA is a non-parametric statisticaltool that transforms a set of variables into few factors called principal components, retainingmaximum information of the original variables. We offer to extract four dominant ratiosexplaining atleast 80% of the total variation in each sector from PCA in two ways, first byfetching the four ratios corresponding to each principal component and second by selectingfrom each of the four categories of ratios on the basis of their communalities. We then usethe SPO model to derive final optimal portfolios constructed from both types of extractions.We named the proposed strategy PCA-SPO(A) in case of deriving the overall four dominantratios and PCA-SPO(B) when each ratio becomes the dominant ratio in its category.

To widen the comparative analysis, we also solve classical SSD (Dentcheva and Ruszczynski2003) model with FRs (named as F-SSD) and without FRs (named as SSD) to reflect theimportance of correct usage of value investment in portfolio selection. While the SSD modelconsiders only mean return values in its objective function, the F-SSD model combines themean return values with the FRs in the objective function. PCA is being utilized when allassets are taken together to select the four dominant ratios in case of the F-SSD model.Also, both the SSD and F-SSD models are optimized on all assets at once. Therefore, an im-provement of the proposed strategies over F-SSD or SSD reflects the importance of utilizingFRs sector wise in asset allocation.

For the purpose of computational analysis, we consider historical data of the FRs andthe closing prices for the stocks listed in the 6 sectors of the S&P BSE 500 market, namelyInformation Technology (IT), Consumer Durable (CD), Energy, Banking and Finance, FastMoving Consumer Goods (FMCG), and Healthcare. We chose a set of total 11 FRs repre-senting four important categories, Profitability, Liquidity, Solvency and Valuation for eachof these 6 sectors for a period of 10 years from April 2004 to March 2014 to filter out domi-nant ratios using PCA. We then solve all the models under the in-sample and out-of-sampleframework over the period of 6 years from April 2014 to March 2020 using a rolling windowscheme.

To make the comparative study robust with respect to the investment period, the out-of-sample analysis is varied over different time periods ranging from 3 months to 24 months. Wesolve a total of five optimization models, SPO, SSD, F-SSD and two models corresponding tothe proposed strategies PCA-SPO(A) and PCA-SPO(B). The performance of the portfoliosgenerated from these models is measured and analyzed on many fronts, including meanreturn, several risk measures and performance ratios. We consider, Conditional Value atrisk (CVaR) at 95% and 97%, Downside Risk, Value at risk (VaR) at 95% and 97% as risk

3

measures and Sharpe, Sortino, STARR and Rachev as performance measures.We find that the portfolios obtained from the proposed strategy PCA-SPO(B) outper-

forms all other models in terms of risk-reward profile by achieving higher values of Sortino,Sharpe, Rachev and STARR ratios and lower risk values in terms of DD, VaR and CVaRin almost all the out-of-sample periods whereas PCA-SPO(A) achieves the highest meanreturn in almost all the periods but fails to impress much on other parameters. It shows thebenefit of considering ratios from each category in the process of portfolio construction. Wealso note down that the F-SSD model suffers with the highest risk and under performs inother measures in comparison to other models, indicating that mere consideration of finan-cial ratios in portfolio selection may not be wise and its correct implementation in form ofsector-wise comparison is even more crucial.

The rest of the paper is structured as follows. Section 2 presents a brief review of literatureon SSD, FA and PCA in portfolio selection. Section 3 describes the SPO model. Section4 elucidates the proposed strategy. Section 5 communicates the sample data, PCA results,the in-sample and out-of-sample performance analysis. Section 6 concludes the paper.

2 Literature Review

We now present a brief survey of literature on SSD, PCA and FA in portfolio selectionprocess.

2.1 Second order Stochastic Dominance

Stochastic Dominance (SD) stems from the theory of majorization (Marshall and Olkin1979) that primarily ranks two real-valued random vectors. SD is popularly known forits relationship with the different classes of utility functions (Levy 1992). Of utmost impor-tance to economics are SD of order one (FSD; Quirk and Saposnik 1962), order two (SSD;Hadar and Russell 1969) and order three (TSD; Whitmore 1970), for they respectively res-onate the behaviour of the class of rational, rational risk-averse, and rational risk-averse andruin-averse investors. Among the three orders, the SSD is widely used in portfolio opti-mization as it describes the preference of all rational, risk-averse investors, characterized bynon-decreasing concave utility functions.

The traditional mean-risk formulations are unable to extract much information fromthe return distribution, and thus some researchers recommended the incorporation of ei-ther more than one risk measures or moments ( Konno et al. 1993; Konno and Suzuki 1995;Sharma and Mehra 2013; Sharma and Mehra 2015; Zhao et al. 2015) or SD (Roman and Mitra2009) in portfolio optimization framework. Apart from being theoretically sound, portfoliooptimization models incorporating SSD criteria from the benchmark portfolio are compu-tational efficient (Fabian et al. 2011; Bruni et al. 2012; Roman et al. 2013; Sharma et al.2017; Sharma and Mehra 2017; Goel and Sharma 2019; Sehgal and Mehra 2020).Dentcheva and Ruszczynski (2003; 2006) used the lower partial moment (LPM) character-ization of SSD in the constraints from the benchmark portfolio. Roman et al. (2006) in-vestigated the applications of SSD in enhanced indexing by implementing the cutting planealgorithm. Fabian et al. (2011) proposed a portfolio optimization model incorporating SSD

4

using tail risk measures. Roman et al. (2013) analyzed the applications of SSD in portfo-lio generation using the reference point method. Fidan Kececi et al. (2016) compared theSSD constrained model with minimum variance and mean-variance models and observedout-performance of the former in terms of high returns and Sharpe ratio. Sharma et al.(2017) relaxed the SSD condition by introducing under-achievement and over-achievementvariables for the purpose of enhanced indexing in view of the almost SD introduced byLeshno and Levy (2002).

Lv et al. (1992) proposed an incremental bundle algorithm using inexact oracle for solvingSSD constrained PO model. Guran et al. (2019) applied the mean-variance PO on theenergy sector stocks by first eliminating inefficient stocks with the help of SSD efficiency test.Goel and Sharma (2019) introduced deviation measure in SSD and explored its applicationto enhanced indexing. Sehgal and Mehra (2020) proposed a robust portfolio model with SSDconstraints. Zhai et al. (2020) designed a multi-constraint SSD model and solved it usingWhale optimization algorithm (WOA). Moreover, extensive empirical results highlightingthe superiority of WOA over other swarm intelligence algorithms are validated in the paper.To widen the application of SSD, Liang et al. (2017) and Singh and Dharmaraja (2017)applied the SD selection criteria in disappointment theory and option pricing, respectively.

2.2 Financial Analysis

Edirisinghe and Zhang (2007; 2008) employed 18 financial ratios to analyze a firm’s finan-cial strength by proposing a generalized Data Envelopment Analysis (DEA) model. Theyfirst selected stocks on the basis of RFSI (relative financial strength indicator) and thenformed an optimal portfolio using the mean-variance model. The RFSI was calculated intwo ways, one using the raw indicators (total assets, long term debt, accounts receivable,net income and total liabilities) from the financial statements and other using the financialratios (profitability, liquidity, leverage, valuation and growth ratios). It was observed thatRFSI calculated using the latter was shown to have higher correlation with stock returnsand the portfolio thus generated performed better. This highlighted the importance of FA inportfolio selection. Some authors (Samaras et al. 2008; Xidonas et al. 2009) used FA to firstfilter stocks and then used Multi-Criteria Decision Making (MCDM) techniques to constructoptimal portfolios. Huang et al. (2014) employed FA using three key indicators- gross profitmargin, return on equity and cash flow growth rate to screen fundamentally strong stocks andthen an optimal portfolio was generated with the help of integrated DEA-Multi-ObjectiveDecision Making (DEA-MODM) model.

Tarczynski and Tarczynska- Luniewska (2017) emphasized on identifying the leading sec-tors in portfolio analysis. They employed several FRs, namely current ratio, return on assets,return on equity, and debt-equity ratio along with other indicators like size of sector to con-struct a fundamental power index (FPI) of sectors. Sharma and Mehra (2017) designed atwo-step SSD portfolio optimization model where the stocks were optimized sector-wise onthe basis of four FRs, namely return on assets, current ratio, debt-asset ratio, and P/E ratioto investigate the importance of sector wise selection. Jothimani et al. (2017) formulated aPCA-DEA model for stock selection. They used several FRs as input and output parame-ters, which were first filtered using PCA and thereafter utilized to select efficient stocks bythe DEA. Galankashi et al. (2020) employed a fuzzy-Analytic Network Process (fuzzy-ANP)

5

approach in portfolio selection process by incorporating multiple financial indicators, likeP/B ratio, P/E ratio, ROA(return on asset), ROE(return on equity), and EPS (earnings pershare). More recently, Liagkouras et al. (2020) incorporated value investment in the form ofsocial and environmental factors into the portfolio optimization framework.

2.3 Principal Component Analysis

Principal Component Analysis (PCA), a subset of Factor analysis is a statistical tool mainlyused for dimensionality reduction in wide variety of domains like computer vision and im-age processing (Hernandez and Mendez 2018), drug discovery and biomedical data (Giuliani2017), and materials science (Shenai et al. 2012). In Finance, PCA is used for time seriesforecasting (Chowdhury et al. 2018), financial risk ranking (Fang et al. 2018), forecastingstock prices (Wang and Wang 2015; Zhong and Enke 2017; Ghorbani and Chong 2020), se-lection of relevant financial ratios for predictive analysis (Olufemi et al. 2012) and stockselection for optimal portfolio generation (Yu et al. 2014; Joldes 2018).

Factor analysis for financial ratios was first applied by Pinches et al. (1973) where theyreduced their data set of bond ratings from 35 variables to 7 variables, retaining around 63%of the total variation. Tan et al. (1997) used factor analysis on 29 ratios for a period of 12years and derived 8 underlying factors for companies listed in Singapore. Ocal et al. (2007)applied factor analysis on the Turkish construction industry and reduced 25 financial ratiosto 5 significant factors. Yap et al. (2013) investigated the application of PCA on 28 financialratios for two industry sectors, namely consumer and trading sector and found that a smallerset of 7 and 9 ratios, respectively were sufficient and industry specific.

Fulga and Dedu (2012) designed a three-step model wherein they first employ PCA andclustering techniques to generate classes of similar stocks. Stocks with minimal Value-at-Risk (VaR) in each class are then filtered and an optimal portfolio is constructed usingmean-VaR model on these filtered stocks. Yu et al. (2014) formulated a PCA-SVM stockselection model. They first implemented PCA to reduce the dimensionality of the training setthat comprised of data of 20 financial ratios from 677 companies and then employed SupportVector Machine (SVM) algorithm for the selection of final stocks. Nadkarni and Neves (2018)combined PCA with a neural network algorithm to generate optimal trading signals.

We observed that either the researchers have totally ignored the importance of FA in thefield of portfolio optimization or even if they did, then they implemented FA using differentFRs depending on their popular usage. However, which ratio or a set of ratios one should useto analyze the objective at hand is a recurring question. Ideally, the selection of financialratios should not merely be based on the popularity of their usage but rather on sometheoretical or empirical evidence. The SPO model (Sharma and Mehra 2017) effectivelyimplements the use of FR sector-wise in the optimization model but uses fixed four ratios,common for all sectors. This paper is a very first attempt to improve the SPO model byimplementing Principal Component Analysis (PCA) for the selection of dominant ratios foreach sector.

6

3 The SPO Model

To make the paper self-sufficient, we begin by first describing the SPO model in brief. Thefollowing notations are used throughout the paper.

3.1 Notations

Γ time horizon of investment

T total number of scenarios of the time horizon Γ

N total number of assets

s total number of sectors

z = (z1, z2, . . . , zN )′ portfolio of N assets; zj denotes fraction of

∈ RN total budget invested in asset j

rj random return from jth asset

rjt return realization from jth asset at scenario t

pt probability of scenario t

Rz random return of portfolio z

Rzt =

N∑

j=1

rjtzj return realization of portfolio z at scenario t

E(Rz) =

T∑

t=1

(

N∑

j=1

rjtzj

)

pt expected value of Rz

FRzdistribution function of portfolio return Rz

3.2 Financial Ratios and sectors

In this subsection, we describe the financial ratios and the economic sectors.

1. Financial Ratios: Benjamin Graham, regarded as the father of fundamental analysis,was the first to idealize the concept of financial ratio analysis (Graham and Dodd 1934;Graham 1949). Financial ratio analysis, a subset of fundamental analysis, analyzes thefinancial statements of a company to determine its overall financial health. Unliketechnical analysis, which is purely based on the market’s past and current trends,fundamental analysis offers a more logical and intellectual decision framework.

Financial Ratios can be broadly classified into four categories: liquidity, profitability,solvency/leverage, and valuation whose brief description is given below:

(a) Profitability Ratio (PR) measures and evaluates the ability of a firm to utilize itsresources to generate profit. Profits generated by a firm are used to fund futuredevelopments and pay off dividends to the shareholders. Therefore, the efficiencyof a firm to make profits is an important factor to look after. A higher value of

7

PR is favourable. Gross/net profit margin, return on assets (ROA), return onequity (ROE), earning per share (EPS) are some of the well known profitabilityratios.

(b) Liquidity Ratio (LR) indicates the ability of a firm to meet its short-term debtobligations. A higher value of LR is preferable. Current ratio, quick ratio, oper-ating cash flow to current liabilities, are few examples of liquidity ratios.

(c) Solvency Ratio (SR) depicts the financial risk involved in terms of long-termborrowings. More borrowings imply higher risk and thus a lower solvency ratiois desirable. Debt-equity/assets ratio, interest coverage ratio, capitalization ratio,are some well known solvency ratios.

(d) Valuation Ratio (VR) helps to determine the relationship between the currentshare price of a company with its intrinsic value. It helps in estimating whethera company is exorbitant or cheaper compared to its growth, earnings and otherprospects. Price-to-earning/sales (P/E), price-to-book ratio (P/B), dividend yield,and enterprise value multiple, are examples of valuation ratios.

2. Sectors: A sector is a large fragment of an economy where companies that are en-gaged in a similar businesses are involved, like Auto, Finance, and Healthcare amongothers. Based on market movement, the sectors are broadly classified as cyclical, de-fensive (non-cyclical), and sensitive. Cyclical and sensitive sectors are the ones thatfollow market trend, while cyclical sectors are highly sensitive to the market peaks andtroughs, the sensitive sectors are comparatively moderate in their correlation to suchmovements. The defensive sectors, on the other hand are those which are least affectedwith the market oscillations. The defensive sectors are more stable in terms of theirearnings than the cyclic and sensitive sectors whose returns are comparatively volatile.Therefore, while comparing stocks on the basis of FA, a firm’s solvency in terms ofdebt-equity/assets ratio or valuation ratios like P/B and P/E ratios demonstrates bet-ter health of a cyclical or sensitive stock and a firm’s earnings figures like return onasset/equity are more relevant for the defensive sectors.

Table 1 describes the major sectors of the Indian economy along with their classification2

and composition 3 in the S&P BSE 500 market. The Finance sector is a combination ofprivate and PSU banks and hence the largest sector under the index. The miscellaneoussector includes those sectors which are either having less than 5 companies, like Tourism, orsectors having diversified businesses.

We now describe the two-step SPO model which optimizes the portfolio on the basis ofFR sector wise.

3.3 Sector Portfolio Optimization Model (SPO)

The SPO model is a two-step PO model (Sharma and Mehra , 2017) that first optimizesthe muliti-objective programs to select stocks sector-wise on the basis of following four fixedFRs:

2https://www.bseindia.com/sensex/IndicesWatch Sector.aspx?iname=BSE500&index Code=173https://www.valueresearchonline.com/stocks/selector/indices/46/bse-500-index/

8

Table 1: Sector composition in S&P BSE 500

S.No. Sectors Classification % weight

1 Finance Cyclic 16.22 Auto(Transport Equipments) Cyclic 6.03 Capital Goods (CG) Cyclic 9.04 Metals Cyclic 5.05 Housing Cyclic 7.26 Chemical & Petrochemical Cyclic 6.87 Information Technology (IT) Sensitive 3.88 Communication Sensitive 2.09 Energy (Oil & Gas) Sensitive 5.410 FMCG Defensive 8.811 HealthCare Defensive 10.012 Power Defensive 1.013 Consumer Durables (CD) Defensive 3.014 Agriculture Defensive 2.015 Textile Defensive 2.416 Media & Publishing Defensive 3.217 Transport services Defensive 1.218 Miscellaneous Mixed 7.0

1. Return on Assets (ROA) from the Profitability Ratio

2. Current Ratio (Liq) from the Liquidity Ratio

3. Debt-asset (Debt) Ratio from the Solvency Ratio

4. P/E Ratio (P/E) from the Valuation Ratio

The selection of above ratios in each category is due to their popular usage in financialanalysis. The optimal stocks found from the first step are then pooled together to generatea final optimal portfolio combining their performance in terms of the mean return and FRs.

Step 1: Solving a multi-objective program for each sector

In Step 1, s optimization models are solved, separately for s sectors, dominating the bench-mark portfolio in SSD rule in constraints while keeping FRs in the objective functions. Letnr be the total number of assets in the rth sector; r = 1, . . . , s and zr = (zri ; i = 1, . . . , nr)be the corresponding weights vector, then the models are multi-objective programs, namedas (SP1)r; r = 1, . . . , s, defined as follows:

(SP1)r Maxnr∑

i=1

(ROA)ri zri , Max

nr∑

i=1

(Liq)ri zri ,

Minnr∑

i=1

(Debt)ri zri , Min

nr∑

i=1

(P/E)ri zri

subject toE(ηr − Rr

z)+ ≤ E(ηr − Y r)+; ηr = yr1, . . . , y

rT

0 ≤ zri ≤ 0.3; i = 1, . . . , nrnr∑

i=1

zri = 1,

9

where yr1, . . . , yrT are the T realizations of Y r, which is a random return of the sectoral

benchmark index and (ROA)ri , (Liq)ri , (Debt)ri , and (P/E)ri are respectively the profitability,liquidity, solvency, and valuation ratios of the ith asset from the rth sector. Constraintszri ≤ 0.3 impose a 30% upper bound on the investment in each asset in every sector.

The s multi-objective programs (SP1)r; r = 1, . . . , s are converted into respective singleobjective programs (WSP1)r; r = 1, . . . , s using the weighted-sum approach, given as:

(WSP1)r Max wr1

nr∑

i=1

(ROA)rizri − wr

2

nr∑

i=1

(Debt)ri zri +

wr3

nr∑

i=1

(Liq)ri zri − wr

4

nr∑

i=1

(P/E)ri zri

subject to

E(ηr −Rzr)+ ≤ E(ηr − Y r)+; ηr = yr1, . . . , y

rT

0 ≤ zri ≤ 0.3; i = 1, . . . , nr

nr∑

i=1

zri = 1.

The weights wrj (j = 1, 2, 3, 4) in the programs (WSP1)r; r = 1, . . . , s are obtained by

analysing the effect of the individual FR on the Sharpe ratio, higher the effect of particularFR on Sharpe ratio, more weight (or importance) is given to that FR.

Step 2: Combining stocks together

Step 2 of the SPO model combines optimal portfolio weights (zr)∗; r = 1, . . . , s derived from sprograms (WSP1)r; (r = 1, . . . , s) to generate final optimal portfolio. Let mr = {i |(zri )∗ >0, 1 ≤ i ≤ nr} be the set comprising of filtered stocks from step 1, then the final optimalportfolio is obtained by solving the following optimization problem:

(SP2) Maxs∑

r=1

∑

j∈mr

((

T∑

t=1

rjtpt

)

+ (zrj )∗

)

zrj

subject toE(η −Rz)

+ ≤ E(η − Y )+; η = y1, . . . , yT0 ≤ zrj ≤ 0.3; j ∈ mr, r = 1, . . . , s

s∑

r=1

∑

j∈mr

zrj = 1,

where Y is the random return of the benchmark index and y1, y2, . . . , yT are the T discretepoint realizations of Y . Step 2 ensures that the stocks having higher weights from Step 1get higher allocation in the final optimal portfolio while dominating the benchmark in SSDcriteria when combined together.

The SPO model incorporates the sector-wise use of FRs in the portfolio construction andhas several advantages over the classical SSD model. However, it ignores the fact that theratios that are found to be critical for one sector may not be so for another sector. This iswhat formed the basis and primarily led to the following proposed strategy.

10

4 Proposed Strategy: An extension to SPO

4.1 Finding the Relevant ratios for each sector using PCA

We aim to improve the SPO model by carefully selecting the critical FRs in its Step 1 foreach sector. To meet the objective, we propose to use Principal Component Analysis (PCA),a non-parametric statistical tool to select critical FRs separately for each sector.

PCA is the buttress of new age data analysis, devised by Pearson (1901) as an analogueof principal axis theorem in mechanics and later developed and termed by Hotelling (1933).PCA has been profusely used in several disciplines, from biomedical sciences to computervision. The central idea of PCA is to compress a large data set of inter-related variables into asmaller set of variables while retaining maximum variation as in the original variables. PCAresults into a new set of factors, called principal components which are uncorrelated and thefirst few components cumulatively explain most of the variation (maximum information) asin the original variables. PCA on financial ratios was first applied by Pinches et al. (1973)with an objective to reduce the number of ratios used for predicting market crashes, bondratings, and company failures.

PCA orthogonally transforms a data set of k random variables, denoted as the randomvector X =

[

X1 X2 · · · Xk

]

into a smaller set of linearly uncorrelated variables, calledprincipal components. Intuitively, PCA can be visualized as fitting a k dimensional ellipsoidto the data, axes of which form the principal components. The longest axis of the ellipsoid isindicative of the direction along which the data points have maximum variation (maximuminformation). If an axis of the ellipsoid is small, then the variation along it is equally smalland thus on deleting the axis (and the corresponding principal component) one does notloose out on a lot of information. The aim is thus to determine the axes of the ellipsoid.

In this paper, the data set of k variables corresponds to the k financial ratios. If theith ratio is denoted by Xi with m independent observations, then X =

[

X1, X2, · · · Xk

]

becomes a matrix of order m× k, X = [Xij ].Steps for determining the Principal Components:

1. Scaling of Data: The mean row of the data matrix X as x =[

x1 · · · xk

]

of order

1 × k, where xj =1

m

m∑

i=1

Xij ∀ j = 1, . . . , k. The mean matrix is then computed as

X = 1Tx,

where 1 denotes a row vector of ones of order m. Subtracting X from the data matrixX gives the scaled data matrix,

Y = X −X

2. Calculating the Covariance Matrix: The covariance matrix is a k× k matrix and

is given by C =1

m− 1Y TY .

3. Computing the principal components: To determine the axes, we compute theeigenvalues and eigenvectors of the covariance matrix C. The eigenvectors of the covari-ance matrix represent the directions of the axes accumulating most of the variation.

11

Table 2: Financial Ratios used in the study

Ratio Label Ratio Name Description

A. Profitability Ratios (larger value is preferable)

NPM Net Profit Margin Net profit/Total revenueROA Return on Assets Net Income/Total assetsCPTI Cash Profit Ratio Cash Profit/Total incomeROE Return on Equity Net income/Average Shareholders’ equity

B. Liquidity Ratios (larger value is preferable)

QR Quick Ratio Liquid Assets/Current LiabilitiesCR Current Ratio Current Assets/Current Liabilities

CCL Cash Ratio Cash & Cash equivalents/Current Liabilities

C. Solvency Ratios (smaller value is preferable)

DER Debt-Equity Ratio Total debt/Shareholders’ equityTLTTA TOL/TNW Ratio Total liabilities/Total tangible assets

D. Valuation Ratios (smaller value is preferable)

PER P/E Ratio Share price/Earnings per share (EPS)PBR P/B Ratio Market price per share/Book value per share

Arranging the eigenvectors in descending order of their corresponding eigenvalues givesthe principal components (PCs). The eigenvalues depicts the variance accounted bythe corresponding PC (eigenvector).

4. Deriving the loadings matrix: After computing the principal components as eigen-vectors, we compute the loadings matrix, comprising of component loadings (or simplyloadings) defined as

Component Loadings=√

eigenvalue× eigenvector

The component loadings denote the covariances (or correlations) between the originalvariables (here, ratios) and the principal components.

Sum of the squared values of the loadings gives the communality of the variable (ratio)with respect to the component solution, which denotes the variance in the original variableaccounted for by the component solution. The extraction of the dominant ratios from PCAcan be done in two ways, one on the basis of component loadings and two, on the basis oftheir communalties as discussed in detail in following section.

4.2 Proposed strategies

For empirical purpose in the present study, PCA is applied to a set of 11 financial ratiosseparately for each sector over a period of 10 years from April 2004 to March 2014. The

12

details of these 11 ratios is given in Table 2. We propose to fetch four components thatexplain atleast 80% of the total ratio variances. These components together are referred toas the component solution. Though there is no thumb rule for extracting the significantoriginal variables from the PCA solution, but several authors (for example, see Yap et al.2013) have considered the loads4 from the loadings matrix as a measure of extraction.

We adopt the following two methods for extracting the dominant ratios from PCs, ex-plained as follows:

(A) On the basis of component loadingsFrom each principal component, the financial ratio with the highest value (absolute)of component loading is extracted. This gives us the dominant ratios for each sector.These ratios may come from any of the category of the financial ratios, PR, LR, SR,and VR. However, we impose a bound on the extraction of maximum of two ratiosfrom each category.

(B) On the basis of communalitiesIn this extraction method, we fetch the financial ratio having the maximum commu-nality from each category, viz., PR, LR, SR and VR. In this way, we have four criticalratios that represent all four categories for each sector.

After the selection of ratios for each sector, the two-step SPO model is employed to obtainfinal optimal portfolios. The portfolio selection strategy using extraction method (A) andextraction method (B), respectively are labelled as PCA-SPO(A) and PCA-SPO(B). Thepurpose of adopting the two extraction methods is to understand the importance and impactof inclusion of critical ratios representing all category of ratios on portfolio performance.

To examine the relevance of applying PCA sector-wise, we also apply PCA on the datawhen all assets are taken together to extract four dominant ratios. Four components thatexplain atleast 80% of the total ratio variances are extracted. We then apply the nominal SSDmodel maximizing the weighted sum of the return function and the functions correspondingto the extracted FRs to obtain the optimal portfolio. We named this model as F-SSD. Letus denote the extracted ratios from PCA for the F-SSD model as R1, R2, R3 and R4, thenfor 0 ≤ α ≤ 1, the F-SSD model is given as follows:

(F-SSD) Max α(FR)z + (1 − α)E(Rz)subject toE(ξ − Rz)

+ ≤ E(ξ − Y )+ ∀ ξ ∈ R

z ∈ Z,

where Y denotes a random return of the benchmark index, and

(FR)z = ±w1

N∑

j=1

(R1)jzj ± w2

N∑

j=1

(R2)jzj ± w3

N∑

j=1

(R3)jzj ± w4

N∑

j=1

(R4)jzj.

4Loads gives the information of the amount of variance of the original variable(ratio), the principalcomponent accounts for

13

The weights w1, w2, w3, and w4 are taken as the proportion of variance the components (PCs)account for, where the sign (“+” for maximization and “−” for minimization) indicates theeffect of that ratio on the return.

For the current framework, we set α = 0.5 giving equal importance to returns and FRs.The F-SSD model incorporates FRs in the SSD framework when all stocks are consideredtogether and does not focus on sector-wise bifurcation. Along with the F-SSD model, wealso solve nominal SSD model which can be easily obtained from the F-SSD model for α = 0.The nominal SSD model maximizes only the return function of the portfolio and thus theassets are not chosen on the basis of their fundamental performance.

5 Empirical Analysis

5.1 Historical Data and sample period

We consider the Indian stock market S&P BSE 500 as our sample data wherein we studyfollowing six sectors in the present empirical analysis.

(A) Cyclic/Sensitive Sectors

1 Energy: India is the 3rd largest consumer of energy globally. With the prolif-erating demand, the energy sector is growing robustly and thereby making thissector conducive for investment.

2 Finance: With present growth rate at 8.5% p.a., the finance sector is intrinsicallystrong and operationally diversified. Mutual Funds Investment and insuranceindustries are crucial components of this sector, contributing to its expandinggrowth.

3 Information Technology (IT): The IT industry has played an important rolein placing India on the international map. With an export of $70 billion, it is oneof the top industry sectors in the country.

(B) Non-Cyclic Sectors

1 FMCG: The FMCG sector is the fourth largest sector in India, with householdand personal care contributing half of its sales. Changing lifestyles, easy accessi-bility and increased awareness are prime factors driving its growth, especially inrural India. The sector has seen a revenue growth of 2.1% in the last decade.

2 HealthCare: India’s healthcare sector is the 3rd largest in the world in terms ofvolume and 14th largest in terms of value. It is highly fragmented and competitive,holding approximately 6-7% of the market share with the leading player.

3 Consumer Durables (CD): The CD sector with a size of $10.9 billion in 2019is one of the key sectors for India to achieve its self sufficiency or the Make-in-India objective. With changing consumer demands towards convenience andautomation, the sector is flourishing and inviting investment opportunities.

14

These sectors together represent approximately 64% of the whole market of S&P BSE500. Bombay Stock Exchange (BSE) is the oldest stock exchange in Asia and the 10thlargest in the world by market capitalization as on March, 2020. On February 19, 2013, BSEpartnered with S&P DOW JONES INDICES and got renamed as S&P BSE “*” (* indicatesvarious defined sets like SENSEX, 100, etc). The S&P BSE 500 comprises of 500 companiescovering all major industry sectors of the Indian economy and depicts nearly 93% of BSE’stotal market capitalization.

We collect the following data for stocks listed under the above mentioned 6 sectors ofS&P BSE 500 from CMIE Prowess IQ Database(https://prowessiq.cmie.com/):

1. The weekly adjusted closing price from April 2014-March 2020.

2. Quarterly data of all the ratios listed in Table 2 from April 2004-March 2019, giving atotal of 40 data points for each asset and each ratio. Thus, for sector r with number ofassets nr, the sample data is a matrix with (nr × 40) rows and 11 columns (for ratios)for proposed strategies and a matrix with 115 × 40 rows and 11 columns in case ofF-SSD.

The weekly return for j-th stock in t-th week is calculated as rjt =pcjt−pcjt−1

pcjt−1

;

j = 1, . . . , N, t = 1, . . . , T, where pcjt and pcjt−1 are respectively, the closing prices in thet-th and (t− 1)-th week. We use R software with SYMPHONY solver on Windows 64 bitsIntel(R) Core(TM) i3-6006U CPU @2.0 GHz processor to solve all optimization models.

Stocks whose price data is missing for more than 2 quarters and/or whose ratios datais missing for more than 4 quarters are deleted from the study. The total number of assetshenceforth in the present study are left to 115 whose sector-wise composition is as follows:

• Energy-14, Finance-32, Information Technology-12,

• FMCG-23, Healthcare-27, Consumer Durables-7.

To analyze the financial benefits of the proposed strategies, we follow a rolling windowscheme of 4 weeks over the time horizon of total 5 years from April 2014 to March 2020with in-sample period consisting of 52 weeks. To make the study robust with respect tothe investment period, we vary the out-of-sample period over 3 months (OSP3), 6 months(OSP6), 9 months (OSP9), 12 months (OSP12), and 24 months (OSP24) while keepingin-sample period fixed as 52 weeks. By sliding the in-sample period by 4 weeks, we get atotal of 51 out-of-sample windows each for OSP3, OSP6 and OSP9, and 49 and 37 windowsrespectively, for OSP12 and OSP24.

The performance of the optimal portfolios obtained from the proposed strategies, PCA-SPO(A), PCA-SPO(B), and the models SPO, SSD, F-SSD along with the benchmark indexS&P 500 BSE are compared on the basis of mean return, three risk measures: downsidedeviation (DD), VaR and CVaR values for 95% and 97% levels of confidence, and the fourperformance measures: the Sharpe, the Sortino, the Rachev, and the STARR ratios. Wereport the values of the Rachev and the STARR ratios for 95% and 97% levels of confidence.

15

Table 3: Component loadings matrix for CD Sector

Ratios PC1 PC2 PC3 PC4 Communality

Liquidity RatiosQR 0.8885 -0.2441 -0.1309 -0.1369 0.8849CR 0.9775 -0.0172 -0.1393 -0.0261 0.9759

CCL 0.5454 0.3073 0.3860 -0.6576 0.9734Profitability RatiosNPM 0.4540 0.8279 -0.0078 0.2826 0.9716ROA 0.2508 0.9159 0.0049 0.2801 0.9802CPTI -0.2493 0.8533 0.2938 -0.0169 0.8769ROE -0.5865 0.7787 0.1791 -0.0373 0.9839

Solvency RatiosDER -0.7159 -0.5552 0.2339 0.0966 0.8848TLTTA -0.9343 -0.2002 0.2488 -0.0675 0.9794

Valuation RatiosPER -0.6339 0.1966 -0.6654 -0.1718 0.9127PBR -0.5358 0.6158 -0.3746 -0.3327 0.9173

Cumm. Variance 0.4356 0.7792 0.8705 0.9401

5.2 In sample outcomes

Recall that four components that explain atleast 80% of the total ratio variances are extractedfor the proposed strategies and as well as for the F-SSD model. Moreover, ratios having thehighest component load are selected from each principal component in case of PCA-SPO(A)and F-SSD whereas ratios having maximum communality from each category are selectedfor PCA-SPO(B).

Tables 3 and 4 respectively, illustrate the values of component loading for CD and En-ergy sectors. The four extracted components (PC1-PC4) are referred to as the componentsolution. We can note that four principal components that are extracted for the sectors CDand Energy together explain respectively, about 94.01% and 90.23% of the total ratio vari-ances. The last column denotes the communalities of the ratios accounted by the componentsolution. The sign of the loading is indicative of the direction of relationship and is thusignored while fetching the ratios from the components.

From Table 3, we observe that the ratios CR, ROA, PER and CCL are found to be thecritical ratios (highlighted in bold) having respective loading as 0.9775, 0.9158, -0.6654, and-0.6576 for PCA-SPO(A) strategy. And for the strategy PCA-SPO(B), the critical ratiosbecome CR, ROE, TLTTA and PBR (bold & italicised) having respective communalities as0.9759, 0.9839, 0.9794, and 0.9173. Similar readings are for the Energy sector from Table 4.

Tables 5 and 6 respectively, explain the ratios selected for both the proposed strategiesPCA-SPO(A) and PCA-SPO(B) corresponding to all sectors. Last columns of the Tablesdescribe the cumulative variance accounted by the component solution for each sector.

Table 7 displays the values of component loadings when PCA is applied on all assets atonce. The dominant ratios are obtained as CR, DER, PBR and CPTI having respectively,

16

Table 4: Component loadings matrix for Energy Sector

Ratios PC1 PC2 PC3 PC4 Communality

Liquidity RatiosQR -0.9267 0.0443 -0.3293 -0.0159 0.9695CR -0.7734 0.3095 -0.4831 -0.0592 0.9309CCL -0.9271 0.1509 -0.3199 0.0512 0.9873

Profitability RatiosNPM -0.7117 0.3077 0.1476 -0.4258 0.8043ROA -0.8114 0.2349 0.4572 -0.1343 0.9406

CPTI -0.8958 -0.0007 0.0191 0.2529 0.8669ROE -0.1725 0.4629 0.7953 -0.0688 0.8813Solvency RatiosDER -0.2436 0.6389 0.2140 0.6076 0.8826

TLTTA 0.5648 0.6838 -0.1598 0.1275 0.8284Valuation RatiosPER -0.3858 -0.8310 0.0843 0.2609 0.9147PBR -0.4787 -0.7479 0.3557 0.0705 0.9200

Cumm. Variance 0.4609 0.6982 0.8357 0.9023

Table 5: Critical Ratios for each sector for the strategy PCA-SPO(A)

Sectors Ratio1 Ratio2 Ratio3 Ratio4 Cummulative Variance

of component solution

Energy CCL PER ROE DER 90.23%Finance ROE CCL PER TLTTA 93.99%FMCG TLTTA NPM PER DER 94.69%CD CR ROA PER CCL 94.01%IT CR PBR ROE ROA 90.81%HealthCare CCL DER PBR PER 90.04%

Table 6: Critical Ratios for each sector for the strategy PCA-SPO(B)

Sectors LR PR SR VR Cummulative Variance

of component solution

Energy CCL ROA DER PBR 90.23%Finance CCL ROE DER PER 93.99%FMCG CCL ROE DER PBR 94.69%CD CR ROE TLTTA PBR 94.01%IT CR ROA DER PBR 90.81%HealthCare QR NPM DER PER 90.04%

17

Table 7: Factor loadings matrix for ALL assets

Ratios PC1 PC2 PC3 PC4

Liquidity RatiosQR 0.85450 0.4003 -0.1891 0.1594CR 0.8561 0.3577 -0.1747 0.2099CCL 0.8149 0.3868 -0.2745 0.0765Profitability RatiosNPM 0.8301 -0.4167 -0.0330 -0.1462ROA 0.6691 -0.6311 0.1598 0.0912CPTI 0.5896 0.2340 -0.4255 -0.5011ROE 0.6559 -0.5815 0.1344 -0.0923Solvency RatiosDER -0.3616 0.7535 -0.1806 -0.2608TLTTA -0.4178 -0.2654 -0.6889 0.1017Valuation RatiosPER -0.2069 -0.5821 -0.3829 -0.4210PBR -0.2629 -0.3390 -0.7060 0.3833Proportion of Variance 0.4054 0.2268 0.1381 0.0697

0.8561, 0.7535, -0.7060 and -0.5011 loads. The last row represents the proportion of varianceeach principal component accounts for and are taken as the weights corresponding to eachratio in the F-SSD model.

We now analyze the financial benefits of the proposed strategies PCA-SPO(A) and PCA-SPO(B) in comparison to the three models, SPO, F-SSD and the nominal SSD along withthe benchmark index S&P BSE 500.

5.3 Out-of-sample analysis

Table 8 records the performance of the portfolios derived from PCA-SPO(A), PCA-SPO(B),SPO, SSD, F-SSD, along with the benchmark index for the out-of-sample period of length3 months. Following observations are drawn from Table 8:

1. The portfolio from PCA-SPO(A) outperforms all portfolios in terms of mean return,Sharpe ratio, and the STARR ratio.

2. The portfolio from PCA-SPO(B) outperforms all other portfolios in terms of all riskmeasures, the Sortino ratio and the Rachev ratios whereas it takes the second bestposition in terms of the Sharpe and STARR ratios after PCA-SPO(A).

3. The F-SSD model turned out to be the most risky as it suffers with the highest valuesof DD, CVaR0.95, CVaR0.97, VaR0.95, and VaR0.97 in comparison to other portfolios,which hints that analysing all stocks together on the basis of FRs may mislead aninvestor. Moreover, not using FRs at all in the selection process can also account forhigh risk as SSD model stands the second most risky model.

18

Table 8: [OSP3] Out-of-sample performance of portfolios for 3 months

Portfolios BM SSD SPO PCA- PCA- F-SSDSPO(A) SPO(B)

Mean Return 0.00289 0.00384 0.00364 0.00471 0.00375 0.00232Sharpe Ratio 0.33423 0.29883 0.42849 0.51429 0.47407 0.13494Sortino Ratio 0.79622 0.66021 0.96805 1.59761 1.60163 0.23782DD 0.00204 0.00389 0.00245 0.00215 0.00155 0.00442VaR0.95 0.00511 0.01015 0.00639 0.00535 0.00373 0.01087VaR0.97 0.00538 0.01113 0.00648 0.00598 0.00542 0.01211CVaR0.95 0.00649 0.01104 0.00807 0.00633 0.00497 0.01250CVaR0.97 0.00718 0.01149 0.00891 0.00681 0.00558 0.01380Rachev0.95 1.74422 1.86926 1.80834 3.02685 3.24614 1.03467Rachev0.97 1.64136 1.91427 1.72559 2.92070 3.22673 0.95072STARR0.95 0.25027 0.23263 0.29389 0.54263 0.49950 0.08409STARR0.97 0.22622 0.22352 0.26619 0.50438 0.44489 0.07617

4. None of the proposed strategies, PCA-SPO(A) and PCA-SPO(B) generated worstresults in any of the performance measures considered in the study.

Table 9 represents the performance for the out-of-sample period of length 6 months, andbased on it, the observations are summarized as follows:

1. The portfolio from PCA-SPO(B) generated the highest value for Sharpe ratio whilesecond highest values for all other ratios maintaining its consistent performance fromOSP3 (Table 8) to OSP6 (Table 9).

2. Unlike in the case of the OSP3 (Table 8), the SPO model in this case of OSP6 performedwell by achieving best values in CVaR, Sortino, Rachev, and STARR ratios.

3. The SSD model and PCA-SPO(A) respectively achieved the highest and the secondhighest mean return values whereas portfolios from F-SSD again suffer with the highestrisk in terms of DD, VaR0.97, VaR0.97, and CVaR0.95.

4. None of the proposed strategies, PCA-SPO(A) and PCA-SPO(B) generated worstresults in any of the performance measures in this case as well. However, the portfoliosfrom PCA-SPO(B) performed better than PCA-SPO(A) in all performance measuresreflecting the advantage of including ratios from each category.

Tables, 10, 11, and 12 exhibit the performance for the out-of-sample periods of length 9,12, and 24 months, respectively. The observations drawn from Tables 10, 11, and 12 are asfollows:

1. The portfolios from PCA-SPO(B) performed the best in all performance measuresexcept in mean return where it achieved the 4th, 3rd, and 3rd best positions for OSP9,OSP12, and OSP24, respectively. Though the portfolios from PCA-SPO(A) achieved

19




best results in terms of mean return over all time periods, they fail to impress much onother performance measures in comparison to other portfolios. These findings highlightthe importance of including ratios from each category in portfolio selection process.However, note that PCA-SPO(A) performs better than the benchmark index in termsof all performance ratios over all out-of-sample periods.

2. The SPO model takes the second best position in the performance analysis, whichagain reflects the importance of considering ratios representing all categories. However,PCA-SPO(B) performed better than SPO indicating the importance of PCA to pickthe critical ratio from each category than simply taking fixed ratios, common for allsectors.

3. The F-SSD model suffered with the highest risk again indicating the penalty of notusing the FRs correctly in the portfolio selection process.

To summarize, the proposed strategy PCA-SPO(B) and the SPO model demonstratesthe overall best and second best results, respectively. Although the strategy PCA-SPO(A)achieves the highest mean return for all out-sample periods, yet it fails to perform well onother parameters, specifically over the 9, 12 and 24 months out-sample periods. However,PCA-SPO(A) never produces worse outcomes in comparison to other models and alwaysgenerated better results than the benchmark index, especially in terms of mean return andthe performance ratios. We also observe that the model F-SSD suffers with the highest riskall the time. Further, the overall analysis can be concluded as follows:

1. The outperformance of PCA-SPO(B) over PCA-SPO(A) indicates the importance ofincorporating the FRs from all category of ratios.

2. The outperformance of PCA-SPO(B) over SPO indicates the importance of selectingthe critical ratio from each category in the portfolio selection process.

20







21




3. The outperformance of PCA-SPO(A), PCA-SPO(B) and SPO over the F-SSD modelhighlights the importance of sector wise selection.

4. Last but not least, the outperformance of PCA-SPO(A), PCA-SPO(B) and SPO fromthe SSD model underlines the importance of including FRs in portfolio selection pro-cess.

The above points clearly indicate the importance of incorporating the critical ratio fromeach category when the selection is done sector wise.

6 Conclusion

The importance of FA in portfolio framework can not be overlooked. However, the rightchoice of ratios and their correct usage is even more crucial, owing to the fact that companiesbelonging to different industry sectors have different operational structures and businessfunctionalities.

The SPO model aptly considers the sector-wise implementation but chooses ratios com-mon for all sectors. This paper is an attempt forward on improvising the SPO model byimplementing PCA for the selection of critical ratios for each sector, extracted in two ways,one from the components (strategy named as PCA-SPO(A)) and other from each categoryof ratios on the basis of communalities (strategy named as PCA-SPO(B)). We conduct thecomparative analysis on the proposed strategies, PCA-SPO(A) and PCA-SPO(B), with themodels F-SSD, SSD, and SPO along with the benchmark index. The rolling window anal-ysis which is carried over varying periods of out-of-sample lengths showed that the optimalportfolios obtained from PCA-SPO(B) resulted in lower values of risk measures and gener-ated better risk-reward profile in terms of Sortino, Sharpe, Rachev, and STARR ratios incomparison to other models on almost all out-of-sample periods.

22

The outperformance of portfolios from PCA-SPO(B) over SPO suggests the importanceof including the dominant ratios than using a fixed set of ratios, common for all sectors. Thestrategy PCA-SPO(A) generated higher mean return over almost all out-of-sample periodsbut was unable to attract much on the other performance measures. The outperformance ofPCA-SPO(B) and SPO over PCA-SPO(A) highlights the importance of consideration of eachcategory of ratios. The F-SSD model results in the highest risk after SSD, underlining thatmere inclusion of fundamental analysis is not sufficient and its correct application sector-wiseis even more crucial.

To sum up, the proposed strategy PCA-SPO(B) is an apt decision making process thatfocuses primarily on the extraction of critical ratios by employing PCA sector wise and thengenerate final portfolio incorporating the two step SPO model. The financial benefits of PCA-SPO(B) over SPO and other models confirms the practical usage of the proposed strategy.Further extensions of the present work are possible by including other important factors liketechnical indicators, sustainability, socio-economic variables, and transaction costs.

AcknowledgmentsThe authors acknowledge the support of the Institute of Management Technology (IMT),Ghaziabad for the access to Prowess IQ Database for the collection of data. The first authorwould also like to thank the Ministry of Human Resource and Development (MHRD), NewDelhi, India for financial support.

References

Arkan T (2016) The importance of financial ratios in predicting stock price trends: A case study in emergingmarkets. Zeszyty Naukowe Uniwersytetu Szczecinskiego Finanse Rynki Finansowe Ubezpieczenia, 1 (1 ):13–26.

Beaver WH (1966) Financial ratios as predictors of failure. Journal of Accounting Research 4 :71–111.

Bruni R, Cesarone F, Scozzari A, Tardella F (2012) A new stochastic dominance approach to enhanced indextracking problems. Economic Bulletin 32 (4 ):3460–3470.

Chen KH, Shimerda TA (1981) An empirical analysis of useful financial ratios. Financial Management,

10 (1 ):51–60.

Chowdhury U, Chakravarty S, Hossain M (2018) Short-term financial time series forecasting integratingprincipal component analysis and independent component analysis with support vector regression. Journalof Computer and Communications, 6 :51–67.

Delen D, Kuzey C, Uyar A (2013) Measuring firm performance using financial ratios: A decision treeapproach. Expert Systems with Applications, 40 :3970–3983.

Dentcheva D, Ruszczynski A (2003) Optimization with stochastic dominance constraints. SIAM Journal on

Optimization 14 (2 ):548–566.

Dentcheva D, Ruszczynski A (2006) Portfolio optimization with stochastic dominance constraints. Journalof Banking and Finance 30 (2 ):433–451.

23

Edirisinghe NCP, Zhang X (2007) Generalized dea model of fundamental analysis and its aplication toportfolio optimization. Journal of Banking and Finance, 31 (11 ):3311–3335.

Edirisinghe NCP, Zhang X (2008) Portfolio selection under dea-based relative financial strength indicators:case of us industries. Journal of the Operational Research Society, 59 (6 ):842–856.

Fabian C, Mitra G, Roman D (2011) Processing second order stochastic dominance models using cutting-plane representations. Mathematical Programming 130 :33–57

Fang L, Xiao B, Yu H, You Q (2018) A stable systemic risk ranking in china’s banking sector: Based onprincipal component analysis. Physica A: Statistical Mechanics and its Applications, 492 :1997–2009.

Fidan Kececi N, Kuzmenko V, Uryasev S (2016) Portfolios dominating indices: Optimization with second-order stochastic dominance constraints vs. minimum and mean variance portfolios. Journal of Risk and

Financial Management, 9 (11 ), DOI https://doi.org/10.3390/jrfm9040011.

Fulga C, Dedu S (2012) Mean-risk portfolio optimization with prior pca-based stock selection. In: TheInternational Workshop Stochastic Programming for Implementation and Advanced Applications, volStochastic Programming for Implementation and Advanced Applications, DOI \textit{10.5200/stoprog.2012.07.}

Galankashi M, Rafiei F, Ghezelbash M (2020) Portfolio selection: a fuzzy-anp approach. Financial Innovation6 (1 ):1–34.

Ghorbani M, Chong EK (2020) Stock price prediction using principle components. PLOS ONE 15 (3 ):1–20

Giuliani A (2017) The application of principal component analysis to drug discovery and biomedical data.Drug Discovery Today, 22 (7 ):1069–1076.

Goel A, Sharma A (2019) Deviation measure in second-order stochastic dominance with an application toenhanced indexing. International Transactions in Operational Research, pp 1–30.

Graham B (1949) The intelligent investor. New York:Harper & Bothers.

Graham B, Dodd D (1934) Security Analysis. New York: Whittlesey House McGraw-Hill Book Co.

Guran C, Ugurlu U, Tas O (2019) Mean-variance portfolio optimization of energy stocks supported withsecond order stochastic dominance efficiency. Finance a Uver, 69 :366–383.

Hadar J, Russell W (1969) Rules for ordering uncertain prospects. The American Economic Review,

59 (1 ):25–34.

Hernandez W, Mendez A (2018) Statistics: Growing Data Sets and Growing Demand for Statistics, UnitedKingdom: IntechOpen., chap Application of Principal Component Analysis to Image Compression.

Hotelling H (1933) Analysis of a complex of statistical variables into principal components. Journal of

Educational Psychology, 24 (7 ):498–520.

Huang CY, Chiou CC, Wu TH, Yang SC (2014) An integrated dea-modm methodology for portfolio opti-mization. Operational Research, 15 :115–136.

Joldes C (2018) The selection of stocks using principal component analysis. In: International Conference onEconomics, Business and Economic Thought (EBET), DOI 10.31410/itema.2018.454

Jothimani D, Shankar R, Yadav S (2017) A pca-dea framework for stock selection in indian stock market.Journal of Modelling in Management, 12 :386–403.

24

Konno H, Suzuki Ki (1995) A mean-variance-skewness portfolio optimization model. Journal of Operational

Research Society of Japan, 38 :173–187.

Konno H, Shirakawa H, Yamazaki H (1993) A mean-absolute deviation-skewness portfolio optimisationmodel. Annals of Operations Research, 45 :205–220.

Leshno M, Levy H (2002) Preferred by all and preferred by most decision makers: almost stochastic domi-nance. Management Science 48 (8 ):1074–1085.

Levy H (1992) Stochastic dominance and expected utility: Survey and analysis. Management Science,

38 :555–593.

Liagkouras K, Metaxiotis K, Tsihrintzis G (2020) Incorporating environmental and social considerations intothe portfolio optimization process. Annals of Operations Research 12., DOI 10.1007/s10479-020-03554-3.

Liang X, Jiang Y, Liu P (2017) Stochastic multiple-criteria decision making with 2-tuple aspirations: amethod based on disappointment stochastic dominance. International Transactions in Operational Re-

search, 25 (3 ):913–940.

Lv J, Xiao ZH, Pang LP (1992) An incremental bundle method for portfolio selection problem under second-order stochastic dominance. Numerical Algorithms, 85 :653–681.

Marshall AW, Olkin I (1979) Inequalities: Theory of majorization and its applications. San Diego: AcademicPress.

Nadkarni J, Neves RF (2018) Combining neuroevolution and principal component analysis to trade in thefinancial markets. Expert Systems with Applications, 103 :184–195.

Ocal M, Oral E, Erdis E, Vera G (2007) Industry financial ratios-application of factor analysis in turkishconstruction industry. Building and Environment, 42 :385–392.

Olufemi A, Fajembola O, Olopete M (2012) Predicting bank failure in nigeria using principal componentanalysis and d-score model. Research Journal of Finance and Accounting, 3 :159–170.

Pearson K (1901) On lines and planes of closest fit to systems of points in space. Philosophical Magazine,

2 :559–572.

Pinches G, Mingo K, Caruthers J (1973) The stability of financial patterns in industrial organizations.Journal of Finance, 28 (3 ):389–396.

Quirk J, Saposnik R (1962) Admissibility and measurable utility functions. Review of Economic Studies,

29 :140–146.

Roman D, Mitra G (2009) Portfolio selection models:a review and new directions. Wilmott Journal, 1 (2 ):69–85.

Roman D, Darby DK, Mitra G (2006) Portfolio construction based on stochastic dominance and targetreturn distributions. Mathematical Programming, 108 :541–569.

Roman D, Mitra G, Zverovich V (2013) Enhanced indexation based on second order stochastic dominance.European Journal of Operational Research, 228 :273–281.

Samaras G, Matsatsnis N, Zopounidis C (2008) A multi criterion dss for stock evaluation using fundamentalanalysis. European Journal of Operational Research, 187 (3 ):1380–1401.

Sehgal R, Mehra A (2020) Robust portfolio optimization with second order stochastic dominance constraint.Computers & Industrial Engineering, 144 :106396.

25

Sharma A, Mehra A (2013) Portfolio selection with a minimax measure in safety constraint. Optimization,

62 (11 ):1473–1500.

Sharma A, Mehra A (2015) Extended omega ratio optimization for risk-averse investors. International Trans-actions in Operational Research, 24 :485–506.

Sharma A, Mehra A (2017) Financial analysis based sectoral portfolio optimization under second orderstochastic dominance. Annals of Operations Research, 256 :171–197.

Sharma A, Agrawal S, Mehra A (2017) Enhanced indexing for risk averse investors using relaxed secondorder stochastic dominance. Optimization and Engineering, 18 :407–442.

Shenai P, Xu Z, Zhao Y (2012) Principal Component Analysis - Engineering Applications,, United Kingdom:IntechOpen., chap Applications of Principal Component Analysis in Materials Science.

Silva A, Neves R, Horta N (2014) A hybrid approach to portfolio composition based on fundamental andtechnical indicators. Expert Systems with Applications, 42 (4 ):2036–2048.

Singh A, Dharmaraja S (2017) Optimal portfolio trading subject to stochastic dominance constraints un-der second order auto regressive price dynamics. International Transactions in Operational Research,

27 (3 ):1771–1803.

Tan P, Koh H, Low L (1997) Stability of financial ratios: A study listed companies in singapore. Asian

Review of Accounting, 5 (1 ):9–39.

Tarczynski W, Tarczynska- Luniewska M (2017) Application of the leading sector identification method inthe portfolio analysis. Acta Universitatis Lodziensis, Folia Oeconomica, 2 :185–200.

Wang J, Wang J (2015) Forecasting stock market indexes using principle component analysis and stochastictime effective neural networks. Neurocomputing, 156 :68–78.

Whitmore G (1970) Third-degree stochastic dominance. The American Economic Review, 60 :457–459.

Xidonas P, Mavrotas G, Psarrasa J (2009) A multi-criteria methodology for equity selection using financialanalysis. Computers and Operations Research, 26 :3187–3203.

Yap B, Mohamad Z, Chong K (2013) The application of principal component analysis in the selection ofindustry specific financial ratios. British Journal of Economics, Management & Trade, 3 (3 ):242–252.

Yu H, Chen R, Zhang G (2014) A svm stock selection model within pca. Procedia Computer Science, 31 :406–412.

Yu L, Wang S, Lai K (2009) Multi-attribute portfolio selection with genetic optimization algorithms. INFOR:

Information Systems and Operational Research, 47 :23–30.

Zhai Q, Ye T, Huang M, Feng S, Li H (2020) Whale optimization algorithm for multiconstraint second-orderstochastic dominance portfolio optimization. Computational Intelligence and Neuroscience, pp 1–19.

Zhao S, Lu Q, Han L, Liu Y, Hu F (2015) A mean-cvar-skewness portfolio optimization model based onasymmetric laplace distribution. Annals of Operations Research, 226 (1 ):727–739.

Zhong X, Enke D (2017) Forecasting daily stock market return using dimensionality reduction. Expert Sys-tems with Applications, 67 :126–139.

26

Sectoral portfolio optimization by judicious selection of ...

Documents

Transcript of Sectoral portfolio optimization by judicious selection of ...