DEA Model for RFSI Relative Financial Strength Index

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Generalized DEA model of fundamental analysisand its application to portfolio optimization

N.C.P. Edirisinghe *, X. Zhang

Department of Statistics, Operations, and Management Science, College of Business Administration,

University of Tennessee, Knoxville, TN 37996, USA

Available online 18 April 2007

Abstract

Fundamental analysis is used in asset selection for equity portfolio management. In this paper, ageneralized data envelopment analysis (DEA) model is developed to analyze a firms financial state-ments over time in order to determine a relative financial strength indicator (RFSI) that is predictive

of firms stock price returns. RFSI is based on maximizing the correlation between the DEA-basedscore of financial strength and the stock market performance. This maximization involves a difficultbinary nonlinear program that requires iterative re-configuration of parameters of financial state-ments as inputs and outputs. We utilize a two-step heuristic algorithm that combines random sam-pling and local search optimization. The proposed approach is tested with 230 firms from various UStechnology-industries to determine optimized RFSI indicators for stock selection. Then, thoseselected stocks are used within portfolio optimization models to demonstrate the usefulness of thescheme for portfolio risk management. 2007 Elsevier B.V. All rights reserved.

JEL classification: C61; C67; G11

Keywords: Portfolio optimization; Fundamental analysis; Relative financial strength; Data envelopment analysis

0378-4266/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.jbankfin.2007.04.008

*

Corresponding author. Tel.: +1 865 974 1684; fax: +1 865 974 2490.E-mail address: [email protected] (N.C.P. Edirisinghe).

Available online at www.sciencedirect.com

Journal of Banking & Finance 31 (2007) 33113335

www.elsevier.com/locate/jbf
mailto:[email protected]:[email protected]


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1. Introduction

Fundamental analysis (FA) is the process of evaluating a public firm for its investment-worthiness by looking at its business at the basic or fundamental financial level, see for

example, Thomsett (1998). It involves examining a firms financials and operations, espe-cially sales, earnings, growth potential, assets, debt, management, products, and competi-tion. FA may also include analyzing market behavior that stresses the study of underlyingfactors of supply and demand, see Doyle et al. (2003) and Piotroski (2000). The main goalis to enhance the ability to predict future security price movement and then use such pre-dictions to design equity portfolios. On the other hand, technical analysis (TA) operateson the theory that market prices at any given point in time reflect all known factors affect-ing supply and demand, as well as a firms relative financial strength. Thus, TA focuses onanalyzing market prices themselves, rather than directly evaluating factors of fundamentalstrength or factors of supply and demand. Strategies based on TA generally utilize a seriesof calculations designed to detect when a price change is likely to occur so that an investorcan manage market positions in the short-term, such as the case in highly leveraged deriv-ative markets. In contrast, FA takes on a more long-term perspective in determining whichfirms are most likely to perform well in the future, based on their fundamental businessstrengths.

The work in this paper complements the approach of fundamental analysis. The objec-tive of our research is to focus only on the publicly-available financial statements of agiven firm and to use them to determine a measure of underlying business strength forthe firm. In determining the underlying financial health of a company, the raw financial

numbers of a firm do not provide the perspective required to differentiate between healthyand unhealthy stocks for investment. In other words, the context provided by a compari-son of a given firm to its industry and to the market as a whole is essential. Therefore, thefocus in this paper is not to evaluate a firms business strength in isolation. Instead, a rel-ative strength indicator is computed by comparing a given firm to many other firms whichare in a similar business segment of the market, such as the industry to which the firmbelongs. The central premise of this research is that market prices have factored in pub-licly-available information about the firm, but the future expectations of price perfor-mance are determined by the perceived business strength of the firm. Thus, this notionis consistent with the efficient market theory, where the price of a stock is assumed to

reflect the knowledge and expectations of all investors since everyone has the same infor-mation about the stock. The aim of this paper is to provide a measurable (objective) metricof that knowledge that is highly correlated with stock price performance. Then, such ametric can be used as a proxy for gauging a firms expected financial performance, andhence the firms future stock price performance. In this sense, a companys financial state-ments (income statement, balance sheet and cash flow statement) become indispensableresources for investment decision making. It must also be stated that this research isnot focused on determining if a stock is undervalued, overvalued, or trading at fair marketvalue, nor does it focus on qualitative market factors that are internal or external to thefirm.

Many quantitative models have been proposed in the literature for stock price predic-tion using financial statements regression models and artificial neural network (ANN)models have been applied, see for instance, Kanas (2001), Quah and Srinivasan (1999),and Thanassoulis (1993). In Kanas (2001), historical financial data is used as inputs and

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stock price is used as output in an ANN model. There are also other approaches based onapplying regression-based techniques using data from financial statements as explanatoryvariables to predict future cash flow or stock performance. Ou (1989) used logistic regres-sion to estimate the probability of an earnings increase in a subsequent year. Graham et al.

(2002) applied ordinary least squares regression to determine a firms market value as alinear function of its earnings and book value.

The work in this paper is motivated by the basic approach of Edirisinghe and Zhang(2007), where financial statement data was used in a data envelopment analysis (DEA)model. However, that work assumed that the analyst is able to categorize financial datainto separate inputs and outputs in a DEA model specification to compute a financialstrength metric. With such an a priori model specification, although measures of highfinancial strength may result, they could display low correlations with market returns.Then, the analyst runs the risk of reaching the inevitable (false) conclusion that underlyingfinancial strength is not factored into market returns, contrary to the efficient markethypothesis. In this paper, we generalize the basic DEA methodology where inputs andoutputs parameters are not fixed a priori, instead they are determined via an optimiza-tion process formulated to maximize correlations between financial strength and marketreturns. This process results in a relative financial strength indicator (RFSI) that is highlypredictive of stock returns.

To the best of our knowledge, a DEA-based financial strength metric for a firm has notbeen directly incorporated within fundamental analysis for stock investments. Data Envel-opment Analysis (DEA) is commonly used to evaluate the relative efficiency of a numberof Decision Making Units (DMUs). The basic DEA model in Charnes et al. (1978), called

the CCR model, has lead to several extensions, most notably the BCC model of Bankeret al. (1984) and the additive model ofCharnes et al. (1985). DEA models have been exten-sively used in performance appraisal in a wide range of applications including financialperformance as well as non-financial performance measurement. In the financial applica-tions of DEA methodology, one particularly appealing idea is to measure managerial effi-ciency of a company by using its financial statements. For example, using certain financialratios as inputs and outputs, DEA is used to evaluate performance of banks (Yeh, 1996),CRAF participants (Bowlin, 2004), defense business segments (Bowlin, 1999), and creditunions (Pille and Paradi, 2002). DEA-based efficiencies and Sharpe ratios are compared toevaluate performance of different hedge funds, see Gregoriou et al. (2005). Alam and

Robin (1998) compute relative technical efficiencies for firms in the airline industry andanalyze their association with corresponding stock price returns. However, their work isbased upon input and output variables that are generally non-financial in nature and theyare typically not found in the publicly-available financial statements.

In all of the above approaches, the underlying DEA model is specified with a fixed(exogenous) set of input and output parameters to compute an efficiency score. Our workcontrasts with the traditional DEA approach in that an input/output categorization isendogenously determined by a model that seeks the highest correlation between stockreturns and efficiency metric. Consequently, our approach leads to identifying the best-performing companies from the poor-performing, i.e., stock screening, for consider-

ation in equity portfolio management. Given that distribution parameters of stock returnsare often subject to estimation error, screening mechanisms such as the proposed RFSI-based stock selection can yield better risk-reward characteristics in portfolio optimization.This is demonstrated by applying the RFSI-based approach to identify favorable stocks as

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candidates for stock portfolio optimization. The resulting portfolios are shown in thispaper to have superior risk-reward performance.

The remainder of the paper is organized as follows. In Section 2, the mathematical for-mulation of a standard DEA model is first provided, along with various financial para-

meters from income statements and balance sheets to be used as inputs and outputs.The generalized DEA approach is then discussed. In Section 3, the RFSI determinationproblem is formulated as a correlation maximization model, and a two-stage (heuristic)solution scheme is developed for its solution. Portfolio selection criteria based on statisti-cal tests of RFSI are covered in Section 4. Section 5 applies the methodology in a casestudy involving 230 firms from six US industries. Using actual data from 1996 to 2002,the RFSI approach is applied to identify firms to include within a mean-variance quadraticportfolio optimization model. The paper concludes with some remarks in Section 6.

2. DEA model

Data envelopment analysis (DEA) is a nonparametric method for measuring the rela-tive efficiencies of a set of similar decision making units (DMUs) by relating their outputsto their inputs and categorizing the DMUs into managerially efficient and manageriallyinefficient. It originated from Farrells, 1957 work, which was later popularized by Char-nes et al. (1978). The CCR ratio model in Charnes et al. (1978) seeks to optimize the ratioof a linear combination of outputs to a linear combination of inputs.

To explain the basic premise of a DEA model, let there be Jindependent DMUs (firms)whose performance (or efficiency) must be evaluated relative to each other. One begins

with a given set of inputs parameters (say, M) and a given set of output parameters(say, N) which are common to all J firms. The relative (managerial) efficiency then mea-sures how well a given firm (in the group ofJfirms) converts its Minputs to the Noutputs,which is computed as the ratio of a certain aggregated output measure to a certain aggre-gated input measure. Such aggregated input (and output) measures are computed by tak-ing a non-negative linear combination of the M inputs (and N outputs). Following thisidea, the input-oriented relative performance (strength or efficiency) fk of some firm k,k= 1, . . . , J, is then defined as the maximized value of the latter ratio, determined overall possible aggregating multipliers such that no firm in the group will attain a relative per-formance measure greater than unity. The CCR model is formulated as follows:

fk : maxu;v

PNn1xonkvnkPMm1ximkumk

s:t:

PNn1xonjvnkPMm1ximjumk

6 1; j 1; . . . ;J

umk; vnkP 0; m 1; . . . ;M; n 1; . . . ;N:

1

For firm j, the (measured) level of input parameter m is (xi)mj, m = 1, . . . , M, while thatof output parameter n is (xo)nj, n = 1, . . . , N. The input and output non-negative multipli-

ers for firm kare denoted by the variables umkand vnk, respectively. The model in (1) yieldsthe maximum achievable efficiency for firm k, denoted fk, provided every other firm is alsoapplying the same aggregating non-negative multipliers in computing their input to outputconversion ratios. fk is termed the DEA efficiency score of firm k. An efficiency score of less



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than one is indicative of that it may be possible to decrease the level of input for the samelevel of output, while a score of 1 indicates the firm is DEA-efficient. By applying (1) toeach firm independently, the respective (maximum) relative efficiency score for each firmis computed. The equivalent linear programming formulation of model (1) is, see Charnes

et al. (1978),

^fk : maxu;v

XNn1

xonkvnk

s:t:XMm1

ximkumk 1

XM

m1ximjumk

XN

n1xonjvnk6 0; j 1; . . . ;J

umk; vnkP 0; m 1; . . . ;M; n 1; . . . ;N:

2

It is easy to show that ^fk fk holds under the non-negativity of the observed data. Moreprecisely, if (xi)mk> 0 for some m = 1, . . . , M, then, ^fk fk holds. Conversely, suppose(xi)mk6 0 for all m = 1, . . . , M. Then, the maximization in (1) is not well-defined, and(2) is infeasible, in which case, we assign a performance strength of ^fk 0. For detaileddiscussions on DEA models that involve negative inputs/outputs, see Lovell and Pastor(1995) and Portela et al. (2004), for instance. The issue of negative data stems from thefact that in the application in this paper, the input and output data come from financialstatements. That is, it is possible that all input parameters for a given firm have non-po-

sitive values, depending on how the input parameters are chosen from financial statements.Such is the case if return on assets and return on equity are chosen as the only inputparameters and if these two parameters are negative for a firm being evaluated.

The input/output parameters for the DEA framework are identified from financialstatements of companies. A total of 18 financial parameters are used, either directly orcomputed, from the quarterly financial statements of a firm, as presented in Table 1. Theseparameters examine a firms fundamental performance through a range of performanceperspectives: profitability, asset utilization, liquidity, leverage, valuation, and growthperspectives.

2.1. The generalized DEA approach

It is important to observe that, in order to apply the DEA model in (2), the M inputparameters and Noutput parameters are required to be explicitly identified a priori. Whilethis may be possible in certain applications (such as production) where input to outputconversion mechanisms are well-understood, our case is different. We must select a setof input and output parameters from the universe of 18 financial parameters describinga firms financial health (see Table 1). The objective of such a selection is that the resultingrelative DEA performance score of a firm can be interpreted as providing a measure of its

underlying financial strength. Such financial strength measures are required to be stronglycorrelated with the market price process, under the efficient market hypothesis. If theinputs and outputs for the DEA model are chosen exogenously (a priori), the resultingDEA performance scores for firms may not be representative of the fundamental financial



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strengths that are rewarded by the financial markets. The generalized DEA approach(GDEA) developed in this paper leaves the selection of inputs and outputs as flexible aspossible in the sense that a proper selection of the latter is sought iteratively to maximize

the correlation of the DEA-based strength evaluation and the stock market performance.This process is best-explained in Fig. 1.

To illustrate the GDEA approach in Fig. 1, consider the universe ofI(=18) parametersthat are potential inputs and outputs. Suppose a given parameter imay be used as an inputand/or output, or not used at all. Furthermore, suppose the level at which a parametermust be specified in the DEA model in (2) is treated as unknown. Consequently, for aparameter i with an observed value xij for firm j, the level at which it enters the modelas an input is denoted by yixij, where the input scaling variable yiP 0. Similarly, the levelat which the parameter i enters as an output for firm j is zixij and the output scaling var-iable ziP 0. Collecting the yi and zi components for all parameters, we define an input

scaling parameter vector by y2 RI and an output scaling vector by z2 RI. An appropriateselection of values for the pair y;z 2 R2I is not a firm-specific issue. Rather, it must bechosen as a property of the industry, i.e., group of firms, so that relative performancescores of firms can be compared to each other within the same industry. More impor-

Table 1Financial parameters used for fundamental analysis

i Parameter Description Perspective

1 Return on equity Net income generated per unit of common shareholders

equity

Profitability

2 Return on assets Net income divided by the total assets Profitability3 Net profit margin Net income a firm makes for every $1 it generates in revenue Profitability4 Receivables turnover Revenues for the period divided by receivables Asset

utilization5 Inventory turnover Revenues for the period divided by inventories Asset

utilization6 Asset turnover Revenue generated per dollar of assets a firm owns Asset

utilization7 Current ratio Total current assets divided by total current liabilities Liquidity8 Quick ratio Total current assets minus inventory divided by total current

liabilitiesLiquidity

9 Debt to equity ratio Long-term debt divided by shareholders equity Liquidity10 Leverage ratio Total assets divided by shareholders equity Leverage11 Solvency ratio-I Total liability divided by total assets Leverage12 Solvency ratio-II Total liability divided by shareholders equity Leverage13 Price to earnings (PE)

ratioStock price divided by net income per share Valuation

14 Price to book ratio Stock price divided by shareholders equity per commonshare

Valuation

15 Earnings per share(EPS)

Net income minus dividends divided by common shares Profitability

16 Revenue growth rate Current quarters revenue divided by the previous quarters

revenue minus one

Growth

17 Net income growth rate Current quarters net income divided by the previous quartersnet income minus one

Growth

18 Earnings per sharegrowth rate

Current quarters EPS divided by the previous quarters EPSminus one

Growth



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tantly, such a performance score must represent the fundamental financial health of a firmthat is predictive of (or highly correlated with) the stock price action. Therefore, the vector(y, z) is to be held fixed when computing efficiencies of all J firms in the group. Under thescaling vector parametrization (y, z), the resulting DEA model is

gky;z : maxu;v

PIi1zixikvikPIi1yixikuik

s:t: PI

i1zixijvikPIi1yixijuik

6 1; j

1; . . . ;J

uik; vikP 0; i 1; . . . ;I;

3

where y is chosen such thatPI

i1yi > 0. gk(y, z) is simply referred to as the (financial) per-formance score of firm k corresponding to the input/output scaling vector pair (y, z). Thefollowing equivalent linear programming model can be used to compute gk(y, z).

gky;z : maxu;v

XIi1

zixikvik

s:t:XIi1 yixikuik 1

XIi1

yixijuik XIi1

zixijvik6 0; j 1; . . . ;J

uik; vikP 0; i 1; . . . ;I:

4

In DEA, the issue of setting a given parameter in both the input and output sets simulta-neously has been addressed in, for instance, Beasley (1995) and Cook et al. (2006). In thecase of a CCR model, when a parameter is used both in inputs and outputs, the resultingDEA efficiency is 1 for each firm. That is,

Proposition 2.1. For some parameter i2 {1, . . . , I}, let yi > 0 and zi> 0. For a firm k beingevaluated, suppose the measured value of parameter i satisfies xik> 0. Then, gk(y,z) = 1holds.

Strength metric

for each firm

Market correlation

with strength

Market correlation

maximized?

Run DEA model

(for each firm)

Set inputs &

outputs fixed

Re-categorize

inputs/outputs

no

yes

STOP

Fig. 1. Schematic of the generalized DEA approach.



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Proof. Can be shown similar to Lemma 1 and Theorem 1 in Cook et al. (2006). h

Therefore, ifyi> 0 and zi> 0 for a parameter iwith data xij> 0 for all firms j, the DEA-based strength score is 1 for all firms. For example, when the parameter i is the current

ratio (see Table 1), the data is always positive for all firms. In such a case, correlationbetween the computed financial strength score and the stock market performance is zero,and thus, such a choice on (y, z) will not maximize the desired strength-market correlation,see Fig. 1. Consequently, to reduce the search space for (y, z) in correlation maximization,we set yizi= 0 for all i= 1, . . . , I. This prohibits a given financial parameter ifrom being inthe inputs and outputs simultaneously.

Definition 2.2. A given vector-pair (y, z) is said to satisfy the complementarity condition ifand only if yizi = 0 for all i= 1, . . . , I. In this case, such a pair is simply referred to as acomplementary pair (y, z).

Therefore, a complementary pair (y, z) allows the categorization of the universe of Iparameters as distinct inputs and outputs. In contrast, Cook et al. (2006) introducedthe notion of flexible measures whereby a new parameter can be considered in the presenceof existing input/output sets. Their model then determines if this new parameter should bean input or an output in order to improve the (maximized) DEA efficiency. In our case, theobjective is to have the highest correlation between DEA efficiencies and the stock marketreturns. Therefore, we take a different approach that allows the complementary vectorpairs (y, z) to play the role of flexible measures in a more generalized setting. For this pur-pose, the domain of (y, z) must be appropriately chosen to force which parameters should

never (or must) be in inputs/outputs.Another important property of the CCR model is its unit-invariance. That is, the DEA-efficiency computed by (1) is independent of the units in which the input and outputparameters are measured, see Ray (2004, pp. 106107) and Lovell and Pastor (1995). Sta-ted in our context,

Proposition 2.3. gk(y,z) is positively homogeneous of degree 0 in (y,z) jointly and sepa-

rately.

Proof. Follows directly from the proof of unit invariance of the CCR-DEA model, see, for

instance, Ray (2004). h

The main implication of Proposition 2.3 is that it restricts the domain of feasiblecomplementary vector pairs (y, z) to a binary space. Along with the complementarity con-dition in Definition 2.2, thus, the feasible domain of the scaling vectors (y, z) in (4) mustsatisfy,

yizi 0;yi;zi 2 f0; 1g; i 1; . . . ;I: 5An equivalent linear transformation of (5), along with the condition that

Piyi > 0, yield

the following Binary Complementary Domain (BCD), denoted by X, for the feasible

choices for (y, z).

BCD : X : y;z :XIi1

yi P 1;yi zi 6 1; yi;zi 2 f0; 1g; i 1; . . . ;I( )

: 6



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Accordingly, for every firm k in the group (i.e., industry), the corresponding financial per-formance score gk(y, z) is determined by the model in (4) for a specified binary complemen-tary vector pair (y, z) 2 X. The goal is to search for (y, z) 2 X such that the performancescore so-computed would be a suitable metric of the underlying financial strength of a

given firm, relative to all firms in the group.When the model in (4) is specified using parameters under the BCD condition in (6) that

requires choosing (y, z) 2 X, it is herein referred to as GDEA under unrestricted BCD, orsimply, unrestricted GDEA version. On the other hand, we also consider a certain restric-tion on the binary complementary domain, based on a practical interpretation of thefinancial perspectives in Table 1, as given next.

2.2. Restricted BCD

The parameters of asset utilization, liquidity, and leverage perspectives can generally beinterpreted as inputs because activities that are measured by these parameters depend on theplanning and operational strategies of a firm. On the other hand, the parameters of profit-ability and growth perspectives are generally considered as outputs because revenue/incomegeneration is a major objective criterion for a firm. The valuation parameters measure howwell the equity markets perceive success of a firm, and they are generally not concernedwith a firms input strategy. Accordingly, in a restrictedGDEA approach, input parametersare only chosen from the perspectives of asset utilization, liquidity, and leverage, while theoutput parameters are chosen only from the profitability, growth, and valuation perspec-tives. This leads to the following restricted binary complementary domain,

Restricted BCD : X : y;z 2 X :X3i1

yi X18i13

yi 0;X12i4

zi 0( )

: 7

Performance of the unrestricted and restricted GDEA versions will be compared withinportfolio optimization using the application reported in Section 5. In the sequel, we willalso compare results with a fixed exogenous input/output categorization, referred to asa base categorization and denoted by (y0, z0),

y0 : f0; 0; 0; 1; 1; 1; 1; 1; 1; 1; 1; 1; 0; 0; 0; 0; 0; 0gz0 :

f1; 1; 1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 1; 1; 1; 1; 1

g:

'8

Thus, the base categorization is completely determined by the interpretation of financialparameters, rather than their usefulness in determining a financial performance score.Note that (y0, z0) 2 X* and this base categorization consists of all 18 financial parametersgiven in Table 1. The interest in (y0, z0) is merely for comparison with (y, z) in X (or X*)that might better represent the underlying financial strength of a firm.

3. Relative financial strength indicator (RFSI)

The process of determining an RFSI requires, first, determining a correlation metric for

the DEA-based performance scores and the stock price returns, for the industry as awhole, for a given vector pair (y, z), and second, designing a suitable iterative procedureto choose (y, z) 2 X (or X*) in an attempt to maximize the latter correlation metric (seeFig. 1).



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Let the DEA-based performance score for a firm k in a given industry be determinedaccording to the model in (4) as gk(y, z), for a specified categorization (y, z) 2 X. The RFSIis developed here for the unrestricted X; for the restricted version of RFSI, X is simplyreplaced with X*. Computing the model in (4) requires the realized values xijof all financial

parameters for all firms. The future value of a parameter i for firm j is a random variable,denoted by Xij. The collection of random variables Xij for i= 1, . . . , I= 18 and j= 1, . . . , Jis X. Realizations of Xij are observed as xij in (published) financial statements of a givenperiod (i.e., quarter). For a future period t of uncertain financial performance, the collec-tion of random variables is the vector Xt : fXtij : 8i; 8jg. Then, the DEA-based relativefinancial efficiency for the industry is represented by the collection of random variablesgt(y, z): = {gj(y, z;X

t) : j= 1, . . . , J}. Once the period t financial statements are observed,with Xt realized as xt, the random vector gt(y, z) is realized as the vector of values{gj(y, z; x

t) : j= 1, . . . , J}.Let Rtj denote the stock price rate of return (RoR) random variable (for future period t)

of firm j, and those for all firms are represented by the random J-vectorR

t : fRtj : j 1; . . . ;Jg. Observed realizations of period t RoR is the vectorrt : frtj : j 1; . . . ;Jg. Consider the pairwise correlations between the two random vec-

tors gt(y, z) and Rt, denoted by the correlation vector Cty;z 2 RJ. Its jth component,for firm j, is given by

Ctjy;z : Corrfgjy;z;Xt;Rtjg; 9

where j= 1, . . . , J. The correlation vector Ct(y, z) is, therefore, a measure of the predictivepower of the DEA-based (financial) efficiency metric on stock price returns for the chosen

industry. Indeed, a positive and significant correlation vector Ct

(y, z) implies that theDEA-based efficiency score is a valuable proxy of the stock market performance of theindustry. Observe that Ct(y, z) for period t depends on the chosen binary complementaryvector (y, z) 2 X. The best industry correlation is thus obtained when one searches for(y, z) 2 X such that an appropriate metric of the vector Ct(y, z) is maximized. Vector normscannot be used as appropriate metrics here because the goal is to seek positive (and large)correlations across all firms in the industry. While more complicated formulae are possi-ble, we use the simple average statistic

Ct

y;z

:

1

JXJj1

Ct

jy;z

;

10

herein termed the industry correlation metric, to search for the highest positive correlationsindustry-wide. Note that the correlation vector Ct(y, z) is unknown for the future period t,and thus, it must be forecasted. To forecast Ct(y, z), we use the historical (observed) samplexs, s = 1, . . . , t 1. Using a history length oft0 periods, Ctjy;z is estimated by the sample

correlation coefficient, given by

ctjy;z : Correlation coefficient between fgjy;z;xsgt1stt0 and frsjgt1stt0 : 11

Then, the industry correlation metricC

t

y;z in (10) is estimated ascty;z : 1

J

XJj1

ctjy;z: 12



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Observe that the statistic cty;z for period t depends on the chosen binary complementaryvector (y, z) 2 X. The best industry-correlation metric is thus obtained when one searchesfor (y, z) 2 X such that cty;z is maximized, i.e., solve the industry-correlation maximiza-tion model

CORMAX : maxy;z

cty;zs:t: y;z 2 X:

13

Let an optimal binary complementary pair solving the above maximization be denoted by(y*, z*). Note that dependence of this pair on the period index t is suppressed. The corre-sponding industry correlation metric, Cty;z in (10), is required to be statistically signif-icant, for if not, the use of the DEA-based financial strength indicator for the givenindustry cannot be validated for investment decision making. Statistical tests for this pur-pose are discussed in Section 4.1. When this industry correlation metric is verified to be

statistically significant, the Relative Financial Strength Indicator (RFSI) for a given firmin the industry is defined as follows.

Definition 3.1. Suppose Cty;z is statistically significant for a given industry, where(y*, z*) is an optimal solution of (13). Then, the Relative Financial Strength Indicator(RFSI) of firm j for (a future) period t, given the observed financial statement data xs fort t0 6 s 6 t 1 for the industry, is defined by

RFSIt;j : Egjy;z;Xtjgjy;z;xtt0; . . . ; gjy;z;xt1; 14

where gj(y*

, z*

; xs

) is computed according to the DEA model in (4) for the input/outputcategorization (y*, z*), and E[ ] denotes the expectation operator.

To simplify the computation of RFSI, the expectation in (14) is estimated by the simplemoving average forecast (of t periods, t6 t0), given as

RFSIt;j; t 1t

Xt1stt

gjy;z;xs: 15

RFSIt;j; t is bounded within 0 and 1, where a value of unity indicates the highest possiblerelative financial strength indicator for firm j, relative to the industry concerned. Also note

that a single input/output categorization (y*

, z*

) of the 18 financial parameters in Table 1 isused in computing the RFSI for all firms in the industry, for the future period t. For futureperiods beyond t, it may be necessary to adapt RFSI to new financial statement observa-tions, by resolving (13) for a revised optimal input/output categorization.

3.1. Two-step heuristic solution method

The CORMAX model in (13) is a difficult optimization problem because evaluation of theobjective function (statistic) cty;z in (12) requires the solution of a sequence of linearoptimization models (4) so that each of the sample correlation coefficients csj

y;z

in

(11) can be computed. Therefore, the objective function in (13) cannot be explicitly writtenin closed-form nor can it be verified to be concave (or pseudo-concave) in the 2 I-dimen-sional decision variable-vector (y, z). Nonconvex optimization is known to be computa-tionally tedious, see for instance, Horst et al. (1995). Moreover, X is a binary solution



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space, i.e., (13) is a binary nonconvex optimization model. Global optimality conditionsfor discrete nonconvex optimization have been studied, e.g. see Larsson and Patrikssony(2005). However, efficient methods are available only for specially structured problemsand/or without integer restrictions, e.g. see Tawarmalani and Sahinidis (2004) and Zhang

et al. (1999).Alternatively, we employ an efficient heuristic solution scheme. The method is a two-

step procedure, which is based on, first, sampling a set of initial (y, z) points from the fea-sible domain X, and then, performing a local search optimization in X for each of thoseinitial sample points. Consider a random sample of (vector) points xs :ys;zs 2 X & R2I, for s 2S, whereS denotes the index set of the sample points. For eachsample point, the objective criterion is calculated and the sample of industry correlationmetric values

fctys;zs : s 2Sg 16is collected. Then, each sample value ctys;zs is improved to a locally optimal value byemploying a non-gradient based local search procedure, starting from the point xs 2 X.The corresponding local optimum is denoted by ~xs : ~ys;~zs. Then, an approximationfor the optimal input/output categorization for the industry is determined by

y;z % arg maxs2S

fct~ys;~zsg: 17

The following procedure is applied to generate a (random) sample point xs 2 X: randomlydraw a set of 2I values from a continuous uniform distribution in [

1,1]. The first I val-

ues are collected to form the I-vector as. The last Ivalues are collected to form the I-vectorbs. Then, the sample point xs = (ys, zs) is defined by the solution of the binary linearprogram

ys;zs : arg maxy;z

fas0y bs0z : y;z 2 Xg; 18

where a prime denotes the transposition of a vector. This process is then repeated for eachs 2S.

3.1.1. Local searchThe non-gradient-based local search procedure is a modification from the Hooke and

Jeeves (HJ) method, see Bazaraa et al. (1993). Given a current solution xp, at some iter-ation p of the local search procedure, the original HJ method performs an exploratorysearch along the coordinate directions. Coordinate directions that improve the objectivefunction are used to define a new iterate. The direction to the new iterate from the startingsolution xp is used to perform a pattern search. This method is adapted to the binarymodel in (13).

For some candidate xp 2 X, the ith coordinate xpi is either 0 or 1. Ifxpi 0, then anexploratory move is allowed only in the positive (xi) coordinate direction. Ifx

pi

1, then

an exploratory move is allowed only in the negative (xi) coordinate direction. Once, a newiterate xp is so-determined, a pattern (line) search is not necessary in our case since thesearch point xp kxp xp 62 X for k 62 {0,1}. The resulting algorithmic steps are asfollows:



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Algorithm-LS

Initialization: Given xs = (ys, zs) 2 X, see (18), determine ctxs.Set p = 1, f

p

ct

xs

, and x(p) = xs.

Step 1: For i= 1, . . . , 2Iand denoting the ith elementary coordinate direction by ei, let

xi : xp ei if xp ei 2 X and fp < ctxp eielse; xi : xp ei if xp ei 2 X and fp < ctxp eielse; xi : xp:

Compute x2I+1 by the XOR (exclusive or or not equal to) operation:

x2I1 : x1 xor x2 xor . . . xor x2I:

Ifx

2I+1

62 X, set x2I+1

= x(p).Step 2: Let xp 1 : arg maxfctxi : i 1 . . . ; 2I 1g.

If ctxp 1 ctxp : Terminate the local search and setxs xp.Else, ifctxp 1 > ctxp, let fp 1 ctxp 1

set p p + 1 and go to Step 1.

4. Portfolio selection using RFSI

Asset allocation is the practice of dividing resources among different categories such asstocks, bonds, real estate, cash equivalents, etc. Often mathematical optimization modelsare used for asset allocation with an intent to reduce risk exposure, since each asset classhas a different correlation to the others. Within a given category, say stocks, the portfoliomanager must choose specific industries to invest in and then specific firms must beselected within those industries. Indeed, the asset allocation and the choice of individualsecurities must occur integrative to portfolio risk management. Furthermore, such portfo-lios must be temporally (say, quarterly) rebalanced to account for the economic and mar-ket evolutions. For integrated risk control with frequent rebalancing under portfoliooptimization, see for instance, Edirisinghe (2007) and the many references therein.

The early work of portfolio optimization dates back to Markowitz (1952), where atrade-off between portfolio expected return and portfolio variance is sought. While manyvariants of this approach have been proposed over the years, the fact remains that a uni-verse of securities must be pre-selected prior to running a portfolio optimization model todetermine optimal portfolio weights. While there is an increased research-focus on riskspecifications (or lack thereof) to capture the investors risk attitude, it is imperative thatan underlying universe of securities be carefully selected and their stochastic elements beaccurately estimated. In fact, the latter two aspects are often the more dominant aspectsin the practice of portfolio management. In this section, we focus on the question of select-ing a universe of securities for portfolio risk management, by applying the RFSI concept

developed in the preceding sections. Then, portfolio optimization is performed to deter-mine optimal weights on individual securities.

Consider a stock portfolio investment problem where a given budget must be allocatedamong stocks in H different industries. Suppose there are Jh stocks (firms) under



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consideration in industry h, h = 1, . . . , H. The fund allocation problem is considered in twostages: in the first stage, the problem is to choose the industries for investment of the givenbudget, and then, to choose specific stocks within those industries as potential candidatesin a portfolio. In the second stage, specific portfolio weights are assigned to the chosen

candidate securities such that the investors risk-return preferences are satisfied. The stageone problem is an asset selection problem and the stage two is a portfolio optimizationproblem. We consider applying a selection process based on RFSI to the stage one. Notethat financial statements of all firms in the Hindustries (thus, a total ofJ0

PHh1Jh firms)

must be used to collect data for the 18 financial parameters in Table 1.Suppose the current time period is t 1 and investment is desired for period t (i.e., the

subsequent quarter). Referring to the method of computing RFSI presented in Section 3,and using the publicly-available financial data xsh for periods s < t, the model for correla-tion maximization in (13) is solved to determine an optimal complementary binary pair(yh*, zh*), for each industry h = 1, . . . , H. The industry correlation metric correspondingto this optimized input/output categorization, Cthyh;zh in (10), must be verified to bestatistically significant. Industry selection for portfolio optimization is based upon the sig-nificance of this correlation, tested via the sample estimate cthyh;zh in (12).

4.1. Statistical tests of correlations

We are concerned with identifying industries that do not provide statistical evidence forRFSI-based predictability of stock returns, i.e., the industry correlation metric Cthyh;zhis not a significant positive value. Consider the following hypothesis test for a minimum

positive correlation (q0) for a given industry h,H0 : Cthyh;zh 6 q0H1 : C

thyh;zh > q0:

)19

The above null hypothesis H0 indicates that DEA-based fundamental financial strength isnot consistent with the efficient market theory for industry h. Note thatCthyh;zh : 1Jh

PJj1C

tj;hyh;zh, see (10), firm-correlations Ctj;hyh;zh are estimated

by ctj;hyh;zh in (11). Consider the following arctan hyperbolic transformation ofctj;hyh;zh:

wj;h : tanh1ctj;hyh;zh 12 loge1 ctj;hyh;zh1 ctj;hyh;zh" #

: 20

Using the results that Ectj;h % Ctj;h and Varctj;h % 1 Ctj;h22, R.A. Fisher (18901962)showed that wj,h is approximately normally distributed,

wj;h % Normal1

2loge

1 Ctj;hyh;zh1 Ctj;hyh;zh

" #;

1

t0 3

; 21

see Tamhane and Dunlop (2000), where t0 is the number of periods used in the estimation

in (11). Next, define the industry-average statistic by

wh : 1Jh

XJhj1

wj;h; 22



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which is the sum of Jh normal random variables, and thus, wh is normally distributed:

wh % Normal 12Jh X

Jh

j1loge

1 Ctj;hyh;zh1 Ctj;hyh;zh

" #; r2

; 23

where the variance of wh is given by

r2 : 1Jht0 3

2

Jh2Xj


16/25

(threshold) value of the right hand side in (29) over all possible h, then, indeed H0 is ac-cepted for the industry h. Define,

w

min

0 : infh w0h : XJh

j1 hj;h Jh;1

q0 < hj;h 0, which is the net

count of positive values in hj,h for j= 1, . . . , Jh, and thus,PJh

j1mj 1; 2; . . . ;Jh. That is,a can take on the set of possible discrete values 1; Jh

Jh1 ;Jh

Jh2 . . . ;Jh

2;Jh

n o. Each of these val-

ues for a defines a distinct KKT point provided the objective function is well-defined atthose points, which is given by

1

2Jh

Xj:mj1

loge1 aq01 aq0

Xj:mj1

loge1 aq01 aq0

" # 1

2Jh

XJhj1

mj

loge

1 aq01 aq0

since the following identity holds:

loge1 q0a1 q0a !

loge1 q0a1 q0a !

0:

Therefore, the objective value associated with each KKT-point is given by 12Jh

Jha

loge1aq01

aq0h i

provided a 6 1q0. Then, the minimum in (32) is obtained by

Z min 12a

loge1 aq01 aq0

: a 1; Jh

Jh 1 ;Jh

Jh 2 . . . ;minfJh; 1=q0g& '

: 34



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Next, noting that the function fx 1x

loge1xq01xq0 is monotonically nondecreasing in x forx 2 [1,1/q0], it follows that a = 1 is indeed the optimal solution in (34), which thus impliesthat each mj= + 1 at the optimum. That is, hj,h = 1, "j, solves the relaxed problem in (32),which yields Z

12

loge1q01

q0 . But, for 0 < q0 < 1, we have

1q0< hj;h

1 < 1

q0for all j.

This leads to the feasible point upper bound on the infimum in (30) as wmin0 6 Z. Combin-ing with Z 6 wmin0 , the proof is completed. h

4.2. Selection criteria for portfolio optimization

The foregoing statistical analysis can be used to determine an industry partition forinvestment. Given a set of industries h = 1, . . . , H for consideration in period t, an invest-ment-worthy (screened) set H is determined under the Industry Selection Criterion givenby

ISC : H : h : wh > j2

loge1 q01 q0

Z

11 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJht0 3

p ; h 1; . . . ;H( )

; 35

where jP 1 is a user-specified (safety) factor. For each industry h 2H, individual stocks jare chosen from the given firms j= 1, . . . , Jh by using RFSI as a selection discriminator.Under Definition 3.1 and referring to the computation of RFSI in (15), for a given indus-try h 2H, evaluate the moving average forecast of t periods,

RFSI

t;j; t

1

tXt1

sttgj

yh;zh;xs

:

36

The subset (of firms) Jh from industry h 2H is chosen for portfolio analysis under theStock Selection Criterion given by

SSC : Jh : fj : RFSIt;j; tP R; j 1; . . . ;Jhg; 37where R* is a prespecified threshold, where 0 < R* 6 1. The stocks in the subset Jh, forh 2H, are then expected to perform well in the stock market with high confidence. Theuniverse of securities for portfolio analysis is thus given by

N : [h2HJh; 38

which is a subset of the original universe of stocks, i.e., jNj6 J0 :PH

h1Jh. Investmentweight to be attached to each stock j 2N is then determined by a portfolio optimizationmodel. There are several models in the literature for this purpose, and the choice of a mod-el is primarily guided by risk/return considerations. Risk specifications are multi-prongedand portfolio optimization models are multi-faceted. For instance, when there are trans-actions and slippage costs of trading, portfolio drawdown characteristics are a major con-cern of risk. Also, when market evolutionary dynamics are nonstationary, multiperiodsequential stochastic decision optimization is shown to yield superior performance com-

pared to static one period models, see Edirisinghe (2007) for details.The focus here is to demonstrate the usefulness of the preceding selection criteria, (ISC)

and (SSC), over the unscreenedset ofJ0 stocks. The specific advantage of our stock screen-ing rules will be that stochastic parameter estimation will be performed in a reduced



18/25

dimension using stocks that are likely to have strong performance. Reduced dimensionstypically lead to smaller statistical estimation errors in RoR parameters. Consequently,a given portfolio optimization model is expected to provide better risk/return performancecharacteristics. To this end, we utilize the standard mean-variance portfolio optimization

framework in a static one period setting. We compare the two models, RFSI-based model(RMV) that uses the universeN and the standard mean-variance model (SMV) using allJ0 stocks. Portfolio weights carry the designation wj for stock j, with a superscript R or Sindicating the type of model used to calculate the weights.

RMV : maxwR

Xj2N

ljwRj k

Xj;k2N

rjkwRj w

Rk 39

s:t:Xj2N

wRj 6 1

wRj P 0;j2N:

SMV : maxwS

XJ0j1

ljwSj k

XJ0j;k1

rjkwSjw

Sk 40

s:t:XJ0j1

wSj 6 1

wSj P 0;j 1; . . . ;J0:In (39) and (40), k is a risk tolerance parameter that trades off portfolio mean return withportfolio variance. Thus, a higher k is indicative of risk-averse portfolio weighting thatstrives for better diversification. Performance comparison of these two models is presentedin the next section.

5. Application of RFSI in the technology sector

The DEA-based relative financial strength indicator, RFSI, is applied in portfolio opti-mization where several (publicly-traded) US companies in various industries are consid-ered. The objective is to validate the use of RFSI-based stock selection as a means ofimproving risk/return performance of optimized portfolios. Quarterly financial statements

of firms during the period 19962002 are used. Of the 28 consecutive quarters, the firstquarter is set aside for the initial calculations of RoR, growth rates etc. Reported resultspertain to a time window oft0 = 27 quarters for industry-correlation maximization in (13).Only the technology sector is used for the experimentation, of which six industries are cho-sen: software & programming (h = 1), communications equipment (h = 2), computerhardware (h = 3), computer networks (h = 4), semiconductors (h = 5), and computer ser-vices (h = 6). Quarterly data for all firms in these 6 industries are electronically obtainedfrom the WRDS (Wharton Research Data Services) database. The financial statementdata, as well as quarterly stock price information, are checked for completeness and onlythose firms with complete data are chosen within each industry. Thus, the usable number

of firms (Jh) in each industry is limited, and they are J1 = 52, J2 = 51, J3 = 14, J4 = 17,J5 = 75, and J6 = 21. Thus, the total number of firms is J0 = 230.

In order to determine the optimal input/output categorization of the 18 financialparameters in Table 1, the two-step heuristic solution method in Section 3.1 is applied,



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where an initial sample (ys, zs), s 2S, is determined using a sample of size jS j 20. Thecorresponding sample of objective correlations in (16) is then computed. For each samplepoint, the objective industry-correlation value cthy;z, being forecasted for period t = 29(i.e., the first quarter of 2003), is improved via the local search procedure in Algorithm-

LS, see Section 3.1.1. The optimal vector pairs (yh*, zh*) corresponding to the largest cor-relation are then obtained, see Table 2. The notation in, out, or represent a givenfinancial statement parameter i is an input, output, or it is not considered, respectively, inan industry h. These results pertain to the unrestricted version that uses X in (6). Thosefor the restrictedversion X* in (7) are in parentheses in Table 2. The local search trajectorycorresponding to the sample point that leads to the reported optimal input/output catego-rization is plotted, for the unrestricted domain X and the restricted domain X*, in Figs. 2and 3, respectively, for each industry.

For industry-optimal (yh*, zh*) categorization, for both cases of unrestricted andrestricted domains, industry-correlation metric cth

yh;zh

is estimated according to (11),

and each industry is tested for statistical significance using the hypothesis test in (19).

Table 2Optimal input/output categorization (yh*, zh*) for RFSI in each industry

Financial parameter (i) Industry (h)

Software Communic. Hardware Networks Semicond. Services

1 in (out) () (out) () () (out)2 () () () out (out) () ()3 (out) () (out) out (out) in () out ()

4 out () (in) out (in) in (in) () out (in)5 out (in) (in) out () () () ()6 () () out () () out () ()7 in (in) (in) out (in) () () out (in)8 () in () in (in) () (in) in (in)9 in (in) () in () in (in) (in) ()

10 (in) in () () in (in) out () ()11 out () () () () in (in) in ()12 in (in) out () () () in (in) in ()13 out () () out (out) () in () (out)14 (out) in (out) (out) out (out) (out) (out)15 (out) out (out) (out) () (out) ()

16 (out) out (out) () () out (out) (out)17 in () () out (out) () () (out)18 out (out) () in (out) () () out ()

Max. Corr. cthyh;zhUnrestricted domain 0.2360 0.1947 0.2424 0.2992 0.2220 0.1856Restricted domain (0.1848) (0.1594) (0.0995) (0.2992) (0.1595) (0.1588)

Test statistic whUnrestricted domain 0.2537 0.2035 0.2551 0.3185 0.2352 0.1951Restricted domain (0.1927) (0.1683) (0.1010) (0.3185) (0.1704) (0.1642)

Base categorization

Corr. cthy0;z0 0.1318 0.0504 0.0328 0.1488 0.0668 0.0416Test statistic 0.1392 0.0521 0.0459 0.1539 0.0677 0.0373ISC critical value 0.1670 0.1674 0.2101 0.2018 0.1592 0.1937



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The test statistic wh in (22) is computed and reported in Table 2. The minimum positivecorrelation is set to q0 = +0.10, which yields w

min0 0:1003, see (31). Setting the level of

significance a = 5% and the safety factor j = 1.2, the resulting critical values for the

1.0-

50.0-

0

50.0

1.0

51.0

2.0

52.0

3.0

53.0

9876543210

#noitaretihcraeslacoL

Industry-correlationmetric

erawtfos snoitacinummoC erawdraH skrowteN srotcudnoCimeS secivreS

Fig. 2. Trajectories of maximizing industry-correlation Unrestricted case.

1.0-

50.0-

0

50.0

1.0

51.0

2.0

52.0

3.0

53.0

9876543210

#noitaretihcraeslacoL

Industry-correlationmetric

erawtfos snoitacinummoC erawdraH skrowteN srotcudnoCimeS secivreS

Fig. 3. Trajectories of maximizing industry-correlation Restricted case.



21/25

ISC criterion in (35) are reported in the same table. Observe that for the optimized indus-try-correlation metric under unrestricted parameter domain X, all six industries are chosenby the ISC criterion, while that under the restricted domain X* leads to rejecting the twoindustries, hardware (h = 3) and services (h = 6).

Correlations corresponding to the exogenous base categorization in (8) are alsoreported in Table 2. Note that the maximized correlations (for quarter 1 of 2003) arestrictly better than those resulting from the base categorization of the 18 financialparameters. Also note that the base categorization fails to pick a single industry for invest-ment, based on DEA-based predictability.

For the industries chosen as above, RFSI indicator is computed according to (36), witht 4. That is, the most recent four quarter moving average is computed for predictingRFSI for quarter 1 of 2003. These RFSI predictions are plotted in Figs. 4 and 5, respec-tively, for unrestricted and restricted cases. Specifying the threshold R* = 0.60 for thestock selection criterion in (37), stocks are chosen for portfolio optimization. As evidentfrom Figs. 4 and 5, only a small fraction of the universe of 230 securities are chosen bythe SSC; for the unrestricted case, 85 securities are selected (jNj 85) and for therestricted case, only 49 securities are selected (jNj 49).

5.1. Portfolio optimization

The model RMV in (39) is executed for several risk tolerance levels using the screenedsubsets of stocks, under the stock selection criterion. Let RMV(u) indicate the model forthe unrestricted case, specified with 85 stocks, and let RMV(r) denote the restricted case,having 49 stocks. In contrast, the model SMV in (40) is specified with the original universeof 230 stocks. In this section, the performance of the model SMV is compared with

0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1

1.1

002051001050ytiruceS

RFSIfor2003Q1

erawtfoS snoitacinummoC erawdraH skrowteN srotcudnoCimeS secivreS

Fig. 4. RFSI predictions for chosen industries Unrestricted case.

http://-/?-http://-/?-


22/25

RMV(u) and RMV(r). Portfolio allocations (i.e., weights) are determined by solving theappropriate quadratic programming models, specified with a model-time period of3-months covering from January 01 to March 31, 2003, herein referred to as the invest-ment horizon. We consider a buy-and-hold strategy, wherein the model-determined

0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1

1.1

002051001050

ytiruceS

RFSIfor2003Q1

erawtfoS snoitacinummoC skrowteN srotcudnoCimeS

Fig. 5. RFSI predictions for chosen industries Restricted case.

%0

%05

%001

%051

%002

%052

%003

%04%53%03%52%02%51%01%5%0

noitaiveDdradnatSdezilaunnA

AnnualizedPortfolioRateofReturn

)u(VMR:detcirtsernU )r(VMR:detcirtseR VMS:dradnatS

3002,13raM-10naJ

Fig. 6. Portfolio (actual) efficient frontiers under RFSI-based and standard models.



23/25

optimal stock allocations are held unchanged during the entire investment horizon. Con-sequently, the resulting portfolios are evaluated (i.e., out-of-sample simulated) on a daily

basis by using the actual price realizations from the investment horizon. That is, portfolioallocations determined at the end of 2002 by the model are simulated using prices fromJanuary 01 to March 31, 2003 to determine the actual portfolio performancecharacteristics.

Performance characteristics are compared across the three different models RMV(u),RMV(r), and SMV. By varying the value of the risk tolerance parameter k in each model,efficient frontiers are traced and plotted in Fig. 6. It is evident that the RFSI-based modelsoutperform the standard mean-variance trade-off optimization, with the unrestricted RFSImodel showing impressive portfolio gains over the restricted version.

During the same investment horizon, the market barometer index, S&P-500 index, dis-

plays an annualized standard deviation of 22.4%. The three model versions are set suchthat each model will provide a portfolio with an annualized standard deviation of (approx-imately) 22.4%. Portfolio evolutions corresponding to this case are depicted in Fig. 7,where the performance of S&P-500 index is also indicated. The daily cumulative returnsare significantly improved in the case of the RFSI-based model using the unrestrictedselection of input/output parameters.

6. Concluding remarks

Capturing fundamental financial strength of publicly traded firms is of paramountinterest to equity portfolio managers. To this end, this article developed a new quantitativemetric, termed the Relative Financial Strength Indicator (RFSI), which is designed to havehigh correlation with stock price returns. The underlying methodology is based on using a

%01-

%5-

%0

%5

%01

%51

%02

%52

166515641463136212611161

#yaD

DailyCumulativeRoR

)u(VMR:detcirtsernU )r(VMR:detcirtseR VMS:dradnatS xedni005P&S

3002,13raM-10naJ %4.22=veDdtS.nnA,seiresemitllaroF

Fig. 7. Portfolio RoR under RFSI-based and standard models.



24/25

generalized version of data envelopment analysis, coupled with selecting inputs and out-puts from financial statements via a well-defined optimization process. The resulting mea-sure of relative fundamental strength is then verified to yield high degree of correlation viastatistical analysis. Consequently, RFSI is a useful tool for selecting investment-worthy

industries and securities for portfolio analysis. With the proposed selection rules, it isshown that portfolios so-optimized have significantly improved risk-return characteristics.To our knowledge, this is the first instance that portfolio optimization is integrated withoptimization-based tools for fundamental analysis of companies.

While significant efforts have been expended in literature toward understanding risk-return characters of portfolios constructed with a given universe of securities, methodsfor selecting such a universe have not been addressed to any appreciable extent. Withthe RFSI methodology proposed in this paper, portfolio managers can objectively validatethe inclusion of a given industry or a given security for possible investment, based on theindustrys or firms fundamental financial strength. The question of diversification ofinvestments can then be addressed within those chosen securities. With limited computa-tional experiments in this paper, the advantages of such an approach is demonstratedusing portfolio variance as the risk measure. In our future work, we will pursue specificchoices for parameters of RFSI computations using data sets that cover a broader market,as well as the effects of using alternative risk measures for portfolio optimization.

Acknowledgements

The authors wish to thank the editor and the anonymous referees for many useful com-

ments that improved the presentation of this paper.

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DEA Model for RFSI Relative Financial Strength Index

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