Equity Valuation with Fuzzy Multicriteria Decision Analysis/(Analise Multicriterio Nebulosa de Empresas).

AutorDuarte, Antonio Marcos, Jr.
  1. Introduction

    Equity valuation is the single most used methodology by equity portfolio managers when selecting assets for investment (Fabozzi & Markowitz, 2011). The traditional equity valuation allows analysts to compare different companies for possible investment based on multiples (i.e., quantitative indicators) obtained, most often, from audited balance sheets and economic projections (Stowe et al., 2014).

    Given the growing sophistication of financial markets, the traditional equity valuation must be enhanced by the introduction of concepts such as corporate governance, risks (liquidity, market and credit) and sustainability. Therefore, equity valuation turns out to be a financial problem characterized by the existence of many quantitative and qualitative indicators that need to be computed and compared simultaneously.

    The Multicriteria Decision Analysis (MCDA) offers methodologies to reach compromise solutions when several criteria are used to compare several alternatives (Belton, Stewart, 2002; Ehgott et al., 2010; Ishizaka, Nemery, 2013; Wallenius et al., 2008). MCDA is suited for equity valuation because it facilitates identifying the most promising investments when quantitative and qualitative indicators are adopted to compare companies.

    Applications of MCDA to areas such as accounting and finance are well documented in the academic literature (Bana e Costa, Soares, 2010; Doumpos, Zopounidis, 2010; Doumpos, Zopounidis, 2011; Doumpos et al., 2016; Gomes et al., 2016; Hallerbach, Spronk, 2002; Lisboa, Duarte, 2013; Steuer, Na, 2003; Steuer et al, 2007; Xidonas et al, 2011; Xidonas et al, 2012; Yu et al, 2014; Zavadskas, Turskis, 2011; Zhang et al., 2014; Zopounidis, Doumpos, 1998; Zopounidis et al., 2015), although concentrated in credit rating, portfolio management and project analysis. In the case of equity valuation, the single attempt to address the problem so far has been documented in Duarte (2018).

    Equity analysts cannot rely solely on historical data. They must forecast the multiples of the companies under analysis to obtain realistic results (Damodaran, 2012, Maginn et al., 2012). This step leads to the introduction of uncertainties into the problem. As an example, let us consider that the rating issued by Moody's (Moody's, 2015) for a company has been published as "Baa". It is reasonable to assume that one year from now the rating will remain at this level, but it may well happen that Moody's revises it upwards (for example, to "A") or downwards (for example, to "Ba"), meaning that uncertainty is naturally present in this credit quality indicator. It becomes necessary to modify MCDA to incorporate uncertainties when applied to the equity valuation problem. In this work we rely on fuzzy mathematics (Bellman, Zadeh, 1970; Kahraman, 2008; Kaufmann, Gupta, 1985; Klir, Yuan, 1995; Zimmermann, 1991). Although there are several multicriteria methods that can be adapted to incorporate fuzziness, in this work, for illustrative purposes, we have chosen the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), originally proposed in Hwang and Yoon (1981).

    Our objective in this work is to propose and illustrate the use of a fuzzy MCDA to the equity valuation of Brazilian companies listed in the BM&FBOVESPA. We incorporate uncertainty in the valuation problem with the use of different membership functions for the fuzzy numbers, with varying levels of uncertainty, including asymmetries, all studied in a series of sensitivity analysis performed under different economic scenarios. We also extend the traditional equity valuation by the introduction of qualitative indicators in the analysis (related to corporate governance, sustainability and credit risk) and stock market data (related to volatility and trading volume), mixing them with the traditional multiples derived from balance sheets (usually related to profitability, liquidity and costs) which are frequently used by financial analysts, according to Stowe et al (2014). Also, we extend the proposal for equity valuation presented in Duarte (2018) in two ways: (a) by allowing risk measures and qualitative indicators to be introduced, given that only multiples were used in that previous work, and (b) by proposing a more general methodology that can be used when comparing companies from different economic sectors, contrary to what was presented in that article, which remained limited to a single economic sector (i.e., financial institutions). We also differ from Duarte (2018) in the fuzzy MCDA used to obtain the numerical examples: we rely on the Fuzzy TOPSIS method, instead of the Fuzzy TODIM method, with the former demanding a simpler computational implementation.

    In terms of organization, in the second section we detail the data and the transformations that need to be imposed to the traditional equity valuation problem to incorporate fuzziness and multicriteria analysis. The third section displays the numerical results, including all studies with their sensitivity analyses. The last section brings in our final comments and future directions for the use of fuzzy mathematics and MCDA in equity valuation. Finally, in the appendix, we detail the multicriteria method used in the numerical analyses for those who may want to replicate our results.

  2. The Methodology: Fuzziness, Multicriteria Analysis and the Data

    The traditional equity valuation can be extended to incorporate fuzziness and multicriteria analysis the following way:

    i. Select all the stocks of interest for possible investment. In the numerical examples presented later we concentrate on the nine largest market capitalizations in BM&FBovespa at the end of 2014: Ambev SA (ABEV3), Banco do Brasil SA (BBAS3), Banco Bradesco SA (BBDC4), BRF SA (BRFS3), Cielo SA (CIEL3), Itau-Unibanco Holding SA (ITUB4), Companhia Brasileira de Distribuido (PCAR4), Petroleo Brasileiro sA (PETR4) e Vale SA (VALE5). It is important to mention that our proposal can handle any number of stocks but, given space limitation in this article, we focus only on these nine for illustrative purposes.

    ii. Identify all the criteria (multiples, stock market data and qualitative indicators) to be used to compare the previously selected stocks. In the numerical examples, we adopt ten criteria for the sake of numerical illustration: five of the most traditional multiples used by stock analysts (Stowe et al., 2014), three risk indicators (market, liquidity and credit) and two related to the organizational structure of the company (corporate governance and sustainability). Table 1 summarizes the ten criteria. It is opportune to mention that our proposal can handle any number of criteria but, given space limitation, we focus only on the ten shown in Table 1, for illustrative purposes.

    iii. Establish the relative importance of the criteria previously defined. This is a critical step when using any MCDA, requiring attention from the decision maker (Figueira et al., 2005). In the numerical examples, we shall experiment with different levels of relative importance, making use of sensitivity analysis to that end.

    iv. Collect and/or compute all the multiples and qualitative indicators for each criteria and company of interest. In this work, we have chosen to use only historical data, all obtained from commercial data feeders (Economatica and Bloomberg) or directly from audited balance sheets provided by the investors' relations area of each company. The values for the criteria summarized in Table 1 are exhibited in Table 2.

    v. Generate scenarios for the possible evolution of each criterion in the future. For example, it is important to understand how a projected scenario can alter the scores (and by consequence, the orderings) of the companies under analysis. We illustrate this step with the use of sensitivity analysis, which is crucial when trying to understand how small changes in the inputs can alter the results, as thoroughly illustrated by the numerical examples presented ahead.

    vi. Apply a multicriteria method combined with fuzzy mathematics to the data to obtain the scores (and by consequence, the orderings). In this article we alter the original TOPSIS method to incorporate fuzzy numbers, finally resulting in a method we refer to simply as Fuzzy TOPSIS from now on. This fuzzy and multicriteria method allows the decision maker to experiment with a myriad of ways to handle uncertainty in the equity valuation problem, as illustrated later. The Fuzzy TOPSIS used in the numerical examples is detailed in the appendix, in order to keep the focus of this article on equity valuation (and not on the mathematical approach adopted in the numerical examples).

  3. Numerical Analyses

    We present in this section the numerical results (including sensitivity analyses) obtained when the methodology presented above was applied to the nine stocks and the data displayed in Table 2.

    The initial step requires defining conversion scales for the three qualitative criteria presented in Table 1. The initial conversion scales adopted are as exhibited in Table 3, Table 4 and Table 5. With respect to the conversion scale for credit ratings, we handle intermediary positions by adding or subtracting 0.5 - for example, given that the conversion value for the rating br.A is set at 7, the ratings br.A+ e br.A--receive values 7.5 and 6.5, respectively. Table 6 summarizes the data after the application of the three conversion scales to Table 2. We anticipate that sensitivity analyses will be performed for each conversion scale later, to help clarify how sensitive the defuzzified scores of the nine companies are to changes in these parameters

    We denote the fuzzy triangular components of the weight vector as

    (([w.sup.a.sub.1]...

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