## 2 Methods and Materials
### The Discounted Cash Flow Method  
As previously mentioned, the DCF model is one of the most trusted methods for valuating companies because of its ability to generate equity for the shareholder [1; 16; 30; 54; 55]. This model computes the EV based on the FCF forecasts. FCF is the amount of money available to cover the debt or to pay back equity holders, and it can be considered as a measure of the financial capacity of a company. It is obtained as follows,  
\[FCF=NI+A+DEP-CapEx-\Delta WC \tag{1}\]  
where \(NI\) is the Net Income, \(A\) is the Amortization, and \(DEP\) is the Depreciation. All of them are recorded at the income statement. The Capital Expenditures is given by \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Market Sector** & **Consumer** & **Consumer** & **Consumer** & **Idustrials** & **Technology** & **Utilities** \\ \hline
**Spain** & Inditex & Baron de Ley & Ferrovial & Telefonica & Endesa \\
**United States** & Melia & DIA & ACS & Amadeus & \\  & Starbucks & WalMart & FedEx & & AEP \\  & – & Coca-Cola & 3 MCo & Microsoft & Edison \\
**Japan** & Toyota & Asahi & Kajima & Sony & Chubi EP \\  & Honda & Shisheido & Shimizu & Rakuten & TEPCO \\
**Brazil** & Grendene & AmDev & Azevedo & Tim & CEMIC \\  & Alparagatas & – & MKV & & CPFL Ene. \\
**India** & Maruti & Dabur & Reilance Infr. & Tata Commu. & - \\  & Mahindra & ITC & – & Bharti Airtel & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Companies analyzed according to market regions and sectors. \(A+DEP\), where \(PPE\) is the Property, Plant, and Equipment of a company, recorded at its balance-sheet, and \(n\) represents any specific year. The Working Capital (WC) is determined by \(WC=CA-CL\), being these values recorded at the balance-sheet as Current Assets and Current Liabilities, respectively. Finally, \(\DeltaWC\) stands for the change of \(WC\) in consecutive years. The FCF forecasts, represented in this work as \(FCF_{n+1}\), are usually calculated for 5 to 10 years at a growth rate (\(r\)) [1; 56; 57]. The first three calculations are based on the year-by-year growth for consecutive \(FCF(r_{FCF_{n}})\), being this ratio complying with \(FCF_{n}=FCF_{n-1}(1+r_{FCF_{n}})\). For these three models, \(r\) is calculated as the mean, median, and Compound Annual Growth Rate (CAGR), of the last 5 years (see Table 3). The CAGR is given by  
\[CAGR=\left(\frac{FCF_{n-1}}{FCF_{n-k}}\right)^{1/(n-k)}-1 \tag{2}\]  
where \(k\) is the number of consecutive years included in the analysis, for the CAGR calculated to be applied to years \(n\) and onward. Finally, we compute the FCF forecast based on the coefficients \((p_{i})\) of polynomials of degree 2 or 3 (see Table 3 for details), which are obtained from the previous five FCFs. Vector \(\mathbf{x}=[n-1,n-2,n-3,n-4,n-5]\) represents the years. The DCF method uses the discount rate WACC for measuring the cost-opportunity of companies to use their own equity (\(E\)) and debt (\(D\)) to finance their activity and generate a cost of capital [24]. The WACC is formulated as follows:  
\[WACC=K_{e}\frac{E}{D+E}+K_{d}\frac{D}{D+E} \tag{3}\]  
and it has the following components. First, the cost of equity (\(K_{e}\)) is the minimum return required by equity holders and it is given by the expression \(Rf+\beta(Rm-Rf)\), where the risk-free rate (\(Rf\)) is generally assimilated to state bonds, \(\beta\) indicates the inherent risk factor of the company in relation to the market, and \(Rm\) is the average profitability of the market where the company is stabilized. Second, the cost of debt holders (\(K_{d}\)) is given by \(int(1-T_{m})\), where \(int\) represents the effective interest rate that the company is paying to support its debt and \(T_{m}\) represents the effective company tax-rate. Third, the percentage that the equity takes in company financing terms is \(\left(\frac{E}{D+E}\right)\). Finally, the equivalent rate from the proportional debt is \(\left(\frac{D}{D+E}\right)\), considering also the applied effective tax-rate.