The EV _F is estimated by assuming that an asset, a company, is worth the discounted value of all the future cash flows it can generate. The cash flows are measured as FCFF following Damodaran’s (2006) definition. Among the members of the discount family, the discounted cash flow model has traditionally been the dominant one in practice ( Jennergren, 2008 ). The results obtained by Imam et al. (2013) indicate that European investment analysts prefer to use cash flow based models in conjunction with non-book value based accrual models.

We will assume that the EV _F will be determined by the previous year’s FCFF, taking perpetuity into account, as Eq. (1) shows.

The choice of this valuation model is influenced by the need to carry out an automatic mass valuation. The main advantage is that the amount of data to be collected is relatively small while the main drawback is the model’s simplicity. Eq. (1) is also used as a way of estimating the continuing value in two-stage DCF models ( Jennergren, 2008 ).

The FCFF are calculated as shown in Eq. (2):

The capital expenditures are obtained by following Eq. (3).

The FCFF are discounted by using the WACC, Eq. (4).

Marques-Perez et al. (2017) stated that the discount rate in agro-industrial firms is an essential tool for appropriate corporate management.

The cost of debt is calculated as an approximation using the financial expenses and the current debt of the company, Eq. (5).

As Petersen et al. (2006) state, the estimation of a target capital structure is a non-trivial issue since market prices on equity and debt are not observable for equity privately-held firms. According to both literature and practitioners, the capital structure in Eq. (4) will take different values which are dependent on three different options.

The first option: the capital structure is taken from the company’s books; this option is sometimes used whenpricingprivately-heldcompanies. Woolley (2009) stated that there are many studies that use the book value of debt and equity. For McLaney et al. (2004) , book value weights may be more objective yet less sensitive to economic reality than market values. However, Damodaran (2006) is not convinced by the arguments of those analysts who continue to use book value weights.

The second option: the capital structure is fixed as the average capital structure of the industry stock market. Since there is no available capital structure of the market for privately-held companies, this is similar to considering the debt ratio when financing investments and it could represent the company’s target capital structure. Vinturella & Erickson (2003) , among others, state that the weights in the cost of capital should represent the firm’s target capital structure. Koller et al. (2010) consider that the industry peer group is a good starting point from which to set the capital structure. McLaney et al. (2004) point out that, given fluctuations in share prices, the weighted average cost of capital (WACC) based on market values can change daily.

The third option: the capital structure is fixed by iterative calculation. This option addresses the so-called circularity problem, which means we cannot know the post-tax WACC without knowing the value of equity, and we cannot know the value of equity without knowing the post-tax WACC. Koller et al. (2010) recommend determining the equity value (for the cost of capital) of privately-held companies either using a multiples approach or through iterative DCF iteratively. Larkin (2011) states that analysts should use the iterative method with the WACC when valuing passive investments. According to Turner (2008) , the circularity problem is important, since a small error in the post-tax WACC calculation can lead to a large error in the FCFF model’s valuation.

Furthermore, the cost of equity (k _e ) will also be obtained in two different ways. The first way is to take the average Return on Equity (ROE) of the agrifood industry as the cost of equity to be used in the WACC formula. For each sample company, the ROE was computed as the NI divided by the book value (BV). Breuer et al. (2014) suggested that one may rely on historical return moments. For Rojo (2014) , ROE based on accounting data seems to be a good instrument with which to analyse the value of an investment project and may be a landmark in the study of the discount rate. Feenstra & Wang (2000) stated that the accounting rate of return, based on accrual concepts and defined as net income divided by book value of equity, is not only a central feature of any basic text on financial statement analysis, but it also figures commonly in the evaluation, by investment analysts, of the financial performance of firms.

The second way is to use the capital asset pricing model (CAPM) model. The cost of equity (k _e ) is typically calculated via the CAPM in both listed companies ( Breuer et al. , 2014 ) and unlisted companies ( Rojo & García, 2006 ; Rojo, 2013 ). By eliciting responses from the finance directors of 193 UK quoted firms, McLaney et al. (2004) found a significant association between the use of the WACC and the use of the CAPM. For Koller et al. (2010) , asset-pricing models that translate risk into expected return are used since the expected rates of return are unobservable. The CAPM is the most common asset-pricing model (Eq. 6) and postulates that the expected rate of return equals the plus the company’s beta times the RP _m . RP _m is obtained as the difference between the E(R _m ) and the R _f following Eq. (7).

The individual beta of each company is obtained by unlevering each food firm beta using the Modigliani & Miller´s (1958) beta formula. Petersen et al. (2006) also report the use of this formula in the valuation of privately-held firms, Eq. (8).

Then, the unlevered average beta of the food industry is computed, as shown in Eq. (9). Koller et al. (2010) recommend using industry rather than company-specific beta.

Finally, the food industry beta is levered by using the capital structure of the individual company, following Eq. (10).

In order to estimate the market risk premium, we have used annual return realizations from the French Cotation Assistée en Continu (CAC), since the company betas are also computed using the CAC as the market index. Based on these data, we have calculated the differences between the total historical CAC returns and the historical 10-year French bonds as risk-free rate to determine historical excess returns. The time series covers the period from 1987 (base year) until the corresponding year of the study. The geometric average of the annual market risk premium has been used ( Koller et al. , 2010 ).

When using the beta for unlisted companies, Damodaran (2006) suggests adjusting it to reflect the total risk rather than the market risk. That implies the assumption that the marginal investor is not a well-diversified investor. The total beta can be obtained by dividing the market beta by the correlation coefficient between each stock and the market index. This operation yields a higher industry beta. Petersen et al. (2006) report that Danish practitioners do not follow the Damodaran approach, but they add an additional 1-3 percentage points to the cost of equity in order to compensate for unsystematic risk. Since our study focuses on the valuation of unlisted companies, we have deemed that the total beta approach should be included as a third option with which to work out the k _e . Unfortunately, we only had enough data from 2009 to 2013; therefore, this option will only cover those years. Moreover, the total beta can be used to ascertain the relative difference in value of the same company for a diversified investor and a non-diversified investor. The existence of the non-diversified investor in unlisted companies is well known ( Damodaran, 2006 ). Petersen et al. (2006) also reported that the valuation of privately-held firms often involves investors who are not well-diversified. Assuming that the owner of an unlisted company is a non-diversified investor, the discount to be used when pricing unlisted companies could be estimated.

The combination of the different options regarding capital structure and cost of equity leads to nine variants of the fundamental valuation model. As the literature shows, each of these nine combinations is deemed to be of possible use by a practitioner when valuing an unlisted agrifood company.

In order to test the difference between fundamental and stock multiples, a bootstrap technique has been used. Bootstrapping is a technique that resamples from the original data set ( Efron, 1979 ; Davison & Hinkley, 1997 ) and allows any lack of normality issues to be avoided. Bootstrap methods have many applications for certain kinds of computations, such as biases, standard errors and confidence limits ( Hesterberg et al. , 2005 ; Chernick & LaBudde, 2014 ). The references to valuation models that include the use of the bootstrap as a basic tool are scarce. Cruz (2012) used bootstrapping and Monte Carlo simulation for decision-making purposes in the assessment of investment projects. Breuer et al. (2014) estimated the discount rate for firm valuation by way of a bootstrap approach.

We have used two different bootstrap approaches: the first one consists of determining the valuation multiples and then bootstrapping them; we have termed it ‘first valuation then bootstrapping’. In the second approach, the bootstrapping of the fundamental variables comes first and, after that, the fundamental valuation is carried out, we have named it ‘first bootstrapping then valuation’.

In the first approach, we have used case resampling to derive the distribution of the mean from a distribution of an n multiple sample. First, we have resampled the data to obtain a bootstrap resample. An example of the first EV/EBIT resample might look like Eq. (11).

EV/EBIT1* is the distribution of the fundamental multiples for each year, while “mi” is the EV/EBIT for company “i”. Note that there are some duplicates, since a bootstrap resample comes from sampling with replacement from the data. The size of the bootstrap resample is equal to the number of observations in the original data set. Then we have computed the mean of this resample and obtained the first bootstrap mean: µ1*. In the same way, a second resample EV/EBIT2* can be obtained, and a second bootstrap mean: µ2* can be computed. This procedure has been repeated 10,000 times to obtain a series of 10,000 bootstrap means; this series is called the empirical bootstrap distribution.

The empirical bootstrap distribution of EV/EBITF would be as shown in Eq. (12). The very same procedure has been carried out for every year for all of the fundamental multiples and stock multiples.

In the second approach, the procedure is slightly different. Each variable of the fundamental model is resampled, the bootstrap mean is obtained and the procedure is repeated 10,000 times. At the same time, the accounting variables (EBIT, EBITDA and Sales) are also bootstrapped. A matrix is obtained, made up of the valuation parameters and the accounting variables as columns and the 10,000 bootstrap means as rows. For each row, the fundamental value is worked out. Using those 10,000 fundamental values and the corresponding accounting variables, the empirical bootstrap distribution for each multiple is built. In the same way, the empirical bootstrap distribution for stock multiples has been determined by bootstrapping stock enterprise value (EVs) and the corresponding accounting variable.

The first approach will provide the empirical distribution of the average multiple, while the second one will provide the multiple distribution of the average company. In the first approach, all the companies are considered to be of equal importance in the industry (in the company group), whereas in the second, the companies with greater value and greater accounting variables (EBIT, EBITDA or Sales) will have a greater pull on the bootstrap empirical distribution of the multiple.

The bootstrap allows the distribution of each multiple mean to be estimated and confidence intervals built. Table 2 summarizes both bootstrap approaches.

The valuation multiples are computed each year and the analysis is first carried out on an annual basis and then by considering a unique time window from 2002 to 2013. The way of working out the valuation multiples is different according to the bootstrap approach. If the ‘first valuation then bootstrap’ approach is taken, the EV _F is estimated and three valuation multiples are calculated, EV/EBIT, EV/EBITDA and EV/Sales, and the corresponding stock valuation multiples are simultaneously obtained. It must be ensured that the observations and the sampling for each valuation multiple and year are the same in terms of stock and fundamental multiples in order to carry out a correct comparison. After applying the nine valuation variants and computing the three valuation multiples, the normality Shapiro-Wilk test ( Shapiro & Wilk, 1965 ) shows that the normality hypothesis is rejected in 82% of the stock multiple cases and 77% of the fundamental multiple cases. Damodaran (2006) also points out that valuation multiples do not follow a normal distribution. Liu et al. (2002) stated that while the skewness is less noticeable for multiples based on forward earnings, it is quite prominent for multiples based on sales and cash flows. The normality test cannot be carried out on the second bootstrap approach, since the variables of the valuation model are bootstrapped first and then the average EV _F and EV _S are computed.

One outlier detection technique has been applied in order to identify those multiples with anomalous measurements. Table 1 shows the maximum for every year and multiple, some of which are extremely high and exert a great influence on the mean. As Vakili & Schmitt (2014) pointed out, a few outliers, if left unchecked, will exert a disproportionate pull on estimated parameters. Removing outliers is a delicate issue but, in the case of valuation multiples, large multiples can greatly affect the average. In this mass valuation process, the application of an automatic criterion for the purposes of detecting and then removing outliers has been deemed unavoidable. In particular, a technique has been used for detecting univariate outliers, specifically the absolute deviation around the median, MAD. Researchers are quite used to detecting the presence of outliers by observing an interval spanning over the mean plus (minus) two or three standard deviations. Leys et al. (2013) relate the problems of using the mean as the central tendency indicator: first, normality is assumed; second, the mean and the standard deviation are greatly affected by outliers and third, there are some issues related to small samples. In this case, the computation of the MAD has been carried out considering a lognormal distribution. The threshold value has been set at ± 3 times MAD. In the first approach, the MAD criterion has been applied on each set of valuation multiples. In the second approach, the MAD has been applied on the variables needed to obtain EV.

Since we have chosen three multiples (EV/EBIT, EV/EBITDA and EV/Sales), built nine valuation variants, taken two bootstrap approaches and the study was carried out over 12 years, we should have 648 bootstrap distributions of the valuation multiples. However, as we had no information on the total beta for 7 years, the total beta model was only applied for 5 years; hence the number of bootstrap distributions and contrasts adds up to 522. Table 3 summarizes the different options on capital structure and cost of equity, and the two approaches used to estimate the fundamental valuation multiples. The models are named “Model x.y.z”. The “x” reference explains the bootstrap approach, the “y” reference the capital structure and, finally, the “z” reference specifies the cost of equity. Fig. 1 displays the general outline of all the empirical models.

The application of the valuation options regarding capital structure and cost of equity is quite straightforward. Nevertheless, computing the capital structure for each company by solving the circularity problem requires some iterative calculations ( Turner, 2008 ). Larkin (2011) details the calculations to be carried out and says that it is extremely simple to execute the iterative method in an Excel spreadsheet. Vélez-Pareja & Tham (2009) also recommend using the spreadsheet to handle circularity.

In our study, taking into account the number of companies and models, two R scripts ( R Core Team, 2016 ) have been written using the secant method and the Newton-Raphson method ( Kaw et al. , 2011 ). Only the solutions of Equity greater than zero are deemed to be right, which implies the removal of some companies with negative solution for the Equity.

In the previous section, several valuation choices have been introduced. The analysis has been carried out on two levels: an annual level (working with annual observations) and a whole period level (working with firm year observations).

The focus of annual level analysis is on determining the number of times there is a statistically significant difference between stock and fundamental multiples. A contrast ratio, made up of the fundamental multiple divided by the stock multiple, has been computed for each type of multiple, model and year. The null hypothesis (H ₀ ) is that the contrast ratio is equal to one. If the contrast ratio is statistically different from one, then there is a significant difference between the fundamental multiple and the stock multiple. The level of significance was set at 95%; this means that if the contrast ratio is outside the 2.5%-97.5% range of the empirical bootstrap distribution, then the average of the multiples is considered to be statistically different, which is to say the null hypothesis is rejected. On the whole, the fundamental multiples are not statistically different from the stock multiples in half of the cases, and when there is a statistical difference, the fundamental models are more likely to undervalue than to overvalue (Table 4). A better insight can be gained by examining the results of the different models.

Table 5 has been built for bootstrap approach 1 (measuring the results of the average multiple) by grouping the model results according to the cost of equity and capital structure. When the cost of equity is fixed as the average agrifood industry ROE an undervaluing bias is introduced in the valuation of agrifood companies. When the industry beta is used to fix the cost of equity by means of the CAPM, the null hypothesis is not rejected in 33% of the cases; in the remaining 67%, fundamental multiples are more likely to be greater than the stock ones, there is an overvaluation bias. When considering that the investor is not diversifying in the Markowitz sense (total beta models), higher stock multiples could be expected since a higher cost of equity is used in the fundamental models. In 71% of the cases, the null hypothesis cannot be rejected and, in the rest of the cases, there is an undervaluation bias.

Likewise, Table 6 gathers the results for bootstrap approach 2, which measures the results of the average company multiple. The results are quite similar to Table 5, the use of the industry ROE or the total Beta tend to cause some undervaluation while the use of the industry beta creates a slight overvaluation.

The different ways of considering the capital structure do not show any clear effect on the value and consequently on the valuation multiples. The selection of the cost of equity seems to be much more influential.

The large number of combinations, 522, can hinder a global vision; for that reason, two graphical analyses have been carried out considering the period 2002-2013.

Figure 2 has been built in order to show the biases introduced by the valuation choices in both approaches. Since we had no data on the total beta for the period 2002-2008, the models that use the total beta have been left out; which is, all the x.y.3 models are not considered. The remaining six valuation models and the two bootstrap approaches have been applied so as to obtain the valuation multiples. Figure 2 is made up of six squared panels. In each panel, a scatterplot of one multiple has been plotted by applying the six valuation models and one of the bootstrap approaches. Each model is represented by a coloured cloud made up of 10,000 dots. The x-coordinate of each panel measures the bootstrap mean of the stock multiple, while the y-coordinate is the bootstrap mean of the fundamental multiple for the very same resampling. A bisector line has been traced to visually ascertain the bias. If a dot cloud is above the bisector line, the fundamental model is overvaluing; on the contrary, if a dot cloud is under the bisector line, the fundamental model is undervaluing. The distance from the cloud to the bisector line displays the strength of the bias. The relative difference between models can be checked as well, since we have used squared panels. Some of the results of the annual analysis are confirmed: using the ROE as the cost of equity induces undervaluation, whereas using the CAPM to fix the cost of equity tends to a slight overvaluation. The method of fixing the capital structure is not very influential, although the clouds from models with circularity structure (1.3.z) are placed slightly higher on the fundamental multiple axis.

The approach 2 models that use the CAPM (models 2.y.2) are clearly the least biased regardless of the method of fixing the capital structure. In this case, the model is much less sensitive and the effect of the circularity selection does not show.

Additionally, the average multiple and the multiple of the average company have been traced as vertical lines in panels 1.y.z and 2.y.z, respectively. Both averages have been calculated taking the company years with positive multiples in the first case, or positive EV _S and positive accounting variables in the second case. Outliers have been removed in both cases.

The variability of the multiples can also be compared in Fig. 2. If the cloud is longer in the direction of one axis than in the other it shows a greater variability in the former. It can be seen that x.y.2 models (the ones that use the CAPM) show greater variability in the fundamental multiples than in the stock multiples, except in the case of EV/Sales. On the contrary, x.y.1 models (the ones that use the ROE to fix the cost of equity) exhibit a similar variability for both kinds of multiples.

With regard to both bootstrap approaches, the first approach shows less variability. In the first approach, only the variability of the numerator and the denominator of the fundamental multiples are considered, whereas in the second one the variability of several parameters of EV are taken into account; the greater number of variables has induced greater variability.

The bootstrap estimates of each multiple are analogous to the idea of the density estimation of the mean. An estimation of the shape of the density function can be obtained by plotting a histogram. As an example, Figs. 3 and 4 show the results of the EV/Sales models for the period 2002-2013; fundamental and stock histograms are plotted together. With these kinds of plots, the variability and the difference between stock and fundamental multiples can be clearly appreciated for each multiple, model and approach. The results match those from Fig. 2. Figs. 3 and 4 also include models x.y.3 (circularity structure); interestingly, the variability in the fundamental multiples of the x.y.3 models is much greater than in the rest.

Assuming that the stock multiple is the true one, a measure of the bias with respect to the stock multiple has been calculated as M _F /M _S -1. If this bias ratio is statistically greater than zero, the fundamental multiples tend to overvalue; on the contrary, if the bias ratio is less than zero, the fundamental multiples tend to undervalue. The average results are shown in Table 7.

The bias ratio confirms the results of the H ₀ contrast in the annual level analysis. The use of ROE as cost of equity introduces a negative bias, that is, an undervaluation of between 25% and 58%. Similarly, when using the total beta to fix the cost of equity, the undervaluation is in the range of 12-52%. When using the industry beta, the bias is slightly positive in almost all the cases.