*Institut Els Alfacs, Dept. Administration and Management. San Carles de la Ràpita (Tarragona) Spain.*

*Universitat Politècnica de València, Dept. Economic and Social Sciences. Camino de Vera s/n, 46022 Valencia, Spain.*

The European agrifood industry is mostly characterized by small and medium enterprises (SMEs); as in 2013, SMEs represented 99.13% of the total number of companies. The valuation of SMEs not listed in any stock market is a complex task since there is not enough information on comparable transactions. When applying discounted cash flow (DCF) models to value private agrifood companies, the capital structure and the cost of equity are two key parameters to be determined. The implications of these parameters in the value of the enterprise are not clear inasmuch as it is not possible to carry out a contrast due, precisely, to the lack of comparables. The main goal of this study is to determine the biases that those two parameters can introduce into the valuation process of an agrifood company. We have used the stock market as a framework wherein to apply a simple fundamental model to the companies of the European food industry in order to obtain three valuation multiples. By means of two bootstrap approaches, the bias induced in the multiples has been assessed for every year from 2002-2013. Results show that the use of the return on equity as cost of equity tends to undervaluation; the use of capital asset pricing model (CAPM) tends to a slight overvaluation, whereas using the total beta induces an undervaluation bias. Moreover, the capital structure shows little influence on the valuation multiples. The conclusions drawn from this paper can be useful for managers and shareholders of privately-held agrifood companies.

_{l}(Levered Beta);

_{F}(Fundamental Enterprise Value);

_{d}(Cost of debt);

_{e}(Cost of Equity);

_{f}(Risk-free rate);

The European agrifood industry is mostly characterized by small and medium enterprises (SMEs); as in 2013, SMEs represented 99.13% of the total number of companies (

In the case of listed companies, discounted cash flow (DCF) models and multiples are widely adopted (

A DCF valuation of privately-held companies implies a series of key decisions regarding capital structure and cost of equity (

To answer these questions, we need to compare the estimated value with the real value. Since there is no real value available for agrifood SMEs, we have taken a group of listed agrifood companies as a benchmark to check the influence of those valuation decisions on the value. Instead of comparing values we have compared multiples. Broadly speaking, the procedure works this way: first, all the agrifood companies listed in the European stock markets are valued by means of DCF obtaining the fundamental enterprise value (EV
_{F}
) for each company. Second, accounting variables are used to compute several valuation multiples, which are termed fundamental multiples. Third, these fundamental multiples are bootstrapped and compared with the corresponding bootstrapped stock multiples in order to contrast the existence of statistical significant differences in each multiple mean. Using multiples instead of values so as to contrast the models has some apparent advantages, such as the better interpretation of relative measures and the possibility of allowing a greater number of contrasts (one ‘enterprise value’ versus several multiples). By introducing different ways of fixing capital structure and cost of equity, the influence on the valuation multiple can be studied. From a practitioner’s point of view, the answers to these questions can help to make valuation decisions and improve the accuracy of valuation multiples with greater insight or at least to know which kind of bias can be introduced by those decisions (

In order to answer these paper questions listed European agrifood companies have been used. Accounting and market data of those companies from 2002 to 2013 have been gathered from the Damodaran website (

According to

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

We will assume that the EV
_{F}
will be determined by the previous year’s FCFF, taking perpetuity into account, as

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.

The FCFF are calculated as shown in

The capital expenditures are obtained by following

The FCFF are discounted by using the WACC,

The cost of debt is calculated as an approximation using the financial expenses and the current debt of the company,

As

The first option: the capital structure is taken from the company’s books; this option is sometimes used whenpricingprivately-heldcompanies.

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.

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.

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).

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 (
_{m}
. RP
_{m}
is obtained as the difference between the E(R
_{m}
) and the R
_{f}
following

The individual beta of each company is obtained by unlevering each food firm beta using the

Then, the unlevered average beta of the food industry is computed, as shown in

Finally, the food industry beta is levered by using the capital structure of the individual company, following

In order to estimate the market risk premium, we have used annual return realizations from the French

When using the beta for unlisted companies,
_{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 (

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 (

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

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

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.

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 (
_{F}
and EV
_{S}
are computed.

One outlier detection technique has been applied in order to identify those multiples with anomalous measurements.

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.

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 (

In our study, taking into account the number of companies and models, two R scripts (

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 (

Likewise,

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.

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

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,

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

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.

The bootstrap technique is applied to take into account the variability of the valuation process and, hence, the variability induced on the valuation multiples. Depending on when the bootstrap is used in the calculation process, the empirical distribution of the mean multiples or the empirical distribution of the multiples of the average company can be obtained. The sign of the bias introduced by the valuation process is essentially the same between the average multiple (approach 1) or the average company multiple (approach 2). Nevertheless the average company ratio is much more robust to outlier’s presence.

Statistical contrasts have been carried out to determine the existence of statistical differences between fundamental and stock multiples. Around 50% of the fundamental multiples are not statistically different from the stock multiples. When pricing privately-held agrifood companies, using the return on equity or the total beta causes undervaluation, whereas using the industry beta causes a slight overvaluation. The quantification of the bias shows that the use of the total beta decreases the value by around 10-50%. The ROE decreases the value by around 25%-58%. These figures could be read as a discount for the lack of diversification, which is a regular case in unlisted companies and SMEs.

The method of fixing the capital structure does not exhibit a clear influence on the valuation multiples.

Fixing the cost of equity by means of the CAPM appears to be the less biased method to value agrifood companies with discounted cash flows; this is consistent with the reviewed literature.

All of these conclusions can be useful for managers and shareholders of privately-held companies when applying for a valuation report or analysing and discussing one. Building empirical distributions of multiples can help to increase the amount of industry information on SMEs and privately-held companies, which would increase the transparency in their valuation process.

Besides the specific conclusions, the main contribution of this paper is the use of the stock market as a framework wherein to apply fundamental valuation together with bootstrapping in order to determine the bias introduced by the choice of some parameters in the DCF valuation model.