3.2.1. Databases, methodology and hypotheses

Heller and Levyne (2014)² carried out statistical studies aiming to evaluate and test the growth potential of stock prices for companies in the CAC 40 index on February 22, 2013. On this date, four elements were taken from the Facset database for each company: the market capitalization, the volatility of shares corresponding to the standard deviation of daily returns over a time-scale of one year, the consensus of brokers over the value of the company and the target value of equity.

Insofar as many brokers use the DCF method to valuate, the insurance companies and banking institutions that make up the stock index were immediately put to one side. If Axa, Crédit Agricole, BNP Paribas and Société Générale can in fact be valuated using a cash flow approach, it nevertheless turns out that in this case, the latter correspond to a surplus of capital that could be transferred to shareholders by taking solvability constraints into account.

Thus, the cash flow can be considered a theoretical dividend that leads brokers to valuate these companies using an adequate method: the dividend discount model. Because of this, the sum of present values of expected free cash flows allows us to directly obtain the value of equity, the future value rate being the cost of capital. In the case of insurance companies such as Axa, the brokers use the target ratio of Solvency 1, in keeping with the target rating of the company. In the matter of banks, they take the equity ratio objective required by Bâle 3 into account. In other words, whether it is insurance companies or banks, no company value is included in the evaluation.

Furthermore, each daily volatility taken from Facset was multiplied by in order to obtain yearly conversions. The risk-free rate taken was that of Treasury Bills at 10 years, namely 2.20% on the evaluation date, or 2.18% in continuous time³. Additionally, the financial debts of each company in the sample were taken from their respective reference document at the close of business 2011.

Nevertheless, insofar as a company has a negative net debt, the use of the DCF method changes from one broker to another. Indeed, some calculate the present value of free cash flows by starting with the cost of capital, while others calculate the WAAC algebraically with a net negative debt. Voluntarily, given these possible divergences, the five companies with a negative net debt at the end of business 2011 – namely, Cap Gemeni, EADS, L’Oréal, STMicroelectronics and Technip – were removed from the studies. Moreover, Renault, whose weight in credit from client consumption is significant in their accounting, was also excluded from the sample. Consequently, the empirical studies focused on the 30 remaining companies of the CAC 40 stock index⁴.

The growth potential of brokers was calculated by relating the target value of equity obtained by the DCF method to the daily market capitalization, then by subtracting 1.

By valuating the sample with the Black–Scholes–Merton approach, the bankruptcy risk of companies – by taking debt maturity and asset volatility into account – is integrated. Thus, the accounting debt, tied to the exercise price and removed from asset cash and cash equivalents to give a net debt, should not be overestimated since it relies on an economic value.

Because of this, the model should reveal the growth potential of the economic value of equity thanks to a new way of dividing the company. In the Black–Scholes–Merton articles, debt is indeed considered a zero-coupon. Consequently, the maturity date of the option corresponds to the residual maturity of the bond. Nevertheless, for numerous companies, the date is made of bonds generating coupons and financial loans from banks.

From a theoretical point of view, an option with different maturities should not be cast aside. However, in order to apply the Black–Scholes– Merton valuation mode, a residual average maturity τ for each of these company debts is assessed.

Table 3.1 provides the calculation details of the economic value of a debt using the Merton method and starting with the following hypotheses: a company value of 2,509, a risk-free rate of 2%, an asset volatility of 30% and an average maturity of 5 years for financial debts.

Table 3.2 shows that the equity value from the DCF method (1,509) is only obtained if the maturity date, that is, the residual average maturity of the financial debt, is zero. Otherwise, the greater the maturity, the higher the time value and the value of equity rises. Moreover, the higher the volatility, the more the probability of a rise in stock prices also goes up, which implies a rise in the value of equity.

In order to apply the model to the data, the company values and their volatility had to be estimated. The volatility of underlying assets corresponds in fact to the volatility of the company value. But the assets are rarely priced, except for company holdings which are not represented in the CAC 40 stock index. Consequently, the estimate of the company value and its volatility are based on the methodology proposed by Hull et al. (2005)⁵ and commonly used by Moody’s notation agency. Using the Ito lemma as a basis:

[3.1]

Thus, by replacing:

• – F by E (for the value of equity);
• – x by EV (for company value);
• – a(x,t) = m.EV;
• – and b(x,t) = σ_EV.EV where σ_V corresponds to the volatility of assets or of the company value.

[3.2]

[3.3]

[3.4]

where σ_E corresponds to the volatility of shares.

Additionally, thanks to the Merton formula:

[3.5]

The values of EV and σ_EV can be obtained using the Excel calculator applied to the following nonlinear system:

[3.6]

[3.7]

To solve this system, the following parameters are considered:

• – E = daily stock capitalization;
• – D = net accounting debt taken from financial reports;
• – τ = the average residual maturity of the debt calculated using the repayment calendar for the debt;
• – σ_E = the volatility of shares taken from Facset and then annualized;
• – r = the continuous risk-free rate of 2.18% corresponding to the OAT French rates at 10 years.

Once the company value and its volatility have been calculated (produced by the calculator), the Black–Scholes–Merton valuation model for each company was applied in order to find the economic value of equity and the net debt. In this case:

[3.8]

with:

[3.9]

where:

• – E: economic value of equity;
• – EV: company value according to broker consensus (DCF);
• – Φ(.): normal standard distribution;
• – D: accounting debt;
• – TA: cash and cash equivalents.

An alternative approach allows us to estimate the value of economic debt for Merton, called B:

[3.10]

[3.11]

[3.12]

[3.13]

is the amount of debt that will be recovered bondholders if the company goes bankrupt.

Then, is the recovery rate and the expected discounted loss, written LGD⁶, which would be absorbed by bondholders in the case of the supposed default by the company. Since Φ(−d₂) is the bankruptcy risk, is the expected discounted loss of earnings. Finally, as used by Moody’s KMV and the risk management hubs at banks in determining the calculations for weighing asset risk:

[3.14]

The three principal parameters of the economic value of the net debt seem to be its maturity (τ), the recovery rate for the level of bankruptcy which includes the risk of bankruptcy, and the weight of its face value, which is expressed as a percentage of the company value (D/EV). In this case, a multiple regression is tested to justify the potential for growth based on the value of equity by Black–Scholes–Merton⁷.

The empirical studies therefore concentrate on the potential for growth in stock prices for the 30 companies listed in the CAC 40, based on target values established by brokers, and on the values obtained using the real options approach. They are compared, systematically, to the market capitalization on the day of the evaluation⁸. In the first case, the target value is the company value, which corresponds to the present value of future free cash flows, determined by brokers, minus the net debt found in financial reports. In the second case, the Black–Scholes–Merton approach proposes a new way of dividing the value of a company obtained by the DCF method between equity values and economic debt.

The hypotheses, with respect to the results that will be obtained, are as follows:

• – the differences in significance between the volatility of shares (σE) and that of assets (σV) should be equal, allowing us to justify taking the volatility of assets into account (instead of shares) in the method of valuation using real options;
• – the differences in significance between the percentage of evolution of stock prices for the companies in the sample using the DCF method and the real options method should be equal, allowing us to justify the relevance of using the real options method;
• – the differences in equal significance between two growth potentials can be explained by the respective debt ratios. This is the reason why it seemed useful to test the ratio of net debt on the company value, calculated on the basis of the accounting net debt, on the one hand, and referring to the economic value (obtained by the Black–Scholes–Merton method), on the other. These ratios are written D/EV and B/EV, respectively. The differences in significance between these ratios should be equal, allowing us to justify using the economic debt.

3.2.2. Equality test for asset and equity volatility and the interpretation of results

The averages of share and asset volatilities are, respectively, 28% and 22%. A difference in the deviation of six points can be the object of a significance test⁹.

This result justifies finding the volatility of assets using the Black– Scholes–Merton method of equity valuation.

3.2.3. Equality test for growth potential of stock prices based on the approach of brokers and Black–Scholes–Merton and the interpretation of results

The average of potential growth based on the target value of brokers and on the equity valuation of Black–Scholes–Merton are, respectively, 7.5% and 13.7%. The difference of the deviation of 6.2 points can be the object of a significance test¹⁰.

The difference between the respective approaches of brokers and Black– Scholes–Merton corresponds to the amount of net debt that is taken from the company value obtained via the DCF method. The justification of such a result resides in the fact that companies in the CAC 40 index are in a healthy financial state. Thus, their bankruptcy risk is very low, close to 0%. In this case, Φ(-d₂) = 0 which means that Φ(d₂) = 1 when d₂ = + ∞. Therefore, d₁ = + ∞ as well, which implies that Φ(d₁) = 1. Using the Black–Scholes– Merton formula as a basis: E = EV – D.e-^rτ. Since τ is relatively low, the value of equity is close to that of EV – D. The justification of the equal growth potentials can be completed by a statistical equality test of the debt ratios, which corresponds to the relationship between net debt and company value.

3.2.4. Equality test for debt ratios based on net debt from the financial states of companies and the recalculation of net debt using the Black–Scholes–Merton approach, and the interpretation of results

The average debt ratios based on the target values of brokers and on the equity valuation method by Black–Scholes–Merton are, respectively, 25% and 18.5%. The deviation of 6.5 points can be the object of a significance test¹¹.

This result allows us to confirm the relevance of using the economic debt in the real options approach.

3.2.5. Regression coefficient to explain growth potential of stock prices

Let us suppose that g is the growth potential of stock prices, RRGD is the recovery rate given default, D/EV is the net accounting debt on the company value and τ is the maturity of the financial debt. Table 3.7 below shows that

The coefficient of determination, R², is approximately 0.8, which is high, but this regression is only significant if the four coefficients are significantly different from 0. Table 3.6 below allows us to test whether the four coefficients are simultaneously equal to 0. According to these hypotheses, RRGD = D/EV = τ = 0, “F stat” follows a Fisher–Snedecor distribution F (k; n-k-1) with k = 3 and n = 28. Therefore, F → F (3;24). The Fisher–Snedecor table gives: P [F > 3.01] = 5%. In other words, if the four coefficients are simultaneously equal to 0, F has a 5% chance of being higher than 3.01. Empirically, t* = 34.62 > 3.01. Thus, with an error risk of 5%, the four coefficients are not simultaneously equal to 0.

Table 3.7 allows us to test if each coefficient a is equal to 0. If a = 0, T stat obeys a Student distribution with n-k-1 degrees of freedom. Here, k = 3 and n = 28. Thus, T → S (24 ). The Fisher–Snedecor table allows us to obtain P [-2.06 < T < 2.06] = 95%. In other words, if a coefficient is equal to 0, T has a 95% chance of finding itself in the interval [-2.06; 2.06]. Through experimentation:

t*(c) = - 5.76 < - 2.06, where c is a constant, t*(RRGD) = 4.71 > 2.06.

t*(D/EV) = -2.44 < -2.06, t*(t) = 3.51 > 2.06. Hence, with an error risk of 5%, none of the four coefficients is equal to 0.

3.3.1. Databases, methodology and hypotheses

In order to carry out statistical tests on a business sector, Heller and Levyne (2016)¹² initially selected 22 companies in the cinema industry. For each company in the sample, the market capitalization, the consensus of brokers on the value of the company (arrived at via the DCF method) and on the target value of equity were taken from the Facset financial database on April 3, 2015. Then, the volatility of shares was calculated. Each of the daily volatilities of the last two years was multiplied by in order to annualize them. To get homogeneous elements, the Cineplex, Cineworld, Europacorp and Mediaset data were converted into dollars. The exchange rates use on April 3, 2015 were the following:

• – EUR = 1.1129 USD (data converted for Europacorp and Mediaset);
• – USD = 1.2519 CAD (data converted for Cineplex);
• – GBP = 1.534 USD (data converted for Cineworld).

The lack of information on the consensus of brokers led to the exclusion of five companies initially present in the sample: BAC Majestic, Gaumont SA, IMAX Corporation, Reading International Inc and Xilam Animation SA. Thus, the empirical studies concentrate on a sample set of 17 companies.

The potential for growth from the brokers was calculated by relating the target value of equity (obtained through the DCF method) to the daily market capitalization, and then subtracting 1. The risk-free rates in continuous time¹³ were obtained by considering the OAT rates of the countries of origin of each company – rates with the same maturity as the average repayment maturity of the company debt in question.

For example, after calculations, the average maturity of debt repayment of Carmike is 9.8 years. Thus, the risk-free rate for this American company was 0.54% – a tax corresponding to the OAT rate for the USA at 10 years on April 3, 2015 (the day that the stock data was collected). It turns out that sometimes, the country that the company comes from does not have a maturity for the debt to be repaid (beyond an OAT rate) corresponding to the same average maturity on the company’s debt. Thus, the risk-free rate used was closer to this initial maturity. For example, the United States did not issue Treasury Bills at nine years. Hence, the risk-free rate used for a company like Viacom, whose average repayment maturity for the debt nominal is 9.2 years, was the one corresponding to the OAT rate for the United States at 10 years, or 0.54%.

The exercise price, moreover, is equal to the amount of accounting debt found in the reference documents of each company at the end of business 2013. To apply the real options valuation method, two other parameters had to be figured: the average residual maturity and the volatility of assets, that is, the volatility of the company value. As in the previous study, the estimation of the company value and its volatility were based on the methodology proposed by Hull et al. (2005)¹⁴. The latter relies on the resolution of a system using the Excel calculator¹⁵.

The empirical studies therefore focus on the potential for growth in stock prices of a sample set of companies in the cinema industry. It is based on the target values established by brokers using the DCF method and the values obtained via the real options approach compared, systematically, to the market capitalization on the day of the evaluation. The Black–Scholes– Merton approach considers the bankruptcy risk of companies through the integration of debt maturity and asset volatility.

The net debt should therefore not be overestimated since it relies on an economic value. Because of this, the model should demonstrate the growth potential of the economic value of equity thanks to the new division of the company value.

The difference between these two growth potentials can be justified by observing the financial leverage calculated using the accounting net debt D or the economic net debt B.

Given the preceding results regarding the CAC 40 stock index, it seemed pertinent to examine one business sector in particular that is made up of more heterogeneous companies in terms of evolution perspectives and the level of risk especially. In this way, the utility of this study is to consider the same hypotheses for the results that will be obtained compared with the preceding study, knowing the significant differences presented above. The hypotheses in question are as follows:

• – the differences of significance between the volatility of shares and that of assets should be equal, allowing us to justify the inclusion of asset volatility (instead of share volatility) in the real options valuation method in a sector where the differences are noteworthy;
• – the differences of significance between the percentage of evolution of stock prices for the companies in the sample using the DCF and the real options methods should be different within a sector where analyst forecasts can be less reliable than for companies in the CAC 40;
• – the differences of significance between the debt ratios based on the accounting net debt, on the one hand, and the economic value (obtained using the Black–Scholes–Merton method), on the other, should be equal, allowing us to justify the use of economic debt.

3.3.2. Equality test for volatility of assets and equity and interpretation of results

The averages of share and asset volatilities are, respectively, 31% and 25%. The difference of the deviation of six points can be the object of a significance test¹⁶.

This result justifies determining the volatility of assets using the equity valuation method by Black–Scholes–Merton.

3.3.3. Equality test for the growth potential of stock prices based on the approach of brokers and Black–Scholes–Merton

The average potential growth based on the target value of brokers and on the valuation of equity by Black–Scholes–Merton are, respectively, 17.5% and 3.8%. The difference in the deviation of 13.7 points can be the object of a significance test¹⁷.

The forecasts of brokers for this business sector are reliable. In other words, their forecasts, especially regarding the level of risk, are coherent. The approach using real options does not bring about a better valuation and is just as relevant as the DCF method.

The justification of the equal growth potentials can be completed by a statistical test for the equal debt ratios which corresponds to the net debt/company value.

3.3.4. Test for equal debt ratios based on net debt from the financial reports of companies and the recalculation of net debts using the Black–Scholes–Merton approach

The averages of debt ratios based on the target values of brokers and on the equity valuation method by Black–Scholes–Merton are, respectively, 21.5% and 19.5%. The deviation of two points can be the object of a significance test¹⁸.

This result allows us to confirm the relevance of using the economic net debt in the real options approach for the cinema industry.