Data

In order to value real investment projects, one must have the primary information that helps quantify the different variables of the valuation model. This study used the following as information sources: the Organization of Producers of Deep Sea Fishing of Cantabria (Organización de Productores de Pesca de Cantabria - OPECA); the fish auction hall of Santander (lonja de Santander), and the Bank of Spain (Banco de España). In this way, based on real information, it was possible to simulate a fictitious investment by a shipowner.

The particular investment up for valuation consisted in the purchase, in 2007, of four gillnet fishing vessels by a shipowner, with fishing licenses, for the capture of hake between 2008 and 2014. The initial payment was of 4 278 818 € (1 069 704.5 € per vessel).

The calculation of the variable costs of the use of an average gillnet fishing vessel of the port of Santander included independent professional services, transportation, premium and insurance, supplies (diesel, oil, phone, water, electricity), other services, other fees, wages and salaries, social security by the company, other financial expenses, purchase of goods (ice, groceries, nets, boxes, naval effects) and expenses in RDI. According to data from OPECA, the total operation costs amounted to 890 323.31 € in 2007, with the final operation cost for every kilogram of hake fished being 3.28 € kg^-1. The average annual growth rate of the operation costs in the last five years was 2.64 %. Based on this growth rate, operation costs were calculated for the rest of the practices of the period analyzed.

Regarding the price of the fishery resource, the average price in 2007 of the auctions carried out in the fish auction hall of Santander was taken as a starting point; it amounted 4.36 € kg^-1 and according to the option valuation model hypotheses, it varied in the later years, following a Brownian geometric process^[2].

The stock of the fishery resource was also calculated, taking as a reference the hake catches auctioned in the fish auction hall of Santander, considering as the amount of catch as an average for the period analyzed the equivalent to the average value observed during the 7 years before the exploitation of the project analyzed. Considering that the investment presented of the acquisition of four vessels would have, depending on its fishing rights and its capacity, the possibility of catching 60 % of the total, a value was established for the amount of resource captured of 359 012.4 kg.

The risk-free rate was considered exogenous with a value of 3.50 %, which corresponds to the average interest rate of the five-year bonuses issued by the Spanish government during the five years before the decision of investment analyzed.

Methods

The investment to be analyzed was carried out under conditions of uncertainty regarding the price of the fishery resource. In each one of the j periods of exploitation of the resource, the shipowner made a similar contingent decision: to decide whether to extract the resource or not at the beginning of the j periods of exploitation, based on the changing economic and biological conditions, which influence on the price of the resource. The shipowner had, in each one of the j periods that the concession lasted, the right, but not the obligation, to extract the resource. This right has a value, known in financial literature as the option to call ^[3].

The analysis included the formula by Black and Scholes (^{Black and Scholes, 1973} and ^{Merton, 1973}) (1). For each j period of exploitation, we obtained the value of the corresponding call, which takes into account the cash flow for each period, and also the option of temporarily abandoning the exploitation of the resource:

where N(.) is the function of distribution of probabilities accumulated from a normal variable with an average of zero and variance of one, which is key in this valuation; T is the time until expiry (7 years, with December 31^st, 2007 as the date in which the valuation of future production was carried out); S is the price of the underlying asset (price of the fishery resource); E is the strike price on the date of expiry of the option (variable exploitation costs); r is the risk-free rate; s is the volatility of the price of the underlying asset, measured as a standard deviation; j is each of the moments under analysis; and e is the exponential.

The calculation of the volatility of the underlying asset used the formula of historical volatility shown below (4), obtaining a value of 8.31 %. Given a sample of "m+1"^[4] consecutive annual prices for hake (S _j ), annual volatility is (^{Hull, 2002}, p. 269):

The price in cash for the resource extracted, or output, S, was given by the following geometric Brownian process (5), as suggested by ^{Hull (2009})^[5]:

where μ: expected value of the change in price of the resource; σ: volatility of the unexpected change in the price of the resource; dz: differential of a Gauss-Wiener or a Brownian process. This process is characterized by the following relation:

The value in the current moment of the exploitation was obtained as the sum of the j call options identified for the total of the j periods of fishing (7):

The difference between each of the real options described lies in their expiry dates (j=1, 2, 3,..., T), moment 1 being the year 2008, and moment T, the year 2014. The value of the exploitation per unit of output was calculated using equation (8) and the global value of the exploitation, using equation (9):

h_j being the amount resource extracted.

The value of the exploitation calculated in this way (per unit and in global terms) included both the basic value and the value of the option of temporary abandonment of the activity, which would resume when the conditions of the price of hake become more favorable.

At first, the convenience rate was negligible to value the decision of investing in the scenario presented, i.e., there were no expectations regarding scarcity of the resource. Later, a sensitivity analysis was performed on the model to contrast the stability of the conclusions obtained, as well as the introduction of the convenience rate, which made it necessary to redefine the equation (1).

The convenience rate is a parameter that reflects the expectations in the market on the future availability of the resource^[6] (^{Murillas, 2004}). When the resource stock is high, the convenience rate tends to be low. Likewise, the rate will increase with the expectations of scarcity.

The calculation of the values of the options, incorporating the effect of the convenience rate (k) made it necessary to redefine the expression (1) to the following:

where N(.) is the function of accumulated distribution of probabilities of a normal variable with an average of zero and a variance of one; T is the time to expiry; S is the price of the underlying asset (price of the resource); E is the strike price on the date of expiry of the option (variable exploitation costs); k is the convenience rate; r is the risk-free rate; s is the volatility of the price of the underlying asset; j is each of the moments under analysis, and e is the exponential.

The value of the option and the traditional methods

The comparison of the value of the exploitation obtained with real options with the one obtained by traditional cash flow discount methods showed that the former was always above the latter, increasing the difference between both values as the expiry date increased (Table 1). This difference is explained by the value of the option of temporary abandonment included in the valuation by real options and ignored by the traditional valuation methods. That is to say that the highest value obtained for the exploitation by real options does not mean that the desire to exploit the fishery increased, but rather that the increase in the value of the opportunity of exploiting the resource was due to the possibility of abandoning it temporarily.

The total cash flows produced by the project by the traditional method, not including the option of temporary abandonment, was 3 329 210.4 € (Table 1), considering the value per unit (€ kg^-1 of hake) of the discounted cash flows obtained for each of the j exploitation periods, as well as the catch of hake in this period. The NPV of the investment project in this case, considering the payment made by the shipowner in 2007 was:

Therefore, in conformity with the traditional evaluation of the project, with a negative NPV the shipowner would not have invested in the purchase of the four vessels, whereas applying the Theory of real options, the shipowner would have undertaken such an investment.

The total value was estimated for the option of temporarily abandoning the hake fishing activity given by the difference between the extended net present value and the basic net present value of the investment project. Thus, the additional value given to the investments by the valuation by real options could be calculated, and therefore, the added value of flexibility (Table 2):

The results showed that the option of temporarily abandoning hake fishing activities meant an additional value of 42.4 % of the cash flows generated by its exploitation. This value is extremely important, since its disregard could give way to the rejection of investment feasible projects if flexibility in its management is considered, and particularly the temporary suspension of activities. Analyzing each period, it was possible to notice that as the temporary horizon of the exploitation increased, the percentage representing the option of temporarily abandoning operations also increase on the current value of the cash flows, since the uncertainty regarding the price of the fishery resource was higher (Table 2).

The comparison of the values of theoretical models with the empirical value observed, in other words, the value reached in the market every year of the period, showed that the value obtained with the methodology of real options was readjusted with greater precision to the empirical value observed in the market between 2008 and 2014 than the discounted cash flow method, which underestimated the value of the exploitation, since it did not consider the value of the option of temporary abandonment. Although the methodology of options did present an overestimation of the real value of the opportunity to exploit the resource, the traditional methods did underestimate it further (Figure 1).

Sensitivity analysis

The model of valuation by real options shown has two strengths. The model does not need estimations of the probability of the possible future prices of the resource, since these are included in the estimation of volatility and in the current price of the resource. Neither does the model need to adjust the type of discount for risk, because do to the fact that valuation is independent to the preferences, one only needs to know the risk-free interest rate (^{Varela, 2003}).

However, given the importance of both variables in the model of valuation of real options, a sensitivity analysis was performed on the results obtained in the light of changes in volatility and the type of risk-free interest.

Higher volatility increases the valuation obtained with options (^{Brealy, et al., 2006}). The explanation for this type of behavior is based on the traditional profitability-risk binomial, according to which both variables move in the same direction for an asset. Also, the more volatile the price of the asset, the higher the probability of important movements taking place in it which favor the value of the option.

In the particular case analyzed, a higher price volatility for hake led to a higher value for exploitation between 2008 and 2014, or in other words, a greater uncertainty related to the price of the fishery resource did not mean a reduction but a generation of value (Table 3). Similar results were obtained by ^{Alonso et al. (2005)} for forestry exploitations.

Based on a scenario with a volatility of 8.3 % we observed how a variation in 15 % in volatility meant a variation of 6.3 % in the final valuation of the fishery exploitation. A variation of 30 % in volatility increased the final valuation by 18.4 %, showing a reduced sensitivity of the valuation model applied to changes in the volatility of prices.

Regarding the risk-free interest rate, the theory of valuation of options foresees a positive relation between this variable and the value ofthe option, since a rise in the interest rate exerts a negative influence on the current strike price, having a positive effect on the value of the option to purchase (^{Fernández and García, 1992}). The sensitivity analysis showed that the higher the value of the risk-free interest rate, the higher the value of the opportunity of exploitation, and therefore increases in the latter were related to increases in the expected profitability, final cause for the generation of greater value for fish exploitation (Table 4). This result is common in literature on options (^{Murillas, 2004}). In particular, based on an average scenario in which the risk-free interest rate was 3.5 %, we could see that the 1.5 % variation in the rise of the interest rate meant an increase in the value of the exploitation of 7.8 %, whereas an equivalent reduction in the interest rate reduced the value of the exploitation by 8.2 %.

Introduction of the convenience yield

The introduction of convenience yields conditioned de valuation of the opportunity of exploitation of the fishery. By rising the convenience yield, the benefit related to the property of the resource stock increased, and the value ofthe operative option decreased^[7] (Table 5). Parameter k can also be understood as a cost of the opportunity to delay the exploitation keeping the option of exploitation (the greater k, and therefore the opportunity cost is, the value of exploitation will be lower). This result is also observed in another type of exploitations (^{Bjerksund and Ekern, 1990}; ^{Murillas, 2004}; ^{Alonso et al., 2005}).

In sum, the sensitivity analysis showed that uncertainty creates values in projects that, like fish exploitation, have flexibility. In this way, we observed that the value of the option increases (decreases) if the volatility and/or interest rate increase (decrease) and the convenience yield decreases (increases).