Introduction

The way in which modern economies consume and produce does not allow giving value to the waste that is generated, and when it is disposed it causes problems of pollution and ecological impact. Waste is a residue of what cannot be or isn’t easy to use, and wasting is not exploiting something property, although technological advances and the pressure to contaminate less and reduce the ecological impact manage for high value products to be obtained from the waste. In fact, a new branch of economy arises, which is waste or circular economy (^{DEFRA, 2012}; ^{Lacy and Rutquist, 2015}).

In the case of México, in 2013 the shrimp capture in open sea was 68 % and in aquaculture, 32 % (^{CONAPESCA, 2014}). Waste is generated in the processing and consumption of shrimp. ^{De Andrade et al. (2012)} estimate that out of each ton of production, 400 kg are wasted and this organic matter is destined to municipal landfills. On the other hand, modern economies have used petroleum excessively, which is now the main source of pollution and climate change. Thus, in all modern economies a great effort is being made to move from oil-economy to bio-economy, which consists in substituting petrochemical products for products obtained from organic matter, such as biofuels, biomaterials, biopolymers, etc. (^{Brambila, 2011a}). Recent scientific studies have shown that a polymer called chitin can be obtained from shrimp shell, which is a waste; it is insoluble in most organic solvents, although through a process of deacetylation it is possible to extract chitosan. Plastic films, gums, gels, threads, and capsules, among others, can be formed from this product (^{Rinaudo, 2006}; ^{Pillai et al., 2009}; ^{Miranda and Lizárraga, 2012}). Chitin is considered to be the second most important biopolymer in the world (^{Támara et al., 2012}; ^{Pradeepa et al., 2014}; ^{Lárez, 2006}).

The price of chitosan ranges between $700 000 to $8 000 000 per ton, depending on the purity and the market (^{Dupont, 2014}). With data from 2014, shrimp is listed at $44 000 per ton in average. This price varies according to the size, the time of the year, whether it is from open sea or aquaculture, from the Pacific or Gulf region, and from states of the republic (^{INEGI, 2014}). With the waste, the shrimp producer has a high-value byproduct and with it the possibility of reducing the ecological impact. The price of the products that substitute petroleum derivatives is volatile because it depends on technological innovation, on the new uses that are found for it and the petroleum price, so that if shrimp enterprises want to take advantage of their waste they must take into account the volatility of shrimp and chitosan prices when making an investment; the latter is correlated with the petroleum price. In order to perform a financial evaluation of a project that participates in a volatile market, it is no longer enough to use traditional methodologies, such as Net Present Value (NPV), Benefit-Cost (B/C) and Internal Rate of Return (IRR), but in addition the methodology of real options must be used (^{Copeland and Antikarov, 2001}; ^{Brach, 2003}; ^{Brambila, 2011b}).

The real options valuation method considers that the management of a project makes decisions throughout its useful life to adapt to changing circumstances of the market and the technology. The management can decide at the time to broaden, reduce, abandon, continue, cease to be monovalent to become polyvalent (^{Brambila et al., 2013}), and to enter new markets, such as biotechnology (^{Álvarez et al., 2012}; ^{Vedovoto and Prior, 2015}). In the specific case of shrimp (Peneus vannamei), the management can decide to postpone, wait to know what is happening in the market, or invest in a shrimp waste processor to obtain and sell chitosan, taking into account the price volatility of both markets. It should be highlighted that the amount of chitosan that can be produced depends on the amount of shrimp obtained.

The objective of this study is to show that a shrimp company can be profitable if there is investment in a waste treatment system to obtain chitosan. In addition to being profitable for the enterprise, the ecological impact is reduced. The working hypothesis is that despite the volatility of the shrimp and petroleum prices, and therefore, of chitosan, it is more profitable to be in the shrimp-chitosan business that to be only a shrimp or chitosan producer.

Materials and Methods

The financial evaluation of a project that has a main product, shrimp in this case, and the use of waste to obtain a byproduct, chitosan, requires understanding the whole shrimp-chitosan system. The shrimp-chitosan process is different from the one just for shrimp or just for chitosan because it allows the recycling of wastes and water. Figure 1 describes the shrimp-chitosan system. The reuse of chitosan is carried out in the seventh link of its productive chain.

Figure 2 exemplifies the shrimp-chitosan system where the shrimp shell is recycled to produce chitosan and water is recycled. In each one of the production systems, organic and inorganic wastes are generated. When recycling water, the environmental impact is reduced. The reutilization of shrimp shell as biomaterial is also shown. With the more dotted thick black arrows, the shrimp-chitosan link of the system is shown, where chitosan is reused and water is recycled. In the long term this system will transform into an integral production system where each one of the wastes from the links will become reintegrated into the global productive chain. What is expected is that the production costs will be reduced with this.

The traditional financial evaluation of a project to estimate its Net Present Value (NPV) consists in estimating the benefits and costs in each period and obtaining a cash flow that is updated to the starting period when using a discount rate that was 12 % in this study. The initial investment is subtracted from the sum in the present value of this flow and the result is the NPV. If this value is positive, it is recommended to invest in the project (^{Baca, 2010}).

where: b_i: Benefit in time i. ci: Cost in time i. I: Initial investment. ∂: Discount rate.

The data to calculate the NPV for shrimp were obtained from Trusts Funds for Rural Development (Fideicomisos Instituidos en Relación con la Agricultura, ^{FIRA, 2009}) for a ship of 16 ton/trip, with an average of six trips per season. The real prices were calculated with data from the National Statistics and Geography Institute (Instituto Nacional de Estadística y Geografía, ^{INEGI, 2014}), ^{FIRA (2009)}, and the National Chamber for Aquaculture and Fishing (Cámara Nacional de Acuacultura y Pesca, ^{CONAPESCA, 2009}). The fixed costs were estimated at $639 135 and the variable ones at $147 276 per season; the initial investment was considered to be $1 200o000. The breakdown and detail of the costs can be reviewed in FIRA’s informative bulletin (2009).

The data to carry out the NPV calculation for chitosan were obtained from ^{Cerón (2010)}. An amount of 24 ton was considered and the real prices were obtained from ^{INEGI (2014)} and ^{Shirai (2011)}. The fixed costs were estimated to be $4 822 683 and the variable ones $1 222 683; the initial investment that was estimated was $2 339 958. The breakdown and detail can be reviewed in ^{Cerón (2010)}. The useful life of the projects was considered to be 10 years. The residual values are in the last year and are included to calculate the corresponding Net Present Values.

The statistical series of the 2000-2011 period of nominal shrimp prices from open sea was taken from the Master Plan for open sea shrimp from Sinaloa (^{CONAPESCA, 2009}). The nominal chitosan prices for the 2000-2011 period were calculated with information from ^{Shirai (2011)} and discounted with the National Consumer Price Index (NCPI) annual accumulated base 2Q Dec 2010, published by the Bank of Mexico (Banco de México, ^{BANXICO, 2014}).

To obtain the real prices for shrimp and chitosan, the National Producer Price Index (NPPI) base August 2014 published by INEGI was used. The formula used to obtain the prices is:

where: PR: Real Price. PN: Nominal Price. INPP: National Producer Price Index.

Given that the shrimp and chitosan prices are very volatile, the financial evaluation of the shrimp-chitosan project must be complemented with the real options methodology. To apply the real options valuation method, it is required to estimate first the volatility of prices. This is done when calculating the continuous growth rate (in natural logarithms) of the prices from one year to another.

where r_t: continuous rate of real price growth. ln: natural logarithm. P_t: real price in the year t. P_t+1: real price in the year t+1.

The volatility is measured with the standard deviation of the continuous rates of real prices.

where x-: average of continuous rates of change. σ: standard deviation of the continuous rates of real prices. t: total from the periods.

With the Net Present Value and the standard deviation of the continuous change rates of real prices, the binomial tree of shrimp is formed.

In the case of the shrimp-chitosan system, the standard deviation was estimated with the Markowitz criteria for an investment portfolio (^{Ross et al., 2005}).

where Γ² : volatility of the shrimp and chitosan portfolio or variance of real prices of shrimp and chitosan. σ₁²: Volatility of real prices of shrimp or variance of real prices of shrimp. δ₁₂: covariance between real prices of shrimp and chitosan. X₁ : proportion of the total investment destined to the shrimp project. σ₂²: volatility of the chitosan prices or variance of the real prices of chitosan. X₂: proportion of the total investment destined to the chitosan project. The proportions of the amounts of shrimp and chitosan were calculated through the following mathematical relations. X ₁ : initial investment to produce shrimp/ Total investment (shrimp and chitosan). X ₂ : initial investment to produce chitosan / Total investment (shrimp and chitosan).

For the calculation of binomial trees with real options, the methodology used by ^{Copeland and Antikarov (2001)}, ^{Brach (2003)} and ^{Brambila (2011b)} was used, who point out that the real options are the right, although not the obligation of exercising an action during the project’s life. In our case, the real option that a shrimp enterprise has is to invest in a chitosan project and form a shrimp-chitosan system with water recycling and the raw material already mentioned. This is a right, but not an obligation. With the standard deviation (σ) the scenarios of the enterprise are formed when the prices are rising (up=e^σ) and when the prices are decreasing (down=e^-σ), where u is what increases the project’s value from a rise in prices, d is what lowers the value of the project from a decrease in prices, and e is the Euler number.

Since there is risk, the project’s value can increase or decrease and the probability of this happening is measured (^{Brach, 2003}; ^{Brambila et al., 2013}):

where p: probability of an increase in the project’s value. 1-p: probability of a decrease in the project’s value. : risk free interest rate (average of real cetes).

A binomial tree of the value of the project is formed when increasing (u) or decreasing (d) in time (Figure 3).

Once all the nodes and a horizon for shrimp production are found, a binomial tree is formed as the real option of integrating the shrimp-chitosan system starting from the second year. For this purpose, the values in the nodes were calculated backwards until reaching the value of the real option of producing shrimp-chitosan (^{Brach, 2003}; ^{Brambila et al., 2013}).

where V_QC: Net Present Value in the current period. p: Probability of an increase in the project’s value. 1-p: probability of a decrease in the project’s value. l : risk free interest rate (average of real cetes). v_u: net present value of an increase in the project’s value from a previous period. v_d: net present value of a decrease in the project’s value from a previous period.

^{Mascareñas et al. (2004)} and ^{Brambila (2013)} indicate that the total net present value of the project will be equal to the traditional net present value plus the value of the real option of being able to opt for the production of chitosan extracted from shrimp shell.

where VAN_TOTAL: total net present value. VAN: net present value or traditional net present value. OR: real option.

The distribution of project values in the last period was estimated, using the formula of binomial probabilities (^{Copeland and Antikarov, 2001}; ^{Brambila et al., 2013}).

where B: probability of being in node n in the moment t. n: number of nodes in the period t. t: period evaluated=1, 2…, 10. p: probability of an increase in the project’s value.1-p: probability of a decrease in the project’s value.

The distribution of values from each period approaches a normal distribution, so the probability of the project having a predetermined value could be estimated, standardizing the values and using the Z tables of a normal distribution.

where: x_i: predetermined value of the project. x: mean value of the project in the period t. σ: standard deviation of the project’s value in the period t.

The NPVs (traditional) are calculated for the case of shrimp and chitosan as separate projects and only the binomial tree of shrimp production was constructed. Then, it is considered that the shrimp enterprise exerts its right to integrate the shrimp-chitosan system in the second year and its binomial tree was built.

Results and Discussion

The NPV (traditional methodology) of the project for producing just shrimp was $ 182 000, so the project should be accepted. The discount rate in this project is 12 %; it takes into account the bank rate and the project’s risk (Table 1 and 2). For the chitosan project, the NPV was $ 1 727 000, so the project should be accepted.

Now, if a shrimp-chitosan system is used with the corresponding recycling and with a real option in the second year of investing in a waste processor to obtain chitosan, the NVP_TOTAL is $ 1 790 000 (Table 3). This result is confirmed in the real options literature for other types of projects (^{Fenichel et al., 2008}; ^{Támara et al., 2012}; ^{Vedovoto et al., 2015}). Furthermore, the case may be that the NPV will be negative and if there are real options the Total Net Present Value can become positive (^{Hannevik et al., 2015}).

The probabilities in each case for the project (shrimp and shrimp-chitosan) to be in the year 10 above or below its initial NPV are contrasted in Table 3. Likewise, the earning values per type of project are reflected. The volatility of the prices of the shrimp project (σ=0.0861) and the volatility of the prices of the shrimp-chitosan project (σ=0.454) are shown simultaneously in the probabilities used for the calculation of each one of the scenarios (binomial tree).

The value of the real option of the decision to invest in the second year was $1 608 000 (Table 4). Investing in a shrimp-chitosan system is economically profitable, technically viable, and the waste and ecological impact are reduced.

Conclusions

Chitosan production from organic shrimp waste is economically profitable in the long term (OR _2YEAR=$1 608 000). In the shrimp productive chain, wastes are an alternative for the generation of earnings additional to the ones obtained with the production. The expectation of products from bioeconomy, such as biomaterials, biofuels, bioplastics, nutraceuticals, and functional foods, among others, obtained from second generation raw material or from solid residues becomes an attractive option for income generation. Therefore, the use of shrimp shell for the extraction of a polymer of high value in the chemical-industrial market becomes a profitable option in shrimp production. It is concluded that chitosan production from shrimp shell wastes is economically profitable and of less ecological impact (less waste); this, despite the volatility of the shrimp and petroleum prices.