Bayesian approach and extreme value theory in economic analysis of forestry projects

Bayesian approach and extreme value theory in economic analysis of forestry projects

Forest Policy and Economics 105 (2019) 64–71 Contents lists available at ScienceDirect Forest Policy and Economics journal homepage:

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Forest Policy and Economics 105 (2019) 64–71

Contents lists available at ScienceDirect

Forest Policy and Economics journal homepage:

Bayesian approach and extreme value theory in economic analysis of forestry projects


Thiago Taglialegna Sallesa, , Denismar Alves Nogueiraa, Luiz Alberto Beijoa, Liniker Fernandes da Silvab ⁎


Instituto de Ciências Exatas - ICEx, Universidade Federal de Alfenas, Rua Gabriel Monteiro da Silva, 700, Alfenas, MG CEP 37130000, Brazil Centro de Ciências Agrárias, Ambientais e Biológicas - CCAAB, Universidade Federal do Recôncavo da Bahia, Rua Rui Barbosa, 710, Cruz das Almas, BA CEP 44380000, Brazil




Keywords: Growth modelling Forest production Net present value Gumbel distribution

Reliable estimates of price and wood yield as well as the calculation of economic criteria that include uncertainty are necessary to make the decision-making process more robust when analysing a long-term activity such as forestry. Through extreme value theory EVT combined with Bayesian inference it is possible to predict probability densities for inputs used in economic evaluation criteria like wood yield and prices. With it, uncertainties regarding the inputs can be taken into account in the economic analysis, improving the way they are obtained. Therefore, this study aimed to use Bayesian approach and EVT to estimate future price and yield to carry out an economic evaluation of a forestry project. Yield, maximum and minimum price were estimated in the form of probability density. Considering 95% of probability, the NPV calculated for the minimum price situation ranged from R$ 2050.11 to R$ 5409.07 ha−1, with a mean of R$ 3771.60 ha−1. The NPV calculated for the maximum price situation ranged from R$ 7766.77 to R$ 9070.29 ha−1, with a mean of R$ 8398.13 ha−1. These results serve as best and worst-case scenarios for managers in the case of a eucalyptus plantation established in 2017 under an outgrower scheme in Brazil. The presented methodology provided good results when estimating the variables of interest. It incorporates probability levels and/or prior information. With it, the economic performance of the project and its risks are better visualized and understood by researchers and managers.

1. Introduction Forest management combines silvicultural practices and business concepts to best achieve an investor's objectives (Bettinger et al., 2009). The allocation of scarce resources is an inherent concept in forest planning and management, and as a result, the strongest arguments in decision-making involve economic analyses (Pirard and Irland, 2007; Amacher et al., 2009; Wagner, 2012). These analyses are used to evaluate possible alternatives to achieve the objectives in order to guide the choice between them (Zhang and Pearse, 2011). In recent years, economic assessments have been used to evaluate the management of forest landscapes for economic and environmental benefits (Mönkkönen et al., 2014; Ninan and Inoue, 2013; Lutz and Howarth, 2014; Monge et al., 2016), measure the returns of carbon sequestration and climate change mitigation by forests (Guitart and Rodriguez, 2010; Köthke and Dieter, 2010; Nijnik et al., 2013; Polglase et al., 2013; Pyörälä et al., 2014) and also evaluate the feasibility of investing in forestry projects (Ying et al., 2010; Stille et al., 2011;

Corato et al., 2013; Akhtari et al., 2014; Frankó et al., 2016). Economic criteria have been used to evaluate investments in evenaged forests in Brazil since the 1990s, especially in the case of eucalyptus plantations. (Fearnside, 1995; Rodriguez et al., 1997; da Silva et al., 1999; Soares et al., 2003; Diaz-Balteiro and Rodriguez, 2006; Vitale and Miranda, 2010; de Souza et al., 2015; Pereira et al., 2018; Simioni et al., 2018; Simões et al., 2018). Criteria that consider the variation of capital over time, most notably the net present value (NPV), are used to carry out this type of study (Knoke et al., 2008; Hauk et al., 2014). When conducting such analysis, it is important to collect all the relevant cost, revenue, growth, and yield information (Bettinger et al., 2009). Wood yield for economic assessments is usually defined based on models fitted to tree growth data coming from the region of interest (e.g. Oliveira et al., 2008; da Silva et al., 2008; South et al., 2010; Oliveira Neto et al., 2013). The fit of growth and yield models for a particular forest species is usually done with data restricted to a local condition and without repetition in time, as in Monte et al. (2009), Hess

Corresponding author. E-mail address: [email protected] (T.T. Salles). Received 6 November 2018; Received in revised form 11 April 2019; Accepted 15 May 2019 Available online 20 May 2019 1389-9341/ © 2019 Elsevier B.V. All rights reserved.

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and Schneider (2010), West et al. (2016), Shater et al. (2011), Ma and Lei (2015) and González-García et al. (2015). There are also studies that use an external reference of wood yield, as data coming from regions similar to the region of interest or models fitted to similar situations to evaluate a project (e.g. Castro et al., 2007; Langholtz et al., 2007; Styles et al., 2008; Berger et al., 2011). In these cases, taking growth information limited to a particular situation as a reference for productivity brings insecurities, since there is little guarantee that the conditions that generated this information will repeat themselves. Still regarding the calculation of economic evaluation criteria, one must also include the sale price of the products, which is usually the price found during the project formulation (e.g. Manzone et al., 2009; Avohou et al., 2011). However, in practice, sales will occur several years after the beginning of the project (time taken for trees to reach the cutting age) and the final price may or may not be the same as previously set. There are two economic extremes in a project that interest managers in this context: a best possible scenario, where the sale price of products is very high, and a worst possible scenario, where the sale price of products is very low. This is because, as tree plantations are long-term projects, the timber market situation during its execution can range from an excellent point, just like before the financial crisis of 2008, to a bad moment, as in the economic crisis that hits Brazil since 2015 (Zeidan and Rodrigues, 2013; Salles et al., 2016; Wade, 2016). Forestry is a good economic alternative for rural producers in Brazil (Faruqi et al., 2018). The problem is that the variability regarding growth models and timber price fluctuation mentioned above disturb the calculation of economic criteria (Holopainen et al., 2010). Pereiro (2006) also states that point estimates of criteria such as the NPV and the internal rate of return (IRR) are not able to account for the uncertainties and flexibility involved in investment opportunities. Reliable estimates of price and wood yield as well as the calculation of economic criteria that include uncertainty are necessary to make the decisionmaking process more robust when analysing a long-term activity such as forestry. The Bayesian methodology may be a good alternative to obtain better estimates of wood growth and yield. In it, each parameter of a model is a random variable with a probability distribution conditioned to prior information coming from situations similar to the one of interest (Box and Tiao, 1973; Hoff, 2009). This allows preliminary knowledge about the behaviour of the growth curve to be incorporated into the fit process. The prior probability distribution is then combined with the information about the observed sample (likelihood function) producing what is called a posterior distribution for the parameters (Ghosh et al., 2006). The extreme value theory (EVT) can be useful to forecast scenarios of extreme price. It is a branch of statistics and probability that studies the stochastic behaviour of extreme events and (Coles, 2001). Longin (2016) comments that in the last decades we have obtained a better understanding of the statistical behaviour of financial extreme flows through EVT. The author also states that the understanding of market behaviour during extreme events is also useful for understanding the market as a whole, under both ordinary and extraordinary conditions. Through EVT combined with Bayesian inference it is possible to predict probability densities for future wood yield and for minimum and maximum prices. Consequently, the product between these two variables would result in a group of probable revenues that allow economic evaluation criteria to also be associated with probability densities. As a result, the uncertainties regarding the inputs of the economic evaluation criteria would be taken into account in the economic analysis, improving the way they are obtained. Therefore, seeking a better way for a manager to calculate in advance the best and worst financial return an investment in forestry can bring, this study aimed to carry out an economic evaluation of a forestry project using Bayesian approach and EVT. Specifically, we sought to present a methodology to obtain prior information and fit a growth and yield model using Bayesian approach; to predict the probability densities of

extreme prices for standing timber through EVT also using Bayesian approach; and, therefore, to better quantify the revenues for the evaluation of a forestry project, as well as to interpret its results. 2. Material and methods Taking into account the interest of investing in forestry in Brazil, the starting point of this study was the definition of a eucalyptus plantation project that could be implemented by small farmers in the state of São Paulo. The choice was due to the fact that this is the most planted forest species in Brazil and, consequently, the one for which more data is available. The economic analysis then aims to encompass the financial performance of the project if executed, as well as its risks. A forestry partnership program, also known as outgrower scheme, was considered for the project. It is an arrangement between farmers and the forest industry where smallholders are responsible for the wood production, with the company guaranteeing its purchase. The smallscale growers have secure sale contracts (market access) and receive technical and financial support, while the company gain access to land or forests suited to commercial forestry and secure future timber supplies (Race et al., 2009). The production of wood would be for industrial processing, since in the state of São Paulo these programs are oriented to obtain raw material for the pulp and paper industry (Eisfeld et al., 2017). In accordance with the common silvicultural practices when planting wood for industrial processing in Brazil, a planning horizon of 6 years was established, considering a spacing of 3.0 × 3.0 m and sale of all standing timber to the company at the end of year 6. The currency used in this study was the Real (official currency in Brazil). This choice was made because the proposed project follows regional characteristics of the eucalyptus wood market, thus not making sense to adopt a foreign currency (da Silva et al., 2007). Besides that, it would be necessary to model the exchange rate for the conversion to another currency such as the dollar or the euro, adding another source of uncertainty in the estimates without benefiting the study results. Assuming a more vertical integration in the partnership program, where the producer is not responsible for the costs of seedlings and harvest, the total investment calculated for the project was R$ 3103.77 ha−1. It was distributed as follows from year 0 to year 6: the initial cost in year zero was R$ 1418.40 ha−1, to be spent on fertilizers, herbicide and formicide (56.22%), labour (5.47%) and machinery/fuel (38.30%); the maintenance costs were equal to R$ 836.82 ha−1 in year one, to be spent on fertilizers, herbicide and formicide (77.53%), labour (5.48%), machinery/fuel (16.05%) and taxes (0.94%); R$ 546.47 ha−1 in year two, to be spent on fertilizers, herbicide and formicide (73.61%), labour (7.87%), machinery/fuel (17.08%) and taxes (1.45%); and R$ 75.52 ha−1 from year three onwards, to be spent on formicide (42.85%), labour (46.69%) and taxes (10.46%). At year 6, the partnership company provides workforce for a clearcut in the property. These costs were an adaptation of the values detailed in Janoselli et al. (2016), adjusted by the IPCA (national consumer price index) for January 2018 (12.83%). Since the spacing in the referred study was 3.0 × 2.5 m, the initial cost (year 0) was decreased by 6.7%, based on the differences seen in the spacing study by Paulino (2012). The opportunity cost of the land was not considered. A discount rate of 9.0% was adopted, since the rates charged in the financing lines of the Brazilian Forest Financing Guide (SFB, 2016) ranged between 8.0 and 9.5%. The net present value was used for the economic evaluation of the project: NPV = ∑j=0kRj(1 + i)−j − ∑j=0kCj(1 + i)−j, where Rj is the total revenue in the year j, Cj is the total cost in the year j, i is the annual discount rate and k is the duration of the project in years. Since the economic performance of a project will occur between the extremes where the price of the wood may be the best or worst possible for the period of sale, the revenues (Rj) were calculated for two situations. In the first situation, it was considered that the wood would be 65

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sold at the maximum price expected for the final year of the project. For the second one, the sale was made considering the minimum price expected for the same period. Both yield and price were estimated in the form of probability density using Bayesian approach. In Bayesian inference, numerical values from the characteristics of a population Y are typically expressed as a parameter θ and the values of a sample from Y constitute a dataset y = {y1, y2, …, yn} (Hoff, 2009). We give the name of prior information to the belief about an amount θ that is unknown before the data y is available. For each numerical value θ, the prior distribution f(θ) describes the belief that θ represents the real characteristics of the population. The parameters inherent to prior distributions, are called hyperparameters, so as to distinguish them from θ. For each θ and y, the distribution f(y|θ) describes the belief that y is the result of interest given that θ is known. So, the distribution given by the product of the prior distribution by the likelihood results in the so-called posterior distribution f(θ|y). It is a quantification of the uncertainty about θ in light of the data. The transition from f(θ) to f(θ|y) is what has been learned from the data y (Ghosh et al., 2006). As a consequence, the revenues (product of price and yield densities) were also obtained in the form of probability density, allowing an HPD95% (highest posterior density) region to be constructed for the NPV. An HPD region of 100(1-α)% is the one where p(θ ∈ HPD) ≥ 1-α, with the condition that all points within the HPD region will have a higher density than those outside (Hoff, 2009).

To obtain the prior distributions of β0 and β1, the model (eq. 1) was fitted via non-linear least squares to the prior information data, providing 14 estimates of each parameter. The hyperparameters average (μ0) and precision (τ02) of their prior distributions were then calculated from these estimates. Since there was no prior information about τ2, its prior distribution was non-informative and its hyperparameters were equal to 0.001 as suggested in Ghosh et al. (2000). More about informative and non-informative prior distributions can be found in (Link and Barker, 2009). The model was fitted by Bayesian approach using Monte Carlo method via Markov Chains (MCMC), as seen in Sorensen and Gianola (2002). This process was carried out in the R software (Core Team, 2016), with coda and R2OpenBUGS packages, integrated with the OpenBUGS software (Thomas, 2017). The results are the posterior probability densities of each parameter. The parameters estimates were the central point value (average) of these densities. The convergence diagnostic of Geweke, Heidelberger and Welch and Raftery and Lewis were used to evaluate the independence and absence of convergence in the results of the MCMC algorithms. They indicate whether the defined number of iterations, thin and burn-in were adequate so there is no evidence of non-convergence of the simulated chains to the posterior densities. For the Geweke criterion, it is considered that there is no evidence of non-convergence when the absolute value of its results is < 1.96. Likewise, the results of the Raftery-Lewis criterion should be close to 1.00 and the p values of Heidelberg-Welch criterion should be > 0.05. Further details on these criteria can be found in Cowles and Carlin (1996). Lastly, the goodness of fit was verified through R2B, a measure similar to the determination coefficient, and the root mean square error. The R2B is calculated by R2B = 1–1/( 2 y2) , where 2 is the precision, being equal to the inverse of the residual variance and y2 is the sample variance, and the root mean square error, calculated by RMSE =

2.1. Estimation of wood yield The data of wood yield obtained by Alvares (2011) (Fig. 1A) were defined as the sample for wood yield in the region of interest. The authors calibrated the physiological process model 3-PG to growth data of Eucalyptus grandis x Eucalyptus urophylla planted in the south region of the State of São Paulo, in spacing 3.0 × 3.0 m. We used five sets of volume data observed in Silva (2005) and nine sets of average volume data estimated by Faria (2007), all of them planted in the State of São Paulo, as the prior information (Fig. 1B). The Schumacher model was used to obtain estimates of wood growth and yield for the project. It is a particular case of the Lundqvist function with the shape parameter equal to 1:

yV = exp (





(yi –yi ) 2 . The probability density of the average wood yield at year 6 was then estimated using the fitted model. 1 n

2.2. Price estimate The term extreme value is defined as the maximum (or minimum) of a data series in a given period (maximum or minimum blocks) or those values that exceed a sufficiently high threshold (POT – Peak over threshold). We worked with the first approach, and the price corresponded to the monthly values found by CEPEA-Esalq (2017) and IEA (2017) for the sale of wood (standing timber) for industrial processing in the State of São Paulo. The monthly data covered from January 2006 to September 2017, totalling 133 observations, which were aggregated into a series of 12 annual minimum and maximum blocks (Fig. 2). This period corresponded to the longest data series available at the time this study was done. The extreme value theory (EVT) (Fisher and Tippett, 1928; Jenkinson, 1955) was used to model the series of maximum and minimum prices. The generalized extreme value distribution (GEV) and the Gumbel distribution were compared through the likelihood ratio


where yV is the volume in m3 ha−1, x is the age in years, βi are the model parameters; and ε is the random error, being ε ~ N(0, σ2). In Bayesian approach, the deviation of the values of a variable from its mean is usually treated in the form of precision (τ2), which corresponds to the inverse of the variance (τ2 = 1/σ2). Thus, the distribution of the sample data was yV ∣ θ~N(exp(β0 – β1xi−1), τ2), where θ = [β0, β1]. The prior distributions of β0 and β1 were N(μ0, τ02) with specific mean and precision for each parameter. For τ2 it was G(a0, b0) since the precision only assumes positive values.

Fig. 1. Wood yield from eucalyptus plantations in the State of São Paulo, used as sample (A) and prior information (B) to fit the Schumacher model.

Fig. 2. Series of sale prices of wood for industrial processing in the State of São Paulo. 66

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test in a preliminary analysis. It was defined that the best option to model the data would be the Gumbel distribution, the case of GEV distribution where ξ → 0. For a series of maximum values, the cumulative distribution function and probability density function are, respectively:

F (x ) = exp –exp –


and f (x ) =


exp –


–exp –

The mean of the posterior density of wood yield at 6 years (Fig. 3A) was 331.81 m3 ha−1, with HPD95% ranging from 321.30 m3 ha−1 to 342.30 m3 ha−1. According to the values obtained by the Geweke, Raftery-Lewis and Heidelberg-Welch convergence diagnostics (Table 1), there was no evidence of non-convergence of the chains estimated by the MCMC method with a number of iterations equal to 12,000, thin equal to 4 and burn-in equal to 5000. Regarding the price estimates, the Durbin-Watson test results indicated that the minimum and maximum series were independent (p = 0.2407 and 0.1076). The Mann-Kendall test resulted in p = 0.0467 for the minimum series and p = 0.1479 for the maximum series. Therefore, the distribution for minimum price was fitted with tendency starting from the first observation. The values observed in the last two periods of the series (corresponding to the return periods 2 and 3) were R$ 39.52 and R$ 39.27 for minimum and R$ 54.00 and R$ 54.00 for maximum. The Gumbel distribution, fitted for 9 observations of minimums, estimated R$ 44.53 and R$ 42.87 for the return periods 2 and 3, with HPD95% ranging from R$ 38.73 to R$ 49.66 and from R$ 36.27 to R$ 49.11. For maximum, the distribution estimated R$ 54.83 and R$ 56.19 for the return periods 2 and 3, with HPD95% ranging from R$ 52.81 to R$ 56.84 and from R$ 53.92 to R$ 58.76. Thus, as the observed values were contained in their respective HPD estimates, the model was considered valid and the estimation of its parameters was updated using all 12 observations (Table 2). The distributions were also fitted with a number of iterations equal to 12,000, thin equal to 4, burn-in equal to 5000 and, according to the values obtained by the Geweke, Raftery-Lewis and Heidelberg-Welch convergence diagnostics (Table 2), there was no evidence of non-convergence of the estimated chains for both. According to the Kolmogorov-Smirnov test (p = 0.6853 and 0.9482), the distributions with the parameters of Table 2 were well fitted to the minimum and maximum prices series. The mean of the posterior density for the minimum price estimated for the return period of 6 years was R$ 34.00, with HPD95% ranging from R$ 25.26 to R$ 42.08 (Fig. 3B). For the posterior density of the maximum price (Fig. 3C), the mean was R$ 57.37 with HPD95% ranging from R$ 54.85 to R$ 60.17. The HPD95% of the revenues (obtained by multiplying the densities of volume and prices) for the minimum price situation ranged from R$ 8392.16 ha−1 to R$ 14,025.48 ha−1, with mean equal to R$ 11,279.29 ha−1. For the maximum price situation, the HPD95% ranged from R$ 17,979.58 ha−1 to 20,165.72 ha−1, with mean equal to R$ 19,038.43 ha−1. Under these circumstances, the NPV calculated for the minimum price situation had HPD95% ranging from R$ 2050.11 ha−1 to R$ 5409.07 ha−1, with mean of R$ 3771.60 ha−1 (Fig. 4A). The NPV calculated for the maximum price situation had HPD95% going from R$ 7766.77 ha−1 to R$ 9070.29 ha−1, with mean of R$ 8398.13 ha−1 (Fig. 4B).

x– (2)

where α is the location parameter and β is the scale parameter. For a series of minimum values, the cumulative distribution function and probability density function are, respectively:

F (x ) = 1 =


exp –exp exp



and f (x ) exp

x– (3)

Thus, let ym be the maximum or minimum value of the sale price of wood over a year, ym ~ Gumbel(α, β). Mendes (2004) explains that, when working with the negative of the original series, the parameters obtained after fitting Eq. (2) correspond to those of Eq. (3) with positive α. Accordingly, the values of the minimum price series were multiplied by −1 to fit the distribution, since we had methods to fit only Eq. (3) by Bayesian approach. Parameters α and β followed a Normal non-informative prior distribution with mean (μ0) equal to 0 and precision (τ02) equal to 0.000001. 2.2.1. Distribution fitting and validation Before fitting the distributions, the independence and stationarity of the minimum and maximum series were verified by the Durbin-Watson and Mann-Kendall tests, respectively. If a series is non-stationary, that is, it has a tendency, the parameter α becomes a linear function in the form α = α0 + α1t where t is the time (Clarke, 2002). When fitting the distributions, in a first step, the first 9 observations (of 12 total), corresponding to the years 2006 to 2014, were used to fit the probability distribution and the last ones, corresponding to the years 2016 and 2017, to validate it. Validation is based on estimates for a given return period. The probable maximum or minimum values for a given return period T are determined by the expression ym(T) = β – α ln {ln[T/(T – 1)]}. As T ≠ 1, the validation was made with the two final observations of the price series. The results were considered satisfactory if the HPD95% region of each ym(T) included its observed real value. After the validation, the fit was updated using all 12 observations (year 2006 to 2017), aiming to predict the minimum and maximum prices in 6 years [ym(T=6)]. In addition, the Kolmogorov-Smirnov test, obtained from the reliaR package (Kumar and Ligges, 2011), was performed to evaluate the fit of the Gumbel distribution to the maximum and minimum series. The fit and estimates for price were also made using MCMC algorithms in the R software integrated with OpenBUGS software. The dependence and absence of convergence of the process were evaluated by the same criteria.

4. Discussion The good performance of the Schumacher model when estimating eucalyptus growth and yield was expected, since the application of this model for dendrometric variables is well documented in the literature (e.g. Tewari et al., 2002; Fontan et al., 2011). The results for wood yield at 6 years (Fig. 3A) are in agreement with what is usually seen in Brazil. Stape et al. (2010) analysed eucalyptus plantations in several regions of the country and the authors found values ranging from 221.65 m3 ha−1 to 444.52 m3 ha−1, with a mean of 337.53 m3 ha−1, for 6-year-old plantations from three different sites that had 1064 to 1186 trees per hectare. A refinement in estimating yield for economic analysis can be done by testing other non-linear models such as Gompertz, Logistic, and Chapman-Richards. In any case, the satisfactory values of RMSE and R2B (Table 1) indicate that using the Schumacher model was sufficient for the proposed objectives.

3. Results After processing the data for the elicitation of prior distributions, the values of prior mean and precision (μ0 and τ02) found for β0 were 6.8026 and 3.3033 respectively. For β1 these values were 5.7388 and 1.2481 respectively. There were good results for the goodness of fit measures when fitting the model to the sample data using the Bayesian approach with these hyperparameters. The R2B was close to 1.00 (Table 1) and RMSE was low, corresponding to < 10% of the mean volume observed at the age of interest (6 years). 67

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Table 1 Results of the Schumacher model fit, with the posterior mean of parameters, lower (LL) and upper (UL) limits for HPD regions, convergence diagnostics criteria, Bayesian coefficient of determination (R2B) and root mean square error (RMSE). Parameter (mean) β0 β1

6.5719 4.6048








6.4770 4.2050

6.6600 5.0550

0.6860 0.4040

1.060 1.000

0.9070 0.9380



Fig. 3. Posterior density of wood yield (A), minimum price (B) and maximum price (C) estimated for year 6 of the evaluated project.

Fig. 4. Probability densities of the net present value (NPV) calculated for the project evaluated in the situations of minimum price (A) and maximum price (B).

For the minimum and maximum prices estimates, the presented method was efficient for a relatively short return period (2 and 3 years) when it comes to the forest sector. Table 2 shows that the precision of the HPD95% region for the parameters of the distribution for minimum price was considerably lower than the precision seen in the parameters for maximum price. We tested a fit with no trend for the series of minimum and it was verified that its addition to the model results in a greater dispersion of the return levels probabilities (Fig. 3C) when compared with estimates of return levels without trend (Fig. 3B). Fernandes et al. (2010) comment that the limits for these probabilities will depend on the size of the sample, the estimation process and the nature of the uncertainties involved. The use of informative prior distributions for the parameters of the Gumbel distribution, as done in Martins and Stedinger (2000), can increase the precision of the estimated return levels. However, at the time this study was carried out, no prior information found by the authors was consistently correlated to the wood price series. Other approaches for estimating the expected value of forest product prices have already been presented, such as Styles et al. (2008) that used an inflation index to project prices, Musshoff (2012) that used a dynamic-stochastic model, Coelho Junior et al. (2013) that used artificial neural networks and Araujo Junior et al. (2016) that used neurofuzzy systems. A future study comparing different methodologies applied to a database with known values for long return periods would be of interest for the forestry sector. The project was economically feasible considering the entire HPD interval of wood yield and the two situations of extreme price. These estimates can be updated as new sample data are obtained. As it happened here, eucalyptus plantations in Brazil are usually economically viable (de Carmo et al., 2011; Coelho et al., 2016; das Virgens et al., 2016). However, they may present negative NPV when implanted in

poor sites as shown in Gonçalves et al. (2017) and Rode et al. (2014). Due to some differences in the inputs, we believe that it is inappropriate to compare the NPV of other studies involving economic analysis of eucalyptus plantations in Brazil with the project presented here. For example, in Rode et al. (2014) the interest rate was 5% and yield did no exceeded 250,0 m3 ha−1. In Queiroz and Silva (2016) the price for standing timber was as low as R$ 15,00 m−3 because it was from a different region in Brazil. In Dobner et al. (2017) the planning horizon was 20 years with timber harvesting happening three times in this period. These differences potentially lead to significantly different results. The Bayesian approach allowed levels of uncertainty to be incorporated into the NPV results (Fig. 4). Sometimes this has been done in the form of Monte Carlo simulations, as mentioned by Hildebrandt and Knoke (2011). The problem in this case is that since the distributions of inputs are usually unknown, uniform distributions or triangular distributions are used with empirical parameters (e.g. Silva et al., 2012; Arnold and Yildiz, 2015). Differently, the probabilities for the NPV inputs in the present study were obtained from distributions that have parameters calculated specifically from the sample data. Another common process in economic studies is the use of sensitivity analysis (e.g. Avohou et al., 2011; Rode et al., 2014). The technique, however, does not incorporate the risk calculation, that is, it does not assign probability levels to the values resulting from the economic evaluation criteria (Hildebrandt and Knoke, 2011). The methodology presented here is not limited to the NPV calculation. Once the probable revenues from the Bayesian approach have been calculated, other criteria such as equivalent annual benefit, benefit/cost ratio and internal rate of return (IRR) can be easily calculated. Using the Bayesian approach has been advantageous in recent decades.

Table 2 Results of the Gumbel distribution fit to the minimum and maximum series of annual wood sale prices, with the value of the posterior mean of the parameters and lower (LL) and upper (UL) limits for their HPD regions and convergence diagnostics criteria. Series

Parameter (mean)


α0 α1 β α β


47.7828 −0.4660 3.1740 53.4194 2.3258






43.4100 −1.1300 1.8360 51.9500 1.2940

52.5900 0.1363 4.9080 54.8000 3.5340

0.4180 0.2920 0.7100 1.1260 0.9790

0.9960 0.9960 1.0100 1.0200 0.9900

0.4530 0.2570 0.5760 0.4810 0.5010


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Galiatsatou et al. (2008), when comparing Bayesian techniques versus conventional techniques, observed that the Bayesian structure offered substantial advantages to analyse data of extreme values. Gelman et al. (1995) mention that this methodology generally produces more precise intervals for parameter estimates than those presented by the frequentist method. In addition to the increase in accuracy, the Bayesian approach may allow for consistent inferences from few observations, as shown by Moeltner and Woodward (2009) and Vul et al. (2014). An analysis of economic performance and risks, such as the one made here, is certainly necessary for investments in forestry. As said by Kangas and Kangas (2004), the characteristics of forest production, such as the dependence of environmental factors and the long term for projects to be profitable brings many uncertainties whose correct treatment is little studied. 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5. Conclusions The Bayesian approach can be used by managers to improve the way they obtain inputs to calculate economic evaluation criteria. It encompasses and represents uncertainties better than other techniques normally seen in this type of analysis. In the proposed project, it gives good results when estimating wood yield for a period of 6 years and, combined with EVT, also gives good results when estimating probable minimum and maximum prices for wood in the same period. The project proposed here is economically feasible in all situations that the study covered. The means of NPV calculated here for the minimum price situation (R$ 3771.60 ha−1) and maximum price situation (R$ 8398.13 ha−1) serve as a best and worst scenario orientation for managers in the case of a eucalyptus plantation established in 2017 under an outgrower scheme in the state of São Paulo. New studies comparing the methodology presented here with other methodologies for price prediction using a larger database should be performed. It is also interesting to investigate the benefits of applying the Bayesian approach when fitting more complex growth and yield models for forests with small size samples. Declarations of interest None. Acknowledgment The authors would like to thank CAPES (Coordenação de Aperfeicoamento de Pessoal de Nível Superior) for the scholarships awarded. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// References Akhtari, S., Sowlati, T., Day, K., 2014. Economic feasibility of utilizing forest biomass in district energy systems – a review. Renew. Sust. Energ. Rev. 33, 117–127. https://doi. org/10.1016/j.rser.2014.01.058. Alvares, C.A., 2011. Mapeamento e Modelagem edafoclimática da Produtividade de plantações de Eucalyptus no Sul do Estado de São Paulo [Mapping and Edaphoclimatic Modeling of Productivity of Eucalyptus Plantations at South of São Paulo State]. Doctoral Thesis. Universidade de São Paulo, Piracicaba, Brazil. http://


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