Rethinking the forestry in the Aquitaine massif through portfolio management

Forest Policy and Economics 109 (2019) 102020

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Rethinking the forestry in the Aquitaine massif through portfolio management

T

Arnaud Z. Dragicevica,b,* a b

IRSTEA — The Center of Clermont-Ferrand, 9 Avenue Blaise Pascal, 63170 Aubière, France AgroParisTech, INRA, IRSTEA, Université; Clermont Auvergne, VetAgro Sup [UMR Territoires], 63170 Aubière, France

ARTICLE INFO

ABSTRACT

Keywords: Bioeconomics Portfolio management CVaR Natural hazards Pine wood nematode

The Aquitaine massif, in the South-West of France, is home to the largest Pinus pinaster monoculture forest in Europe. In light of the possibility of occurrence of severe natural hazards, such as the pine wood nematode, what could or should be done with the land historically dedicated to the culture and the exploitation of maritime pine? In order to address this critical question, we make use of the well-known portfolio management and apply it to the spectrum of opportunities that can be seized in the Aquitaine region. The optimal land-use allocations are based on two models from the literature on portfolio theory, i.e. the Markowitz mean-variance model and the conditional-value-at-risk (CVaR) model. Historical data and Monte Carlo simulated data are both considered in the study. According to our results, low levels of risk imply the preponderance of one asset, whereas severe levels of risk are further reflected in greater portfolio diversification. Yet, greater diversity does not necessarily imply uniform allocation of assets. Moreover, the outputs from the CVaR assignment programs, based on data sampled from normal and non-normal distributions, attach greater deal of importance to the risk minimization. In practical terms, we find that extreme losses lead to mixed-asset portfolios that are inclusive of market gardening and of grain production. Therefore, investments in forestry should not be abandoned but rather enriched by agroforestry practices. As for the eventual production of solar energy, the investment turns out to be an economic bet.

1. Introduction The presence of Pinus pinaster in the circum-Mediterranean region is attested since the Roman Empire. The Aquitaine massif, in the SouthWest of France, is home, since the 19th century, to the largest maritime pine monoculture forest in Europe. The tree species had been planted in this locality for industrial purposes. The Landes forest occupies roughly a total surface area of 10,000 square kilometers. Although it is owned up to 90% by private forest owners, 75% of the area belongs to less than 20% of large institutional landowners (Service Nature Forêt, 2008). In 2015, the maritime pine harvesting for lumber production amounted to 3,331 thousands of cubic meters in the Nouvelle-Aquitaine administrative region. The harvest of the tree species for industrial purposes was of 2,293 thousands of cubic meters (Agreste, 2017).1 The sum of these figures represents a percentage of 56.41 of total silvicultural harvesting in the territory. A series of extreme weather events – hurricane Martin in 1999;

hurricane Klaus in 2009; storm Xynthia in 2010 – provoked enormous amounts of windthrow in the region. Many living trees have then been turned into dead broken or uprooted woody debris. Negative effects from natural disasters have also been exacerbated by insect pest damages. As a result, the largest European reforestation project to date has been implemented (GIS GPMF, 2014). Another serious threat to the French South-Western territory is the probability of arrival of Bursaphelenchus xylophilus (de la Chesnais, 2015), also known as the pine wood nematode (PWN), which is an invasive pest of pine forests (Mallez et al., 2013). The risk of PWN spread is both a conjunction of natural spread and of timber trade (Robertson et al., 2011). The microscopic worm is now widely distributed in North America and in Asia. As for the European continent, PWN was first detected in the Portuguese subregion of the Setúbal Peninsula (Mota et al., 1999). Additional outbreaks have since been recorded in the center of Portugal and in Spain (Abelleira et al., 2011; Robertson et al., 2011; Fonseca et al., 2012). The parasite provokes the pine wilt disease through a complex

IRSTEA — The Center of Clermont-Ferrand, 9 Avenue Blaise Pascal, 63170 Aubière, France. E-mail addresses: [email protected], [email protected]. 1 The only available figure related to the uses of fuelwood concerns all tree species produced in the region. In 2015, the harvest volume was in the amount of 1,074 thousands of cubic meters. ⁎

https://doi.org/10.1016/j.forpol.2019.102020 Received 20 July 2019; Received in revised form 27 August 2019; Accepted 4 September 2019 Available online 01 November 2019 1389-9341/ © 2019 Elsevier B.V. All rights reserved.

Forest Policy and Economics 109 (2019) 102020

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biological process that combines the nematode, an insect vector and a susceptible tree. The disease is considered to be dramatic, for it typically kills affected trees within a few weeks to a few months (Donald et al., 2003). If we now consider the worst-case scenario, in which the parasite enters the Landes area and provokes the death of the pine forest, what could or should be done with the geographic zone historically allotted to the culture and to the exploitation of maritime pine? In order to address this critical question, we make use of the well-known portfolio management and apply it to the spectrum of land-use opportunities that can be theoretically seized in the Aquitaine region. The possibility of restarting the plantation and the exploitation of Pinus pinaster, as well as the possibility of broadening the land-use to agricultural crops such as cereals, not to mention that market gardening or the production of photovoltaic solar energy could be introduced as well, are part of all the alternatives that are envisaged in our managed portfolio of activities. We must first recognize that the consideration of a portfolio of activities is no novelty. In Macmillan (1992), the farmer's decision problem was approached, in the manner of a von Thünen spatial economy, as a mean-variance problem under both equality and inequality portfolio constraints. Later on, Hyytiäinen and Penttinen (2008) suggested that portfolio optimization had potential to be of assistance in planning the forest clearcutting. By means of portfolio theory, other studies considered land-use portfolios and the benefits of land-use diversification, such that, under the constraint of economic viability, a portion of land was intended for a specific agricultural or silvicultural use (Castro et al., 2015; Djanibekov and Khamzina, 2016). The examination of risks and returns associated with payments for ecosystem services has also been the subject of research (Matthies et al., 2015). As an illustration, the optimization of the landscape portfolio, devoted to allocating incentives to rehabilitate degraded agricultural lands, led to high compositional diversity which, in turn, supported multiple ecosystem services vital for human well-being (Knoke et al., 2016). Our wish is to put portfolio management into service – in the form of a decision-making tool which does not take account of spatial constraints – for land allocation of the Aquitaine basin covered by the pine forest.2 A widely used measure of risk is variance or standard deviation. Since Markowitz (1952, 1959), the trade-off between the expected return of a portfolio of assets and its combined variance has been at the core of portfolio management. A specific weighted combination of assets is selected to calculate the risk-return optima, that is, so as to minimize the portfolio variance subjected to a given target return (Mostowfi and Stier, 2013). In biotechnical variance-based models, based on productivity-risk tradeoffs, the portfolio selection was employed with the purpose of determining the combinations of tree species (Dragicevic et al., 2016; Raes et al., 2016; Brunette et al., 2017a; Brunette et al., 2017b). Notwithstanding its success, the Markowitz model has been the subject of various criticism soon after its inception. For instance, the mean-variance admissible set may be dominated under stochastic dominance, such that mean-lower partial moments ought to be used instead (Bawa, 1975; Lee and Rao, 1988). Other works emphasized that decision makers frequently associate risk with failure to reach a specified target rate of return. In the case in point, risk simply corresponds to any deviation below that target (Fishburn, 1977; Harlow and Rao, 1989). Just as much, variance is seen as inappropriate for the highly skewed fat-tailed distributions (Mausser and Rosen, 2000). Those can be observed in commodity markets (González Pedraz, 2017), where the probability of extreme losses is much larger than what is predicted by the normal distribution. As a result, the minimum-variance portfolio is inefficient with respect to unexpected losses (Arvanitis et al., 1998).

Value-at-risk (VaR) is defined as the maximum potential loss that should be achieved, over a given period, with a given probability (Manganelli and Engle, 2001). It is a quantile of the distribution of losses associated with the asset holding. It solely reflects the information contained in the distribution tail. Unfortunately, VaR as a threshold disregards extreme losses beyond the quantile. Put differently, it does not address how large can these expected losses be when a harmful small probability event occurs (Bertsimas et al., 2004). Likewise, it fails to be subadditive, be it a property that enables to achieve reduction, through asset diversification, in the portfolio risk (Yamai and Yoshibo, 2002). Therefore, Artzner et al. (1997) introduced the conditionalvalue-at-risk (CVaR), also called the expected-shortfall model, to overcome the limits encountered with VaR. In short, CVaR measures the average losses in states beyond the VaR level. In order to address the topic of optimal land-use design, the present study is conducted with respect to indexed annuities, rather than per annum rates of change of returns, translated to euros per unit sold. The present value of indexed annuities measures, at a given point of time, the series of discounted future periodic payoffs, also referred to as discounted indexed cash-flows. The primary reason for choosing indexed annuities is because the allocation of initial monetary units cannot be undertaken in absence of market proxies. Without even referring to the limited availability of data, providing proof that various market returns are representative of the global market investment – pertaining to the diversity of assets collated in our groundwork – is presently out of reach. Following the literature on portfolio theory, the optimal land-use allocations in the Aquitaine region have been obtained from the abovequoted models, i.e. the Markowitz mean-variance model and the CVaR model. Historical data and Monte Carlo simulated data are both considered in the study. According to our results, low levels of risk imply the preponderance of one asset, whereas severe levels of risk are further reflected in greater portfolio diversification. Yet, greater diversity does not necessarily imply uniform allocation of assets. Moreover, the outputs from the CVaR assignment programs, based on data sampled from normal and non-normal distributions, attach greater deal of importance to the risk minimization. In practical terms, we find that extreme losses lead to mixed-asset portfolios that are inclusive of market gardening and of grain production. Therefore, investments in forestry should not be abandoned but rather enriched by agroforestry practices. As for the eventual production of solar energy, the investment turns out to be an economic bet. The paper proceeds as follows. While Section 2 presents the portfolio frameworks, Section 3 reports the results from the simulation examples. Section 4 discusses the results achieved through optimization. In Section 5, we offer our concluding remarks. 2. Methodology and materials After the presentation of two well-known models in portfolio management, the section touches upon the calculation of discounted indexed annuities. Following the explanation on the chosen type of Monte Carlo simulations, the section also brings to the attention, through predetermined natural risk factors, the coverage of extreme non-market risks. 2.1. Mean-variance framework Following Dragicevic et al. (2016), Brunette et al. (2017a) and Brunette et al. (2017b), consider an investor that has at his or her disposal a set of N assets, which correspond, for a given geographical area, to different types of agricultural, forestry or energy productions. The investor would like to invest in those risky assets and expects a positive return, in form of a composite present value of indexed annuities, from the overall investment. An asset is considered to be risky because its return – dependent on

2 Despite the face that the Landes forest is mainly a pine tree forest, it should be recalled that the area also comprises broadleaved species such as oak, beech, chestnut and locust.

2

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the changes in prices – can be volatile over time. The upward trend of the market provides with increasing returns. On the contrary, when the market is bearish, the volatility with a declining trend exposes the investor, in comparison with the initial investment, to economic losses. By combining the assets, in a way that the combination takes account of their characteristics, the investor seeks to minimize these potential losses. His or her choice on what to invest in is then represented by an N × 1 vector array of asset allocations or weights w = (w1,…,wN)′. The standard mean-variance optimization problem is defined as

min

w w

w

subject to w r = r w1=1

(1)

where w w represents the portfolio risk to be minimized, with weights w = (w1,…,wN)′ and the covariance matrix of returns . The first constraint is the portfolio expected return, obtained from N assets r = (r1,…,rN), set equal to a target return r decided by the investor. The second constraint w′1 = 1 denotes the vector array of ones and corresponds to the standard budget constraint. Solving the minimization problem provides us with information on the optimal weights of different assets that minimize the portfolio risk for a given level of portfolio return. The efficient frontier of the mixedasset portfolios is then the superior segment of a parabola originating from the linear combination of assets.

Fig. 1. CVaR scheme for the assessment of the distribution of losses at the (1 − ε)-quantile and beyond.

min

w,

subject to ys

r VaR1

(r )

( r ) f (r )dr

(w, ) =

+

1 r

( r

) f (r )dr

ys s=1

rw , s = 1, …, S ys 0, s = 1, …, S (4)

2.3. Discounted indexed annuities

(2)

One of the main issues raised in the study is to deal with the potential changes in land use, due to natural hazards, in the event of a collapse in the production of maritime pine. The former revolves around a bundle of six assets that can be theoretically envisaged as a global land-use allocation project in the administrative departments of Landes, Gironde and, in smaller measure, in Lot-et-Garonne. Those include the productions of fruits, vegetables, cereals, maritime pine and – through solar panels – the production of electricity. With 9,483 agri-food companies present on the Aquitanian territory, the region generated, in sales, thirty billion euros in 2013. Agriculture represents the first industrial sector in the region. It is the 3rd largest agricultural sector in France (DRAAF, 2019). Together with the farming of asparagus, carrots, tomatoes, green beans, leeks, zucchini and of mushrooms, the local market gardening includes the productions of kiwi, apples, strawberries and of prunes (Loc'Halle Bio, 2013). As regards the cereals, we mainly focus on the production of wheat. Although most of the arable land in the Landes is sown with maize,3 wheat enters in the crop rotation with rapeseed on parcels which irrigation proves difficult (Duboscq, 1973). As a matter of fact, the Nouvelle-Aquitaine administrative region is already the leading maize producing region in France, with a harvest of 4,226 million tons in 2016 (FranceAgriMer, 2017d). It is only the 5th largest wheat producing region in France – with a harvest of 3,275 million tons in 2016 –, which

The expression being integrated is the expected value of the portfolio loss at (1 − ε) -quantile of the loss distribution, where ε ∈ [0.00,1.00] represents the confidence level. Rockafellar and Uryasev (2000) showed that CVaR was a tractable risk measure. It thus enables to define a portfolio of VaR measures through the vector of risk levels (Francq and Zakoïan, 2013). Instead of using the CVaR function defined above, the first authors propose to use an alternative auxiliary objective function as a conditional expectation with respect to the portfolio loss, that is,

F1

S

The set of two first constraints restricts the auxiliary variables such that ys ≥ max {−r′w − ξ,0}. The third constraint is the portfolio return obtainable at a given level of CVaR. The last constraint corresponds to the budget constraint. Fig. 1 shows a tool schema for evaluating the expected losses beyond the (1 − ε)-quantile. In view of the underlying risks, the variables in the optimization problem are the weights in the land-use portfolio distribution, which have to be in reasonable proportions (Mausser and Rosen, 2000). Therefore, solving the minimization problem structures the portfolio of activities in such a way that a target return is achieved under the CVaR constraints of potential tail losses.

The CVaR from a portfolio is a function of uncertain returns of different assets and their weights in the portfolio. It considers the levels of losses that exceed the expectations of the investors by more than a given percentage (Pettenuzzo et al., 2016). Following Pachamanova and Fabozzi (2010), consider a set of weights w, in form of a vector of exposures to risk factors, such that the portfolio return amounts to rp = r′w, with r′ the transposed vector of expected returns obtained from the risk factors. CVaR is then a function of portfolio weights that determine the probability distribution of rp with density function f. It can be written in the following form

CVaR1 (r ) =

1 S

rw r w1=1

2.2. Conditional-value-at-risk framework

1

, y

+

(3)

The value of ξ corresponds to the portfolio VaR value. However, due to −r − ξ ≥ VaR1−ε(r), with −r ≥ ξ, the minimum value of the auxiliary function now equals CVaR, as the (1 − ε)-quantile of the distribution takes all possible losses into account. Provided the difficulty in estimating the joint probability density function of returns of all assets in the portfolio, a set of scenarios s = 1, …, S – all equally likely to occur – is used in its place, where the historical data observed at a given time step represents a scenario. They provide with auxiliary decision variables ys = y1, …, yS, which are meant to linearize the piecewise function in the definition of the CVaR risk metric. They also measure the portfolio losses in excess of VaR (Topaloglou et al., 2010). The portfolio CVaR minimization problem becomes

3 The following study considers the prices of cereals, which consist of undifferentiated types of grain.

3

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opens up some additional investment opportunities in the contours of the Gironde department. In light of the differentiated market prices of standard lumber and industrial-oriented wood uses of Pinus pinaster, which theoretically depend on the tree diameter put up for sale, we have decided to distinguish between these two possible types of investment. The main uses of maritime pine sawn timber are joinery (moldings, flooring, skirting boards and paneling) and the manufacture of furniture and factory and structural lumber. The species is also used for the fabrication of products for outdoor fittings (cladding and street furniture). Likewise, it serves in the field of packaging and for the manufacture of pallets (FrenchTimber, 2018). In order to treat the cash-flow streams equally, simulations have been performed in terms of present values of indexed annuities, per unit sold in euros, discounted at rate i. Their actuarial notation relative to agricultural productions – such as cereals, fruits and vegetables – is in the following form

rx , t = (px , t

cx,t )

1

(1 + i ) i

40

i (1 + i) 40 (1 + i) 40 1

difference between the gross grain prices stored between 2004 and 2015 (Joubert, 2015) and the costs of wheat production recorded between 2001 and 2016 (FranceAgriMer, 2017c). The gross margin of solar panels has been assessed by subtracting the estimated cost of solar panels per watt (EnergySage, 2018) from the modified electricity prices per kilowatthour in France (Eurostat, 2017). The silvicultural expected margins for lumber and for industry-oriented timber were obtained from the database on timber prices, held from 1966 to 2008 by the French newspaper, entitled La forêt privée, dedicated to private forest owners and industry players. The costs per cubic meter of Pinus pinaster were estimated from the study of Chevalier and Henry (2012).6 Due to different endpoints of the respective time series to which we had access, all the missing data have been filled through five-year moving averages. Yet, the unavoidable synchronization of time series affects the correlations between the gross margins of different assets. In order to validate the consistency between available data and lacking data constructed from moving averages, we have computed the statistical significance apropos of the correlations observed in time intervals 1990 − 2007 and 2008 − 2016.7 Through the calculation of z-statistics, the values of which have been found to range from −1.73 to 0.49, we observe no significant difference between the coefficients, at a level of significance of 0.05, which corresponds to the threshold level of ± 1.96. The data series subject to optimization techniques should therefore provide us with reliable outputs. In the case of mean-variance optimization, the standard deviation around the portfolio return – reflecting the market volatility – was considered to be a measure of risk. In the case of CVaR minimization, the risk was identified as the expected value of the portfolio loss in states beyond the VaR level, within a 95% confidence level, where each year represented one possible scenario.

(5)

where px, t corresponds to the market price in euros of agricultural asset x in a given year t and cx, t its production cost in euros. The discount rate has been set at the level of 3%. For comparison purposes with a forestry management plan, the present value of indexed annuities is weighted by the silvicultural rotation factor, such that the representative investor evaluates the overall rotations of cereal crop and of market gardening productions at a forty-year length. The calculation of the present value of indexed annuities from timber sales, in euros per meter cubed per year, is as follows

ry, t = (py, t

c y, t )

1

(1 + i) i

40

i (1 + i ) 40 (1 + i ) 40 1

(6)

2.4. Non-market risks

where py, t corresponds to the market price, in euros per meter cubed in a given year t, of silvicultural asset y. Its yearly production cost in euros per meter cubed is given by cy, t. According to expert judgment, the average rotation length of a silvicultural management plan applied to Pinus pinaster is defined to be of 40 years (Augusto et al., 2015). The present value of indexed annuities from the investment in photovoltaic electricity is defined as

rz , t = (pz , t

cz , t )

1

(1 + i) i

40

i (1 + i ) 40 (1 + i ) 40 1

A projection on non-market risks cannot be handled from historical data. Consequently, we consider these types of risks by generating random sampling through the Monte Carlo method. Its aim is to bear in mind the sensitivity of optimal allocations to the period of historical data used to compute the series of returns (Mills and Hoover, 1982). Even if a single method does not exist, many simulations (Abelleira et al., 2011) model a system as a probability density function; (Agreste, 2017) repeatedly sample from that function; and then (Allen and Gale, 1997) compute the statistics of interest (Harrison, 2010). We decide to consider a normal distribution, nonetheless defined over the statistics – such as mean and standard deviation – issued from the time series at disposal, from which we randomly sample a series of fictitious data. Despite the fact that normal distribution underestimates both the frequency and magnitude of extreme negative events (Sheikh, 2009), Gaussian distribution allows for negative market returns (Ho and Lee, 2004), what happens to have been observed in the indexed annuities computed from the real time series. In order to take account of the risks at stake, these data are then weighted by predetermined risk factors. For example, due to natural risks such as storms, fires or biotic attacks, a major destruction of a forest is considered to occur every 70 years. Indeed, stand destroying fires with mixed severity underburns are presumed to occur every 12 to 70 years (Boise National Forest, 1997). The risk on forest production

(7)

where pz, t corresponds to the electricity price, in euros per kilowatthour, of solar panel z in a given year t and cz, t the yearly cost of installation and functioning in euros per kilowatthour. Although solar panels can operate shortly after their fitting – with the life span of a solar panel at full capacity being of 20 years – the discounting has been applied with respect to the rotation of forestry production currently in place. The optimization was performed from discounted indexed annuities obtained as an assessment in euros of per annum gross margins.4 The latter have been computed from the equations formulated above. Those of fruits and vegetables have been obtained by weighting, by a gross margin of 34.42%5 (Crédit Agricole, 2017), the retail prices of their representative baskets sold to the French hypermarkets and supermarkets from 2008 to 2016 (FranceAgriMer, 2017a; FranceAgriMer, 2017b). The gross margin of cereals has been calculated as the

6 It should be mentioned that weighting the present values of indexed annuities with respect to the longest rotation length does not modify the results meaningfully, that is, it mostly impacts the hundredths. 7 The first time interval corresponds to the weightings of the consumer price index (basis 2015) of all households in metropolitan France (https://www. insee.fr). The second time interval corresponds to the available prices. Those have formerly been homogenized with respect to the shortest interval issued from FranceAgriMer (2017a,b).

4

We do not parameterize the historical data, such that they would follow a fixed probability distribution, for the methodology artificially creates a strong dependence to the model. 5 Indicative average value developed from the INSEE data “Income statement and balance sheet data for natural persons.” Mass distribution takes into account the commercial margin, be it the difference between the selling and the purchasing prices. 4

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was then evenly distributed, by a factor of 1 , on the data generated by 28 the Monte Carlo method. This is equivalent to risk smoothing over the period considered in the rotation length. Likewise, by reason of extreme weather events, the risk of power generation shutdown was evenly distributed, by a factor of 1 , on the simulated data. This implies that a 40 storm capable of destroying the solar panel park takes place every 100 years. From the foregoing, total monetary losses from a catastrophic event have been excluded from the model. The mechanism of intertemporal risk smoothing has been applied because a natural catastrophe in a fixed geographical area corresponds to a non-diversifiable type of risk.8 At a given point in time, the latter affects all assets under consideration in a similar way. In such cases, risks can be averaged over time (Allen and Gale, 1997).

with respect to both return and variance at an asymptotic significance of p < 0.01. That is, a cut in fruit production engenders increases in both the portfolio return and risk. On the other hand, the weights of lumber and industrial timber increase proportionally with both portfolio characteristics, for a perfect positive correlation of 1.00 has been estimated at p < 0.01. We then conducted a linear regression in order to refine the analysis on which assets contribute to minimizing the portfolio variance the most. The exercise enabled us to obtain the standardized coefficients, presented in Table 1, which compare the effects of each individual asset on the portfolio level of risk. The estimation shows that only one asset significantly affects (p < 0.01) the level of risk that is tolerated for a certain level of return. Indeed, fruits contribute to minimizing the overall risk (β = − 1.00; t = − 6,636.26). Forasmuch as timber appears preeminent in the optimal allocations at very high levels of variance, it does not come up as explanatory for ensuring the minimization of risk. In any case, the results show coherence with Fig. 2b, where fruits dominate the portfolio composition in eight out of twelve scenarios.

3. Results In what follows, we present the results obtained from the computation of optimal portfolios, coming from the observed time series, based on the mean-variance and CVaR frameworks. The last part is devoted to the optimal outcomes using the Monte Carlo simulations. The study is also driven by the econometric analysis on the outputs derived from correlation coefficients and linear regressions.

3.2. Conditional-value-at-risk framework Fig. 3a. outlines the characteristics of the optimal portfolios obtained by means of the CVaR model. The figure unveils a similar configuration, characterized by a linear segment, where the frontier is both convex and concave. This time, an increase of the portfolio expected loss is obtained with a linear return increase.9 As we now take a look on the optimal distribution of assets in the portfolios minimized with respect to the CVaR, Fig. 3b. depicts allocations significantly different from those obtained with the Markowitz model. We observe greater portfolio diversification, but also rather stationary optimal allocations throughout the levels of risk.10 This output comes from the constraints on the auxiliary variables bounded by zero, the existence of which is akin to the control, through penalties, of the deviation of portfolio weights (Kourtis, 2015). In that sense, stability comes from the fitting of the linear combination of optimal weights. A closer look reveals two patterns. When the tail losses are expected to be very low (CVaR ≤ 240), a large weight of industry-oriented timber along with marginal shares of vegetables and cereals are privileged in the optimal allocations. At levels of expected tail losses defined to be moderate or high (240 < CVaR ≤ 1,320), the upholding of stability of the optimal weights along the risk interval should be stressed. Whether expected tail losses are slightly elevated or become very important, their minimization depends practically on the same distribution of assets as before. The slight difference deals with an increase in vegetables, a strong fall of industrial timber, as well as with a great

3.1. Mean-variance framework Setting a series of target returns yielded a range of optimal portfolios, obtained by means of the Markowitz mean-variance model, minimizing the overall market risk. The combinations of optimal levels of risk and return give rise to an efficient frontier illustrated in Fig. 2a. Such as commonly outlined in the literature on portfolio management, the superior segment of a parabola represents the efficient frontier of mixed-asset portfolios. As can be noticed, the higher the risk, the higher the expected return from the land allocation. This is mainly due to the fact that investments with significantly high expected returns also have high risks (Hull, 2006). The risk-free asset being chosen as the minimum return found among the set of assets in the portfolio subject to optimization, its presence yields an efficient frontier in form of a straight line, which is thus both convex and concave. Fig. 2b. illustrates the optimal allocations of assets in the portfolios. Two patterns can be identified. The first one corresponds to moderate low levels of risk (σ ≤ 2.23) and shows that the optimal portfolio is mainly composed of fruits. Despite the prominence of fruits and the presence of lumber, industry-oriented timber succeeds in emerging modestly. While the quasi-exclusivity of investment in fruit production can be observed with low variance, the production of Pinus pinaster only attains such weights under high variance. Indeed, as the level of risk increases (σ ≥ 2.53), the early hegemony of fruit production is progressively replaced by greater portion of timber production. We then observe a predominance of lumber and, to a less than proportional extent, increasing portions of industrial timber. Thereby, the Landes forestry turns out to be not only an activity yielding strong returns, but also that of implicitly tolerating exposure to market volatility. The unexpected trait of this graphic representation comes from the absence of three other assets within the reach of investors. In regards to the substantial level of risk, we note that the inclusion of other assets in the computation of optimal portfolios reduces the portion of maritime pine to approximately 70%. Table 1 shows the Pearson r-coefficients of correlation between the portfolio return and the optimal weights. Likewise, the table displays the correlation coefficients between the portfolio variance and the optimal portions. From the statistical point of view, we see that two assets stand out. On the one hand, fruits have a perfect negative relationship

9

Despite being unusual, the result is encountered in the literature on rewardto-shortfall ratios relating the constant excess return in comparison with the risk-free return (Pederson and Satchell, 2002; Kühn, 2006). From the mathematical viewpoint, the explanation comes from the parabola, the equation of which is polynomial. Such as observed in the calculation of the discounted indexed annuities, the polynomial has a repeated real number solution. Its discriminant is thus null and the roots of both the tangent and the parabola coincide. 10 Our initial work considered the rates of change of the returns rather than discounted indexed annuities. The outcomes obtained from the CVaR optimization then gave rise to significant swings in the optimal allocations along the risk scale. This issue has been previously discussed by He and Litterman (1999). They suggested to apply the Black-Litterman model, according to which the market allocation and the investor's attitude toward risk should be jointly considered. Finance professionals do not necessarily share their opinion. The investigation conducted by Bertsimas et al. (2004) led to the conclusion that, whenever the quantile values tend to zero, the mean-shortfall estimates become more volatile than the mean-variance estimates.

8 This is because we did not distinguish between the threats to the forest production.

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Fig. 2. Fig. 2a: Efficient frontier of the optimal portfolios, minimizing the portfolio risk, obtained from the Markowitz mean-variance model. By means of the standard deviation (σ), the x-axis represents different levels of risk. The y-axis measures the average portfolio returns (r). Fig. 2b: Compositions of optimal portfolios, minimizing the portfolio risk, obtained from the Markowitz mean-variance model. By means of the standard deviation (σ), the x-axis represents different levels of risk. The y-axis measures the optimal weights of different assets (w). Table 1 Econometric analysis on the mean-variance outputs. Asset

Fruits Vegetables Cereals Pinaster (lumber) Pinaster (industry) Solar panels

Return

Variance

Standardized coef.

Student's t-test

r-coef.

p-value

r-coef.

p-value

β-value

std. error

t-statistic

p-value

−1.00***

< 0.01

−1.00***

< 0.01

−1.00

0.00

−6,636.26

< 0.01

1.00*** 1.00***

< 0.01 < 0.01

1.00*** 1.00***

< 0.01 < 0.01

*** Statistical significance at a 0.01 level.

Fig. 3. Fig. 3a.: Efficient frontier of the optimal portfolios, minimizing the portfolio risk, obtained from the CVaR model. By means of CVaR (95%), the x-axis represents different levels of risk. The y-axis measures the average portfolio returns (r). Fig. 3.b.: Compositions of optimal portfolios, minimizing the portfolio risk, obtained from the CVaR model. By means of CVaR (95%), the x-axis represents different levels of risk. The y-axis measures the optimal weights of different assets (w).

increase in cereals. In addition, the investment in photovoltaic electricity neither surfaces nor achieves a breakthrough at any level of risk. Altogether, the appreciation of moderate expected tail losses favors the conifer exploitation, with a 13% share to be allotted between cereals and vegetable gardening. Table 2 indicates the Pearson r-coefficients of correlation between the portfolio return or CVaR and the optimal distribution. Unlike the

previous case, the table shows no significant relationship between the portfolio characteristics and the optimal weights of assets at p < 0.01. The linear regression, the standardized coefficients of which are shown in Table 2, enables us to partly infirm the results obtained with the Pearson correlation coefficients. The asset that should negatively affect the increase of CVaR is fruits (β = − 0.51; t = − 0.77). On the contrary, vegetables should spread the risk (β = 1.08; t = 1.62). The 6

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Table 2 Econometric analysis on the CVaR outputs. Asset

Fruits Vegetables Cereals Pinaster (lumber) Pinaster (industry) Solar panels

Return

CVaR

Standardized coef.

Student's t-test

r-coef.

p-value

r-coef.

p-value

β-value

std. error

t-statistic

p-value

0.48 0.60 0.52 −0.55 −0.49 0.48

> 0.01 > 0.01 > 0.01 > 0.01 > 0.01 > 0.01

0.48 0.60 0.52 −0.55 −0.49 0.48

> 0.01 > 0.01 > 0.01 > 0.01 > 0.01 > 0.01

−0.51 1.08

0.66 0.66

−0.77 1.62

> 0.01 > 0.01

analysis of negative t-statistics requires that we examine their absolute values against the critical t-value. The degrees of freedom used in the regression model being of three, we find a value of 4.54 at a probability level of 0.01. As the statistics do not exceed the critical value, vegetables do take part in canceling the minimization of the expected tail losses.

coefficients partly validate the results issued from the correlation analysis. As a matter of fact, the only explanatory variables disclosed by the linear target path are fruits and vegetables. The increase in the portfolio risk is revealed by their respective estimates, at p < 0.01, of β = 0.13 (t = 13.35) and of β = 0.90 (t = 94.68). The estimates neither take account of solar energy production nor of cereals as conditioning variables. The characteristics of the CVaR optimal compositions, obtained from the Monte Carlo simulations, are to be found in Fig. 5a and b. Like in the case obtained from historical data, the shape of the efficient frontier is linear. As for the optimal compositions, the big picture presents more diversified allocations than in the Markowitz case. The figure points up three patterns. At moderate levels of expected tail losses (CVaR ≤ 720), the optimal portfolios are only composed of fruits. Beyond this level of risk (CVaR ≥ 840), vegetables take advantage to the detriment of fruits. At great levels of risk (CVaR ≥ 1,080), proportionate weights of lumber, timber dedicated to industrial purposes and of solar panels are well positioned in the ranking of optimal allocations. The surprise comes from the presence of electricity production at severe levels of risk, while the asset is favored, in the Markowitz model, at low to moderate levels of risk (Fig. 5). Figures from Table 4 show that all portfolio assets but vegetables and cereals have a significant relationship with both the portfolio return and risk at p < 0.01. More thoroughly, while timber production and solar panels contribute to increasing the return and the expected tail losses, fruits aim at reducing them. Another striking trait that should be highlighted comes from the set of correlation coefficients, the absolute values of which happen to be identical and equal to 0.76. The estimates from the linear regression, shown in Table 4, partially disprove the results obtained via Pearson correlations. They bring to the

3.3. Monte Carlo method The characteristics of the optimal Markowitz outputs, obtained from the Monte Carlo simulations, are respectively depicted in Fig. 4a and b. Contrary to the previous case, the efficient frontier is steeper at low levels of risk (σ ≤ 1.52), which in terms of optimal allocations corresponds to a progressive replacement of solar panels by market gardening. A tiny portion of grain production can also be noted. Beyond this level of risk (σ ≤ 1.80), a clear pattern of diminishing portions of fruits and of increasing portions of vegetables can be detected. At high levels of risk, the progress of cereals in the optimal portfolios is worth considering. By applying natural risks to timber production, the total absence of wood assets is clearly discernible. This could be explained by data generated from the normal distribution that disadvantages low return-low risk assets. Surprisingly, the investment in solar power is not affected by such updating of the indexed annuities. We learn from Table 3 that vegetables have a strong positive relationship with both the portfolio return and variance at a probability level of p < 0.01. Reciprocally, solar panels have a strong negative impact, with 99% of confidence, on joint optimal portfolio attributes. What is more, Table 3 shows the standardized coefficients issued from the linear regression. Given that the global portfolio variance depends on market gardening, grain production and on solar panels, the

Fig. 4. Fig. 4a.: Efficient frontier of the optimal portfolios, minimizing the portfolio risk, obtained from the Monte Carlo simulations by applying the Markowitz mean-variance model. By means of the standard deviation (σ), the x-axis represents different levels of risk. The y-axis measures the average portfolio returns (r). Fig. 4b.: Compositions of optimal portfolios, minimizing the portfolio risk, obtained from the Monte Carlo simulations by applying the Markowitz mean-variance model. By means of the standard deviation (σ), the x-axis represents different levels of risk. The y-axis measures the optimal weights of different assets (w). 7

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Table 3 Econometric analysis on the mean-variance outputs. Asset

Fruits Vegetables Cereals Pinaster (lumber) Pinaster (industry) Solar panels

Return

Variance

Standardized coef.

Student's t-test

r-coefficient

p-value

r-coefficient

p-value

β-value

std. error

t-statistic

p-value

0.84 0.99*** 0.82

> 0.01 < 0.01 > 0.01

0.79 0.99*** 0.86

> 0.01 < 0.01 > 0.01

0.13 0.90

0.01 0.01

13.35 94.68

< 0.01 < 0.01

−0.99***

< 0.01

−0.98***

< 0.01

*** Statistical significance at a 0.01 level.

Fig. 5. Fig. 5a: Efficient frontier of the optimal portfolios, minimizing the portfolio risk, obtained from the Monte Carlo simulations by applying the CVaR model. By means of CVaR (95%), the x-axis represents different levels of risk. The y-axis measures the average portfolio returns (r). Fig. 5b: Compositions of optimal portfolios, minimizing the portfolio risk, obtained from the Monte Carlo simulations by applying the CVaR model. By means of CVaR (95%), the x-axis represents different levels of risk. The y-axis measures the optimal weights of different assets (w).

fore that fruits do account in the minimization of CVaR (β = − 1.08; t = − 3.57). Vegetables arrive just behind with β = − 0.36(t = − 1.31). Without being significant with 99% of confidence, the absolute values of t-statistics, compared with the critical value of 4.54 at 0.01, permit confirming that they affect the risk minimization.

4.1. Historical data With respect to historical data, minimizing low to moderate levels of risk, measured through the portfolio variance, non-timber assets behave as land-use substitutes toward timber production. Whereas lumber unveils very high levels of return and risk, industry-oriented timber is characterized by a low level of return for a relatively limited level of risk. Therefore, the minimization of a low portfolio risk relies on fruits. Higher risks depend on the complementarity between the assets. This hypothesis is supported by the Pearson r-coefficient at p < 0.01 (r = 1.00). In detail, their correlation coefficient shows a maximum correlative relationship toward the portfolio performance. Accordingly, their proportional replacement succeeds in minimizing all levels of variance. Given that we are on the superior segment of the frontier parabola, higher risk implies higher expected return. In order to attain the desired levels of returns, lumber becomes the ongoing replacement for fruits.

4. Discussion We now make a comparative analysis between the attributes of discounted indexed annuities observed from historical data and from the Monte Carlo simulated data. The additional econometric analysis enables us to bear interest on the eventuality of complementarity, or on that of substitutability, among the assets generated, by the optimization programs, within the optimal sets. Table 4 Econometric analysis on the CVaR outputs. Asset

Fruits Vegetables Cereals Pinaster (lumber) Pinaster (industry) Solar panels

Return

CVaR

Standardized coef.

Student's t-test

r-coefficient

p-value

r-coefficient

p-value

β-value

std. error

t-statistic

p-value

−0.87*** 0.35 0.63 0.76*** 0.76*** 0.76***

< 0.01 > 0.01 > 0.01 < 0.01 < 0.01 < 0.01

−0.87*** 0.35 0.63 0.76*** 0.76*** 0.76***

< 0.01 > 0.01 > 0.01 < 0.01 < 0.01 < 0.01

−1.08 −0.36 0.09

0.30 0.28 0.23

−3.57 −1.31 0.38

< 0.01 > 0.01 > 0.01

***Statistical significance at a 0.01 level. 8

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With respect to the expected tail losses, assets happen to be complements as well as substitutes. More specifically, the industry-oriented production of Pinus pinaster behaves as a substitute toward other assets. This hypothesis is supported by the Pearson r-coefficient at p < 0.01 (r = {−0.92,−0.99}). Furthermore, the Pearson r-coefficients show the complementarity between vegetables and cereals with r = 0.96 at p < 0.01. It can be underlined that all non-wood assets show low levels of returns and of risks. In summary, in case of CVaR, the portfolio risk is minimized, along the risk scale, by dint of a combination of land uses – the mix of which penalizes deviations from the early optimal allocations – endowed with an overall substitutability with respect to timber. We can see that, unlike the CVaR model, the Markowitz meanvariance optimization translates in the combination of assets drastically opposite in properties. Albeit the CVaR optimization yields greater portfolio diversification, an overwhelming share of one asset at a certain level of risk must be underscored. This implies that the minimization of the portfolio risk conducted from data following a nonnormal distribution outweighs the constraint of attaining a target level of return. Otherwise, lumber should have unconditionally had the biggest portion in every optimal distribution of assets. If we now take a look at the past timber market characteristics, such that the investor keeps the same risk profile, the optimization would lead to a fall in timber production down to 53%. When the portfolio risk takes the form of CVaR, keeping the same risk profile as that visible from the silvicultural production currently in place would lead to abandoning 13% of wood production. Whatever the form of risk incurred by an investor, it can be stated that the exploitation of Pinus pinaster is, on average, favored at moderate levels of risk.

increase in the portfolio return, be it at the expense of greater tail losses in expectation, investments in lumber, industry-oriented timber and in solar energy provide both low returns and risks. To this extent, they moderate the excessive risk-taking in the portfolio of activities. Unlike the Markowitz mean-variance optimization performed on historical data, that from Monte Carlo data is further prone to variability, for it combines a higher number of assets. The CVaR optimization is, beyond a threshold level of risk, reflected in greater portfolio diversification. Provided that the minimization of extreme risks relies on assets characterized by both low return and risk, it means that, once again, the constraint of attaining a target level of return counts less than the minimization of the great expected tail losses. Alternatively, investment in market gardening should have been privileged at all levels of risk. Keeping the same risk profile would lead to abandoning the production of timber, for the returns obtained from silvicultural activities do not compensate the expected tail losses. In consequence, should the variance reduction be the main objective, maintaining the silvicultural activities is not optimal. When the CVaR is taken into account as a risk metric, nor is forestry recommended. 5. Conclusion Conclusive remarks give decisive guidance on the practical application of our results. They also question the benefit of the portfolio approach. As a short remainder, we have considered the optimal landuse apportionment in the current forest territory of the Aquitaine massif. Would the pine wood nematode spread on a wide scale – not forgetting that other natural hazards might occur once again –, which combination of assets could be undertaken by considering the tools developed within the portfolio theory? Indeed, the return on investment in forestry being particularly long, the series of natural disasters have made forest owners pessimistic, some of whom have decided to retrain or to sell their land (Nicolas, 2009). Accordingly, landowners in the region could think their investment strategies through such a portfolio approach. Our results first show that the problem-solving built from data on record, which did not follow a normal distribution, comes with a relatively lower portfolio diversification in both mean-variance and CVaR frameworks. We also find that minimizing the portfolio risk often plays a key role in the optimal compositions computed from abnormal-data distributions. Judging by the allocations obtained from simulated normal distributions, finding the minimum of a risk function is no less important than the constraint of reaching a portfolio target return. In a context consistent with reality, i.e. that of data following non-normal distributions with an emphasis on potential losses supported by real facts, the CVaR framework based on historical data is most probably the least arbitrary one. Would timber production be in serious jeopardy, market gardening and cereals appear, in a diversified portfolio of activities, as plausible complements to Pinus pinaster plantations. The related findings do not imply that forestry activities should be abandoned. Although the Landes forest has neither been planted nor developed as an economic game of chance, the maritime pine production could be included within a wider portfolio of activities. This calls for reflecting upon the possibility to allocate, in case of moderate levels of expected tail losses, 13% of area to agroforestry. This land use management system corresponds to a frame of reference in which trees are grown around or among crops or pastureland. The idea is all the more interesting, for the combination of agriculture and forestry can sometimes lead to a better use of inputs (Terreaux and Chavet, 2005) and to higher revenues (Burgess and Rosati, 2018) than their separate practices. With the introduction of extreme non-market risks, forestry activities should not be abandoned would market gardening and photovoltaic solar energy production be included in the land-use allocation. The investments in vegetables of 17% and in electricity production of

4.2. Simulated data As for the simulated data yielded by the Monte Carlo method, in which the inputs were forced to follow the normal distribution, the results differ from the historical case study. At low and moderate levels of portfolio variance, the substitutability between solar panels and all the other assets present in the optimal portfolio is verified by means of the Pearson r-coefficients with r = {−0.74,−0.96} at 99% of statistical confidence. The investment in solar energy comes with both low return and risk. Market gardening products and grain production behave as complements. The highest correlation coefficient of 0.90 (p < 0.01) is observed between vegetables and cereals. However, the most privileged assets at high levels of portfolio risk are fruits and vegetables. This can be explained by their high levels of returns and risks. Due to their very high level of variance, cereals never achieve a breakthrough. In spite of being non-risky, the absence of wood assets in the optimal portfolios can be explained by their null contribution to the portfolio return.11 The consideration of the superior segment of the frontier parabola involves the progressive replacement of a low-return low-risk asset by high-return moderate-risk assets. On the subject of CVaR outputs, the first observation is about lower stability than that obtained with respect to data following a non-normal distribution. This being said, the stability is visible through mono-asset investment opportunities. When it comes to the optimal portfolios' properties, the statistical results are different. The latter show low substitutability between fruits and other assets, with r-coefficients of −0.44 and of 0.68 at p < 0.01. Pearson r-coefficients of correlation also provide evidence of perfect complementarity between solar panels and timber production (r = 1.00; p < 0.01). Owing to these features, the minimization of expected tail losses comes from a combination of three similar assets. While fruits and vegetables provide an immense 11 Let us recall that the values come from a random sampling, such that extreme values of indexed annuities can be chosen from the density function. In return, this impacts the average features of that asset.

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28% ought to succeed in minimizing severe low-probability risks. While on the subject, the Lot-et-Garonne administrative department is already underway of being hemmed in by a photovoltaic park project of great scale (Gagnebet, 2018). On another note, the energy production can be seen as an investment bet, for it is favored in respect to indexed annuities issued from a normal distribution. In point of fact, applying the central limit theorem on energy assets is based on very questionable foundations (Weron, 2009). The major limitation of the present work revolves around the modeling of disasters, such as the outbreak of Bursaphelenchus xylophilus. During the optimization of portfolios, we have simply added ancillary risk factors to data generated by the Monte Carlo simulations. This shortage can be motivated by the fact that outbreaks are usually discovered tardily (Epanchin-Niell et al., 2014). Moreover, the work undertaken by Mallez (2014) draws attention to the unavailability of probabilities of the nematode arrival, as well as to the lack of evaluation of the levels of statistical confidence, such that the stochastic processing of its behavior is for now impossible to do. In due time, hazard functions dependent on forest inventory and on spatial resolution – based on tree age, stand type and on geolocalized coordinates – should be implemented so as to overcome the lack of refinement in the computation of discounted indexed annuities. These extensions would also permit supplying complexity to the spatial allocations of risky assets at stake. Another boundary issue relates to time series chosen for the historical method. We have computed the discounted indexed annuities from data to which we have had access. Those were mainly generalized public databases. Taking into account the accounting systems of agricultural cooperatives and of energy companies, while considering state aid measures provided to them, would permit calculating more accurate margins obtained by the agribusiness and the green-energy industry. The final criticism that may be made concerns the lack of sensitivity analysis on the natural risks that could be further investigated. The risk weighting factors and the uniform smoothing over the period under consideration lead to conclusions that could be seen as a simplification of the full range of intensities of threatening natural phenomena. This brings us to contemplate working more by crossing multidisciplinary approaches. In a future research work, an extended analysis would consist in taking account of other investment assets, such as cattle breeding within woodland grazing, as well as extending silviculture to mixedwood species plantations, in the sense that mixed forests enhance both tree productivity (Morin et al., 2011) and forest stability against natural hazards (Mayer et al., 2005; Schütz et al., 2006; Griess and Knoke, 2011). Options such as energy forestry, where the cost-effective production of renewable electricity is obtained by using crops as fuel (Journal Officiel de l'Union européenne, 2003), could just as much be taken into consideration.

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