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Cut or keep: What should a forest owner do after a windthrow?
Forest Ecology and Management 461 (2020) 117866
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Cut or keep: What should a forest owner do after a windthrow? a,⁎,1
Claudio Petucco a b c
b
c
, Pablo Andrés-Domenech , Lilian Duband
T
Environmental Research and Innovation (ERIN) Research Department, Luxembourg Institute of Science and Technology, Luxembourg BETA, AgroParisTech, France Office National des Forêts (ONF), France
ARTICLE INFO
ABSTRACT
Keywords: Faustmann Land expectation value Optimal rotation age Risk Windthrow Forestry Pine Plantation
After a windstorm hits a forest stand the forest owner has two possibilities: clear cut and start a new rotation or let the standing trees grow until maturity. We compute the Land Expectation Value (LEV) under the risk of windthrow and introduce an endogenous rule to account for this decision. We compare the result with the default alternative of systematic clear felling and replanting (commonly assumed in existing literature). We have calibrated our model to represent maritime pine plantations in south-western France. Results show that by following this rule, payoffs may increase by up to 90% when the economic risk (i.e. probability of having a windstorm) is high. About 75% of the time it is profitable to keep the standing trees until maturity. Although payoffs are sensitive to economic risk, the optimal rotation age does not change much with significant increases in the risk probability.
1. Introduction Windstorms are one of the most important risks for forest management. The impacts of catastrophic events and the risk of total destruction on the optimal rotation age have been addressed by previous works focussing on even-aged forests (e.g. Martell, 1980; Routledge, 1980; Reed, 1984). In these earlier works three major simplifying assumptions have been used to obtain analytical expressions for the model’s optimal solutions: (i) The probability of a catastrophic event is exogenous; (ii) whenever a catastrophic event hits the forest stand, it is completely destroyed as a result, hence damages are also exogenous; and (iii) the salvage value after the catastrophe is equal to zero. While the assumption of exogeneity of a catastrophic event is rather easy to justify, the other two are harder to meet in the real world. Therefore, several studies have tried to relax these assumptions, often through numerical methods. Valsta (1992) relaxed the assumption of total destruction of the stand by using an uniformly-distributed random variable to represent the fraction of the stand destroyed, and assumed a salvage value. Haight et al. (1995) introduced an age-dependent damage risk and a salvage proportion. Successively, Thorsen and Helles (1998) computed the optimal rotation length and thinning regime using an endogenous damage function that depends on the forest’s characteristics as well as on the forest
management (i.e. thinning operations). While Thorsen and Helles (1998) and Haight et al. (1995) assumed an exogenous land value, Loisel (2011) and Loisel (2014) computed the optimal rotation age and thinning regime that maximise the land expectation value (LEV) – hence taking into account the value of successive rotations – under the risk of destructive events while accounting for the salvage value of windthrown timber. In an evaluation context, rather than in an optimisation one, Bright and Price (2000) proposed a simple and elegant way to compute the expected LEV under the risk of hazards whose realisation implies the end of the rotation. A similar methodology was applied by Deegen and Matolepszy (2015) to compute site-dependent optimal rotation ages and expected LEV based on historical management data in Germany. All these studies have assumed total clearing of the stand after a windthrow, independently of the level of damage. However, following a windthrow, it might be profitable to keep the standing trees until the “prescribed” end of the rotation. The continuation of the rotation after a disturbance was firstly discussed by Xu et al. (2016a,b), who analysed the forest management problem under the risk of two sequential disturbances. They tested three alternative strategies: (i) the systematic clearing and replanting of the stand after the first disturbance (the common assumption in previous literature); (ii) keeping the standing trees after the first disturbance and cutting and replanting after the second disturbance; (iii) keeping the standing trees also after the second disturbance2 and harvest the stand at
Corresponding author. E-mail address: [email protected] (C. Petucco). 1 The contribution of Claudio Petucco in this paper is the result of his PhD thesis work carried out at the former Laboratory of Forest Economics (INRA, AgroParisTech), now BETA (UMR 1443, Université de Lorraine, Université de Strasbourg, AgroParisTech, CNRS, INRA). 2 Xu et al. (2016a,b) limited their analysis to two disturbances, though they argued that the model could be extended to the n-disturbance case. ⁎
https://doi.org/10.1016/j.foreco.2020.117866 Received 9 August 2019; Received in revised form 15 November 2019; Accepted 1 January 2020 0378-1127/ © 2020 Elsevier B.V. All rights reserved.
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maturity. Their results showed that ignoring the continuation options could lead, under some circumstances (i.e. disturbances’ arrival rates, survival probabilities and post-disturbance tree-growth losses), to suboptimal harvest decisions and land expectation values. In their work, Xu et al. (2016a,b) assumed and compared the outcomes of fixed continuation strategies that are kept constant along the planning horizon. However, whether keeping the standing trees is profitable or not will ultimately depend on the cost structure as well as the characteristics of the stand after the windstorm. In this paper we relax the hypothesis of a fixed continuation strategy and introduce a condition to determine when it is profitable to continue with the remaining trees until maturity and, alternatively, when it is best to cut them immediately and replant the whole stand. We assume (i) an exogenous rate of return for windstorms as in previous works, as well as (ii) an endogenous damage function determining the fraction of windthrown trees after a storm; (iii) a positive salvage value for the windthrown trees, and (iv) explicit clearing costs. In Section 2 of this paper we present the theoretical model and in Section 3 we introduce the cut-or-keep decision rule. In Section 4 we calibrate our model with real data for maritime pine in Landes Region in France and run a Monte Carlo simulation. Sections 5 and 6 are devoted to presenting and discussing the results.
The volume harvested is denoted by q and is equal to:
q (a ) =
V0 V (a ) =
Let us assume a forest owner who plants a mono-specific and singleaged forest stand3 at the beginning of the planning horizon, t = 0 . The forest owner begins by planting at that time, and the plantation costs are represented by CP (0) . The age of the forest stand – denoted by a (t ) – is equal to zero both at the initial period (i.e. a (0) = 0 ) and the year after every harvest. Time is discrete and the stand’s age, a (t ) , is measured (i.e. discretised) in one-year steps. Denote by R the rotation length, that is, the age (in years) at which the stand is harvested (i.e. clear-cut). The forest owner decides when to harvest the stand, after which he/she replants and starts with a new plantation of the same type. By choosing the rotation length, R, the forest owner maximises the land expectation value (LEV) which is the infinite sum of discounted net returns from a forest plantation with an infinite number of rotations (Faustmann, 1849). In a deterministic setting, LEV can be written as follows: R
n= 0
R
,
d (a) =
CV q
CF ,
1
(
gv 1
if a = 0 V (a 1) V
)+1
if a > 0,
(4)
d0 d¯ e
if a = 0 {gd [a] 1/ }
if a > 0,
(5)
where d 0 is the diameter of newly planted trees, and d¯ is the maximum diameter that can be achieved in the stand in the long run. Parameter gd > 0 represents the diameter’s maximum growth rate, and > 0 is a curvature parameter. The forest dynamics are described by expressions (4)-(5). See Figs. 1b for its graphical representation. 2.2. The stochastic storm disturbances Following Reed (1984), we model windstorms as random events with a homogeneous Poisson distribution, i.e. windstorm events are assumed independent from one another and occur randomly in time. We consider different types of windstorms, s = 1, …, S , each characterised by a unique rate of return s and maximum wind speed ws . Unlike previous works, we do not consider windstorms as events resulting in total destruction of the stand. Rather, whenever a windstorm hits the stand, the size of the damage is variable and will typically depend on the wind speed, ws , the dominant height, h, and a number of
(1)
where n is an index that takes into account the succession of rotations in time (e.g. n = 0 for the first rotation, n = 1 for the second rotation, and so forth), r is the discount rate and (R) represents the net timber revenues, or stumpage value. These revenues are a function of the trees’ diameter and, therefore, of their age4. The stumpage value, at any given age a, is defined as a function of timber prices and clear-felling costs:
(a) = P (d (a)) q (a)
V a
0
where V0 is the volume of the seedlings planted, V > 0 is the carrying capacity of the stand in terms of volume (i.e. the volume towards which the stand converges in the long run) and gv > 0 is the volume’s maximum growth rate. Note that, in a deterministic setting, one may use age or time interchangeably in expressions (3) and (4). Since the age at which the harvest takes place is known and equal to R, at any given age a, time t may be written as: t = n (R + 1) + a . In a stochastic setting (as in Section 2.2), however, the harvest may take place before maturity and this equivalence does not hold any more. In such case, it is more convenient to write the model’s variables as a function of age. Henceforth, all the relevant variables are presented as a function of age, a. In expression (2) we saw that the stumpage value depends on stem price which in turn depends on the average stem diameter. The averagetree diameter dynamics are represented by the expression below:8
2.1. Stand dynamics and harvesting
CP (0) + (R)(1 + r ) (1 + r )n (R + 1)
(3)
where V (a) is the standing volume at any given age a. Expression (3) tells us that harvest takes place at the end of the rotation alone. We further assume that whenever the harvest takes place, it does at the end of the year6, i.e. forest growth and disturbances within a specific time period occur before the harvesting operations take place. The standing volume in the forest V (a) evolves according to the following dynamics:7
2. The model
max LEV (R) =
V (a) if a = R 0 elsewhere,
(footnote continued) simplifying assumption that CV is constant and not a function of the standing volume or the average tree diameter. As a matter of fact, empirical observation shows that extraction and felling costs are quite similar for a wide range of parameters; that is, for extraction rates comprised between 30 to 100 cubic metres per hectare and for stands whose trees have an average tree volume in the range of 1 to 5 cubic metres. 6 Trees grow from March to October and the harvest, in our model, takes place in December, when growth is sensibly smaller. 7 The volume dynamics are obtained from the discretisation of a continuous logistic growth model. 8 We make the simplifying assumption that all trees are equal and have the same diameter.
(2)
where the timber price, P (·), is a function of the average-tree diameter, d (a) , which, in turn, is a function of age; CV are the extraction and felling costs per unit of volume5; and CF are the fixed harvesting costs. 3 This very well represents the silviculture of maritime pine in Landes that we will analyze later on. 4 Timber revenues are also a function of timber quality. For simplicity, the issue of timber quality is not addressed in this paper. 5 Although extraction and felling costs are a function of both the total volume harvested and the volume of the average tree harvested, we make the
2
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Fig. 1. Forest dynamics functions (1a–c) calibrated for maritime pine plantations in Landes (France). The storm-damage function (1d) was calibrated to represent the maximum wind speed observed during the windstorms in 1999 and 2009 (i.e. 180 km h 1) in Landes.
stand characteristics such as the degree of recency since the last harvest, the plantation density and other technical factors, e.g. the slope of the stand and the type of planting. The stand’s dominant height, h, is a function of age and evolves according to the following expression:
h (a ) =
h0 h¯ e
windthrow is an increasing function of both the wind speed and the dominant height (i.e. fh > 0 and f ws > 0 ). We decided to restrict the arguments of the damage function to these two variables for simplicity, but other variables may be included for the sake of greater realism (Meilby et al., 2001). The exact functional specification used in this paper is an extension of the one given by Piton (2002):
if a = 0 {gh a 1/ k }
if a > 0,
(6)
f h (a), ws =
where h 0 is the height of newly planted trees; h¯ > 0 is the maximum height that can be achieved in the stand in the long run; gh > 0 is the maximum growth rate of the tree height; and k > 0 is a curvature parameter (Fig. 1c). Windstorm damages are measured by the fraction of trees broken or windthrown, that we denote by . This fraction of trees lost due to windthrow is given by:
0 if h (a) < h , (ws, h (a)) = f (h (a), ws ) elsewhere.
e{
+ h (a) + (ws ) }
1 + e{
+ h (a) + (ws ) }
,
(8)
where , , and are positive parameters. In Fig. 1, we illustrate the forest dynamics and the damage function calibrated for a maritime pine plantation in the Landes forest. The parameters and their values are presented in detail in the Appendix. 3. Cut-or-keep rule Most papers modelling windstorms and wind damage make the simplifying assumption that whenever a storm hits the stand either all or none of the trees are windthrown. A few papers allow for a fraction of the trees to be windthrown, but after the windstorm the remaining standing trees are felled –by default– and a new rotation begins. This assumption is both restrictive and unrealistic, but makes modelling simple and analytical solutions possible. For the sake of realism, in our
(7)
According to expression (7), whenever a windstorm hits the forest stand, we observe damages only if the stand’s dominant height is greater than the threshold height h, below which the stand is supposed not to have enough wind load to suffer any damage (Landmann et al., 2010). Conversely, above h, the fraction of trees destroyed by the 3
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model, only a fraction of the standing trees will be lost after a windstorm. So a relevant question to ask is: What should the forest owner do after a windthrow with the remaining (i.e. standing) trees? Suppose a windstorm occurs when the stand is a years old and a fraction of the standing trees is windthrown as a consequence. The forest owner then faces two options: (i) clear cut the remaining standing trees, sell all the timber in the market9, and start with a new rotation by replanting the stand (i.e. cut); or alternatively (ii) let the remaining standing trees grow until maturity or a new windstorm arrives (i.e. keep). The revenues from these two options are gathered and compared in the following expression: ST
a, V +
salvage (a )
E ( ST (R)) + (1 + r ) R a
+
the end of the rotation. Given that we are in a stochastic setting it is necessary to compute its mathematical expectation. Since we assume that the price function11 and costs do not change in time, we can write:
E(
ST
(·) V (a )
CF ,
(10)
where P (d (a )) is the price of timber, which is a function of its diameter d, and CC represents the cost of clearing the windthrown trees. (0 < < 1) accounts for the fact that a fraction of the Parameter windthrown trees are damaged and cannot be sold in the market. Since only a fraction of the total volume is windthrown, and this fraction is denoted by (·) ; the product (·) V stands for the total volume windthrown. The fixed extraction costs CF also apply to windthrown trees. If harvested just after a windstorm, the stumpage value of the standing (and undamaged) trees, ST (a , V ) , is given by the following expression: ST (a ,
V ) = (P (d )
CV ) (1
(·)) V (a ).
(12)
a, V +
E (LEV (R)) 1+r
E ( ST (R)) E (LEV (R)) + . (1 + r ) R a (1 + r ) R a + 1
(13)
The model described above cannot be solved analytically because of both the non-linear nature of the stand dynamics and the use of the cutor-keep condition. Hence, we solve the model numerically using Monte Carlo technique with MATLAB. The code is organised as shown in Table 1. For a given rotation age, the model simulates the evolution of the stand for 300 years projecting the stand growth, harvesting operations and windstorms. The code then sums all the discounted revenues and costs generated by the forest management and the disturbances12. Given that we consider stochastic disturbances, we repeat the stand evolution 10, 000 times, and record the approximation of E(LEV ) for each repetition. In order to find the optimal rotation age, we run this code for all rotation ages between 30 and 65 years. The optimal rotation age is the one giving the highest average E(LEV ) across the 10, 000 trials. The model is calibrated to represent a maritime pine plantation in Landes (France). The volume, tree-height and diameter-growth parameters were computed using the non-linear least-square curve fitting on the Lemoine and Decourt (1969) maritime pine yield tables. For simplicity, in each simulation we considered only one storm type. We use a windstorm’s maximum wind speed of 180 km h−1, in line with the wind speed observed in the region during the last two episodes of 1999 and 2009. In our simulations we perform a sensitivity analysis for different return rates: we tested (i) one windstorm expected every 100 years ( s = 0.01 ); (ii) one windstorm every 50 years ( s = 0.02 ); and (iii) one windstorm every 25 years ( s = 0.04 ).13 When a windthrow is observed, we apply decision rule (13). However, the careful reader will note that E(LEV (R)) is endogenous, since it depends both on the rotation length and on wind disturbances. In order to compute the RHS and the LHS of (13), we need to know E(LEV ) .
(1 + r )
CC )
CF ,
4. Model calibration and Monte Carlo simulations
the salvage value of the (same) windthrown trees, salvage ; and the mathematical expectation of the land expectation value, E(LEV ) . Note that E(LEV ) needs to be discounted R a + 1 years which is when the next rotation will be planted: The remaining trees will be ready for felling at age R, within R a years; however, since replanting takes place the year after the harvest, E(LEV ) has to be discounted one additional year. Note also that the expectation term is needed because there could be additional windstorms before the harvest, thus affecting the harvested volume and, therefore, the pay-off. The salvage value of windthrown trees, salvage , is given by the following expression:
= (P (d (a ))
CV ) E (V (R))
This particularly simple expression allows to determine when it is profitable to replant the whole stand after a windstorm (LHS > RHS); and when it is profitable to let the remaining trees grow until maturity (LHS < RHS). The forest owner is indifferent when the payoff is equal (i.e. LHS = RHS).
(9)
where in the left-hand side (LHS) of (9) we have the payoffs obtained from cut, that is, the stumpage value of the standing trees, denoted by 10 ST (a , V ) ; the salvage value from the windthrown trees , salvage ; and the mathematical expectation of the land expectation value, E(LEV ), which is the expected intrinsic value of the land without trees (i.e. its price in the market). This last term has to be discounted one year because replanting, by assumption, can only take place during the year following the harvest. In the right-hand side (RHS) of expression (9) we have the payoffs from keep, that is, the revenues obtained when the remaining standing trees are allowed to grow until maturity E( ST R(R))a ;
salvage (a )
= (P (d )
where the expected stumpage value is a function of the expected volume, E(V (R)), that can be computed recursively as a function of (i) future expected damages that are themselves a function of age and therefore of time, and of (ii) the storm’s rate of return s . Finally, expression (9) simplifies to the following cut-or-keep condition:
E (LEV (R)) 1+r
E (LEV (R)) , salvage (a ) + (1 + r ) R a + 1
ST (R))
(11)
Note that in expression (11) parameter is missing as there is no loss in the standing trees since they suffered no damage. Note also that the fixed extraction costs CF are missing in expression (11). These costs are to be paid only once: Whenever the harvest takes place. If the owner decides to harvest both the standing trees and the windthrown trees, then CF is not to appear in both expressions, else one would be paying the same cost twice. This is so when the harvesting of the windhtrown trees and the harvesting of the remaining standing trees happen at the same time, i.e. the case in the LHS of (9). Back to the right-hand side (RHS) of expression (9), let us recall that ST (R) is the stumpage value of the standing trees that are harvested at
11
It is important to note that prices are not constant but rather a function of diameter, however this function does not change across time, i.e. in this paper we do not analyse the impact of supply and demand effects on the price of timber. 12 The discounted revenues and costs converge towards zero as time evolves and become negligible after a certain time. We use a planning horizon of 300 years because, with a discount rate of 3%, the gains and losses within such horizon account for more than 99.9% of the total discounted value, i.e. by arbitrarily limiting the time horizon at 300 years we approximate E(LEV ) with an error smaller than 0.1%. 13 Note that in this paper we model windstorms that hit the forest stand, which is not the same as a windstorm hitting a whole massif: It may well happen that a storm hits a forest massif but the stand is spared. This explains why the probability of a windstorm hitting the stand is sensibly lower than that of hitting the whole massif.
9
The timber from both windthrown and standing trees. Trees windthrown by a storm are often damaged and their commercial value is sensibly lower, though far from negligible. 10
4
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Table 1 Sketch of the Monte Carlo algorithm used to determine the optimal rotation age and the land expectation value. fix the storm’s rate of return, the discount rate and the economic parameters for each rotation length R [30, 65] compute the forest dynamics for 300 years, with a harvest every R years check, for each time period, if there is a windstorm: if so compute the damages (·) : if gains from clearing and replanting > gains from waiting until end of rotation: then harvest the standing trees, compute profits and replant: set a (t + 1) = 0 ; else if gains from clearing and replanting gains from waiting to rotation end: (·)) V (t ) ; then compute the new standing volume: V (t ) = (1 sum the discounted revenues and costs for the 300-year horizon (LEV) repeat the 300-year simulation 10,000 times for every rotation length average LEV across the 10,000 replications for every R select the rotation length that maximizes expected land expectation value.
However, E(LEV ) cannot be computed correctly without plugging in the correct values on both sides of (13). To overcome this problem we ran an iterative algorithm, i.e. we ran the program several times using the LEV vector computed in the previous run to feed expression (13) in the successive one. More precisely, in the first run we obtained LEV in a deterministic setting. This value is a first approximation of E(LEV ) that was used to feed expression (13) in the second run. After running the program a second time we obtained as output a new and different set of values for E(LEV ) that were used as inputs for the third run, and so on until convergence was reached. To summarize, the following approximation of the cut-or-keep condition was used: ST
a, V +
LEV (R) 1+r
E ( ST (R)) LEV (R) + , (1 + r ) R a (1 + r ) R a + 1
Table 2 Comparison of optimal rotation lengths and LEVs for different windstorm return rates, without decision rule and with the cut-or-keep rule. No rule Windstorm return rate 0 0.01 0.02 0.04
Cut-or-keep rule
RNR
E(LEVNR )
R CK
E(LEVCK )
46 44 44 43
2,047 1,205 485 −753
46 44 44 43
2,047 1,437 937 135
Difference
E(LEVCK ) -E(LEVNR ) 0 232 452 888
* Optimal rotation ages (R) are given in years. ** Land Expectation Values (E(LEV ) ) are given in EUR per hectare.
(14)
probability of having a windstorm of the above-mentioned characteristics. The second and fourth columns give the optimal rotation age (R) for the no-rule case (NR) and the case in which the cut-or-keep (CK) rule is applied, respectively. The third and fifth columns show the Land Expectation Value for the no-rule case and the cut-or-keep case respectively. And the sixth column gives the difference between the values of E(LEV ) displayed in the third and fifth column. The results in Table 2 show that the E(LEV ) of a stand decreases as the storm’s rate of return increases: E(LEV ) is very sensitive to changes in the windstorm’s probability and for rates equal to 4% and beyond the economic return is negative or very low. The optimal rotation age is also a decreasing function of the windstorm probability: The greater the risk the lower the rotation age. Reducing the rotation age is thus a good hedging strategy for two reasons. First, as the standing volume increases, so does the loss in case of a windthrow, and by cutting more often (i.e. by lowering the rotation age) the expected loss is reduced in case of a catastrophic event. The second reason why reducing the rotation age reduces the risk is to be found in expression (7), according to which the greater the height, the greater the wind load, and thus the greater the probability of a windthrow. Qualitatively speaking these results are in line with the literature (Reed, 1984; Haight et al., 1995; Thorsen and Helles, 1998; Loisel, 2014), however it is interesting to note that the changes in the optimal rotation age are mild, and this even when changes in the windstorm probability are substantial. Table 2 also displays the optimal rotation age and E(LEV ) when the cut-or-keep decision rule is applied after a storm. When the probability of having a windstorm is equal to zero, the cut-or-keep decision rule never applies (i.e. the rule is inactive and E(LEV ) in the first row is the same in both cases). However, as the windstorm probability increases so does the difference in profits between the cut-or-keep case and the norule case. For instance, for a windstorm probability of 1%, a forest owner that decides optimally when to cut and when to continue after a windstorm may earn 20% more (EUR 232 per hectare) than another owner who – by default – cuts the whole stand after a windthrow and replants it anew afterwards. When the yearly probability of having a windstorm is 2% the gain may amount to roughly 90% (EUR 937 per hectare as opposed to EUR 485 per hectare). The reason why the
where LEV (R) , rather than E (LEV (R)) , was used. In the second run LEV (R) was approximated by its deterministic counterpart. In the third and successive runs LEV (R) was proxied by the outcome of the previous run. The iterative process stopped when the LEV-vectors obtained by two consecutive runs were identical. Convergence was obtained after 4, 5, and 7 iterations for the scenarios characterised by 0.01, 0.02 and 0.04 windstorm’s return rate, respectively. Regarding the other data used in the simulations, the cost of planting is assumed equal to EUR 1 per seedling, with 1,250 seedlings planted per hectare. The unitary cost of harvesting and the cost of salvage extraction are both EUR 8 per m3 . We assumed a fixed cost of management operations equal to EUR 200 per hectare. Timber prices were taken from the French forest sector magazine La fôret privée which has collected timber prices since 1958 (Chevalier et al., 2011). The data used were the maximum observed prices per diameter class for years 2009–2014. We corrected these prices by deflating them first and, then, we averaged them across years within each diameter class. Successively, each price was coupled with the mid point of its corresponding diameter class. In this way we defined a set of points in the pricediameter space. Last, we fitted a logistic function to these points by using the non-linear least-square curve fitting. The price function as well as the model’s parameters are presented in greater detail in the Appendix. 5. Results Using Monte Carlo technique we ran simulations for a number of scenarios with different windstorm return rates (i.e. 0%, 1%, 2% and 4%). For all these scenarios we computed both the (approximated) mathematical expectation of the Land Expectation Value (E(LEV ) ) and the optimal rotation age. Two cases of figure were considered: (i) when no decision rule is applied, which means that after a windstorm the whole stand is, by default, cut down and then replanted; and (ii) with the cut-or-keep decision rule presented above, that checks after every windthrow whether the remaining standing trees should be left to grow until maturity or rather fully cut down and replanted. The results obtained are summarized in Table 2. From left to right: The first column in Table 2 stands for the yearly 5
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summarized in the lower part of Table 3. Three values have been analysed: the benchmark value of EUR 200 per hectare and the alternatives EUR 0 per hectare and EUR 400 per hectare. For all three cases we have kept the windstorm probability fixed and equal to its benchmark value. The results in Table 3 show that the probability to continue is quite sensitive to these fixed costs. The probability of keeping, though high in all cases, is a decreasing function of the fixed harvesting costs.
Table 3 Outcomes of the cut-or-keep decision rule expressed in percentage of “continue” decisions (keep) and “harvest & replant” decisions (cut), under different scenarios of windstorm return rate and fixed costs. Rate of return
Fixed costs (EUR/ha)
N. of decisions
Continue (keep)
Harvest & Replant (cut)
0.01 0.02 0.04
200 200 200
22795 45183 90150
78.5% 74.9% 73.7%
21.5% 25.1% 26.3%
0.02 0.02 0.02
0 200 400
45697 45183 45476
81.2% 74.9% 72.1%
18.8% 25.1% 27.9%
5.1. What should a forest owner do after a windthrow? In the previous section, it was showed that, after a windthrow, it is profitable to keep a maritime pine stand until maturity more than 70% of the time. In this section, we explore more closely the factors that explain this decision to keep the stand until maturity. After a windthrow, the forest owner may take the decision to cut or keep the stand based on a number of indicators, e.g. current age of the stand, average tree diameter, standing volume before and after the storm, percentage of trees windthrown. We have analysed the outcomes of our simulations for these indicators and only two allow to explain the decision to cut or keep: the current age of the stand and the standing volume after the storm.14 Fig. 2a shows a plot of the results obtained for our benchmark case (r = 3%, CF = 200 EUR ha 1, R = 44 years, = 0.02 ). Every point in Fig. 2a represents a decision taken after a windthrow. Whenever an observation (point) is plotted above the curve (blue dots) the stand was kept. Likewise, whenever a point is located below the curve (red circles) the stand was harvested. The plot of cut and keep decisions in Fig. 2a gives two disjoint regions, meaning that all the relevant information needed to take a sound decision is embedded in the age of the stand and the standing volume after the windthrow (i.e. the system behaves in a markovian way with respect to these two variables). The plot also shows that as the age of the stand increases the owner has a greater incentive to cut the stand after a windthrow. We can interpret the figure as follows: The likelihood to cut the stand after a windthrow is age-dependent and increases monotonically with the age of the stand, up to a threshold age (a ) beyond which, if a windthrow happens, the owner will never find it profitable to keep the stand. For our benchmark case, a equals 37 years. Hence, whenever a windthrow takes place in a stand whose trees are 38 years of age or above, the owner will harvest the stand immediately afterwards, and regardless of the standing volume after the windstorm. At this point the careful reader will have noticed that the 75% probability that was found previously for our benchmark case was an overall probability to continue and was obtained by pooling all the decisions together (i.e. 75% of the dots in Fig. 2a are above the curve). A corollary is that by applying the cut-or-keep condition to a specific point (a given age and a given post-storm volume) any forest owner facing a windthrow can improve upon this 75%. By making use of the information available in Figure 2a, the forest owner will take a decision that is correct, a priori, 100% of the time. As mentioned before, age and diameter convey the same information (i.e. all the relevant information), for this reason they may be used interchangeably: They both separate all points (decisions) in two disjoint regions as shown in Fig. 2b. Other indicators (e.g. windthrown volume, percentage of trees windthrown) neglect relevant information, therefore they are not fully discriminating: See Figs. 2c and 2d. For instance, whether the most recent storm windthrew few or many trees is not sufficiently informative to take a sound decision with respect to the profitability of keeping the remaining standing trees until maturity. This expected profitability ultimately depends on what is left after the windstorm rather than on what was removed; and on the future expected value of what was left after the windthrow, rather than on the current value of what is left after it.
* All the values are computed for the benchmark rotation age of 44 years.
difference in payoffs increases with increases in the storm probability is explained by the fact that the number of cases in which the owner is confronted to the post-storm decision of cutting or keeping also increases; which in turn increases the absolute frequency of wrong decisions by an owner who chooses to cut by default. We have tested the frequency with which, ceteris paribus, the owner will have greater payoffs by continuing with the remaining standing trees (keep), rather than by immediately harvesting the whole stand (cut). To be more precise, for each scenario (10,000 simulations ran every time for every rotation age) we have computed the amount of times the cut-or-keep rule was evaluated and, among these, the proportion in which the rule prescribed to continue or, alternatively, to harvest and replant. Table 3 shows the results for each windstorm probability and for the benchmark value of 44 years. The number of times that the rule was applied was 22,795 times for 1%, 45,183 times for 2% and 90,150 for 4%. A few comments are in order before proceeding forward: First, we consider only windstorms taking place at stand ages beyond 9 years old (i.e. when the stand has a dominant height above 5 m) since storms that take place at earlier ages are supposed to have no impact on the stand due to the stand’s weak windload. Second, it is very rare, though not impossible, that more than one windstorm hits the stand during the same year. Whenever that happens, the cut-or-keep condition is not applied after every single windstorm but rather only once: at the end of the year. So where it reads that the rule was applied, say, 22,795 times for 10,000 simulations with R = 44 it more precisely means that between ages 9 and 44 there were 22,795 observations (years) for which at least one windstorm event was observed. The precision made, and according to Table 3, the probability to keep (i.e. let the remaining trees grow until maturity after a windstorm event) is above 70% regardless of the scenario analysed. In other words, after a windthrow, keeping the remaining standing trees until maturity is an economically-sound decision more than 70% of the time. This result greatly challenges the assumption made by some papers that support the default strategy of clear-cutting the stand after a windthrow. The results in Table 3 also seem to suggest that, ceteris paribus, there is a monotonous relation between the windstorm probability and the probability to keep: The greater the probability of a windthrow, the smaller the probability of increasing profits by keeping the stand. The decision to cut or keep is affected not only by the windstorm probability, but also by the fixed harvesting costs. Whenever there is a windthrow the owner has to pay some fixed costs to harvest the stand. If the owner decides to remove both the windthrown and the standing trees these costs are only paid once. However, if the owner decides to keep the stand and let the standing trees grow until maturity, the question then becomes: Does the additional growth of the remaining trees justify paying these costs twice, (now and at the maturity date)? The qualitative answer is: It depends on how important these costs are and how far the maturity date is. The smaller the costs and the further from maturity the more likely to continue. Conversely, as the fixed costs of harvesting increase the likelihood of keeping the stand until maturity is reduced. We have performed a sensitivity analysis for different values of the fixed harvesting costs (measured in EUR per hectare). The results are
14 Since the current age and the tree diameter have a biunivocal relationship in our model, current age and tree diameter may be used interchangeably.
6
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Fig. 2. Cut and keep regions for different indicators. Every point represents a decision taken after a windthrow for our benchmark case (r = 3%, CF = 200 EUR ha 1, R = 44 years, = 0.02 ). Red circles ( ) represent “cut” decisions, blue points ( ) show “keep” decisions. Subfigure (2a) and Subfigure (2b) give two disjoint regions, meaning that the two indicators in the vertical and horizontal axis convey all relevant information for a sound decision. This is not the case for the indicators in Subfigure (2c) and Subfigure (2d) which neglect relevant information and generate overlapping “cut” and “keep” regions (dashed area).
To clarify this last point, it is important to recall that a forest stand may endure several windstorms during its life and still be profitable to be kept until maturity. This is particularly true if windthrows take place at a young age when damages are relatively small and the standing trees have time to grow (i.e. the stand has time for its value to pick up). Likewise, as the stand grows older its wind load increases and so do the damages in case of a windthrow. To summarize, it may be profitable to keep a stand which suffered two windstorms at young ages (e.g. ages 10 and 15), but it is not profitable to keep a stand that suffered a single windstorm at an old age (i.e. ages 38 and above).
way of hedging against risk and the decision to continue or not may be understood as an option that may be exercised in some circumstances. Our results show that in most cases (roughly 75% of the time) it is profitable to exercise this option, i.e. continue with the standing trees until maturity. Applying this simple decision rule may result in sound economic gains (up to 90% in some cases) with respect to the – often recommended – rule of thumb of full harvesting and replanting after a storm. We also show that, in the case of maritime pine plantations, an increase in the windstorm risk has a very strong impact on payoffs but not necessarily on the rotation age. The reason why the rotation age is not so affected by risk is that it is more profitable to wait until the diameter is large enough to obtain important timber prices even at the expense of greater exposure to risk. The decision to cut or keep the remaining trees of a stand after a storm depends on (i) the fixed harvesting costs, (ii) the probability of windthrow, (iii) the stand’s age, and (iv) the distance to maturity. In the first case, the owner is more likely to cut immediately when the fixed harvesting costs are more important. With respect to the probability of windthrow (second case), our results suggest that, at optimum, as the windthrow probability increases the owner has a greater interest to cut the stand more often after a storm. For the third, it is more profitable to keep the stand for lower values of
6. Discussion Whenever a windstorm hits a forest stand it may create severe economic damages. The importance of these damages is a function of the wind speed and the trees’ height. We model the decision problem that a forester faces after a windstorm hits the stand: The owner may decide to remove both the windthrown trees and the standing trees and replant or, alternatively, continue with the standing trees until maturity. We propose a new decision rule that allows to compare the economic gains of each option. The proposed decision rule works like a 7
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the stand’s age. This effect seems to be independent of, fourth, the fact that the closer to maturity the more likely to harvest the remaining trees of a stand after a windthrow. There are a number of simplifying assumptions that call for a critical interpretation of the results. To keep the model simple and tractable we have focused mainly on the impact of the decision rule. We thus did not account for price variations, i.e. timber price functions and extraction costs are fixed in our model. Also, we modelled forest owners as risk-neutral profit maximizers rather than risk-averse agents (see e.g. Valsta, 1992) and did not discuss intertemporal substitution issues (Couture et al., 2016). Further, we did not consider thinnings in this setting, in order to keep the modelling simple. Finally, we modelled forest management following the traditional clear-cut system observed in many plantation forests as opposed to continuous-forest-cover systems that may result in higher payoffs (Tahvonen, 2009; Tahvonen, 2016; Jacobsen et al., 2016; Rämö and Tahvonen, 2017). That being said, it is unclear whether uneven-aged forests are less prone to windthrow than even-aged ones (Mason, 2002; Gardiner et al., 2005; Hanewinkel et al., 2014; Díaz-Yañez et al., 2017). Future works should address the risk of windthrown for uneven-aged forests (Couture et al., 2016) and the post storm optimal management. In this paper we analysed a single-species plantation in a specific context (i.e. maritime pine plantations in Landes Region). Yet, there are a number of results that may be generalised to other species and other contexts: First, the use of the cut-or-keep condition may greatly increase the profitability of forest owners facing these adverse storm events. Second, the cut-or-keep condition is particularly useful when dealing with species whose growth (volume wise) is slow and whose prices increase in a non-linear way as the diameter increases, i.e. the case of broadleaved trees. In such case, the option keep is expected to yield greater payoffs in a wider range of situations. Third, the profitability – for the forest owner – of using the cut-or-keep condition is increases with economic risk: The gains, both in absolute and relative terms, are an increasing function of the probability of facing a
catastrophic event. Fourth, economic risk has a very strong impact on payoffs, a mild impact on the rotation age, and a low impact on the propensity to keep the stand after a catastrophic event: In our case this propensity is always above 70% across the different scenarios analysed which suggests that the cut-or-keep condition is a profitable and recurrent tool to improve forest management under risk. CRediT authorship contribution statement Claudio Petucco: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Pablo Andrés-Domenech: Conceptualization, Methodology, Writing original draft, Writing - review & editing, Supervision. Lilian Duband: Methodology, Formal analysis. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors appreciate the helpful suggestions and comments of Robert Haight (USDA Forest Service – Northern Research Station), Anne Stenger (Université de Lorraine, Université de Strasbourg, AgroParisTech, CNRS, INRA, BETA) and Yves Ehrhart (LERFob, AgroParisTech – INRA). The authors would also like to thank Alexandra Niedzwiedz at the Forest Economics Observatory for the support with the empirical data. The UMR 1443 is supported by a grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (ANR-11-LABX-0002-01, Lab of Excellence ARBRE).
Appendix A The parameters of the model are presented in Table 4. Table 4 Model parameters. Parameter
CP (a (t ) = 0) CP (a (t ) r CV
0)
Description
Value
Unit of measure
Planting cost
1,250
EUR ha
Planting cost
Discount rate Unitary harvesting cost
CF
Fixed harvesting costs
V V0
Asymptotic stand volume Initial stand volume
gv d¯ d0 gd
Maximum volume growth rate Asymptotic diameter
h¯ h0 gh k
h
CC ws
0
0.03 8
0, 200, 400
684.332 38.111 0.126 68.885
Initial diameter Diameter growth parameter Diameter curvature parameter Asymptotic dominant height
0 33.02 1.115 44.056
Initial height Height growth parameter Height curvature parameter Storm damage function parameter Storm damage function parameter Storm damage function parameter
0 13.075 0.841 −3.26 0.07
1.02672 × 10 12 5.4701 5 0.49 8
Storm damage function parameter Damage threshold height Share of marketable windthrown trees Unitary clearing cost Storm’s rate of return Wind speed
0.01, 0.02, 0.04 180
8
1
EUR ha 1 Dimensionless EUR m
3
EUR ha
m3 ha
ha
1
1
1
m3 ha 1 Dimensionless cm cm Dimensionless Dimensionless m
m Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless m Dimensionless
EUR m 3 ha 1 Dimensionless km h
1
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Fig. 3. Observed maritime pine prices (average 2009–2014) and the estimated price function.
In Fig. 3 we present the average prices for the mid points of each diameter class and the fitted price function. Timber prices were taken from the French forest sector magazine La fôret privée which has collected timber prices since 1958 (Chevalier et al., 2011). In this data set, prices are recorded each year for different diameter classes. Prices were deflated and then averaged across five years (2009–2014).
ouest de la France. Technical report, ENGREF, Ecole nationale du génie rural, des eaux et des forêts. FRA, Nancy. Loisel, P., 2011. Faustmann rotation and population dynamics in the presence of a risk of destructive events. J. For. Econ. 17 (3), 235–247. Loisel, P., 2014. Impact of storm risk on Faustmann rotation. For. Policy Econ. 38, 191–198. Martell, D.L., 1980. The optimal rotation of a ammable forest stand. Can. J. For. Res. 10 (1), 30–34. Mason, W., 2002. Are irregular stands more windfirm? Forestry 75 (4), 347–355. Meilby, H., Strange, N., Thorsen, B.J., 2001. Optimal spatial harvest planning under risk of windthrow. For. Ecol. Manage. 149 (1), 15–31. Piton, B., 2002. Facteurs de la sensibilité au vent des peuplements. Etude a partir des données de l’IFN relatives aux tempêtes de décembre 1999 (Master’s thesis), FifEngref. 131 p. Rämö, J., Tahvonen, O., 2017. Optimizing the harvest timing in continuous cover forestry. Environ. Resour. Econ. 67 (4), 853–868. Reed, W.J., 1984. The effects of the risk of fire on the optimal rotation of a forest. J. Environ. Econ. Manage. 11 (2), 180–190. Routledge, R., 1980. Effect of potential catastrophic morality and other unpredictable events on optimal forest rotation policy. For. Sci. 26 (3). Tahvonen, O., 2009. Optimal choice between even-and uneven-aged forestry. Nat. Resour. Modeling 22 (2), 289–321. Tahvonen, O., 2016. Economics of rotation and thinning revisited: the optimality of clearcuts versus continuous cover forestry. For. Policy Econ. 62, 88–94. Thorsen, B.J., Helles, F., 1998. Optimal stand management with endogenous risk of sudden destruction. For. Ecol. Manage. 108 (3), 287–299. Valsta, L.T., 1992. A scenario approach to stochastic anticipatory optimization in stand management. For. Sci. 38 (2), 430–447. Xu, Y., Amacher, G., Sullivan, J., 2016a. Amenities, multiple natural disturbances, and the forest rotation problem. For. Sci. 62 (4), 422–432. Xu, Y., Amacher, G.S., Sullivan, J., 2016b. Optimal forest management with sequential disturbances. J. For. Econ. 24, 106–122.
References Bright, G., Price, C., 2000. Valuing forest land under hazards to crop survival. Forestry 73 (4), 361–370. Chevalier, H., Gosselin, M., Costa, S., Bruciamacchie, M., Paillet, Y., 2011. Volatilité des cours du bois par essence et qualité: perspectives pour la gestion forestiere. Revue Forestiere Française 63 (4), 456. Couture, S., Cros, M.-J., Sabbadin, R., 2016. Risk aversion and optimal management of an uneven-aged forest under risk of windthrow: a Markov decision process approach. J. For. Econ. 25, 94–114. Deegen, P., Matolepszy, K., 2015. Economic balancing of forest management under storm risk, the case of the Ore Mountains (Germany). J. For. Econ. 21 (1), 1–13. Díaz-Yañez, O., Mola-Yudego, B., González-Olabarria, J.R., Pukkala, T., 2017. How does forest composition and structure affect the stability against wind and snow? For. Ecol. Manage. 401, 215–222. Faustmann, M., 1849. Berechnung des Werthes, welchen Waldboden, sowie noch nicht haubare Holzbestände für die Waldwirtschaft besitzen. Allgemeine Forst- und JagdZeitung 441–455. Gardiner, B., Marshall, B., Achim, A., Belcher, R., Wood, C., 2005. The stability of different silvicultural systems: a wind-tunnel investigation. Forestry: Int. J. For. Res. 78 (5), 471–484. Haight, R.G., Smith, W.D., Straka, T.J., 1995. Hurricanes and the economics of loblolly pine plantations. For. Sci. 41 (4), 675–688. Hanewinkel, M., Kuhn, T., Bugmann, H., Lanz, A., Brang, P., 2014. Vulnerability of uneven-aged forests to storm damage. Forestry 87 (4), 525–534. Jacobsen, J.B., Jensen, F., Thorsen, B.J., 2016. Forest value and optimal rotations in continuous cover forestry. Environ. Resour. Econ. 1–20. Landmann, G., Danjon, F., Brunet, Y., Meredieu, C., 2010. Expertise collective scientifique et technique à visée prospective sur l’avenir du massif forestier landais – Critère C1: Vulnérabilité aux tempêtes. INRA, Rapport d’experts http://prodinra.inra.fr/record/ 41301. Lemoine, B., Decourt, N., 1969. Tables de production pour le pin maritime dans le sud-
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Cut or keep: What should a forest owner do after a windthrow? a,⁎,1
Claudio Petucco a b c
b
c
, Pablo Andrés-Domenech , Lilian Duband
T
Environmental Research and Innovation (ERIN) Research Department, Luxembourg Institute of Science and Technology, Luxembourg BETA, AgroParisTech, France Office National des Forêts (ONF), France
ARTICLE INFO
ABSTRACT
Keywords: Faustmann Land expectation value Optimal rotation age Risk Windthrow Forestry Pine Plantation
After a windstorm hits a forest stand the forest owner has two possibilities: clear cut and start a new rotation or let the standing trees grow until maturity. We compute the Land Expectation Value (LEV) under the risk of windthrow and introduce an endogenous rule to account for this decision. We compare the result with the default alternative of systematic clear felling and replanting (commonly assumed in existing literature). We have calibrated our model to represent maritime pine plantations in south-western France. Results show that by following this rule, payoffs may increase by up to 90% when the economic risk (i.e. probability of having a windstorm) is high. About 75% of the time it is profitable to keep the standing trees until maturity. Although payoffs are sensitive to economic risk, the optimal rotation age does not change much with significant increases in the risk probability.
1. Introduction Windstorms are one of the most important risks for forest management. The impacts of catastrophic events and the risk of total destruction on the optimal rotation age have been addressed by previous works focussing on even-aged forests (e.g. Martell, 1980; Routledge, 1980; Reed, 1984). In these earlier works three major simplifying assumptions have been used to obtain analytical expressions for the model’s optimal solutions: (i) The probability of a catastrophic event is exogenous; (ii) whenever a catastrophic event hits the forest stand, it is completely destroyed as a result, hence damages are also exogenous; and (iii) the salvage value after the catastrophe is equal to zero. While the assumption of exogeneity of a catastrophic event is rather easy to justify, the other two are harder to meet in the real world. Therefore, several studies have tried to relax these assumptions, often through numerical methods. Valsta (1992) relaxed the assumption of total destruction of the stand by using an uniformly-distributed random variable to represent the fraction of the stand destroyed, and assumed a salvage value. Haight et al. (1995) introduced an age-dependent damage risk and a salvage proportion. Successively, Thorsen and Helles (1998) computed the optimal rotation length and thinning regime using an endogenous damage function that depends on the forest’s characteristics as well as on the forest
management (i.e. thinning operations). While Thorsen and Helles (1998) and Haight et al. (1995) assumed an exogenous land value, Loisel (2011) and Loisel (2014) computed the optimal rotation age and thinning regime that maximise the land expectation value (LEV) – hence taking into account the value of successive rotations – under the risk of destructive events while accounting for the salvage value of windthrown timber. In an evaluation context, rather than in an optimisation one, Bright and Price (2000) proposed a simple and elegant way to compute the expected LEV under the risk of hazards whose realisation implies the end of the rotation. A similar methodology was applied by Deegen and Matolepszy (2015) to compute site-dependent optimal rotation ages and expected LEV based on historical management data in Germany. All these studies have assumed total clearing of the stand after a windthrow, independently of the level of damage. However, following a windthrow, it might be profitable to keep the standing trees until the “prescribed” end of the rotation. The continuation of the rotation after a disturbance was firstly discussed by Xu et al. (2016a,b), who analysed the forest management problem under the risk of two sequential disturbances. They tested three alternative strategies: (i) the systematic clearing and replanting of the stand after the first disturbance (the common assumption in previous literature); (ii) keeping the standing trees after the first disturbance and cutting and replanting after the second disturbance; (iii) keeping the standing trees also after the second disturbance2 and harvest the stand at
Corresponding author. E-mail address: [email protected] (C. Petucco). 1 The contribution of Claudio Petucco in this paper is the result of his PhD thesis work carried out at the former Laboratory of Forest Economics (INRA, AgroParisTech), now BETA (UMR 1443, Université de Lorraine, Université de Strasbourg, AgroParisTech, CNRS, INRA). 2 Xu et al. (2016a,b) limited their analysis to two disturbances, though they argued that the model could be extended to the n-disturbance case. ⁎
https://doi.org/10.1016/j.foreco.2020.117866 Received 9 August 2019; Received in revised form 15 November 2019; Accepted 1 January 2020 0378-1127/ © 2020 Elsevier B.V. All rights reserved.
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maturity. Their results showed that ignoring the continuation options could lead, under some circumstances (i.e. disturbances’ arrival rates, survival probabilities and post-disturbance tree-growth losses), to suboptimal harvest decisions and land expectation values. In their work, Xu et al. (2016a,b) assumed and compared the outcomes of fixed continuation strategies that are kept constant along the planning horizon. However, whether keeping the standing trees is profitable or not will ultimately depend on the cost structure as well as the characteristics of the stand after the windstorm. In this paper we relax the hypothesis of a fixed continuation strategy and introduce a condition to determine when it is profitable to continue with the remaining trees until maturity and, alternatively, when it is best to cut them immediately and replant the whole stand. We assume (i) an exogenous rate of return for windstorms as in previous works, as well as (ii) an endogenous damage function determining the fraction of windthrown trees after a storm; (iii) a positive salvage value for the windthrown trees, and (iv) explicit clearing costs. In Section 2 of this paper we present the theoretical model and in Section 3 we introduce the cut-or-keep decision rule. In Section 4 we calibrate our model with real data for maritime pine in Landes Region in France and run a Monte Carlo simulation. Sections 5 and 6 are devoted to presenting and discussing the results.
The volume harvested is denoted by q and is equal to:
q (a ) =
V0 V (a ) =
Let us assume a forest owner who plants a mono-specific and singleaged forest stand3 at the beginning of the planning horizon, t = 0 . The forest owner begins by planting at that time, and the plantation costs are represented by CP (0) . The age of the forest stand – denoted by a (t ) – is equal to zero both at the initial period (i.e. a (0) = 0 ) and the year after every harvest. Time is discrete and the stand’s age, a (t ) , is measured (i.e. discretised) in one-year steps. Denote by R the rotation length, that is, the age (in years) at which the stand is harvested (i.e. clear-cut). The forest owner decides when to harvest the stand, after which he/she replants and starts with a new plantation of the same type. By choosing the rotation length, R, the forest owner maximises the land expectation value (LEV) which is the infinite sum of discounted net returns from a forest plantation with an infinite number of rotations (Faustmann, 1849). In a deterministic setting, LEV can be written as follows: R
n= 0
R
,
d (a) =
CV q
CF ,
1
(
gv 1
if a = 0 V (a 1) V
)+1
if a > 0,
(4)
d0 d¯ e
if a = 0 {gd [a] 1/ }
if a > 0,
(5)
where d 0 is the diameter of newly planted trees, and d¯ is the maximum diameter that can be achieved in the stand in the long run. Parameter gd > 0 represents the diameter’s maximum growth rate, and > 0 is a curvature parameter. The forest dynamics are described by expressions (4)-(5). See Figs. 1b for its graphical representation. 2.2. The stochastic storm disturbances Following Reed (1984), we model windstorms as random events with a homogeneous Poisson distribution, i.e. windstorm events are assumed independent from one another and occur randomly in time. We consider different types of windstorms, s = 1, …, S , each characterised by a unique rate of return s and maximum wind speed ws . Unlike previous works, we do not consider windstorms as events resulting in total destruction of the stand. Rather, whenever a windstorm hits the stand, the size of the damage is variable and will typically depend on the wind speed, ws , the dominant height, h, and a number of
(1)
where n is an index that takes into account the succession of rotations in time (e.g. n = 0 for the first rotation, n = 1 for the second rotation, and so forth), r is the discount rate and (R) represents the net timber revenues, or stumpage value. These revenues are a function of the trees’ diameter and, therefore, of their age4. The stumpage value, at any given age a, is defined as a function of timber prices and clear-felling costs:
(a) = P (d (a)) q (a)
V a
0
where V0 is the volume of the seedlings planted, V > 0 is the carrying capacity of the stand in terms of volume (i.e. the volume towards which the stand converges in the long run) and gv > 0 is the volume’s maximum growth rate. Note that, in a deterministic setting, one may use age or time interchangeably in expressions (3) and (4). Since the age at which the harvest takes place is known and equal to R, at any given age a, time t may be written as: t = n (R + 1) + a . In a stochastic setting (as in Section 2.2), however, the harvest may take place before maturity and this equivalence does not hold any more. In such case, it is more convenient to write the model’s variables as a function of age. Henceforth, all the relevant variables are presented as a function of age, a. In expression (2) we saw that the stumpage value depends on stem price which in turn depends on the average stem diameter. The averagetree diameter dynamics are represented by the expression below:8
2.1. Stand dynamics and harvesting
CP (0) + (R)(1 + r ) (1 + r )n (R + 1)
(3)
where V (a) is the standing volume at any given age a. Expression (3) tells us that harvest takes place at the end of the rotation alone. We further assume that whenever the harvest takes place, it does at the end of the year6, i.e. forest growth and disturbances within a specific time period occur before the harvesting operations take place. The standing volume in the forest V (a) evolves according to the following dynamics:7
2. The model
max LEV (R) =
V (a) if a = R 0 elsewhere,
(footnote continued) simplifying assumption that CV is constant and not a function of the standing volume or the average tree diameter. As a matter of fact, empirical observation shows that extraction and felling costs are quite similar for a wide range of parameters; that is, for extraction rates comprised between 30 to 100 cubic metres per hectare and for stands whose trees have an average tree volume in the range of 1 to 5 cubic metres. 6 Trees grow from March to October and the harvest, in our model, takes place in December, when growth is sensibly smaller. 7 The volume dynamics are obtained from the discretisation of a continuous logistic growth model. 8 We make the simplifying assumption that all trees are equal and have the same diameter.
(2)
where the timber price, P (·), is a function of the average-tree diameter, d (a) , which, in turn, is a function of age; CV are the extraction and felling costs per unit of volume5; and CF are the fixed harvesting costs. 3 This very well represents the silviculture of maritime pine in Landes that we will analyze later on. 4 Timber revenues are also a function of timber quality. For simplicity, the issue of timber quality is not addressed in this paper. 5 Although extraction and felling costs are a function of both the total volume harvested and the volume of the average tree harvested, we make the
2
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Fig. 1. Forest dynamics functions (1a–c) calibrated for maritime pine plantations in Landes (France). The storm-damage function (1d) was calibrated to represent the maximum wind speed observed during the windstorms in 1999 and 2009 (i.e. 180 km h 1) in Landes.
stand characteristics such as the degree of recency since the last harvest, the plantation density and other technical factors, e.g. the slope of the stand and the type of planting. The stand’s dominant height, h, is a function of age and evolves according to the following expression:
h (a ) =
h0 h¯ e
windthrow is an increasing function of both the wind speed and the dominant height (i.e. fh > 0 and f ws > 0 ). We decided to restrict the arguments of the damage function to these two variables for simplicity, but other variables may be included for the sake of greater realism (Meilby et al., 2001). The exact functional specification used in this paper is an extension of the one given by Piton (2002):
if a = 0 {gh a 1/ k }
if a > 0,
(6)
f h (a), ws =
where h 0 is the height of newly planted trees; h¯ > 0 is the maximum height that can be achieved in the stand in the long run; gh > 0 is the maximum growth rate of the tree height; and k > 0 is a curvature parameter (Fig. 1c). Windstorm damages are measured by the fraction of trees broken or windthrown, that we denote by . This fraction of trees lost due to windthrow is given by:
0 if h (a) < h , (ws, h (a)) = f (h (a), ws ) elsewhere.
e{
+ h (a) + (ws ) }
1 + e{
+ h (a) + (ws ) }
,
(8)
where , , and are positive parameters. In Fig. 1, we illustrate the forest dynamics and the damage function calibrated for a maritime pine plantation in the Landes forest. The parameters and their values are presented in detail in the Appendix. 3. Cut-or-keep rule Most papers modelling windstorms and wind damage make the simplifying assumption that whenever a storm hits the stand either all or none of the trees are windthrown. A few papers allow for a fraction of the trees to be windthrown, but after the windstorm the remaining standing trees are felled –by default– and a new rotation begins. This assumption is both restrictive and unrealistic, but makes modelling simple and analytical solutions possible. For the sake of realism, in our
(7)
According to expression (7), whenever a windstorm hits the forest stand, we observe damages only if the stand’s dominant height is greater than the threshold height h, below which the stand is supposed not to have enough wind load to suffer any damage (Landmann et al., 2010). Conversely, above h, the fraction of trees destroyed by the 3
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model, only a fraction of the standing trees will be lost after a windstorm. So a relevant question to ask is: What should the forest owner do after a windthrow with the remaining (i.e. standing) trees? Suppose a windstorm occurs when the stand is a years old and a fraction of the standing trees is windthrown as a consequence. The forest owner then faces two options: (i) clear cut the remaining standing trees, sell all the timber in the market9, and start with a new rotation by replanting the stand (i.e. cut); or alternatively (ii) let the remaining standing trees grow until maturity or a new windstorm arrives (i.e. keep). The revenues from these two options are gathered and compared in the following expression: ST
a, V +
salvage (a )
E ( ST (R)) + (1 + r ) R a
+
the end of the rotation. Given that we are in a stochastic setting it is necessary to compute its mathematical expectation. Since we assume that the price function11 and costs do not change in time, we can write:
E(
ST
(·) V (a )
CF ,
(10)
where P (d (a )) is the price of timber, which is a function of its diameter d, and CC represents the cost of clearing the windthrown trees. (0 < < 1) accounts for the fact that a fraction of the Parameter windthrown trees are damaged and cannot be sold in the market. Since only a fraction of the total volume is windthrown, and this fraction is denoted by (·) ; the product (·) V stands for the total volume windthrown. The fixed extraction costs CF also apply to windthrown trees. If harvested just after a windstorm, the stumpage value of the standing (and undamaged) trees, ST (a , V ) , is given by the following expression: ST (a ,
V ) = (P (d )
CV ) (1
(·)) V (a ).
(12)
a, V +
E (LEV (R)) 1+r
E ( ST (R)) E (LEV (R)) + . (1 + r ) R a (1 + r ) R a + 1
(13)
The model described above cannot be solved analytically because of both the non-linear nature of the stand dynamics and the use of the cutor-keep condition. Hence, we solve the model numerically using Monte Carlo technique with MATLAB. The code is organised as shown in Table 1. For a given rotation age, the model simulates the evolution of the stand for 300 years projecting the stand growth, harvesting operations and windstorms. The code then sums all the discounted revenues and costs generated by the forest management and the disturbances12. Given that we consider stochastic disturbances, we repeat the stand evolution 10, 000 times, and record the approximation of E(LEV ) for each repetition. In order to find the optimal rotation age, we run this code for all rotation ages between 30 and 65 years. The optimal rotation age is the one giving the highest average E(LEV ) across the 10, 000 trials. The model is calibrated to represent a maritime pine plantation in Landes (France). The volume, tree-height and diameter-growth parameters were computed using the non-linear least-square curve fitting on the Lemoine and Decourt (1969) maritime pine yield tables. For simplicity, in each simulation we considered only one storm type. We use a windstorm’s maximum wind speed of 180 km h−1, in line with the wind speed observed in the region during the last two episodes of 1999 and 2009. In our simulations we perform a sensitivity analysis for different return rates: we tested (i) one windstorm expected every 100 years ( s = 0.01 ); (ii) one windstorm every 50 years ( s = 0.02 ); and (iii) one windstorm every 25 years ( s = 0.04 ).13 When a windthrow is observed, we apply decision rule (13). However, the careful reader will note that E(LEV (R)) is endogenous, since it depends both on the rotation length and on wind disturbances. In order to compute the RHS and the LHS of (13), we need to know E(LEV ) .
(1 + r )
CC )
CF ,
4. Model calibration and Monte Carlo simulations
the salvage value of the (same) windthrown trees, salvage ; and the mathematical expectation of the land expectation value, E(LEV ) . Note that E(LEV ) needs to be discounted R a + 1 years which is when the next rotation will be planted: The remaining trees will be ready for felling at age R, within R a years; however, since replanting takes place the year after the harvest, E(LEV ) has to be discounted one additional year. Note also that the expectation term is needed because there could be additional windstorms before the harvest, thus affecting the harvested volume and, therefore, the pay-off. The salvage value of windthrown trees, salvage , is given by the following expression:
= (P (d (a ))
CV ) E (V (R))
This particularly simple expression allows to determine when it is profitable to replant the whole stand after a windstorm (LHS > RHS); and when it is profitable to let the remaining trees grow until maturity (LHS < RHS). The forest owner is indifferent when the payoff is equal (i.e. LHS = RHS).
(9)
where in the left-hand side (LHS) of (9) we have the payoffs obtained from cut, that is, the stumpage value of the standing trees, denoted by 10 ST (a , V ) ; the salvage value from the windthrown trees , salvage ; and the mathematical expectation of the land expectation value, E(LEV ), which is the expected intrinsic value of the land without trees (i.e. its price in the market). This last term has to be discounted one year because replanting, by assumption, can only take place during the year following the harvest. In the right-hand side (RHS) of expression (9) we have the payoffs from keep, that is, the revenues obtained when the remaining standing trees are allowed to grow until maturity E( ST R(R))a ;
salvage (a )
= (P (d )
where the expected stumpage value is a function of the expected volume, E(V (R)), that can be computed recursively as a function of (i) future expected damages that are themselves a function of age and therefore of time, and of (ii) the storm’s rate of return s . Finally, expression (9) simplifies to the following cut-or-keep condition:
E (LEV (R)) 1+r
E (LEV (R)) , salvage (a ) + (1 + r ) R a + 1
ST (R))
(11)
Note that in expression (11) parameter is missing as there is no loss in the standing trees since they suffered no damage. Note also that the fixed extraction costs CF are missing in expression (11). These costs are to be paid only once: Whenever the harvest takes place. If the owner decides to harvest both the standing trees and the windthrown trees, then CF is not to appear in both expressions, else one would be paying the same cost twice. This is so when the harvesting of the windhtrown trees and the harvesting of the remaining standing trees happen at the same time, i.e. the case in the LHS of (9). Back to the right-hand side (RHS) of expression (9), let us recall that ST (R) is the stumpage value of the standing trees that are harvested at
11
It is important to note that prices are not constant but rather a function of diameter, however this function does not change across time, i.e. in this paper we do not analyse the impact of supply and demand effects on the price of timber. 12 The discounted revenues and costs converge towards zero as time evolves and become negligible after a certain time. We use a planning horizon of 300 years because, with a discount rate of 3%, the gains and losses within such horizon account for more than 99.9% of the total discounted value, i.e. by arbitrarily limiting the time horizon at 300 years we approximate E(LEV ) with an error smaller than 0.1%. 13 Note that in this paper we model windstorms that hit the forest stand, which is not the same as a windstorm hitting a whole massif: It may well happen that a storm hits a forest massif but the stand is spared. This explains why the probability of a windstorm hitting the stand is sensibly lower than that of hitting the whole massif.
9
The timber from both windthrown and standing trees. Trees windthrown by a storm are often damaged and their commercial value is sensibly lower, though far from negligible. 10
4
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Table 1 Sketch of the Monte Carlo algorithm used to determine the optimal rotation age and the land expectation value. fix the storm’s rate of return, the discount rate and the economic parameters for each rotation length R [30, 65] compute the forest dynamics for 300 years, with a harvest every R years check, for each time period, if there is a windstorm: if so compute the damages (·) : if gains from clearing and replanting > gains from waiting until end of rotation: then harvest the standing trees, compute profits and replant: set a (t + 1) = 0 ; else if gains from clearing and replanting gains from waiting to rotation end: (·)) V (t ) ; then compute the new standing volume: V (t ) = (1 sum the discounted revenues and costs for the 300-year horizon (LEV) repeat the 300-year simulation 10,000 times for every rotation length average LEV across the 10,000 replications for every R select the rotation length that maximizes expected land expectation value.
However, E(LEV ) cannot be computed correctly without plugging in the correct values on both sides of (13). To overcome this problem we ran an iterative algorithm, i.e. we ran the program several times using the LEV vector computed in the previous run to feed expression (13) in the successive one. More precisely, in the first run we obtained LEV in a deterministic setting. This value is a first approximation of E(LEV ) that was used to feed expression (13) in the second run. After running the program a second time we obtained as output a new and different set of values for E(LEV ) that were used as inputs for the third run, and so on until convergence was reached. To summarize, the following approximation of the cut-or-keep condition was used: ST
a, V +
LEV (R) 1+r
E ( ST (R)) LEV (R) + , (1 + r ) R a (1 + r ) R a + 1
Table 2 Comparison of optimal rotation lengths and LEVs for different windstorm return rates, without decision rule and with the cut-or-keep rule. No rule Windstorm return rate 0 0.01 0.02 0.04
Cut-or-keep rule
RNR
E(LEVNR )
R CK
E(LEVCK )
46 44 44 43
2,047 1,205 485 −753
46 44 44 43
2,047 1,437 937 135
Difference
E(LEVCK ) -E(LEVNR ) 0 232 452 888
* Optimal rotation ages (R) are given in years. ** Land Expectation Values (E(LEV ) ) are given in EUR per hectare.
(14)
probability of having a windstorm of the above-mentioned characteristics. The second and fourth columns give the optimal rotation age (R) for the no-rule case (NR) and the case in which the cut-or-keep (CK) rule is applied, respectively. The third and fifth columns show the Land Expectation Value for the no-rule case and the cut-or-keep case respectively. And the sixth column gives the difference between the values of E(LEV ) displayed in the third and fifth column. The results in Table 2 show that the E(LEV ) of a stand decreases as the storm’s rate of return increases: E(LEV ) is very sensitive to changes in the windstorm’s probability and for rates equal to 4% and beyond the economic return is negative or very low. The optimal rotation age is also a decreasing function of the windstorm probability: The greater the risk the lower the rotation age. Reducing the rotation age is thus a good hedging strategy for two reasons. First, as the standing volume increases, so does the loss in case of a windthrow, and by cutting more often (i.e. by lowering the rotation age) the expected loss is reduced in case of a catastrophic event. The second reason why reducing the rotation age reduces the risk is to be found in expression (7), according to which the greater the height, the greater the wind load, and thus the greater the probability of a windthrow. Qualitatively speaking these results are in line with the literature (Reed, 1984; Haight et al., 1995; Thorsen and Helles, 1998; Loisel, 2014), however it is interesting to note that the changes in the optimal rotation age are mild, and this even when changes in the windstorm probability are substantial. Table 2 also displays the optimal rotation age and E(LEV ) when the cut-or-keep decision rule is applied after a storm. When the probability of having a windstorm is equal to zero, the cut-or-keep decision rule never applies (i.e. the rule is inactive and E(LEV ) in the first row is the same in both cases). However, as the windstorm probability increases so does the difference in profits between the cut-or-keep case and the norule case. For instance, for a windstorm probability of 1%, a forest owner that decides optimally when to cut and when to continue after a windstorm may earn 20% more (EUR 232 per hectare) than another owner who – by default – cuts the whole stand after a windthrow and replants it anew afterwards. When the yearly probability of having a windstorm is 2% the gain may amount to roughly 90% (EUR 937 per hectare as opposed to EUR 485 per hectare). The reason why the
where LEV (R) , rather than E (LEV (R)) , was used. In the second run LEV (R) was approximated by its deterministic counterpart. In the third and successive runs LEV (R) was proxied by the outcome of the previous run. The iterative process stopped when the LEV-vectors obtained by two consecutive runs were identical. Convergence was obtained after 4, 5, and 7 iterations for the scenarios characterised by 0.01, 0.02 and 0.04 windstorm’s return rate, respectively. Regarding the other data used in the simulations, the cost of planting is assumed equal to EUR 1 per seedling, with 1,250 seedlings planted per hectare. The unitary cost of harvesting and the cost of salvage extraction are both EUR 8 per m3 . We assumed a fixed cost of management operations equal to EUR 200 per hectare. Timber prices were taken from the French forest sector magazine La fôret privée which has collected timber prices since 1958 (Chevalier et al., 2011). The data used were the maximum observed prices per diameter class for years 2009–2014. We corrected these prices by deflating them first and, then, we averaged them across years within each diameter class. Successively, each price was coupled with the mid point of its corresponding diameter class. In this way we defined a set of points in the pricediameter space. Last, we fitted a logistic function to these points by using the non-linear least-square curve fitting. The price function as well as the model’s parameters are presented in greater detail in the Appendix. 5. Results Using Monte Carlo technique we ran simulations for a number of scenarios with different windstorm return rates (i.e. 0%, 1%, 2% and 4%). For all these scenarios we computed both the (approximated) mathematical expectation of the Land Expectation Value (E(LEV ) ) and the optimal rotation age. Two cases of figure were considered: (i) when no decision rule is applied, which means that after a windstorm the whole stand is, by default, cut down and then replanted; and (ii) with the cut-or-keep decision rule presented above, that checks after every windthrow whether the remaining standing trees should be left to grow until maturity or rather fully cut down and replanted. The results obtained are summarized in Table 2. From left to right: The first column in Table 2 stands for the yearly 5
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summarized in the lower part of Table 3. Three values have been analysed: the benchmark value of EUR 200 per hectare and the alternatives EUR 0 per hectare and EUR 400 per hectare. For all three cases we have kept the windstorm probability fixed and equal to its benchmark value. The results in Table 3 show that the probability to continue is quite sensitive to these fixed costs. The probability of keeping, though high in all cases, is a decreasing function of the fixed harvesting costs.
Table 3 Outcomes of the cut-or-keep decision rule expressed in percentage of “continue” decisions (keep) and “harvest & replant” decisions (cut), under different scenarios of windstorm return rate and fixed costs. Rate of return
Fixed costs (EUR/ha)
N. of decisions
Continue (keep)
Harvest & Replant (cut)
0.01 0.02 0.04
200 200 200
22795 45183 90150
78.5% 74.9% 73.7%
21.5% 25.1% 26.3%
0.02 0.02 0.02
0 200 400
45697 45183 45476
81.2% 74.9% 72.1%
18.8% 25.1% 27.9%
5.1. What should a forest owner do after a windthrow? In the previous section, it was showed that, after a windthrow, it is profitable to keep a maritime pine stand until maturity more than 70% of the time. In this section, we explore more closely the factors that explain this decision to keep the stand until maturity. After a windthrow, the forest owner may take the decision to cut or keep the stand based on a number of indicators, e.g. current age of the stand, average tree diameter, standing volume before and after the storm, percentage of trees windthrown. We have analysed the outcomes of our simulations for these indicators and only two allow to explain the decision to cut or keep: the current age of the stand and the standing volume after the storm.14 Fig. 2a shows a plot of the results obtained for our benchmark case (r = 3%, CF = 200 EUR ha 1, R = 44 years, = 0.02 ). Every point in Fig. 2a represents a decision taken after a windthrow. Whenever an observation (point) is plotted above the curve (blue dots) the stand was kept. Likewise, whenever a point is located below the curve (red circles) the stand was harvested. The plot of cut and keep decisions in Fig. 2a gives two disjoint regions, meaning that all the relevant information needed to take a sound decision is embedded in the age of the stand and the standing volume after the windthrow (i.e. the system behaves in a markovian way with respect to these two variables). The plot also shows that as the age of the stand increases the owner has a greater incentive to cut the stand after a windthrow. We can interpret the figure as follows: The likelihood to cut the stand after a windthrow is age-dependent and increases monotonically with the age of the stand, up to a threshold age (a ) beyond which, if a windthrow happens, the owner will never find it profitable to keep the stand. For our benchmark case, a equals 37 years. Hence, whenever a windthrow takes place in a stand whose trees are 38 years of age or above, the owner will harvest the stand immediately afterwards, and regardless of the standing volume after the windstorm. At this point the careful reader will have noticed that the 75% probability that was found previously for our benchmark case was an overall probability to continue and was obtained by pooling all the decisions together (i.e. 75% of the dots in Fig. 2a are above the curve). A corollary is that by applying the cut-or-keep condition to a specific point (a given age and a given post-storm volume) any forest owner facing a windthrow can improve upon this 75%. By making use of the information available in Figure 2a, the forest owner will take a decision that is correct, a priori, 100% of the time. As mentioned before, age and diameter convey the same information (i.e. all the relevant information), for this reason they may be used interchangeably: They both separate all points (decisions) in two disjoint regions as shown in Fig. 2b. Other indicators (e.g. windthrown volume, percentage of trees windthrown) neglect relevant information, therefore they are not fully discriminating: See Figs. 2c and 2d. For instance, whether the most recent storm windthrew few or many trees is not sufficiently informative to take a sound decision with respect to the profitability of keeping the remaining standing trees until maturity. This expected profitability ultimately depends on what is left after the windstorm rather than on what was removed; and on the future expected value of what was left after the windthrow, rather than on the current value of what is left after it.
* All the values are computed for the benchmark rotation age of 44 years.
difference in payoffs increases with increases in the storm probability is explained by the fact that the number of cases in which the owner is confronted to the post-storm decision of cutting or keeping also increases; which in turn increases the absolute frequency of wrong decisions by an owner who chooses to cut by default. We have tested the frequency with which, ceteris paribus, the owner will have greater payoffs by continuing with the remaining standing trees (keep), rather than by immediately harvesting the whole stand (cut). To be more precise, for each scenario (10,000 simulations ran every time for every rotation age) we have computed the amount of times the cut-or-keep rule was evaluated and, among these, the proportion in which the rule prescribed to continue or, alternatively, to harvest and replant. Table 3 shows the results for each windstorm probability and for the benchmark value of 44 years. The number of times that the rule was applied was 22,795 times for 1%, 45,183 times for 2% and 90,150 for 4%. A few comments are in order before proceeding forward: First, we consider only windstorms taking place at stand ages beyond 9 years old (i.e. when the stand has a dominant height above 5 m) since storms that take place at earlier ages are supposed to have no impact on the stand due to the stand’s weak windload. Second, it is very rare, though not impossible, that more than one windstorm hits the stand during the same year. Whenever that happens, the cut-or-keep condition is not applied after every single windstorm but rather only once: at the end of the year. So where it reads that the rule was applied, say, 22,795 times for 10,000 simulations with R = 44 it more precisely means that between ages 9 and 44 there were 22,795 observations (years) for which at least one windstorm event was observed. The precision made, and according to Table 3, the probability to keep (i.e. let the remaining trees grow until maturity after a windstorm event) is above 70% regardless of the scenario analysed. In other words, after a windthrow, keeping the remaining standing trees until maturity is an economically-sound decision more than 70% of the time. This result greatly challenges the assumption made by some papers that support the default strategy of clear-cutting the stand after a windthrow. The results in Table 3 also seem to suggest that, ceteris paribus, there is a monotonous relation between the windstorm probability and the probability to keep: The greater the probability of a windthrow, the smaller the probability of increasing profits by keeping the stand. The decision to cut or keep is affected not only by the windstorm probability, but also by the fixed harvesting costs. Whenever there is a windthrow the owner has to pay some fixed costs to harvest the stand. If the owner decides to remove both the windthrown and the standing trees these costs are only paid once. However, if the owner decides to keep the stand and let the standing trees grow until maturity, the question then becomes: Does the additional growth of the remaining trees justify paying these costs twice, (now and at the maturity date)? The qualitative answer is: It depends on how important these costs are and how far the maturity date is. The smaller the costs and the further from maturity the more likely to continue. Conversely, as the fixed costs of harvesting increase the likelihood of keeping the stand until maturity is reduced. We have performed a sensitivity analysis for different values of the fixed harvesting costs (measured in EUR per hectare). The results are
14 Since the current age and the tree diameter have a biunivocal relationship in our model, current age and tree diameter may be used interchangeably.
6
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Fig. 2. Cut and keep regions for different indicators. Every point represents a decision taken after a windthrow for our benchmark case (r = 3%, CF = 200 EUR ha 1, R = 44 years, = 0.02 ). Red circles ( ) represent “cut” decisions, blue points ( ) show “keep” decisions. Subfigure (2a) and Subfigure (2b) give two disjoint regions, meaning that the two indicators in the vertical and horizontal axis convey all relevant information for a sound decision. This is not the case for the indicators in Subfigure (2c) and Subfigure (2d) which neglect relevant information and generate overlapping “cut” and “keep” regions (dashed area).
To clarify this last point, it is important to recall that a forest stand may endure several windstorms during its life and still be profitable to be kept until maturity. This is particularly true if windthrows take place at a young age when damages are relatively small and the standing trees have time to grow (i.e. the stand has time for its value to pick up). Likewise, as the stand grows older its wind load increases and so do the damages in case of a windthrow. To summarize, it may be profitable to keep a stand which suffered two windstorms at young ages (e.g. ages 10 and 15), but it is not profitable to keep a stand that suffered a single windstorm at an old age (i.e. ages 38 and above).
way of hedging against risk and the decision to continue or not may be understood as an option that may be exercised in some circumstances. Our results show that in most cases (roughly 75% of the time) it is profitable to exercise this option, i.e. continue with the standing trees until maturity. Applying this simple decision rule may result in sound economic gains (up to 90% in some cases) with respect to the – often recommended – rule of thumb of full harvesting and replanting after a storm. We also show that, in the case of maritime pine plantations, an increase in the windstorm risk has a very strong impact on payoffs but not necessarily on the rotation age. The reason why the rotation age is not so affected by risk is that it is more profitable to wait until the diameter is large enough to obtain important timber prices even at the expense of greater exposure to risk. The decision to cut or keep the remaining trees of a stand after a storm depends on (i) the fixed harvesting costs, (ii) the probability of windthrow, (iii) the stand’s age, and (iv) the distance to maturity. In the first case, the owner is more likely to cut immediately when the fixed harvesting costs are more important. With respect to the probability of windthrow (second case), our results suggest that, at optimum, as the windthrow probability increases the owner has a greater interest to cut the stand more often after a storm. For the third, it is more profitable to keep the stand for lower values of
6. Discussion Whenever a windstorm hits a forest stand it may create severe economic damages. The importance of these damages is a function of the wind speed and the trees’ height. We model the decision problem that a forester faces after a windstorm hits the stand: The owner may decide to remove both the windthrown trees and the standing trees and replant or, alternatively, continue with the standing trees until maturity. We propose a new decision rule that allows to compare the economic gains of each option. The proposed decision rule works like a 7
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the stand’s age. This effect seems to be independent of, fourth, the fact that the closer to maturity the more likely to harvest the remaining trees of a stand after a windthrow. There are a number of simplifying assumptions that call for a critical interpretation of the results. To keep the model simple and tractable we have focused mainly on the impact of the decision rule. We thus did not account for price variations, i.e. timber price functions and extraction costs are fixed in our model. Also, we modelled forest owners as risk-neutral profit maximizers rather than risk-averse agents (see e.g. Valsta, 1992) and did not discuss intertemporal substitution issues (Couture et al., 2016). Further, we did not consider thinnings in this setting, in order to keep the modelling simple. Finally, we modelled forest management following the traditional clear-cut system observed in many plantation forests as opposed to continuous-forest-cover systems that may result in higher payoffs (Tahvonen, 2009; Tahvonen, 2016; Jacobsen et al., 2016; Rämö and Tahvonen, 2017). That being said, it is unclear whether uneven-aged forests are less prone to windthrow than even-aged ones (Mason, 2002; Gardiner et al., 2005; Hanewinkel et al., 2014; Díaz-Yañez et al., 2017). Future works should address the risk of windthrown for uneven-aged forests (Couture et al., 2016) and the post storm optimal management. In this paper we analysed a single-species plantation in a specific context (i.e. maritime pine plantations in Landes Region). Yet, there are a number of results that may be generalised to other species and other contexts: First, the use of the cut-or-keep condition may greatly increase the profitability of forest owners facing these adverse storm events. Second, the cut-or-keep condition is particularly useful when dealing with species whose growth (volume wise) is slow and whose prices increase in a non-linear way as the diameter increases, i.e. the case of broadleaved trees. In such case, the option keep is expected to yield greater payoffs in a wider range of situations. Third, the profitability – for the forest owner – of using the cut-or-keep condition is increases with economic risk: The gains, both in absolute and relative terms, are an increasing function of the probability of facing a
catastrophic event. Fourth, economic risk has a very strong impact on payoffs, a mild impact on the rotation age, and a low impact on the propensity to keep the stand after a catastrophic event: In our case this propensity is always above 70% across the different scenarios analysed which suggests that the cut-or-keep condition is a profitable and recurrent tool to improve forest management under risk. CRediT authorship contribution statement Claudio Petucco: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Pablo Andrés-Domenech: Conceptualization, Methodology, Writing original draft, Writing - review & editing, Supervision. Lilian Duband: Methodology, Formal analysis. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors appreciate the helpful suggestions and comments of Robert Haight (USDA Forest Service – Northern Research Station), Anne Stenger (Université de Lorraine, Université de Strasbourg, AgroParisTech, CNRS, INRA, BETA) and Yves Ehrhart (LERFob, AgroParisTech – INRA). The authors would also like to thank Alexandra Niedzwiedz at the Forest Economics Observatory for the support with the empirical data. The UMR 1443 is supported by a grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (ANR-11-LABX-0002-01, Lab of Excellence ARBRE).
Appendix A The parameters of the model are presented in Table 4. Table 4 Model parameters. Parameter
CP (a (t ) = 0) CP (a (t ) r CV
0)
Description
Value
Unit of measure
Planting cost
1,250
EUR ha
Planting cost
Discount rate Unitary harvesting cost
CF
Fixed harvesting costs
V V0
Asymptotic stand volume Initial stand volume
gv d¯ d0 gd
Maximum volume growth rate Asymptotic diameter
h¯ h0 gh k
h
CC ws
0
0.03 8
0, 200, 400
684.332 38.111 0.126 68.885
Initial diameter Diameter growth parameter Diameter curvature parameter Asymptotic dominant height
0 33.02 1.115 44.056
Initial height Height growth parameter Height curvature parameter Storm damage function parameter Storm damage function parameter Storm damage function parameter
0 13.075 0.841 −3.26 0.07
1.02672 × 10 12 5.4701 5 0.49 8
Storm damage function parameter Damage threshold height Share of marketable windthrown trees Unitary clearing cost Storm’s rate of return Wind speed
0.01, 0.02, 0.04 180
8
1
EUR ha 1 Dimensionless EUR m
3
EUR ha
m3 ha
ha
1
1
1
m3 ha 1 Dimensionless cm cm Dimensionless Dimensionless m
m Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless m Dimensionless
EUR m 3 ha 1 Dimensionless km h
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Fig. 3. Observed maritime pine prices (average 2009–2014) and the estimated price function.
In Fig. 3 we present the average prices for the mid points of each diameter class and the fitted price function. Timber prices were taken from the French forest sector magazine La fôret privée which has collected timber prices since 1958 (Chevalier et al., 2011). In this data set, prices are recorded each year for different diameter classes. Prices were deflated and then averaged across five years (2009–2014).
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