Low Carbon Economy, 2010, 1, 25-28
doi:10.4236/lce.2010.11004 Published Online September 2010 (http://www.SciRP.org/journal/lce)
1
An Analytical Optimal Strategy of the Forest Asset
Dynamic Management under Stochastic Timber
Price and Growth: A Portfolio Approach
Jianwu Xiao*, Wenxing Kang, Shaohua Yin, Hong Zhai
Central South University of Forestry & Technology, Changsha, China.
Received February 29th, 2010; revised September 28th, 2010; accepted September 30th, 2010.
ABSTRACT
Considering the valuation of forest stands based on revenue from wood sales, concession policy (such as carbon sub-
sidies) and associated costs, the paper focuses on the stochastic control model to study the forest asset dynamic manage-
ment. The key contribution is to find the optimal dynamic strategy about harvesting quantity in the continual and multi-
ple periods in conditions of stochastic commodity price and timber growth by using portfolio approach. Finally, an
analytical optimal strategy is obtained to analyze the quantification relations through which some important conclu-
sions about the optimal forest management can be drawn.
Keywords: Forest Management, Analytical, Stochastic Price and Growth, Portfolio, Carbon Subsidies
1. Introduction
Forest ecosystem harbors a large potential for carbon
sequestration and biomass production. When the public
good benefits of carbon sequestration are considered, the
cash flows from forest management include not only
timber value but also carbon subsidies. It is necessary to
study how the carbon credit payment influences the deci-
sions of forest harvesting.
Graeme Guthrie and Dinesh Kumareswaran [1] con-
sidered the effect of carbon credit payment schemes on
forest owners’ harvest decisions by using a real options
model. They studied two possible payment schemes: one
where the government rents the carbon sink, in which
case the carbon credit payment is proportional to the
current carbon stock and another where the government
buys the carbon sink, in which case the carbon credit
payment will be proportional to the change in the carbon.
They referred to rental scheme as the tree-based carbon
credit payment scheme but did not give the detail of con-
trast.
According to this classification, we found an analytical
optimal dynamic strategy about harvesting quantity in
conditions of stochastic commodity price and timber grow-
th under the buying scheme by using portfolio approach
[2]. In this paper, we will focus on the rental scheme to
found the strategy about harvesting quantity and compare
with the tow results to draw some conclusions about the
selection of carbon subsidies schemes.
The paper is organized as follows. A portfolio model,
including carbon sequestration under stochastic wood
prices and growth, is introduced in Section 2. In Section
3 we solve the model and obtain the analytical optimal
strategy of the forest asset dynamic management by ap-
plying stochastic control method in portfolio field. Fi-
nally, Section 4 contains some conclusions.
2. The Stochastic Control Model
2.1. The Stochastic Prices
Under stochastic prices, if the harvest is delayed until the
next period, the owner will face uncertainty over whether
prices will be higher or lower than the current period.
According to the geometric Brownian motion, suppose
the price of timber, P ($/m3), characterized by the fol-
lowing stochastic differential equation:
1,dPPdtPd 1



 (1)
where both the expected percentage growth rate

(drift term) and the volatility coefficient 1

are exoge-
nously given positive constants and 1

is Brownian
motion, t is the current time [3-5].

26 An Analytical Optimal Strategy of the Forest Asset Dynamic Management under Stochastic Timber Price and
Growth: A Portfolio Approach
2.2. The Stochastic Growth
Define I (m3) as inventory of timber (or biomass volume),
It + 1 and It as the stocking levels at age of t + 1 and t,
as the control variable (the quantity of cutting)
of the time t which is depended on the market price P and
stocking level I,
(,,)qPIt
(,(,, ))
t
g
tI qPIt
q
as the timber growth
function with I and . The relationship of all these va-
riables follows:
1(,(, ,))(, ,), tt t
I
IgtIqPItqPIt
  (2)
If we assume that growth is governed by the stochastic
process, the timber volume I satisfies the stochastic dif-
ferential equation:
22
22
()
[(1)]
dIIq dtdqdt
Iqdtd

,


 
  
]
(3)
The inventory growth rate,[(1)
I
q

  , depends on
the cutting rate policy q(P, I, t) and can be either negative
or positive. The parameter

corresponds to the inven-
tory growth rate as a percentage of the residual inventory.
The coefficient 2

is the volatility parameter repre-
senting the uncertainty over the inventory growth rate
and 2

is Brown motion [6-8]. nia
Under these assumptions, the forest owner's optimal
problem is to determine the control strategies
at the different age t for the maximization of a so-called
value function V during the period0, where 0is
the age of the mature forest that is permitted to be cut,
and can be noted as 0 for simplification, and T is the final
time when forest is harvested in one time or the expira-
tion year of the forest lease (e T),
We assume that management decisions can be imple-
mented only at fixed time points01 n
*(,,)qPIt
t
( )}
ett
tqI.
T
(,)tt T
min{TT
, ......,tt
,
t

, e.g.,
only on a yearly basis, and action payments are also re-
ceived at these specified time points only.
2.3. The Value Function
Under the optimality portfolio principle, the value of
forest is composed of log market value and standing for-
est value, and the value function V about time t can be
decomposed into the sum of the immediate profit

and the expected discounted continuation value:
()EV
*0
[0,min( ,)]
(,,)max{(,,)
[(,,), ]}.
T
t
qIq
tt ttt t
VPIte Pqtdt
EV PIttP I




 


 (4)
where q is the maximum annual cutting permission of
policy,

is the resource manager’s risk-free discount
rate, and the immediate profit

is composed of amen-
ity value of standing forest (A) and log value (B) minus
the operating costs of harvesting and management (C)
(including the appropriate amount of carbon credits that
must be purchased back once the harvest is performed).
Especially, if we consider the payments arising from
carbon sequestration and suppose that carbon payments
are received from any increment in standing volume, the
CO2 prices is a constant number C, and the parameter γ
(a conversion factor) states how many tons of CO2 are
sequestered in 1 m3 of wood, the value of standing forest
will be as follows:
P
(, )()
CC
A
PI PIq


 (5)
The log value is
(,, )(1),BPqPq



 (6)
where

corresponds to the tax rate of revenues.
The cost function is given by the quadratic equation
2
01 2
() ,Cqaaq aq  (7)
where 0 is the fixed cost, 1 is the variable cost, and
2 is the quadratic term reflecting increasing marginal
cost. The quadratic functional form is not critical, and is
chosen solely for its algebraic simplicity.
a a
a
Then, the cash flow from forest [4,8,9] is
2
01 2
(,,)( ,)(,,)()
()(1) (
C
C
PqtAPIB Pq Cq
PIqPqaaq aq


 

),
(8)
So, the value function under the above assumptions
follows the equation:
(,,)VPIt

*0
[0,min( , )]
2
01 2
max{[() (1)
()][(,,)
T
t
C
qIq
tt ttt t
ePIqPq
aaqaqdtEVP It tPI




 
 
 

,]}.
(9)
3. An Analytical Solution
According to the above model, the classical tools of sto-
chastic optimal control and maximum principle [10,11]
lead to the following Hamilton-Jacobi-Bellman (HJB)
equation:
22 2
2112
2
01
[(1)]
11
22
{() [(1)]},
tIP
PP IIPI
t
CC
VVIqVPV
PVV PV
ePIaP Paqaq

 



 
 

2
(10)
We denote it simply as:
22 2
2112
2
01 2
[(1)]
11
22
(),
tI
PP IIPI
t
VVIqVPV
PVV PV
eqq



 

P

 




a
(11)
where
00
11
22
,
(1 ),
,
C
C
PIa
PP
a


 




 

(12)

An Analytical Optimal Strategy of the Forest Asset Dynamic Management under Stochastic Timber Price and 27
Growth: A Portfolio Approach
From (11), the optimal policy satisfies
*
q
*
1
2
1[(1 )],
2
t
I
qVe




 (13)
Putting (13) in (11), we obtain another partial differen-
tial equation:
22
2
2
112 0
2 2
211
22
1
2
1
2
(1 )
(1 ).
424
tI PPP
t
II PI
tt
II
VV IVPVPV
VPVe
eVV e

2



 
 




 




(14)
We try to find a solution to (14) in the following way:
(,),
t
VefPI
 
 (15)
Introducing this in (14), we obtain:
22
2
2
2
112
2
2
11
0
22
1
22
1(1
24
(1 ),
24
IP PP
II PII
I
fIfPf Pf
fPf f
f
 


 

 



2
)

(16)
Namely, as that:
2
22
1
2
1
12
2
2
22 2
2
2
22 2
111
22
1(1)
224
()(1) (1)(1 )
2
1(1)
24
()(1)2 4
24
IIIC I
C
IPI
PPP
CCC
fIffPIf
a
Pa
2
02
2
I
f
Pf Pf
a
PfPfP
a
PaPaP aaa
P
aa


 





 
 



 

a

(17)
Decompose (,)
f
PI as:
()(),
f
gI hP (18)
Introducing this in (17), we obtain:
2
22 1
1
22
2
2
222 1
2
22
222
1102
2
()(1) 1(1)
24 2
(1)(1 )2
2
()(1)
1(1)
24 2
24
2.
4
C
III I
CIP
C
PP
CC
Pa
I
I
gg gg
aa
PIPgh Ph
a
Pa
Ph PP
aa
PaP aaag
a
 






 
 
 








(19)
Assume that
2
112 22
(),.( ),
g
IkIlhPkPlPm  (20)
Introducing this in (18), we obtain:
2
2
1
11
2
2
1
11
22
22
222221 2
2
1
2
2
22 2
1102
2
(1 )
(2)[ 4
[](1) (1 )
2][
24
(1)(1 )
22][2
2
[](1)
]2
2
24
0,
4
C
C
C
CC
kkPI k
a
Pa kl
aa
kkkPlk
a
Pa Pm
a
PaP aaa
a


 


l

 




 
 
 

 



(21)
It is not difficult to decide 11 22 by the
method of undeterminated coefficients, such as
2
,,. ,,klklm
1,(2 0)
2
C
P
k




 (22)
We are not usually interested in the value function, but
rather in the optimal strategy. From (12), (13), (15), (18)
and (22), it is given by:
*
1
2
121
[(1 )]
22
C
qPP
a




 
a
(23)
4. Conclusions
It is not difficult to draw some conclusions about the
quantity of harvesting from (22):
1) The quantity is decided by the log market price (P),
the concession value or the carbon price (PC), the timber
growth average speed (

), the discount rate (

), the tax
rate (

), the carbon transform coefficient (γ), and the
costs coefficients (1and 2), but it has nothing with the
gross of present forest (I), the fixed cost (0), the price
volatility (
a a
a
1

), and the timber growth volatility (2

).
2) It is obvious that the more expensive timber price is
and the lower tax rate is, the more trees will be cut down.
3) Normally, two times of the discount rate is larger
than the timber growth rate (2

 ), so the more con-
cession value there is, the less trees will be cut down. It
means that the forest concession or the carbon credit
payment schemes can discourage deforestation. Other-
wise, if the timber growth average speed is faster than
two times of the discount rate, the owner will harvest
more forest to pursuit more wood sales value, with the
fast-growing forestry as an example. In another word,
only policies such as subsidies from government are not
able to satisfy the forest owners’ profits. So it is impor-
tant to improve the management and technology to in-
crease the growth rate of forest.
4) Obviously, the more cost of harvesting, the less

28 An Analytical Optimal Strategy of the Forest Asset Dynamic Management under Stochastic Timber Price and
Growth: A Portfolio Approach
trees will be cut down. But we find that the fixed cost
does not influence it. The cost is also can be regarded as
the penalty for the destruction.
5) As said above, the harvesting strategy has nothing
with the gross of present forest, so the control of the port-
folio is suitable for any scope of forest regardless of the
amount.
In a word, we believe many more interesting conclu-
sions will be brought under the different assumptions by
using the portfolio approach, and welcome more resear-
chers to take part in the field.
5. Acknowledgements
The author is pleased to acknowledge the financial support
of Science Foundation of Science & Technology Depart-
ment, Forestry Department, and Social Science Research
Funds in Hunan Province, the Key Research Institute of
Philosophies and Social Sciences in Hunan Universities,
Central South University of Forestry & Technology, China,
and Eurasia–Pacific Uninet, Austria. The author would li-
ke to express sincere thanks to Prof. Franz Wirl who
chairs Industry Economy, Energy Economy and Envi-
ronment Economy in Faculty of Business, Economics
and Statistics, University of Vienna.
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