THE OPTION VALUE OF FOREST CONCESSIONS IN AMAZON RESERVES

1
THE OPTION VALUE OFFOREST CONCESSIONS IN
AMAZON RESERVES
Katia Rocha Ajax R. B. Moreira Leonardo
Carvalho Eustáquio J. Reis
katia_at_ipea.gov.br
- IPEA - Institute for Applied Economic Research
of Brazilian Government
2
Forest Lease in Legal Amazon - Overview

• Brazilian Government
• Planning to implement
• Natural Forest concession
• in Legal Amazon
• Legal Amazon 500 millions hectares
• Volume estimated 60 billions m3 of wood
• Annual production 25 millions m3 of wood
  
• Area for logging 3 of Legal Amazon
• Discussion Increasing logging area up to 12
  
• Legal process Analyzed by the Brazilian
  congress.

3
Forest Lease in Legal Amazon - Overview

• Participation on
• international market
• 4 of global exportations
• Expansion over next decade
• - gradual exhaustion of the Asian forestry
  resources -
• Regulatory Policies
• the minimum inventory held in the lease area,
• the maximum extraction rates allowed,
• the use of environmental handling techniques

4
Environmental and Economical Issues

• We use Real Option Valuation based on the
  Economic Market Value of concession
• Focus on Expected Cash Flows coming from
  Timber Harvest
• For Real Options Valuation based on social
  benefits
• Conrad (1997) Ecological Economics Analysis
  On the option value of old-growth forest The
  case of Headwaters Forest old-growth coast
  redwood
• Real Option Valuation based on Stochastic
  Amenity flow the sum of non-timber benefits
  (wildlife habitat, flood control and visitation)

5
Introduction to the Model

• Extend Morck, Schwartz and Stangeland
  (JFQA-1989) The Valuation of Forestry Resources
  under Stochastic Prices and Inventories with
  some modifications
• Mean Reverting Process for Timber Price
  instead of GBM
• Uncertainty over Current Timber Inventory
  (volume of biomass) - use of spatial
  econometric models -
• Comparisons between ROT and NPV
• Minimum Timber Inventory has to be preserved
• Extraction cost is a linear function of
  production
• Dynamic Programming Approach instead of
  Contingent Claims
• Brazil does not have Environmental Commodities
  such as Forest
• Products ? Risk Premium estimate and Arbitrage
  Theory become
• a difficult task ? Risk Premium estimates is the
  current work

6
Spatial Econometric Model

• Realistic Assumption
• The amount of timber in the lease area
• - current inventory (biomass) - is
  uncertainty
• We have only sample data identified as points
  
• (1 ha - small area) in any Amazon location
• ? We have to Estimate the Probability
  distribution
• of logging volumes in concession
  areas.

?
?
?
?
Sample data
?
?
Concession Area
7
Spatial Econometric Model

• The volume distribution is specified in a
  spatial model
• Relates the density of biomass (b) with the
  density of
• neighboring areas, and explanatory variables
  (x) which
• are measured for the whole area.
• The explanatory variables -x- considered are
• Geological and Ecological factors such as
• kind of soil, vegetal cover, altitude, distance
  from the sea
• Climatic factors including
• rainfall and mean temperature per quarter of
  the year.

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Concession Value under Uncertainty over Current
Timber Inventory

• f ( I ) probability distribution function
• estimated for the current timber inventory
  in the area.
• The Option value - F(P,t) - considering the
  uncertainty
• over the current Timber Inventory

where F( P , I , t ) is the Option Value for all
possible Current Inventories
f ( I ) LN ( 25m3/ha, 0.41 - associated normal-
)
9
Timber Prices

• Timber price time series data
• Mahogany Brazilian logs
• Hardwood logs - Malaysia
• (International Financial Statistics - IMF)
• Softwood Logs - USA
• (International Financial Statistics - IMF)

10
Timber Prices
Price (jan 82 / jan 01) - /m3 - in real prices
of 95

Stationary Process
11
Timber Price Stochastic Process.

• We model timber prices as Arithmetic Mean
  Reversion Process (MRP)

dP changes in price P Timber Price (/m3)
long-run equilibrium mean sP volatility
parameter h reversion speed
Stationary Process - Natural choice for
commodities Assumption ? Price Level is
sufficiently high therefore negative prices have
very small probability
12
Timber Price Estimates

• AR(1) process ?Pt a bPt et
  etN(0,?2)
• Mahogany and USA Softwood logs present unit
  root processes (b0) which is not reasonable.
• Therefore we consider Malaysian data that better
  describes timber prices process.

13
Timber Inventory Stochastic Process

• We use the standard Stochastic Differential
  Equation
• from the population ecology literature

dI changes in Inventory I Inventory of
Timber in the leasehold (m3/ha) mI average
growth rate in of timber inventory held (
p.a.) sI volatility parameter in of timber
inventory held ( p.a.) q quantity of timber
produced (m3/ha.year) - cutting rate policy

q control variable that will be managed
optimally
14
Stochastic Dynamic Programming ApproachThe
Bellmans Equation

• Concession value- F(P,I,t) maximizing the
  expected
• profit function throughout the lifetime of the
  lease

P,I state variables q control variable
quantity of timber produced p(q) instantaneous
free cash flow C (q) cost function c1.q T
Lifetime of the concession r discount factor
15
Optimality Equation

• After Itos Lemma , Concession Value -
  F(P,I,t)
• follows the PDE of parabolic type in two
  dimensions (P I)

subject to the appropriated boundary conditions
explained next

• Analytical solution are rare
• Numerical solution is always available
• We use Finite Difference Method - Explicit type

16
Boundary Conditions and Constraints - MSS (1989)

• F( P , I , t T ) 0 Null value at
  the expiration
• F ( P 0 , I , t ) 0 Null value if
  price drops to zero
• F( P , I 0 , t ) 0 Null value if
  the timber is over

• For very
  high prices the value is
• proportional to the inventory held
• 
  Reflector barrier due to the
• geographic limitation

• 0 lt q(P,I,t) lt qmax Constraint on
  production capacity
• q ( P , I lt Imin , t ) 0 Regulatory policy
• bellow a certain level of inventory (Imin) the
  harvest is not allowed

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Concession Value F (/ha) X Time to Maturity (T)
at t 0
T30 ? 9 T15 ? 71 T5 ? 310
Up to 15 yrs to maturity there is no
significant increase
18
Price Uncertainty sensitivity analysis
Option ? FPP .?P 2
F PP lt 0
F PP gt 0
19
Inventory Uncertainty sensitivity analysis
Option ? FII.?I 2
Min. Inventory held
NPV 0
F II gt 0
F II lt 0
20
ROT X NPV
Option Value (/ha) at t 0 , for base case
153
32
ROT No Uncertainty over Current Timber
14
ROT Uncertainty over Current Timber
NPV

Uncertainty over Current Inventory reduces
concession in ? 12
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Concluding Remarks

• Higher Values
• Concession Value is 153 higher for the base
  case comparing to NPV results.
• Duration for the Concession
• Increasing the exploitation time up to 15 years
  does not increase significantly the Concession
  Value.
• Uncertainty over Current Timber Inventory
• Uncertainty over Current Timber Inventory reduces
  the Concession Value by roughly 12
• Option is very sensitive to discount risk (
  30)
• Estimate the risk premium is the next research

22
Support Material
23
Model Results
24
Model Results
25
Stochastic NPV

• for I(t) ? I_min and p(q) gt 0

F(P,I,t)

• otherwise

F(P,I,t) 0
26
Real Options on Renewable Resources -Literature-

• Conrad (1997)
• Analysis On the option value of old-growth
  forest
• The case of Headwaters Forest old-growth coast
  redwood
• Ecological Economics
• The first to apply Real Options to value Forest
  
• resources based on social benefits
• Stochastic Amenity flow the sum of non-timber
  
• benefits (wildlife habitat, flood control
  and visitation)
• Optimal policy
• To Harvest if Amenity Net value of Standing
  Timber

27
Real Options on Renewable Resources -Literature-

• Robert Pindyck (1984)
• Uncertainty in the Theory of Renewable
  Resources Markets
• Review of Economic Studies
• Deterministic Prices and Stochastic Inventories
• Price is function of aggregate extraction rate
• Extraction Cost is a convex function of
  inventory
• Inventory uncertainty reduces the lease value

28
Real Options on Renewable Resources -Literature-

• Morck, Schwartz and Stangeland (1989)
• The Valuation of Forestry Resources under
• Stochastic Prices and Inventories
• The Case of a White Pine Forest Lease in Alberta,
  Canada
• Journal Financial and Quantitative Analysis
• Stochastic Prices and Inventories
• Price is uncorrelated to extraction rate or
  inventory
• -small firm assumption-
• Extraction Cost is quadratic function
• Price uncertainty increases the lease value

29
Concession Value (/ha) X discount rate (r -
year)
43
Base Case
-32
Option is very sensitive to discount
risk Estimate the risk premium is the next
research
30
Numerical Techniques

• Stochastic Optimization Problems can be solved
  by
• Simulation Processes
• Monte Carlo simulation with Optimization
  Method
• Lattice Methods
• Binomial Method
• Trinomial Method
• Solving the Partial Differential Equation
• Analytical Solutions Black Scholes
• Numerical Solutions Finite Difference Method

31
Finite Difference Method

• Implicit form
• The PDE can be solved indirectly by solving a
  system
• of simultaneous linear equations
• Convergence is always assured
• Explicit form
• The PDE can be solved directly using the
  appropriated
• boundary conditions and proceeding backward
  in time
• through small intervals until find the
  optimal path
• q(P,I,t) to every t.
• Convergence is assured for specifics size of
  increments
• - interval length -

32
Finite Difference - Explicit Method

• It consists of transforming the continuos
  domain of P, I
• and t (state variables) by a network or mesh
  of discrete
• points.
• The PDE is converted into a set of finite
  difference
• equations
• Each unknown value is function of known values
  of the
• subsequent period - backward procedure
• unknown value t
  known values t1
• The function represents weights and acts as
  probabilities

Function
probabilities
33
Finite Difference - Explicit Method
.
.
.
known values
F( P , I , t ) F( iDP, jDI , nDt )
i
p-
p
p0
P i DP
Grid
.
.

DP
?
.
.
.
unknown value
interval length for P
.
.
probabilities


j
Dt
DI
t
I j DI
interval length for I
interval length for t
34
Discretization Process

• Discretization to Lease Value
• F( P , I , t ) F( iDP, jDI , nDt ) F i,j,t
• Partial derivatives are approximated by
  following
• difference equations
• FPP F i1,j , t1 - 2F i,j,t1 F
  i-1,j,t1 / (DP)2
• FP F i1,j,t1 - F i-1,j,t1 / 2DP
  central difference
• FII F i,j1,t1 - 2F i,j,t1 F i,j-1,t1
  / (DI)2
• FI F i,j1,t1 - F i,j-1,t1 / 2DI
  central difference
• Ft F i,j,t1 - F i,j,t / Dt
  forward-difference

35
Finite Difference - Explicit Method

• Substituting the approximations into the PDE