IPEA - Institute for Applied Economic Research of Brazilian Government

REAL OPTIONS IN ENVIRONMENTAL ECONOMY

“Environmental Investment Decisions - The Case of Natural Forest Concession in Legal Amazon”

KATIA ROCHAkatia@ipea.gov.br

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.

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

Forest Lease in Legal Amazon - Overview

Concern : Economic Market Value of concession

Focus on Expected Cash Flows and Option Values

Environmental commodities has been suggested :

Forest Products Harvest allowances - analogy to the “pollution allowances” used in USA. These allowances add value and options to the leaseonwer

Real Options can quantify social benefits coming from:

kidnapping of carbon contribution to global climatic stability water balance maintenance preservation of biodiversity

Environmental and Economical Issues

Forest Lease as Real Option Forest Lease is a Capital Investment Opportunity

Long time horizon - usually 30 years

High uncertainty about timber prices and inventory

Option : Right but not the obligation to proceed the harvest

Decisions:

When management should proceed the harvest ?

What is the optimal cutting rate policy?

Harvest decision is an instantaneous irreversible decision

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

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

The present model is similar to Morck, Schwartz and Stangeland (1989) with slightly modification :

Comparisons between ROT and NPV are performed

Regulation Policies are included

Uncertainty over Initial Inventory - use of spatial econometric models -

Extraction cost is a linear function - realistic assumption

Further work - to model changes in timber price as a Mean-Reverting Process - standard for commodities

Introduction to the Model

Formulating a Profit Maximization Inter-temporal Problem

ROT : inter-temporal maximization procedure under uncertainty considering the options available to managers

Maximization tool :

Bellman´s Equation - Stochastic Dynamic Programming

split the decision sequence into two parts :

immediate profitLease = Value

Optimal action today is the one that maximizes the Value

expectations over all the future profits throughout the lifetime of the lease - continuation value -

Today Future

+

Bellman´s Equation - Stochastic Dynamic Programming -

]Fd[dt

1)t,u,x(max)t,x(F

~

u

dt

Fd)t,u,x(maxE)t,x(F

~

u

1 )

2 )

Although 1 < 2 we have to use the expectations about the

future. Therefore alternative 1 is the correct optimization procedure

Comparing Optimization Procedure

A

B

Max = (A+B). 1/2

C

D

FutureOutcomes

q

q

q

q

qA*

qB*

q*

Max = (C+D). 1/2

q*

Optimal control DOES NOT exist Optimal control EXIST - q*

prob = 0.5

prob = 0.5

prob = 0.5

prob = 0.5

Procedure 2 Procedure 1

Procedure 2FutureOutcomes Procedure 1

>

Management Decisions / Flexibilities :

“When to harvest ?” Option to Delay or Defer

“How much to harvest ?” Option to Expand or Contract

These flexibilities add an extra Value - Option Premium - to the Lease.

Lease = NPV + Option Premium ( Real Option Value )

Option Premium

Price and Inventory Uncertainties

1 - The market price of timber: “How the prices will be on the next decade or year ?”

2 - The amount of timber inventory in the leasehold:

“How the timber inventory will grow in the lease area? Fast or Low?”

Loss of timber inventory : burnings,

limitations on cutting valuable species

Increase of timber inventory: discovery of new or valuable species

The best you can do is to add some uncertainty

to your forecast on timber price and inventory time

evolution processes

{

Timber Prices

Wood price time series data :

Brazilian logs of medium value

Hardwood logs from Malaysia (International Financial Statistics - IMF)

Both data present similar volatility 30 % p.a.

Hardwood Log Prices

0

100

200

300

400

500

600

jan/77 jan/80 jan/83 jan/86 jan/89 jan/92 jan/95 jan/98

Hardwood-Logs Malaysia Mahogany-Logs Brasil

Price (jan ‘77 / jan ‘99) - $/m3 - in real prices of ‘95

Mean Reverting Process appears to be a good guess

0

100

200

300

400

500

600

jan/82 jan/85 jan/88 jan/91 jan/94 jan/97 jan/00

Softwood-Logs USA Hardwood-Logs Malaysia

USA Lumber X Malaysia HardwoodPrice (jan ‘82 / apr ‘00) - $/m3 - in real prices of ‘95

Similar Pattern for US Lumber and Malaysia HardwoodHardwood Conv.Yield could be approximated by Lumber Conv. Yield

Timber Price as a Stochastic Process. Basic approach is to use Stochastic Differential Equations - SDE

We use the standard Geometric Brownian Motion (GBM) :

Pdz PPdt PdP

dP = changes in priceP = Timber Price ($/m3) P = average growth rate in % of price (% p.a.) P =volatility parameter in % of price (% p.a.)

Percentage changes in prices (dP/P) are normally distributed with mean P t, and variance P

2 t.Variance grows as time passes by : Non-Stationary Process

Timber Inventory as a Stochastic Process

dw Idt qI dI I*

I dI = changes in Inventory

I = Inventory of Timber in the leasehold (m3/ha)

I = average growth rate in % of timber inventory held (% p.a.)

I = volatility parameter in % of timber inventory held (% p.a.)

q* = quantity of timber produced (m3/ha.year)

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

q* = control variable that will be managed optimally

Mathematical Formulation for the Lease Value

Lease value is calculated by maximizing the expected profit function throughout the lifetime of the lease

dte)q(CqP max t .rTt

0t

**ttt

*q

P = Timber Price q* = quantity of timber produced T = Lifetime of the lease C (q*) = cost function r = risk-free interest rate

Revenues Costs Discount Factor

Contingent Claims Approach The Lease Value can be viewed as a Contingent Claim on the underlying timber

Dynamic Programming with risk-neutral drift (r-k) discounting by the risk-free rate Contingent Claims Approach

Continuos time finance assumptions :

1 - There are futures markets for timber - Forest Products on CME - Futures and Options on Lumber -

2 - The convenience yield (k) is proportional to the spot price of timber.

Convenience yield can be calculated using the relationship between future price and spot price :

t)kr(e SpotFuture

Contingent Claims Approach Set up an instantaneous riskless portfolio using hedge

hedge : long and short positions over the underlying variable ( timber price)

Market risk associated to stochastic changes in inventory is not appraised by the market. Risk is uncorrelated with market.

Risk premium is zero

Market risk associated to stochastic changes in inventory is not appraised by the market. Risk is uncorrelated with market.

Risk premium is zero

The hedge eliminates all market risk associated to timber priceThe hedge eliminates all market risk associated to timber price

The riskless portfolio must earn the risk-free rate (r) to avoid arbitrage possibilities.

Lease Value Differential Equation After application of Ito´s Lemma , the Lease Value -F(P,I,t) follows the Partial Differential Equation (PDE) of parabolic type in two dimensions (P & I) :

subject to the appropriated boundary conditions “explained next”

Similar to B&S equation except by the terms in red Analytical solution are rare Numerical solution is always available We use Finite Difference Method

tPPP22

P FF P )kr(F P 2

1

*)q(C*Pq rF

IIII22

I F *qIF I σ2

1

Boundary Conditions and Constraints 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

minP

II P

F

0I

F

max II

0 < q*(P,I,t) < qmax Constraint on production capacity

q ( P , I < Imin , t ) = 0 Regulatory policy bellow a certain level of inventory (Imin) the harvest is not allowed

For very high prices the value is proportional to the inventory held

Reflector barrier due to the geographic limitation

Lease Value using NPV

Traditional Capital Budgeting (NPV):

0

dte max)qC(-maxqePNPV

Tt

0t

rtt)kr(0

subject to the constraint :

q( P , I < Imin , t ) = 0 Regulatory policy

Free Cash Flows = Revenues - Costs

, if FCF > 0

, if FCF < 0

Higher price uncertainty increase the Lease ValueHigher price uncertainty increase the Lease Value

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

Price ( $/m3 )

volatility = 0.2 volatility = 0.3 volatility = 0.4

Price uncertainty sensitivity analysis on Lease Value ($/ha) at t = 0 (T=30)

Option orLease Value FPP.P

2

F PP > 0

Boundary Condition Effect (Price)

0

500

1000

1500

2000

0 40 80 120 160 200

Price - $/m3

volatility = 0.2 volatility = 0.3 volatility = 0.4

F (P = 0) = 0

P = 0 P =

Parallel LinesEqual derivatives

Inventory uncertainty sensitivity analysis on Lease Value ($/ha) at t = 0 (T=30)

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35 40

Inventory ( m3/ha )

volatility = 0.08 volatility = 0.15 volatility = 0.2

Region AF II > 0

Inventory uncertainty produces different effects on the Lease value

Min. Inventory held 12.5

Option orLease Value FII.I

2

Region BF II < 0

NPV = 0

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35 40

Inventory ( m3/ha )

volatility = 0.08 volatility = 0.15 volatility = 0.2

Boundary Condition Effect (Inventory)

FI = 0F (I = 0) = 0

I = 0 I =

NPV = 0

Equal derivatives

ROT & NPV outcomesLease Value ($/ha) at t = 0 , for I = 25m3/ha and T = 30 years

0

50

100

150

200

250

0 10 20 30 40 50 60

Price ( $/m3 )

NPV ROT

+115 %

+96 %

+103 % NPV

ROT

Effect of Regulatory policy on Lease Value ($/ha) at t = 0 (T=30)

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30 35 40

Inventory ( m3/ha )

Inventory held = 0 Inventory held = 6.25 Inventory held =12.5

Lease Value increases as Regulation becomes less intense

Base Case

+34 %

+69 %

Interest rate sensitivity analysis on Lease Value ($/ha) at t = 0 (T=30)

0

50

100

150

200

250

300

350

400

450

0 10 20 30 40 50 60

Price ($/m3)

r = 5% p.a. r = 10% p.a. r = 15% p.a.

+112 %+57 %

Base Case

Optimal cutting rate policy

0

8

16

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

q* (r = 5 % p.a.) q* (r = 10 % p.a.) q* (r = 15 % p.a.)

P* = $16r = 15 % p.a.

P* = $12r = 5 % p.a.

Threshold - P* - increases as interest rate increases NPV rule : Static Threshold P* = Total Cost = $12

Cutting rate policy q* at t = 0, T=30 , for Total Cost = $12/m3

P* = $14r = 10 % p.a.

qmax

Model Results

Sensitivity Analysis relative to Initial ConditionsAlternatives NPV ROT %Base Case I0 = 25, P0 = 30 39.6 80.3 +103 Poor Density of Timber I0 = 15 29.5 33.4 +13High Density of Timber I0 = 35 39.6 111 +180Higher Price P0 = 60 104.7 245.1 +134Lower Price P0 = 10 0 10.6 +

Model Results

Sensitivity Analysis relative to UncertaintiesAlternatives NPV ROT %

Base Case P = 0.3,I = 0.08 39.6 80.3 +103

(-)Price Uncertainty P = 0.2 39.6 47.2 +19

(+)Price Uncertainty P = 0.4 39.6 124.5 +214

(-)Inventory Uncertainty I = 0.01 39.6 82.5 +108

(+)Inventory Uncertainty I = 0.15 39.6 68 +72

Concluding Remarks

Higher Values

Lease Value is 100 % higher for the base case

Different Thresholds

Threshold P* for harvest varies by an amount of 15 % relative to interest rate. NPV produces static Threshold (Costs)

Analysis about the Regulatory Policy

Reducing the regulatory limit in 50 % , leads to an increases of 34% in the Lease value. NPV is not able to quantify this change

The application of ROT instead of NPV leads to :

Support Material

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

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 -

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 t+1

The function represents weights and acts as “probabilities”

“probabilities”

Function

i

t

.

.

.. .

jI

P

I = j I

P = i P

.t

Finite Difference - Explicit Method

.. .

“probabilities”

.

p- p+p0

known values

unknown value?

Grid

F( P , I , t ) = F( iP, jI , nt )

interval length for Iinterval length for t

interval length for P

Discretization Process

Discretization to Lease Value :

F( P , I , t ) = F( iP, jI , nt ) = F i,j,t

Partial derivatives are approximated by following difference equations :

FPP = [ F i+1,j , t+12F i,j,t+1 + F i1,j,t+1 ] / (P)2 ;

FP = [ F i+1,j,t+1 F i1,j,t+1 ] / 2P ; central difference

FII = [ F i,j+1,t+1 2F i,j,t+1 + F i,j1,t+1 ] / ()2 ;

FI = [ F i,j+1,t+1 F i,j1,t+1 ] / 2I ; central difference

Ft = [ F i,j,t+1 F i,j,t ] / t ; forward-difference

t1

r

2i)kr(

i21

p

22P

i

t1

r

2i)kr(

i21

p

22P

i

t1

r

I2qIj

j21

p

22I

j

t1

r

I2qIj

j21

p

22I

j

t1

r

t1

jip

22I

22P0

Finite Difference - Explicit Method

t

1r

)q(Cost)q.Pi(Fp

FpFp

FpFpF

1t,j,i0

1t,1j,ij1t,1j,ij

1t,j,1ii1t,j,1iit,j,i

Substituting the approximations into the PDE :

Spatial Econometric Model

?

Realistic Assumption : The amount of timber in the lease area - initial inventory or biomass - is not known completely

We have sample refers to places – identified as points since they are small areas (1 ha)

Estimate the Probability distribution of logging volumes in concession areas.

?

??

??

Concession Area

Spatial Econometric Model The volume distribution is specified in a spatial model

Relates the density of biomass (b) with the density of neighboring regions, and explanatory variables (x) which are measured for the whole area.

The explanatory variables 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.

Lease Value and the Uncertainty in Initial Inventory

Define : as the probability distribution of the timber inventory in the lease area.

The Lease value -F(P,t)- with uncertainty in Initial Inventory :

dI)I().t,I,P(F)t,P(F

where F( P , I , t ) is the Lease Value with known Initial Inventory

,

Uncertainty in Initial Inventory

0

20

40

60

80

100

120

140

20 22 24 26 28 30 32 34 36 38 40

Price - $/m3

Known Inventory volatility = 0.4 volatility = 0.5

Initial Inventory Distribution : Lognormal (25m3/ha , volatility)

Lease Value - $/ha - in t = 0

Known Value

-14%

-21%

Uncertainty in Initial Inventory reduces the Lease Value