Optimal Rotation
Optimal Rotation
Biological vs. Economic Criteria
 What age should we harvest timber?
 Could pick the age to yield a certain size
 Or could pick an age where volume in a stand is maximized
 Or pick an age where the growth rate is maximized
 Our focus will be on finding the rotation that maximizes
economic returns
How do we find that?
 Determine the age that maximizes the difference
between the present value of future revenues and future
costs
 We first simplify the problem
 Only interested in commercial returns
 Only one type of silvicultural system-Clearcutting (evenaged)
 Start with an existing timber stand
Harry Nelson 2011
Volume and Value Increase with
Age
Volume or value of
timber ($/ha/yr or
m3/ha/yr)
value
volume
Age (years)
Average and Incremental Growth in Value and
Volume
Average growth
and marginal
(incremental)
growth (m3/ha/yr)
Marginal or
incremental
growth
Average
growth
Age (years)
Relationship Between Maximum Marginal
Growth and Average Growth in Value
Volume or value of
timber ($/ha/yr or
m3/ha/yr)
Value or p(t)
Volume or Q(t)
Average growth
and marginal
(incremental)
growth ($/ha/yr)
Average value of the
stand or p(t)/t
Marginal or incremental
growth in value or ∆p
Age (years)
Optimal Rotation for a Single Stand
Key idea is to weigh the marginal benefit of growing the
stand another year against the marginal cost of not
harvesting
The marginal benefit of waiting to harvest a year is the increase in value of the
stand
The marginal cost is what you give up in not harvesting now is the opportunity to
invest those funds-or the opportunity cost
As long as you earn a higher return “on the stump”, it makes sense to keep your
money invested in the timber
When the rate falls below what you can earn elsewhere, then harvest the timber
and invest it where it can earn the higher return
Optimal Rotation for a Single Stand
Rate of growth
in the value of
timber (%/yr)
Change in value/Total value
or ∆p/p(t)
i
T*
Introducing Successive Rotations
 In the previous example only considered
the question of how best to utilize capital
(the money invested in growing the timber
stand)
 We now turn to the problem of deciding the
optimal rotation age when we have a series
of periodic harvests in perpetuity
 We assume each rotation will involve identical
revenues and costs
 And we will start off with bare land
Harry Nelson 2010
Optimal Rotation for a Series of Harvests
What then is the present value of a series of recurring harvests
every 60 years (where p=Revenues-Costs)?
p
p
p
p
60
120
180
240
Perpetual Periodic Series
–(pg. 129 in text)
Harry Nelson 2011
Associated Math
V0=
Vs=
p
(1 + r)t - 1
p
(1 + r)t - 1
This is the formula for calculating the
present value of an infinite series of
future harvests.
Pearse calls this “site value”. It can also
be called “Soil Expectation Value (SEV)”,
“Land Expectation Value (LEV)”, or
“willingness to pay for land”.
If there are no costs associated with producing the
timber, Vs then represents the discounted cash flow-the
amount by which benefits will exceed costs
Land Expectation Value
Present value of a
series of infinite
harvests, excluding
all costs
Evaluated at the
beginning of the
rotation
p
Vs=
(1 + r)t - 1
So if I had land capable of
growing 110 m3/ha at 100
years, and it yielded $7
per m3, evaluated at a
discount rate of 6% that
would give me a value of
$42.26/ha
Harry Nelson 2011
Associated Math
So in order to maximize LEV the goal is to pick the rotation age (t*)
that maximizes this value.
p
Vs=
(1 + r)t* - 1
At 90 years, only 109 m3/ha and
worth $6 per m3, but LEV is
higher-$49.17
This can be done in a spreadsheet by putting in different rotation ages
and seeing which generates the highest value
Harry Nelson 2011
Calculating Current Value and Land Expectation
Value at Different Harvest Ages
Harvest Age
Volume
(ys)
Net
(m3/ha)
Revenues/m3
Value ($/ha)
LEV
10
29
0
$0
$0.00
20
46
0
$0
$0.00
30
61
0
$0
$0.00
40
74
1
$74
$32.71
50
85
2
$170
$50.24
60
94
3
$282
$57.65
70
101
4
$404
$58.40
80
106
5
$530
$54.97
90
109
6
$654
$49.17
100
110
7
$770
$42.26
110
109
8
$872
$35.12
120
106
9
$954
$28.30
130
101
10
$1,010
$22.13
140
94
11
$1,034
$16.76
150
85
12
$1,020
$12.25
160
74
13
$962
$8.57
170
61
14
$854
$5.65
180
46
15
$690
$3.39
LEV maximized at 70 years
Harry Nelson 2011
Comparison with Single Rotation
The problem now becomes determining what age given successive harvests
The idea is still the same-calculate the benefit of carrying the timber stand
another year against the opportunity cost
The difference here is that
instead of evaluating only
the current stand you now
look at the LEV, which
takes into account future
harvests
Vs(t*)=
P(t*)
(1 +
r)t*
-1
=
Vs(t*+1)=
(1 + r)t+1* - 1
∆P
P(t)
P(t*+1)
r
=
1 -(1+r)-t
Faustmann Formula
∆P
P(t)
Annual
costs &
returns
Incremental growth
in value or ∆p/p(t)
Incremental increase in
cost or r/1-(1+r)-t
T*
Rotation age (t)
r
=
1 -(1+r)-t
This result-where the marginal benefit
is balanced against the marginal cost
of carrying the timber-is known as the
Faustmann formula
You end up harvesting sooner relative
to the single rotation
The economic logic is that there is an
additional cost-land.
By harvesting sooner is that you want
to get those future trees in the ground
so you can harvest sooner and receive
those revenues sooner
Harry Nelson 2011
Modifying the Math
The formula can be modified to include other revenues and costs
Here recurring annual revenues and costs are included in the 2nd
term
Vs=
p
(1 + r)t* - 1
+
a -c
r
Harry Nelson 2011
Further Modification
Imagine you have a series of intermittent costs and revenues
over the rotation -how do you calculate the optimal rotation
then?
Reforestation-Cr
0
Pre-Commercial Thin -Cpct
Commercial thinning -
Harvesting -
net revenue (NRt)
net revenue (NRh)
20
50
80
You can compound all the costs and revenues forward to a
common point at the end of the rotation-this then becomes p
P=
(1 + r)80 *Cr
Vs=
+ (1 + r)60*Cpct
p
(1 + r)t - 1
+ (1 + r)30*NR +
t
NRh
Impact of Different Factors
 Interest rate
 Higher the interest rate the shorter the optimum
rotation
 Land Productivity
 Higher productivity will lead to shorter rotation
 Prices
 Increasing prices will lengthen the optimal rotation
 Reforestation costs
 Increase will increase the optimal rotation length
Harry Nelson 2011
Amenity Values and Non-Monetary Benefits
Growth in value
without amenity
values
Rate of growth in
the value of
timber (%/yr)
Growth in value
with amenity values
“Perpetual rotation”
Growth in value with
amenity values
i or MAR
Rotation age
Rotation age
In this case you’d never harvest
Harry Nelson 2011
How Does the Rule Affect Harvest
Determination?
 How does the rotation rule apply when we
extend it to the forest?
 Start with the assumption of a private owner
maximizing value
 Imagine applying the optimal rotation age
to two types of forests
 In one forest all the stands are the same
age so all the harvest would take place in
one year with no harvests until the stands
reached the optimal age again
Harry Nelson 2011
“Normal” forest
In another forest the stands are divided
into equal-sized areas and there is a
stand for each age class-so that each
year one stand is harvested
In this case the harvest levels would be
constant (assuming everything else such
as prices and costs remained constant)
Why Private Harvest Levels Are Unlikely
to be Constant
 Stands vary in size and productivity
 Markets are changing
 So harvest levels are likely to fluctuate
 May also be specific factors that influence the owner
(size constraints, etc.)
Regulating Harvests on Public
Land
 Harvest rules on public land have historically been
concerned with maximizing timber yield
 Historic concern has been that cyclical markets would
lead to variations in harvesting, employment, and
income for workers
 Goal has been to smooth out harvest levels and maintain
harvests in perpetuity
Harvesting policies in Canada
Sustained yield (or non-declining even flow)
has been preferred approach as it was
originally seen as contributing to community
stability and maintaining employment
Established on basis of growth rate for a given
age
Usually done as a volume control (AAC
determination)
 Alternative is area control
Several Important Consequences
Where mature forests exists affects the
economic value of forestry operations
Can be long-term effects on timber supply
Changes how we evaluate forestry
investments
Fall Down Effect
Historically transition from old growth
(primary forest) to sustained yield
This approach yields the “fall-down” effect
Hanzlick formula-based on proportion of old
growth and mean annual increment
associated with average forest growth
 AAC = (Qmature /T*) + mai
 where Qmature equals amount of timber greater than harvest age T*
Harry Nelson 2011
Fall Down Effect
Allowable Cut Effect
 Cost of improving the stand -$1000 per hectare
 Result-doubling of growth (an additional 995 cubic
metres)
 Standard cost-benefit:
 Discounted Benefit: $13,187/1.0558=$778
 Cost: $1000
 So NPV =-$222; B/C = 0.78
Introducing ACE
 If you can take additional volume over the 58 years…
($13,187/58)
 Then it looks quite different
 Using a formula-the present value of a finite annuity
 NPV = ($13,187/58)*((1.05)58-1)/.05*(1.05)58
 Or $4,546