1. Optimal Rotation
Optimal Rotation

2. 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

3. 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 (even- aged)
Start with an existing timber stand

4. Volume and Value Increase with Age
Harry Nelson 2011
Volume and Value Increase with Age
Volume or value of timber ($/ha/yr or m3/ha/yr)
value
volume
Age (years)

5. 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)

6. 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)

7. 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

8. 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*

9. 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

10. Optimal Rotation for a Series of Harvests
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)

11. Harry Nelson 2011
Associated Math
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”. p
V0=
(1 + r)t - 1 p
Vs=
(1 + r)t - 1
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

12. 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

13. 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
At 90 years, only 109 m3/ha and worth $6 per m3, but LEV is higher-$49.17
Vs=
(1 + r)t* - 1
This can be done in a spreadsheet by putting in different rotation ages and seeing which generates the highest value

14. Harry Nelson 2011
Calculating Current Value and Land Expectation Value at Different Harvest Ages
LEV maximized at 70 years

15. Comparison with Single Rotation
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
P(t*)
P(t*+1)
Vs(t*)= =
Vs(t*+1)=
(1 + r)t* - 1
(1 + r)t+1* - 1 ∆P r =
P(t)
1 -(1+r)-t

16. Faustmann Formula ∆P r = P(t) 1 -(1+r)-t
Annual costs & returns
Incremental growth in value or ∆p/p(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
Incremental increase in cost or r/1-(1+r)-t T*
Rotation age (t)

17. 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 p
a - c +
Vs= r
(1 + r)t* - 1

18. 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?
Commercial thinning -
net revenue (NRt)
Harvesting -
net revenue (NRh)
Reforestation-Cr
Pre-Commercial Thin -Cpct 80 20 50
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 +
(1 + r)60*Cpct
(1 + r)30*NRt
NRh p
Vs=
(1 + r)t - 1

19. 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

20. Amenity Values and Non-Monetary Benefits
Harry Nelson 2011
Amenity Values and Non-Monetary Benefits
Growth in value without amenity values
“Perpetual rotation”
Rate of growth in the value of timber (%/yr)
Growth in value with amenity values
Growth in value with amenity values
i or MAR
Rotation age
Rotation age
In this case you’d never harvest

21. How Does the Rule Affect Harvest Determination?
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

22. 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)

23. 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.)

24. 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

25. 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

26. 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

27. 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*

28. Harry Nelson 2011
Fall Down Effect

29. 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

30. 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