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 (even-

aged) Start with an existing timber stand

Volume and Value Increase with Age

Volume or value of timber ($/ha/yr or m3/ha/yr)

Age (years)

volume

value

Harry Nelson 2011

Average growth and marginal (incremental) growth (m3/ha/yr)

Age (years)

Average growth

Marginal or incremental growth

Average and Incremental Growth in Value and Volume

Relationship Between Maximum Marginal Growth and Average Growth in Value

Volume or value of timber ($/ha/yr or m3/ha/yr)

Age (years)

Volume or Q(t)

Value or p(t)

Average growth and marginal (incremental) growth ($/ha/yr)

Marginal or incremental growth in value or ∆p

Average value of the stand or p(t)/t

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

T*

Rate of growth in the value of timber (%/yr)

i

Change in value/Total value

or ∆p/p(t)

Optimal Rotation for a Single Stand

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

p

60 120 180 240

Perpetual Periodic Series– (pg. 129 in text)

What then is the present value of a series of recurring harvests every 60 years (where p=Revenues-Costs)?

Optimal Rotation for a Series of Harvests

p p p

Harry Nelson 2010

V0=p

(1 + r)t - 1

Vs=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

Associated Math Harry Nelson 2011

Land Expectation Value

Present value of a series of infinite harvests, excluding all costs

Evaluated at the beginning of the rotation

Vs=p

(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

Vs=p

(1 + r)t* - 1

So in order to maximize LEV the goal is to pick the rotation age (t*) that maximizes this value.

This can be done in a spreadsheet by putting in different rotation ages and seeing which generates the highest value

Associated Math Harry Nelson 2011

At 90 years, only 109 m3/ha and worth $6 per m3, but LEV is higher-$49.17

Harvest Age (ys)

Volume (m3/ha)

Net Revenues/m3 Value ($/ha) LEV

10 29 0 $0 $0.0020 46 0 $0 $0.0030 61 0 $0 $0.0040 74 1 $74 $32.7150 85 2 $170 $50.2460 94 3 $282 $57.6570 101 4 $404 $58.4080 106 5 $530 $54.9790 109 6 $654 $49.17

100 110 7 $770 $42.26110 109 8 $872 $35.12120 106 9 $954 $28.30130 101 10 $1,010 $22.13140 94 11 $1,034 $16.76150 85 12 $1,020 $12.25160 74 13 $962 $8.57170 61 14 $854 $5.65180 46 15 $690 $3.39

Calculating Current Value and Land Expectation Value at Different Harvest Ages

LEV maximized at 70 years

Harry Nelson 2011

Vs(t*)=P(t*)

(1 + r)t* - 1Vs(t*+1)=

P(t*+1)

(1 + r)t+1* - 1

=r

1 -(1+r)-t

∆P

P(t)

Comparison with Single RotationHarry Nelson 2011

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

=

Incremental growth in value or ∆p/p(t)

Incremental increase in cost or r/1-(1+r)-t

Annual costs & returns

Rotation age (t)

=r

1 -(1+r)-t

∆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

T*

Faustmann Formula

Modifying the MathHarry Nelson 2011

Vs=p

(1 + r)t* - 1+

a - c

r

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

Reforestation-Cr

Commercial thinning -

net revenue (NRt)

0 20 50 80

P = (1 + r)80 *Cr + (1 + r)60*Cpct+ (1 + r)30*NRt

+ NRh

Imagine you have a series of intermittent costs and revenues over the rotation -how do you calculate the optimal rotation then?

Pre-Commercial Thin -Cpct

Harvesting -

net revenue (NRh)

You can compound all the costs and revenues forward to a common point at the end of the rotation-this then becomes p

Further ModificationHarry Nelson 2011

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

Growth in value without amenity values

Growth in value with amenity values

Rotation age

Rate of growth in the value of timber (%/yr) Growth in value with

amenity values

Rotation age

“Perpetual rotation”

i or MAR

Amenity Values and Non-Monetary Benefits

Harry Nelson 2011

In this case you’d never harvest

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)

Harry Nelson 2011

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*

Fall Down Effect

Harry Nelson 2011

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