Optimal Rotation-Pt 2

February 27, 2014

John Maynard KeynesBorn: June 5, 1883, Cambridge, United KingdomDied: April 21, 1946, East Sussex, United Kingdom

Adam SmithBorn: June 5, 1723, Kirkcaldy, United KingdomDied: July 17, 1790, Edinburgh, United Kingdom

David RicardoBorn: April 18, 1772, London, United KingdomDied: September 11, 1823, Gatcombe Park, United Kingdom

Alfred MarshallBorn: July 26, 1842, Bermondsey, London, United KingdomDied: July 13, 1924, Cambridge, United Kingdom

Ronald CoaseBorn: December 29, 1910, Willesden, London, United KingdomDied: September 2, 2013, Chicago, Illinois, United States

Feedback

Post lecture slides before class Identify equations and graphs to be used in problem

sets Examples of how to use them Better labeling of graphs

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

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=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

Reforestation-Cr

Commercial thinning -

net revenue (NRt)

0 20 50 80

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

Pre-Commercial Thin -Cpct

Harvesting -

net revenue (NRh)

Further ModificationHarry Nelson 2011

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

For periodic costs and revenues over the rotation:

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

For problem setHarry Nelson 2011

For problem setHarry Nelson 2011

Vs=p

(1 + r)t* - 1+

a - c

r

Recurring annual revenues and costs can be are included in a 2nd expression

Adding Carbon

It is not the biological side that makes C accounting complex-it is the market side.

• Baseline

• Leakage

• Buffer

• Harvest

What happens if we manage for Carbon?

Carbon payment schemes pay for either C sequestered or avoided C emissions.

In forestry focus has been on sequestration (trees are efficient C storage mechanisms)

Impact of Different Factors

Interest rate Higher the interest rate the shorter the optimum

rotation

Impact of Different Factors

Interest rate Higher the interest rate the shorter the optimum

rotation Land Productivity

Higher productivity will lead to shorter rotation

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

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

What if there are other values?

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

Rotation age (t)T*

i*

Annual costs & returns

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

Assessing Risk

No selection

Selection

No infestation

No infestation

infestation

infestation Selection worksSelectio

n didn’t

workP=0.2

P=0.2

P=0.8

P=0.8

P=0.3

P=0.7

p. 124 in text

Cost of outcome: S1 Cost of outcome: S2

Cost of outcome: NS2

Cost of outcome: NS1

Cost of outcome: S3

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

From Chapter 8, 163-65

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

Using ACE as an incentive

Experience with ACE