Optimal Rotation

1. Optimal Rotation Optimal Rotation

2. 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 Biological vs. Economic Criteria

3. 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 How do we find that?

4. 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. 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 Introducing Successive Rotations

10. 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 240 60 120 180 • 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. Present value of a series of infinite harvests, excluding all costs • Evaluated at the beginning of the rotation Land Expectation Value 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. 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 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 growth in value or ∆p/p(t) 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 0 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. 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 Impact of Different Factors

20. 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. Harry Nelson 2011 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 How Does the Rule Affect Harvest Determination?

22. Harry Nelson 2011 • 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) “Normal” forest

23. 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.) Why Private Harvest Levels Are Unlikely to be Constant

24. 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 Regulating Harvests on Public Land

25. 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 Harvesting policies in Canada

26. 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 Several Important Consequences

27. 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

28. Harry Nelson 2011 Fall Down Effect

29. 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 Allowable Cut Effect

30. 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 Introducing ACE