WikiHome

Forest Valuation and the Net Present Value Concept in Forestry Economics

Guillermo Navaro
CATIE

Net Present Value Concept as a Tool for Analysis in the Investment Theory

Investment theory uses banking and financial mathematics of interest as instruments for analyzing the intertemporal behavior of the investor.  Therefore, it is important to review the Net Present Value (NPV) concept as a tool in forest economics, which deals with the problems of choice in intertemporal production of forest.  The NPV is the present value of positive payments minus the present value of negative payments made at different points in time[1].  All these future values are discounted to the present by using the investor’s interest rate.  The NPV expresses a value in terms of the tradeoff between present and delayed consumption.  This means the individual by borrowing and lending (saving) reallocates the consumption over time within the limitations of current and future income.

The theory indicates that the rational man would choose the investment yielding the largest present value because it is equivalent to a larger choice[2].  The NPV is the keystone model for finding the investor’s value of any asset yielding a future cash flow[3].  The NPV is a powerful tool for valuing forest resources (trees and land) based on the future monetary benefits and costs, all discounted to the present using the investor’s minimum acceptable rate of return (MAR)[4].

Modeling the Investor’s Capital Cost through Time

In economics, two exponential functions have been traditionally used to model the growth or the cost of money over time.  One is a generalized exponential function and the other is the natural exponential function of base e.  The generalized exponential function of base b,a base that in economics is represented by an interest factor, has a periodic independent variable t, which appears in the role of an exponent, and has traditionally assumed the value of positive time integers as discrete periods[5].  However this function is not limited to this restriction[6].  This discrete exponential function is f(t) = (1+i)^t of base (1+i) > 1, and it has been a good model to express the growth of capital.  It is equal to 1 in the present (t = 0).  The constant i represents a discrete or simple interest rate that represents the growth as a percent per unit of time.  The important characteristic of this exponential function of base (1+i), known as the discrete discount factor, is that the exponent t and any other multiple of t shares the same vertical intercept at the point t = 0[7].

In addition, the natural exponential function of base e (e=2.71828...) has been a preferred base because it has the remarkable property of being its own derivative, a fact that would reduce the work of differentiation to no work at all.  This is the mathematical convenience in the use of e.  The constant r, as a difference with i, is an exponential rate of growth at a particular point in time, and it is understood as a continuous or instantaneous interest rate.  However r’s magnitude, like i, has the connotation of so many percent per unit of time t, say, per year if t is measured in years.  Therefore, the continuous concept of this formula is measured also in a discrete time, period or fractions of this period because growth, by its very nature occurs only over a period of time[8].

From the mathematical standpoint, discounting is calculating a present value at the point in time t = 0 at the intersection of the t axis from a future period t.  Likewise, compounding, as the inverse process, calculates a future value at the end of a period from a value that is located on the y-axis at the point where t = 0.  Moreover, it should be understood that the continuous and the discrete concept refer to the process of compounding and discounting.  This is not due to the concept of time because in two formulas the time is understood as discrete or periodic.  Both formulas used periods called years, month or days.

                            1/ (1 + i)^t = e^-rt                             iff  r  = ln(1 + i)   or  i =  e^r – 1            (1)

If the NPV calculated with discrete and continuous exponential models are equal, it has to fulfill one of the following two requirements: (1) the interest rate r, which simulates a continuous rate of growth, should be equal to the expression ln(1+i) of the discrete interest rate, (2) the discrete interest rate i would be equal to e^r-1 as shown in expression (1).  For discount rates lower than 10%, i would have values quite close to r, but when i is greater than 10%, i is progressively greater than r[9].  Therefore, if a unique percentage is assigned to i and r, the NPV, calculated with the continuous factor e^-rt, will yield a smaller NPV than with the discrete factor (1+i)^-t.

 

If properly indicated, the interest rate is, either discrete i or continuous r.  If the relation is known between both of these, then either one of the two discounting models can be used.  These two interest rate models have been a source of confusion in forest economics.  In several articles in forest economics, the interest rate has been represented without regarding the type of interest rate that is used, and in many other cases it is implied that i = r.  As an example, Chang (1984) presented the Faustmann formula[10] with both discrete and continuous discounting factors as a teaching aid for forest economics students.  The problem is that he used a unique interest rate that was called r for the two discounting models, which results in a source of confusion for the reader, creating the illusion that there is no difference between the interest rates of the two discounting systems[11].

 

Mathematical Techniques of the NPV: Discounting Payment Series

The NPV can discount a single sum or the cash flow of a productive activity during the asset useful productive span.  There are certain investment activities that can be modeled as a series of equal payments occurring and discounted at regular intervals[12].  The concept of rotation in even-aged (EAC) forestry and of cutting cycles in uneven- aged selection (UAS) systems can be simplified and modeled in regular intervals using the classical theory of forest economics. 

The NPV can be calculated as a discounted payment series either with terminating or perpetual payments.  In order for the mathematics of this discounting periodic series to work, several rules must be followed according to Klemperer (1996): Payments must be equal and occur at regular intervals.  These two intervals are called periods.  No payments occur at point-0 in time, which is sometimes referred to as year 0 in some introductory textbooks[13].  The first payment occurs at the end of the first period.  Payments must all have the same sign, positive or negative.

 

Present Value of a Perpetual Periodic Series

If payments are regularly separated by more than one year in an infinite series they are called a perpetual periodic series.  In equation (2), each t-year period is equal in length, p is a future value that occurs at the end of the t-year period.  This amount p is discounted to the present by the discounting factor (1+i) ^t and t recalls the period needed to bring p to the present.  What about the 1 subtracted from this discounting factor in the denominator?  This unit represents in relative terms all future incomes that will occur every t-years assuming identical periods in perpetuity, which by definition represents the asset value in relative terms. (2):

                     V₀  =  p / (1 + i)^t – 1                     (2)

In forestry there is usually a sequence of activities within the rotation or cutting cycle that represents the cash flow composed of positive and negative payments between the point in time 0 and the end of the t-year.  This range, where this cash flow is enclosed, is defined as the period t in the investment calculation.  This period is equal to the physical growing period in even aged and uneven-aged silvicultural systems when the calculation of the NPV is made at the beginning of the growing period.  In this case, is the calculation understood before or just after planting?  The reader should keep this question in mind because it belongs to the main discussion of this paper. 

Now, since this formula works by discounting a single periodic payment p that occurs at the end of every period t, all payments within this range must be compounded to the end of the rotation or period using the same interest rate factor for calculating a t-year net income.  Once this future value p has been found, then the formula can be directly applied.  Formula (3) represents this rationale[14]:

                                         T                             T

                    NPV_infinity =  ( ∑ R_t (1 + i)^T-t -∑ C_t (1 + i)^T-t )  /  ((1 + i)^t – 1)          (3)

                                       t=1                         t=1

Likewise, to be consequent all payments of the cash flow within the t-year period have to be made at the end of their respective time units[15], in forestry usually years, and with some tropical forest investments the opportunity cost should be accounted in months.

 

Efficiency Criteria for Investment Analysis in Forestry

Practitioner forest economist and managers plan for investments using capital budgeting techniques.  Capital budgeting is that branch of applied economics that involves decisions with respect to how money is invested in forestry projects, which usually require investment capital, followed by further production costs in order to receive incomes in later years[16].  Capital budgeting deals with ranking alternative projects in order to find investment alternatives for the same asset, which can maximize its value.  Five criteria are mainly used for ranking projects: Net Present Value, Land Expectation Value, Benefit Cost Ratio, Payback Period, and Internal Rate of Return.

The Net Present Value (NPV), also known as Net Discounted Value (NDV) or Net Present Worth (NPW), of an investment is the present value of its revenues minus the present value of its costs where T is the horizon of the investment.  NPV considers the benefits and costs over the entire span of the project.  According to the decision rule of the NPV, an investment is acceptable if the NPV ³ 0[17].  A remark about this criterion is that it is difficult to interpret which one is more profitable for investments with different scales.

The Land Expectation Value (LEV) is the maximum an investor can invest in the land asset and still earn the MAR on the invested capital.  The decision criterion is to accept the investment if the LEV ³ the land market price or the calculated LEV for an alternative land-use investment on the same piece of land[18].  This criterion is used for land base investments; however, it could be adapted for other types of assets.

The Benefit/Cost Ratio (B/C), also called the profitability index, is the present value of revenues divided by the present value of costs using the investment’s MAR.  The decision rule states that an investment is accepted if the B/C ³ 1[19]

The Payback Period is a method that identifies the period in which the capital invested in the project is recovered directly from the benefits produced within this period.  The project with the shortest payback period is selected.  However, the main critic of this method is that benefits and cost occurring after the payback period are ignored.  The payback period is not an investment efficiency criterion, but a financial one because it does not consider the value in the present but how to solve cash availability in the shortest time.  This criterion should be used in relation to the NPV and the LEV criteria[20].

The Internal Rate of Return (IRR) is the interest rate at which the present value of revenues equals the present value of costs.  The IRR decision rule accepts a project if the IRR ³ investment’s MAR.  There are several technical and theoretical problems with the use of the IRR.  The first one is that the IRR is more difficult to calculate, the second is the multiple roots problem and the assumption with respect to reinvestment and additional investment[21].  Moreover, the IRR differ from the NPV or the LEV decision criteria in that the IRR is an average rate while the others are marginal rates in respect to T.

A drawback for using the NPV and all the other derived techniques (B/C, IRR, PBP) is that they do not take into account the opportunity cost of the asset value, and only use the derived cash flow throughout the investment horizon.  Thus, the LEV model is a stronger investment criterion because it takes into account the asset by considering the cash flow horizon in perpetuity.  Moreover, some of the techniques presented are not real investment criteria but financial criteria.

 

Choice of the Discount Rate in Forest Economics

The choice of the discount rate in investment analysis is one of the most important problems in investment analysis because the discount rate considerably affects the magnitude of the calculated value.  Moreover, its impact is specially noticed in long-term projects, which requires large initial investment spending such as in forestry.  Thus, it is very important to make the right selection because it is also a subjective descriptor of the investor’s unique characteristics in respect to the type of investment, individual preference of present, and particular budget level, etc.

In addition, forest investments, which have especially long horizons, have an interesting characteristic.  They have an intergenerational distribution aspect in which the income is redistributed among different generations.  When the felling takes place today, the present generation receives the benefits of a forest investment that was made by earlier generations long ago; likewise, when planting takes place today, the benefits that result will be obtained by future generations.  Any model that uses discounting analysis (NPV, C/B or LEV) results in a discrimination of the future generations because multiplying distant costs and revenues from the future by a discount factor reduces the quantities to insignificant figures. 

The higher the discount rates, the smaller the amount over a shorter period of time; thus, reducing the present value of future benefits and cost to practically nothing[22].  However, investment models that use discounting are unlikely to determine a real value from long horizon investments, rather they are models that allow us to rank projects or make management decisions such as the optimal time to harvest a forest product.  The choice of the discount rate not only determines whether a project is accepted or not, it also affects the optimal rotation, and the ranking of the projects.  Forestry is very sensitive to the discount rate chosen for the analysis that projects the time preference of the intergenerational investor.

What should be the discount rate used for private forestry investments?  In this case, the question of the discount rate is related to the cost of capital, and whether capital rationing is a problem or not.  When capital rationing is used, the interest rate should reflect the internal opportunity cost of the forgone alternative investment[23].  Often the next investment opportunity will give a higher return than forestry, especially in temperate regions.  In most cases, the discounted value based on the other financial opportunity will result in a negative figure.  With this approach, the forest owner mainly holds a forest investment based on the non-accounted for amenities of the forest.

Several aspects have to be considered in determining the rate of return of the alternative investment. First, the expected alternative rate of return should be based on the history of this type of investment.  Second, the risk of the alternative investment has to be the same as in forestry.  Third, forestry investments are often partly taxed or promoted through incentives; therefore, the after-taxed yield is what is relevant. Fourth, It should be made clear which rate of return is used in the analysis of the nominal or real rates of interest.  It is not recommended to include inflation when discounting cash flows over long time horizons[24].

In the case of poor or subsistence farmers in some developing countries that do not have available money capital to invest elsewhere, their opportunity for investing outside the farm does not represent a real opportunity cost.  Investing in forestry will usually be at the cost of current and future agricultural production.  In this case the consumption rate of interest is the discount rate to be used based on the elasticity of marginal utility of consumption, expected income growth, and pure time preference.  The marginal utility of consumption and pure time preference suggest the use of higher interest rates for small farmers than for large ones[25].  In the case of expected income growth, farmers with upward trends of income growth will have high discount rates and vice versa[26].

 

Criteria for Optimal Rotations in Forestry

This article deals with the rotation problem seen from the investment theory standpoint.  Therefore, it is important to explain several criteria used for selecting optimal rotations in forestry.  The rotation idea originates from the development of the even-aged clear-cut (EAC) system, and it is defined as:

“The period of years required to establish and grow timber crops to a specified condition of maturity”[27]

It is also complemented by the following idea:

“A rotation is the planned number of years between establishment and clear felling…”[28]

 
The felling or cutting cycle is the term of the rotation applied to periodic selection systems. The advantage of this system is that it favors regeneration of light demanders or shade tolerant trees depending on the intensity and the size of the gaps[29].

Several rotation criteria can be identified depending on the type of forest management objectives.  These criteria have been derived from the management objectives.  Therefore the length of the rotation depends on several factors. The most traditional rotation criteria are based on factors oriented towards timber production.  There are 9 main rotation criteria [30] reported in forest economics literature:

The Biophysical rotation is basically based on the life expectation of the crop in which natural regeneration is assured.  The susceptibility to diseases, plagues and other climatic events like windfall are some of the factors affecting these criteria.

The Technical rotation is achieved when the maximum amount of wood is produce to satisfy a particular product demand.

The Maximum Volume Production (MVP) rotation maximizes the mean annual increment (MAI).

 The Maximum Revenue rotation maximizes the mean annual revenue and in this rotation the timber price is taken into account.   This rotation is based on the maximum MAI, but is longer than the MVP rotation.

 The Forest Rent rotation takes into account not only timber prices (revenues), but also costs.  However, this rotation does not account for interest on invested money and the land costs.

 The Land Expectation Value (LEV) rotation maximizes the NPV of all future rotations starting with bare land.  It is assumed that subsequent rotations are equal regarding periodic costs and revenues.  The rotation that maximizes the LEV is the one in which marginal costs are equal to marginal revenues.  In other words, the forest stand should be harvested when the value growth percent of the timber and the land in respect to time is equal to the selected interest rate of the invested capital[31].

 The Net Present Value (NPV) rotation is similar to the LEV rotation; however, it only includes revenues and costs within the planning horizon (one rotation).  Valuing the land asset under forest production, would be an incorrect solution because this method only accounts for the timber and not for the land [32].  The NPV rotation is longer than the LEV rotation.

 The Financial Maturity rotation is also based on marginal growth of the value of the forest (trees and land) in respect to density not time.  Considering several densities, the optimal rotation is achieved if the rate of increase per year between one growing stock density option and the marginal growth or increase of the next period (marginal revenues)[33] is equal to the investor’s rate of return.

 The Internal Rate of Return (IRR) rotation is a technique used when there is no knowledge of the investor’s internal rate of return.  In this case the rotation that is selected, is the one that yields the highest IRR.

References

Chang, S. J. [1984]: Determination of the Optimal Rotation Age: A Theoretical Analysis.  Forest Ecology and Management.  8: 137-147

Chiang, A. C. [1984]: Fundamental Methods of Mathematical Economics.  McGraw-Hill Co.  Singapore. 788p.

Davies, K. P. [1954]: American Forest Management.  McGraw-Hill. U.S.A. 482p.

Filius, A. M. [1992]: Investment Analysis in Forest Management: Principles and Applications. Department of Forestry, WAU. 192p

Gregersen, H. & Contreras A. [1992]: Economic Assessment of Forestry Project Impacts. F.A.O. Forestry Paper  No.106, Rome. 134p.

Gregory, R. [1987]: Resource Economics For Foresters. John Wiley & Sons, Inc. U.S.A. 457p.

Gregory, R. [1987]: Resource Economics For Foresters. John Wiley & Sons, Inc. U.S.A. 457p.

Hoekstra, D. A. [1985]: Choosing The Discount Rate For Analyzing Agroforestry Systems/Technologies from a Private Economic Viewpoint. Forest Ecology and Management, 10: 177-183.

Johansson, P.-O. & Löfgren, K.-G. [1985]: The Economics of Forestry and Natural Resources. Basil Blackwell Ltd., U.K. 292p.

Klemperer, W. D. [1996]: Forest Resource Economics and Finance. McGraw-Hill Series in Forest Resources. U.S.A. 551p.

Kula, E. [1988]: The Economics of Forestry: Modern Theory and Practice.  Timber Press. Portland 185p.

Matthews, J. D. [1989]: Silvicultural Systems.  Oxford University Press. Oxford. 284p.

Speidel G. [1984] Forstliche Betribswirtschaftslehre.  Verlag Paul Parey Hamburg u. Berlin 226p.

Posted 4 September 2007