202 Peterson and Straka: Cash Flow Analysis Formulas for Urban Trees and Forests contingent upon the tree reaching a certain “minimum size.” An example would be the “windbreak” ability of a tree in a windstorm. First, the tree would need to reach a minimum size to have windbreak ability and second, the benefit would oc- cur periodically, not every year. The MSDPCF formula is [7] where nat is the number of years for which the se- ries occurs, t is the length of the time period, and nv is the number of years in the future the series begins. Present Value of Patterned Terminating Periodic Series (PTPS) Urban trees may have several systematic, “stacked” cash flows, where one cash flow is “stacked” onto another. A cash flow of a smaller magnitude (i.e., the base series) may occur on a fre- quent basis, but necessitate a cash flow of a larger magnitude (i.e., the stacked series) on an infrequent basis. An example would be the soil enhancement benefit of trees. Fertilization might be reduced on an annual basis (the base series) and soil aera- tion might be reduced every 10 years (i.e., the stacked series). In this case, the larger cash flow is stacked onto the pattern of the smaller cash flow, and the following formula should be used: [8] Urban Tree Site Value (UTSV) In traditional forestry literature, land expectation value, or bare land, value is calculated for land in permanent timber production (Klemperer 1996). This methodology can be used to calculate the PV of any perpetual cash flow-producing in- vestment (Straka and Bullard 1996). This means a site value for an urban tree can also be calculated by compounding the PV of the tree’s cash flows to the end of its rotation (defined as its viable life on the site) and assessing this over a perpet- ual time frame. The following formula accomplishes this: [10] where UTSV is the urban tree site value with a perpetual time ho- rizon, while present value is the present value of all benefits and costs of the tree for one rotation, and n is the length of the rotation. Other Available Constructions Other formulas can be applied to urban tree cash flow analysis as well. For example, definite integration of dis- counted linear or nonlinear functions can show PV of cash flows with a functional increase or decrease over a particu- lar period of time (Sartoris and Hill 1983). Sound barrier benefits are an example of this. Transition matrices can be used to calculate risk, given that certain information about the probability of risk is specified (Kaye and Pyke 2003). Marginal analysis can be used to look at the cumulative ef- fect of cash flows with the same time series to reduce the overall number of DCF calculations (White et al. 1998). the base series occurs, t1 is the length of the time period for the base series, n2 is the number of years the stacked series occurs, and t2 is the length of the time period for the stacked series. where a1 is the cash flow of the base series, a2 of the stacked series, i is the interest rate, n1 Present Value of Minimum Size Delayed Pat- terned Terminating Cash Flows (MSDPTCF) Like other benefits or costs that do not begin until a minimum tree size is reached, patterned terminating benefits or costs need be discounted back to year zero. A systematic pruning of a tree on two levels is an example of this calculation; for ex- ample, minor pruning every five years and major pruning ev- ery twenty years. If so, the following formula should be used [9] is the cash flow is number of years RESULTS A series of examples was developed to illustrate the use of both standard and specialized valuation formulas. To stan- dardize the models, the examples were developed around a planted white oak (Quercus alba) in Atlanta, Georgia, U.S. It is estimated that a planted white oak has a lifespan of about 120 years, during the first 90 of which it is struc- turally sound (Burns and Honkala 1990). White oak trees reach a size of significant canopy coverage around 10 years of age. Thus, energy savings for the white oak begins at age 10. In 2010, the nursery price, plus planting, for white oak was USD $70. For simplicity later in discussion, this will be called the “planting cost.” Tree removal was assumed to cost $406 for a tree greater than 60.96 cm in diameter (ATSC 2011). Tree removal will be in real terms and not increased with inflation. All of the figures were calculated at a 5% real discount rate. This is a reasonable discount rate based on past consumer price index data (U.S. Dept. of Labor 2011). and n2 is the number of years the stacked series occurs, t2 length of the time period for the stacked series, nv1 is the num- ber of years the base annuity is away from year zero, and nv2 where a1 is the cash flow of the base series, a2 the stacked series, n1 series occurs, t1 is the number of years for which the base is the length of the time period for the base series, is the is the number of years the stacked annuity is away from year zero. ©2011 International Society of Arboriculture is the cash flow of Five Standard Formulas (SSD, TA, PA, TPS, and UTSV) The five formulas are regularly used in urban tree and forest valuation by appraisers using the income approach (McPherson 2007). All have an assumption that the first payment is at the end of the first year or time period. Annual and periodic payments or benefits are assumed to occur at the end of the first year or time period, respectively. A key characteristic of the “special” formu-
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