Arboriculture & Urban Forestry 37(5): September 2011 las is that the timing of the first payment often differs from this standard case. In a nonstandard or special case, an annual series first payment may be, for example, at year 10 instead of year one. Or a periodic series of payments every 20 years, for example, may have a first payment at year 35 instead of year 20. The dis- counting formulas often account for a first payment that occurs earlier or later than assumed in the standard formula situation. A simple example using a 5% interest rate was used to illus- trate the use of each formula (Table 1). A planting cost of $70 occurs at year zero (the present) and a one-time removal cost of $406 occurs at year 90. These are single sum problems, using Formula 1 the present values of the planting and removal costs are $70 and $5.03, respectively. Trees around a residence en- hance privacy and provide additional home security protection; the study authors assumed this tree’s benefit was worth a uni- form $90 per year. This benefit lasted for the average expected lifetimes of existing trees, or 90 years. This would be a ter- minating annuity problem; using Formula 2, the present value of these annual benefits was $1,777.70. Another benefit might occur every 10 years; for example, stormwater mitigation, start- ing at year 10 and ending at year 90. This would be a termi- nating periodic series and would be solved with Formula 4 to produce a value of $314.07. Finally, it was assumed that the tree increased property taxes by $10, and that this cost would last in perpetuity. This cash flow was valued at $200 with Formula 3. Note that the cash flows are summed in the Table. A sum of an investment’s discounted benefits and costs is a net present value (NPV). The UTSV was the value of the same tree on- site on a perpetual basis, rather than just one 90-year lifetime. Carbon Sequestration (FRIA and PA Calculations) The carbon sequestration ability of a tree increases as its size increases. Eventually, the tree reaches a maximum size but is still sequestering carbon. If the rate of sequestration is known, and increases at a geometric rate, the present value of 203 the cash flow series can be calculated using Formula 5. Table 2 shows a simplified scenario for carbon sequestration; in the first year the benefit is $1, and this value increases by 4% an- nually until age 70. The study authors assumed that after age 70 the tree’s growth becomes negligible, but used the age 70 value to represent carbon sequestration for the rest of perpetu- ity (because even after removal, the tree will still hold a fixed amount of carbon, assuming it ends up in building products or some other permanent use). The study authors used Formula 3 to value benefits past age 70. The FRIA calculation was the present value of a $1 benefit that increased 4% annually to be worth $15.57 at age 70. The value of all the benefits past age 70 is $311.40 (Formula 3), but that value must be discounted for 70 years and becomes a present value of $10.23. Carbon sequestrating in this example produces a benefit of $59.05. Energy Savings (MSDACF Calculation) American Forests (2001) showed that in Atlanta, GA, 0.4 ha of land with tree coverage would reduce natural gas usage by $13.67. A tree must reach a certain minimum size before its crown is able to provide shade. In the provided example (Table 3), the designated minimum size is reached at year 10. Both planting costs and removal costs were single sums. Single sums were discounted using Formula 1. The annual savings repre- sented a terminating annual series and were discounted us- ing the standard formula for the present value of a terminat- ing annual series to a present value of $268.14 (Bullard and Straka 1998). However, since the first savings occur at the be- ginning of year 10, the $268.14 must be discounted for nine years and then become a present value at year 0 of $172.74. Formula 6 allowed for the delayed annual series to be direct- ly calculated as a PV of $172.74. The “special case” formula simplified the calculation and required only a single formula. Note that PV was the value of a tree over a single lifetime and UTSV was the value of that same tree on a perpetual basis. Table 1. Illustration of standard discounting formulas at a 5% interest rate. Formulas used include single sum discounting (SDS), terminating annuity (TA), perpetual annuity (PA), terminating periodic series (TPS), and urban tree site value (UTSV). Currency is in U.S. dollars. Treatment Plant tree Annual benefit Periodic benefit Perpetual cost Remove tree Net present value (over 90 years) UTSV (in perpetuity) Frequency Single time Annual Every ten years Annual Single time Years 0 1–90 10, 20, …90 1, 2, 3, …∞ 90 Amount ($) (70.00) 90.00 200.00 (10.00) (406.00) PV ($) (70.00) 1,777.70 314.07 (200.00) (5.03) 1,816.74 1,839.53 Table 2. Carbon sequestration model at a 5% interest rate illustrating fixed rate increasing annuity (FRIA) and perpetual annuity (PA). Currency is in U.S. dollars. Item Plant tree Sequester Sequester Remove tree Net present value (over 90 years) UTSV Frequency Single time Annual Annual Single time Years 0 1–70 70 + 90 Amount ($) (70.00) 1.00–15.57 15.57 (406.00) PV ($) (70.00) 48.82 10.23 (5.03) (15.98) (16.18) ©2011 International Society of Arboriculture
September 2011
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