Average Annual Equivalent Damage

Equivalent Annual Damage and System Performance in the Plan Evaluation Framework 

USACE water resource development projects typically involve many alternatives, require several years to plan and install, and provide benefits and sometimes deferred installation costs for many years after implementation. They typically incur the greater proportion of their costs early during the construction phase. In that case, benefits would accrue over an extended period, often increasing over time.

Recognizing that a dollar in hand is not worth the same as a dollar 1, 10, or 25 years hence, the problem confronting the water resource development analyst is to convert unevenly distributed benefit (and cost) streams to comparable or equivalent measures.

Benefits from each alternative that differ in total magnitude but that are made at different dates may be made equivalent to one another and thus enables comparisons to be made. Discounting and compound interest procedures using the same discount rate, base year and time frame for all alternative benefits and costs allows us to make such comparisons.

The base year is the first year in which the project is expected to become operational, and the period of analysis is defined as the time frame, beginning with the base year, for which project benefits and costs are considered. For most USACE studies, the period of analysis is typically 50 years since predicting futures beyond that frame involves much greater uncertainty. When comparing alternatives, the same base year and period of analysis should be used for all plans being evaluated.

The expected annual benefits streams are the estimates of the annual benefits that are expected to occur during the construction period and over the period of analysis. Generally, benefits are only anticipated after plan implementation, but for some projects benefits can also occur during the construction period. The analytical problem is to convert the varying benefit streams to their equivalent and comparable average annual measures over a common period of analysis.

Prior to converting the benefit and cost streams to average annual equivalent measures, the present worth of the streams must first be determined. The present worth for each stream is the single value in the base year that is equivalent to the many payments that would accrue from that stream. Discounting is the procedure used to reduce future values, those occurring during the project life, to their present worth in the base year. Compound interest procedures are used to determine the present worth of benefits accrued during the construction period, prior to the base year. The present worth value for the entire benefit stream is dependent on the magnitude, number, and timing of individual payments as well as the appropriate discount rate used in the analysis.

Equivalence concept and steps.

The average annual equivalent benefit, generally termed the average annual benefit, is defined as the amortized value over the period of analysis of the present worth in the base year of the benefit stream. The average annual or amortized value, therefore, is a constant amount of benefit occurring each year during the period of analysis. This constant stream of benefits is equivalent to the present worth in the base year of the entire benefit stream. Equivalent annual damage calculations are performed first by interpolating expected annual damage values between existing and future time periods. Next, for each year in the period of analysis, the present value of EAD is computed. The present value of EAD for each year is then summed yielding total present value of EAD for the period of analysis (usually fifty years). Finally, using the federal discount rate, total present value of EAD is amortized over the period of analysis to yield equivalent annual damages. This process is completed for the without condition and repeated for each with project condition. Equivalent annual damages for each with project condition is compared to equivalent annual damages without project to determine project EAD reduced (benefits). 

USACE policy dictates that discounting is to be used to convert future monetary values to present values. For USACE studies, analysts calculate present values using the discount rate established annually for the formulation and economic evaluation of plans for water and related land resources. In effect, the discount rate is the ratio between the value of a future payment and its present worth at the base year in USACE water resource development studies.

Each year’s EAD, which is assumed to accrue at the end of the year, is discounted back to the beginning of the base year (yielding its present worth) if the benefits accrue after implementation. If benefits accrue prior to the entire project being installed (prior to base year) then each year's prior EAD is discounted forward to the base year. 

To discount back to the beginning of the base year, we can use the present worth single payment factor equation given by 

where P is the present worth, F is a sum of money at the end of n periods from the present worth date that is equivalent to P with interest i (also the discount rate) and n is the number of periods.

Sometimes plans are implemented in phases and measures to reduce risk become operational before the project is completed (prior to the base year).  For example, the nonstructural portion of a plan may be completed prior to the levee portion of the plan is completed and therefore damages reduced by the nonstructural measures will occur pre-base year. To discount benefits accrued forward to the base year, we can use the compound interest formula given by 

To amortize total present value over the period of analysis using the federally prescribed discount rate, the present worth of annuity factor is applied and is given by the equation below. The sum present value is calculated using the function above and then is amortized by dividing the sum present value by the present value interest factor of annuity. 

Take a look at the spreadsheet below to follow along the calculation of AAED.