Annual Worth and Equivalent Uniform Annual Worth

The annual worth is the net of all the benefits and costs incurred over a one-year period. Therefore, we present the net of all the different benefits and costs incurred at different points of time in a one-year period with one number, and we call it the annual worth. For a system whose life is longer than one year, this number will be different for different years. For systems having more than one year of life, we can calculate a single virtual number that represents an equivalent annual net benefit or cost for the duration of the system life. This virtual number is called the equivalent uniform annual worth (EUAW) and is equal to the total benefit and cost of the system as if it was spread evenly throughout the years of its life. We can express this in a different way. The net present worth ofthe system, calculated as if its net benefit or cost for each year was the calculated EUAW, is the same as the net present worth of the same system using the real values of costs and benefits at their real time of occurrence.

The simplest process for calculating this number takes two steps.

Step 1:

All the costs and benefits are transferred to the present year using equation 2.2a to calculate the NPW.

Step 2:

Multiplying the NPW by a factor called capital recovery factor converts it to EUAW. This multiplier is

Or as in previous cases

(A/P, i, n) is the capital recovery factor, and its value for any i and n can be found in compound interest tables. This is the same as spreading the NPW of a project over the life of the project. When the EUAW of a system or project is a positive number, it indicates that the project is economically viable or profitable. The advantage of this method is that we need not worry about the unequal lives or the unequal initial investment of the two systems being compared. They are taken into account automatically through the mathematics of the analysis method.

If the EUAW method is used for choosing among more than two alternatives, we simply have to calculate the EUAW of all of them and choose the one with the highest EUAW.

When a bank gives you a loan to buy a house, it spreads the loan over the next 15 or 30 years. In effect, your mortgage payments are the EUAW of the loan.

Example 4.1

Find the EUAW for the project of example 1.3. Step 1:

Draw the cash flow diagram and calculate the NPW:

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We have already done this and we know that NPW = 326.61

We can either use equation 4.1a or expression 4.1b. Using the latter, EUAW = 8 26.61 (A/P, 10,4) = 826.51 (.3155) = 260%

This means that the NPW of 826.61 is the same as an annual value of 260.8 for four years. The use of spreadsheet will give us the same result.

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Example 4.2

A $120,000 house is bought by making a $20,000 down payment and obtaining a loan from the local bank at an interest of10% for 30 years. What is the annual payment?

We can use equation 4.1a or expression 4.1b to solve this problem. Using expression 4.1b EUAW = A=P{A/P.I0,30)

Using the compound interest rate tables EUAW = 1M,0OO (0.1061) = 10.610

The spreadsheet solution is

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