## Present Value of Uniform Annual Series

If the same benefits and/or costs occur for every period, e.g., every year as in Fig. 2.1, then in this case, the present worth is called uniform series present worth factor.

Where A is the annual cost or benefit, the present value of the uniform series can be calculated by multiplying A and equation 2.3 a or can be obtained from the compound interest tables using expression 2.3b.

Example 2.1

Calculate the net present value of the leasing project of example 1.3 using the netted cash flow diagram of Fig. 2.2a and assuming an interest rate of10%.

Fig. 2.2a

We have to calculate the preset (time zero) value of all the costs and benefits. The simplest and therefore the longest way to do this is to break the above cash flow into three components shown in Fig. 2.2b.

Step 1:

The \$24,000 cost occurs at time zero, so its present value is \$24,000. Now, we have to obtain the NPW of the other elements of the cash flow.

Step 2:

The \$4600 benefit is received for three years (n=3). We can then use equation 2.3 a to calculate the equivalent value of the total of these three payments referred to at time zero.

Fig. 2.2d i is 10%, that is 0.1 and n=3, hence present value at time zero is

We could have used expression 2.3b. AM600

From the compound interest tables, the page for i=10%, row n=3 and column for P/A, we obtain Hierefore, PV= 4600 * 2.487 = 11440

This is the same value calculated before. The small difference is due to rounding of the numbers. Step 3:

Calculate the present value of the \$19,600 at year 4. For this, we use equation 2.2a.

Fig. 2.2e pvb = t96o0 * (h-0.1)"1 = 13387.1 Step 4:

Calculate total cost, total benefit, and net worth. Total Cost = 24,000

Total Benefit = 11439.52 + 13387.1 = 24826.62

Net Worth of the Project = 826.62

Again, we could have used the expression 2.3b and used the page for 10% from the compound interest rate tables. We would have obtained the same answer.

There are many different ways to arrive at this number. The above is the simplest way and is prone to less error. As you, the reader, gain more experience, you will develop shortcuts with which you are more comfortable. You will then be able to solve some problems with only one line of arithmetic. The trick is to break the netted cash flow into components with which you feel comfortable.

The Excel spreadsheet calculation of the net present worth for this problem is shown below. The Quattro Pro spreadsheet expression to be used is also shown.

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