### EXAMPLES - Uniform Series That Are Shifted

**EXAMPLE 1**

The offshore design group at Bechtel just purchased upgraded CAD software for $5000 now and annual payments of $500 per year for 6 years starting 3 years from now for annual upgrades. What is the present worth in year 0 of the payments if the interest rate is 8% per year?

**Solution**

The cash flow diagram is shown in Figure 3–4. The symbol PA is used throughout this chapter to represent the present worth of a uniform annual series A , and P'A represents the present worth at a time other than period 0. Similarly, P T represents the total present worth at time 0. The correct placement of P'A and the diagram renumbering to obtain n are also indicated. Note that P'A is located in actual year 2, not year 3. Also, n = 6, not 8, for the P/A factor. First find the value of P'A of the shifted series.

**EXAMPLE 2**

Recalibration of sensitive measuring devices costs $8000 per year. If the machine will be reca-

librated for each of 6 years starting 3 years after purchase, calculate the 8-year equivalent

uniform series at 16% per year. Show hand and spreadsheet solutions.

**Solution by Hand**

Figure 3–5 a and b shows the original cash ﬂ ows and the desired equivalent diagram. To convert the $8000 shifted series to an equivalent uniform series over all periods, first convert the uniform series into a present worth or future worth amount. Then either the A/P factor or the A/F factor can be used. Both methods are illustrated here.

**Present worth method.**(Refer to Figure 3–5 a .) Calculate P'A for the shifted series in year 2, followed by PT in year 0. There are 6 years in the A series.

**Future worth method.**(Refer to Figure 3–5 a .) First calculate the future worth F in year 8.

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