### Calculations for Uniform Series That Are Shifted

**shifted series**. In this case several methods can be used to ? nd the equivalent present worth P . For example, P of the uniform series shown in

**Figure 3–1**could be determined by any of the following methods:

• Use the P/F factor to find the present worth of each disbursement at year 0 and add them.

• Use the F/P factor to find the future worth of each disbursement in year 13, add them, and then find the present worth of the total, using P/F ( P/F , i ,13).

• Use the F/A factor to ? nd the future amount F/A ( F/A , i ,10), and then compute the present

worth, using P/F ( P/F , i ,13).

• Use the P/A factor to compute the “present worth” P3 = A ( P/A , i ,10) (which will be located in year 3, not year 0), and then ? nd the present worth in year 0 by using the ( P/F , i ,3) factor.

Typically the last method is used for calculating the present worth of a uniform series that does not begin at the end of period 1. For

**Figure 3–1,**the “present worth” obtained using the P/A factor is located in year 3. This is shown as P3 in F

**igure 3–2**. Note that a P value is always located m1 year or period prior to the beginning of the first series amount. Why? Because the P/A factor was derived with P in time period 0 and A beginning at the end of period 1. The most common mistake made in working problems of this type is improper placement of P . Therefore, it is extremely important to remember:

The present worth is always located one period prior to the ﬁ rst uniform series amount when using the P/A factor.

To determine a future worth or F value, recall that the F/A factor derived in Section 2.3 had the F located in the same period as the last uniform series amount.

**Figure 3–3**shows the location of the future worth when F/A is used for

**Figure 3–1**cash flows.

The future worth is always located in the same period as the last uniform series amount when using the F/A factor.

It is also important to remember that the number of periods n in the P/A or F/A factor is equal

to the number of uniform series values. It may be helpful to renumber the cash ﬂ ow diagram to

avoid errors in counting.

**Figures 3–2 and 3–3 show Figure 3–1**renumbered to determine n = 10.

As stated above, several methods can be used to solve problems containing a uniform series that is shifted. However, it is generally more convenient to use the uniform series factors than the single-amount factors. Specific steps should be followed to avoid errors:

1. Draw a diagram of the positive and negative cash flows.

2. Locate the present worth or future worth of each series on the cash flow diagram.

3. Determine n for each series by renumbering the cash flow diagram.

4. Draw another cash flow diagram representing the desired equivalent cash flow.

5. Set up and solve the equations.

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