Calculations for Uniform Series That Are Shifted

When a uniform series begins at a time other than at the end of period 1, it is called a 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 Figure 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.