Lecture 8
ENGR 140
Combining Factors
Most, real world, cash flows do not fit the structure of the models presented for determining equivalence. However, the models can be combined to calculate equivalence for most cash flows.
Shifted Uniform Series
When a uniform series begins at a time other than at the end of period 1, it is called a shifted series. An example of a shifted series is illustrated by the cash flow diagram below.
Determining the present worth of the uniform series can be accomplished using a variety of approaches. The two most efficient approaches , which each require two steps, will be presented here.
Method 1: Determine the “present worth” of the uniform series (P/A), which produces an equivalent single value at year 2. Next, determine the present worth of the single value at year 2 using the P/F function.
- “P” = $3,000(P/A, 7%, 4)
- P = “P”(P/F, 7%, 2)
The steps are illustrated in the following cash flow diagrams.
Method 2: Determine the future value of the uniform series (F/A), which will produce a single value at year 6. Next, determine the present value of the single value at year 6 (P/F).
- F = $3,000(F/A, 7%, 4)
- P = F(P/F, 7%, 6)
The steps are illustrated in the following cash flow diagrams.
Determining an equivalent uniform series from a shifted uniform series
To determine an equivalent uniform series from a shifted uniform series, a present worth or future values must first be determined. Once one of those values has been determined using either method 1 or 2, an equivalent uniform series can be determined using the methods from Chapter 2:
A = P(A/P, i%, n) or A = F(A/F, i%, n)
Using Excel functions to evaluate shifted uniform series
The net present value (NPV) function in Excel can be used to determine the present worth of an irregular cash flow such as a shifted uniform series. The syntax for the NPV function is:
=NPV(i%, second cell : last cell) + first cell
To use the NPV function, first put all values for the cash flow in a column (or row) in the spreadsheet. Remember to put zeros in cells for periods with zero cash flows. The first cell contains the cash flow for period zero and must be listed separately for NPV to calculate the present worth correctly.
For the previous example, the data entry is:
Year / Cash Flow0 / 0 / i = / 7%
1 / 0
2 / 0 /
3 / $3,000
4 / $3,000
5 / $3,000 / NPV = / $8,876
6 / $3,000
To determine an equivalent uniform series in Excel, the =PMT function is used in conjunction with the =NPV function used above. The =PMT function can either reference the cell containing the =NPV result or the =NPV function can be imbedded in the =PMT function.
=PMT(i%, n, cell with NPV result)
- or -
=PMT(i%, n, NPV(i%, B4:B9) + B3)
Adding another layer of complexity – random single payments
Adding random single payments to the shifted uniform series considered earlier changes the cash flow diagram to the following:
To determine an equivalent P, F, or A for the illustrated cash flow handle all components separately, determine the desired equivalency, then sum the equivalencies to determine the desired equivalency for the all of the cash flow components.
Example, present worth of the illustrated cash flow:
P = P of shifted uniform series + P of year 1 future value + P of year 5 future value
P = $3000(F/A, 7%, 4)[P/F, 7%, 6] + $5000(P/F, 7%, 1) + $4000(P/F, 7%, 6)
A similar method can be used to determine the future value, F. to determine the equivalent uniform series value, first determine P or F, then convert to A using (A/P) or (A/F).
Shifted Gradients
The equivalent P, F, and A can be calculated for shifted gradients using a process similar to the process for determining equivalencies of shifted uniform series. Begin by determining the “present worth” of the gradient at the period two years prior to the start of the gradient series (recall that a gradient starts at the end of period 2). Next determine the present worth at the actual year zero using the P/F function. Finally, the future worth or uniform series equivalencies of the present worth value can be calculated using F/P or A/P respectively, if desired.
To determine the equivalent present worth of the cash flow with the shifted gradient, calculate the present worth of each cash flow component and sum them.
P = A(P/A, 10%, 7) + G(P/G, 10%, 5) (P/F, 10%, 2)
The first component determines the present worth equivalent of the $5,000 per year uniform series. The second component determines the “present worth” at year 2 of the gradient then determines the present worth of the year 2 value.
P = $5,000(4.8684) + $250(6.8618) (0.8264)
P = $24,342 + $1,418 = $25,760
Reading: Chapter 3
Homework: 3.3, 3.9, 3.14, 3.19,3.22, 3.27, 3.35Due:Wednesday, October 12.
Quiz: Wednesday, October 12
1Ohlinger