## Pl PFPiN

Figure 3.1

Single Receipt at End of Period N

so that the compound amount factor is

A handy way of thinking of the notation is (reading from left to right): "What is F, given P, i, and N?"

The compound amount factor is useful in determining the future value of an investment made today if the number of periods and the interest rate are known.

The present worth factor, denoted by (P/F,i,X), gives the present amount, P, that is equivalent to a fumre amount, F, when the interest rate is i and the number of periods is N. The present worth factor is die inverse of the compound amount factor, (F/P,i,X). That is, while the compound amount factor gives the fumre amount, F, that is equivalent to a present amount, P, the present worth factor goes in the other direction. It gives the present worth, P, of a fumre amount, F Since (F/P,i,X) = (1 + /)-\

The compound amount factor and the present worth factor are fundamental to engineering economic analysis. Their most basic use is to convert a single cash flow that occurs at one point in time to an equivalent cash flow at another point in time. When comparing several individual cash flows which occur at different points in time, an analyst would apply the compound amount factor or the present worth factor, as necessary, to determine the equivalent cash flows at a common reference point in time. In this way, each of the cash flows is stated as an amount at one particular time. Example 3.1 illustrates this process.

Although die compound amount factor and the present worth factor are relatively easy to calculate, some of the other factors discussed in this chapter are more complicated, and it is therefore desirable to have an easier way to determine their values. The compound interest factors are sometimes available as functions in calculators and spreadsheets, but often these functions are provided in an awkward format that makes them relatively difficult to use. They can, however, be fairly easily programmed in a calculator or spreadsheet.

A traditional and still useful method for determining the value of a compound interest factor is to use tables. Appendix A at the back of this book lists values for all the compound interest factors for a selection of interest rates for discrete compounding periods. The desired compound interest factor can be determined by looking in the appropriate table.

### EXAMPLE 3.1

How much money will be in a bank account at the end of 15 years if Si00 is invested today and the nominal interest rate is 8% compounded semiannually?

Since a present amount is given and a future amount is to be calculated, the appropriate factor to use is the compound amount factor, (F/P,i,X). There are several ways of choosing / and N to solve this problem. The first method is to observe that, since interest is compounded semiannually, the number of compounding periods, A', is 30. The interest rate per six-month period is 4%. Then

The bank account will hold S324.34 at the end of 15 years.

CHAPTER 3 Cash Flow Analysis

Alternatively, we can obtain the same results by using the interest factor tables. F = 100(3.2434) (from Appendix A) = 324.34

A second solution to the problem is to calculate the effective yearly interest rate and then compound over 15 years at this rate. Recall from Equation (2.4) that the effective interest rate per year is r \m 1 + — - ■

m j where i, = the effective annual interest rate r = the nominal rate per year vi = the number of periods in a year i, = (1 + 0.08/2)2 - 1 = 0.0816 where r = 0.08 m = 2

\Mien the effective yearly rate for each of 15 years is applied to the future worth computation, the future worth is

Once asrain, we conclude that the balance will be S324.34.B

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