Single-Payment Factors (F/P and P/F)

 

The most fundamental factor in engineering economy is the one that determines the amount of money F accumulated after n years (or periods) from a single present worth P, with interest compounded one time per year (or period). Recall that compound interest refers to interest paid on top of interest. Therefore, if an amount P is invested at time t = 0, the amount F1 accumulated 1 year hence at an interest rate of i percent per year will be

where the interest rate is expressed in decimal form. At the end of the second year, the amount accumulated F2 is the amount after year 1 plus the interest from the end of year 1 to the end of year 2 on the entire F1.

The amount F2 can be expressed as

Similarly, the amount of money accumulated at the end of year 3, using Equation (2-1), will be

Figure (2-1): Cash flow diagrams for single-payment factors: (a) find F, given P, and (b) find P, given F.

From the preceding values, it is evident by mathematical induction that the formula can be generalized for n years. To find F, given P,

The factor  is called the single-payment compound amount factor (SPCAF), but it is usually referred to as the F/P factor. This is the conversion factor that, when multiplied by P, yields the future amount F of an initial amount P after n years at interest rate i. The cash flow diagram is seen in Figure (2–1 a).

Reverse the situation to determine the P value for a stated amount F that occurs n periods in the future. Simply solve Equation (2-2) for P.

The expression  is known as the single-payment present worth factor (SPPWF), or the P/F factor. This expression determines the present worth P of a given future amount F after n years at interest rate i. The cash flow diagram is shown in Figure (2–1 b).

Note that the two factors derived here are for single payments; that is, they are used to find the present or future amount when only one payment or receipt is involved.

A standard notation has been adopted for all factors. The notation includes two cash flow symbols, the interest rate, and the number of periods. It is always in the general form (X/Y, i, n). The letter X represents what is sought, while the letter Y represents what is given. For example, F/P means find F when given P. The i is the interest rate in percent, and n represents the number of periods involved.

Using this notation, (F/P ,6%,20) represents the factor that is used to calculate the future amount F accumulated in 20 periods if the interest rate is 6% per period. The P is given. The standard notation, simpler to use than formulas and factor names, will be used hereafter.

Table (2–1) summarizes the standard notation and equations for the F/P and P/F factors.

Table (2–1): F/P and P/F Factors: Notation and Equations

Factor

 

Standard Notation

Equation

Excel

Notation

Name

Find/Given

Equation

with Factor Formula

Function

(F/P, i, n)

Single-payment

compound amount

F/P

F = P(F/P, i, n)

F = P (1 + i)n

= FV(i%,n,,P)

(P/F, i, n)

Single-payment

present worth

P/F

P = F(P/F, i, n)

P = F (1 + i)-n

= PV(i%,n,,F)

For spreadsheets, a future value F is calculated by the FV function using the format

A present amount P is determined using the PV function with the format

 

Sandy, a manufacturing engineer, just received a year-end bonus of $10,000 that will be invested immediately. With the expectation of earning at the rate of 8% per year, Sandy hopes to take the entire amount out in exactly 20 years to pay for a family vacation when the oldest daughter is due to graduate from college. Find the amount of funds that will be available in 20 years by using:

1)      hand solution by applying the factor formula and tabulated value

2)      a spreadsheet functions.

 

 

The cash flow diagram is the same as Figure (2–1a). The symbols and values are

P = $10,000     F = ?     i = 8% per year     n = 20 years

1)      Factor formula: Apply Equation (2-2) to find the future value F. Rounding to four decimals, we have

Standard notation and tabulated value: Notation for the F/P factor is (F/P,i%,n).

2)      Spreadsheet: Use the FV function to find the amount 20 years in the future. The format is that shown in Equation (2-4); the numerical entry is = FV(8%,20,,10000).