Equivalence Calculations Involving Series with PP ≥ CP 

When the cash flow of the problem dictates the use of one or more of the uniformseries or gradient factors, the relationship between the compounding period, CP, and payment period, PP, must be determined. The relationship will be one of the following three cases:

Type 1. Payment period equals compounding period, PP = CP.

Type 2. Payment period is longer than compounding period, PP > CP.

Type 3. Payment period is shorter than compounding period, PP < CP.

The procedure for the first two types is the same. Type 3 problems are discussed in the following section. When PP = CP or PP > CP, the following procedure always applies:

Step 1. Count the number of payments and use that number as n. For example, if payments are made quarterly for 5 years, n is 20.

Step 2. Find the effective interest rate over the same time period as n in step 1. For example, if n is expressed in quarters, then the effective interest rate per quarter must be used.

Use these values for n and i (and only these!) in the factors, functions, or formulas. To illustrate, Table (3-5) shows the correct standard notation for sample cash-flow sequences and interest rates. Note in column 4 that n is always equal to the number of payments and i is an effective rate expressed over the same time period as n.

       Table (3-5): Examples of n and i Values Where PP = CP or PP > CP

For the past 7 years, a quality manager has paid $500 every 6 months for the software maintenance contract on a laser-based measuring instrument. What is the equivalent amount after the last payment, if these funds are taken from a pool that has been returning 10% per year compounded quarterly?

The cash flow diagram is shown in Figure (3-3). The payment period (6 months) is longer than the compounding period (quarter); that is, PP > CP. Applying the guideline, determine an effective semiannual interest rate. Use Equation (3-2) or Table (3-3) with r = 0.05 per 6-month period and  m = 2 quarters per semiannual period.

The value i = 5.063% is reasonable, since the effective rate should be slightly higher than the nominal rate of 5% per 6-month period. The number of semiannual periods is n = 2(7) = 14 The future worth is

Figure (3-3): Diagram of semiannual payments used to determine F, Example (3-4)