Equivalence Calculations Involving Series with PP < CP 

If a person deposits money each month into a savings account where interest is compounded quarterly, do the so-called interperiod deposits earn interest? The usual answer is no. However, if a monthly payment on a $10 million, quarterly compounded bank loan were made early by a large corporation, the corporate financial officer would likely insist that the bank reduce the amount of interest due, based on early payment. These two are examples of PP < CP, type 3 cash flows. The timing of cash flow transactions between compounding points introduces the question of how interperiod compounding is handled. Fundamentally, there are two policies: interperiod cash flows earn no interest, or they earn compound interest. The only condition considered here is the first one (no interest), because many realworld transactions fall into this category.

For a no-interperiod-interest policy, deposits (negative cash flows) are all regarded as deposited at the end of the compounding period, and withdrawals are all regarded as withdrawn at the beginning. As an illustration, when interest is compounded quarterly, all monthly deposits are moved to the end of the quarter, and all withdrawals are moved to the beginning (no interest is paid for the entire quarter). This procedure can significantly alter the distribution of cash flows before the effective quarterly rate is applied to find P, F, or A. This effectively forces the cash flows into a PP = CP situation, as discussed in Section 5.

Rob is the on-site coordinating engineer for Alcoa Aluminum, where an under- renovation mine has new ore refining equipment being installed by a local contractor. Rob developed the cash flow diagram in Figure (3-4a) in $1000 units from the project perspective. Included are payments to the contractor he has authorized for the current year and approved advances from Alcoa’s home office. He knows that the interest rate on equipment “field projects” such as this is 12% per year compounded quarterly, and that Alcoa does not bother with interperiod compounding of interest. Will Rob’s project finances be in the “red” or the “black” at the end of the year? By how much?

With no interperiod interest considered, Figure (3-4a) reflects the moved cash flows. The future worth after four quarters requires an F at an effective rate per quarter such that PP = CP = 1 quarter, therefore, the effective i = 12%/4 = 3%. Figure (3-4b) shows all negative cash flows (payments to contractor) moved to the end of the respective quarter, and all positive cash flows (receipts from home office) moved to the beginning of the respective quarter. Calculate the F value at 3%.

Rob can conclude that the on-site project finances will be in the red about $357,600 by the end of the year.

Figure (3-4): (a) Actual and (b) moved cash flows (in $1000) for quarterly compounding periods using no interperiod interest,, Example (3-5)