Formulation Equivalence Calculations Involving Only Single Amount Factors

There are many correct combinations of i and n that can be used when only singleamount factors (F/P and P/F) are involved. This is because there are only two requirements: (1) An effective rate must be used for i, and (2) the time unit on n must be the same as that on i. In standard factor notation, the single-payment equations can be generalized.

Thus, for a nominal interest rate of 12% per year compounded monthly, any of the i and corresponding n values shown in Table (3-4) could be used (as well as many others not shown) in the factors. For example, if an effective quarterly interest rate is used for i, that is, (1.01)³-1 = 3.03%, then the n time unit is 4 quarters in a year.

Table (3-4): Various i and n Values for Single- Amount Equations Using r =12% per Year, Compounded Monthly

Alternatively, it is always correct to determine the effective i per payment period using Equation (3-2) and to use standard factor equations to calculate P, F, or A.

Sherry expects to deposit $1000 now, $3000 4 years from now, and $1500 6 years from now and earn at a rate of 12% per year compounded semiannually through a company-sponsored savings plan. What amount can she withdraw 10 years from now?

Only single-amount P and F values are involved (Figure (3-2)). Since only effective rates can be present in the factors, use an effective rate of 6% per semiannual compounding period and semiannual payment periods. The future worth is calculated using Equation (3-5).

An alternative solution strategy is to find the effective annual rate by Equation (3-2) and express n in years as determined from the problem statement.

Figure (3-2): Cashflow diagram, Example (3-3)