## Compound Interest Factors for Annuities

The next four factors involve a series of uniform receipts or disbursements that start at the end of the first period and continue over N periods, as illustrated in Figure 3.2. This pattern of cash flows is called an annuity. Mortgage or lease payments and maintenance contract fees are examples of the annuity7 cash flow pattern. Annuities may also be used to model series of cash flows that fluctuate over time around some average value. Here the average value would be the constant uniform cash flow. This would be done if the fluctuations were unknown or deemed to be unimportant for the problem.

The sinking fund factor, denoted by (A/FJ,N), gives the size, A, of a repeated receipt or disbursement that is equivalent to a future amount, F, if the interest rate is i and the number of periods is .V. The name of the factor comes from the term sinking fund. A sinking fund is an interest-bearing account into which regular deposits are made in order to accumulate some amount.

The equation for the sinking fund factor can be found by decomposing the series of disbursements or receipts made at times 1, 2, . . . , N, and summing to produce a total future value. The formula for the sinking fund factor is

Figure 3.2 Annuity Over N Periods

or, as an alternative representation

The sinking fund factor is commonly used to determine how much has to be set aside or saved per period to accumulate an amount F at the end of Nperiods at an interest rate i. The amount F might be used, for example, to purchase new or replacement equipment, to pay for renovations, or to cover capacity expansion costs. In more general terms, the sinking fund factor allows us to convert a single fumre amount into a series of equal-sized payments, made over N equally spaced intervals, with the use of a given interest rate i.

The uniform series compound amount factor, denoted by (F/A,i,N), gives the future value, F, that is equivalent to a series of equal-sized receipts or disbursements, A, when the interest rate is i and the number of periods is N. Since the uniform series compound amount factor is the inverse of the sinking fund factor,

The capital recovery factor, denoted by (A/P,i,N), gives the value, A, of the equal periodic payments or receipts that are equivalent to a present amount, P, when the interest rate is and the number of periods is N. The capital recovery factor is easily derived from the sinking fund factor and the compound amount factor:

The capital recovery factor can be used to find out, for example, how much money must be saved over N future periods to "recover" a capital investment of P today. The capital recovery factor for the purchase cost of something is sometimes combined with the sinking fund factor for its salvage value after A7 years to compose the capital recovery formula. See Close-Up 3.1.

The series present worth factor, denoted by (P/A,i,N), gives the present amount, P, that is equivalent to an annuity with disbursements or receipts in the amount, A, where the interest rate is / and the number of periods is .V. It is the reciprocal of the capital recovery factor: