An interest rate is a standardized measure of either: (1) the cost of borrowing money or (2) the return for lending money for a specified period of time (usually one year), such as 12% annual percentage rate (APR).

First consider the term "interest" from the perspective of a borrower. In this case, "interest" is the difference between the amount of money borrowed and the amount of money repaid. Interest expense is incurred as a result of borrowing money. On the other hand, interest revenue is earned by lending money.

For example, the amount of interest expense, as a result of borrowing $1000 on January 1, 20XX, and repaying $1120 on December 31, 20XX is $120 ($1120 $1000). The lender, on the other hand, received $1120 on December 31, 20XX in exchange for lending $1000 on January 1, 20XX, or a total of $120 in interest revenue. Thus, with regard to any particular lending event, interest revenue equals interest expense.

The formula used to calculate the amount of interest is:

interest = principal × interest × rate time [1]

where:

principal amount of money borrowed

interest rate = percent paid or earned per year

time = number of years

Equation (1) can be rewritten as:

interest rate = interest × principal [2]

where:

time = one year

The principal is also known as the present value. The interest rate in equation (2) is called the annual percentage rate or APR. APR is the most useful measure of interest rate. (In the remainder of this discussion, the term "interest rate" refers to the APR.)

Equations (1) and (2) are useful in situations that involve only one cash flow (a single-payment scenario). Many economic transactions, however, involve multiple cash flows. For instance, a consumer acquires a good or service and in exchange promises to make a series of payments to the supplier. This type of transaction describes an annuity. An annuity is a series of equally spaced payments of equal amount. The annuity formula is:

present value of annuity = annuity payment × annuity factor _i,n [3]

where:

present value of annuity = value of the good or service received today (when the exchange transaction is finalized)

annuity payment = amount of the payment that is made each period

annuity factor = a number obtained from an ordinary annuity table that is determined by the interest rate (i) and the number of annuity payments (n).

An analysis of the effect of changes in interest rates requires controlling (or holding constant) two of the other three variables in equation (3).

The term "future cash flow(s)" describes cash that will be received in the future. Holding the number of payments and the amount of each payment constant, the present value of future cash flows is inversely related to the interest rate. Holding the number of payments and present value of the future cash flows constant, the amount of each payment is directly related to the interest rate. Holding the present value of the future cash flows and the amount of each payment constant, the number of payments is directly related to the interest rate. In summary, everything else held constant, increases in the interest rate (1) increase the amount of each payment, or (2) increase the number of payments required, or (3) decrease the present value of the future cash flows.

In order to understand the effect of changes in interest rates from a consumer's perspective, we first examine borrowing transactions in which the present value of the future cash flows and the number of payments are fixed. Consider, for instance, a thirty-year mortgage or a four-year auto loan. In each case, the effect of an increase in interest rates is an increase in the amount of the home or auto payment. This is shown in Table 1.

Well-known lending interest rates include the prime rate, the discount rate, and consumer rates for automobiles or mortgages. The discount rate is the rate that the Federal Reserve bank charges to banks and other financial institutions. This rate influences the rates these financial institutions then charge to their customers. The prime rate is the rate banks and large commercial institutions charge to lend money to their best customers. While the prime rate is not usually available to consumers, some consumer loans (such as mortgage lines of credit) are priced at "prime 2 percent; that is, a consumer will pay 2 percent over the prime rate to borrow money. When the Federal Reserve raises the discount rate, typically banks raise the prime rate and consumers pay higher interest rates.

Individuals lend money by investing in debt instruments, such as Treasury bills and bonds. In this scenario, the investor receives periodic payments (annuity payments) and a lump sum when the debt instrument matures. This stream of cash flows is valued as follows:

market value = annuity payment × annuity factor _i,n + maturity value × present value factor _i,n [4]

where:

market value = value of the debt instrument

annuity payment = amount of the payment that is made each period; it is equal

Effect of Changing Interest Rates On the Amount of Monthly Payments

Borrow $100,000 for home purchase		Borrow $20,000 for auto purchase
Interest Rate	30-Year Mortgage Payment	Interest Rate	4-Year Auto Loan
6%	$599.55	7%	$478.93
8%	$733.76	10%	$507.25

to the interest rate stated on the debt instrument multiplied by the face value of the debt instrument

annuity factor = a number obtained from an ordinary annuity table that is determined by the interest rate (i) and the number of annuity payments (n).

maturity value = amount received by the investor when the instrument matures, also known as the face value of the debt instrument

present value factor = a number obtained from a present value table that is determined by the interest rate (i) and the number periods until maturity (n).

When an investor purchases a debt instrument, the following factors are "fixed": (1) the amount of each annuity payment, (2) the amount of the maturity value, and (3) the number of periods until maturity (this is also the number of annuity payments that will be received in the future). As interest rates increase, the market value of the investment will decrease; that is, the price of debt securities is inversely related to the market rate of interest. This is shown in Table 2.

The investors who keep the investment until the debt instrument matures will receive the market rate of interest on their investment from the date of purchase. The investor who sells their investment prior to maturity will receive the market rate of interest on the investment until it is sold. At that time, this investor

Effect of Changing Interest Rates on the Value of an Investment in Debt, Holding n constant

$20,000 Maturity Value Bonds paying 8% (stated) annual interest, Due in 25 years		$20,000 in Treasury Bills paying 0% Interest, Due in 90 Days
Market Interest Rate	Market Value of the Bonds	Market Interest Rate	Market Value of the Treasury Bills
6%	$25,113	6%	$19,711
8%	$20,000	8%	$19,619
10%	$16,369	10%	$19,529

will also receive either a gain or a loss due to changes in the market value of this investment. If market interest rates decrease, the investor will receive a gain. If market interest rates in crease, the investor will receive a loss on the value of the investment.