Calculating Interest Rates and APR

In its simplest form, the interest rate on a loan is calculated as the dollar amount of interest charged divided by the amount of money borrowed. For example, if you borrow $1,000 that must be repaid at the end of one year along with $60 in interest, you're being charged an interest rate of $60/$1,000, or 6 percent. So, the general formula for calculating the interest rate on a loan is:

i = I/P
in which
i is the APR
I is the dollar amount of interest paid, and
P is the amount borrowed

In real life most loans are significantly more complicated than this example. They often involve a term of other than one year, and many loans are repaid with a series of payments rather than a single one. Some loans require periodic interest payments to be made by the borrower until a predetermined date, when the entire amount borrowed becomes due. Because the specifics of loans can vary dramatically, it's worthwhile to familiarize yourself with some of the more important conventions and details that will affect the interest rate.

First, interest rates are quoted on an annual (or yearly) basis. Certain financial variables, including inflation, investment returns, and interest rates, are virtually always computed and quoted in this manner. Therefore, an adjustment is required to annualize the interest rate for loans of other than one year. Suppose, for instance, you borrow $1,000 and sign a loan agreement that requires you to repay $1,060 in three months. You're paying the same dollar amount of interest required by the one-year loan agreement discussed above, but in this instance you have use of the borrowed money for only three months rather than a full year. The interest rate on this loan must be annualized to account for the shortened term. To accomplish this, you must divide the number of months the loan is scheduled to be outstanding (in this case, three months) into the number of months in a year (which is, of course, always twelve). The result (12/3 = 4) is multiplied by the interest rate calculated from the previous formula. Thus, the annualized rate for the three-month loan is ($60/$1,000) x (12 months/3 months), or 6 percent times four, or 24 percent. The same amount borrowed, the same dollar amount of interest paid, for one-fourth the time, equates to four times the interest rate. Likewise, if $60 in interest is charged for a $1,000 loan that is to be repaid in two years, the annualized interest rate would be ($60/$1,000) x (12 months/24 months), or 3 percent. You must always make certain that you and the lender are using the same assumptions when discussing the interest rate of a proposed loan.

A loan's repayment schedule may also have a major impact on the interest rate you pay. For instance, let's suppose you borrow $2,400 for one year. At the end of the year, you must repay the amount borrowed plus $240 in interest. This arrangement permits you to retain control of the entire amount borrowed ($2,400) for the full year. A requirement to pay interest or a portion of the principal prior to a loan's maturity will cause an increase in a loan's effective rate of interest. Assume instead, for a moment, that an alternative loan agreement to borrow the same $2,400 requires you to make twelve monthly payments of $220 each. You'd still repay a total of $2,640, the same as with the single-payment obligation, but this schedule requires a payment of interest and a gradual repayment of the principal during the term of the loan. The monthly principal repayments cause you to control less and less of the borrowed money over the course of the loan. A loan agreement in which the total amount of interest is added to the amount borrowed, and that sum divided by the number of payments to determine the size of each payment, is known as the add-on interest method.

Given the choice of paying $2,640 at the end of the year or $220 per month for one year, which is better? It's actually to your advantage to choose the single payment of $2,640 because you pay the same dollar amount of interest ($240) but have, on average, control of a larger amount of the money you borrowed. Add-on interest, therefore, increases the effective cost of a loan.

Interest collected on a loan's front-end increases the effective interest rate, as well. Certain installment loans, known as discount loans, are structured to allow the lender to collect the total amount of the interest at the time that the loan is made. A discount loan actually charges interest on money that will not be made available to the borrower. Suppose you borrow $1,500 for two years at a quoted interest rate of 8 percent. The lender deducts $240 in interest charges (8 percent x $1,500 x 2 years) and gives you the balance, which amounts to $1,260. The loan obligates you to pay twenty-four monthly installments of $62.50 each. Therefore, quoting 8 percent interest for this loan is misleading because monthly payments require you to repay interest charges and a portion of the principal each month. Repaying principal throughout the term of the loan means that you don't have the use of all you've borrowed for the full term. Additionally, interest is being charged on $1,500 even though you actually only received $1,260 from the lender. So, discounted interest increases the effective interest rate by requiring you to pay interest on more money than you receive.

Interest can be calculated on a simple or a compounded basis. Let's say, for example, that you're interested in obtaining a $4,000 loan for three years. One bank offers you the loan at 12 percent annually with principal plus accumulated interest of $1,440 (12 percent x $4,000 x 3 years) to be paid at the end of three years. Interest that's calculated only on the original principal of a loan is known as simple interest. A different bank offers a loan at the same stated rate but requires you to pay $40 interest each of the thirty-six months. The last interest payment is to be accompanied by repayment of the original $4,000 principal. Paying interest monthly, or, alternatively, having the lender calculate interest in each subsequent month on both principal and accumulated interest is known as compounding. Compounding not only increases the effective rate of interest earned by savers and investors, it also increases the cost of borrowed money.

Creditors sometimes use different loan balances to calculate interest charges. It's already been shown that they sometimes calculate interest on the full amount borrowed even though installment payments will gradually reduce the balance that's outstanding over the life of the loan. Credit card companies also utilize several methods to calculate the interest charged to cardholders. Some issuers credit any payments that you've made before calculating the finance charges, and it's to your advantage that this be done because the credit reduces the loan balance on which interest is calculated. Other credit card issuers calculate interest for a given period without regard for payments that you may have made during the period. This, of course, works to your disadvantage because a higher loan balance is used to calculate your interest charges. Still other credit card companies charge interest that's based on the average daily balance of your borrowing. It's, therefore, easy to see that the interest you're charged on a credit card balance depends to a large degree on the calculation method used by the issuer.

Annual percentage rate

Because several different calculations can be used to arrive at an interest rate quotation, it's not at all uncommon for consumers to become confused when comparing loans or credit card offers. There is, however, an accepted method for computing the cost of borrowing money. A standardized calculation allows you to compare the interest rates being offered by different lenders. Without a standardized method, it's difficult to know for certain whether the lowest quoted rate is, in fact, the lowest actual rate. This standard measure of the cost of borrowing is known as the annual percentage rate (APR), and its formula for calculation is:

i = 2 x n x I / P(N + 1) 
in which
i is the APR
n is the number of payment periods in one year
I is the total financing charges (mostly interest)
P is the principal (the amount borrowed), and
N is the number of scheduled payments

The Truth-in-Lending Act requires that all creditors, including banks and other financial institutions, car dealers, retailers and credit card companies, provide a borrower with a loan's total finance charges and annual percentage rate. These two pieces of information will give you much of the knowledge you'll need to compare the offerings of various lenders. Remember, interest rates are only comparable when they've been calculated in the same manner.

Let's look at one more loan. Suppose you're offered a $5,000 loan to be repaid in six monthly installments of $900 each. Total repayments amount to $5,400 (6 x $900), which means that you'll be charged $400 in interest ($5,400 in total amount payments less $5,000 that you originally borrowed). The lender tells you that the loan's interest rate is a very attractive 8 percent ($400/$5,000). Is this an accurate quotation, or is the lender trying to pull a fast one?

Well, the truth is that you should run from this lender with as much speed as you can physically muster, and here's why. First, the lender is requiring you to repay the loan in installments but quoting an interest rate based on your having the full amount of the loan available for the entire six months. Also, the lender has failed to annualize the rate to adjust for the scheduled payoff in six months instead of one year. With that in mind, let's calculate the annual percentage rate using the formula above:

i = 2 x 12 x $400 / $5,000(6 + 1) = $9,600 / $35,000 = 27.4 percent

Quite a bit different from the lender's quoted rate, wouldn't you say? So, don't take their word for it. Always run the numbers yourself, or have someone you trust run them for you. Having the right information can save you a lot of money

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