Interest Rate Calculations
Date: Wed, 03/07/2007 - 09:37
Interest represents the price borrowers pay to lenders for credit over specified periods of time. The amount of interest paid depends on a number of factors: the dollar amount lent or borrowed, the length of time involved in the transaction, the stated (or nominal) annual rate of interest, the repayment schedule, and the method used to calculate interest.
If, for example, an individual who borrows $1,000 for one year at 5 percent and repays the loan in one payment at the end of a year may pay $50 in interest, or some other amount, again depending on the calculation method used.
Simple Interest
The various methods used to calculate interest are basically variations of the simple interest calculation method. The basic concept underlying simple interest is that interest is paid only on the original amount borrowed for the length of time the borrower has use of the credit. The amount borrowed is referred to as the principal. In the simple interest calculation, interest is computed only on that portion of the original principal still owed.
Example 1
Suppose $1,000 is borrowed at 5 percent and repaid in one payment at the end of one year. Using the simple interest calculation, the interest amount would be 5 percent of $1,000 for one year, or $50, since the borrower had use of $1,000 for the entire year.
When more than one payment is made on a simple interest loan, the method of computing interest is referred to as "interest on the declining balance." Since the borrower only pays interest on that amount of original principal that has not yet been repaid, interest paid will be smaller the more frequent the payments. At the same time, of course, the amount of credit at the borrower's disposal is also smaller.
Example 2
Using simple interest on the declining balance to compute interest charges, a 5 percent, $1,000 loan repaid in two payments  one at the end of the first half-year and another at the end of the second half-year would accumulate total interest charges of $ 37.50. The first payment would be $500 plus $25 (5 percent of $1,000 for one-half year), or $525; the second payment would be $500 plus $12.50 (5 percent of $500 for one-half year), or $512.50. The total amount paid would be $525 plus $512.50, or $1,037.50. Interest equals the difference between the amount repaid and the amount borrowed, or $37.50. If four quarterly payments of $250 plus interest were made, the interest amount would be $31.25; if 12 monthly payments of $83.33 plus interest were made, the interest amount would be $27.08.
Example 3
When interest on the declining balance method is applied to a 5 percent, $1,000 loan that is to be repaid in two equal payments, payments of $518.83 would be made at the end of the first half-year and at the end of the second half-year. Interest due at the end of the first half-year remains $25; therefore, with the first payment the balance is reduced by $493.83 ($518.83 less $25), leaving the borrower $506.17 to use during the second half-year. The interest for the second half-year is 5 percent of $506.17 for one-half year, or $12.66. The final $518.83 payment, then, covers interest of $12.66 plus the outstanding balance of $506.17. Total interest paid is $25 plus $12.66, or $37.66, slightly more than in Example 2.
This equal payment variation is commonly used with mortgage payment schedules. Each payment over the duration of the loan is split into two parts. Part one is the interest due at the time the payment is made, and part two  the remainder  is applied to the balance or amount still owed. In addition to mortgage lenders, credit unions typically use the simple interest/declining balance calculation method for computing interest on loans. A number of banks also offer personal loans using this method.
Other Calculation Methods Add-on interest, bank discount, and
Other Calculation Methods
Add-on interest, bank discount, and compound interest calculation methods differ from the simple interest method as to when, how, and on what balance interest is paid. The "effective annual rate" for these methods is that annual rate of interest which, when used in the simple interest rate formula, equals the amount of interest payable in these other calculation methods. For the declining balance method, the effective annual rate of interest is the stated or nominal annual rate of interest. For the methods described below, the effective annual rate of interest differs from the nominal rate.
Add-on interest
When the add-on interest method is used, interest is calculated on the full amount of the original principal. The interest amount is immediately added to the original principal, and payments are determined by dividing principal plus interest by the number of payments to be made. When only one payment is involved, this method produces the same effective interest rate as the simple interest method. When two or more payments are to be made, however, use of the add-on interest method results in an effective rate of interest that is greater than the nominal rate. The interest amount is calculated by applying the nominal rate to the total amount borrowed, but the borrower does not have use of the total amount for the entire time period if two or more payments are made.
Example 4
Consider, again, the two-payment loan in Example 3. Using the add-on interest method, interest of $50 (5 percent of $1,000 for one year) is added to the $1,000 borrowed, giving $1,050 to be repaid; half (or $525) at the end of the first half-year and the other half at the end of the second half-year.
Recall that in Example 3, where the declining balance method was used, an effective rate of 5 percent meant two equal payments of $518.83 were to be made. Now with the add-on interest method each payment is $525. The effective rate of this 5 percent add-on rate loan, then, is greater than 5 percent. In fact, the corresponding effective rate is 6.631 percent. This rate takes into account the fact that the borrower does not have use of $1,000 for the entire year, but rather use of $1,000 for the first half-year and use of about $500 for the second half-year.
To see that a one-year, two equal-payment, 5 percent add-on rate loan is equivalent to a one-year, two equal-payment, 6.631 percent declining balance loan, consider the following. When the first $525 payment is made, $33.15 in interest is due (6.631 percent of $1,000 for one-half year). Deducting the $33.15 from $525 leaves $491.85 to be applied to the outstanding balance of $1,000, leaving the borrower with $508.15 to use during the second half-year. The second $525 payment covers $16.85 in interest (6.631 percent of $508.15 for one-half year) and the $508.15 balance due.
In this particular example, using the add-on interest method means that no matter how many payments are to be made, the interest will always be $50. As the number of payments increases, the borrower has use of less and less credit over the year. For example, if four quarterly payments of $262.50 are made, the borrower has the use of $1,000 during the first quarter, around $750 during the second quarter, around $500 during the third quarter, and around $250 during the fourth and final quarter. Therefore, as the number of payments increases, the effective rate of interest also increases. For instance, in the current example, if four quarterly payments are made, the effective rate of interest would be 7.922 percent; if 12 monthly payments are made, the effective interest rate would be 9.105 percent. The add-on interest method is sometimes used by finance companies and some banks in determining interest on consumer loans.
The reason I've posted this is because I've noticed that interes
The reason I've posted this is because I've noticed that interest rates have been calculated incorrectly several times on this site . . . . It's not always as simple as 100 x 15% = Total Owed.
We all need to remember that normal people didn't write the pdl laws . . . . The legislature for each state did, and they don't do anything simply.
Compound interest When the compound interest calculation is us
Compound interest
When the compound interest calculation is used, interest is calculated on the original principal plus all interest accrued to that point in time. Since interest is paid on interest as well as on the amount borrowed, the effective interest rate is greater than the nominal interest rate. The compound interest rate method is often used by banks and savings institutions in determining interest they pay on savings deposits "loaned" to the institutions by the depositors.
Example 7
Suppose $1,000 is deposited in a bank that pays a 5 percent nominal annual rate of interest, compounded semiannually (twice a year). At the end of the first half-year, $25 in interest (5 percent of $1,000 for one-half year) is payable. At the end of the year, the interest amount is calculated on the $1,000 plus the $25 in interest already paid, so that the second interest payment is $25.63 (5 percent of $1,025 for one-half year). The interest amount payable for the year, then, is $25 plus $25.63, or $50.63. The effective rate of interest is 5.063 percent, which is greater than the nominal 5 percent rate.
The more often interest is compounded within a particular time period, the greater will be the effective rate of interest. In a year, a 5 percent nominal annual rate of interest compounded four times (quarterly) results in an effective annual rate of 5.0945 percent; compounded 12 times (monthly), 5.1162 percent; and compounded 365 times (daily), 5.1267 percent. When the interval of time between compoundings approaches zero (even shorter than a second), then the method is known as continuous compounding. Five percent continuously compounded for one year will result in an effective annual rate of 5.1271 percent.
Awesome work, Goudah! Thanks for all the hard work and starting
Awesome work, Goudah! Thanks for all the hard work and starting this topic. Pdl laws are very complex and the calculations need proper understanding. I am sure this topic is going to help many. I have included this in my bookmarks and will be referring it to others.
