Appendix A: The Calculation of Interest and APR

Appendix A: The Calculation of Interest and APR

The objective of this appendix is to explain how the annual percentage rate of charge (APR) is calculated. We begin with a review of the formulae by which simple and compound interest are calculated.

A.1 Simple and compound interest

Simple interest is where the total interest charge is based only on the initial amount borrowed, the length of the loan and the interest rate charged. If the following symbols are used:

S = The sum borrowed, known as the principal. t = The time over which the money is borrowed, known as the term. r = The interest rate expressed as a percentage (so a rate of 0.05 used for calculations is the same as 5 percent) I = The total amount of interest that must be paid.

Then the interest payable on a loan can be calculated using the following formula:

I=S*r*t

(1)

For example, for $1,000 borrowed for a term of two years at an interest rate of 10 percent per annum, the total interest payable at the end of two years would be:

$1,000 * 0.1 * 2 = $200.

Where compound interest is applied, the term is divided up into a number of intervals of equal length. At the end of each interval, interest is

221

222 Appendix A: The Calculation of Interest and APR

calculated and added to the outstanding debt. In the next interval interest is calculated on the total sum outstanding; that is, the original sum, plus the interest that has already accrued from previous intervals. If n is used to represent the number of intervals over which interest is to be applied, then the following formula can be used to calculate the total compound interest accruing over the term of the loan:

I = S * [(1 + r)n ? 1]

(2)

Using the same example as before, but this time with compound interest applied at the end of each year (so the loan comprises of two intervals, each of one year); then the total interest payable will be:

1,000 * [(1 + 0.1)2 ? 1) = $210.

Compound interest is often quoted over one interval of time, such as a year, but calculated and applied to the outstanding debt over shorter intervals; such as a day or month. In this case, the interest rate for each sub-interval is calculated as:

rn = (1 + r)t/n ? 1

(3)

Where:

r = The interest rate over time t. n = The number of intervals within t, for which interest is to be calculated. rn = The interest rate over time t/n.

For example, if a lender quotes an annual interest rate of 9.9 percent, but applies compound interest to accounts on a monthly basis, the monthly equivalent rate is:

(1 + 0.099)1/12 ? 1 = 0.7898 percent per month.

For the reverse situation, where the interest rate for one interval is known, but you want to know what the equivalent rate is over a term equal to some multiple of this interval, the appropriate formula is:

r = (1 + rn)t ? 1

(4)

Appendix A: The Calculation of Interest and APR 223

So if the monthly interest rate, rn is 1.0 percent per month then the equivalent annual rate of interest is:

(1.01)12 ? 1 = 12.68 percent per year.

Note that simply multiplying the monthly rate by 12 leads to an underestimate of the equivalent annual rate.

A.2 Calculating interest for fixed term amortizing loans and mortgages

For fixed term amortizing loans, such as a repayment mortgage, the outstanding debt decreases over time. Consequently, the interest charged each interval decreases, reflecting the reduced debt. With each subsequent payment the borrower is paying off more of the principal and less interest. What people normally want to know is; given an initial sum to be repaid over a fixed term and at a fixed rate of interest, what will the monthly payments be? In this situation the value of each payment can be calculated as:

P

=

S*

r * (1 + r)t (1 + r)t ?1

(5)

Where:

P = The payment each interval, with the first payment made at the end of the first interval. S = The principal (the loan amount). r = The interest rate charged over each interval (e.g. each day, month or year). t = The term of the loan; that is, number of intervals over which the agreement runs.

The total amount payable over the entire term of the loan is then simply P * t, and the total interest charge over the term can be calculated as:

I = (P * t) ? S.

(6)

So for a $50,000 mortgage repaid over 240 months (20 years) at a monthly interest rate of 0.4789 percent (5.9 percent per annum

224 Appendix A: The Calculation of Interest and APR

using formula 4) the regular monthly payment (using formula 5) will be:

P

=

$50,000

*

0.04789 * (1 + 0.004789)240 (1 + 0.04789)240 ?1

= $350.950518

and the total amount payable will be: $350.950518 * 240 = $84,228.10.

and the total interest payable over the lifetime of the agreement can be calculated using (6):

($350.950518 * 240) ? $50,000.00 = $34,228.10.

In practice, the standard payment would be rounded down to $350.95 and the first payment adjusted to include the outstanding $0.10; that is, $351.05. Also note that calculations should be made using at least six decimal places to ensure a suitable level of accuracy is maintained.

For balloon loans, such as interest only mortgages, interest is the same for each interval and calculated using the simple method defined in (1). For the $50,000 loan, the interest paid each month would be:

$50,000 * 0.004789 = $239.45.

Therefore, the total sum repaid over the term of the agreement would be:

240 * $239.45 + $50,000 = $107,468.

Of which $57,468 ($107,468 ? $50,000) is interest.

A.3 Calculating interest for credit cards

In theory, interest on a credit card is calculated in exactly the same way as any other type of credit agreement. What makes credit card charges so difficult for people to understand is that different interest rates are applied to different parts of the balance at different times. The result is that two credit cards that technically charge the same rate of interest can result in different amounts being charged for the same

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amount of debt. Some of the differences in the way that card providers calculate interest are as follows:

? Some lenders will start charging interest from the date that the card was used to make a purchase. Others will only begin to charge interest on the date that the transaction registers on the account. This can be one to two days after the transaction occurred, due to the time it takes for it to be processed by the card network.

? Some lenders calculate interest separately for each transaction. If the balance is not settled in full by the due date, then interest is charged on each transaction from the day it occurred to the statement date following the due date. Others choose to apply the average balance method. An average daily balance is calculated for the entire statement period and interest is calculated on this average figure. With the average balance method the date that payments are made to the account are important because the earlier a payment is made the lower the average balance will be, resulting in a lower interest charge.

? When someone makes a part payment, leaving part of their balance to revolve, interest can be charged on the entire balance or only on the amount outstanding after a payment has been made. If a customer receives a statement showing a balance of $1,200 and then pays $900 by the due date, a balance of $300 will revolve to the next statement. Interest can be charged on $1,200 or $300.

? When a balance has revolved from a previous statement, but is then settled in full by the next due date, three options are available. The first is to charge no interest because the balance has been paid in full. The second is to charge interest only on the sum that revolved. The third is to charge interest on the full statement balance. For example, a balance of $500 revolves from the previous statement period (this includes any interest from the previous period that would have been charged because the balance revolved). Another $100 is spent using the card so that the total balance appearing on the new statement is $600. The $600 is then paid in full by the due date. Some lenders would charge no interest because the balance was paid in full. Some would charge interest on the $500 that revolved, others on the full $600.

A secondary issue is payment hierarchies. Many revolving credit products allow different types of transaction to be charged at different interest

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