Fundamentals of business mathematics: interest is normally expressed in the form of an annual percentage rate, but not every loan lasts exactly one year, of course, so a mathematical technique is needed to calculate an actual charge.
| Author | Scarlett, Bob |
| Position | Paper CO3 |
Interest is a fee paid by a borrower of assets to the owner of those assets as a form of compensation for using them. It may be calculated on a "simple" basis, whereby it's charged on the original principal only, or on a "compound" basis, whereby it's charged on the principal plus the interest that has accumulated on each stated payment date.
Say we are lending [pounds sterling]100 for a year and charging interest at 1 per cent a month. At the end of the first month, [pounds sterling]1 in interest is charged ([pounds sterling]100 x 1%) and that amount is added to the principal. The principal becomes [pounds sterling]101. At the end of the second month, [pounds sterling]1.01 in interest is charged ([pounds sterling]101 x 1%) and the principal becomes [pounds sterling]102.01. This sequence goes on until the end of month 12, with interest being charged at 1 per cent a month on a compound basis. The amount repayable at the end of the term will be [pounds sterling]112.68. This figure may be arrived at through the following calculation: [pounds sterling]100 x [1.01.sup.12] = [pounds sterling]112.68, using the formula S = X[(1 + r).sup.n], where S is the future value of a sum, X, invested for n periods and compounded at r per cent.
To put it another way, the interest rate of 1 per cent a month corresponds to an annual percentage rate (APR) of 12.68 per cent (that is, [l x [1.01.sup.12]] - 1).
It's also possible to work in the reverse direction, with an annual rate being expressed as an equivalent monthly compound rate. For example, say we are lending [pounds sterling]100 for a year at an APR of 15 per cent chargeable on a monthly basis. A 15 per cent annual rate corresponds to a monthly rate of 1.1715 per cent. This figure can be worked out using the following calculation: ([1.15.sup.1/12]) - 1, or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
If we lend [pounds sterling]100 for a year with interest chargeable at 1.1715 per cent a month, the amount repayable at the end of month 12 will be [pounds sterling]100 x [1.0117153.sup.12] = [pounds sterling]115.
When banks lend (or take deposits), it's usual for the associated interest rate to be expressed as an APR but with interest charged on a daily basis. The interest rate may be a fixed amount for the whole term or it may be a variable rate linked to some benchmark. Large loans may be charged at a variable rate linked to the London inter-bank offered rate (Libor). This is an independently...
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