Interest Rates

Maths Help suggests:

The APR is intended to give a truer indication of the interest rate being applied (typically to a loan).

Suppose you borrowed £100 for one year, and at the end of the year you repaid £120. The annual interest would therefore be 20% "flat rate".

However, if you repaid £10 per month over the year, you might think that by repaying £120 in total the interest is still 20%. But this is not the case, because you have not had the benefit of the £100 for the whole year.

By paying off the loan in installments, the amount you have borrowed is reducing each month. The £20 interest you are paying is in fact more than 20% of the money owed throughout the year. The APR will take this into account, and it will be a figure higher than 20% when worked out.

A bit of background revision before we go any further...
If an amount A is borrowed and the annual interest rate is P%, then the amount owed after one month is

To calculate the APR in the example above, where £100 is repaid by 12 monthly installments of £10, you need to work out the amount outstanding at the end of each month. By the end of the 12th month, this amount must be zero.

Suppose the APR = P%. Then at the end of month 1 you will have an outstanding balance of

At the end of month 2 you will have an outstanding balance of

A pattern seems to be building up. To see this more clearly, use the following substitution:

We then have that after 1 month, the amount owed is
100x - 10
After 2 months, the amount owed is
(100x - 10)x - 10
= 100x² - 10x - 10

Continue this pattern recursively month by month. You will end up with a very complicated equation with the value at the end of month 12 being equal to zero:
100x¹² - 10x¹¹ - 10x¹⁰ - 10x⁹ - 10x⁸ - 10x⁷ - 10x⁶ - 10x⁵ - 10x⁴ - 10x³ - 10x² - 10x¹ - 10 = 0

You will need some computer package (or a numercial/graphical technique) to solve this.
The actual solution is x = 1.02923 (5 d.p.)

Remember that x was introduced as the substitution for the 12th root of the annual multiplier.
Therefore (1 + P/100) = x¹² = 1.02923¹² = 1.413 (3 d.p.)

Thus the APR = 41.3%

• Note that the APR is roughly twice the annual "flat" rate.
• Note also that the above calculation is greatly simplified. We have assumed that a year consists of 12 equal months. In practice, the APR would be calculated by taking the exact number of days between payments into account. The equation would then be even more complicated. A spreadsheet could be used to set up the general model, and the value of P varied until the required result is achieved.