True Rate

We can now see from these figures that a rate of 1% per month produces the same end result as a rate of 12.683% per annum – not 12%.  We can therefore say that the equivalent interest rate expressed per annum is 12.683% pa.  This is called the true annual rate of interest and is exactly equivalent to saying the true monthly rate is 1%. 

If we continue compounding over three years, the end result would be as follows: -

I know I promised to avoid too much maths apart from a special section, but the following bit is quite fundamental and I can not resist telling you about it now.

The basic formula used to calculate the figures shown above is quite simple and is this: -

Future Value = Present Value x (1 + i) ⁿ  

where the superscripted n means “to the power of”.  In other words we multiply (1 + i) by itself n times.  i is the fractional interest rate and n is the number of time periods. 

So the future value after 36 months is: -

£1,000 x (1 + 1%) ³⁶  = 1,000 x 1.01 ³⁶  = £1,430.77 

Instead of calculating 1% for 36 months, the same result can also be calculated as 12.683% per year for 3 years: -

£1,000 x  (1 + 12.683%) ³  = £1,430,77

Either way the result is the same.  But using the annual rate requires fewer calculations, once you know what the true annual rate is.

In short compounding the true annual rate over three years produces the same answer as compounding the monthly rate over thirty-six months.