Nominal is not true

Whilst it is tempting to say 1% per month is 12% per annum, clearly it is not true.  However lenders do use the term Nominal rate.  12% pa nominal can mean 1% per month provided it is stated as such.  It could equally mean 3% compounded quarterly.  The effect of more frequent compounding is higher overall interest, so we need to know more than just a rate.  Whilst financial Institutions often quote a nominal interest rate, without knowing the compounding frequency, it means very little. 

So a correctly defined interest rate must also indicate the compounding frequency or the number of rests per annum.  12% per annum nominal, compounding monthly (ie twelve rests per annum) means 1% per month true and is mathematically equivalent to 12.683% per annum true.  Since the true rate is approximate (there are actually more decimal figures) it is often more convenient to just say 12% pa nominal payable monthly. 

Lenders are seldom precise about this.  In short, be sure you know just what it means when an interest rate is quoted.    Why do we want to know the true annual rate?  Why is it so important? The practical answer is for comparison purposes.  We need a consistent standard to compare all loans, regardless of their compounding frequency.  Quoting rates on a true, annual basis is the most familiar standard and common usage has made it prevalent.   The later section on Technical Bits provides more details about the mathematics.

The importance of the compounding frequency  
Interest can compound at many other frequencies as well as monthly. The Nominal rate used in all of the earlier examples was 12% per annum.  Figure 4 illustrates the true annual rates for various other compounding frequencies, but where the nominal rate is the same in every case @ 12% per annum.

This table highlights the fact that unless one knows the compounding frequency, the nominal rate is meaningless.  Moreover the difference is certainly not trivial. 

Figure 4  
12% per annum nominal interest with different compounding frequencies.

Compounding Frequency

Rest Periods per annum

True annual Rate

1,000 over 30 years

Annual

1

12.000000

29,960

Half-yearly

2

12.360000

32,988

Quarterly

4

12.550881

34,711

Monthly

12

12.682503

35,950

Daily

365

12.747462

36,577

Hourly

8,760

12.749592

36,597

Continuously

Infinite

12.749685

36,598

The true rate rises more slowly at the higher compounding frequencies and fortunately trends towards a maximum.  The practical effect of growing 1,000 over 30 years is shown for perspective.

Over 100 years the difference between daily and annual compounding is even more staggering:

        1,000 @ 12% pa compounding annually produces 83,522,266

        1,000 @ 12% pa compounding daily produces 162,434,128

That is a difference of almost 79 million, using the same nominal rate, but with different compounding frequencies almost double the annual figure!  You can now see the importance of identifying the true rate.

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