Canadian mortgages: (Note: a similar approach must be followed in other cases when payment frequency is different from the number of compounding periods.) Canadian disclosure regulations are a bit peculiar. See page 143-144 in the textbook (132-133 in the 1st edition) for a discussion on how Canadian mortgage rates are quoted and how the effective annual rate, the monthly rate and the monthly payments are calculated. (In summary, Canadian banks are required to quote the semiannually compounded APR of the mortgages they offer to borrowers.) Example: if the quoted mortgage rate in Canada is 9%, then we know that the effective annual interest rate (i.e. the interest rate the bank will actually earn over a one year period) is EAR = (1 + APR/m)m = (1 + 9%/2)2 – 1 = 9.2025%. Trouble is, the bank needs to know how much to charge per month since mortgage payments are made on a monthly basis. So what is the effective monthly rate? It is the monthly rate that – when compounded over twelve months – will give us the same future value by the end of the year as the effective annual rate (EAR): $100 (1 + reff. monthly )12 = $100 (1 + EAR) (1 + reff. monthly )12 = (1 + EAR) 1 + reff. monthly = (1 + EAR)1/12 reff. monthly = (1 + EAR)1/12 – 1. So in our case:
reff. monthly = (1 + 9.2025%)1/12 – 1 = 0.73631%
So in cases when the number of payments per year is not the same as the number of compounding periods, we need to determine the effective monthly rate in two steps: first find the EAR and then use EMR = (1 + EAR)1/12 to find the monthly rate.
Loans where the number of payments per year equals the number of compounding periods (most car loans etc.): In this case the effective monthly rate is simply the quoted rate (the APR) divided by 12. Example: if a car loan requires monthly payments and it is advertised with an APR of 9% (compounded monthly), then we know that the effective monthly rate is simply reff. monthly = APR/12 = 9%/12 = 0.75%. This is the monthly interest rate we need to use in order to calculate the monthly payment. (See an example of a monthly payment calculation below.) If we are still interested in the effective annual interest rate then we can calculate it as before (but notice that we use monthly instead of semiannual compounding): EAR = (1 + APR/m)m = (1 + 9%/12)12 – 1 = 9.3807%. Do we get the same result by compounding the effective monthly rate over twelve months? Of course: (1 + 0.75%)12 – 1 = 9.3807%. Notice that while the quoted APRs on the mortgage and the car loan are the same, both the annual (EAR) and the effective monthly rates are higher on the car loan than on the mortgage as a result of the more frequent compounding.
Note: I have always found that the APR is easiest to think of in the following way. A bank decides that it will pay its depositors 0.75% per month in interest. Now those of us who know about the time value of money and compounding will realize that investing $1,000 at a monthly rate of 0.75% interest and compounding every month will increase our wealth to $1,000 (1 + 0.75%)12 = $1,093.807 by the end of the year. In other words the interest we receive is $93.807, so we earn an effective annual rate of 9.3807%, which is greater than simply twelve times the monthly rate: 9.3807% > 0.75% 12 = 9%.
Since most people have not taken a finance course at university, the average banking customer will be confused by the above compounding process, so when the bank posts the quoted rate, for simplicity they multiply the monthly rate by twelve to get what is called the Annual Percentage Rate (APR for short and also known as the quoted annual rate). While this is clearly only approximately correct and always underestimates the Effective Annual Rate, there is an…