APR, What is it?

A textbook definition of Annual Percentage Rate, APR, is as follows. Charges imposed on a borrower to obtain a mortgage, expressed on an annualized basis as an interest rate. The APR includes the annual interest rate (AIR), loan fees and points. In the USA the APR is meant to disclose the total cost of borrowing. In Canada the APR is defined in section 6 of the Federal Interest Act in an ambiguous manner and is generally understood to mean effective interest rate (EIR). In Canada, Ontario specifically, the total cost of borrowing, TCOB is calculated as the new EIR because of loan fees and points.

Here is an American APR example (taken from a printed booklet) showing why you should be very careful acting upon “rules of thumb”. Which is better? Loan#1, a $100,000, 30 year
loan at 10% plus 3 points or Loan#2, a $100,000, 30 year loan at 10.25% plus 1 point.

The example in the booklet is wrong because this simple
rule of thumb was used. Loan #1 is better for the borrower
using the APR as the yardstick. Loan #1 has an APR of
10.3657% and Loan #2 an APR of 10.3716% which is
greater resulting in more interest being charged.

Before the APR can be appreciated it sometimes helps
to understand other names for the various interest rates
used in the industry.  See below for an explanation.

An annual interest rate can be referred to or written as,

12 % annual interest rate

12 % annual interest rate, compounded monthly

The latter phrase is the one you will see printed in the newspapers, and the front of mortgage payment booklets, and is usually called the nominal interest rate. If the annual interest rate is quoted along with the compounding method then it is customarily called the nominal interest rate.

In order to compare interest rates the compounding method must be stated along with the rate. For every annual interest rate (AIR) there is a corresponding effective interest rate that depends upon the compounding method. For example, an American mortgage with an annual interest rate of 12% with monthly compounding has an effective interest rate of 12.6825%. A Canadian mortgage with an annual interest rate of
12% with semi-annual compounding has an effective interest rate of 12.36%. The Canadian mortgage is a better deal for the borrower because the effective interest rate (EIR) is lower. Bottom line, if a Lender offers you a 12% rate with either monthly or semi-annual compounding, you should take the semi-annual compounding option as it will cost less in interest
because of the lower effective interest rate. The borrower always pays more in interest with a more frequent compounding method. Using mathematical vernacular, Effective Interest Rates, EIR, are always considered to be on an annual basis. The effective interest rate is always greater than the nominal rate.
When the compounding method is annually the effective interest rate and the nominal rate are identical.

The advantage of quoting effective interest rates is one does not need to be concerned about the type of compounding. An effective interest rate, EIR, can stand on its own merit. If someone offered you a one year loan of $1000 at an effective interest rate of 12.36% you would owe the lender $1,123.60 at the end of the year. This same loan could have been given to you at 11.7106% with monthly compounding or 12% with semi-annual compounding. The two loans are identical because they both have an effective interest rate of 12.36%. In other words an annual interest rate of 11.7106% with monthly compounding is exactly equivalent to an annual interest rate of 12% with semi-annual compounding. Both loans require you to pay back $1,123.60 at the end of the year. In this screenshot you can see that both loans due to deemed reinvestment (negative amortization schedule) lead to the same thing for the lender, because the EIR is the same!

In summary the APR does not mean effective interest rate, EIR. The APR has absolutely nothing to do with the effective interest rate, EIR, by definition. In financial circles the EIR is meant to mean one thing and one thing only, … the actual interest rate at the end of the year because of the compounding frequency, nothing more nothing less.

In loan #1, the APR is nothing more than the new annual interest rate of 10.3657% because of the $3000 (3 points is 3% of $100,000 = $3000). Changing the Principal to $97,000 and recalculating the annual interest rate gave a new rate of 10.3657%. The borrower is making monthly payments based on $100,000 even though the net amount left is $97,000 thus the rate in reality is 10.3657%. Because of the up front $3000
charge the annual interest rate became 10.3657%.

In loan #2, the APR is nothing more than the new annual interest rate of 10.3716% because of the $1000 (1 points is 1% of $100,000 = $1000). Loan#2 is the more expensive loan (because of the APR method) because the APR is 10.3716% which is greater than 10.3657%. So much for the books
generalized rule of thumb that incorrectly shows that Loan#2 is better. In fact Loan #1 is better by $1,257 in less interest after 6 years (comparing accumulated interests).

comparing accumulated interests

When the APR is meaningless!
If the calculation of the mortgage’s APR is based upon the 360 day year, aka, the bankers year then the APR is of value. A $100,000 loan at 6% using monthly compounding has an interest cost of $500 the very first month. The interest factor is .005 per month and is CONSTANT each month as the year is assumed to be divided into 12 equal 30 day months or 0.5% per month.

If the loan or mortgage is an exact day monthly payment based upon a 365 day year or an exact day monthly payment based upon a 360 day year, ..the APR is not accurate. The algebraic formulas cannot take into account the changing monthly interest factor. Consider a $300,000 mortgage at 6% for 30 years. Taking into account total fees of $3,000, the lender that uses a 365 day year monthly schedule will collect $1,144 more in interest over the 30 years (comparing spreadsheet interest), yet both lenders would quote you an APR of 6.094%. This screenshot demonstrates the point.