Among the many three-letter acronyms (TLAs) that always seem to pop up in personal finance, two of the more important are APR (annual percentage rate) and EAR (effective annual rate). Both tell you something about how much interest you'll pay for credit or a loan. They differ in that APR is based upon simple interest, whereas EAR takes into account the compounding of interest.
Calculating Annual Percentage Rate
APR is a standardized interest rate you calculate as the interest you pay over the life of a loan on a principal amount divided by the principal amount, and then adjusting for a one-year period. The principal amount is the amount of money you borrow, including fees added to the loan amount (but not fees you pay separately). The APR formula is:
APR = ((((Fees + Interest)/P) / n) x T) x 100,
where P is the initial principal balance, n is the number of times interest is compounded per time period and T is the number of time periods.
The formula can be simplified by labeling (((Fees + Interest)/Principal) / n) as the daily periodic interest rate, where the period is one day (that is, n = 1). Therefore:
APR = (Daily Periodic Interest Rate x 365) x 100.
APR represents simple interest because it ignores the effects of compounding.
For example, Credit Card A has a daily periodic interest rate of 0.06273%. When you multiply by 365 and 100, you get an APR of 22.9%.
The Role of Compounding
Compounding occurs when the interest you incur is added to the principal balance of a loan. Compound interest is the result of paying interest on interest, which increases the total interest you'll have to pay. Interest can be compounded at various intervals, including annually, semi-annually, quarterly, monthly, daily or continuously.
The general formula for calculating the amount of compound interest on a loan is:
A = (P x (1 +R/n) nT),
where A is the interest amount.
Credit card interest is usually compounded daily. The appropriate formula for compound interest on credit cards is:
A = (P x (1 +R)365).
For example, a loan balance of $1,000 and a daily interest rate of 0.06273% will cost the following interest amount:
A = $1,000 x (1.0006273) 365 = $1,257.21.
Effective Annual Rate
The EAR is more realistic than APR when you want to know how much interest you'll pay after adjusting for the compounding of interest. The formula is:
EAR = (1 + periodic rate)number of compounding periods) - 1).
All other things being equal, EAR gets bigger as you increase the number of compounding periods per year. You obtain the maximum EAR by using continuous compounding.
Assuming a daily compounding period, the formula simplifies to:
EAR = (1 + daily periodic interest rate)365) - 1).
For example, assume a daily periodic interest rate of 0.06273%:
EAR = (1.06273%)365 - 1 = 25.721%.
Converting EAR to APR
If you already know the EAR, you can calculate the APR using this formula:
APR = n x ((EAR+1)1/n-1)
where n is the number of compounding periods. For daily compounding, it simplifies to:
APR = 365 x (EAR + 1)1/365 -1
For example, if EAR = 25.721%. then
APR = 365 x (1.25721) 1/365 -1 =365 x 0.06273% =22.9%.
You can see that compounding adds (25.721% - 22.9%), or 2.821%, to the cost of the loan.
Saving vs. Borrowing
APR is commonly used to standardize borrowing or savings rates so that they can be compared on an equal footing. You'll always see loans and credit cards disclosing their APRs in their advertising and their loan agreements. However, lenders can manipulate APRs somewhat by choosing which fees to include when they calculate their rates.
When you deposit money into a savings account, money market account or certificate of deposit, you'll frequently see EAR quoted. The reason is simple enough – EAR is larger than APR and thus more enticing to savers. EAR is also more correct because it recognizes the action of compounding to grow your money faster.