Loan Terms and APRDate: 02/18/97 at 02:28:00 From: Danny Schlee Subject: Annual percentage rate What is the formula for determining an APR? Thanks! Date: 02/18/97 at 12:25:03 From: Doctor Mitteldorf Subject: Re: Annual percentage rate Dear Danny, There is no formula for determining APR. The definition of APR is: that loan rate which makes the present value of all loan payments equal to the net proceeds to the borrower on the loan date. Explanation: It is conventional to quote interest rates along with a "compounding period." Mathematically, the natural compounding period is 0, i.e., continuous compounding. Interest rate quoted for continuous compounding is called a "true rate." You can quote the same situation by telling how much growth that true rate produces over a 1-year period. This is called a "yield." The formula for connecting true rate to yield is: yield = exp(true_rate)-1 Where exp means "e raised to the power of", and e = 2.71828..., which is the base of natural logarithms. "Loan rate" is an alternative to both "true rate" and "yield." Since most loans are paid on a monthly schedule, a "loan rate" is usually quoted as a true rate compounded not for 1 year but just for 1 month. The formula is: loan_rate = 12 * (exp(true_rate/12)-1) Now about present values: Present value of a payment of d dollars on a date t years from now is: present value = d * exp(-true_rate*t) Normally, the present value of all the loan payments equals the amount of the loan. But there are often costs and fees and points paid up front at the time a loan is written. Then the bank computes the payments from the gross amount of the loan, but the borrower only gets the net amount. APR is the rate that the bank would have had to use to get that same loan payment schedule if it had been computing based on the (lower) net proceeds of the loan, rather than the (higher) gross amount. There is a formula for computing present value from the interest rate, but it is not invertible - there is no inverse formula for computing interest rate from the present value; hence there is no formula for APR, but a smart trial-and-error scheme can calculate APR efficiently by guessing, computing present value, learning from how far your computed present value is from the target, and guessing again. -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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