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Loan Terms and APR

Date: 02/18/97 at 02:28:00
From: Danny Schlee
Subject: Annual percentage rate

What is the formula for determining an APR?


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.

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
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Associated Topics:
High School Interest

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