Over my career, people have asked me for the formula for
monthly payments on a loan. So, I'm giving it to you along with its derivation.
It is very handy for car loans, mortgages, & other loans.
Let
A=amount of the loan, r%= annual percentage rate (APR)(decimal),
n= number of
periodic payments over the life of the loan,
j%= percentage rate per period
(r divided by # of periods per year),
P= amount of each periodic payment
(usually monthly).
Note: The derivation uses lots of algebra &
factoring along with the properties mentioned above. I've included it in this
discussion for those of you who are thinking about becoming math majors or who
just can't stay away from math problems.
The
Derivation:
After the 1st period, you will owe:
A+jA=A(1+j)
After the 1st payment, you will owe: A(1+j)-P (subtract
payment)
After the 2nd period, you will owe:
[A(1+j)-P] +
j[A(1+j)-P]=[A(1+j)-P](1+j)=A(1+j)2-P(1+j)
After the 2nd
payment, you will owe: A(1+j)2-P(1+j)-P (subtract
payment)
After the 3rd period, you will owe:
A(1+j)2-P(1+j)-P+j[A(1+j)2-P(1+j)-P]=[A(1+j)2-P(1+j)-P](1+j)=
A(1+j)3-P(1+j)2-P(1+j)
After
the 3rd payment, you will owe:
A(1+j)3-P(1+j)2-P(1+j)-P (subtract
payment)
Continuing in this matter, a pattern develops.
So, after
the nth payment, we have:
A(1+j)n-P[(1+j)(n-1)+(1+j)(n-2)+...+1]
(factor out -P after first term)
The quantity inside the [ ] is a
geometric series with a common ratio of (1+j).
Look at it this way:
1+(1+j)+(1+j)2+...+(1+j)(n-2)+
(1+j)(n-1)
Whose sum can be taken to be:
S=[1-(1+j)n]/[1-(1+j)]
This is the sum of n terms of a
geometric series with common ratio of 1+j.
(1 minus last term divided by 1
minus the common ratio))
Substituting this in the expression (after the
nth payment) we get:
A(1+j)n-P[1-(1+j)n]/(-j).
This will be zero, when your loan is paid off.
So, setting the
expression = 0,
we get; A(1+j)n = P[(1+j)n-1]/j,
which gives P = Aj(1+j)n/[(1+j)n-1].
dividing top
& bottom by (1+j)n, we get;
P = Aj /
[1-(1+j)(-n)]<-----this is it! tuck it away in your
files!
In words, multiply your loan by the APR divided by # of
payments per yr
(this is the periodic interest rate, j)
divide the
result by one minus (1+ that periodic interest rate) raised to the -n power,
where n = the number of periodic payments you make.
Note: It would be
nice, but not necessary, to understand this derivation.
However, the end
result is a handy formula for future use. I used it several times to check the
correctness of my monthly payments on my car loans.
Hope this wasn't
torture!