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In Reply to: BMW of Riverside: Bad @ math! posted by Fatta on September 04, 1999 at 19:54:55:
Interest payments for loans are typically based on the recursive equation which has the following closed form solution:
Y(k) = [Y(0)*(1+i)^k] - {[(1+i)^k - 1]/i}*p
Y(k) => amount you owe at any time period k
Y(0) => initial loan amount
k => payment period (weekly,monthly,...)
i => interest/period
p => payment
Example:
car cost = $42000
down payment (20%)= $8400
APR = 8% = 0.08
TERM = 36 months
Y(0) = 42000 - 8400 = $33600
i = 0.08/12 = 0.0067 => period is monthly
0 <= k <= 36 months
p = ???.??
p needs to be a value such that after 36 months you don't owe anything. Hence:
0 = [Y(0)*(1+i)^36] - {[(1+i)^36 - 1]/i}*p
p = {[Y(0)*(1+i)^36]*i}/[(1+i)^36 - 1]
p = {[33600*(1.0067)^36]*0.0067}/[(1.0067)^36 - 1]
p = 42679*0.0067/0.2702
p = ~1052.90 (the last payment will be $0.07 more)
I've used this equation for years and it has never been wrong. As a matter of fact, I know it by heart so that I can always check what I owe at any point in time on any loan provided I have a calculator handy.
If someone did the following, then they would not be taking into account the compounding nature of interest.
$33600*1.08/36 = $1008
As you can easily see, the delta is ~$53.
If you replace the first "-" sign in the equation with a "+" sign, you can determine how an investment will grow with regular contributions "p". I do this calculation alot:)
The "rules" associated with interest as in the "rule of 88" or "rule of 72" are based on this closed form solution of the recursive equation
David
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