Interest calculation, please help

[anatta]

If someone loans you $10,000 at 4% interest, and you are to pay 400/mo, how many months are you going to pay. At 0 its 25 months. Assume the normal parameters that a car loan company would use, like how its compounded or whatever. Thanks and be well and happy.

[Nottom]

For payments of $402 you will pay it off in 26 months according to this site .

Over such a short time frame, interest (especially a low rate like 4%) isn't really going to have much time to compound on you so aren't really going to add many payments.

Here is the formula for calculating a payment:
P = principal
r = interest rate
m = length of loan in months

P ( r / 12 )
-------------------------

(1 - ( 1 + r / 12 )^-m )

Feel free to try and solve for m, but it looks like some nasty math with logarithms.

[BruceZ]

[ QUOTE ]
If someone loans you $10,000 at 4% interest, and you are to pay 400/mo, how many months are you going to pay. At 0 its 25 months. Assume the normal parameters that a car loan company would use, like how its compounded or whatever. Thanks and be well and happy.

[/ QUOTE ]

I'm sure there are calculators all over the place which do this, and financial types probably know the formula, but let's see if we can derive it from first principles.

Let P = the principal = $10,000
Let i = the monthly interest = 0.04/12
Let m = the monthly payment = $400
And n = the number of months to pay off loan

After the 1st month, our balance will have grown to P*(1 + i) and we will pay m, so our balance will be P*(1 + i) - m. We compute the new balance at the end each month as the previous month's balance times (1 + i) minus m:

month 2: [P*(1 + i) - m]*(1 + i) - m
= P*(1 + i)^2 - m*(1 + i) - m

month 3: [P*(1 + i)^2 - m*(1 + i) - m]*(1 + i) - m
= P*(1 + i)^3 - m*(1 + i)^2 - m*(1 + i) - m

...

month n: P*(1 + i)^n - m*sum{k = 0 to n-1} (1 + i)^k = 0

We set this equal to zero since after n months the loan will be paid off. Now we solve for n. First we sum the geometric series as sum{k = 0 to n-1} x^k = (1 - x^n) / (1 - x). See derivation of this at the bottom. [img]/images/graemlins/diamond.gif[/img]

P*(1 + i)^n = m*[1 - (1 + i)^n] / -i

-P*i/m * (1 + i)^n = 1 - (1 + i)^n

(1 - P*i/m)*(1 + i)^n = 1

Taking logs of both sides:

ln(1 - P*i/m) + n*ln(1 + i) = 0

n = -ln(1 - P*i/m) / ln(1 + i)

n = -ln[1 - 10,000*(0.04/12)/400] / ln(1 + .04/12)

n =~ 26.15 months This means we make 26 monthly payments of $400 and one final payment of $58.76. Paying it off in exactly 26 months would require monthly payments of $402.17.


[img]/images/graemlins/diamond.gif[/img] Derivation of geometric series used above:

S = sum{k = 0 to n-1} x^k

S = 1 + x + x^2 + x^3 + ... + x^(n-1)

x*S = x + x^2 + x^3 + ... + x^(n-1) + x^n
----------------------------------------------------------
S - x*S = 1 - x^n

S = (1 - x^n) / (1 - x) [img]/images/graemlins/diamond.gif[/img]

[BruceZ]

[ QUOTE ]
For payments of $402 you will pay it off in 26 months according to this site .

Over such a short time frame, interest (especially a low rate like 4%) isn't really going to have much time to compound on you so aren't really going to add many payments.

Here is the formula for calculating a payment:
P = principal
r = interest rate
m = length of loan in months

P ( r / 12 )
-------------------------

(1 - ( 1 + r / 12 )^-m )

Feel free to try and solve for m, but it looks like some nasty math with logarithms.

[/ QUOTE ]

Solving that for m gives the equation that I derived for the number of months, and solving my equation for the monthly payment gives your equation. Your m is my n, and my m is the monthly payment.

[AaronBrown]

You can approximate problems like this in your head if the interest rate is low or the time is short.

At $400 per month, it will take you 25 months to pay off the principal ($10,000/$400 = 25). The principal decreases from $10,000 to zero over the period, so on average it's about $5,000 (it's actually a bit higher because the decline is not linear, but ignore that). 4% interest on $5,000 is $200 per year, or $417 over 25 months (2 1/12 years). So that's about one extra payment to cover the interest, 26 months.