#1
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Interest calculation, please help
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.
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#2
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Re: Interest calculation, please help
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. |
#3
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Re: Interest calculation, please help
[ 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] |
#4
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Re: Interest calculation, please help
[ 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. |
#5
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Re: Interest calculation, please help
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. |
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