Principal or Interest?Date: 08/08/2002 at 07:18:31 From: S. Jackson Subject: New Car Loan If I have a new car loan for $20,000 at 9% simple interest, how do I calculate how much of that monthly payment is going to the principal versus the interest? I plan on making additional principal payments monthly and want to keep up with the balance of the principal for an early payoff. Let's say the loan is for 5 years (60 months). So we are working with $20,000 loan for 60 months at 9% interest. Thank you. Sue Jackson Date: 08/08/2002 at 08:39:09 From: Doctor Jerry Subject: Re: New Car Loan Hi Sue, If at the beginning of any month, after you've made whatever payment you wish, you still owe the bank A dollars, then during the coming month your debt will increase by the amount of the interest, which will be A*9/(12*100)=0.0075*A. The notation a_0 means a sub 0. amount borrowed: a_0 (dollars) interest rate: r (like 7%) number of months the loan is for: N payments are made at the end of each month, in the amount x you borrow the money at the first day of some month let p_0 be the amount you owe the bank at time t=0 (months) let p_j be the amount you owe the bank at time t=j (months) Now, p_0 = a_0 p_1 = a_0 + a_0*r/(12*100)-x=a_0(1 + r/1200) - x (amount owed one month ago, plus interest accrued on this amount, minus payment) For convenience, let 1+r/1200 = w. So, p_1 = a_0*w - x p_2 = p_1 + p_1*r/(12*100) - x = p_1*w - x=a_0*w^2-x*w - x p_3 = p_2 + p_2*r/(12*100) - x = p_2*w - x = a_0*w^3 - x*w^2 - x*w - x So, p_j = a_0*w^j - x[w^{j-1} + w^{j-2}+...+ w + 1] Now, w^{j-1} + w^{j-2}+...+ w + 1 = (1-w^j)/(1-w), finite geometric sum. So, p_j = a_0*w^j -x*(1-w^j)/(1-w). We want to choose x so that p_N=0. Solving for x we find x = a_0*w^N(1 - w)/(1 - w^N) For your situation, in which a_0 = 20,000, N = 60, and r = 9, we find x = 415.17. This is the monthly payment. Just below I list the time (1 month, 2 months, and 3 months), the amount owed, the amount paid on principal, and the amount on interest. The sum of the latter two must be the payment. So, we need only calculate the second of these. 1, 19734.83, 150.00, 265.17 2, 19467.68, 148.01, 267.16 3, 19198.52, 146.01, 269.16 To calculate the interest for the first month, do this: take the amount owed at the beginning of the month (20000) and calculate the interest: 20000*(9/100)*(1/12) = 150 For the interest during the second month, take the amount owed at the first of this month (20000-265.17 = 19734.83) and calculate the interest: 19734.83*(9/100)*(1/12) = 148.01 It goes on like this. I hope that this has been of some help. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 08/08/2002 at 17:07:38 From: S. Jackson Subject: Thank you Thank you. I really appreciate it. I will work it into my schedule on the spreadsheet and go from there. Again, thanks a lot! Sue Date: 08/09/2002 at 07:32:21 From: Doctor Jerry Subject: Re: Thank you Hi Sue, Just one more thing. If you make an extra payment with the regular payment, and the extra payment goes to reduce the principle, I think that you can handle this on your spreadsheet by subtracting that amount from the principal used on the next line (on which the interest for the coming month is calculated). - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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