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I have a loan for a principal amount of \$117,000.00 at 9.75% interest and a total amount due (including the interest for 84 months) of \$191,805.60. I had 84 months to pay it off, or I could pay it off early, without penalties. I have made 61 payments totalling \$139,287.40.

How do I calculate the principal balance due, not including any further interest?

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this question is not fit for asking here. – noname1014 Mar 30 '12 at 15:37
@TaoHong洪涛 Why not? How would you solve such a problem? I would use math. – Graphth Mar 30 '12 at 15:42
I'm a bit confused. Is the 9.75% a monthly or annual interest? compound of course, right? – user2468 Mar 30 '12 at 15:42
This is financial mathematics. I am teaching a class where we do problems like this. – Graphth Mar 30 '12 at 16:10
@Mindy: There are a few things one needs to know. The most important is whether the $9.75$ is the nominal yearly rate, or the effective annual rate. If nominal, then interest rate is likely $9.75/12$ monthly, compounded monthly, which means effective annual rate is higher than $9.75$. One will also have to know whether the $61$ payments were all equal. – André Nicolas Mar 30 '12 at 17:00

Payments: First off, I assume the payments are equal. This is standard for loans, though I'm sure there are cases where it is different. If 84 payments total \$191,805.60, then each payment must be \$2283.40. I will also assume the first payment is 1 month after the loan is taken out, which is also pretty standard.

Interest rate: Next, we need to know about the interest rate. Is it an annual effective rate of 9.75% or is it a nominal (meaning in name only) rate 9.75% compounded monthly? The latter is a fancy way of saying the interest rate for each month is 9.75/12% = 0.8125% = 0.008125. I can do the problem both ways. The answers should be relatively close to each other. If the interest rate is an annual effective rate, then let $j$ be the monthly rate. We would have $(1 + j)^{12} = 1.0975$, or $j = \sqrt[12]{1.0975} - 1 \approx 0.0077830371$

Answer: 61 payments have been made, which means there are 23 left. Let us assume the 61st payment was made today, so the 62nd is due in exactly one month. Then, the outstanding loan balance is equal to the present value today of the remaining payments.

Assuming a nominal 9.75% interest rate, convertible monthly: $$\begin{align*} &2283.4 \cdot (1.008125^{-1} + 1.008125^{-2} + \cdots + 1.008125^{-23}) \\ &= 2283.4 \cdot \frac{1.008125^{-1} - 1.008125^{-24}}{1 - 1.008125^{-1}} \\ &= 2283.4 \cdot \frac{1 - 1.008125^{-23}}{0.00825} \\ &= 47658.00 \end{align*}$$

Assuming an annual effective rate of 9.75%, so the monthly rate is $j \approx 0.0077830371$:

$$2283.4 \cdot \frac{1 - 1.0077830371^{-23}}{0.0077830371} = 47915.88$$

Note, these are about \$257 apart, which isn't all that big relative to \$48,000.

Inconsistency: However, there is something wrong with this problem. The amount of the loan should be equal to the present value of all of the future payments, at the time the loan is made. Since the loan is for \$117,000, the present value of all the 84 payments, at the exact time the loan is made, should also be \$117,000. However, it turns out that it is actually:

$$2283.4 \cdot \frac{1 - 1.008125^{-84}}{0.00825} = 138620.63$$

or for the other interest rate:

$$2283.4 \cdot \frac{1 - 1.0077830371^{-84}}{0.0077830371} = 140413.40$$

Neither of these are anywhere near \$117,000, so something is going on here. That is, we don't have all the information, or some of the information we have is wrong. I used a financial calculator to solve for the monthly interest rate in the case of 84 equal monthly payments of \$2283.40, with the first a month from now, where the loan is for \$117,000, and the answer is 1.282503786%. This is the monthly interest rate which translates to a yearly rate of $$1.01282503786^{12} - 1 = 0.1652339966$$ In other words, it corresponds to a 16.5% annual effective interest rate.

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