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I have been stuck on this one for hours ... not too great at math can someone help. Thanks.

Isaac borrowed $\$4000$ at $11.5\%$ compounded quarterly $5.5$ years ago.

One year ago he made a payment of $\$1500$. What amount will extinguish the loan today?

I've tried a bunch of different approaches, none were right.

From what I understand we should calculate the FV for $4.5$ years when $PV=\$4000$ then subtract $\$1500$ from answer and calculate FV for one more year.

But still no luck..

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First: The "value" of the loan after $4.5$ years would be $$ FV(4.5) = 4,000\left(1 + \frac{0.115}{4}\right)^{4\cdot 4.5}. $$ Now then after the $4.5$ years you would subtract $1,500$ from the debt and the add one years extra interest to what is left over. This will give you the future value after the $5.5$ years:

$$ (FV(4.5) - 1500)*(1 + \frac{0.115}{4})^{4\cdot 1} $$

So this is equivalent to making a new loan of $FV(4.5)$ and then calculate what that load is "worth" after $1$ year with $11.5\%$ compounded quarterly.

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It sounds like your idea is correct. The current value of loan should be given by: $4000(1+ \frac{.115}{4})^{4 \times 5.5} - 1500(1+ \frac{.115}{4})^{4 \times 1}$

If you aren't getting the right answer, it could be a problem with using the calculator. Some can be difficult to use, for example, if they don't let you see what you are typing.

You also don't need to turn it into a multi step problem by calculating the value at 4.5 years, subtracting 1500, and then calculating the interest for the final year after everything else. If you calculate the current value of everything at once, you can save yourself some work. It is fairly easy to see that both must be equivalent if you draw a time diagram of the payments.

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  • $\begingroup$ Thanks guys, finally got the answer. I guess I was making a mistake in my calculations somewhere. $\endgroup$ – user100115 Oct 15 '13 at 1:51
  • $\begingroup$ The right answer is $5,782.29. $\endgroup$ – user100115 Oct 15 '13 at 1:51
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Various methods will all work, and are all based on the same two principles:

1) You move money earlier and later in time using the compound interest formula

2) You can only add or subtract money amounts if they are at the same point in time.

So in this case you have many options, some of which have been given above. Another may be:

Move the partial payment to the date of the original loan, subtract from the original balance, and move what's left to today.

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$$ $4,000*1.115^{18} = $28,379.68.$$ Paid off $$ $1,500 $$ Left to repay: $$ $26,879.69 $$ waited a year: $$ $26,879.69*1.115^4 = $41,545.47.$$

Unless, you mean, 11.5% per annum, compounded quarterly, then the interest would be: $$1 + (0.115/4) $$ Rather than simply $1.115$.

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  • $\begingroup$ $11.5\%$ is the nominal quarterly interest, not the effective quarterly interest; that is the convention. $\endgroup$ – Tyler Oct 15 '13 at 1:57
  • $\begingroup$ Uhh, not quite. $11.5\%$ is the nominal annual rate, compounded quarterly. The quarterly interest rate is $2.875\%$, and the effective annual rate is $12.01\%$ $\endgroup$ – DJohnM Oct 15 '13 at 2:02
  • $\begingroup$ Right, sorry... $\endgroup$ – Tyler Oct 15 '13 at 2:03
  • $\begingroup$ I did mention at the end that he didn't actually explicitly say it was per annum, it merely says 11.5% compounded quarterly, I don't know why I have been downvoted. I made this answer directly after the other two, to give a basic answer in terms of the non-per annum perspective. $\endgroup$ – Display Name Oct 15 '13 at 2:18

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