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Suppose that you took a mortgage of 100000 on a house to be paid back in full by 10 equal annual installments, each consisting of the interest due on the outstanding balance plus a repayment of a part of the amount borrowed. If you decided to clear the mortgage after eight years, how much money would you need to pay on top of the eight installment, assuming that a constant annual compounding rate of 6% applies throughout the period of the mortgage?

I know how to find the payment amount if we had to pay the mortgage back in 10 years: $C=\frac{100,000}{PA(6\%, 10)} = 13,586.80$ where C= payment amount, PA= present value of annuity and $PA(n, r)= \frac{1-(1+r)^{-n}}{r}$
But I am confused on how to figure how much money I need to pay on top of the eigth installment if I decide to clear the mortgage after eight years instead of ten?
My textbook says to do $PA(6\%, 2) \times 13,586.8 = 24,909.93$ but I don't understand why this is correct?

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To illustrate how it works: Use excel pmt function. Generate the series. Add a column for extra payment and set all entries to 0. Change how you calculate the next period starting balance to subtract the excess payment (which you set to 0). After you made the 8th payment, you'll see the balance for the next period/s. That's what you'll need to clear.

If you still can't see it, you can use excel's goal seek to generate the excess that will zero out starting balance for period 9.

Good luck!

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  • $\begingroup$ I did it on excel and now I understand what is going on, you are paying 13,586.80 per payment for 8 years then after 8 years the remaining balance to be paid is 24,909.93. What I don't understand is why $PA(6\%, 2) \times 13,586.80=24,909.93$, basically why does multiplying $PA(6\%, 2)$ by the payment amount give you the remaining balance after 8 years? $\endgroup$ – idknuttin Feb 15 '16 at 0:38
  • $\begingroup$ @idknuttin That's basically how it's defined. It's a mistake to try to think of this as something reasonable. It's a convention. Aliens would do it differently. $\endgroup$ – Matt Samuel Feb 15 '16 at 1:04
  • $\begingroup$ Are you saying that what you don't understand is why it is $24,909.93 and not $27,173.60 (=2*13586.8)? If so, this is why: you are expected to pay $27,173.60 at the end of the term (2 payments). Because you are paying it off early, you have to give it a present value. Present value is equivalent to to discounting the amount you owe by the interest rate (and according to the terms and conditions of the interest rate - simple, compounded, continuously compounded or other). $\endgroup$ – zevij Feb 15 '16 at 9:14
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At the beginning of year $k$ there will be $10-k+1$ payments remaining. The amount of each payment is the remaining principal divided by $PA(6\%,10-k+1)$. Thus after 8 years (the beginning of the ninth year) the installment amount is equal to $P/PA(6\%,2)$ and we need to pay $P$ to pay off the mortgage. Computing $P$ is then done as in the book.

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  • $\begingroup$ "The amount of each payment is the remaining principal divided by $PA(6\%,10−k+1)$" the book multiplied the payment amounts by $PA(6\%, 10-k+1)$? Also, $13,586.80$ is the amount to pay per payment if you pay the mortgage over 10 years, not the remaining balance after 8 years? $\endgroup$ – idknuttin Feb 14 '16 at 23:09
  • $\begingroup$ @idknuttin the convention is that after every payment the principal goes down by the amount that makes the original yearly payment still correct. After the first year we have a 9 year mortgage and the same payment amount, and in order for the payment to be the same the principal must be different. $\endgroup$ – Matt Samuel Feb 14 '16 at 23:11

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