why does this answer on paying a mortgage two years earlier make sense?

Question

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?

Answer 1

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!

Answer 2

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.