Not that it’ll necessarily be convincing, but here’s a proof that extra payments reduce the lifetime of a loan. This sort of material is covered in any introductory book on financial mathematics; if you’re looking for one, I recommend Broverman’s Mathematics of Investment and Credit.
If you have a loan of amount L with n payments and an effective monthly interest rate j, the payment R is given by Lj/(1 - v[sup]n[/sup]), where v = 1/(1 + j). The quantity (1 - v[sup]n[/sup])/j shows up often enough that there’s a special symbol for it. I’m not going to try to reproduce it here, so I’ll write it as a(n, j). a(n, j) is the value at time 0 of payments of $1 at times 1, 2, …, n with an interest rate of j between payments. There’s a similar symbol, s(n, j), which is the value of that same series of payments at time n. s(n, j) = ((1 + j)[sup]n[/sup] - 1)/j.
For an ordinary 30-year mortgage with monthly interest rate j and payments due at the end of each month, the loan amount is given by Ra(360, j). If you’re making an extra payment at the end of every year, the loan amount is given by Ra(m, j) + Ra(m, j)/s(12, j). The factor of 1/s(12, j) in the second term lets me pretend that there’s an extra payment every month, which means that I’m working with the same interest rate everywhere. It’s not strictly necessary, but it makes things easier.
Since the loan amount is equal to both Ra(360, j) and Ra(m, j) + Ra(m, j)/s(12, j), it follows that Ra(360, j) = Ra(m, j) + Ra(m, j)/s(12, j). The payment amount cancels out, and we’re left with a(360, j) = a(m, j) + a(m, j)/s(12, j). Therefore, a(m, j) = a(360, j)s(12, j)/(1 + s(12, j)). But a(m, j) = (1 - v[sup]m[/sup])/j, so you can substitute that in and solve for m. I get that m = -ln(1 - ja(360, j)*s(12, j)/(1 + s(12, j)))/ln(1 + j). You could expand out the other interest symbols, but it gets to be a messy expression pretty quickly, and there are even simple calculators that know how to evaluate them, so it’s not really worth it.
So, how does this work out numerically? The effective monthly rate j is calculated by taking your quoted rate and dividing by 12. If you have a quoted interest rate of 5%, making an extra payment at the end of every year brings the term of the loan down to 304.2 months, which is just over 25 years. If you have a quoted interest rate of 6%, the extra payment brings the term down to 24.66 years. As your interest rate increases, the extra payment method will make the term of your loan decrease, and by a significant amount.