# When will I have paid or owe $X on my mortgage?

Because of the effects of amortization, the percent paid on the balance is not equal to the percent of payments made. For instance, if you have a 15-year mortgage and make paymets for 3 years, you will not have paid off 20% of the loan balance at the end of those 3 years even though you have made 20% of the payments. Amortization means that a greater portion of your initial payments goes toward the interest owed on the loan, while payments made near the end of the borrowing period are mostly devoted to the loan principal.

If you want to figure when you will have paid $X of the principal, or when you will owe only $X of the principal, you must take amortization into account. The calculator on the left will figure the number of years and months when you will have paid or owe a certain amount. You can also apply the formulas described below.

### Monthly Payment and Amortization Equations

If the amount borrowed is*P*, the annual interest rate

*i*(expressed as adecimal), and the number of years in the lending period

*N*, then the monthly payments

*M*are given by the formula

M = (Pi/12)(1 + i/12)

^{12N}/[(1 + i/12)

^{12N}- 1].

For any given month

*d*in the borrowing period, let B(d) be the amount of the monthly payment that goes toward the principal, and let J(d) be the amount that goes toward the interest. The equations for B(d) and J(d) are

B(d) = (M - iP/12)(1 + i/12)

^{d-1}

J(d) = M - (M - iP/12)(1 + i/12)

^{d-1}

As you can see, B(d) + J(d) = M. Also, as d increases, B(d) becomes larger while J(d) becomes smaller.

To find the total principal paid up to month d, you must compute the sum

B(1) + B(2) + ... + B(d).

Using geometric series, the sum is

B(1) + B(2) + ... + B(d)

= (M - iP/12)[(1 + i/12)

^{0}+ (1 + i/12)

^{1}+ ... (1 + i/12)

^{d-1}]

= (M - iP/12)[(1 + i/12)

^{d}- 1]/(i/12)

= (12M/i - P)[(1 + i/12)

^{d}- 1]

If you set this equation equal to a fixed amount, you can solve for d using algebra.

### Example

A $100,000 home loan is financed for 15 years at an annual interest rate of 6.6%. After how many months will the owners have paid off $60,000 of the loan?The first step is to set P = 100000, N = 15, and i = 0.066 to solve for M. This gives

M = 100000(0.0055)(1.0055

^{180})/[1.0055

^{180}- 1]

= $876.61

Now you must solve the following equation for d:

60000 = (12M/i - P)[(1 + i/12)

^{d}- 1]

60000 = (159383.63 - 100000)[(1.0055)

^{d}- 1]

60000 = (59383.63)[(1.0055)

^{d}- 1]

1.01037935 = (1.0055)

^{d}- 1

2.01037935 = (1.0055)

^{d}

Ln(2.01037935) = d*Ln(1.0055)

Ln(2.01037935)/Ln(1.0055) = d

127 = d

After 127 months, the owners will have paid off $60,000 of the loan balance. This is equivalent to 10 years and 7 months.

© *Had2Know 2010
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