Suppose you have finished 3 years’ payment on a 5-year loan of $9,952 at 8% annual interest, for a car. As we saw in Sample Problem 1.3in(Compound Interest Loans and Payments), your payments were $200 per month.

Think about the remaining 24 months. Your situation is as though you had just taken a loan of $A at 8% per annum, where

                  A(151/150)²⁴/150 = 200[(151/150) ²⁴−1],

so 1.17288A = 150×200×0.17288, A = 4,421.94.

We say your equity in the loan is $9,952 - $4,421.94 , about $5,530.

You might think that, after making payments for three-fifths of the payment period, you would own 60% of your car. However your equity is a little less than that amount: around 55.5%.

The difference is much greater on longer-term loans. For example, suppose you take out a 30-year house loan for $100,000 at 8% per annum, with equal monthly payments of $733.76 (as we calculated above). After three-quarters of the term— 270 of the 360 payments have been made—it is as if you had just taken a loan at 8% per annum with principal $A, where

            A(151/150)⁹⁰/150 = 733.76[(151/150)⁹⁰−1],

that is

                     A×.012123295= 600.578

so A = 49,539.20 and your equity is $50,460.80.