Explanation

A loan, as we are using the term here, can be understood by combining thinking about an ordinary annuity and thinking about compound interest.

Basically, when you take out a loan, you are going to pay it back with regular payments. That works quite a bit like an ordinary annuity; your group of payments are going to have a future value when they are done. That future value will be all the money that you pay plus the interest you earn with your payments over the time of the loan.

How much interest will you earn on your payments? Well, that would be the amount of money the bank would have earned if the bank would have invested the loaned sum at the loan rate over the term of the loan. In fact, the bank did make an investment. They invested in you.

So, a loan is like an ordinary annuity in that it has regular payments, except you do not collect the future value of that annuity. You give it to the bank.

The future value of the 'annuity' (loan) is equal to the value of a compound interest investment over the loan's time period at the same rate.

So, the bank gives you an amount of money and asks you to pay it back at regular intervals. Your payments act like an annuity and have a future value. As the bank sees it, that future value is equal to the value of an compound interest investment (an investment that the bank has made in you) compounded at the rate and payment period over the term of the loan.

By equating the future value of an annuity with the future value of a compound interest investment, both with the same interest rate and payment schedule, we get the following mathematics:

Formula symbols

A | Amount to be loaned |

R | Amount of payment |

r | Annual percentage rate (APR) |

k | Interest periods per year |

i = r/k | Percentage rate per interest period |

n | Number of payments |

The following equation relates the above quantities:

A = R((1 - (1 + i)^{-n}) / i)