Annual Interest Rate: | |

Loan terms (months): | |

Principal Amount: | |

Cost of loan: |

*Note:*
An Auto loan, and many other consumer loans, use the Rule of 78
to calculate the balance in cases where you wish to pay off the loan
early. In that case, the outstanding balance is *more* than
is calculated by simple interest. That's how things are: banks are
allowed to penalize you slightly for paying off these loans early.
It makes no sense to payoff these sorts of loans after about halfway
through.

The formula for these loans looks complicated, but it reveals an interesting fact, which has much meaning. Making a simple interest loan and paying it off in installments is mathematically equivalent to making two seperate deals:

- make a simple interest loan which you pay off with a lump sum payment at the end of the loan, letting the interest on the initial principal accumulate,
- open a savings account which pays the same interest as the
the loan, and make monthly installments to the account.
At the end of the loan, the amount in the savings account equals
the amount due on the loan, so you pay off the loan with the account
balance, closing both.
The amount due on the loan, with accrued interest, is:

Principal * ( 1 + interest ) ** Term

that is, the principal compound with its interest for Term times. The amount saved in the bank is:Payment * ( 1 + interest ) ** (Term - 1) + Payment * ( 1 + interest ) ** ( Term - 2 ) + ... + Payment

that is, add up all the payments, applying individually to each, how the interest compounds as the payment sits in the bank.The hard part is making sense of the large sum of payments compounded each by a different factor. Mathematicians know the formal power series:

1 / ( 1- t ) = 1 + t + t**2 + ...

This is an infinite sum. This series is truncated by subtracting itself from itself after it has been shifted up by the desired amount:(t ** N) * 1 / ( 1 - t ) = t**N + t*(N+1) + ...

So the payment sum is also written:Payment * ( 1 / ( 1 - t ) ) - Payment * (t**Term) * 1 / ( 1 - t )

wheret = 1 + interest

Setting Principal with accrued interest equal to savings with accrued interest, and simplifying some algebra:

Principal * (1+interest)**Term = Payment * ( (1+interest)**Term - 1 ) / interest

This is a formula used in the JavaScript function embedded in this page.Burton Rosenberg

Math and Computer Sci

Univ of Miami

August 1997