The amount one borrows with a promise to return it later is called the principal. Usually, the borrower is required to return the principal with interest. In the simplest case, the whole debt is returned at the end of the specified period. Let the interest over that period be 12% or .12. Borrowing 1000 goats one will have to return 1000 + 1000·.12 = 1120 goats. With the principal of 500 bushels of wheat one will have to return additional 500·.12 = 60 bushels.

The assumption above was that the interest owed is evaluated at the time of the final (and only) payment. This may not be the case. For example, the interest may be evaluated twice before the payment is due. Assume, as before, that the interest for the whole period is 12%. Naturally, for the half that period it is only 6%. The principal of 1000 will thus grow by the middle of the specified period to 1000 + 1000·.06 = 1060. However, if at this point in time, the debt is not returned, another 6% are assessed during the remaining half of the period. However, now this 6% apply to the accrued debt of 1060: 1060 + 1060·.06 = 1123.60. Were the interest evaluated only once, the debt would be 1120. When it is evaluated twice the amount owed grows; the effective interest becomes 12.36% instead of 12%. Such interest is known as compound. In evaluating the compound interest it is important to take into consideration the frequency of interest evaluation. In our example,

It is convenient (and is usually the case) to cite the yearly interest: interest assessed at the end of the year during which a debt is owed. Let q stand for the yearly interest expressed as a fraction rather than a percent. If the interest is 5%, q = 0.05. If the interest is 7.5%, q = 0.075, and so on. Assume the principal is P and that the interest is compound N times a year. At each of N of those evaluations, the q/N-th fraction of the debt is added. At the first evaluation the debt grows from P to P + P·q/N = P(1 + q/N). On the second evaluation, the interest q/N applies to the newly evaluated debt of P(1 + q/N) which leads to P(1 + q/N) + P(1 + q/N)·q/N = P(1 + q/N)². After the N-th evaluation, the actual debt will grow to P(1 + q/N)^N.

Instead of talking of debts, we may consider lending money (e.g., to a bank on a savings account). An untouched principal of P at the yearly interest of q will grow to P(1 + q) at the end of the first year, it will become P(1 + q)² at the end of the second year, and P(1 + q)³ at the end of the third, and so on. If at the end of every year you contribute an additional amount of p, then, at the end of the first year, your account will have P(1 + q) + p. At the end of the second year, the interest applies to that whole amount to give you P(1 + q)² + p(1 + q). Should you decide to make regular deposits of p, you'll get P(1 + q)² + p(1 + q) + p. At the end of the third year, you'll get P(1 + q)³ + p(1 + q)² + p(1 + q) + p. After M years, your principal P and regular, yearly deposits of p will grow to

With a known formula for the sum of a geometric series this simplifies to

Back to the debt, assume you borrow the principal P at the rate of q for M years. How much should you return each year to make good on your debt? Denote that amount as p. At the end of the first year, your debt will be P(1 + q) - p. At the end of the second, P(1 + q)² - p(1 + q) - p, and so on. Similarly to the above, at the end of the M-th year the debt will be

or

We expect this amount to become zero; whence

If the interest is compound several times a year, q must be replaced by the effective yearly interest. As we learn from the Passover story, it pays to know your interests.

In the applet below, interest is expected as a percentage rate: insert 12 for 12%, 13.5 for 13.5%, etc. It can be used as a loan or a mortgage calculator or to verify your investment strategy.