An old adage has it that those who understand compound interest are more likely to collect it, those who don't more likely to pay it.
J. A. Paulos, A Mathematician Plays the Stock Market
Basic Books, 2003, p. 86

Interest Rate Calculations

In Exodus XII we read:

  1. And the children of Israel had done according to the word of Moses; and they had asked of the Egyptians vessels of silver, and vessels of gold, and garments.
  2. And the Lord had given the people favor in the eyes of the Egyptians, so that they gave to them what they required; and they emptied out Egypt.

On the first night of Passover, we are reminded of the ten plagues that afflicted the Egyptians (Exodus VII-XII: Blood, Frogs, Vermin, Wild Beasts, Pestilence, Boils, Hail, Locusts, Darkness, Plague of the First-Born.) Seeing G-d's hand in everything that was happening in Egypt, it is hard to imagine that the parting action alone had no divine origin or was even spontaneous. And indeed many years beforehand, G-d made a promise to Abraham (Genesis XV):

  1. And He said to Abram, Know for sure, that your seed shall be a stranger in a land not theirs, and they will make them serve, and they will afflict them four hundred years.
  2. And also that nation whom they shall serve, will I judge; and afterward shall they go out with great possessions.

So this is very surprising that according to legend (Talmud Bavel, Sanhedrin), after Alexander the Great captured Egypt and Jerusalem, a delegation of Egyptian priests approached him with the request that the Jews should be forced to return the goods they took along upon the exodus from Egypt. Moreover, they referred to Exodus XII.35 to support their request. In response, Gviha Ben-Psisa reminded the Egyptians that the Jewish people had been slaving in Egypt for 430 years without adequate compensation. In the wilderness of Sinai (Numbers I), the Jewish tribes (not counting Levites) numbered in excess of 600,000 males 20 years and up. Ben-Psisa offered to trade all the Egyptian goods in return for a fair compensation plus the interest accrued during all these years. The Egyptians chose to forfeit their claim.

I am not sure about the rates at those remote times but, say, 17%-19% charged by most credit cards today may easily make a huge hole in one's budget in a much shorter time frame.

1% (one percent) of any amount is 1 hundredth part of that amount. 100% of a quantity is that whole quantity. Other percents are evaluated proportionally. For example, 5% of 700 is five hundredth of 700, i.e., 700·5/100 = 35. 15% of 2 is 2·15/100 = 3/10 = 0.3. (For more on per cent, see What Is Percent?)

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,

 
PrincipalInterestNumber of times
interest is
compound
The DEBT
    
10001211120.00
10001221123.60
10001241125.50
100012121126.82

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 have

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

P(1 + q)M + p(1 + q)M-1 + ... + p(1 + q)2 + p(1 + q) + p

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

P(1 + q)M + p[(1 + q)M - 1]/q

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

P(1 + q)M - p(1 + q)M-1 - ... - p(1 + q)2 - p(1 + q) - p

or

P(1 + q)M - p[(1 + q)M - 1]/q

We expect this amount to become zero; whence

p = qP((1 + q)M)/[(1 + q)M - 1]

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.


If you are reading this, your browser is not set to run Java applets. Try Firefox and declare the site http://www.cut-the-knot.org as trusted in the Java setup.

Interest Rate Calculations


What if applet does not run?

Note: A JavaScript loan calculator is available elsewhere.


Related material
Read more...

  • What Is Fraction?
  • What Is Percent?
  • Percent Bloopers
  • Diluted Paint
  • Correlation and Causation: Misuse and Misconception of Statistical Facts
  • Benford's Law and Zipf's Law
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