# Guide to Calculating Interest Rates

QUESTION: I am responsible for the investment and safety of another person’s finances, so I pay careful attention to such things as newspaper advertisements placed by banks and savings and loan associations listing their rates on certificates of deposit and other investments. After doing this for a while, I have come to the conclusion that the rates and yields posted by the various financial institutions are confusing and even misleading. Some tout their rates and method of compounding while others focus on their annual yield. Just when I think I’ve figured out which is the best deal, I realize I’m wrong. I am only interested in arriving at the total amount credited to the account at the end of the term. But when I ask the bank or S&L; people, they tell me they don’t know how to arrive at that answer because it’s all done by their computer. Do you know where I can find a table or formula to arrive at the information I need?--G.W.C.

ANSWER: The Federal Reserve Bank of New York has a free publication that is just what you need. Called “The Arithmetic of Interest Rates,” the booklet guides readers on a journey through the interest-rate maze. Starting with some simple formulas--principal, multiplied by rate of interest, multiplied by time equals simple interest, for example--the booklet explains how to calculate simple and compound interest. Several examples are scattered throughout the booklet to make this confusing subject easier to understand.

Here, for example, is the bank’s explanation of compound interest, followed by the formula it uses and a sample problem and solution.

“Compound interest,” the publication explains, “is the amount paid or earned on the original principal plus the accumulated interest. With interest compounding, the more periods for which interest is calculated, the more rapidly the amount of interest on interest and interest on principal builds.”

Therein lies an important lesson for consumers trying to compare rates on investments or loans: The method and frequency of compounding are just as important as interest rates themselves in calculating the earning power of money, which is what you’re really after.

Compounding can be done annually (in which case, the interest on a one-year investment is no different than simple interest), quarterly, monthly or even daily. To add to the complexity, “annual” for purposes of calculating interest can be 360 days, 365 days or even 366 days.

In formula language, calculating compound interest may be reduced to F = P(1+R), raised to a power that represents the term of the loan or investment. In other words, the total future repayment value of a loan or the total pay-back of an investment (F) is equal to the principal (P), multiplied by the sum of one plus the annual percentage rate (R) after the sum is multipled by itself for as many years as the loan or investment runs.

Confused? Here is a sample application of that formula from the booklet. The problem: How much interest will be paid on a \$3,000 loan for six years at 10% per year, compounded annually?

To arrive at the answer, which is \$2,314.68, add 1 to the percentage rate, which gives you 1.10. Then, multiply 1.10 by itself six times, the number of compounding periods. Multiply the resulting sum (1.771561) by \$3,000, the principal of the loan. You will arrive at \$5,314.68, which is the lump-sum amount that must be repaid at the end of the six years. To determine how much of that number represents interest on the loan, simply subtract the principal (\$3,000) from \$5,314.68 to get \$2,314.68.

Interest that is compounded daily is somewhat more difficult to figure because you have to add two calculations to the formula. The annual interest rate must be divided by the number of daily compound periods each year and the number of years of the loan or investment must be multiplied by the number of calendar days each year. Remember, the number of daily compound periods per year isn’t necessarily the same as the number of calendar days in a year.

Here is the booklet’s sample problem: What will a \$500 savings deposit grow to in six years at 5 1/2% per year, compounded daily?

The answer is \$698.66. To get that figure, divide the interest rate (0.055) by 360 days (we’re assuming that the financial institution uses a 360-day year, because many do) and add 1 to the result for a sum of 1.000152778. Multiply that figure by itself 2,190 times (six years times 365 days). The result is 1.397322433. Then, multiply that number by \$500 to get \$698.66.

There are probably a few math enthusiasts who would get a charge out of multiplying a 10-digit number by itself 2,190 times using a calculator or even pencil and paper. But practically speaking, it would be much easier and faster to turn to a table for help. The booklet provides this as well.

In its compound interest rate table, the Federal Reserve Bank of New York lists the future value of a \$1 investment for a range of annual percentage rates (1% to 18%) and a range of time periods (1 to 60).

(Please note that the interest rate in the table is the annual percentage rate, which also is known as the annual yield or the APR. This is important since many financial institutions cite an interest rate that is something other than the annual yield and is virtually meaningless for consumers trying to compare interest rates. Institutions are required by law to provide consumers with the annual percentage rate, so insist on getting it.)

Back to the table. Few of us would ever invest a single dollar, of course. But the numbers in the table need only be multiplied by the amount of the investment to be applicable to any situation.

For example, say you want to know the amount of principal and interest that you would be entitled to receive on a 3-year, \$2,000 certificate of deposit that yields 10% and is compounded annually. You would simply turn to the table, find the intersection of the columns for 10% yield and three periods, and jot down the number that appears there, 1.331. Then, multiply that by \$2,000 to arrive at \$2,662.

If the compounding period is something other than yearly, simply make the two adjustments detailed earlier. Also, there is a separate table showing what a \$1 deposit would grow to if the interest is compounded daily.

The booklet also demonstrates how to calculate the monthly payment on a conventional home mortgage (and provides a table showing finance charges on installment loans) and the effective yield to maturity of Treasury securities.

To get a copy of this free, 33-page booklet, Southern California readers may simply contact the Los Angeles branch of the Federal Reserve Bank of San Francisco. The phone number is (213) 683-8358. The address is: Public Information, Federal Reserve Bank of San Francisco, Los Angeles Branch, P.O. Box 2077, Los Angeles, Calif. 90051.

Readers elsewhere may write to: Public Information Department, Federal Reserve Bank of New York, 33 Liberty Street, New York, N.Y. 10045, or call its publications number, (212) 791-6134.