## Interest

Interest problems employ the rules of percentage problems but include the additional factor of time.

The interest (I) is the amount of money paid for the use of money.

The rate (R) is the percent charged on the basis of one year's use of the money.

The time (T) is the number of years, months and days over which the money is used. Note especially that 30 days are considered a month and, in some cases, 360 days are considered a year.

The amount (A) is the sum of the principal and the interest.

To find the interest for any given period of time, multiply the principle by the rate by the time.

Formula:I = P x R x T

Example: Find the simple interest on \$900 for 2 years at 6%.

I = ?,   P = \$900,   B = 6%,   T = 2
\$900 x .06 = \$54,   \$54 x 2 = \$108 interest - Answer

To find the amount, add the interest (I) to the principal (P).

Formula: A = P + I

Example: Find the interest and amount of \$400 for 3 years, 3 months and 10 days at 6%.

I = P x R x T and A = P + I.

 \$400 x .06 x 3 = \$72.00 \$400 x .06 x 1/4 = 6.00 \$400 x .06 x 10/360 = .66 2/3 \$78.67 interest - Answer A = P + I = \$400 + \$78.67 = \$478.67 amount - Answer

Example: A real estate agent sold a piece of property
for \$50,000. His commission is 7.5%.

P = \$50,000,   B = 7.5%   P = ?
\$50,000 x .075 = \$3,750.00 commission - Answer

### Indirect Cases of Interest

To find the rate when the principle, interest and time are given, divide the total interest by the time to get the amount of the interest for one year; then divide the quotient by the principal.

Example: What must be the rate of interest on \$400 to produce \$25 in 6 months?

\$25 / .5 = \$50 interest for 1 year,
\$50 / 400 = 12.5% rate of interest - Answer

To find the time when the interest, the rate percent and time are given, divide the interest by the time to get the interest for one year; then divide this by the rate.

Example: How much money will you have to lend to get \$240 interest at 6%, if you lend it for 6 months?

\$240 / .5 = \$480 interest for 1 year,
\$480 / .06 = \$8,000 - Answer

[ Note: There are alternative methods for solving interest problems of the above type. See if you can figure them out. ]

To find the principal when the amount, rate percent and time are given, divide the given amount by the amount of \$1 for the given time at the given rate.

Example: How much would you have to deposit at 4% in order to withdraw \$819 at the end of 1 year and 3 months?

\$1 at 4% for 1 year will amount to \$1.04.
\$1 at 4% for 1.25 years will amount to \$1.05.
The number of dollars that will amount to \$819 is...
\$819 / \$1.05 or \$780. - Answer

[ Note that when dates are given for an interest-bearing period, the time from a given date in one month to the same date in any other month is sometimes figured as even months of thirty days each. Thus, from February 5 to March 6 would be reckoned as 27 days, but extending the time to the next day, March 5, makes it 30 days or one month. ]

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### The 6% Method

When the rate of interest is 6%, use is often made of what is known as the 6% method or the 60-day method. By this method the time is reduced to multiples or fractions of 60 days or of 6 days.

Rule:

#### 1.

To find interest for 60 days by the 6% method, move the decimal point in the principal two places to the left.

#### 2.

To find interest for 6 days, move the decimal point in the principal three places to the left.

#### 3.

To find interest for other intervals of time, multiply or divide as may be necessary, the amounts which are thus found.

Example: What is the interest on \$500 for 5 months and 8 days at 6%?

 5 months = 60 days + 60 days + 30 days. 8 days = 6 days + 2 days. \$5.00 int. for 60 days. \$5.00 int. for 60 days. \$2.50 int. for 30 days. .50 int. for 6 days. .17 int. for 2 days. \$13.17 total int. - Answer

EXPLANATION:
Since the interest for 1 year is 0%, the interest for 2 months or 60 days is 1% and the interest for 6 days is 1/10 of that for 60 days, or .1%. Taking the given example, the time is divided as shown. Each 60-day period earns 1% or \$5.00 of interest. The 30-day period earns 1/2 of \$5.00 or \$2.50. For 6 days the interest is 1/10 of 1%, or \$0.50 and for 2 days it is 1/2 of 1/10 of 1% or \$0.162/3, which is changed to \$0.17. Adding these figures gives the total interest for the whole period.

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### Accurate Interest

The rule of the 30-day month and the 360-day year is most commonly used in business, but the United States Government and the Federal Reserve Bank, as well as certain commercial banks under certain conditions, pay interest on the basis of a 365-day year. Interest on this basis is known as accurate interest or exact interest.

In figuring this kind of interest the exact number of days in the period is computed. Each day is figured as 1/365 of a year. The total amount of interest paid is naturally somewhat smaller than it would be on the basis of a 360-day year.

 Jan. 17 days 200   x   5 / 100   x   28 / 365   =   \$112 / 100 Feb. 11 days 28 =   \$1.12 interest - Answer

EXPLANATION:
As interest starts on the 14th day of January, there are 17 interest-bearing days left in the month. These added to 11 days in February make a total of 28 days. Multiplying the principal by the rate by 28/365 produces the interest.

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### Bank Discount

When a bank lends money against a promissory note it collects the interest charges in advance by deducting them from the full value or face of the note. This is called discounting the note. The amount thus deducted is called the discount. The remainder left when the discount has been deducted from the face of the note is called the net proceeds and is what the borrower receives. When the note falls due at maturity, the borrower pays back to the bank it's full face value.

Example: A 90-day note for \$400 at 6% is discounted when made. Find the discount and the net proceeds.

Interest for 60 days = \$4.00
Interest for 90 days = \$6.00 discount,
\$400 - \$6 = \$394 net proceeds.

Example: A 90-day note for \$400 at 6%, dated April 15, is discounted May 15. Find the net proceeds.

Term of discount = 90 days = 30 days = 60 days.
Interest for 60 days = \$4.00 discount.
\$400 - \$4 = \$396 net proceeds. Answer

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### True Discount

True discount (not to be confused with bank discount) is the deduction made from the face value of an obligation payable at some future date in order to determine what the obligation is worth at the present time.

Present worth is the sum of money which, if invested at the same rate as that which applies to the given obligation, will equal the face value of the debt when it becomes due.

Rule:

#### 1.

To find present worth divide the face value of the indebedness by the amount of \$1 for the given time at the given rate.

#### 2.

To find true discount subtract present worth from face value.

True discount may apply to obligations bearing either simple or compound interest.

Example: What is the present worth of an obligation for \$500 payable 2 years and 6 months from now and bearing interest at 6% not compounded. Also what is the true discount?

.06 x 2.5 = .15 interest rate for 2.5 years.
\$1.00 x .15 = \$.15 interest on \$1 for 2.5 years.
\$1.00 + .15 = \$1.15 amound on \$1 for 2.5 years.
\$500 / \$1.15 = \$434.78 percent worth. -Answer
\$500 - \$434.78 = \$65.22 true discount.-Answer

EXPLANATION:
Since \$1 would grow to \$1.15 in 2 years and 6 months at 6% simple interest, to find out how many dollars would grow to \$500 we divide \$500 by \$1.15 and get \$434.78 as the required amount. The difference between this and \$500 is the true discount. The example may be checked by multiplying \$434.78 by .15 to arrive at \$65.22 interest at 6% for 2.5 years, corresponding with the same amount calculated as true discount.

Since a sound bond on the day when it matures is worth neigher more nor less than its face value, it follows that as such a bond approaches maturity its value on the stock market tends more to coincide with present worth as determined by the principles of true discount.

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### APR

#### Annual Percentage Rate

APR can be defined as "the yearly cost of a loan" expressed as a percentage."

The APR includes, as a percent of the principal, not only the interest that has to be paid on a loan, but also some other costs.

Mortgage loans - The APR usually includes the "points" and may include title fees, escrow fees. For more information, on mortgages and related fees, click here.

Credit card debt - The APR is the annual interest rate you will be expected to pay if you carry over a balance, take out a cash advance, or transfer a balance from another card. For more information, on credit cards and related fees, click here.

#### Periodic Rate

Because “APR” is the abbreviation for annual percentage rate, and because credit card issuers send out their bills monthly, they do not use the APR to calculate the finance charges you owe in a given billing cycle.

Instead, they use what’s called the periodic rate. This is the APR divided by 12 (as in 12 months). So if your card has an APR of 12 percent, the periodic rate would be 1 percent.

That explains why you’ll see periodic rate for new charges on your credit card bills, along with the APR for your outstanding balance.

### APY

#### Annual Percentage Yield

APY is defined as "the effective annual return. The APY is calculated by taking one plus the periodic rate raised to the number of periods in a year. For example, a 1%-per-month rate would offer an APY of 12.68%."

Most interest compounds much more often than once a year. Typically, it's more like monthly, weekly, or daily.

Simply put, the difference between APR and APY is that APR is the annual rate of interest without taking into account the effects of compounding.

APR Formula:   APR = Periodic Rate x Number of Periods in a Year

Example: A credit card company charges 1% interest each month. What is the APR?

1% x 12 months = 12% - Answer

EXPLANATION:
As mentioned above the APR differs from APY, which takes into account compound interest. The APY for a 1% rate of interest compounded monthly would be [(1 + 0.01) / 12 – 1 = 12.68%] 12.68% a year. If a credit card borrower only carries a balance on their credit card for one month's period they will be charged the equivalent yearly rate of 12%. However if they carry that balance for the year, the effective interest rate becomes 12.68% as a result of the compounding each month.

APY Formula:   APY = (1 + APR / n) n – 1

n = the number of compounding periods

Example: You open a savings account with \$1.000 at a bank that pays 5% APR. (You do not add or subtract any money from that account.) How much would you have at the end of one year?

APY = (1 + .05 / 365) 365 – 1
APY = (1 + .00137) 365 – 1
APY = 1.00137365 – 1
APY = 1.051267 – 1
APY = 051267 or 5.1267% - Answer

Banks prefer to quote APY to individuals who want to open savings accounts or CD's since it looks better.

### APR and APY - Converting Back and Forth Using a Spreadsheet

It’s very simple to do with the aid of Microsoft Excel or Open Office Calc, the very fine freebie spreadsheet software program. The Open Office software is available here.

Converting APR To APY

Useful for evaluating how much you’re actually paying on credit card debt.

• In cell A1, type APR
• In cell A2, type # times a year
• In cell A3, type APY
• In cell B3, type =(1+B1/B2)^B2-1
... Or as an alternative: Use the "Power" function. Type =POWER((1+(B1/B2));B2)-1
Next,
• Type the desired APR value into cell B1.
• Type the number of times a year the interest is compounded into B2.
(Usually, it's daily for a credit card. If so, type in 365).
• The APY will appear in B3.
Format cell B3 as a "percentage".
(To accomplish that, just right click on B3, click the "Format Menu",
then "Cell" and then select "Percentage".)

Converting APY To APR

Useful for figuring out how much a savings account is really paying you.

• In cell A1, type APY
• In cell A2, type # times a year
• In cell A3, type APR
• In cell B3, type =((1+B1)^(1/B2)-1)*B2
Next,
• Type the desired APY value into cell B1.
• Type the number of times a year the interest is compounded into B2.
(Usually, it’s monthly. If so, type in 12).
• The APR will appear in B3.
Format cell B3 as a "percentage".
(To accomplish that, just right click on B3, click the "Format Menu",
then "Cell" and then select "Percentage".)

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### Compound Interest

Compound interest is interest which for each successive interest period is figured on a base that represents the original principal plus all the interest that has accrued in previous interest periods. In other words, compounding interest means that, in addition to the principle in an account, the interest earns interest.

To compute compound interest add the interest for each period to the principal before figuring the interest for the next period.

Example: What is the compound interest on \$5000 for 4 years at 6% compounded annually?

 \$5000 original principal .06 rate \$300 interest for 1st year 5000 \$5300 new principal, 2nd year .06 \$318 interest for 2st year \$5300 \$5618 new principal, 3rd year .06 \$337.08 interest for 3rd year \$5618 \$5955.08 new principal, 4th year .06 \$357.30 interest for 4th year \$5955.08 \$6312.38 amount at end of 4th year \$5000.00 original principal \$1312.38 Compound interest for 4 years - Answer

EXPLANATION:
We first find the interest on the original principal (\$5,000) for one year. This amounts to \$300. We add this to \$5,000, making the new principal at the start of the second year \$5,300. One year's interest on \$5,300 comes to \$318, which we add to \$5,300. to arrive at \$5,618 as the new principal at the start of the third year. We continue in this way until we reach the end of the fourth year when we find that principal and interest together amount to \$6,312.38. From this we subtract the original principal of \$5,000 leaving \$1,312.38 as the compounded interest for the entire period of 4 years. To avoid awkward four-place decimals in figuring compound interest multiply the cents in the principal separately (you will usually be able to do it mentally) and if the result comes to more than one cent, add this to the first figure you write down.

Money will double itself at compound interest in 35.003 years at 2%, in 23,450 years at 3%, in 17.675 years at 4%, in 14.207 years at 5%, in 11.896 years at 6%, in 10.245 years at 7%. These figures are based on annual compounding of interest. If interest is compounded semi-annually or quarterly the periods will be only very slightly shorter. at simple interest money doubles itself in 25.000 years at 4%, in 20 yers at 5%, in 16.667 years at 6%, in 14.286 years at 7%.