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Compound Interest


    what you'll learn...

overview

Compound interest is made easy with the following.

maturity value in n time periods A=P(1+R)n
=P(1+R)(1+R)(1+R)...n times
Note: R is the (interest rate in percentage)/100.

The interest is calculated on the principal for each of the time period in n. That is, the principal for the second time period is the maturity value of first time period, and so on for subsequent time periods.

Once the above is understood, the equation is easily derived.

reinvesting interest

A borrower takes a loan from lender. The loan is for 2000 coins with 10% interest per annum for 3 years.

The interest for the first year is 2000×10100=200
The interest for the second year is 2000×10100=200
The interest for the third year is 2000×10100=200

There are two choices in paying the interest.
 1.  The borrower can pay the interest every year, OR,
 2.  the borrower can pay the maturity value at the end of 3 years.

In deciding between these two choices, the following consideration is made.

If the interest is paid every year, then the lender can use the interest amount to make more money.

If, at the end of first year, the interest 200 coins is paid to lender, then the lender can loan the interest and earn additional interest on that loan too.

If the borrower wants to pay back the principal and the interest only after the 3 year period, then the borrower should compensate on the additional interest, the lender would make on the interest from first year.

compound

A borrower takes a loan from a lender. The loan is for 2000 coins with 10% compound interest per annum for 3 years.

The interest for the first year is 2000×10100=200
The principal for the second year is the principal and interest, 2000+200=2200

The interest for the second year is 2200×10100=220
The principal for the third year is the principal of second year and interest, 2200+220=2420

The interest for the third year is 2420×10100=242
The maturity value in the third year is, 2420+242=2662

This interest calculation compensates the lender for the additional interest she would earn in lending the interests from first and second years.

another example

A customer deposits 2000 coins in a bank. The interest is 10% per annum for 3 years.

The interest for the first year is 2000×10100=200
The interest for the second year is 2000×10100=200
The interest for the third year is 2000×10100=200

There are two choices in paying the interest.
 1.  The bank can pay the interest every year, OR,
 2.  the bank can pay the maturity value at the end of 3 years.

In deciding between these two choices, the customer considers that:

If the interest is paid every year, then the customer can deposit the interest which will provide additional interest.

If, at the end of first year, the interest 200 coins is paid to the customer, then the customer can deposit the interest and earn interest on that deposit too.

If the bank wants to pay back the principal and the interest only after the 3 year period, then the bank should compensate on the additional interest the customer would make on the interest from first year

compound interest

A customer deposits 2000 coins in a bank. The compound interest is 10% per annum for 3 years.

The interest for the first year is 2000×10100=200
The principal for the second year is the principle and interest, 2000+200=2200

The interest for the second year is 2200×10100=220
The principal for the third year is the principle in second year and interest, 2200+220=2420

The interest for the third year is 2420×10100=242
The maturity value in the third year is, 2420+242=2662

This calculation compensates the customer for the additional interest she would earn in depositing the interests from first and second years.

definition

The word "compound" means: made up of several parts.

Compound Interest : At the end of every interest period, the interest is added into principal is compound interest. The interest for the next period is paid on the revised principal.

examples

A borrower takes a loan of 2000 coins at 12% per annum interest. The interest is compounded every month.

Note that the interest rate is given for yearly.
But the compounding of interest is for every month.

The interest per month is
12/12 monthly which is 1%


A borrower takes a loan of 2000 coins at 12% per annum interest. The interest is compounded every month.
The interest for the compounding interval (1 month) is 1%.
The interest for the first interval =2000×1100 =20
The interest for the second interval =2020×1100 =20.20
The interest for the third interval =2040.20×1100 =20.402

terms

The word "interval" means: a time gap between two events.

The word "frequency" means: rate at which something occur repeatedly.

Compounding Interval : The time period for which the interest is calculated and added to the principal.

Compounding Frequency : The compounding interval given as a frequency form, like monthly, quarterly, annual, etc.

reference list

Students may find it difficult to memorize all the formulas for compound interest. This is listed for reference and is explained later. No need to memorize any of them.

Principal P
Interest Rate R
Number of time periods n
Maturity value or amount A

Maturity value A=P(1+R/100)n
Principal P=A/(1+R/100)n

summary

Compound interest: One need not memorize any formulas. Quickly follow through the story to recall the formula on the fly.

 •  Interest is calculated as percentage of the principal (Interest I=PR/100 for every time period. Note: only the first interest duration is considered.)

 •  Interest at the end of first time duration is added to the principal. (A=P+PR100 =P(1+R/100))

 •  The borrower has to return the principal and the interest, which together is the maturity value.

Principal for the second time period P1=P

Maturity Value in first time period A1 = Principal P+ Interest P×R/100

Principal for the second time period P2 = Principal P+ Interest P×R/100 =P(1+R/100)

Maturity Value in second time period A2 = Principal P2+ Interest P2×R/100=P2(1+R/100) =P(1+R/100)2

maturity value in n time periods A=P(1+R/100)n

There are 4 variables (A, P, R, n) in this equation. In a problem, 3 of these 4 variables are given and this formula is a form of equation of one variable (algebra) to solve for the unknown variable.

Outline

The outline of material to learn "commercial arithmetics" is as follows.

Note: Click here for the detailed ouline of commercial arthmetics.

  •   Ratio, Proportion, Percentage

    →   Comparing Quantities

    →   Introduction to Ratio

    →   Ration & Fraction Differences

    →   ProportionsP

    →   Percentages

    →   Conversion to percentage

  •   Unitary Method

    →   Introduction to Unitary Method

    →   Direct Variation

    →   Inverse Variation

    →   DIV Pair

  •   Simple & Compound Interest

    →   Story of Interest

    →   Simple Interest

    →   Compound Interest

  •   Rate•Span=Aggregate

    →   Understanding Rate-Span

    →   Speed • Time=Distance

    →   Work-rate • time = Work-amount

    →   Fill-rate • time = Filled-amount

  •   Profit-Loss-Discount-Tax

    →   Profit-Loss

    →   Discount

    →   Tax

    →   Formulas