maths > commercial-arithmetics

Compound Interest

what you'll learn...

overview

Compound interest is made easy with the following.

maturity value in $n$$n$ time periods $A=P{\left(1+R\right)}^{n}$$A = P {\left(1 + R\right)}^{n}$
$=P\left(1+R\right)\left(1+R\right)\left(1+R\right)...n$$= P \left(1 + R\right) \left(1 + R\right) \left(1 + R\right) \ldots 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$$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$$2000$ coins with $10%$ interest per annum for $3$$3$ years.

The interest for the first year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 200$
The interest for the second year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 200$
The interest for the third year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 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$$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$$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$$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$$2000$ coins with $10%$ compound interest per annum for $3$$3$ years.

The interest for the first year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 200$
The principal for the second year is the principal and interest, $2000+200=2200$$2000 + 200 = 2200$

The interest for the second year is $2200×\frac{10}{100}=220$$2200 \times \frac{10}{100} = 220$
The principal for the third year is the principal of second year and interest, $2200+220=2420$$2200 + 220 = 2420$

The interest for the third year is $2420×\frac{10}{100}=242$$2420 \times \frac{10}{100} = 242$
The maturity value in the third year is, $2420+242=2662$$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$$2000$ coins in a bank. The interest is $10%$ per annum for $3$$3$ years.

The interest for the first year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 200$
The interest for the second year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 200$
The interest for the third year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 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$$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$$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$$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$$2000$ coins in a bank. The compound interest is $10%$ per annum for $3$$3$ years.

The interest for the first year is $2000×\frac{10}{100}=200$$2000 \times \frac{10}{100} = 200$
The principal for the second year is the principle and interest, $2000+200=2200$$2000 + 200 = 2200$

The interest for the second year is $2200×\frac{10}{100}=220$$2200 \times \frac{10}{100} = 220$
The principal for the third year is the principle in second year and interest, $2200+220=2420$$2200 + 220 = 2420$

The interest for the third year is $2420×\frac{10}{100}=242$$2420 \times \frac{10}{100} = 242$
The maturity value in the third year is, $2420+242=2662$$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$$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$$12 / 12$ monthly which is $1%$

A borrower takes a loan of $2000$$2000$ coins at $12%$ per annum interest. The interest is compounded every month.
The interest for the compounding interval (1 month) is $1%$.
The interest for the first interval $=2000×\frac{1}{100}$$= 2000 \times \frac{1}{100}$ $=20$$= 20$
The interest for the second interval $=2020×\frac{1}{100}$$= 2020 \times \frac{1}{100}$ $=20.20$$= 20.20$
The interest for the third interval $=2040.20×\frac{1}{100}$$= 2040.20 \times \frac{1}{100}$ $=20.402$$= 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$$P$
Interest Rate $R$$R$
Number of time periods $n$$n$
Maturity value or amount $A$$A$

Maturity value $A=P{\left(1+R/100\right)}^{n}$$A = P {\left(1 + R / 100\right)}^{n}$
Principal $P=A/{\left(1+R/100\right)}^{n}$$P = A / {\left(1 + R / 100\right)}^{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$$I = P R / 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+P\frac{R}{100}$$A = P + P \frac{R}{100}$ $=P\left(1+R/100\right)$$= P \left(1 + R / 100\right)$)

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

Principal for the second time period ${P}_{1}=P$${P}_{1} = P$

Maturity Value in first time period ${A}_{1}$${A}_{1}$ = Principal $P+$$P +$ Interest $P×R/100$$P \times R / 100$

Principal for the second time period ${P}_{2}$${P}_{2}$ = Principal $P+$$P +$ Interest $P×R/100$$P \times R / 100$ $=P\left(1+R/100\right)$$= P \left(1 + R / 100\right)$

Maturity Value in second time period ${A}_{2}$${A}_{2}$ = Principal ${P}_{2}+$${P}_{2} +$ Interest ${P}_{2}×R/100$${P}_{2} \times R / 100$$={P}_{2}\left(1+R/100\right)$$= {P}_{2} \left(1 + R / 100\right)$ $=P{\left(1+R/100\right)}^{2}$$= P {\left(1 + R / 100\right)}^{2}$

maturity value in $n$$n$ time periods $A=P{\left(1+R/100\right)}^{n}$$A = P {\left(1 + R / 100\right)}^{n}$

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

Outline