BUSINESS MATH Compound interest

Compound interest

Compound interest is interest that is paid on both the principal and also on any interest from past years.

The formula used to calculate compound interest is:

M = P( 1 + i )ⁿ

M is the final amount including the principal.

P is the principal amount.

i is the rate of interest per year.

n is the number of years invested.

Formula:

P is the principal (the initial amount you borrow or deposit)

r is the annual rate of interest (percentage)

n is the number of years the amount is deposited or borrowed for.

A is the amount of money accumulated after n years, including interest.

When the interest is compounded once a year:

A = P(1 + r)n

However, if you borrow for 5 years the formula will look like:

A = P(1 + r)⁵

Frequent Compounding of Interest:

What if interest is paid more frequently?
Here are a few examples of the formula:

Annually = P × (1 + r) = (annual compounding)

Quarterly = P (1 + r/4)⁴ = (quarterly compounding)

Monthly = P (1 + r/12)¹² = (monthly compounding)

Example 1

$1000.00 to invest for 3 years at rate of 5% compound interest.

M = 1000 (1 + 0.05)³ = $1157.62.

Example 2

$100,000 principal amount with a 6% interest rate, compounded annually for three years.

Year 1

$100,000 X .06 for one year is $6000 interest.

Year 2

$106,000 X .06 =$6360 interest.

Year 3

Starting with $112,360 accumulated X .06 = $6742 interest.

At the end of year 3 we have $119,102. As you can see, compound interest definitely beats simple interest for return.

1.    As a mathematical formula: This is a straight formula, but a bit trickier as we need to raise a number by a power.

Principal X (1 + Periodic Rate) ^ Number of Periods = Future Amount

$100,000 X (1 + .06) ^ 3 = Future Amount

$100,000 X (1.06 x 1.06 x 1.06) = Future Amount

$100,000 X 1.19 = $119,100 rounded off.

Excercise

1. $1000 invested with compound interest at a rate of 15% per year for 9

years.

2. $400 invested with compound interest at a rate of 3% per year for 2 years.

3. $1250 invested with compound interest at a rate of 5% per year for 4

years.

4. $1400 invested with compound interest at a rate of 9% per year for 6

months.

5. $300 invested with compound interest at a rate of 25% per year for 8

years.

6. $600 invested with compound interest at a rate of 4% per year for 10

years.

7. $750 invested with compounded interest at a rate of 19% per year for 13

years.

8. $100 invested with compounded interest at a rate of 10% per year for 10

years.

9. $250 invested with compounded interest at a rate of 4% per year for 4

years.

10. $4250 invested with compounded interest at a rate of 5% per year for 3

years.

Solutions to Above Exercise

Compound Interest

Solutions

1. M = $3517.88

2. M = $424.36

3. M = $1519.38

4. M = 2347.94

5. M = $1788.14

6. M = $888.15

7. M = $7197.34

8. M = $259.37

9. M = $292.46

10. M = $4919.9

Simple Interest

Principal X Rate X Time = Interest Amount

$100,000(Principal) X 0.08(8% Rate) X 1 Year (Time) = $8000 Interest

Principal X {1 + (Rate X Time)} = Total Amount

1.    $100,000 X {1 + (.08 X 1)} = $100,000 X 1.08 = $108,000

2.    Let's do that again for three years:

Here we'll multiply the .08 (8%) rate times 3 years to equal .24.

$100,000 X {1 + .24} = $124,000