Compound Interest

Compound Interest is when the interest is added to the principal at regular intervals.
The interest is said to be compounded.

The Compound Interest rule is:

A = P ( 1 + i ) ⁿ

where
A = amount after investment time period
P = principal or amount invested
i = interest rate (when the interest is compounded or added to the bank account; as a decimal)
n = number of times that the interest is compounded or added to the account)

This rule is really a series of Simple Interest calculations (Interest = Principal × Interest Rate × Time) as shown in the following long example. A shorter method using the Compound Interest rule is also shown below.

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Example One - Compound Interest

I have $100 in a savings bank account. The bank pays 10% interest each year. This interest is added to my bank account each year. How much will I have after 3 years?

Answer (LONG METHOD WITHOUT THE RULE):
These calculations are a series of yearly simple interest calculations.
Interest after 1st year
= Principal × Interest Rate × Time
= $100 × 10% × 1yr
= 100 × 0.1 × 1
= $10

Amount after 1st year
= $100 + $10
= $110

Interest after 2nd year
= $110 × 10% × 1yr
= $11

Amount after 2nd year
= $110 + $11
= $121

Interest after 3rd year
= $121 × 10% × 1yr
= $12.10

Amount after 3rd year
= $121 + $12.10
= $133.10

Answer (SHORT METHOD WITH THE RULE):
P = $100
i = 10% = 0.1
n = 3 yr

A = P × ( 1 + i ) ⁿ
A = 100 × ( 1 + 0.1 ) ³
A = $133.10

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Example Two - Compounded Yearly

I have $100 in my bank account. The bank pays interest of 12% p.a. (per annum = each year) compounded yearly. How much will I have after 5 years?

Answer:
P = $100
r = 12% = 0.08
n = 5 yr

A = P × ( 1 + i ) ⁿ
A = 100 × ( 1 + 0.12 ) ⁵
A = $176.23

Example Three - Compounded Monthly

I have $100 in my bank account. The bank pays interest of 12% p.a. compounded monthly. How much will I have after 5 years?

Answer:
P = $100
i = 12% ÷ 12 months = 1% = 0.01
n = 5 yr × 12 months = 60

A = P × ( 1 + i ) ⁿ
A = 100 × ( 1 + 0.01 ) ⁶⁰
A = $181.67

Notice that you calculate compound interest monthly, you divide i by 12 and multiply n by 12.

Example Four - Compounded Quarterly

Compounded Quarterly is compounding 4 times per year.

I have $100 in my bank account. The bank pays interest of 12% p.a. compounded quarterly. How much will I have after 5 years?

Answer:
P = $100
i = 12% ÷ 4 = 3% = 0.03
n = 5 yr × 4 quarters = 20

A = P × ( 1 + i ) ⁿ
A = 100 × ( 1 + 0.03 ) ²⁰
A = $180.61

Notice that you calculate compound interest quarterly, you divide i by 4 and multiply n by 4.

Example Five - Compounded Weekly

I have $100 in my bank account. The bank pays interest of 12% p.a. compounded weekly. How much will I have after 5 years?

Answer:
P = $100
i = 12% ÷ 52 = 0.0023
n = 5 yr × 52 weeks = 260

A = P × ( 1 + i ) ⁿ
A = 100 × ( 1 + 0.0023 ) ²⁶⁰
A = $182.09

Notice that you calculate compound interest weekly, you divide i by 52 and multiply n by 52.

Example Six - Inflation

Inflation is the change of money's value over time. A Big Mac costs $4 now. If our country's inflation rate is a steady 3% each year, what will the same Big Mac cost 10 years into the future?

Answer:
P = $40
i = 3% = 0.03
n = 10 yr

A = P × ( 1 + i ) ⁿ
A = 4 × ( 1 + 0.03 ) ¹⁰
A = $5.38

Example Seven - Appreciation

Appreciation is the increase in an item's value over time. A 1967 Shelby Mustang cost $4000 when new. Its annual appreciation rate is approximately 9%. What would you expect to pay for it 2017?

Answer:
P = $4000
i = 9% = 0.09
n = 50 yr

A = P × ( 1 + i ) ⁿ
A = 4000 × ( 1 + 0.09 ) ⁵⁰
A = $297 430

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Example Eight - Population Growth

The Philippines has a fast-growing population with an annual increase of approximately 1.7 %. If its population in 2012 was 98 million, what is the expected population in 2020?

Answer:
P = 98 000 000
i = 1.7% = 0.017
n = 8 yr

A = P × ( 1 + i ) ⁿ
A = 98 000 000 × ( 1 + 0.017 ) ⁸
A = 112 148 559 people approximately