What is Compound Interest?

Interest is the cost of borrowing money, or the earnings from lending or investing. It can be classified into two types: simple interest and compound interest.

Simple Interest vs. Compound Interest

• Simple Interest is calculated only on the original principal. The formula is:Interest=Principal×Rate×Time\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}Interest=Principal×Rate×TimeExample: Borrowing $100 at 10% annual simple interest for two years results in:100×10%×2=20100 \times 10\% \times 2 = 20100×10%×2=20So, the total interest is $20.
• Compound Interest, on the other hand, is calculated on both the principal and the accumulated interest.
  Example: Borrowing $100 at a 10% annual compound interest rate for two years:
  • Year 1: 100×10%=10100 \times 10\% = 10100×10%=10 New balance = $110
  • Year 2: 110×10%=11110 \times 10\% = 11110×10%=11 Total interest after two years = $21 (compared to $20 for simple interest).

Since compound interest grows exponentially, it benefits investors but can also increase debt if unpaid.

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Compounding Frequency

Interest can compound at different intervals: daily, monthly, quarterly, or annually. More frequent compounding leads to higher total interest.

Example: A 10% annual interest rate compounded semi-annually (twice a year) means each half-year has a 5% interest rate:

• First 6 months: $100 × 5% = $5
• Next 6 months: ($100 + $5) × 5% = $5.25
• Total interest: $10.25 (slightly higher than $10 for annual compounding).

Savings accounts and CDs typically compound annually, while mortgages and credit cards compound monthly.

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Compound Interest Formulas

Basic Compound Interest Formula

At=A0(1+r)nA_t = A_0 (1 + r)^nAt​=A0​(1+r)n

where:

• A0A_0A0​ = Principal (initial amount)
• AtA_tAt​ = Total amount after time ttt
• rrr = Interest rate
• nnn = Number of compounding periods

Example: A $1,000 savings account with 6% annual interest compounded once per year for two years:At=1000(1+6%)2=1123.60A_t = 1000 (1 + 6\%)^2 = 1123.60At​=1000(1+6%)2=1123.60

For different compounding frequencies, the formula adjusts to:At=A0×(1+rn)ntA_t = A_0 \times \left(1 + \frac{r}{n}\right)^{nt}At​=A0​×(1+nr​)nt

where nnn is the number of times interest is compounded per year.

Example: The same $1,000 with 6% daily compounding:At=1000×(1+6%365)365×2=1127.49A_t = 1000 \times \left(1 + \frac{6\%}{365}\right)^{365 \times 2} = 1127.49At​=1000×(1+3656%​)365×2=1127.49

Continuous Compounding

At=A0ertA_t = A_0 e^{rt}At​=A0​ert

where e≈2.718e \approx 2.718e≈2.718.

Example: The maximum possible interest on a $1,000 deposit at 6% compounded continuously for two years:At=1000e0.12=1127.50A_t = 1000 e^{0.12} = 1127.50At​=1000e0.12=1127.50

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The Rule of 72

A quick way to estimate how long an investment takes to double with compound interest:Years to double=72Annual Interest Rate(%)\text{Years to double} = \frac{72}{\text{Annual Interest Rate} (\%)}Years to double=Annual Interest Rate(%)72​

Example: At an 8% annual return, money doubles in:728=9 years\frac{72}{8} = 9 \text{ years}872​=9 years

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History of Compound Interest

• First recorded by Babylonians and Sumerians (~2400 BCE).
• Many societies, including Romans, Christians, and Muslims, considered it usury (unethical lending).
• In 1683, mathematician Jacob Bernoulli discovered the mathematical constant eee while studying compound interest.
• Leonhard Euler later refined eee to approximately 2.718, making continuous compounding calculations possible.

Compound interest remains one of the most powerful financial concepts, benefiting investors while also increasing long-term debt for borrowers.