What is Compound Interest?
Compound interest is the interest on an investment or loan that is calculated based on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which is calculated only on the principal, compound interest grows exponentially over time.
Continuous Compounding (Compounded Infinitely)
If interest is compounded continuously, the formula is:
Where:
e≈2.718 approx (Euler’s number)
Derivation of Compound Interest Formula Using Simple Interest
To understand compound interest, we first need to understand simple interest.
. Deriving Compound Interest from Simple Interest
-
Step 1: Understanding the Interest Calculation
- Suppose you invest P (Principal) at an annual interest rate of r% compounded yearly.
- At the end of 1st year, interest is calculated on P and added to it.
- This process repeats every year, leading to an increasing principal amount.
Step 2: Finding the Amount for Each Year
After 1 Year:
- Interest for the 1st year = (r/100) × P
- New principal after 1 year = P + (r/100) × P
- Factorizing:
Step 4: Special Cases
-
If Interest is Compounded Half-Yearly:
- Since interest is applied twice a year, the rate per half-year is r/2, and the number of time periods becomes 2n.
- Since interest is applied twice a year, the rate per half-year is r/2, and the number of time periods becomes 2n.
-
If Interest is Compounded Quarterly:
- Interest is applied 4 times per year, so the rate per quarter is r/4, and the number of time periods becomes 4n.
- Interest is applied 4 times per year, so the rate per quarter is r/4, and the number of time periods becomes 4n.
Real-Life Applications of Compound Interest
-
Bank Savings & Fixed Deposits – Your savings grow exponentially over time.
-
Loans & Mortgages – Interest on loans accumulates faster due to compounding.
-
Investments (Stocks, Mutual Funds, Retirement Plans) – Long-term investments benefit significantly from compound growth.
-
Inflation & Depreciation – Inflation compounds over the years, affecting real purchasing power.
-
Business Growth & Financial Planning – Helps in projections and decision-making.
Why is Compound Interest Powerful?
-
The earlier you invest, the more you benefit from compounding.
-
Small contributions can grow into significant amounts over time.
-
It is the key to long-term wealth creation.
One more interesting term related to deprivation
The compound interest formula is typically used to calculate growth, but in the case of deprivation or decreasing values (such as depreciation, depletion, or decay), a similar formula is used with a negative rate.
Formula for Decreasing Compound Interest (Depreciation/Decay)
