Compound Interest

Chapter

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INTRODUCTION TO COMPOUND INTEREST

Compound interest is a method where interest is added not only to the initial principal but also to the accumulated interest from previous periods. This process, also known as capitalization of interest, means that in each compounding period, the new basis for calculating interest becomes the sum of the principal and the previously earned interest. Because of this, the value of an investment does not increase linearly but exponentially.

BASIC FORMULA

Let:

P be the initial principal (the initial amount of money),
r be the annual interest rate (in decimal form),
n be the number of times that interest is compounded per year,
t be the time the money is invested or borrowed for, in years,
A be the future value of the investment/loan after 't' years,

then the formula is:

**A = P * (1 + r/n)^(n*t)**

If interest is compounded once per year (annually), then n = 1, and the formula simplifies to:

**A = P * (1 + r)^t**

CALCULATION OF INTEREST EARNED

The amount of interest (I) that the investment has earned is:

I = A – P

CALCULATION EXAMPLE (ANNUAL COMPOUNDING)

Let's invest €1,000 at a 5% annual interest rate, with annual compounding, for a period of 3 years:

P = 1000
r = 0.05 (5%)
n = 1 (compounded annually)
t = 3 years

A = 1000 * (1 + 0.05/1)^(1*3) = 1000 * (1.05)³ = 1000 * 1.157625 = €1157.63 I = 1157.63 – 1000 = €157.63

EXAMPLE WITH MORE FREQUENT COMPOUNDING (QUARTERLY)

If interest is compounded quarterly (n = 4):

P = €1,000
r = 0.05
n = 4
t = 3 years

A = 1000 * (1 + 0.05/4)^(4*3) = 1000 * (1 + 0.0125)¹² = 1000 * (1.0125)¹² (1.0125)¹² ≈ 1.1607545 A ≈ 1000 * 1.1607545 ≈ €1160.75

With a higher frequency of compounding, the final amount is also slightly higher.

FINAL NOTE ON GROWTH

Due to the compounding effect of interest capitalization, the value of an investment increases exponentially over time. This means that a longer period and more frequent compounding significantly increase the total return. This very principle of compound interest is the foundation of long-term saving and investing.

CONCLUSION

Compound interest allows interest itself to earn interest, leading to accelerated growth of an investment. This principle is key in financial mathematics, as we encounter it in savings, loans, and long-term investments.

Simple and Compound Interest

Explanation