Compound Interest

Compound interest is the “interest-on-interest” method of calculation and one of the most important concepts in financial investing. Understanding the mathematics of compound interest is understanding the application of geometric sequences.

What Is Compound Interest?

Simple interest: Interest is computed only on the principal; interest does not earn more interest. Compound interest: Interest is added to the principal and continues to earn interest, i.e., “interest-on-interest”.

Deriving the Compound Interest Formula

Let the principal be $P$, the annual interest rate be $r$, with annual compounding:

• End of year 1: $P(1+r)$
• End of year 2: $P(1+r)(1+r) = P(1+r)^2$
• End of year 3: $P(1+r)^3$
• End of year $n$: $P(1+r)^n$

This is a geometric sequence with first term $a_1 = P(1+r)$ and common ratio $q = 1+r$.

Compounding More Than Once a Year

If interest is compounded $m$ times per year, with an interest rate of $\frac{r}{m}$ each time, then after $n$ years:

$A_n = P\left(1 + \frac{r}{m}\right)^{mn}$

As $m \to \infty$ (continuous compounding):

$A_n = P \cdot e^{rn}$

Practical Applications

Example 1: Bank Deposit

Deposit 10000 yuan at an annual rate of 3% with annual compounding. What is the amount after 10 years?

Solution:

$A_{10} = 10000 \times (1 + 0.03)^{10} = 10000 \times 1.03^{10} \approx 13439 \text{ yuan}$

Example 2: Investment

Invest 50000 yuan with an annualized return of 8%. How many years until it doubles?

Solution:

Suppose it doubles after $n$ years. Then:

$50000 \times 1.08^n = 100000$ $1.08^n = 2$ $n = \frac{\ln 2}{\ln 1.08} \approx 9.01 \text{ years}$

Example 3: The Rule of 72

Rule of 72: Under compound interest, the number of years for the principal to double $\approx \frac{72}{\text{annual interest rate (\%)}}$

For example, with an annual rate of 8%, the doubling time $\approx \frac{72}{8} = 9$ years, very close to the precise value of 9.01 years!

The Power of Compound Interest

Suppose you invest 1000 yuan per month with an annualized return of 10%. What is the total after 30 years?

This is a geometric series problem:

$S = 1000 \times \frac{1.1^{360} - 1}{1.1^{1/12} - 1} \approx 2,260,000 \text{ yuan}$

Total invested: $1000 \times 360 = 360,000$ yuan Final amount: about 2.26 million yuan Profit: about 1.86 million yuan (more than 5 times the principal!)

Practice Problems

Exercise 1

Deposit 20000 yuan at an annual rate of 4% with annual compounding. What is the amount after 5 years?

Exercise 2

Invest 100000 yuan and hope to reach 200000 yuan in 10 years. What annualized return is needed?

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Summary

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