Compound Interest & Financial Math: Formula, Rule of 72 & Growth

Table of Contents

1. The Discrete Compound Interest Formula
2. The Rule of 72 Doubling Shortcut
3. Continuous Compounding A = Pe^(rt)
4. Step-by-Step Worked Financial Examples
5. Accounting for Inflation & Real Purchasing Power

Albert Einstein reportedly called compound interest the "eighth wonder of the world." Unlike simple interest which earns returns strictly on the original principal, compound interest earns returns on both principal and previously accumulated interest over time.

1. The Discrete Compound Interest Formula

• Formula: A = P(1 + r/n)ⁿᵗ • A = Future Value ($) • P = Initial Principal ($) • r = Annual Interest Rate (decimal, e.g. 0.07 for 7%) • n = Compounding periods per year (12 for monthly, 365 for daily) • t = Time duration in years

2. The Rule of 72 Doubling Shortcut

Estimate how many years it takes for an investment to double in value by dividing 72 by the annual interest rate $r$:

• Doubling Time (Years) ≈ 72 / Interest Rate (%) • Example: At 8% annual return, money doubles in approx 72 / 8 = 9 years!

3. Continuous Compounding A = Pe^(rt)

• Continuous Formula: A = P · eʳᵗ (using Euler's constant e ≈ 2.71828)

4. Step-by-Step Worked Financial Examples

Worked Example 1: $10,000 at 8% Compounded Monthly

Question: Calculate the future value of $10,000 invested at 8% annual interest compounded monthly for 10 years.

Solution: $P = 10000$, $r = 0.08$, $n = 12$, $t = 10$.

$$A = 10000 left(1 + rac{0.08}{12} ight)^{12 imes 10} = 10000(1.006667)^{120} = $22,196.40$$

Total interest earned: $12,196.40!

Worked Example 2: Monthly Recurring Investments (DCA)

Question: Investing $500/month at 7% annual interest for 30 years.

Solution Formula: $FV = PMT imes rac{(1 + r/n)^{nt} - 1}{r/n}$.

Result: Total contributed = $180,000. Future Value = $609,985.50! Compound growth accounts for over $429,000 of the total!

5. Accounting for Inflation & Real Purchasing Power

To compute the real inflation-adjusted return rate $r_{real}$, use Fisher's equation:

• Real Return Rate: r_real ≈ Nominal Rate - Inflation Rate • Exact Fisher Equation: (1 + r_real) = (1 + r_nominal) / (1 + i_inflation)

6. The Power of Dollar-Cost Averaging & Tax-Deferred Growth

Investing a fixed dollar amount at regular intervals (such as $500 monthly into an index fund) leverages compound growth while smoothing out market volatility.

• Annuity Future Value: FV = PMT × [ (1 + r/n)ⁿᵗ - 1 ] / (r/n) • Example: $500/month at 8% annual return for 35 years: Total Contributions: $210,000 | Future Portfolio Value: $1,146,940!

7. Annual Percentage Yield (APY) vs. APR

Annual Percentage Rate (APR) does not account for compounding within the year. Annual Percentage Yield (APY) reflects the true annual return: APY = (1 + r/n)ⁿ - 1.

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Frequently Asked Questions (FAQs)

What is simple interest vs compound interest?

Simple interest pays interest only on initial principal. Compound interest pays interest on principal PLUS previously accumulated interest.

How does compounding frequency affect total return?

More frequent compounding (daily vs monthly vs annually) results in slightly higher total yields due to faster interest reinvestment.