Compound Interest Formula: Complete Guide with Every Variation

Formula 1: Basic Compound Interest

The standard compound interest formula calculates the future value of a lump-sum investment that compounds at regular intervals. This formula is the standard compound interest equation used in financial mathematics, as described by Investopedia. For a comprehensive overview of how compound interest works beyond just the formula, see our complete compound interest guide.

Variable Definitions

Worked Example: $10,000 at 6% for 10 Years (Monthly Compounding)

Your $10,000 grows to $18,193.97. The total interest earned is $18,193.97 - $10,000 = $8,193.97.

How the Formula Derives from Simple Interest

Simple interest uses the formula A = P(1 + rt), which adds interest linearly. Compound interest modifies this by dividing the rate into n periods (r/n), then raising to the total number of periods (nt). This creates exponential growth because each period's interest is calculated on the new, larger balance.

For a single compounding period (n=1, t=1), compound interest equals simple interest: A = P(1 + r). The difference emerges when n > 1 or t > 1, because accumulated interest starts earning its own interest. For a deeper comparison, see our simple vs. compound interest guide.

Understanding Each Variable in Depth

Mastering the compound interest formula requires a thorough understanding of what each variable represents and how it affects your calculations. As explained by Khan Academy's finance tutorials, each component plays a distinct role in determining your investment's growth trajectory.

Principal (P): Your Starting Point

The principal represents your initial investment before any interest accrues. This is the foundation upon which all compound growth builds. Whether it is $1,000 in a savings account or $100,000 in a retirement fund, the principal establishes the base amount that will multiply over time. The larger your principal, the more absolute dollars you earn in interest, though the percentage return remains the same. Many investors focus on building a larger principal through strategies like lump-sum investing from bonuses, inheritance, or tax refunds.

Interest Rate (r): The Growth Multiplier

The interest rate must always be expressed as a decimal in the formula. Convert percentages by dividing by 100: a 7.5% rate becomes 0.075. This rate represents the annual percentage rate (APR) before accounting for compounding frequency. The rate has a dramatic exponential effect on long-term growth—even a 1% difference can translate to tens of thousands of dollars over a 30-year period. Historical stock market returns average around 10% nominally, while high-yield savings accounts currently offer 4-5% APY.

Compounding Frequency (n): The Acceleration Factor

The variable n determines how often interest gets added to your principal within a year. Common compounding frequencies include:

More frequent compounding means interest begins earning its own interest sooner, accelerating growth. However, the difference between daily and monthly compounding is minimal compared to the difference between annual and monthly compounding.

Time (t): The Most Powerful Variable

Time in years is the exponent's multiplier, making it the most powerful factor in the formula. Because compound interest is exponential, doubling your time horizon does more than double your returns. A $10,000 investment at 7% grows to $19,672 in 10 years, $38,697 in 20 years, and $76,123 in 30 years. This mathematical reality explains why financial advisors consistently emphasize starting early—time cannot be bought, borrowed, or recovered once it has passed.

Formula 2: Compound Interest with Regular Contributions

Most investors don't just make a one-time deposit. The formula expands to include periodic monthly contributions using the future value of an annuity. Understanding this variation is essential for anyone building a long-term savings or investment plan:

Where PMT is the contribution per period. If you contribute monthly and compounding is monthly, PMT is your monthly deposit amount.

For Beginning-of-Period Contributions (Annuity Due)

The only difference is the extra (1 + r/n) multiplier, which accounts for each contribution earning one additional period of interest.

Worked Example: $5,000 Initial + $300/Month at 7% for 25 Years

Total contributed: $5,000 + ($300 × 300) = $95,000. Interest earned: $271,657.11 - $95,000 = $176,657.11. That means 65% of the final balance came from compound interest alone.

Multiple Worked Examples: Different Scenarios

Understanding the compound interest formula becomes clearer when you see it applied to various real-world situations. The following examples demonstrate how different combinations of principal, rate, time, and contributions affect outcomes. For more detailed scenarios, explore our compound interest examples guide.

Example 1: Emergency Fund in a High-Yield Savings Account

Scenario: You deposit $15,000 into a high-yield savings account offering 4.5% APY, compounded daily. How much will you have after 3 years?

Your emergency fund earns $2,164.05 in interest while remaining fully liquid and FDIC insured.

Example 2: Retirement Savings with Employer Match

Scenario: You contribute $500/month to your 401(k), your employer matches 50% ($250/month), and you expect 8% annual returns over 35 years. Starting balance: $0.

Total contributed: $750 × 420 = $315,000. Interest earned: $1,368,742.50. The employer match and compound interest transform modest monthly savings into a substantial retirement fund.

Example 3: College Savings (529 Plan)

Scenario: You start a 529 plan for your newborn with $2,500 and add $200/month. Assuming 6% returns, what will be available when they turn 18?

Total invested: $2,500 + ($200 × 216) = $45,700. Growth from compound interest: $38,744.25.

Formula 3: Continuous Compounding

Continuous compounding represents the mathematical limit as the number of compounding periods approaches infinity. It uses Euler's number (e ≈ 2.71828):

This formula is derived by taking the limit of (1 + r/n)^(nt) as n approaches infinity, which equals e^(rt).

Worked Example: $10,000 at 5% for 10 Years (Continuous)

Compare this to daily compounding ($16,486.65) — the difference is only $0.56 over 10 years. Continuous compounding is mainly used in theoretical finance and options pricing (the Black-Scholes model), not in everyday banking.

The Mathematics Behind Continuous Compounding

The continuous compounding formula emerges from a fundamental limit in calculus. As compounding frequency increases toward infinity, the expression (1 + r/n)^n approaches e^r, where e is Euler's number (approximately 2.71828). This mathematical constant appears throughout nature and mathematics, from population growth models to radioactive decay.

The derivation involves the limit definition: e = lim(n→∞)(1 + 1/n)^n. Substituting r/n for the rate per period and taking the limit yields the elegant formula A = Pe^(rt). While no real-world financial institution compounds continuously, this formula serves as the theoretical upper bound for compound growth and is essential in:

• Options pricing: The Black-Scholes model uses continuous compounding for theoretical option values
• Bond mathematics: Zero-coupon bond pricing often uses continuous compounding
• Academic finance: Many theoretical models assume continuous compounding for mathematical convenience
• Actuarial science: Insurance calculations sometimes use continuous models

Continuous vs. Discrete Compounding Comparison

The table demonstrates diminishing returns as compounding frequency increases. Moving from annual to monthly adds $285.49, but moving from daily to continuous adds only $0.75.

Formula 4: Future Value of Annuity (Monthly Contributions Only)

When you want to calculate the growth of regular contributions without an initial principal, you use the future value of an ordinary annuity formula. This is the second component of the full contribution formula, isolated for clarity:

This formula calculates how much your regular contributions will grow to, assuming each payment is made at the end of each period. The SEC's investor education resources emphasize the importance of understanding this formula for retirement planning.

Understanding the Annuity Factor

The bracketed portion [((1 + r/n)^(nt) - 1) / (r/n)] is called the "future value annuity factor" or "annuity accumulation factor." It represents the total multiplier applied to each payment after accounting for all the compounding periods. Financial calculators and spreadsheets often have this built in as the FV function.

Worked Example: $500/Month for 30 Years at 7%

Total contributed: $500 × 360 = $180,000. Interest earned: $429,985.00. Your contributions more than tripled due to compound interest.

Annuity Due vs. Ordinary Annuity

The formula above assumes payments at the end of each period (ordinary annuity). If you make payments at the beginning of each period (annuity due), multiply the result by (1 + r/n):

For the same $500/month example above, an annuity due yields: $609,985.00 × 1.005833 = $613,543.67. The extra $3,558.67 comes from each payment having one additional month to compound.

Formula 5: CAGR (Compound Annual Growth Rate)

The CAGR formula works in reverse: given a starting value, ending value, and time period, it tells you the annualized growth rate:

Worked Example: Investment Grew from $10,000 to $25,000 in 8 Years

CAGR smooths out year-to-year volatility and gives you a single annualized rate. CAGR is widely used by investment professionals and is featured in SEC educational materials as a standardized measure of investment performance. It is the standard way to compare investment performance across different time periods and asset classes. Note that CAGR differs from average annual return when returns are volatile.

Formula 6: APY vs APR Formulas and Conversions

Understanding the difference between APY (Annual Percentage Yield) and APR (Annual Percentage Rate) is crucial for comparing financial products accurately. For a comprehensive exploration, see our APY vs APR explained guide.

Where r is the APR (stated annual rate) and n is the compounding frequency.

Worked Example: 5% APR Compounded Daily

A 5.00% APR with daily compounding yields 5.127% APY. The higher the base rate and the more frequent the compounding, the larger the gap between APR and APY. Banks are required by the Truth in Savings Act (Regulation DD) to disclose APY to consumers. Always compare APY when evaluating savings accounts or CDs.

When to Use Each Metric

• APY for savings: Always compare savings accounts, CDs, and money market accounts using APY
• APR for loans: Loan costs are typically expressed as APR, though the true cost depends on compounding
• Credit cards: Compare using APR, but remember daily compounding makes effective rate higher

APR to APY Conversion Table

Formula 7: Rule of 72

The Rule of 72 is a quick estimation formula for determining how long it takes an investment to double:

Why 72?

The exact formula for doubling time is ln(2)/ln(1+r), which equals approximately 0.6931/r for small r. Multiplying both numerator and denominator by 100 gives 69.31/rate. The number 72 is used instead because it is divisible by more numbers (2, 3, 4, 6, 8, 9, 12) making mental math easier, and it provides slightly better accuracy for rates in the 6-10% range that most investors encounter.

Solving for Different Variables

The basic formula can be rearranged to solve for any unknown variable:

Solving for Principal (How Much to Invest Now)

Example: To have $50,000 in 15 years at 6% compounded monthly: P = 50,000 / (1.005)^180 = 50,000 / 2.45409 = $20,374.16

Solving for Rate (What Return Do I Need?)

Example: To grow $10,000 to $30,000 in 12 years with monthly compounding: r = 12[(30,000/10,000)^(1/144) - 1] = 12[3^(0.00694) - 1] = 12 × 0.00766 = 9.19%

Solving for Time (How Long Will It Take?)

Example: How long for $15,000 to reach $40,000 at 7% compounded monthly: t = ln(40,000/15,000) / (12 × ln(1.005833)) = ln(2.6667) / (12 × 0.005816) = 0.9808 / 0.06979 = 14.05 years

Complete Formula Reference Table

This comprehensive table summarizes every compound interest formula covered in this guide, along with when to use each one:

Using These Formulas in Spreadsheets

You can implement all compound interest formulas in Excel or Google Sheets:

The built-in FV() function is the most convenient for calculations with contributions. Note that Excel uses negative signs for cash outflows (deposits), so you prefix P and PMT with a minus sign.

How Each Variable Impacts Your Results

Understanding the relative importance of each variable helps you focus your financial strategy on the factors that matter most.

The Sensitivity of Time (t)

Time is the most powerful variable in the compound interest formula. Doubling your investment period does far more than doubling your interest rate. Here is why:

Doubling the time period from 20 to 40 years produces almost the same result as doubling the interest rate. Both outperform simply doubling the starting principal. This mathematical reality is why financial advisors emphasize starting early above all else.

The Rate-Frequency Interaction

Higher interest rates magnify the benefit of more frequent compounding. At 2% annually, switching from annual to daily compounding adds just $2 per year on $10,000. At 12%, the same switch adds $63 per year. This interaction matters for high-yield environments: when rates are elevated, the compounding frequency on your savings accounts and CDs becomes a more significant factor.

When Contributions Dominate

For most working professionals, the contribution term (PMT) eventually dominates the formula. If you start with $5,000 and add $500 per month at 7% for 30 years, your final balance is approximately $612,438. Of that, only $30,024 came from compounding on the original $5,000. The remaining $582,414 came from your contributions and the compound growth on those contributions. This demonstrates why consistently increasing your savings rate matters more than chasing higher returns once you have built the habit.

Common Mistakes When Using the Formula

1. Forgetting to Convert the Rate to a Decimal

The formula uses r as a decimal: 7% = 0.07, not 7. Using 7 instead of 0.07 will give you absurdly large results.

2. Mismatching Compounding Frequency and Contribution Frequency

If contributions are monthly but compounding is daily, you need to adjust the contribution formula. The simplest approach is to use monthly compounding (n=12) when calculating monthly contributions, which gives a very close approximation.

3. Using the Wrong n Value

Common n values: daily = 365, monthly = 12, quarterly = 4, semi-annually = 2, annually = 1. Using the wrong n changes your result significantly over long periods.

4. Ignoring Inflation

The formula gives nominal (not inflation-adjusted) results. To estimate real returns, subtract the expected inflation rate from your interest rate before calculating, or apply the formula separately for inflation adjustment.

5. Assuming Constant Returns

The formula assumes a fixed interest rate. Real investments like stocks have variable returns. The formula gives you what a constant rate would produce, which is useful for planning but not a guarantee.

6. Confusing Total Return with Interest Rate

When using the formula for stock market projections, people often use the total return figure (e.g., 10% for the S&P 500) without accounting for inflation. The formula works with whatever rate you input, so if you want inflation-adjusted projections, use a real return rate (typically 6-7% for equities). Using nominal returns produces nominal future values that overstate purchasing power.

7. Forgetting About Taxes

The compound interest formula calculates gross returns. In taxable accounts, interest and investment gains are subject to federal and state taxes. A 5% return in a taxable account for someone in the 24% federal bracket yields an after-tax return of approximately 3.8%. For long-term planning, it's more realistic to apply an after-tax rate in the formula, unless you're modeling tax-advantaged accounts like 401(k)s or Roth IRAs where compounding occurs on the full pre-tax or tax-free amount.

Choosing the Right Formula for Your Situation

With six formulas available, selecting the right one depends on your specific scenario:

• One-time investment, fixed rate: Use the basic formula A = P(1 + r/n)^(nt). This applies to CDs, savings accounts, or modeling a lump-sum investment.
• Regular deposits over time: Use the contribution formula with PMT. This fits 401(k) contributions, monthly savings plans, or any scenario with recurring deposits.
• Theoretical or academic analysis: Use continuous compounding A = Pe^(rt). This is standard in options pricing, mathematical modeling, and physics applications.
• Evaluating past performance: Use CAGR = (FV/PV)^(1/t) - 1. This works for comparing investments, analyzing portfolio returns, or benchmarking against an index.
• Comparing financial products: Use APY = (1 + r/n)^n - 1. This normalizes different compounding frequencies into a single comparable number.
• Quick mental estimates: Use the Rule of 72. This requires no calculator and gives surprisingly accurate results for rates between 5% and 12%.

In practice, most personal finance calculations only require the first two formulas. The basic formula handles existing savings, and the contribution formula handles ongoing investment plans. Our compound interest calculator implements all six and lets you switch between them with a single click.

Frequently Asked Questions