The Rule of 72 Explained: Quick Mental Math for Investors

What Is the Rule of 72?

The Rule of 72 is a simple formula for estimating how long it takes for an investment to double in value at a fixed annual rate of return. The formula is:

Years to Double = 72 ÷ Annual Interest Rate

For example, if your investment earns 6% per year, it will take approximately 72 ÷ 6 = 12 years to double. If it earns 10%, doubling takes about 72 ÷ 10 = 7.2 years. The calculation works with any interest rate and requires nothing more than basic division you can do in your head.

The rule also works in reverse. If you want to double your money in 8 years, you need a return of 72 ÷ 8 = 9% per year. This reverse application is particularly useful when setting investment goals or evaluating whether a financial product can realistically meet your objectives.

As the Investopedia overview of the Rule of 72 notes, this shortcut has been used by investors for centuries and remains one of the most practical mental math tools in personal finance. For precise calculations, you can always use our doubling time calculator or the main compound interest calculator.

Why the Rule of 72 Works: The Mathematics Behind It

The Rule of 72 is not arbitrary — it is a carefully chosen approximation grounded in logarithmic mathematics. Understanding why 72 works so well will deepen your appreciation for this elegant shortcut and help you know when to trust it.

The exact formula for calculating how long it takes an investment to double comes from the compound interest equation. If you invest $P at an annual rate r (expressed as a decimal), the future value after t years is P(1+r)^t. To find when this equals 2P (double your money), you solve:

2 = (1 + r)^t

Taking the natural logarithm of both sides gives: t = ln(2) / ln(1+r). Since ln(2) is approximately 0.693, and for small values of r, ln(1+r) is approximately equal to r, the formula simplifies to roughly 0.693/r. Multiply by 100 to work with percentages instead of decimals, and you get 69.3/r%.

So why do we use 72 instead of 69.3? Three reasons make 72 superior in practice:

• Divisibility: The number 72 has an exceptionally large number of factors — it divides evenly by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36. This makes mental math much easier since common interest rates like 6%, 8%, 9%, and 12% divide cleanly into 72.
• Discrete compounding adjustment: The 69.3 derivation assumes continuous compounding. Real investments typically compound annually, quarterly, or monthly. The upward adjustment to 72 compensates for this discrete compounding and actually improves accuracy for rates between 5% and 12%.
• Sweet spot accuracy: At around 8% — roughly the historical average return of the stock market — the Rule of 72 is nearly perfect. This is not coincidental; it is where the approximation was optimized to perform best.

For a complete derivation of the underlying mathematics, including how compounding frequency affects results, see our compound interest formula guide.

How to Use the Rule of 72: Quick Mental Math Examples

The true power of the Rule of 72 lies in its speed. Here is how to use it for instant financial insights in everyday situations, no calculator required.

Basic Doubling Time Calculations

Simply divide 72 by the annual interest rate to get the approximate years to double:

• High-yield savings account at 5%: 72 ÷ 5 = 14.4 years to double
• Stock market average at 10%: 72 ÷ 10 = 7.2 years to double
• Bond returns at 4%: 72 ÷ 4 = 18 years to double
• Aggressive growth fund at 12%: 72 ÷ 12 = 6 years to double

Finding the Required Rate of Return

If you have a timeline goal, divide 72 by the number of years to find the rate you need:

• Double your money in 5 years: 72 ÷ 5 = 14.4% annual return required
• Double your money in 10 years: 72 ÷ 10 = 7.2% annual return required
• Double your money in 15 years: 72 ÷ 15 = 4.8% annual return required

Comparing Investment Options Instantly

When evaluating two investments, the Rule of 72 immediately reveals which grows faster:

Investment A offers 6% returns (72 ÷ 6 = 12-year doubling), while Investment B offers 9% (72 ÷ 9 = 8-year doubling). Over a 24-year period, Investment A doubles twice (4x original), while Investment B doubles three times (8x original). That 3% difference means Investment B produces twice the final wealth.

Understanding Inflation's Impact

The rule works for any compound growth — including the erosion of purchasing power:

• At 3% inflation: 72 ÷ 3 = 24 years for prices to double (your money loses half its purchasing power)
• At 6% inflation: 72 ÷ 6 = 12 years for prices to double

This is why keeping money in a 0% checking account during inflationary periods is so costly — you are guaranteed to lose purchasing power over time. The SEC’s guide to savings and investing emphasizes the importance of earning returns that outpace inflation.

Rule of 72 vs Rule of 70 vs Rule of 69: Which Should You Use?

The Rule of 72 is the most popular, but two alternative versions exist. Each has its optimal use case, and understanding the differences will help you choose the right tool for any situation.

The Rule of 69.3 (Exact for Continuous Compounding)

Since ln(2) = 0.693, dividing 69.3 by the interest rate gives the mathematically exact doubling time for continuous compounding. This version is preferred in academic settings, derivative pricing, and any context where continuous compounding is assumed. However, 69.3 is awkward for mental math and less accurate for real-world investments that compound discretely.

The Rule of 70 (Best for Low Rates)

The Rule of 70 is a compromise — easier to compute than 69.3 and more accurate than 72 for interest rates below 5%. Economists often prefer the Rule of 70 when discussing GDP growth, inflation, or population growth rates, which typically fall in the 1-4% range.

The Rule of 72 (Best for Investment Rates)

The Rule of 72 excels in the 5-12% range where most investment returns fall. Its exceptional divisibility (72 = 2^3 x 3^2) makes it ideal for mental math, and its slight upward adjustment compensates well for annual and monthly compounding.

Bottom line: Use the Rule of 70 for rates under 5%, and the Rule of 72 for rates between 5% and 15%. For anything more precise, use our doubling time calculator or the compound interest calculator.

Examples at Various Interest Rates

Let us walk through detailed examples across the spectrum of common interest rates, comparing the Rule of 72 estimate to the mathematically exact answer.

Low Interest Rate Example: 2% Savings Account

A traditional savings account earning 2% APY:

• Rule of 72 estimate: 72 ÷ 2 = 36 years
• Exact calculation: ln(2) / ln(1.02) = 35.00 years
• Error: +1 year (2.9% overestimate)

The Rule of 72 overestimates slightly at low rates. For greater precision at 2%, the Rule of 70 gives 35 years — exactly matching the actual result.

Moderate Rate Example: 6% Bond Fund

A diversified bond fund averaging 6% annually:

• Rule of 72 estimate: 72 ÷ 6 = 12 years
• Exact calculation: ln(2) / ln(1.06) = 11.90 years
• Error: +0.1 years (0.8% overestimate)

At 6%, the rule is highly accurate — off by just about 5 weeks over an 12-year period.

Investment Sweet Spot: 8% Returns

The historical average return of balanced portfolios:

• Rule of 72 estimate: 72 ÷ 8 = 9 years
• Exact calculation: ln(2) / ln(1.08) = 9.01 years
• Error: -0.01 years (virtually perfect)

At 8%, the Rule of 72 is essentially exact. This is not coincidental — the rule was optimized for this range.

Stock Market Average: 10% Returns

The long-term average return of the S&P 500:

• Rule of 72 estimate: 72 ÷ 10 = 7.2 years
• Exact calculation: ln(2) / ln(1.10) = 7.27 years
• Error: -0.07 years (1% underestimate)

Still highly accurate — the error is less than one month over 7+ years.

High Growth Example: 15% Returns

An aggressive growth stock or emerging market fund:

• Rule of 72 estimate: 72 ÷ 15 = 4.8 years
• Exact calculation: ln(2) / ln(1.15) = 4.96 years
• Error: -0.16 years (3.2% underestimate)

At higher rates, the Rule of 72 increasingly underestimates doubling time. For rates above 15%, consider using 72 + (rate - 8)/3 as an adjusted formula, or simply use a calculator.

The Mathematical Origin: Why 72?

The Rule of 72 is a mathematical approximation derived from the natural logarithm. The exact formula for the time it takes to double an investment at a continuously compounded rate r is:

Exact Doubling Time = ln(2) ÷ ln(1 + r)

Since ln(2) ≈ 0.6931, the exact doubling time at a rate r (expressed as a decimal) is approximately 0.6931 ÷ r. Multiply both sides by 100 to express r as a percentage, and you get 69.31 ÷ r%. So why do we use 72 instead of 69.31?

The answer lies in two factors. First, 72 is a highly divisible number — it divides evenly by 1, 2, 3, 4, 6, 8, 9, and 12, making mental math much easier. Second, the exact formula assumes continuous compounding, but most real-world investments compound annually, monthly, or daily. The adjustment from 69.31 to 72 compensates for this discrete compounding and actually improves accuracy for interest rates in the 5–12% range, which is exactly where most investment returns fall.

For a deeper dive into the underlying mathematics, our compound interest formula guide explains the complete derivation and how different compounding frequencies affect the result.

Accuracy of the Rule of 72

How close does the Rule of 72 come to the exact answer? The table below compares the Rule of 72 estimate with the precise doubling time (calculated using the exact formula with annual compounding) across a range of interest rates. Data for historical rates can be referenced through the Federal Reserve FRED database.

As you can see, the Rule of 72 is remarkably accurate between 5% and 12%, with errors of just a few weeks at most. At 8%, the rule is essentially perfect. At very low rates (1–2%) or very high rates (15%+), the estimates drift slightly, but they remain useful for quick mental approximations.

Quick Reference Table: Common Interest Rates

Bookmark this table for instant reference. It covers the most common interest rates you will encounter in savings accounts, bonds, and equity investments.

Limitations: When the Rule of 72 Is Less Accurate

While the Rule of 72 is an incredibly useful tool, it has limitations that every investor should understand. Knowing when the rule breaks down will help you decide when to reach for a calculator instead.

Very Low Interest Rates (Below 3%)

At rates under 3%, the Rule of 72 consistently overestimates doubling time. For example, at 1%, it predicts 72 years, but the actual time is 69.66 years — an error of over 2 years. For precision at low rates, use the Rule of 70 instead, or better yet, our exact doubling time calculator.

Very High Interest Rates (Above 15%)

At rates above 15%, the Rule of 72 increasingly underestimates doubling time. At 25%, it predicts 2.88 years, but the actual time is 3.11 years — nearly 3 months off. For high-rate scenarios (like credit card debt calculations), consider using an adjusted formula: use (72 + rate/3) instead of just 72.

Variable Interest Rates

The Rule of 72 assumes a constant rate of return. Real investments fluctuate year to year. A portfolio that averages 8% might have years of +25% and -10%. While the rule still provides a reasonable long-term estimate for the average rate, short-term results will vary significantly.

Compounding Frequency

The Rule of 72 is calibrated for annual compounding. More frequent compounding (monthly, daily) slightly reduces actual doubling time. The difference is typically small — about 1-2% faster with monthly versus annual compounding — but for maximum precision, account for compounding frequency using our compound interest calculator.

Taxes and Fees

The rule does not account for taxes or fees. If your investment earns 10% but you pay 2% in fees and 1.5% in taxes, your effective rate is only 6.5%. Always apply the Rule of 72 to your after-tax, after-fee rate for realistic estimates. The SEC’s compound interest guidance emphasizes accounting for all costs when projecting returns.

Inflation Adjustment

The rule calculates nominal doubling time, not real (inflation-adjusted) doubling time. To find how long until your purchasing power doubles, subtract the inflation rate from your return before applying the rule. At 10% returns with 3% inflation, your real return is 7%, so purchasing power doubles in about 72 ÷ 7 = 10.3 years rather than 7.2 years.

The Rule of 69.3 and Rule of 70: Alternatives

While the Rule of 72 is the most popular, two alternative versions exist for different situations:

The Rule of 69.3 uses the exact value of 100 × ln(2) = 69.3 in place of 72. This version is mathematically precise for continuous compounding, which makes it ideal for academic settings or when working with continuously compounded rates (common in options pricing and some fixed-income calculations). However, it is harder to divide mentally and less accurate for annual or monthly compounding.

The Rule of 70 splits the difference. It is easier to divide mentally than 69.3 and more accurate than 72 at low interest rates (1–4%). Some economists prefer the Rule of 70 when discussing GDP growth or inflation rates, which typically fall in the 1–4% range.

In practice, the Rule of 72 remains the best all-around choice for typical investment scenarios because it is both easy to compute and highly accurate in the 6–10% return range where most equity investors operate.

Practical Applications Beyond Investing

The Rule of 72 is not limited to investment returns. Anywhere a quantity grows (or shrinks) at a compounded percentage rate, the rule applies. Here are several practical applications as discussed in Investopedia’s time value of money guide:

Inflation erosion: At 3% annual inflation, the purchasing power of your money is cut in half in about 72 ÷ 3 = 24 years. This means $100,000 in today’s dollars will only buy $50,000 worth of goods in 2050. Understanding this helps you set savings targets that account for the real cost of future expenses.

GDP doubling: If a country’s economy grows at 3.5% per year, its GDP doubles in roughly 72 ÷ 3.5 = 20.6 years. At 7% growth (like China during its rapid development phase), GDP doubles in about 10 years. Economists frequently cite the Rule of 72 when discussing long-term economic projections.

Debt doubling: Credit card debt at 24% APR doubles in just 72 ÷ 24 = 3 years if you make no payments. This is why high-interest debt is so dangerous — compounding works against you with the same mathematical force it works for you in savings.

Population growth: At a 1.1% annual growth rate (approximately the current global average), the world population would double in about 72 ÷ 1.1 = 65 years. Demographers and policy planners regularly use this mental shortcut.

Our guide on compound annual growth rate (CAGR) explores how these growth rates translate into long-term trends across different asset classes and economic indicators.

Using the Rule of 72 for Different Investments

Here is how the Rule of 72 applies to common investment vehicles, helping you quickly compare their growth potential. Current rate data is available from the SEC’s investor education portal:

• High-yield savings account (5.0% APY): 72 ÷ 5 = 14.4 years to double. Safe, but slow growth.
• Bond index fund (4–5% average): 72 ÷ 4.5 = 16 years to double. Moderate growth with lower volatility.
• Stock market index fund (10% historical average): 72 ÷ 10 = 7.2 years to double. Higher volatility, but the fastest compounder for long-term investors.
• Real estate (7–8% average with leverage): 72 ÷ 7.5 = 9.6 years to double. Returns vary significantly by market and property type.
• Inflation-adjusted stock returns (7%): 72 ÷ 7 = 10.3 years for your purchasing power to double. This is the number that matters most for retirement planning.

By quickly computing doubling times, you can immediately see why long-term investors heavily favor stocks: your money doubles roughly every 7 years in the stock market versus every 14 years in a savings account. Over a 40-year career, that is the difference between roughly 5–6 doublings (32–64× your money) versus 2–3 doublings (4–8× your money). Use our comprehensive compound interest guide to learn more about how different rates and timeframes affect your wealth.