The Math Behind Retirement: 4% Rules, Compounding, and Sequence of Returns

Two retirees start with $1,000,000. Both withdraw $50,000 a year. Both get exactly the same ten annual returns, just in opposite order. After ten years, one has about $855,000 left. The other has about $653,000.

Same returns, same withdrawals, a $200,000 difference. While you're saving, the order of returns doesn't matter. Once you start withdrawing, it matters a lot. That's the core math of retirement.

Order Matters Once You Withdraw

With a single lump sum and no withdrawals, multiplication is commutative:

1,000,000 × 0.80 × 0.90 × 1.15 = 1,000,000 × 1.15 × 0.90 × 0.80

The order doesn't matter. But withdrawals break the pure product. Each year becomes:

balance = (balance − withdrawal) × (1 + return)

Now a loss early on hits a large balance and is locked in by withdrawals before the recovery arrives. Here's the example in full, using the returns −20%, −10%, +15%, +10%, then six years of +7%:

This is sequence-of-returns risk. The first five to ten years of retirement matter disproportionately.

Arithmetic vs. Geometric Returns

Those ten returns average 3.7% arithmetically. The compound (geometric) return is only 3.17% a year.

Why the gap? Losses and gains aren't symmetric. A 50% loss followed by a 50% gain leaves you at:

1.50 × 0.50 = 0.75 → a 25% loss

The geometric mean is always less than or equal to the arithmetic mean, and the gap grows with volatility. A useful approximation:

Geometric ≈ Arithmetic − σ² / 2

Where σ is the standard deviation of returns. Volatility is a drag on compound growth, even when the average looks fine. The statistics formulas page covers both means and standard deviation.

The 4% Rule: Where It Came From

In October 1994, financial planner William Bengen published "Determining Withdrawal Rates Using Historical Data" in the Journal of Financial Planning. He tested every rolling 30-year retirement period in U.S. market history since 1926, using a stock/bond portfolio.

His finding: a retiree who withdrew 4% of the starting balance in year one, then raised the dollar amount with inflation every year, never ran out of money within 30 years. That included someone who retired just before the Great Depression.

In 1998, three professors at Trinity University (Philip Cooley, Carl Hubbard and Daniel Walz) published a similar analysis, now known as the Trinity Study. The "4% rule" became a planning shorthand.

The Math in One Line: The 25× Rule

Flip the 4% rule around and you get a savings target:

Nest egg needed = Annual spending / 0.04 = Annual spending × 25

Need $40,000 a year from your portfolio? You need about $1,000,000. Need $60,000? About $1.5 million. You can test these numbers with the 4% rule calculator.

Why 4% and Not 7%?

If stocks have historically returned around 7% after inflation, why not withdraw 7%? Three reasons, and all three are mathematical:

1. Sequence risk. A bad first decade can permanently shrink the base that later gains compound on.
2. Volatility drag. Geometric returns fall below average returns, as shown above.
3. Inflation adjustments. Your withdrawals grow each year while the portfolio might not.

The 4% figure isn't the expected safe amount. It's roughly the worst-case historical safe amount. In most historical periods, a 4% retiree ended 30 years with more money than they started with.

Compounding on the Way In

Retirement math has two phases. In the accumulation phase, the compound interest formula dominates:

FV = PMT × ((1 + r)^n − 1) / r

Saving $500 a month at 7% a year for 40 years grows to roughly $1.3 million, though you only contribute $240,000. Start 10 years later and the same monthly savings reaches only about $610,000. The last decade of compounding does more than the first three combined. We dig into this in The Mathematics of Compound Interest.

Two Concepts Worth Knowing

Monte Carlo Simulation

Modern planners run thousands of randomized return sequences and report a probability of success. It's a direct application of probability: instead of one historical path, you sample the distribution of possible futures.

Required Minimum Distributions

In the U.S., tax-deferred accounts eventually require withdrawals based on an IRS life expectancy divisor. Your balance divided by the divisor for your age gives the minimum. Try it with the RMD calculator.

Quick Answer: What Is the 4% Rule?

The 4% rule says you can withdraw 4% of your starting retirement portfolio in year one, then adjust that dollar amount for inflation each year, with a high historical chance of the money lasting 30 years. It comes from William Bengen's 1994 study of U.S. market history.

Try Them Yourself

• 4% Rule Calculator: test withdrawal rates against your balance
• Retirement Calculator: project savings to retirement age
• RMD Calculator: required minimum distributions by age
• Social Security Break-Even Calculator: when delaying benefits pays off
• Compound Interest Calculator: model the accumulation phase
• Statistics Formulas: means, variance and standard deviation

Run your own numbers twice: once assuming a strong first decade, once assuming a weak one. The gap is the best argument for planning conservatively.