How Periodic Withdrawals Break the Commutative Property of Investment Returns

Retail investment platforms and accumulation-phase advisors frequently project future wealth by assuming a steady average return—often 7% or 8% annually. They assert that as long as the long-term average holds, a retirement portfolio will easily sustain a standard withdrawal rate.[2]

The historical evidence contradicts this assumption entirely once a portfolio transitions from accumulation to distribution. The moment periodic cash outflows begin, the mathematical rules governing compound growth fundamentally change. The order in which market returns arrive becomes the decisive factor in whether a portfolio lasts 30 years or runs dry in 15.[2]

This vulnerability is known as sequence of returns risk. It demonstrates that two retirees can experience the exact same average annual return over the exact same time horizon, yet arrive at terminal wealth figures separated by millions of dollars simply because their returns arrived in a different order.

Why the Math Changes in Retirement

During the accumulation phase, investment returns obey the commutative property of multiplication. This fundamental mathematical law states that the order of factors does not change the product, meaning that multiplying a sequence of numbers yields the same result regardless of how they are arranged.[1]

In a static portfolio with no cash flows, a 20% gain followed by a 10% loss yields the exact same final balance as a 10% loss followed by a 20% gain. The sequence is irrelevant. An investor saving for retirement can safely rely on average returns to project their future balance.[1][2]

But periodic withdrawals break this commutative property. When a retiree sells shares to generate cash during a market downturn, they permanently remove capital from the portfolio. Those liquidated shares are no longer available to participate in the eventual market recovery.

"Every withdrawal sells shares. Sell them after a crash and you have to sell more shares to raise the same dollars," notes Brad Roth of Thor Financial Technologies. "Those shares are gone before the recovery shows up."

The Cost of a Bad Sequence

To quantify the impact, consider a $1,000,000 portfolio invested in the S&P 500 between 2000 and 2019. Over those two decades, the index delivered an average annual total return of roughly 7.7%, with a compound annual growth rate of 6.1%.

If an investor took no withdrawals, the commutative property applies perfectly. Running the actual historical sequence forward—starting with the dot-com crash in 2000—leaves the portfolio at $3.25 million. Running the exact same 20 years of returns backward, starting with the strong gains of 2019, yields the identical $3.25 million.[1]

Now introduce a fixed $50,000 annual withdrawal at the start of each year. In the forward sequence, the retiree immediately faces three consecutive years of severe losses: -9.1% in 2000, -11.9% in 2001, and -22.1% in 2002.

Because the retiree is pulling $50,000 out of a shrinking asset base, the portfolio is decimated before the subsequent bull market begins. By the end of the 20-year period, the forward-sequence portfolio has dwindled to just $273,000.

If that same retiree experienced the exact same returns in reverse order, the outcome transforms. The early years feature strong gains, allowing the portfolio to easily absorb the $50,000 withdrawals while continuing to compound. The backward-sequence portfolio finishes the 20-year period at $1.85 million.

The Retirement Danger Zone

This $1.57 million divergence illustrates why average returns are a dangerous metric for retirees. Sequence risk is heavily concentrated in the first five to ten years of retirement, creating a distinct window of vulnerability that dictates the portfolio's long-term viability.[2]

During this early phase, the portfolio is at its maximum value relative to the withdrawal amount. A 20% market decline on a $1,000,000 portfolio erases $200,000 in capital. If that same 20% decline occurs in year 25, when the portfolio has been drawn down to $400,000, the absolute loss is only $80,000.[2]

"A decline in year one is not the same as a decline in year fifteen," explains a 2026 analysis by FamilyVest. "Early losses matter more because the portfolio has more years left to support spending and less time to heal before additional withdrawals continue."

If a severe bear market strikes late in retirement, the portfolio has already benefited from decades of compounding. The sequence risk has largely passed, and the retiree's remaining life expectancy is shorter, meaning the diminished capital only needs to survive a few more years.

Conversely, an early bear market forces the portfolio to attempt a recovery from a severely depleted base. A 30% loss requires a 43% gain just to break even. When a 5% withdrawal is layered on top of that loss, the required recovery gain climbs past 50%.

The Multiplier Effect of Inflation

Sequence risk is further compounded by inflation, which acts as a secondary, invisible withdrawal. The standard retirement planning framework assumes that the retiree increases their absolute dollar withdrawal every year to maintain purchasing power, regardless of what the market is doing.[2]

If inflation spikes to 5% during an early-retirement bear market, the retiree is forced to liquidate an even larger number of depressed shares just to buy the same groceries. This accelerates the depletion of the asset base precisely when it is most vulnerable.[2]

During the stagflation of the 1970s, retirees faced the worst possible sequence: plummeting equity values paired with double-digit inflation. Portfolios were crushed from both sides, proving that sequence risk involves the timing of both market returns and consumer price spikes.[2]

Rethinking Safe Withdrawal Rates

The structural threat of sequence risk is what prompted William Bengen to develop the "4% rule" in 1994. Bengen, an MIT graduate and financial planner, realized that using average historical returns to set withdrawal rates was causing advisors to recommend dangerously high distributions.

Bengen tested withdrawal rates against the worst return sequences in modern financial history, including the Great Depression and the 1970s. He found that a 4% initial withdrawal, adjusted annually for inflation, was the maximum rate that survived every historical sequence over a 30-year horizon.

However, the safe withdrawal rate is not a static law of physics. It fluctuates based on the macroeconomic environment at the exact moment a retiree begins taking distributions. High equity valuations and low bond yields at the retirement date amplify sequence risk.[2]

In 2021, Morningstar researchers re-evaluated Bengen's framework using forward-looking return forecasts rather than historical data. They concluded that the safe starting withdrawal rate had temporarily dropped to 3.3% due to inflated asset prices and suppressed fixed-income yields.

"Our latest estimate is a 3.9% starting withdrawal rate," Morningstar reported in late 2025, noting that higher bond yields provided a stronger buffer against early equity market declines.

Bengen himself recently revised his own models. He suggested that broader diversification across international and small-cap equities could push the safe withdrawal rate to 4.7% for modern retirees.[2]

Defending Against the Sequence

Because investors cannot predict or control the sequence of market returns, financial planners use structural defenses to mitigate the risk. The most common approach is the "bucket strategy," which segments the portfolio by time horizon to protect near-term cash flows.

A retiree might hold two to three years of living expenses in cash, certificates of deposit, or short-term Treasury bills. If the equity market crashes in year one, the retiree draws income from the cash bucket, leaving the stock portfolio untouched until prices recover.

Another defense is dynamic spending. Rather than blindly adjusting withdrawals upward for inflation every year, retirees establish guardrails. If the portfolio's value falls below a certain threshold, the retiree skips the inflation adjustment or takes a modest pay cut.

Implementing guardrails—taking less in down markets and setting limits during bull markets—can significantly increase the initial safe withdrawal rate. This flexibility prevents the retiree from selling shares at the absolute bottom of a market cycle.

The mathematics of sequence risk prove that wealth preservation is a fundamentally different discipline than wealth accumulation. Averages build the nest egg during the working years, but the sequence of returns determines how long that capital survives the drawdown phase.[2]