Side Lesson 28: Monte Carlo — Projecting Retirement and the Sequence-of-Returns Trap

---

1. Why This Is Important

Every retirement projection you see online — the bank's planner, the 401(k) provider's "you will have $X at 65" widget, the financial advisor's slide deck — runs the same arithmetic. Plug in a starting balance, an assumed return, a horizon, push the button, get one number. "At 7% per year for 30 years, your $1M becomes $7.6M."

That number is a lie. Not because the math is wrong — 1.07^30 does equal 7.61 — but because the markets you actually live in are not a constant-return savings account. They deliver +30% in some years and -25% in others. The calendar matters: getting the bad years first versus last can be the difference between dying with money in the bank and running out of it at age 78. The single-number projection silently assumes the average returns arrive in a smooth, well-behaved order. They never do.

Four reasons to learn Monte Carlo simulation properly:

The same average produces wildly different outcomes. Two

retirees with identical 30-year average returns can finish with $0 or $4M depending only on the order in which the returns arrived. This is the single most under-appreciated risk in personal finance.

The Bengen 4% rule is one number derived from one technique —

historical bootstrap on US data 1926-1976. Modern updates from Pfau (2020), Kitces, and Morningstar using lower forward-looking returns put the safe rate closer to 3.0-3.5%. If your plan assumes 4% you may be 50 basis points too aggressive.

**The 95th-percentile-bad outcome is 30-40% lower than the

median.** Planning to the median is planning for 50/50 success. Monte Carlo gives you the distribution — the shape and tail — not just a point estimate.

Vol-tail-wags-dog. The tail is not a curiosity; it

is the entire game in withdrawal phase. A 30% drawdown in accumulation phase is a buying opportunity. The same drawdown in year three of retirement is a permanent reduction in lifetime spending capacity.

---

2. What You Need to Know

2.1 Assumed Return + Assumed Vol vs. Monte Carlo

The traditional retirement projection is a single line:

$$ W_T = W_0 \cdot (1 + \mu)^T - C \cdot \frac{(1+\mu)^T - 1}{\mu} $$

where $W_0$ is your starting balance, $\mu$ is the assumed return, $T$ is the horizon, and $C$ is the annual withdrawal. Plug in $W_0 = \$1\text{M}$, $\mu = 7\%$, $T = 30$, $C = \$40k$ and the formula spits out roughly $\$3.5\text{M}$ ending balance. Done.

The problem is that $\mu$ is not a number, it is a distribution. The S&P 500's long-run nominal return is roughly 10% with a standard deviation of 16%. In any single year the return is most often somewhere between -22% and +42% (one-sigma band). Over 30 years the average clusters near 10%, but the path — the order in which the years arrive — varies enormously.

Monte Carlo simulation replaces the single line with **thousands of randomized lines.** In each simulation:

Generate 30 random annual returns (or 360 monthly) from a

distribution matching your assumed return and vol.

Compound them through a path with the same starting balance and

contribution/withdrawal schedule.

Record the ending balance.

After 1,000 to 10,000 simulations you have a distribution of outcomes. The 50th percentile is your median expected wealth. The 5th and 95th percentiles bracket the realistic range. The fraction of paths that ended above zero is your success probability.

A 95% success rate on a $40k/yr withdrawal plan is much more honest than "your expected balance is $3.5M." The first number tells you the plan works; the second tells you only that on average it works, while leaving you to discover for yourself that on average you only die once.

2.2 Sequence-of-Returns Risk — The Killer

Two retirees both retire at 65 with $1M and the same plan: withdraw $40k inflation-adjusted per year. Both experience the *exact same 30 annual returns.* The only difference is the order. Retiree A gets a bad first decade; Retiree B gets a bad last decade.

Mathematically they have identical arithmetic mean returns — the calendar can be reshuffled freely without changing the average. But the wealth paths are completely different. The reason is brutal: **withdrawals from a depressed portfolio sell more shares than withdrawals from an inflated portfolio.** A $40k withdrawal at $700k forces you to liquidate 5.7% of remaining shares; the same $40k at $1.4M is only 2.9%. The shares sold during a drawdown are permanently gone — they cannot participate in the eventual recovery.

Run a constructed example: first decade averages -5% per year, last two decades average +13% per year. Arithmetic mean over 30 years hits 7%. Run the opposite ordering: +13% for 20 years then -5% for 10. Same mean, same volatility, same end-of-history return.

• Sequence A (bad early): Portfolio bleeds out fast. Year-10

balance roughly $278k. The recovery returns kick in too late — the portfolio runs to zero around year 24, six years before the end of the 30-year plan. Retiree A dies broke.

• Sequence B (bad late): Portfolio compounds aggressively for

20 years to roughly $8M, then bleeds during the bad final decade but ends with multi-million-dollar dynastic wealth. Retiree B leaves an inheritance.

Same mean. Same vol. Same withdrawals. Same starting balance. **The difference between dying broke and leaving generational wealth is the calendar order alone.** This is sequence-of-returns risk, and no single-number projection captures it.

The retirees most exposed to sequence risk are those in the first five to ten years of withdrawal, when the balance is largest and withdrawals are nibbling at the principal. After year 15-20 the sequence risk attenuates — surviving portfolios are usually large enough that further drawdowns cannot ruin them.

2.3 The Bengen 4% Rule and Its Modern Updates

In 1994 William Bengen, a financial planner in southern California, published Determining Withdrawal Rates Using Historical Data in the Journal of Financial Planning. He ran a then-novel exercise: **bootstrap 30-year retirements from the actual historical sequence of US returns 1926-1976.** For each starting year he asked the same question: what is the highest constant inflation-adjusted withdrawal rate that survived all 30 years?

The answer was 4.15% for a 50/50 stock-bond portfolio. Bengen rounded down to 4.0% as a "safe" rule of thumb. The rule rapidly displaced earlier industry conventions (most of which assumed withdrawal rates of 6-7% based on average returns, ignoring sequence risk).

The 4% rule is calibrated to **the worst historical 30-year period in US data,** not the median. The worst case in Bengen's window was a retirement starting in 1966, just before the stagflation decade of 1968-1981. That sequence — bad first 15 years, recovery from 1982 onward — is the canonical sequence-of-returns nightmare. A portfolio that survived 1966-1995 will, by Bengen's logic, survive any other 30-year window in the historical sample.

The Trinity Study (Cooley, Hubbard, Walz 1998) ran the same exercise on a wider date range and slightly different portfolios and confirmed the 4% number with 96% historical success rate at 30 years on a 50/50 stock-bond mix. The number stuck.

But: the 4% rule is historical-bootstrap-derived, which embeds two huge implicit bets:

The historical equity premium of ~6.5% will persist.

The historical 30-year-Treasury yield of ~5% will persist.

In April 2026 neither assumption is on solid ground. Forward-looking expected returns from sober sources (Vanguard, BlackRock, GMO, Research Affiliates) cluster around 5-7% nominal for US equities and 4-5% for 10-year Treasuries — well below the historical numbers Bengen used. Wade Pfau's 2020 update *Safe Withdrawal Rates: A Comprehensive Examination* recalculated the Bengen exercise using forward-looking expected returns and found the safe rate falls to 3.0-3.5%. Morningstar's 2024 State of Retirement Income report converged on 3.7% for new retirees under current valuations.

The practical translation: if you used the 4% rule to set your nest-egg target, you may be undersized by 15-30%. A retiree who wanted $80k/year with the 4% rule needed $2M. Under a 3.5% rule the target is $2.29M — a 14% increase in the required nest egg. Under 3.0% it is $2.67M — a 33% increase.

2.4 Bootstrap vs. Parametric Monte Carlo

There are two flavours of Monte Carlo, and the choice matters more than people realise.

Parametric MC. Assume returns follow a parametric distribution — most often Normal with assumed $\mu$ and $\sigma$. Generate random samples from the distribution. Run forward.

• Pros: simple, fast, infinitely-extendable, the math is clean.
• Cons: the distribution assumption is wrong. Real equity

returns have fat tails (Week 42 — kurtosis of monthly S&P 500 returns is roughly 7-15, not the 3.0 a Normal distribution predicts). Parametric MC under the Normal assumption systematically underestimates the probability of catastrophic outcomes.

Bootstrap MC. Sample with replacement from the actual historical sequence of returns. No distributional assumption — the data generates the distribution.

• Pros: captures fat tails, captures observed correlation structure

between asset classes, captures observed serial dependence (mean reversion at long horizons).

• Cons: limited by the historical sample. The US dataset has only

about three independent 30-year windows in it. The future may not resemble the past.

A practical compromise: block bootstrap with 5- to 10-year blocks. This preserves observed within-block correlation and serial structure while still allowing recombination. It is the technique Pfau, Kitces, and most academic researchers default to.

The retail tools (Vanguard's projection tool, Schwab's, the typical robo-advisor) usually run parametric Normal MC because it is faster and easier. The underestimate of tail risk is real but typically small at 30-year horizons (the central limit theorem helps), and the difference between 3% and 5% safe withdrawal rate is dominated by the return assumption, not the sampling method.

2.5 Reading the Output — Probability of Success

The Monte Carlo output you actually use is not the median ending wealth, it is the success rate — the fraction of simulations where the plan does not run out of money before the end of the horizon.

Industry conventions:

Success rate	Verdict
≥ 95%	Plan is over-funded. Consider spending more or retiring earlier.
85-95%	Solid plan. Normal target.
70-85%	Plan is risky. Build flexibility into withdrawals.
< 70%	Plan is broken. Save more, work longer, or spend less.

Two crucial nuances:

Success rate is binary; it does not tell you whether the

failed plans failed at year 23 (recoverable with part-time work) or at year 5 (catastrophic). The conditional-on-failure analysis is just as important as the headline number.

The 5th-percentile ending balance is what you should compare

against the 50th percentile, not against the goal. A plan with a 95% success rate but a 5th-percentile ending balance of $50k leaves you with no margin in the bad scenarios. A plan with 90% success but a 5th-percentile ending balance of $1M is far more robust to surprises.

The interactive lab on this page exposes both numbers — the success rate and the 5/50/95 percentile terminal wealth. Practice reading them as a pair.

2.6 Common Pitfalls

Three modelling mistakes that show up in nearly every retail retirement projection:

Assuming Normal returns. Real equity returns are negatively

skewed and fat-tailed. A Normal model with $\sigma = 16\%$ says a single-year -32% return (-2 sigma) happens once every 44 years. The empirical record has it happening every 15-20 years on average, plus the occasional -45% (1931, 2008-09 cumulative). See Week 42.

Assuming constant volatility. Vol clusters. The 2017 calm

regime had realized vol of 6%; 2008 had 40%. Models that hold $\sigma$ constant at 16% smooth over the regime shifts entirely and hide the joint event of "your portfolio drops 35% in the year you start withdrawing." Vol-of-vol is a real risk that parametric MC ignores.

Ignoring correlation regime shifts. The 2003-2021 negative

stock-bond correlation made 60/40 portfolios look bulletproof in simulation. The 2022 correlation flip to +0.3 broke every 60/40 backtest in real time. Models that assume a static correlation matrix between asset classes will under-state crisis correlations and over-state diversification benefits — exactly the wrong direction for retirement planning.

The fix for all three is the same: **use bootstrap or block-bootstrap on real historical data,** and stress-test the output by re-running with the worst observed historical regime as the starting point. Ask the planner: "what is the success rate if I retire into another 1966?"

2.7 Building Sequence-Risk Robustness Into a Plan

Three structural defences against sequence-of-returns risk that work in real life:

Cash buffer / two-bucket strategy. Hold 2-3 years of expenses

in cash + short Treasuries outside the equity sleeve. In a bear market, withdraw from the buffer instead of selling depressed equity. Refill the buffer in good years. Reduces forced liquidation at lows; small drag on long-run return (~30-50bps).

Variable withdrawal rules. Instead of $40k inflation-adjusted

forever, let withdrawals scale with portfolio balance. The simplest version (Guyton-Klinger guardrails) cuts withdrawals by 10% if the current rate exceeds 5% of the portfolio after a bad year, and ratchets up by 10% if it drops below 4%. Pfau's research finds this lifts the safe initial rate from 3.5% to 4.5-5% with the same success probability.

Equity glidepath. Start retirement at 40% equities, ramp up

to 70% over the first 15 years. This is counterintuitive — most people de-risk over time — but it explicitly reduces exposure to the highest-sequence-risk window and increases it during the late-retirement window when sequence risk has attenuated. Kitces and Pfau (2014) showed it improves success probability by 2-4 percentage points on the same withdrawal rate.

The barbell, applied to retirement: keep two years of cash ironclad, let the equity sleeve be aggressive. The bond/cash side is for survival; the equity side is for return. Mixing them at 60/40 throughout retirement is the worst of both worlds — too much volatility for the early years, too little growth for the late years.

---

3. Common Misconceptions

1. "Monte Carlo gives precise answers." It gives a probability distribution, which is much more honest than a point estimate, but the distribution is only as good as the input assumptions. Garbage return assumptions produce garbage Monte Carlo output.

2. "If I average 7% per year I will hit the 7% line." No. The median of a 30-year 7%-mean path is roughly the 7% line, but your particular path will diverge from the median by a factor of 2-3x in either direction.

**3. "Sequence risk only matters in the first few years of retirement."** It matters most then, but a bad decade in years 5-15 is also fatal. The risk window extends roughly through year 15-20 of withdrawals.

4. "More simulations means more accurate results." Beyond about 1,000-5,000 simulations the Monte Carlo error is dominated by model error (wrong distribution, wrong correlations) rather than sampling error. Running 100,000 sims of a misspecified model just gets you wrong answers with more decimal places.

5. "The 4% rule is a law of nature." It is one specific result from one specific bootstrap on one specific dataset. Modern updates using forward-looking returns put the safe rate at 3.0-3.5%.

6. "If my plan has a 95% success rate I'm safe." Success rate is binary. The 5% of failed paths can fail catastrophically (year 5) or marginally (year 28). Always look at the conditional-on- failure year of ruin alongside the headline rate.

7. "Monte Carlo accounts for everything." It accounts for return volatility within the assumed model. It does not account for: model risk, regime shifts, your own behavioural failures (panic-selling in March 2020), longevity uncertainty, healthcare cost shocks, or divorce. Real retirement planning is Monte Carlo plus stress tests plus margin.

8. "Lowering the withdrawal rate fixes the plan." It fixes the math. It does not fix the lifestyle problem of having saved for an $80k/yr retirement and now needing to live on $60k/yr. Plan-fixing should preferentially happen before retirement, not after.

9. "I'll just adjust if returns are bad." Sometimes you can, sometimes you cannot. Health, family obligations, and the labour market all constrain late-life income flexibility. Variable withdrawal rules (Guyton-Klinger) build this adjustment into the plan structurally rather than relying on improvisation.

10. "Monte Carlo replaces backtesting." It complements backtesting. Backtests show what the past actually did; Monte Carlo shows what could happen given assumed dynamics. A plan that survives both a bootstrap of 1929-1959 and a parametric MC at forward-looking returns has been stress-tested two ways. A plan that only survives one has not been.

---

4. Q&A

Q1: How many simulations should I run?

1,000 to 10,000. Below 1,000 the percentile estimates are noisy (the 5th percentile in a 100-path simulation has a wide confidence interval). Above 10,000 the marginal information is small, and runtime grows linearly. Most retail tools run 1,000; institutional planners run 10,000 or more.

Q2: What return assumption should I use?

For US equities in April 2026: 5-7% nominal expected return (Vanguard CMA: 4.5-6.5%; BlackRock: 6-7%; GMO 7-year: ~3% real for US large-cap, more for international and value). For bonds: roughly the current 10-year yield (4.0-4.3% in April 2026) plus 0-50bps for credit risk. Do not use the historical 10% S&P average — it embeds the 1980-2000 valuation expansion that is unlikely to repeat.

Q3: What volatility assumption?

S&P 500 long-run vol is roughly 16% nominal. A balanced 60/40 portfolio is roughly 10-11%. The lab on this page defaults to 15%, which is a touch optimistic for 100% equities but realistic for the tilted-equity portfolios most retirees actually hold.

Q4: Should I model inflation?

Yes. The simplest method is to deflate everything to today's dollars: use real expected return (subtract 2-2.5% expected inflation from the nominal) and keep withdrawals fixed in today's dollars. The output is then directly comparable to your current spending. The lab does this implicitly — change the inputs to real returns and the picture works.

Q5: How do I handle taxes in the projection?

For most projections, tax is a constant drag — a $40k pre-tax withdrawal from an IRA at 22% bracket is $31k spending. You can either (a) reduce the withdrawal goal to after-tax, or (b) use after-tax expected returns. For tax-mixed portfolios (taxable + 401k + Roth) the order of withdrawals matters and is best handled by a tax-aware projection tool (NewRetirement, ProjectionLab, or the IRS's actuarial spreadsheets).

Q6: What about Social Security?

Treat it as a fixed inflation-adjusted income stream that reduces the required portfolio withdrawal. A $30k/yr Social Security benefit on an $80k spending need leaves $50k from the portfolio — which is a 5% withdrawal rate on $1M, much more aggressive than the 4% headline. The portfolio is doing the marginal work; SS is the floor.

Q7: Should I use parametric or bootstrap MC?

Block bootstrap is the academic standard. Most retail tools use parametric Normal because it is faster. The difference at 30-year horizons is modest (1-2 percentage points on success rate). The return assumption matters far more than the sampling technique.

Q8: How often should I re-run the simulation?

Annually at minimum, after any major life change, and after any market move greater than 15% in either direction. The plan that made sense in 2019 (8% expected returns, 0% rates) does not make sense in 2026 (lower equity returns, 4% rates). Static Monte Carlo is a snapshot; the plan is a process.

**Q9: What is the worst sequence in US history I should stress-test against?**

Retirement starting in 1966. Stagflation 1968-1981, the lost decade to 1982 in real terms, then a recovery. A 60/40 portfolio withdrawing 4% inflation-adjusted from a 1966 retirement survived 30 years but with very low ending balance. A 5% rate failed. The 1966 cohort is Bengen's worst case and the canonical sequence-of-returns nightmare.

Q10: Where can I run a serious Monte Carlo for free?

The interactive lab on this page is the educational version. For a real plan: Portfolio Visualizer (free, runs both parametric and bootstrap MC at retail level), NewRetirement (free tier; tax-aware), ProjectionLab (paid; the most flexible). Avoid single-number projection tools — if it does not show you a distribution, it is not modelling sequence risk at all.

Q11: Does the 95th-percentile-bad outcome really matter?

Yes. That is the outcome where you are still alive at 90 with $50k remaining. Planning to the median means accepting a 50% chance of running out of money before death — which is a catastrophic outcome you would not knowingly accept anywhere else in life. The whole point of running MC is to see the bottom of the distribution and design the plan to survive it.

Q12: What's the takeaway in one sentence?

Single-number retirement projections are wrong and dangerous; Monte Carlo replaces them with an honest distribution of outcomes; the sequence in which returns arrive matters as much as their average; the modern safe-withdrawal rate is closer to 3.5% than 4.0%; and the cash-buffer / variable-withdrawal / glidepath defences let you push that rate back up while keeping success probability high.