Retirement Planner: Method

Method: Projection, Monte Carlo, Withdrawal Rules and the Historical Studies

What the tool answers

A retirement plan has two phases: an accumulation phase, in which savings grow and contributions are added, and a decumulation phase, in which the portfolio pays the part of the spending that other income (Social Security, a pension, an annuity) does not cover. The tool asks whether the plan survives to the planning age, with what probability, and what would have to change (contribution, spending, retirement age, withdrawal rule) to reach a chosen probability. It does so three ways: a deterministic projection at the expected return, a Monte Carlo simulation of thousands of return sequences, and the rolling-start historical test of Bengen (1994) and the Trinity study on Robert Shiller's long return record.

Conventions and timing

Time runs in whole years from the current age. Contributions are paid at the end of each year before retirement and grow at the entered rate (a nominal growth, as with a salary). Withdrawals are taken at the start of each retirement year, from the retirement age to the year before the planning age, so a plan from 65 to 95 makes 30 withdrawals; this is the timing of Pfau (2011) and of Guyton and Klinger (2006). The portfolio is rebalanced to the entered allocation every year. All amounts are in one currency; "today's money" divides a nominal amount by the price level accumulated since the current age. Fees are taken from the return, 1 + r_net = (1 + r)(1 − fee). Spending and other income are entered in today's money: spending is indexed to inflation, the other income to its own cost-of-living adjustment (equal to inflation keeps its real value; 0 is a fixed nominal pension). A flat tax rate on withdrawals grosses the net need up by 1/(1 − tax); other income is taken as after tax.

Return assumptions

Each asset class has an arithmetic mean μ_i and a standard deviation σ_i of its annual return, and the classes have pairwise correlations ρ_ij. The rebalanced portfolio's arithmetic mean is Σ w_i μ_i and its variance Σ_iΣ_j w_i w_j ρ_ij σ_i σ_j. Parameters given in real terms are converted to nominal before the simulation, 1 + r_nom = (1 + r_real)(1 + i) and σ_nom = σ_real(1 + i), so that spending can be indexed to inflation explicitly; with constant inflation the two formulations are identical. The arithmetic mean is what a single year is expected to return; the growth rate of a long sequence is the geometric mean, which for a lognormal return equals exp(μ_ln) − 1 with μ_ln = ln(1 + μ) − ½ ln(1 + σ²/(1 + μ)²), about σ²/2 below the arithmetic mean. The parameter sets offered are Pfau's (2011) Table 1, the Ibbotson SBBI real returns 1926–2010 (stocks 8.70 % with a 20.39 % standard deviation, intermediate-term government bonds 2.52 % / 6.84 %, Treasury bills 0.69 % / 3.90 %, stock–bond correlation 0.08), and two sets computed from Shiller's data (nominal S&P Composite and 10-year Treasury total returns, 1871–2022 and 1926–2022). All are historical averages; they are the starting point of a discussion, not a forecast, and the sensitivity table shows what one percentage point of return does.

Deterministic projection

With a constant net return r every year, the balance at retirement is the future value of today's savings, W₀(1 + r)ⁿ, plus the future value of the growing contribution stream, C[(1 + r)ⁿ − (1 + g)ⁿ]/(r − g), the geometric-gradient series of Blank and Tarquin. The inflation-indexed withdrawal that exhausts the balance exactly at the planning age follows from the annuity-due formula in real terms: with the real return i = (1 + r)/(1 + inflation) − 1 and n retirement years, X = B i/[(1 + i)(1 − (1 + i)⁻ⁿ)] in today's money. The table then follows the balance year by year under the chosen withdrawal rule; the same engine that runs the Monte Carlo paths runs this single path, so the two cannot disagree. The projection is also shown at the geometric mean, which is close to the median outcome of the simulation: a projection at the arithmetic mean overstates what a typical sequence delivers.

Monte Carlo simulation

In the lognormal model each year's portfolio return is drawn from a lognormal distribution of 1 + r with the portfolio's arithmetic mean and standard deviation, independently from year to year, the approach of Pfau (2011); the random numbers are seeded, so the same inputs reproduce the same result and every sensitivity case re-uses the same draws (common random numbers), which makes the differences between cases real rather than simulation noise. In the bootstrap models whole historical years (stock return, bond return and that year's inflation together) are drawn at random from Shiller's record or from a pasted history, in blocks of a chosen length so that runs of good and bad years can be kept together; the historical correlations, the fat tails and the inflation–return relationship are then preserved without a distributional assumption, at the price of never producing a year worse than the record. A lifetime succeeds when the portfolio pays every withdrawal to the planning age and fails in the year a withdrawal exceeds the balance. The outputs are the probability of success, the percentiles of the balance by age (the fan), the distribution of the terminal balance and of the age at which the failing lifetimes run out, and, for rules that vary the income, the probability that the real income never fell below the spending need and the lowest income relative to the need.

Safe withdrawal rate

Bengen (1994) defined the safe rate as the first-year withdrawal, as a percentage of the balance at retirement, that can be indexed to inflation every year without exhausting the portfolio within the planning horizon on any historical sequence; the Trinity study (Cooley, Hubbard and Walz 1998) tabulated success rates instead, and Pfau (2011) computed the maximum withdrawal rate of every simulated path and reported the rate sustained with a chosen failure probability. The tool uses Pfau's definition. For a fixed-real withdrawal the balance is a linear function of the rate, B_t = B_r(a_t − w d_t) with a_t the growth factor of an untouched portfolio and d_t the compounded sum of the inflation factors, so each path's maximum rate is exact, w_max = min_t a_t/(d_t + G_t), and the rate sustained with probability p is the (1 − p) quantile of the maxima. The rate applies to each path's own balance at retirement, so it separates the retirement-phase risk from the accumulation-phase risk; the plan's implied rate, the first-year need divided by the median simulated balance at retirement, is printed beside it. The same linearity gives the contribution needed: the balance is linear in the annual contribution, each path's smallest sufficient contribution is max_t (X_t − b_t)/e_t over the withdrawal years, and the contribution that succeeds with probability p is the p quantile across paths (computed for the fixed real spending rule).

Withdrawal rules

• Fixed real amount. The net spending need, indexed to inflation; the portfolio pays it until it cannot. Success means never being depleted.
• Bengen's rule. The first withdrawal is the entered initial rate times the balance at retirement, then indexed to inflation; the spending need is ignored by the rule and used only to judge the income. Bengen's 1994 finding, on Ibbotson data 1926–1976 with 50 % stocks and 50 % intermediate-term Treasuries and end-of-year withdrawals, was that 4 % never exhausted a portfolio before 33 years (the 1966 start), 3 % lasted 50 years from every start, and 4.25 % could fail in 28 years; he recommended 50 to 75 % stocks.
• Fixed percentage of the balance. The rate times the current balance every year: the portfolio can never be depleted, but the income falls and rises with the market, so the tool reports the probability that the real income stayed at or above the need and the lowest income of a lifetime.
• Guyton–Klinger guardrails (Guyton and Klinger 2006). The initial withdrawal is the spending need. Each year it is indexed to the previous year's inflation (the withdrawal rule, in the authors' modified form: no increase following a year with a negative portfolio return when the current withdrawal rate would exceed the initial rate, and no make-up later). The capital preservation rule cuts the withdrawal by 10 % when the current withdrawal rate has risen more than 20 % above the initial rate, except in the final 15 years of the horizon; the prosperity rule raises it by 10 % when the rate has fallen more than 20 % below the initial rate; the adjusted amount becomes the base for the next year. The authors' portfolio management rule (which assets fund the withdrawal) needs several asset classes and is not modelled; the inflation rule (6 % cap) is dropped, as the authors did once the capital preservation rule was in force. With all rules the authors found maximum initial rates of 5.2–5.6 % at 99 % success and 6.5 % at 90 % for a 65 % equity S&P 500 portfolio over 40 years on 1973–2004 data, against 5.3 % at 99 % without the guardrails (their Table 3); the price is a lower income in the bad sequences, which the tool reports as the lowest and the final-year income relative to the need.

Sequence of returns

Two sequences of returns with the same arithmetic and geometric means end with the same wealth for an investor who neither adds nor withdraws, but not for a retiree: a loss early in retirement is taken on a large balance that is also being drawn down, and the later gains act on what is left. The tool replays the lifetime whose terminal balance is the median with its retirement-phase returns in reverse order and charts both. Bengen's "black hole" retirees of 1929, 1937 and 1969 and his "stars" of 1949 and 1958 are the historical faces of this effect.

Historical sequences

The rolling-start test puts a retiree at the start of every year of Shiller's record (1871 to the last year that allows a full horizon) with a unit portfolio, the chosen allocation rebalanced yearly, and a first-year withdrawal at the plan's initial rate indexed to each year's actual inflation; it counts the windows that lasted and the shortest portfolio life, as in Bengen's Figure 1 and the Trinity study's Table 3. The studies used Ibbotson's data (Bengen: intermediate-term Treasuries, starts 1926–1976, end-of-year withdrawals; Cooley, Hubbard and Walz: long-term high-grade corporate bonds, returns 1926–1995, 41 thirty-year windows), so the rates on Shiller's 10-year Treasury series are close to but not identical with theirs; the Trinity Table 3 values the tool quotes are 95 % for 4 % over 30 years from 50/50 stocks/bonds, 98 % from 75/25, 100 % at 3 % for every allocation with at least 25 % stocks.

Planning horizon

The planning age can be entered or taken from the NCHS United States Life Tables, 2020 (Tables 2 and 3): the retirement age plus the expectation of life at that age (17.0 years for a 65-year-old man, 19.8 for a woman) plus a margin, rounded up. Half of the people reaching that age live longer than its expectation, which is why Bengen suggested life expectancy plus 5 or 10 years and why a planning age of 95 is common for a 65-year-old.

Reading the results and limits

The probability of success is the probability under the stated assumptions, not a forecast: it moves by about ten points for one percentage point of expected return (see the sensitivity table), and historical parameter sets are generous by the standards of current yields. The lognormal model has no serial dependence and no valuation effect; the bootstrap has only the years that happened. Taxes are a flat rate, Social Security is a fixed real or nominal stream, the allocation is constant through life (no glide path), and spending is constant in real terms (no decline with age, no long-term-care shock). The tool is a planning aid; a financial plan should be revisited as the balance, the market and the person's circumstances change, which is what the Guyton–Klinger rules formalise.