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Retirement plan at 40 for a 65–95 horizon: Monte Carlo success probability, Pfau's maximum withdrawal rate and the Bengen / Trinity rolling-start test

Tool	Retirement Planner
Source	Pfau, W. D. (2011). Capital Market Expectations, Asset Allocation, and Safe Withdrawal Rates. MPRA Paper 32973, National Graduate Institute for Policy Studies, Tokyo. Tables 1–3; Bengen, W. P. (1994). "Determining Withdrawal Rates Using Historical Data." Journal of Financial Planning, Vol. 7, No. 4, pp. 171–180; Cooley, P. L., Hubbard, C. M., and Walz, D. T. (1998). "Retirement Savings: Choosing a Withdrawal Rate That Is Sustainable." AAII Journal, Vol. 20, No. 2, pp. 16–21, Table 3.
Level	Professional

Problem

A 40-year-old with 200,000 saved contributes 20,000 a year (growing 2 %/yr) until 65 and then spends 60,000 a year in today's money to age 95 from a 60/40 stock/bond portfolio with the 1926–2010 real-return parameters of Pfau's Table 1 (stocks 8.70 % / 20.39 %, bonds 2.52 % / 6.84 %, correlation 0.08), 2.5 % inflation and 0.5 % fees. Find the expected-return projection, the probability that the money lasts, the safe withdrawal rate for 90 % success and the contribution that would reach 90 %; compare the method with Pfau's Table 3 and the historical studies.

Inputs to Enter

Ages 40 / 65 / 95; saved 200,000; contribution 20,000/yr growing 2 %; spending 60,000/yr (today's money); no other income; tax 0 %; fees 0.5 %/yr
Parameter set Pfau (2011) SBBI 1926–2010 real; 60 % stocks, 40 % bonds; inflation 2.5 %; rule: fixed real amount; lognormal model, 10,000 paths, seed 2026, target success 90 %

Open the Retirement Planner tool (loaded with this example)

Answers

Portfolio: arithmetic mean 8.884 % nominal (6.228 % real), SD 13.07 %, geometric 8.108 %; net of fees 8.339 %
Deterministic: balance at 65 = 200,000 · 1.08339^25 + 20,000 (1.08339^25 − 1.02^25)/(0.08339 − 0.02) = 3,300,714 (1,780,374 in today's money); sustainable annuity-due withdrawal 118,428/yr in today's money; terminal balance 4,629,579 (today's money)
Monte Carlo: probability of success 79.9 % (numpy, 1,000,000 paths: 79.54 %); median terminal balance 2.35 M in today's money (numpy 2,359,371); 5th percentile 0
Safe withdrawal rate for 90 % success 4.05 % of the balance at retirement (numpy 4.01 %); the plan's need implies 3.82 % on the median balance of 1.57 M; contribution needed for 90 % success 28,800/yr (numpy 29,007)
Pfau Table 3 reproduced by the same engine: 30 years / 10 % failure / 45 % stocks → 4.3 %; 30 / 5 % / 37 % → 3.9 %; 20 / 10 % / 38 % → 5.7 %; 40 / 10 % / 58 % → 3.7 %
Historical sequences on Shiller's data (60/40, 3.82 %, 30-year windows 1871–1993): 95.9 % succeed; Trinity window 1926–1995 at 4 %: 95.1 % (Cooley, Hubbard and Walz Table 3: 95 % for 50/50, 98 % for 75/25)

Notes

The tool's default inputs are this case; press Submit. The parameters are historical averages and the case is illustrative (Pfau's own warning). Monte Carlo values vary by a few tenths of a point with the seed; the test script (test_retirement_planner.php) checks them against the 1,000,000-path numpy run in References/Retirement/ and the Pfau cases to 0.1 point. Bengen's 33-year worst case (4 %, 50/50, 1966) comes out at 28 years on Shiller's 10-year Treasury series with his end-of-year timing.