Revisiting the 4% Withdrawal Rule Using Monte Carlo Simulations with Random Market Declines

Tamimi, Nabil; Sebastianelli, Rose; Rajan, Murli; Rocco, Vincent

doi:10.2478/fprj-2024-0001

Figures & Tables

Weibull Distribution of S&P 500 Historic Returns.Note: the Weibull distribution (mean = 11.60%, standard deviation = 18.77%) is used to simulate yearly equity growth rates. This distribution is skewed to the left, which allows for the simulation of large negative values, thereby mimicking historical returns.

Logistic Distribution of US T-Bonds Historic Returns.Note: the Logistic distribution (mean = 4.40%, standard deviation = 7.39%) is used to simulate US T-Bonds returns. The shape of the logistic distribution is similar to that of the normal distribution but with relatively longer tails.

Logistic Distribution of Baa Bonds Historic Returns.Note: Like the distribution of US T-bonds, the Logistic distribution (mean = 6.75%, standard deviation = 7.33%) is used to simulate Baa bonds returns.

Lognormal Distribution of Historic Inflation Rates.Note: The lognormal distribution is used to simulate inflation rates. The distribution is positively skewed, with a mean inflation rate of 3.70% and a standard deviation of 2.66%. Simulated inflation rates allow for an adaptive withdrawal strategy that responds to the prevailing inflation environment.

Single Simulation Run with Two Market Crashes and 15% Portfolio Decline in Each Instance.Note: This figure illustrates a single trial example for a 70% equities – 30% bonds allocation where two simulated market crashes (i.e., portfolio declines) occur, one at age 67 and the other at age 89. Each decline results in a 15% decrease in portfolio value. The simulated returns on assets and inflation rates are generated from their respective distributions, as indicated by their mean values. The ending balance is calculated by deducting the withdrawal amount from the beginning balance and then adjusting it with the simulated returns. The adjusted ending balance checks whether the ending balance column is negative. If the value is negative, then the adjusted ending column corrects the value to zero. Otherwise, a negative ending balance will be carried over to the next period’s beginning balance.

Projected Retirement Portfolio Value at Age 93: Median and Standard Deviation Across Different Asset Allocations and Market Crashes.Note: The figure displays the median and standard deviation performance metrics for each asset within a hypothetical portfolio subject to both a decline and an increased frequency of market crashes.

Autocorrelation Functions for SP 500.Note: The figure displays the autocorrelation function for various market assets using lag periods ranging from 1 to 12 years.

Projected Retirement Portfolio Value at Age 93: Average Balance and Probability of Fund Running Out of Across Different Asset Allocations and Market Crashes

Two Market Crashes

Allocation: 50% Stocks - 50% Bonds			Allocation: 60% Stocks - 40% Bonds			Allocation: 70% Stocks - 30% Bonds

Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93	Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93	Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93

5%	$2,572,924	22%	5%	$3,440,807	21%	5%	$4,448,100	20%
10%	$2,098,119	27%	10%	$2,866,918	25%	10%	$2,773,931	24%
15%	$1,684,465	33%	15%	$2,341,817	30%	15%	$3,110,075	28%

Three Market Crashes

Allocation: 50% Stocks - 50% Bonds			Allocation: 60% Stocks - 40% Bonds			Allocation: 70% Stocks - 30% Bonds

Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93	Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93	Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93

5%	$1,998,895	27%	5%	$2,691,420	25%	5%	$3,537,160	24%
10%	$1,427,056	37%	10%	$2,009,766	32%	10%	$2,685,524	30%
15%	$968,235	46%	15%	$1,445,437	41%	15%	$1,976,380	38%

Four Market Crashes

Allocation: 50% Stocks - 50% Bonds			Allocation: 60% Stocks - 40% Bonds			Allocation: 70% Stocks - 30% Bonds

Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93	Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93	Portfolio Decline	Average Ending Balance	Probability of Fund Running Out of Money at Age 93

5%	$1,526,137	33%	5%	$2,133,393	30%	5%	$2,777,253	29%
10%	$953,182	46%	10%	$1,380,912	41%	10%	$1,903,689	38%
15%	$569,125	59%	15%	$850,908	53%	15%	$1,233,732	48%

Pairwise Pearson Correlation Coefficients

Sample 1	Sample 2	Correlation (r)	P-Value
UST. Bond	S&P500	0.02	0.87
Baa Bonds	S&P500	0.40	0^**
Inflation	S&P500	−0.16	0.12
Baa Bonds	US T. Bond	0.60	0^**
Inflation	US T. Bond	−0.09	0.42
Inflation	Baa Bonds	−0.20	0.05^*

Revisiting the 4% Withdrawal Rule Using Monte Carlo Simulations with Random Market Declines

Figures & Tables

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Projected Retirement Portfolio Value at Age 93: Average Balance and Probability of Fund Running Out of Across Different Asset Allocations and Market Crashes

Pairwise Pearson Correlation Coefficients

Paradigm

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