Researcher

Charles Hoover

Abstract

This study evaluates alternative simulation methods beyond the commonly used Geometric Brownian Motion (GBM) to assess the risk associated with withdrawal rates from retirement portfolios. The primary goal is to determine which models best capture market dynamics and offer better predictive accuracy. The research suggests that GARCH models, particularly GJR-GARCH, provide more reliable risk assessment compared to traditional GBM. Through both historical backtesting and forward-looking testing, the study finds that GBM tends to underestimate withdrawal rates and misrepresent risk, while GARCH-based models capture volatility clustering and perform better in modeling portfolio behavior.

1. Introduction and Motivation

Retirees often question how much they can safely withdraw from their investment portfolios without the risk of depletion. The standard approach to evaluating withdrawal risks is through GBM simulation, which assumes normality in portfolio returns and independent price movements (Markov Process). However, market returns often exhibit excess kurtosis, skewness, and volatility clustering, making GBM an inadequate model.

To address these limitations, the study explores alternative volatility-based models such as:

  • Historical Block Bootstrapping (to account for non-normal distributions)
  • GARCH (Generalized Autoregressive Conditional Heteroskedasticity) (to model conditional volatility)
  • GJR-GARCH (to account for asymmetric volatility effects)

The study aims to determine which model best predicts withdrawal risk, ensuring retirees make informed decisions.

2. Literature Review

Key references in the field include:

  • Bengen (1994): Established the 4% withdrawal rate rule for sustainable withdrawals over a 30-year period.
  • Cooley, Hubbard, and Walz: Expanded on sustainable withdrawal rates.
  • Spitzer, Strieter, and Singh: Investigated bootstrapping approaches but did not consider block bootstrapping.

Prior studies mostly focused on empirical withdrawal rates rather than simulating long-term portfolio behavior using alternative risk models.

3. Exploratory Data Analysis

Data was collected from:

  • S&P 500 Index and DJI Corporate Bond Index (since April 1915).
  • Inflation data from the Consumer Price Index (CPI).
  • Portfolio composition: 60% S&P 500, 40% Bond Index.

Key findings:

  • Returns are non-normal: They exhibit negative skewness and high kurtosis (fat tails).
  • Volatility clustering is present: Large return fluctuations tend to cluster over time.
  • Heteroskedasticity (changing volatility over time) is confirmed via Engle’s test.

4. Simulation Models

4.1 Traditional Geometric Brownian Motion (GBM)

  • Assumes log-normal returns and constant volatility.
  • Standard model for Monte Carlo simulations.
  • Issue: GBM fails to capture real-world market behavior (volatility clustering, non-normality).

4.2 Historical Block Bootstrapping

  • Improves upon GBM by sampling blocks of historical returns instead of assuming a normal distribution.
  • Key Advantage: No parametric assumptions.
  • Survival rate (4% withdrawal rate): 89% (better than GBM but still underestimates).

4.3 GARCH(1,1) Model

  • Introduced to capture volatility clustering by modeling dynamic volatility.
  • Survival rate (4% withdrawal rate): 94%.
  • More accurately reflects real-world financial market dynamics than GBM.

4.4 GJR-GARCH Model

  • Improves upon GARCH(1,1) by introducing asymmetric volatility (negative returns increase future volatility more than positive returns).
  • Survival rate (4% withdrawal rate): 93%.
  • Most effective model in predicting withdrawal risk.

5. Methods of Performance Evaluation

  • Survival Curve Analysis: GJR-GARCH closely matches the actual survival curve, outperforming GBM.
  • Back-Testing Predicted Withdrawal Rates: GARCH models predict higher, more realistic withdrawal rates.

6. Key Results

  • GBM significantly underestimates withdrawal success rates.
  • GARCH and GJR-GARCH outperform GBM by capturing volatility clustering and producing accurate survival curves.

7. Conclusion and Future Research

Key Takeaways:

  • GARCH models are superior to GBM for modeling portfolio risk.
  • GBM is too conservative, leading to lower withdrawal rate estimates.
  • GJR-GARCH best captures market dynamics, providing realistic risk assessment.

Future Research Directions:

  • Apply these models to different asset classes.
  • Improve parameter estimation for better risk assessment.
  • Explore multi-asset portfolios to test model robustness.