1 Simulation in Determining Optimal Portfolio Withdrawal Rates from a Retirement Portfolio Michael Tucker Professor of Finance Fairfield University * Please do not quote without permission

2 Simulation and Retirement Many studies examine risk and retirement: –Ameriks et al. 2001, Bengen 1994, Cooley 2003, Gyton & Klinger 2006, Stout & Mitchell 2006, Young 2004, Pye 2000 –Returns are simulated –Different strategies are tested –Probability of running out of money is examined

3 Milevsky & Robinson (2005) Heuristic formula using the gamma distribution to estimate the probability of running out of money during retirement. Assumptions are a fixed withdrawal rate in real dollars for the retirement portfolio

4 Formula (probability of Stochastic PV>Wealth 0 ) First term is alpha: (2*return per yr as pctge)+4*(nat log of (2))/(life expectancy)/(variance of returns)+nat log of 2/(life expectancy) - 1 beta: (variance of returns+nat log of 2/(life expectancy)/2 given a drawdown (payout pctge) of initial wealth

5 Calculating Probability of Ruin

6 Excel Version Producing Probability of Ruin From Milevsky Inputs

7 Milevsky’s Derivation Detailed in The Calculus of Retirement Income Appears to be without issues. Milevsky uses Stochastic Present Value to gauge risk of bankruptcy

8 Estimating Risk of Bankruptcy with Simulations Stochastic Future Value (SFV) is used: r n = real return generated by simulation for period n W n-1 = real wealth S = fixed real dollar withdrawal rate, N = life expectancy at retirement.

9 Replication of Milevsky Table 3 from: – Milevsky, Moshe and Chris Robinson, A sustainable spending rate without simulation, Financial Analysts Journal, v. 61, n6, Nov/Dec 2005, –Risk of bankruptcy from retirement age at different withdrawal rates with mean return of 5% and σ =12%.

10 Probability (Percentage) of Bankruptcy Statistical probability calculated cell) and then applying NORMDIST in Excel for each simulation, saving outcome. Count: macro counts iterations per 10,000 simulations where ending value<0. Pctge is count/10,000. Can use RiskTarget(target cell,0).

11 Comparing Milevsky to Simulation Pctges of Bankruptcy Pattern is similar to comparison with previous chart (NORMDIST vs Count). Could Milevsky be assuming distribution of outcomes is different than it actually is?

12 Simulation Problem with Distribution? Are stock returns normally distributed?

13 Distribution of Large Company Real Stock Returns (Ibbotson Associates) Normal was best fit of actual data

14 Lognormal vs. Normal Distribution to Simulate Finance research may assume lognormality for stock returns. This doesn’t describe the output of ending value retirement savings and as can be seen the bankruptcy count pctges are nearly identical

15 Does Lognormal Make a Bad Situation Worse? This further justifies using the normal distribution to limit skewness at least to some degree

16 Distribution of Output Distribution of output from one of the simulations – 3,000 simulations (computer memory balked at 10,000). Skewness is apparent. choice for fit was had to subdue skewness to make this fit.

17 Milevsky and Gamma Milevsky’s heuristic assumes Gamma distribution Does Inverse Gaussian (also called Wald distribution) that is the best fit (and not perfect fit) for data mean M’s stats are prone to error? Count of events in simulation is best measure under uncertain distributions and statistical applications

18 Optimal Portfolios and Bankruptcy Risk Compare risk of bankruptcy for portfolios ranging from 100% stock to 100% bonds with different market conditions. and Milevsky’s Heuristic advise similar strategies and identify similar risks?

19 Worst mkt returnstd dev stock4.28%17.48% bond-2.24%7.60% Pctge of bankruptcies rises after bond allocations top 50%. Bonds had very poor real returns (negative). But Milevsky’s graph portends riskier portfolios than the simulation.

20 Best Mkt Under the best mkt conditions bankruptcy is very rare as a pctge of 10,000 simulations – not even hitting 3% with all bonds. Milevsky’s curve rises more quickly w/bond assets again showing more risk in general. return std dev stock bond

21 On the other side of the curve Milevsky’s heuristic underestimates bankruptcy risk when withdrawals increase which was shown earlier.

22 Annual Bankruptcy Risk Using worst mkt data and 50/50 portfolio mix the annual cumulative bankruptcy risk (simulated count, Milevsky prediction) shows heuristic with much higher estimates until year 27. Heuristic overestimates early years.

23 Conclusions? Simulations outcomes are not necessarily of the same distribution as inputs. Caution in using normal statistics. Milevsky’s heuristic is “in the ballpark” when compared with pctge of bankruptcies.