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Fairfield University* Please do not quote without permission

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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

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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

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Formula (probability of Stochastic PV>Wealth0)

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) - 1beta:(variance of returns+nat log of 2/(life expectancy)/2given a drawdown (payout pctge) of initial wealth

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Calculating Probability of Ruin

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Excel Version Producing Probability of Ruin From Milevsky Inputs

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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

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Estimating Risk of Bankruptcy with Simulations

• Stochastic Future Value (SFV) is used:

)))()1(( 11

SWr n

N

nn

rn = real return generated by simulation for period nWn-1= real wealthS = fixed real dollar withdrawal rate,N = life expectancy at retirement.

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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, 89-100.

– Risk of bankruptcy from 50-80 retirement age at different withdrawal rates with mean return of 5% and σ =12%.

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Probability (Percentage) of Bankruptcy

Statistical probability calculated as @Riskmean(target cell), @Riskstddev(target 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).

Statistical Prob of Bankruptcy20 Yr Retirement 5% Mean, 12% SD Using

NormDist, Count

0%

10%

20%

30%

40%

50%

60%

70%

80%

withdrawal rate

Pro

b (

Pct

ge)

of

Ban

kru

ptc

ystat prob ofbankruptcy

count

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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?

Pctge of Bankruptcy (Count and Milevsky)

20 Yr Retirement 5% Mean, 12% SD

0%

10%

20%

30%

40%

50%

60%

70%

80%

2% 3% 4% 5% 6% 7% 8% 9% 10%

withdrawal rate

Pro

bab

ilit

y o

f B

ankr

up

tcy

milev

count

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Simulation Problem with Distribution?

• Are stock returns normally distributed?

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Distribution of Large Company Real Stock Returns 1926-2004

(Ibbotson Associates)

Normal(.0917, .204)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

-0.4

-0.3

-0.2

-0.1 0.0

0.1

0.2

0.3

0.4

0.5

0.6

< >90.0%-0.244 0.427

Lognorm2(1.0917, .204)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

>5.0% 90.0%2.130 4.167

Normal was best fit of actual data using @Risk

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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

Bankruptcy Counts with Lognormal and Normal Simulation 20 Yr Retirement 5%

Mean, 12% SD

0

2000

4000

6000

8000

withdrawal rate

Ban

kru

ptc

y C

ou

nt

Per

10

,000

Iter

atio

ns

lognorm2

norm

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Does Lognormal Make a Bad Situation Worse?

This further justifies using the normal distribution to limit skewness at least to some degree

Skewness of Lognormal and Normal Simulation

20 Yr Retirement 5% Mean, 12% SD

0

0.5

1

1.5

2

2.5

withdrawal rate

Ban

kru

ptc

y C

ou

nt

Per

10

,000

Iter

atio

ns

lognorm2

norm

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Distribution of Output

InvGauss(48270209,68480973) Shift=-5409143

Val

ues

x 10

^-8

Values in Millions

0.0

0.5

1.0

1.5

2.0

2.5

-50 0 50 100

150

200

>5.0% 5.0%90.0%5.9 121.6

Distribution of output from one of the simulations – 3,000 simulations (computer memory balked at 10,000). Skewness is apparent. Second @Risk choice for fit was lognormal.

@Risk had to subdue skewness to make this fit.

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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

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Optimal Portfolios and Bankruptcy Risk

• Compare risk of bankruptcy for portfolios ranging from 100% stock to 100% bonds with different market conditions.

• Do @Risk and Milevsky’s Heuristic advise similar strategies and identify similar risks?

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@Risk Model Prob of Ruin with 4.0% Withdrawal Beginning Age 65 Using Worst

Stock Returns (max cv: 1956-1981)

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

Pctge of portfolio in Stock

Pro

b o

f R

uin

Mil

Risk

Worst mkt return std dev

stock 4.28% 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.

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Best Mkt@Risk Model Prob of Ruin with 4.0%

Withdrawal Beginning Age 65 Using Best Stock Returns(min cv 1975-00)

0.0%2.0%4.0%6.0%8.0%

10.0%

Pctge of portfolio in Stock

Pro

b o

f R

uin

Mil

Risk

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 0.11751 0.14410

bond 0.05700 0.13933

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On the other side of the curve

@Risk Model Prob of Ruin with 9.0% Withdrawal Beginning Age 65 Using Worst

Stock Returns (max cv: 1956-1981)

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

Pctge of portfolio in Stock

Pro

b o

f R

uin

Mil

Risk

Milevsky’s heuristic underestimates bankruptcy risk when withdrawals increase which was shown earlier.

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Annual Bankruptcy Risk

Cumulative Risk of Bankruptcy Worst Mkt Data

0%10%20%30%40%50%60%70%

10 12 14 16 18 20 22 24 26 28 30

Year

Pct

ge

Ban

kru

pt

risk

mil

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.

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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.