Simulation in Determining
Optimal Portfolio Withdrawal
Rates from a Retirement Portfolio
Michael Tucker
Professor of Finance
Fairfield University
* Please do not quote
without permission 1
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
2
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
3
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) - 1
beta:
(variance of returns+nat log of 2/(life expectancy)/2
given a drawdown (payout pctge) of initial wealth
4
Calculating Probability of Ruin
5
Excel Version Producing Probability of Ruin
From Milevsky Inputs
6
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
7
Estimating Risk of
Bankruptcy with Simulations
• Stochastic Future Value (SFV) is used:
N
 ((1  r )(W
n 1
n
n 1
)  S ))
rn = real return generated by simulation for period n
Wn-1= real wealth
S = fixed real dollar withdrawal rate,
N = life expectancy at retirement.
8
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%.
9
Probability (Percentage) of
Bankruptcy
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 Bankruptcy
20 Yr Retirement 5% Mean, 12% SD Using
NormDist, Count
Prob (Pctge) of Bankruptcy
Statistical probability
calculated as
@Riskmean(target cell),
@Riskstddev(target cell)
and then applying
NORMDIST in Excel for
each simulation, saving
outcome.
80%
70%
60%
stat prob of
bankruptcy
count
50%
40%
30%
20%
10%
0%
2%
4%
6%
8%
%
10
withdrawal rate
10
Comparing Milevsky to Simulation
Pctges of Bankruptcy
80%
70%
60%
50%
milev
count
40%
30%
20%
10%
9%
10
%
8%
7%
6%
5%
4%
3%
0%
2%
Probability 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
withdrawal rate
11
Simulation Problem with
Distribution?
• Are stock returns normally distributed?
12
Distribution of Large Company Real
Stock Returns 1926-2004
(Ibbotson Associates)
Lognorm2(1.0917, .204)
Normal(.0917, .204)
0.7
2.0
1.8
0.6
1.6
0.5
1.4
1.2
0.4
1.0
0.3
0.8
0.6
0.2
0.4
0.2
0.1
<
90.0%
-0.244
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
-0.5
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0.0
-0.4
Normal
was best
fit of
actual
data
using
@Risk
>
0.427
5.0%
90.0%
2.130
>
4.167
13
Lognormal vs. Normal Distribution
to Simulate
8000
6000
lognorm2
norm
4000
2000
0.
1
8
0.
0
6
0.
0
4
0.
0
2
0
0.
0
Bankruptcy Count Per
10,000 Iterations
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
withdrawal rate
14
Does Lognormal Make a Bad
Situation Worse?
2.5
2
1.5
lognorm2
norm
1
0.5
0.
1
9
8
0.
0
7
0.
0
5
6
0.
0
0.
0
4
0.
0
3
0.
0
0.
0
2
0
0.
0
Bankruptcy Count Per
10,000 Iterations
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
withdrawal rate
15
Distribution of Output
InvGauss(48270209,
68480973) Shift=-5409143
1.5
1.0
200
150
50
0.0
0
0.5
100
@Risk had to subdue
skewness to make this
fit.
2.0
-50
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.
Values x 10^-8
2.5
Values in Millions
5.0%
5.9
90.0%
5.0% >
121.6
16
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
17
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?
18
@Risk Model Prob of Ruin with 4.0%
Withdrawal Beginning Age 65 Using Worst
Stock Returns (max cv: 1956-1981)
100.0%
Prob of Ruin
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.
80.0%
60.0%
Mil
Risk
40.0%
20.0%
0.0%
10 90 80 70 60 50 40 30 20 10 0%
0% % % % % % % % % %
Pctge of portfolio in Stock
Worst mkt
return
std dev
stock
4.28%
17.48%
bond
-2.24%
7.60%
19
Best Mkt
@Risk Model Prob of Ruin with 4.0%
Withdrawal Beginning Age 65 Using Best
Stock Returns(min cv 1975-00)
Prob of Ruin
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.
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
Mil
Risk
10
8
7
6
5
4
2
1
4
0% 8% 6% 4% 2% 0% 8% 6% %
Pctge of portfolio in Stock
return
std dev
stock
0.11751
0.14410
bond
0.05700
0.13933
20
On the other side of the curve
100.0%
Prob of Ruin
Milevsky’s
heuristic
underestimates
bankruptcy risk
when
withdrawals
increase which
was shown
earlier.
@Risk Model Prob of Ruin with 9.0%
Withdrawal Beginning Age 65 Using Worst
Stock Returns (max cv: 1956-1981)
80.0%
60.0%
Mil
Risk
40.0%
20.0%
0.0%
10 90 80 70 60 50 40 30 20 10 0%
0% % % % % % % % % %
Pctge of portfolio in Stock
21
Annual Bankruptcy Risk
Cumulative Risk of
Bankruptcy Worst Mkt Data
Pctge Bankrupt
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.
70%
60%
50%
40%
30%
20%
10%
0%
risk
mil
10 12 14 16 18 20 22 24 26 28 30
Year
22
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.
23