Enhanced Monte Carlo Simulation
Introduction
Traditional Monte Carlo simulation involves generating a random series of
hypothetical numbers dispersed around a specified mean and standard deviation.
Typically, one would choose an average yearly return and standard deviation
similar to what has been seen in the past and generate a new sequence numbers
that looks like what has been seen in the past but differs from year to year.
For example, if the S&P 500 averages 12% yearly returns and the standard deviation
is 21% per year then one might plug these numbers into the Monte Carlo simulator
and see how the hypothetical portfolio would have performed over the years.
This can be done hundreds or thousands of times to generate a distribution of
results.
There are some problems with this approach, however. The following
twopart series outlines my approach to Monte Carlo simulation which I believe
has significant advantages over conventional Monte Carlo methodology.
Part I
My simulator is based on the Monte Carlo method but with one very important
exception. Monte Carlo methodology does not allow for "reversion to the mean" 
in other words the tendency of long periods of underperformance to usually be followed by
out performance and vice versa. Monte Carlo simulators don't correct for this,
i.e. a string of high returns is just as likely to be followed by another string
of high returns and vice versa. This tends to lead to overly pessimistic
determinations for "safe" withdrawal rates (SWRs) for retirement
portfolios.
What I did was create a random number generator where the standard deviation of
returns was approximately equal to the historical norm. I set the average return
to whatever number I want (more on this later). If the random number sequence is
not within a certain tolerance I set then it is thrown out, either because the
mean or standard deviation are off or there is no mean reversion tendency from the number
sequence. This typically means rejecting around 9799% of the number sequences.
The tendency to mean revert is manifest by the long term standard deviation (SD)
of a return
sequence being smaller than would be predicted from the short term SD. For
example, if the one year SD is 18% then you would expect the 30 year SD to be
18/sqrt(30) or about 3.3%. In practice, however, it is about half that for most
equity markets, including ours, likely because of mean reversion. To correct for
inflation I use only real returns for the simulator.
I usually generate multiple random sequences of 75 years which, if not rejected
for reasons described above, are used for a simulated very long retirement (I
have made it 55 years long since I am retired in my 40's and have designs on
living to 100  haha!).
I arbitrarily pick the middle 55 years of the 75 year period for the simulated
retirement. I run this 1000 times and look a the results for various withdrawal
rates.
For example, historically a 75:25 mix of the S&P500:fixed income gave a compound
annual return of 5.6% with a one year SD of 15.2% and a 40 year SD of 0.76%. If
I run these portfolio factors through my simulator I get 1000 75year periods
where the compound annual return ranged from 4.4% to 6.8% and the one and 40 year
SD's are within a 10% tolerance window (i.e. 15.2% +/ 1.52% and 0.76% +/
0.076%). For a simulated 4% withdrawal rate I come up with an 89% success rate
for 55 years, not too far from the 95% success rate actually observed from
19272001. Here is how that looks graphed out (red dots = financial ruin at 4%):
So what if future real equity returns are in the 34% range as predicted by the
dividend discount model and virtually all other valuation models at the time of
this writing (summer 2003)? Well, if I run
the same 75:25 mix through the simulator changing only the average return for
the S&P500 to reflect a 3.5% real return we get a very different looking chart
for a 4% withdrawal rate:
Here the portfolio returns range from 2.3 to 4.7% but the SD's are kept the
same. For fixed income I used intermediate bonds in both simulations 
both times I plugged in the historical compound annual growth rate (CAGR) of 2%
(real) for the bond returns. The
failure rate is 52%!
So what I'm saying here is that if we get around 3.5% and 2% returns for stocks
and bonds, respectively, along with historical average volatility you would have
a roughly 50:50 chance of going broke in less than 55 years with a 4%
inflationadjusted annual
withdrawal rate from a conventional 75:25 portfolio mix.
Part II
Simulation would seem to be a
good way to get a handle on how reliable a SWR (safe withdrawal rate)
determination is. In other words, what is the standard deviation of , say, a 4% withdrawal
rate calculation.
Here's a stab at it. First I looked at what we actually saw for the last 130
years for a 55 yr. withdrawal period:
Actual portfolio values (75:25 S&P500:Commercial paper, 0.2% annual
expenses) observed since 1871:
Compound annual real return for portfolio: 5.7%
Std. dev. of portfolio for 1, 10, and 40 years: SD1 = 13.6%, SD10
= 3.13%, SD40 =
1.08%
Success rate = 86% @ 4% annual inflationadjusted withdrawal for 55 years
(109 overlapped periods, some partial)
Mean safe withdrawal rate (SWR) : 5.68% (109 overlapped periods, some
partial)
Now for my simulations:
First simulation (1000 separate 75 year runs, middle 55 years of each run used,
same rate of return and volatility as seen historically):
Compound annual real return for portfolio: avg = 5.7% (4.6  6.9%)
SD1, SD10, SD40: same as historical values above (+/ 10% tolerance)
Success rate = 87% @ 4% annual withdrawal
Mean safe withdrawal rate (SWR) : 5.93% (SD = 1.81%)
Here is the scatter plot of the simulation:
As you can see, the 4% success rate and mean SWR are very similar for the
historical data and my simulated results, giving me confidence that the
simulation method is sound.
The main item of interest to me is the SD of the simulation's mean SWR, 1.81%.
What this means to me is if we have the same average returns and volatility
(SD1, 10, & 40) in the future compared to the past then a retiree could expect
only an 68% chance that his SWR will be between 4.12 and 7.74%. This leaves a
roughly 16% chance that the SWR may turn out to be over 7.74% and, more
importantly, nearly a 16% chance that it will be below the 4% number that is
bandied about as being "100% safe" according to some who cite historical studies
and assume that "the future will be no worse than the past".
In fact, both the historical results and my simulation show that there is nearly
the predicted 16% chance of failure at a 4.12% withdrawal rate (mean minus one
SD). Here are the failure rates I get with withdrawal rates of 27% in the
simulation:
SET 1
7%: 72.1%
6%: 54.1%
5%: 31.4%
4%: 13.3%
3%: 2.0%
2%: 0.0%
These numbers vary a little from run to run but not by much. 1000 nonoverlapped
periods seems to be adequate. Anything more is overkill in my opinion since no one knows
what future returns will look like. I feel that
there is is good reason to believe that returns will less in the future.
This would lower the SWR a lot.
Also clear to me is that the historical data is woefully inadequate for
establishing an accurate longterm SWR since it consists of only about 130 years
with considerable data overlap. In fact, there are barely two nonoverlapping 55
yr. periods vs. 1000 in my simulation. I used to cringe when I'd see a statement
on Internet discussion boards like "a 4% withdrawal rate for 30 years is 96.56% safe". That sort
of 2 or 3 place decimal notation ascribes WAY too much precision to any such
conclusion.
While I think my simulator gives me a very good idea of what a SWR would be at a
given portfolio rate of return and volatility it is limited by the inability to
know what these numbers will be in the future. If for example, the S&P500
returns an inflationadjusted 3.5% per year for the next few decades which seems likely to smart
people such as Wm Bernstein, Warren Buffet, etc. then the failure rates would look
something like this:
SET 2
7%: 95.8%
6%: 89.6%
5%: 74.7%
4%: 50.8%
3%: 22.0%
2%: 2.4%
Compare this to Set 1 above and you'll see the difference that only a 2.2% per
year difference in returns can make in the SWR realized by the investor.
Similarly, modest changes in portfolio volatility also can make a big
difference. Here I increased the yearly portfolio SD by 2% but left the rate of
return at the same historical annual arithmetic average of 6.7% which yields a
geometric (compound) average return of about 5.4% (lower than the 5.7% cited
above because of increased volatility). The failure rates:
SET 3
7%: 74.7%
6%: 58.4%
5%: 42.5%
4%: 23.3%
3%: 6.3%
2%: 0.1%
These numbers look better than Set 2 but still show that a slightly more
volatile portfolio would have roughly double the number of failures at 4%
compared to the historical norm (23.3% vs. 13.3%).
Last edited:
06/28/2004
