A Critique of Monte Carlo
A deterministic retirement calculator is computer software that uses parameters representing
the user's financial situation and plans to project spending during retirement. A few of
these parameters are expectations of environmental forces beyond on the retiree's
A Monte Carlo Retirement Calculator is a deterministic calculator that is run many
times with randomly generated investment return values and inflation values to project
the retirement situation in a variety of economic environments.
There are certain criteria by which a Monte Carlo retirement calculator can be measured:.
- Framing the Problem: Monte Carlo retirement calculators compute when retirement
funds are exhausted, based on a specified rate of withdrawal. Another approach is maximizing
the amount of money that is available for spending based on a fixed life expectancy.
- Comprehensive Model: The progressive Federal personal income tax significantly affects the
management of retirement funds. Omit it, as most Monte Carlo retirement calculators do,
and the calculations are irrelevant.
- The Input Domain: The choice of how the random values for the model are generated
has a big impact on the utility of the model.
This paper explores these criteria and then discusses how the Monte Carlo option of
The Optimal Retirement Planner (ORP)  addresses them.
Retirement Calculators Defined
A retirement calculator is computer software that uses input parameters
reflecting the user's age, financial situation and retirement plan choices to compute
how the retirement savings will be spent over retirement.
A comprehensive retirement
calculator will have the input parameters shown in Appendix A. Ockham's Razor needs
to be kept at hand during parameter list definition. Parameter and features
that apply to only a tiny minority of users or that have very little impact
on the models results need to be culled.
For purposes of this paper the following terms are used:
- Calculator or retirement calculator is a calculator that uses a set
of parameters to generate a report describing the spending over time.
- MC calculator is a Monte Carlo retirement calculator which runs a calculator many time
with randomly generated values for exogenous variables.
- Optimizer is an implementation of a retirement planner using
linear optimization, sometimes known as linear programming LP).
- Stochastic LP is linear programming software that been extended
to solve an LP model many times with randomly generated exogenous values. 
- Software includes any or all of the above.
- Model is the mathematical expression of the logic and rules
that are implemented by the software.
The retirement software should be both a strategic planner and
an educational tool.
- As a strategic retirement planner it should compute results that accurately reflect
the real world situation at hand,
meaning the age when the money runs out or the amount of money
available for spending, should be realistic. From this the strategic outline
of retirement can be derived.
- As an instructional tool the retirement calculator illuminates issues that
affect retirement. This may be on the input form and help documents where, for example,
the usefulness of a reverse mortgage is brought to the users attention. Or it may
be in the computed results where withdrawals are made
from more than one account in a year or an IRA to Roth IRA partial rollover
Dangerous Side Effects
The appearance of the parameters shown in Appendix A on the software's input parameter
form leads to the conclusion that the software is using them in its computer
modeling. Parameters listed here but absent from a particular software
indicate that the retirement calculator is missing required features and will not
produce results that are truly representative of the user's situation. For example, most
users of retirement software own their own homes and the asset value of their home will
have a substantial equity buildup by the time they decide to sell it. When they choose to sell their
home will have a huge impact on their retirement picture.
An incomplete model representation leads predictably to results with one of two unwanted side effects:
- Overly optimistic results will fail to accurately predict when the money will run out
leaving the retirees to move in with their heirs.
- Overly pessimistic results will understate the amount of money in the estate leaving
the heirs with a windfall while the retirees had lived unnecessarily frugally.
Monte Carlo Simulation
Wikipedia defines Monte Carlo simulation as follows:
Monte Carlo methods are a class of computational algorithms that rely on repeated random
sampling to compute their results. Monte Carlo methods are often used when simulating physical
and mathematical systems. There is no single Monte Carlo method; instead, the term describes a
large and widely-used class of approaches. However, these approaches tend to follow a particular pattern:
- Define a domain of possible inputs.
- Generate inputs randomly from the domain, and perform a deterministic computation on them.
- Aggregate the results of the individual computations into the final result.
Framing the Problem
The thought processes of retirement planning is evolutionary and goes through these steps:
A big difference between a calculator and an optimizer is that the optimizer's reports includes a
plan for phased withdrawals from all three savings accounts (IRA, Roth IRA, After-tax). In many
cases these are parallel withdrawals from more than one account in the same year. Calculators
depend on the user to specify the withdrawal strategy which is usually something on the order of:
- Preserve the capital and live off the interest and dividends: This approach does not require a model.
This concept is soon discarded because it does not account for the impact of inflation and rather than enjoying
the benefits of saving for retirement it causes the retiree to live too frugally and leave too much
to the heirs. A good retirement plan will spend down the assets as time goes by.
- The 4% rule: The annual withdrawal is fixed at 4% of the sum of the money
in all asset accounts on the day of retirement.
This is the amount of money available for spending during the first year of retirement.
Subsequent years use the same amount after adjustment for inflation.
This approach was first popularized by the Trinity Study and continues to be recommended
by many financial advisors and is the basis
for most retirement calculators. Alternatively, some retirees set their annual spending
at 75% (or some such fraction) of their pre retirement annual spending.
However done, the annual spending is required by a conventional retirement calculators which
computes the year that the money runs out (called the year of ruin by some authors)..
This not a useful approach because the chosen values
are arbitrary. How does one predict the total amount of assets the day of retirement without
doing market forecasting. Is the day of retirement more meaningful that the week before
or the month after retirement? This date is purely arbitrary! The 4% spending rate is
arbitrary as well, founded on somebody's educated guess or based on academic studies that
use overly simplified models that fail to take into account the complexities of
retirement factors. Kotlikoff  delves deeper into these Rules of Thumb
and ends up characterizing them as Rules of Dumb. Sharp, et al  demonstrate that
the 4% rule is inefficient to the point costing the retiree real money.
Dogma repeated often enough becomes truth.
ORP regularly produces
retirement plans with spending rates north of 6%. For most retirees this is a substantial amount of money
to be taken out of the estate and spent on annual consumption.
Conventional retirement calculators, Monte Carlo or otherwise, compute the year of ruin
because that is all such calculators are capable of.
- Optimization: Optimize for maximum money available for spending, after taxes are paid and inflation is
accounted for, throughout retirement. Calculators, both stochastic and deterministic, require
the user to enter his spending rate and the calculator computes the year of ruin.
Optimization stands this concept on its head: Optimization requires the user enter her life expectancy
and the optimizer computes the amount of money available for annual spending. Life expectancy
is based on general state of health and genetic factors, Alternatively the IRA actuarial tables can
The superiority of optimization over this Rule of Thumb is discussed in .
- Spend all of the After-tax account first.
- Spend the IRA next.
- Spend the Roth IRA last.
Although MC software solves the same model many times, with different exogenous variable values
the basic software needs to be just as comprehensive as deterministic software. All retirement planning
- Model the Federal and state progressive
income tax codes (i.e. the more you earn the higher the tax rate).
- Model IRAs and Roth IRAs as accounts separate for the standard brokerage account.
- Model the retiree's and spouse's tax advantaged retirement accounts separately.
- Taxes on IRA withdrawals combined with taxes on Social Security Benefits will significantly
reduce the after tax amount of money available for spending.
- The retiree's house is a significant capital asset and its sale will significantly
effect the amount of money available for spending over the life of the plan.
- Although not in effect for 2009, the IRA minimum withdrawal requirement with effect
spending plans in future years.
These are just some of the factors that make a difference in formulation of strategic
spending plans for retirement. More important than modeling these factors is computing
the interaction between them in arriving at a final solution.
The stochastic optimizer, running in Monte Carlo mode, uses the same model that it uses for a
normal solution; it just solves it
many times with random asset return values and without the formal reports. It is the nature
of optimization that its solutions with be at least as good as a calculator's and almost
always better. There are solutions that optimization will compute that cannot be reached
by a calculator.
Random Value Generation
Monte Carlo software assumes that the two exogenous
parameters, rate of inflation and return
on asset investment, are beyond the retiree's influence. Therefore these two parameters
((Domain of possible inputs) are omitted
from the user settable parameters. The Monte Carlo Simulator uses
randomizing process to generate values for these parameters
for each trial of the retirement calculator.
There are at least four different ways of computing
the random values.
- Uniform Distribution: Any number within a specified range has an equal chance of being used.
Except for games of chance the uniform distribution generator has no know use in financial
- Normal Distribution: The domain of selection is bell shaped about some selected average value
and selected width (as measured by its standard deviation).
Some calculators allow the user to select the average and width. Use of the normal
distribution means that when two or more values are being generated that they are totally independent and
do not correlate with each other.
Nawrocki  writes that investment returns are not
normally distributed. Further, there are many situation where there will be trends in the
data being simulated that are not captured by the normal distribution. For example in a bear
returns after a down year typically may have another down year. In a bull market investment returns
may show several up years in a row.
The chosen average and width has a big impact on the calculator's results.
- Random Walk: The values are chosen randomly from a domain of historical data.
A particular year is randomly selected and the historical values for all generated
variables are chosen for that year. This deals with the problem of correlated values, e.g.
investment returns and inflation. Malhotra  demonstrates with simulation results that
the values generated by the Random Walk method are unlike anything
actually produced by the stock market.
1926 to the present is a popular time period for historical values. A more realistic choice
would be 1946 to the present. The.Full Employment Act of 1946  put the Federal government,
and particularly the Federal Reserve, directly into the business of stabilizing the U.S. economy.
The economic environment has been different since 1946 than it was before World War II so this domain of
asset returns and inflation rates is more representative of what will occur in the future than
those from the Great Depression.
- Random Interval: A starting and ending year are randomly selected and then the values
supplied to the calculator are the same values in the same order as the selected historical period.
Random intervals overcomes the trend problem and correlation problem noted with normal distributions. See Otar
However, this approach still has the flaw that it is based on United States market history, 1900
(or some such year) to the present. Since this only one of many historical sequences it is
antidotal in some sense.
Evensky  says it is vital to select a random value generator that will replicate the distribution of values
in the future. This is of course impossible. The best that can be done is to replicate
the past which carries no guarantee for the future. Evensky and Nawrocki make the case
that the distribution of the random numbers are the Achilles' Heal of the Monte Carlo Method.
A calculator is used to solve the model with a combination of user inputs
and random values (the deterministic computation).
The user specifies an amount that she hopes will be available for spending during retirement
and the MC calculator makes anywhere from 2,000 to 10,000 trials. Each trial is done with new randomly selected values.
Each solution is recorded at the conclusion of each trial. After all trials are completed
the final result is displayed.
The final result is a probability that the retiree will not run out of
money before the end of the plan. Probabilities between 90% and 99% are deemed safe.
A far more unpredictable exogenous variable is the United States Congress meddling with the US Personal
Income Tax Code. This is an infrequent but completely unpredictable event. The resultant tax tables
are completely unknown until the legislation is passed. This random event is not modeled by any
retirement calculator, Monte Carlo simulator or otherwise.
Finally, an additional complexity is that asset returns and inflation rates are correlated to some
degree. Thus, the random values generated for asset returns are influenced by the rate of inflation
for the same year. Asset returns are not independent of each other.
The Optimal Retirement Planner (ORP)
In preceding sections it is shown that the optimizing retirement calculator frames the retirement problem
differently than conventional calculators and that it includes superior detail in its model. In this section
the technique that ORP/MC uses to generated its randomly selected asset return values is described.
Generating random asset values for a retirement calculator should honor two aspects of the problem:
- The generated values should be historically reasonable.
- Values generated for annual asset returns need to take into account the fact that
the sequential annual asset return values are not independent of each other in that down years tend to follow preceding
down years and likewise for up years.
ORP's method of generating random annual return values is a follows.
The first step is
to give a historical bias to the randomly generated annual investment return.
S&P 500 annual returns from 1955 to the present were collection from yahoo.com. This is done
by creating two lists, Pos and Neg. Pos contains returns where the preceding year was positive.
Neg contains returns where the preceding year was negative. For details see Appendix B.
ORP generates random investment returns for all years being modeled beginning with current
year. When it generates a new value it looks at the sign of the value for the preceding year
and randomly selects a value from Pos if the previous year was positive or from Neg if the
previous year was negative. (See Appendix C for the details.)
The random values are scaled by the user's market investment policy as
specified in the investment return boxes on the input form. A conservative investment policy
will cause the coefficient to be scaled down and an aggressive investment policy will cause
the coefficient to be scaled upwards, increasing volatility. (See Appendix D for the details.)
ORP/Monte Carlo repeats his process many times to obtain many solutions. It creates and solves
models until the average Amount Available for Spending After Taxes stops changing. If
this doesn't happen ORP will stop after 500 stochastic iterations.
Astute ORP users will observe that standard ORP will compute a value for amount available for spending
that is larger than ORP/MC's average value for the same set of parameters. This is variance drag.
Bernstein  defines variance drag succinctly as:
Varying standard deviation (SD) also varies return, via so-called "variance drag." Say you begin with an annual
return of 10% each and every year (zero SD). You will of course wind up with a 10% annualized
return over the long haul. But crank in a normally-distributed random term with 20% SD,
and you find that your range of annualized returns falls to a median of about 8%.
Variance drag can be justified in the real world. Standard ORP assumes an actively managed
portfolio which has less volatility than the stock market. ORP/MC assumes that
the retiree's portfolio is at the mercy of the market.
This version of ORP/MC does not randomize inflation for two reasons:
- Since the 1980's the Fed has become an inflation hawk and price changes have been less
volatility then asset returns. Thus inflation does not have that much impact on
- For technical reasons modeling inflation in ORP is more difficult than asset returns
so inflation will have to wait for the next release.
Monte Carlo Retirement Calculators are only as good as their components:
- Domain of random numbers. Nawrocki states that Monte Carlo simulation techniques are not
valid for financial modeling because realistic random values cannot be generated.
- Deterministic computation: Most of the deterministic calculators underlying the Monte Carlo
calculators lack the
parameters and features listed in Appendix A. Therefore the deterministic results are
unlikely to realistically represent the retiree's true retirement situation.
Most calculators will err by being overly pessimistic,
over estimating the time until ruin.
The most serious problem is that conventional Monte Carlo retirement calculators answer the wrong question.
The retirement question they attempt to answer is:
When will the money run out?
The relevant question is:
How much money can I spend each year so that my money will last a lifetime?
All Monte Carlo calculators need to be used judiciously as such they give an assessment of the
risk faced by the retiree. ORP/MC is better than most because of how it frames the problem,
the detail of the model and its historical perspective in the generation of the random, annual investment
Appendix A: Important Retirement Calculator Input Parameters
These twenty one
parameters provide values necessary to compute a comprehensive
retirement plan. Changing the value of any one of these variables produces a significant
in the resultant computed retirement plan.
- Retiree and Spouse:
These accounts that have to be manage separately because age differences between
the retiree and spouse
combined with the tax law require that the accounts be treated individually.
- Current Age of Retiree and Spouse
- Tax-deferred Account Balances (401K IRA, 403B, Profit Sharing, etc.)
- Roth IRA Balances
- Tax-deferred Account Annual Contributions
- Roth IRA Account Annual Contributions.
- Annual Social Security Benefits
- Annual Pension
- Post Retirement Earned Income.
- Age When Earned Income is to stop.
- As a Couple:
These are parameters apply to the family and don't need individual treatment.
- After-tax Account Balance - conventional brokerage account.
- After-tax Account Annual Savings.
- Value of Illiquid Asset, e.g. the couples home.
- Age for Sale of the Illiquid Asset
- Reverse Mortgage Parameters
- State income tax code specifications..
- Optional Parameters:
Depending upon the calculator these parameters will be specified by the
user or computed by the retirement calculator.
- Annual Spending in Retirement.
- Age That Plan is to End -- Life expectancy
- Exogenous Parameters
- Estimated Rate of Inflation
- Estimated Investment Rate of Return.
- Built in features
- Federal Progressive Income Taxes
- Required Minimum Distribution for Tax-deferred Account.
Appendix B: Domain of Exogenous Annual Asset Return Input Values
The 700 annual investment return values for the S&P 500 wee accessed from Yahoo.com.
They were sorted by value
to produce the frequency distribution in Table 1:
Table 1 Frequency Distribution
|-0.4||3 ||Frequency of<= -.4
|-0.3||8 || Frequency of <= -.3 & > -.4
Table 2 Summary Statistics
This frequency is kind of a normal with a definite skew to the positive side.
To create ORP's asset return domain the original list, sorted by year is
directed into two other lists.
List Pos contains returns where the previous year's return was positive.
i.e. Pos list contains r(j) if r(j-1) > 0.
Likewise the Neg list contains returns where the previous years return was negative:
i.e. Net list contains r(j) if r(j-1) <= 0.
The two lists have these characteristics:
Table 3: Summary
for List Pos
Table 4: Summary
for List Neg
List Pos and list Neg are both sorted by value so that addressing each
with a uniform random number from the c language uniform random number generator
will be accessing a more or less normal distribution of investment returns.
Appendix C: Computational Algorithm
The idea is to compute a random annual investment return for each year j.
Compute model's return m for year j as:
For j=1, randomly select m(1) from the combined list.
Set m(j) = Pos(k) if m(j-1) > 0 where k is a random integer 1<= k <= 501, j > 2.
Set m(j) = Neg(k) if m(j-1) < 0 where k is a random integer 1 <= k <.= 195, j > 2.
Appendix D: Incorporating the User's Investment Policy
According to the statistics presented in Table 2 the average annual return for the S&P is 8%, or .08.
The user has specified her expected annual return on the ORP parameter form.
A specified return R < .08
is a conservative investment policy with a low volitility.
Expected returns R >= .08 are for an aggressive portfolio with a high volatility.
Each m(j) is adjusted to reflect the users expectations (hopes).
m(j) is scaled to reflect this using the scale R/.08, where
.08 is the average return of the combine list of returns and R is the user specified
expected asset returns. Note that if R < .08 then R/.08 is
less than 1.0 and return volitity is reduced. Similarly for R >.08.
Therefore the generated value is m`(j) = (R/.08) * m(j).
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