# OneFPA > Journal > Will The True Monte Carlo Number Please Stand Up?

*by Moshe A. Milevsky and Anna Abaimova*

The recent best-selling book by Lee Eisenberg, *The
Number *(Free Press, 2005), has focused national attention
on a sum of money retirees need to live comfortably for the rest of
their lives. According to Eisenberg, depending on one's desired
lifestyle and income aspirations, this number ranges from a few
hundred thousand dollars to the millions. Many readers of the book
have been surprised at how high this number can actually be. In
fact, a PBS show that aired in mid-May of this year called "Can
Americans Afford to Retire?" is just another indication of the
national preoccupation with financial sustainability toward the end
of life.

Of course, as most investment advisors have known for years, a retirement number—if it actually exists—is vague and imprecise, as it depends on many economic unknowns, especially future equity market returns. After all, this number must be invested somewhere in order to produce income, and the portfolio return process is inherently random. This is precisely the reason so-called retirement income Monte Carlo (RIMC) simulations have become ubiquitous in financial planning. Numerous Web sites, software packages, and vendors have devoted themselves to computing "the probabilistic number"—namely, the odds that a given nest-egg amount will last for the rest of a retiree's life. Unfortunately—as we have found out the hard way—these estimates can be all over the map.

We set out to investigate the extent to which RIMC simulators—at least the ones that are widely available to the public—provide different answers to this question: Will my number last?

To do this in a quasi-scientific manner, we hypothesized a 55-year-old single male who is contemplating early retirement with a nest egg of $500,000, from which he would like to withdraw $25,000 (adjusted for inflation each year). His entire nest egg was assumed to be invested and rebalanced in a portfolio of diversified equities which was projected to earn an arithmetic average 7 percent (after inflation) each year, which is equivalent to a geometric average of 5 percent with a standard deviation or volatility of 20 percent. Our test case retiree was assumed to be in average health and with no other source of retirement income. Obviously this situation might not represent the typical user of an RIMC simulator—since, at the very least, their desired $25,000 per year should be supplemented by Social Security—but our objective was to design a benchmark case that could be used across programs. We tried a variety of other cases and the results—which we will describe in a minute—were generally the same.

Despite the inherent uncertainty, we embarked on this particular exercise with a pretty good idea of what the output or "the probabilistic number" should be. There are unique situations in which it is possible to obtain the probability of running out of money before running out of life, with no simulations required. In fact, the above mentioned test case we constructed is an ideal candidate.

Let us explain how one can get analytic solutions to a
randomness problem by examining a slightly different question. What
is the probability the total return from the S&P 500 index will
be greater than 5 percent in 2006? There are a number of
philosophical approaches to dealing with such a question. One is to
literally "dump" the one-year historical returns of the S&P 500
index in a "hat," sample from this hat a large number of times, and
then count the frequency with which the sample average exceeded 5
percent. This approach is at the heart of Monte Carlo. But as many
mathematicians know, anther approach is to fit a curve (for
example, the normal distribution) to long-term historical returns
and then evaluate the curve—that is, compute the tail
probability—at the 5 percent mark. Some readers will
recognize this as the difference between parametric (the latter)
and non-parametric (the former) approaches to forecasting.
Interestingly, when we adopt the parametric approach and assume a
constant withdrawal rate over a random retirement horizon, we also
can recover an analytic expression.^{1} In other
words, in simple cases we know the odds of retirement ruin and
sustainability without generating any random numbers—just
like we know that the odds of tossing four heads in a row on a fair
coin is 1/16.

Thus, using this analytic methodology the sustainability—for a 55-year-old who starts withdrawing a constant inflation adjusted $25,000 from a $500,000 retirement nest egg—is 75 percent, which means that the probability of ruin is 25 percent. Remember that although Monte Carlo generators are based on randomized returns or spinning roulette wheels, the randomly generated results should only differ from each other by a statistical margin of error. In fact, that is the point of a properly designed Monte Carlo simulation. The odds should eventually converge to the truth. Yet our preliminary analysis of over ten different calculators—six of which are publicly available and listed in Table 1—revealed a wide range of results for our 55-year-old retiree. The lowest sustainability number was 48 percent and the highest was 88 percent.

## Why Were the Output Results So Different?

Our first reaction was to blame the programmer or manufacturer of the RIMC for building a faulty product. Like all high-tech gadgets on the market, some are better than others, but that is a simplistic, knee-jerk reaction. The true reason for the divergence of results is more complicated and subtle. In fact, we traced the range of results to a number of factors, which leads us to the following insights.

1. Every sponsor has their own view on the dynamics of equity
markets and future stock prices. Some believe that equity prices
obey random walks with no dependence (that is, zero serial
correlation) from year to year, while others take a subjective view
on the so-called equity premium and thus implicitly opine on the
market's current valuation levels. Along the same lines, different
sponsors have their own correlation assumptions for asset classes.
Some define equity/stocks as essentially being the S&P 500
index while others take a broader Russell 3000 view of portfolio
allocation. All of this makes an obvious and important
difference.

2. Some sponsors consider retirement to have a random horizon based
on various population or annuity mortality tables while others use
Social Security Administration projections. Other sponsors or RIMC
programs pick an arbitrary 25 or 35 years of retirement and ignore
this important source of longevity randomness altogether. In some
cases, a 90th percentile of the mortality table is used as a fixed
horizon. Of course when these RIMC generators are used for couples
as opposed to individuals, the results can be even more
diverse.

3. Some programs have sophisticated models for the term structure
of interest rates (the yield curve), which drives bond returns,
while others assume a basic random walk process similar to the one
used for equity prices. Indeed, it is debatable whether a
sophisticated three-factor model of the LIBOR (London Interbank
Offered Rate) swap curve is any better than a coin toss in
predicting bond yield 25 years from now. But then again, one is
left with an uncomfortable sense that something is missing when the
RIMC simulation results don't care that the yield curve is on the
verge of inverting or that long-term interest rates remain
stubbornly low. In other words, everyone has their own answer to
the question: how does a bond fund behave?

4. The cost of living in retirement is often confused with the
general level of macroeconomic inflation in the economy. Note that,
for many years, the Bureau of Labor Statistics has been computing a
CPI-E (Consumer Price Index), an experimental inflation index for
retirees, which differs from the CPI-U (urban) or the CPI-W (wage
earners). The projected increase in a retiree's annual expenditures
might only be loosely correlated to the widely quoted CPI. In this
vein, some programs explicitly (stochastically) model the evolution
of inflation and then increase withdrawal rates by this amount each
year, while others implicitly model the real after-inflation
evolution of portfolio returns. While both methodologies have their
merits, the end results obviously will differ.

5. One factor that does affect the dispersion of results, but for
which we can provide no justification, is the use of a small number
of scenarios or random numbers. While running the requisite 100,000
scenarios that would provide a minimal margin of error is
computationally not feasible within most real-time engines, running
as little as a few hundred scenarios can be woefully inadequate.
Tails will be ignored and results will be biased by optimism.

## What Should Be Done About All This Randomness?

Some critics might view this as a critical flaw in the Monte Carlo methodology. We obviously don't think the baby should be thrown out with the bath water. Other participants in the industry might not view this divergence of estimates as a problem or concern at all and will rightfully claim that every software developer or "quant" is entitled to their own opinion on how to model asset classes, mortality, or serial dependence. In fact, it is not our objective to criticize any particular set of assumptions per se. Obviously, every Monte Carlo vendor or developer can justify their own assumptions and approximations. But, akin to widely reported economic forecasts, or even weather predictions themselves, the market will eventually reward those who get it right and punish those who are consistently wrong or provide unrealistic predictions. Indeed, we do have some sympathy with this free-market view.

Unfortunately, we do not believe the nascent retirement income planning industry can afford the luxury of ignoring the divergence of Monte Carlo results. First of all, many outsiders who are unfamiliar with the subtleties of simulations might erroneously interpret the dispersion of results as a flaw in the Monte Carlo methodology itself, instead of as a powerful technique for measuring and explaining risk. More importantly, we suspect that one of the key players in this industry—namely, security regulators such as the NASD, Securities and Exchange Commission, or even state insurance departments—will remain deeply skeptical of Monte Carlo forecasts and illustrations. Remember that these groups are frequently called upon to approve marketing and presentation material containing results of these simulations. If they continue to observe statistically divergent results on seemingly identical inputs—or even worse, results that are biased in the favor of the product that is implicitly being promoted—they will (justifiably) continue to frustrate the efforts of those who believe Monte Carlo is an ideal pedagogical tool.

In our opinion, one possible solution to this conundrum would be for the relevant stakeholders in the retirement income industry to develop a set of calibration tests (or certification procedures) for which all RIMC programs would generate the same results, within a statistically tolerable margin of error. This approach has been used quite successfully in the field of solvency and capital requirements, where the insurance or banking regulator allows a financial entity to use their own internal models to set capital requirements with the proviso that certain statistical calibration points must be met. We envision the same idea in the retirement income industry. The analytic formulas we described above—for some basic cases—can be used to compute sustainability and ruin probabilities with full precision. This suggestion is akin to the age-old method of testing a weight scale by using dummy one-, five-, and ten-pound weights to calibrate accuracy. There is obviously no guarantee the scale will tell the truth when used with any other weights, but the odds are pretty good.

In sum, while attempting to impose an industry-wide standard or benchmark for retirement income Monte Carlo results might be fraught with difficulty, we believe developers and promoters of RIMC engines should calibrate their output to a number of baselines scenarios. Obviously, the industry as a whole stands to benefit from less randomness in Monte Carlo.

*Moshe A. Milevsky, Ph.D., is the executive director of the
IFID Centre and associate professor of finance at the Schulich
School of Business at York University in Toronto, Canada.*

*Anna Abaimova is a research associate at the IFID Centre and
a recent graduate of the Schulich School of Business.*

**Endnote**

- For readers who are interested in precise what this formula
looks like and how this formula is derived, see chapter 9 of The
Calculus of Retirement Income: Financial Models for Pension
Annuities and Life Insurance by Moshe A. Milevsky, Cambridge
University Press (2006).

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