How risky is your retirement income risk model?

Patrick J. Collins, Ph.D., CLU, CFAa, Huy Lam, CFAb,*,
Josh Stampfli, MA, EESORc

aSchool of Management, University of San Francisco, 2130 Fulton Street, San Francisco, CA 94117, USA
bSchultz Collins, Inc., 455 Market Street, #1250, San Francisco, CA 94105, USA

c151 Rock Creek Lane, Scarsdale, NY 10583, USA

Abstract

Adequately sustaining lifetime income is a critical portfolio objective for retired investors. This
article provides a brief review of various retirement income modeling approaches including historical
back testing, Monte Carlo simulations, and other more advanced risk modeling techniques. Implau-
sible assumptions underlying common risk models may mislead investors concerning the risk and
return expectations of their retirement investment strategies. We compare risk models, evaluate their
credibility, and demonstrate how an oversimplified model may distort the risks retired investors face.
Differences in sustainability rates are stark: 4% failure at the low end versus 49% failure at the high
end. The article ends with general comments regarding model risk and practitioner investment advice.
© 2015 Academy of Financial Services. All rights reserved.

Keywords: The 4% rule; Monte Carlo simulation; Portfolio sustainability; Retirement income; Risk modeling

1. Introduction

Sustainability of adequate lifetime income is a critical portfolio objective for retired
investors. Commentators often define sustainability in terms of (1) a portfolio’s ability to
continue to make withdrawals throughout the applicable planning horizon, or (2) a portfo-
lio’s ability to fund a minimum level of target income at every interval during the planning
horizon. The first approach focuses on the likelihood of ending with positive wealth, or, if
wealth is depleted before the end of the planning horizon, on the magnitude and duration of

* Corresponding author. Tel.: �1-415-354-3704; fax: �1-415-291-3015.
E-mail address: huy@schultzcollins.com (H. Lam)

Financial Services Review 24 (2015) 193–216

1057-0810/15/$ – see front matter © 2015 Academy of Financial Services. All rights reserved.



the shortfall; the second focuses on the likelihood of consistently meeting all period-by-
period minimum cash flow requirements.

Risk models help advisors assess a portfolio’s ability to provide adequate cash flow
throughout retirement. Conclusions about cash flow sustainability are usually reached by
determining the likelihood that withdrawals (fixed amounts, percentage of corpus, or “dy-
namic”) can be maintained for either deterministic or stochastic time periods under various
asset allocations and longevity assumptions. Expressed in terms of a Venn diagram, portfolio
success lies at the intersection of the three elements in Fig. 1:

“Sustainability” differs from the concept of “feasibility.” Feasibility depends on an actuarial
calculation to determine if a retirement income portfolio is technically solvent—current market
value of assets equals or exceeds the stochastic present value of the cash-flow liabilities. If the
current market value of assets is less than the cost of a lifetime annuity, the targeted periodic
withdrawals exceed the resources available to fund them. In short, the portfolio violates the
feasibility condition. Determination of the feasibility of retirement income objectives is not
subject to model risk because the determination rests on current observables—annuity cost versus
asset value—rather than on projections of financial asset evolutions and the distribution of
longevity. Although it is important to track both risk metrics—sustainability and feasibility—as
part of prudent portfolio surveillance and monitoring, the remainder of this article focuses on the
sustainability or shortfall probability risk metric.

Fig. 1.

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2. Sources of retirement income model risk

Probability assessments are only as good as the models upon which they are based—that
is to say, assessments are prone to “model risk.” In general, model risk arises from several
sources:

1. Variables of Interest: Projected model outcomes may differ considerably depending
on the range of input variables. Health shocks, inheritance expectations, life insurance
availability, and other variables may or may not improve the calculated probability of
a successful retirement investment and consumption strategy.

2. Model Sensitivity to Changes in the Value of Input Variables: Output can be
notoriously sensitive to small changes in input values; likewise, compounding over
long planning horizons can produce large differences in outcome values and likeli-
hoods given small changes in input values.

3. Model Structure: Deterministic inputs will likely project outcomes different than
those generated by a model that treats investment, inflation, and longevity variables
stochastically. The nature of the model’s covariance matrix may be an additional
source of estimation error.

4. Model Assumptions: The choice of utility function can influence model output. For
example, assumption of Constant Relative Risk Aversion may rank outcomes differ-
ently from those flowing from models assuming a Hyperbolic Risk Aversion func-
tion.1 Likewise, the functional form chosen to generate inflation or investment returns
will often influence investment recommendations.

Econometricians often discuss model risk in terms of specification error. Errors may arise
as a result of including irrelevant variables in the model, failure to incorporate relevant
variables, and inaccurate estimation of input variable values. Specification errors may
account for different models producing dissimilar outputs when considering the same
problem. This is an underlying reason why any single retirement income risk model may be
unable to provide a good assessment of retirement risk.2

This article focuses on postretirement shortfall risk from assessments derived from
modeling investment, longevity, and inflation related risks. Model-based probability assess-
ments rely, in part, on outputs generated by computer algorithms that approximate, with
varying degrees of accuracy, the processes that drive financial asset price changes. Thus, any
assessment of the sustainability of a retirement income investment program should not over
rely on outputs produced by a single risk model; and, when using model outputs to monitor
the portfolio, practitioners should take care to ascertain that the model is academically
defensible.

Beyond the above-listed sources of model risk are two other considerations:

1. Bonini’s Paradox: models that explain the workings of complex systems are seemingly
impossible to construct. As a model of a complex system becomes more complete, it
becomes less understandable; for it to be more understandable it must be less complete
and, therefore, less accurate. When a model becomes accurate, it is just as difficult to
understand as the real-world processes it represents.

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2. Ambiguity in how the investor should preference rank heterogeneous outcomes:
outcomes from equally credible models may differ significantly even when each model
uses identical inputs and input values. Interpretation of calculated results becomes
difficult and there is no clear “winning strategy” or preferred solution path.

3. Modeling approaches

There are a number of modeling approaches to ascertain the likelihood that a portfolio’s
investment strategy is suitable to its cash flow requirements—that is, estimating the proba-
bility that a jointly determined asset allocation/retirement spending strategy is sustainable
throughout the planning horizon absent significant, and possibly difficult to implement,
midcourse corrections:

Y Analytic formulae (closed form solutions usually within a life-cycle model context)
Y Historical back testing of empirical returns
Y Bootstrapping (reshuffled historical returns)
Y Monte Carlo simulation (assuming a two-parameter normal or lognormal distribution)
Y Simulating non-normal distributions (Student’s t, Pareto, truncated Levy flight,

gamma, logistic, exponential, etc.)
Y Vector auto-regression
Y Regime-switching simulation models.

A brief discussion of each method follows.

3.1. Analytic formula

The analytic formulae approach attempts to solve the sustainability question by (1)
describing retirement planning as complex systems of equations, (2) transforming descriptive
formulae with algebraic manipulation to achieve closed formed solutions, and (3) plugging
in assumed values for the independent variables to arrive at a final conclusion. Most
formulaic solutions pile assumption upon assumption regarding the functional forms and
parameterized values of the numerous variables included in the model. To make the
mathematics tractable, many models assume that stock returns are independent, identically
distributed, and, as a consequence, that the underlying distribution of stock returns is stable.
Input variables may include rates of returns and volatilities for financial assets, inflation rate
behavior, interest rate term structure, and the form of an investor’s utility function. Specific
input variables are often estimated using econometric techniques that, although critically
important to an assessment of a model’s credibility, are, nevertheless, tangential to the focus
of this article.3 Generally speaking, most studies of stock price change reject the hypothesis
that the return series is normally distributed, with the most often cited deficiency as a failure
to capture the “volatility of volatility.”4

Finding the closed formed solutions to analytic formula models is a daunting task that
often requires applying highly sophisticated integral calculus or solving intricate partial
differential equations.5 Huang, Milevsky, and Wang (2004), for example, use a formulaic

196 P.J. Collins et al. / Financial Services Review 24 (2015) 193–216



approach to conclude that an inflation adjusted withdrawal rate equal to 3.33% of a
65-year-old investor’s starting portfolio value exhibits a 95% sustainability rate.

Despite its mathematical elegance, application of the analytic formula approach to sus-
tainability analysis has been quite limited. Solutions often seem enigmatic, and generally
require a complex array of equations. Another weakness of using analytic formulas to arrive
at retirement success rates lies in the fact that analytic models often ignore randomness in the
independent variables. Asset returns are the prime example. For instance, to assert that an
individual willing to run out of money after 10 years can withdrawal nearly four times as
much as one determined to preserve capital forever, Orszag (2002) assumes a constant dollar
return of 3% in his equations. Returns, however, are not constant; and the sequence of returns
can play a major role in portfolio depletion rates.

3.2. Historical back testing

Perhaps the simplest approach for determining the sustainability rate of a retirement
income plan is Historical Back Testing (also known as Rolling Period Analysis or
Overlapping Period Analysis). As the name implies, this approach relies on a sufficiently
long set of historical returns data. The historical returns used are the actual returns an
investor’s portfolio would have experienced, given its asset allocation. Many retirement
income risk models specify a withdrawal strategy throughout the planning horizon– often
a fixed 20, 25 or 30 years. A commonly evaluated strategy is the 4% rule that withdraws
an annual inflation-adjusted amount equal to 4% of the portfolio’s initial dollar value.
The historical back testing method tests the success or failure of the retirement plan for
each unique planning horizon in the data set of historical returns. The number of unique
periods is determined by rolling up the start date of each planning horizon by a single
increment of time. For example, Bierwirth (1994) begins his analysis in 1926 and uses
a one year rolling window to calculate 42 unique 27-year rolling periods, ending in 1992.
Each sample period is “unique” by virtue of the fact that its start year drops out of the
data set as a new ending year enters the data set. Intervening years, however, continue
to appear in multiple samples. Assuming that the past is indicative of the future, the
historical model calculates the likelihood of retirement income sustainability by dividing
the number of successful planning periods by the total number of rolling periods for any
given asset allocation/retirement spending strategy combination. The combination with
the highest success rate is considered optimal when optimality is measured by the
likelihood that the unmodified or “autopilot” withdrawal strategy is sustainable over the
applicable horizon.

The acceptable retirement income sustainability rate is highly subjective, and depends on
investor circumstances and risk tolerance. However, for the purposes of this discussion, we
will benchmark model outputs relative to the 75% guideline of Cooley, Hubbard, and Walz
(2011). That is to say, a retirement risk model incorporating an asset management strategy
exhibiting a 75% or greater likelihood of success is acceptable to a retired investor.6 The key
question is: how credible is the success probability derived from a particular risk model; or,
when is a model’s 75% or greater success rate not really indicative of a 75% or greater
likelihood for success?

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Historical back testing is easy to understand, and is a simple way to calculate relatively
accurate assessments of what would have happened.7 However, an investor relying on such
an approach should proceed with caution. Decisions based solely on historical data force an
investor to have faith in the highly dubious assumption that future returns will mimic realized
past returns.8 Furthermore, as stated, the rolling period method overweighs observations in
the middle of the time period relative to observations occurring at the beginning or end. Such
over weighting creates statistical bias. Extreme observations found in overweighted middle
time horizons can cause clusters of failed sustainability. O’Flinn and Schirripa (2010)
attribute one such cluster of failures in their study of withdrawal plans to significant inflation
during the 1970s and 1980s. Although modeling the serial and cross correlation of asset
returns is desirable, the rolling period approach fails to provide a sufficient number of
independent samples for credible portfolio sustainability testing. A looping method is one
way to deemphasize the importance of the middle observations in the data set (Davis, Wicas,
and Kinniry, 2004). Instead of stopping at the last observation date, the looping method
carries calculations back to the beginning of the historical return sequence. However, the
looping method distorts the value of the autocorrelation statistic in the historical dataset and
presents its own statistical difficulties. Although interesting, the pure history model fails to
provide assurance that past conditions are sufficiently similar to current conditions so that
they act as conditions precedent.

3.3. Bootstrapping

Retirement income risk models sometimes use a bootstrap approach to develop a broader
set of financial asset returns. Bootstrapping is a process that develops return sequences by
randomly drawing, typically with replacement, from the historically realized set of returns.
Randomly drawn sequences serve as possible future economic paths for testing the sustain-
ability of retirement spending. A large number of economic paths can be bootstrapped, thus
providing a larger set of scenarios for testing sustainability than is possible with the historical
back testing process.

Bootstrapping, depending on the structure of the risk model, can either preserve correla-
tions across asset classes—by making random draws which take a period’s realized returns
across two or more asset classes (“cross-sectional” random draws)—or eliminate correla-
tions across asset classes by taking random draws of asset returns from differing periods. It
is beneficial to preserve the covariance of asset class returns; and so studies based on
bootstrapping often preserve cross-sectional correlations. Spitzer (2008), for example, uses
the bootstrapping method to discern the best withdrawal rate and asset allocation over a
pre-specified time horizon given an acceptable portfolio sustainability rate. However, unlike
historical back testing, the bootstrapped scenarios do not preserve serial correlations evident
in empirical returns, because the bootstrapped returns for each time period are independent
draws from the set of historical outcomes.

Much like the rolling period technique, bootstrapping requires a long history of asset
returns. Without a long history, the sequences created from a small set of possible outcomes
will be too similar, with the result that the retirement income risk model performs sustain-
ability tests on scenarios that do not credibly reflect potential future economies. Even with

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a large history, sampled outcomes cannot differ from the pre-specified set of observables;
therefore, propagated series over rely on the past to predict the future.

3.4. Monte Carlo simulation: Normal distribution

The Monte Carlo method further expands the set of possible outcomes for sustainability
testing. It overcomes the limitation of relying on realized past returns as the basis of potential
outcomes inherent in both the historical back testing and bootstrapping routines. Monte Carlo
simulators generate sequences of potential economic paths by drawing random samples from
probability density functions meant to represent the true underlying distribution of financial
asset returns. This allows for a much greater range of potential outcomes. Most commonly
used Monte Carlo simulation engines assume asset class returns adhere to the well-known,
bell-shaped normal, or log-normal, probability distribution that is often parameterized by the
historical mean and variance. Furthermore, a well behaved correlation matrix is used to
preserve cross correlation in simulated outcomes. Using a simple two asset class Monte
Carlo simulation model with log-normally distributed returns, Klinger (2011) reports sus-
tainability rates above 85% for all retirement strategies he analyzed.

Klinger’s results, however, are based on the assumptions that stock returns average 6.9%
per annum with a standard deviation of 15.7%, and that bonds return 6.6% on average with
a standard deviation of 2.4%. The values of distribution parameters in simple Monte Carlo
simulation models have been a point of contention.9 Blanchett and Blanchett (2008) point out
that sustainability rates derived from simple Monte Carlo simulations are very sensitive to
changes in the assumed expected rates of returns and standard deviations. However, the
debate over what values constitute the most reliable parameters for the mean and standard
deviation may be a secondary concern if asset price returns are not normally distributed.
Moreover, much like the bootstrapping method, simple Monte Carlo simulations destroy
serial correlations evident in historical asset price returns. As we shall see, more complex
simulation engines can imitate the auto-correlated nature of returns and the time-varying
behavior of risk; but for now, we turn our attention to the topic of non-normally distributed
returns.

3.5. Monte Carlo simulation: Non-normal distributions

The normality assumption implies that returns are stationary, symmetric and, at reasonable
values for the standard deviation statistic, have low probabilities of realizing extreme
deviations from the mean. Simulated returns based on the Gaussian distribution exhibit, on
average, neither skewness nor excess kurtosis. Statistical analysis of historical returns by Lee
(2009), however, indicate that realized returns are slightly skewed (asymmetric) and have
higher likelihood of extreme events than predicted by a normal distribution (leptokurtic,
fat-tailed, or heavy-tailed). However, if the bell curve does not accurately represent the true
underlying distribution of returns, what other stable distributions can Monte Carlo engines
use? Levy and Duchin (2004) fit monthly historical asset returns to eleven different proba-
bility distributions and infer that the logistic distribution is the best fit for many asset classes.
Athavale and Goebel (2011) simulate asset returns based on 10 different distributions (Beta,

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Extreme, Gamma, Laplace, Logistic, Lognormal, Pert, Rayleigh, Wakeby, and Weibull) to
test the 4% withdrawal rule. They conclude that a 4% withdrawal rate tested in non-normal
distributions generally results in a lower sustainability rate when compared to test results
using a simple Monte Carlo method that assumes distributional normality. The major
econometric weakness of simulating one or more stable distributions, however, is that each
distribution assumes that periodic returns are independent and stationary with the result that
the model fails to capture autocorrelation.

3.6. Vector auto-regression

Simulations using non-normal distributions address some problematic assumptions; but
such simulations fail to account for autocorrelation in asset price returns. The time-dependent
nature of asset prices manifests itself through momentum and mean reversion in the return
series. More complex simulations based on vector autoregressive processes, however, can
better reflect serial correlations. Pang and Warshawsky (2009), for example, use a Vector
Autoregressive simulation model to compare six retirement plans using combinations of
mutual funds and annuities. In addition to addressing the serial correlation issue, simulations
based on vector autoregressive models can also incorporate what Campbell and Viceira
(2005) call “state variables,” which are variables useful for forecasting asset returns. How-
ever, the vector autoregressive approach is often complex and, for investment advisors,
difficult to implement. Furthermore, the coefficients of the vector autoregressive equation
must be estimated, and adding numbers of economic variables significantly reduces the
precision of estimated parameters (Campbell and Viceira, 2005). With additional suitable but
complex extensions, a vector autoregressive model can model heteroskedasticity and, there-
fore, capture a portfolio’s time-varying risk.10

3.7. Regime switching

An alternative to complex vector autoregressive conditional heteroskedastic models is a
regime switching model.11 Regime switching models assume returns come from two, or
more, sets of probability distributions—one representing asset price behavior during states
of normalcy, and the other(s) during financial crises. When financial crises occur, markets are
afflicted with a flight to liquidity and with the contagion of fear. Expected returns fall,
volatility increases, and correlations converge towards one.12 The model we present in this
article generates asset returns from two separate market regimes [Bull and Bear], with
inflation modeled as an autoregressive process.

Assuming a two-state model, for each simulated return path, the regime switching
algorithm calculates the probability either that the underlying economy will transition to the
other regime in the next period, or a corresponding probability (1 – probability of switching)
that the economy will remain in the same regime. Both Ang and Bekaert (2004), and
Kritzman, Page, and Turkington (2012), for example, utilize a variable Markov process with
a constant transition probability. As simulated economies evolve through bull and bear
markets of various lengths and magnitudes, the Ang and Bekaert model generates the

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volatility cluttering and correlation breakdowns that characterize asset price behaviors in
turbulent market conditions.

As stated, an advantage of a regime switching approach is its ability to capture dynamic
correlation and time-varying risk premia. Thus, instead of using average unconditional
correlation values determined by the historical data, the approach applies correlation values
conditioned on the economic state of nature. For example, over the entire sample period, an
asset class may exhibit a mean of 10% and a standard deviation of 20%. However, during
bull markets, the parameter values may be �18% mean and 15% standard deviation; while,
during bear markets, the parameter values may be �23% and 25% respectively. Thus, simply
using the unconditional mean, standard deviation and correlation values for the aggregate
historical period cannot capture realistic asset price behavior.

4. Inflation and model risk

Given the number of modeling approaches, it should not be surprising to find that there
is a correspondingly wide range of model outputs. However, the investor, or advisor, may
base decision making on the output from only one type of modeling approach; and,
furthermore, may not realize that even this single-perspective view of retirement risk may
flow from a model that incorporates oversimplified assumptions regarding critical factors
such as inflation, investment costs, and rebalance frequency. Here is the critical point:
variations in a model’s mathematical structure and input assumptions can lead to outputs
suggesting drastically different conclusions regarding the suitability of current asset man-
agement policies to a client’s financial objectives. For example, even within the simplistic
Rolling Period Analysis approach, using historical returns spanning different time periods or
varying the size of the rolling window can generate substantially different success or failure
probabilities. Understanding such sensitivities is essential to discerning the trustworthiness
of outputs from a retirement income risk model.

To illustrate model risk under a sustainability risk metric, we present outputs from our
proprietary risk modeling system which has the capacity to illustrate simulated outputs under
varying asset management and modeling approaches. The initial model incorporates only a
few basic variables and reflects a simple bell-curve structure for the distribution of future
investment returns. We then utilize risk models that incorporate more variables and that
allow for greater modeling flexibility. We demonstrate how an oversimplified model—many
of which form the basis for normative articles in the financial press—may seriously distort
the risks faced by retired investors.

Consider a simple, annually rebalanced, two asset class portfolio, allocated 70% to U.S.
equities and 30% to U.S. bonds. Initially, the model ignores fees, taxes, and transaction costs.
The model assumes normally distributed asset returns parameterized by historical averages
and standard deviations. Portfolio price evolutions are multivariate normal, where the
process derives from a single variance/covariance matrix assuming static (average historical)
correlation values for all future economies. The initial portfolio value is $1,250,000 with an
annual inflation-adjusted withdrawal of $60,000 for exactly 30 years. The portfolio consists
of only two asset classes and, therefore, is not well diversified.

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To best illustrate the risk of relying on over-simplistic retirement income risk models, we
focus on the inflation variable. A common way to incorporate inflation into a risk model is
to assume a constant rate. However, retirement income risk models are hypersensitive to the
level of assumed inflation with the result that success or failure probability assessments may
differ widely. Table 1 exhibits the inflation adjusted results of our two-asset class model
under three different levels of fixed inflation: 3%, 4%, and 5%. A one percentage point
difference in the assumed rate generates a marked divergence in ending portfolio values. At
the median,13 the model assuming 3% inflation yields an ending value worth roughly a
million dollars more in goods and services than the model assuming 4% inflation. In terms
of the 75% probability of sustainability benchmark, the outcomes of both the 3% and 4%
inflation models are acceptable (sustainability rate � [1 – bankruptcy rate]). The 3% inflation
input yields a 91% sustainability rate, which is fully 9% higher than the 4% inflation input’s
82% sustainability rate.

Inflation, however, is not constant; and, dealing with inflation in such an oversimplified
manner produces implausible outputs. More credible risk models treat inflation as a random
variable. Alongside the three constant inflation outputs in Table 1 are two additional outputs,
one based on inputting inflation starting at its long term average (4.32% during the period
from 1973 to 2012), and the other based on inputting inflation starting at its previous 12
month average (1.74% in 2012). Furthermore, the enhanced risk model generates paths of
future inflation by treating it as a stochastic variable exhibiting a mean reversion factor. The
inflation process has an expected value (drift) factor, and an innovation (diffusion) factor.
The result is an output that accounts for serial correlation in a random but “sticky” time series
of inflation rates. Because the 2012 rate is so much lower than its historical average, the
stochastic model of inflation using 2012 inflation as its starting point leads to a more
plausible estimation of a portfolio’s sustainability rate. This result, in turn, contrasts with the
results generated by a stochastic process starting off in the historically average inflationary
environment.14

We direct the interested reader to the 2006 study by Paul Kaplan (2006) for additional
discussion of how success probabilities differ depending on the retirement income
risk model’s inflation assumptions. Kaplan models inflation in three ways: constant,
single-period lagged auto-correlated process, and two-period lagged auto-correlated
process.

Table 1 Various models of inflation

Risk model with inflation Constant
3%
inflation

Constant
4%
inflation

Constant
5%
inflation

Stochastic
long term
average
inflation

Stochastic
previous
12 months
inflation

Ending wealth at the 50th percentile $2,417,712 $1,347,812 $665,967 $1,050,470 $1,423,391
Ending wealth at the 30th percentile $1,232,951 $499,252 $33,002 $129,020 $397,651
Ending wealth at the 10th percentile $71,029 $0 $0 $0 $0
Bankruptcy 9% 18% 29% 26% 21%
Assets ever � $750k inflation adjusted 27% 43% 57% 51% 44%

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5. Illustrating risk

5.1. A simple two-asset class model

Having illustrated how five distinct inflation behaviors yield significantly different as-
sessments of portfolio sustainability, we now turn our attention to the assumptions under-
lying the modeling of asset price evolutions. In our earlier discussion, we introduced the
Monte Carlo method for generating the distribution of future asset prices. Risk models using
simple Monte Carlo simulations assume normally distributed asset returns parameterized
with historic means and standard deviations. The simple Monte Carlo method also utilizes a
historical correlation matrix to account for the co-movement of asset prices. We designate
this model the “NH” model, for “Normal Historical.” It is a model commonly used by
financial advisors.15

Next, we modify the historical parameters of the basic Monte Carlo model to conform to
the single-index, Capital Asset Pricing Model’s assumption that all investments have the
same long-term expected real Sharpe Ratio in efficient markets. This modification assumes
that expected returns plot on the capital market line. However, the model maintains the
assumption of distributional normality. We designate this model NE, for “Normal Efficient.”
Although the NH and NE models differ in the values they use to parameterize the normal
distribution assumption, both assume time-invariant parameters.

Finally, we consider modeling variations that mitigate many statistical difficulties arising
when assuming distributional normality. We previously demonstrated that it is possible to
originate economic evolutions from non-normal distributions using a number of techniques:
bootstrapping, Monte Carlo simulations sampling from non-normal distributions, vector
autoregressive models, and regime switching simulations. Despite the fact that each approach
has its pros and cons, we prefer the regime switching approach when cumulating dollar
values over long planning horizons. Mary Hardy (2003), a prominent Canadian actuary,
stresses the importance of using credible risk models when cumulating dollar values over
lengthy planning horizons. She provides strong support for using a regime-switching model.
After an in-depth survey of various modeling tools and techniques, she concludes that the
best way to approximate the range of future portfolio dollar value within an asset/liability
matching context is through a two state regime-switching lognormal model.

Unlike simple Monte Carlo simulations, return series produced in a regime switching
engine exhibit all of the following empirical asset price behaviors: skewness, fat-tails,
autocorrelation, volatility clustering, and dynamic correlations. The first of our regime
switching models assumes an investor with an agnostic view of the capital markets. This
means that there is no attempt to predict whether the immediately forthcoming returns will
start in either a bull market or a bear market. Outputs, therefore, do not depend on the
accuracy of the investor’s forecasting ability. This variation of the risk model randomly
selects the underlying initial state of the economy where the selection of an initial bull or bear
market state reflects historical relative frequency. We identify this randomly selected initial-
state model “BB” for Bull/Bear.

However, if the investor wishes to impose a market viewpoint, there is an opportunity to
specify the forthcoming initial underlying economic state from which financial asset returns

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are generated. For example, if the investor strongly believes that asset prices will rise in the next
period, we start our regime switching procedure in a bull market, and we identify the non-
randomly selected initial-state model as the “Bull” model. On the other hand, if the investor
wishes to reflect a pessimistic outlook for forthcoming returns; or, if the investor simply wishes
to see how starting retirement in adverse economic conditions impacts portfolio values, then we
start the return-generating process in a bear market. We identify this non-randomly selected
initial-state model as the “Bear” model.

Putting these models into the context of retirement planning, we generate return evolutions
for an annually rebalanced, two-asset class portfolio, without accounting for taxes, fees, and
other investment costs. Inflation follows a stochastic process using a trailing 12 month
average starting value. We present the inflation-adjusted outputs generated for this “simple”
portfolio under the five distinct asset price models introduced above:

1. Multivariate normal distribution historical model (NH)
2. Multivariate normal efficient returns model (NE)
3. Regime switching model market agnostic (BB)
4. Regime switching model bull market prediction (Bull)
5. Regime switching model bear market prediction (Bear)

The risk-metric of interest is the likelihood that a $1.25 million portfolio distributing an
inflation-adjusted $60,000 annually (4.8% of its initial value) will become fully depleted—
that is, bankrupt—before the end of the planning horizon.16 We designate the series of model
outputs by the symbol “S,” for their streamlined inputs, in Fig. 2. Fig. 2 also presents outputs
from models we will discuss later.

Projected portfolio sustainability rates for the five models of the S series clearly depict
a wide range of possible outcomes. Although the commonly used NH model’s bank-
ruptcy rates are less than 25%, the investor is left to ponder the extent to which this

Fig. 2.

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favorable picture is merely an artifact of a risk model that fails to incorporate critical
elements of asset pricing behaviors. The NE model modifies parameters fit directly from
historical data; it produces a 6% higher portfolio risk profile. The market agnostic
retirement income portfolio risk model (BB) indicates that the risk of portfolio depletion
by myopically following a fixed asset allocation/4.8% withdrawal strategy is signifi-
cantly greater than suggested by the simple Monte Carlo model. Not surprisingly, by
starting the simulations off in a predicted Bull market environment, the bankruptcy rate
decreases— but only by 3% over the agnostic view of future markets. It is still 6% higher
than projected by the simple Monte Carlo model. On the other hand, having a pessimistic
view of near term market returns drastically increases portfolio bankruptcy rates to
approximately 49%. This outcome is illustrative of return sequence risk faced by
retires.17 If a retiree wishes to impose a market viewpoint on his or her investment
strategy—always a dangerous and uncertain proposition; or, if a retiree wishes insight
into financial asset performance in “worst-case” economies, the Bear model option
illustrates a distribution of future results reflecting implied pessimism.

5.2. A diversified portfolio

Luckily, investors do not live in a two asset class world. The portfolio returns generated
in our simplified set of models suffer from unsystematic market risk. A savvy investor would
diversify away the unsystematic risk inherent in their portfolio, and move it to, say, a 14 asset
class allocation.18 Doing so allows us to illustrate the effect of diversification within each of
the five return-generating models. In Fig. 2, we label the new series “D,” for diversified. By
taking advantage of broad-scope diversification, three of the five models in series D exhibit
acceptable sustainability rates, compared to just one in series S. Furthermore, each diversi-
fied portfolio in model series D has a lower bankruptcy rate compared with their two asset
class counterparts in series S.

However, it is interesting to see the relative benefits of diversification abate in models
incorporating regime-shifting BB methodologies.19 When measured by the improvement in
failure rates, there is only a diversification advantage of just 4% in the BB approach versus
a 9% improvement in the NH model. The relative diversification benefits diminish in a
regime shifting model. Turbulent market conditions tend to raise the correlation between
asset classes above their long term historic averages, thus reducing the benefit of diversifi-
cation. Neither the NH or NE models can replicate such conditions since their returns
generating mechanism uses only a single correlation matrix.

5.3. Longevity

Thus far, the model series assumes a fixed 30-year planning horizon. This assumption is
not realistic for individual investors with uncertain life spans. A 30-year planning horizon
may overstate shortfall risk for many post age 65 retired individuals. To get a more realistic
view of risk, the uncertain nature of longevity is integrated into the next series of simulations.
We term the series “DL” for a diversified portfolio reflecting an uncertain life span. The
lengths of simulated trials are no longer preset at 30 years. Rather, the distribution of life

205P.J. Collins et al. / Financial Services Review 24 (2015) 193–216



span reflects the Society of Actuaries’ mortality table for the subpopulation of high-income,
white collar retirees—the group most likely to use the services of a financial advisor.
Longevity risk (the likelihood of outliving resources) is a stochastic variable, not simply an
average. The distribution of life span for this population group differs significantly from the
distribution of life span for the general population. Inputting Social Security general popu-
lation mortality data, for example, decreases the failure rate probabilities significantly.
Although beyond the scope of this discussion, the issue of expected versus actual life span
is important when modeling retirement income portfolios.20 One result is that the uncertainty
of an investor’s remaining lifetime increases with age despite a reduction in expected
longevity. In fact, actuarial life expectancy is conditional on attained age—the longer you
live, the longer you are expected to live.

The DL model assumes a 68 year old female investor in excellent health. Average life
expectancy for such an investor is roughly 19 years (mean � 18.7; median � 19.1); and,
therefore, we expect bankruptcy rates to fall. Trial length is the lesser of 360 months (30
years), the date of portfolio depletion (bankruptcy), or the month of death, whichever event
comes first. As Fig. 2 shows, the mortality-adjusted time horizon lowers bankruptcy risk
dramatically. All five market models incorporating longevity exhibit portfolio sustainability
rates in excess of the 75% acceptability benchmark.

An investor using a simple Monte Carlo simulation program anticipates that there is an
approximately 4% chance that the investment strategy, operating across time without mod-
ifications, will be unable to meet critical needs. An investor with a pessimistic near-term
market outlook, anticipates an approximately 25% chance that the retirement portfolio will
be unable to provide the target lifetime income. The empirical data underlying each model’s
output is exactly the same. This means that the differences in risk measurement is due solely
to the structure of the retirement income risk models—that is, model risk. This does not
imply that the model generating worst-case results is the most credible. However, it does
suggest that inappropriate investment advice may be offered to investors based on outputs
from overly simplistic models.

5.4. Portfolio frictions (fees and taxes)

The simulations presented to this point have ignored investment costs such as trading
commissions, custodial/trustee fees, mutual fund/ETF expenses, investment advisor fees, and
so forth. The next series of simulations reflect the costs of investing for a diversified portfolio
paying management fees21 and transaction costs. Fig. 3 labels the series “DFL”—diversified
portfolio paying fees and expenses and reflecting uncertain life span. As seen in Fig. 3,
incorporating fees and transaction costs increases bankruptcy rates.

Just as death is certain, so are taxes. Taxes, however, are particularly difficult to model
because of (1) myriad nuances within the tax code, and (2) variations in investor tax
circumstances. The type of investment account often determines the nature and extent of
taxation. Additionally, the assets themselves may be tax exempt; interest and dividend
income are often taxed differently than capital gains income; turnover rates may determine
whether long or short term capital gains rates apply, and so forth. Nonetheless, taxes are a
cost factor either for setting the threshold target budget or for determining the drag on

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portfolio growth. Usually, investors calculate their income needs by including an estimate of
the taxes that they must pay. However, advisors may, from time-to-time, need to incorporate
taxes directly into the retirement income model.

Rather than ignore taxes, we have made some simplistic assumptions that allow us to
evaluate, to a reasonable extent, the impact of taxes. Our model maintains a constant tax
regime in that it does not forecast changes in tax law over the investor’s lifetime. We do not
include an allocation to municipal bonds, and we assume that no assets are held in tax
sheltered accounts such as 401(k)s, 403(b)s, IRAs, and so forth.22 Interest income is taxed
at the ordinary income rate—assumed to be 20%, and all dividends are tax qualified and are
taxed at a long-term rate of 15%. Long-term rates are also used for the taxation of investment
gains when assets are sold for both withdrawals and rebalancing. Assets are taxed at a blend
of short and long-term rates, depending on the annual turnover rates. We assume that an
asset’s initial cost basis is half that of value in the start year, and that portfolio investment
positions are low cost, low turnover, passive indexed funds. The model further assumes that
tax loses and or tax payments are accounted for monthly. Both operations are reasonably
consistent with the notion of prepaying estimated taxes quarterly.

Fig. 3 also compares previous pretax results to results from a model encompassing
taxation. In this case, the investor specifies that she requires a $5,000 minimum monthly
income net of income tax liabilities. We label the new outputs “DFTL” for a diversified
portfolio paying investment fees, paying taxes, and incorporating an uncertain planning
horizon by virtue of investor mortality. Because taxes are an additional cost, the DFTL
bankruptcy rate probabilities all plot above the previous outputs that ignored taxes.

The effect of our tax model is quite significant: more than doubling the failure rates in the
normal distribution models, and raising bankruptcy rates from 4 to 7 percentage points in the
regime switching models.

Fig. 3.

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5.5. Summary of results

Fig. 4 puts all five previously examined series together on a single graph. Our example
started with the investor, holding a streamline two-asset class portfolio (the S series),
demanding a constant retirement income stream for 30 years. Bankruptcy rates from those
simulations were somewhat alarming. The investor then sought help from an investment
advisor who recommended diversifying the portfolio. The benefits of diversification are
evident in the lower failure rates in the D series. However, diversification is less beneficial
in the regime switching models. This result is not surprising given the tendency for
correlation values to move towards one in times of market turbulence. However, it is unlikely
that a retired investor will need income for exactly 30 years. Incorporating the variability of
life span into the model yields results presented in the DL series. The drastic fall in failure
rates across all risk models indicates the importance of accounting for uncertain life span.
When unknown life span is considered, bankruptcy rates fall to a third of their level in the
NH and NE risk models: 12% to 4% and 16% to 5%, respectively. Failure rates are only cut
in half in the BB and Bull models: 26% to 14% and 22% to 11%, respectively. The Bear
model exhibits the least benefit from the force of mortality: a 40% reduction in shortfall rates
from 43% to 25%. The DL series depicts the lowest level of failure rates among model
outputs. Mortality affects sustainability rates in the normal-distribution models much more
so than it does in the regime switching models.

Investment advice is not free and asset management generates trading cost; therefore, we
further expand the models to include investment costs and advisory fees. The DFL series
incorporates the force of mortality into a diversified portfolio model decremented for
investment costs. The DFL model is a credible retirement income risk model when the
investor’s budget is defined in pretax dollars. However, retirement income may come
primarily from a trust established for the benefit of the investor. This may result in a tax

Fig. 4.

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liability that must be paid from the trust portfolio rather than by the individual investor.
Therefore, we extend our models to cover taxation in the DFTL series. Taxation has a lesser
effect on sustainability rates under regime switching models than under models assuming
normality in the return distribution.

Fig. 4 clearly illustrates that the most simple and commonly used risk model, the NH
model, understates risk given any set of underlying inputs or assumptions. On the other hand,
assuming markets start out in a bear regime may overstate the risk of retirement failure.
Nevertheless, it is always interesting to examine the “worst case” environment. The differ-
ences in sustainability rates are stark: a 4% failure rate at the low end versus a 49% failure
rate at the high end. Decisions based on implausible risk models may not be appropriate and,
thus, may mislead investors in assessing the risks and return expectations of their retirement
investment strategies.

6. Utility

Discussions concerning strategies to enhance portfolio sustainability differ from
discussions concerning how to optimize aggregate utility of consumption for a retired
investor with finite resources. Portfolio sustainability, when defined as the ability to fund
a minimum periodic target income, implies a state preference utility function. That is to
say, a retired investor may have a strong preference for avoiding periods during which
consumption falls below a minimum acceptable threshold. Such an investor is willing to
sacrifice greater utility in higher-portfolio-value states in favor of assuring a minimum
standard of living in lower-portfolio-value states. Aggregate utility—summed over all
consumption/investment states—takes a back seat to assuring, at least, a minimum living
standard in each state. Optimization of expected utility, in most retirement income risk
models, is a probability-weighted value taken over the entire distribution of outcomes—
that is, over all possible economic states from depression to prosperity. Summing utility
values over all states assumes a separable utility function. Furthermore, conclusions
derived from optimization procedures may differ drastically from those drawn from
sustainability analysis. For example, an optimal withdrawal rate in a study by Finke,
Pfau, and Williams (2012) requires that the utility-maximizing investor accept only a
43% sustainability rate. Tomlinson (2012) notices similar observations. Typically, com-
mentators tracking shortfall risk metrics would consider such an optimal withdrawal
strategy to be unacceptable.

Fortunately, whenever a threshold level of consumption must be maintained, the two
approaches sometimes coincide. The investor may apply a utility penalty—negative utility—
for failing to meet threshold income requirements. Some retirement income risk models
impose an additional penalty for exceeding a periodic income or ending wealth target.23 The
risk model imposes a penalty for overachieving under the supposition that target financial
objectives could have been met at a lower level of risk. In terms of consumption objectives,
some commentators view surplus ending wealth merely as a missed opportunity to enjoy a
higher standard of living throughout retirement.

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7. Conclusion

The normal (bell curve) distribution is not a good fit for most financial asset return series.
Quantifying investment risk by the first two moments of multivariate symmetric distributions
(Gaussian, Student’s t, etc.) is often misleading. Furthermore, Monte Carlo simulations based
on a normal distribution cannot realistically capture asymmetry in the distribution (skew) or
the frequency and magnitude of tail-risk events (leptokurtosis). Risk models inputting return
distributions with differing assumptions, operating under different stochastic processes, often
produce significantly different results. To mitigate deficiencies, we use a hybrid autoregres-
sive, regime-switching risk model using two state-dependent normal distributions with
separately calculated means and variances. The distributions, according to the Markov
transition probabilities, capture the frequency and magnitudes of outlier results better than
distributions produced by single-parameter input variables. Given a large number of simu-
lation paths, our retirement income risk model provides a rich set of future asset returns.
Although no investment risk model can predict the future, one hallmark of a credible model
is that it enables investors to make good decisions within a wide range of possible futures.24

Success or failure should never be evaluated in terms of just a single model—nor in terms
of just a single metric. A model, in many respects, is just one person’s—that is, the model
builder’s—opinion about how the future may unfold.

Given the computer power currently available to investment advisors, and given the
approximately 30 years of research into econometric topics such as time series analysis, what
accounts for the propensity to use oversimplified risk models? Perhaps the financial advice
profession suffers from what Paul Kleindorfer (2010) terms “Legitimation.” He defines the
term as follows: “. . . a credible anticipation of being held accountable not just for outcomes
but for the logic that led to them will have predictable effects on the nature of the choice
process itself.” A disgruntled investor’s demands for explanations regarding why and
how an advisor’s recommendation went sour may lead advisors to “play it safe.” In the
context of this article, advisors may tend to use only models commonly used throughout
the profession—for example, historical back testing or Monte Carlo simulation models—so that
if the investment strategy fails to produce its intended result, the advisor can take comfort in
the fact that most other advisors also got it wrong: “. . . the mere thought of making choices
of consequence under conditions of ambiguity and ignorance calls out for company.”
Undoubtedly, there are other explanations for the financial industry’s slow adoption of
academic advances in risk modeling. However, if the central focus of each investor remains
the intelligent consideration of risk/return tradeoffs, then the tools of the advisory profession
should be evaluated in terms of their ability to indicate the consequences of portfolio
management elections.

A model is an imperfect representation of a more complex reality. In this case, there are
(at least) two embedded “model risks” to consider:

1. Investors are interested in forecasts of a price change process. However, the time series
of asset prices is not statistically stationary (i.e., exhibits the potential for infinite
variance). It is only by differentiating the logarithm of prices on a period-by-period
basis that the creation of a stationary series of returns is possible. That is to say, it is

210 P.J. Collins et al. / Financial Services Review 24 (2015) 193–216



only possible to model asset returns; but an investor measures wealth based on asset
prices. This is a subtle but important distinction. Returns are based on the single
historical path of price changes, the realization of which is merely a manifestation of
an unknown price generating process. Simulation analysis greatly broadens our per-
spective about possible future outcomes; but any model of such a process must remain
only a crude approximation of reality. Indeed, calculation of investment return is a
function of the measurement interval (yearly, monthly, daily, intraday, continuous
time) and, at the limit, may be meaningless in a statistical context.

2. The single historically realized return path for each asset class may be “representative”
of the unknown price generating process; or, may merely be an outlier result unlikely
to persist. For example, an asset allocation tilt towards small and value stocks is
justified based on historical return data. If the premium for investing in small and value
stocks reflects a reward for systematic risks, then investors have some confidence that
they will continue to be rewarded for making these investments. If, however, the
premium for such investments is merely an artifact of a chance historical price change
process, then investors may be increasing risk as they move their asset allocation
deeper into the small/value gradient. Furthermore, investment volatility is measured by
the variance statistic—the squared difference between actual returns and average
return. However, if the historical return path is not representative, then the concept of
average becomes meaningless and statistical measures are not illuminating.

Beyond this, a savvy investor should be aware of the limitations of basing decisions on the
outcome of a risk model. If optimal decisions are model-dependent, how does an investor
make the best decision when the outputs of various models differ significantly? This means
that the investor must consider both the credibility of each risk model as well as the economic
consequences of the various choices that the risk models present. Investors are rewarded for
taking prudent and calculated risks. Investors may use historical data to make inferences
concerning the interrelationships between asset allocation, risk, and reward. However, in
designing and implementing a portfolio, it is always wise to remain aware of uncertainties
in both data and the risk models that incorporate it. This is why it is important to track both
sustainability and feasibility as part of a prudent assessment of retirement income strategies.
Past performance is not a guarantee of future results. We are grateful to an anonymous
referee for a series of helpful comments.

Notes

1 Technically, CRRA is nested in hyperbolic risk aversion functions. The critical
distinction is between relative and absolute risk aversion. Hyperbolic risk aversion
functions [HARA] encompass decreasing absolute risk aversion [DARA]—as wealth
increases the investor is more comfortable committing dollars to the risky asset;
increasing absolute risk aversion [IARA]—as wealth increases the investor pulls back
on the number of dollars put at risk; and constant absolute risk aversion CARA]—as
wealth increases the investor keeps the same amount of dollars at risk. Whenever the
applicable risk metric defines the percentage of wealth put at risk, the risk model

211P.J. Collins et al. / Financial Services Review 24 (2015) 193–216



incorporates a relative risk aversion measure; whenever the applicable risk metric
defines the level of dollar wealth put at risk, the risk model incorporates an absolute
risk aversion measure. It is, however, the rare investor who remains indifferent to
changes in the level of wealth when evaluating investment and distribution strategies.
This is a central criticism of incorporating relative risk aversion into a retirement
income model.

2 The Society of Actuaries and The Actuarial Foundation’s review of a cross-section of
financial planning software determined that “. . . programs vary considerably regard-
ing when the user runs out of assets, if at all. Because of this finding, the study
recommends that people run multiple programs, use multiple scenarios within pro-
grams, and rerun the programs every few years to reassess their financial position”
(Turner and Witte, 2009).

3 An excellent review of econometric issues in the modeling of asset price returns is
Carol Alexander’s (2008) four volume series Market Risk Analysis.

4 See, for example, A. G. Karolyi (2001) “Why Stock Return Volatility Really Matters,”
Strategic Investor Relations.

5 Analytic models often assume lognormality when returns are measured in discrete
time; when returns are measured in continuous time, the models often assume that
returns follow a geometric Brownian motion process. Many analytical models must
incorporate geometric Brownian motion as a pre-condition to using the techniques of
integral calculus. Over the small intervals serving as units of time for continuous
finance models, return differences between normal and non-normal distributions are
minimal and seemingly inconsequential. Aggregation of results over time often relies
on the Central Limit Theorem’s tenet that the mean of a sequence of normal random
variables is, itself, a normal random variable that, at the limit, exhibits the mathemat-
ical property of convergence. However, for longer term planning horizons, assuming
normality in the distribution of investment returns may have severe economic conse-
quences if returns are, in fact, not normally distributed. Additionally, the Central Limit
theorem characterizes the distribution of sample means and provides only limited
insight into the value of the variance statistic.

6 See DiCarlo Jr. and Fast (2008) for a survey of opinions found in financial advice
literature regarding the acceptability of various levels of portfolio shortfall risk. The
article focuses primarily on standards of prudence for management of trust-owned
investment portfolios.

7 We note additional complications surrounding continuous index availability through-
out the planning period as well as portfolio drift in the absence of constant rebalancing
to the designated asset allocation target.

8 McGoun (1995) argues that the empirical distribution of financial asset price returns is not
a measure of risk. It is merely a measure of historical realizations that may or may not be
applicable to the current economic situation. McGoun’s article presents a history of risk
measurement by economists. It provides a good theoretical basis for a monitoring and
surveillance system using current observables to supplement shortfall risk measures.

9 Milevsky and Abaimova (2006) observe that different commercially-available Monte
Carol simulation programs produce different solutions even when given the same inputs.

212 P.J. Collins et al. / Financial Services Review 24 (2015) 193–216



10 Research by Kopcke, Webb, Hurwitz, and Li (2013) compares outputs generated by
three Vector Autoregressive models.

11 For an introduction to Markov chain transition probability matrices, see Laverty,
Miket, and Kelly (2002). Hybrid models utilize a regime switching mechanism in
conjunction with a vector autoregressive conditional heteroskedastic process. For
example, a variety of autoregressive processes are incorporated into regime switching
models by Hamilton and Susmel (1994). Litzenberger and Modest (2010) develop a
model with eight different states in which the financial asset behavior within each state
is modeled by a normal distribution with state-dependent means and standard devia-
tions. Ameriks, Caplin, and Van Nieuwerburgh (2008) utilize a four-state health model
to examine issues in retirement income planning that seeks to preserve a threshold
income level. The matrix is a Markov chain transition matrix with an age-varying,
one-period” transition probability. The evolution of health status is also an important
variable in the model described by Gupta and Li (2013).

12 Smith and Gould (2006) discuss the differences between unlucky draws from a stable
probability distribution versus a substantial change (for the worse) in the probability
distribution itself.

13 The 50th percentile value of the distribution when ranking outputs (trials) according
to terminal value from low (1st percentile) to high (100th percentile).

14 Our risk model also incorporates CPI into the variance/co-variance matrix of asset
classes used to generate asset price evolutions. This means that there is a complex
interaction between increases in CPI—that are more probable under a mean-reve-
rsionary process when the inflation rate is below its historical average—and asset price
evolutions which may be negatively correlated to inflation rate increases.

15 Since the realized path of history is a single vector of results—a sample of
one—the NH model produces a credible distribution of investment outcomes only
under the assumption that future investment conditions will mirror previous
economic environments.

16 In this example, we arbitrarily fix the planning horizon at 30 years. We later expand
the example to incorporate the force of mortality.

17 The risk that the combination of (1) unfavorable portfolio returns realized early in
retirement and, (2) portfolio withdrawals will deplete the portfolio’s dollar value to the
point where future favorable returns operating on smaller dollar values are insufficient
to offset the early losses.

18 The diversified portfolio invests in broad range of domestic and developed foreign
equities with a tilt towards value oriented and small capitalization stocks. For fixed
income, the portfolio invests in diversified pools of high quality domestic and foreign
bonds with short and intermediate maturities. Other holdings in the portfolio include
positions in diversified baskets of US REITs (Real Estate Investment Trusts) and
emerging market stocks.

19 An early study of the effect of diversification on portfolio sustainability is Collins,
Savage, and Stampfli (2000). This article presents results from simulating multi-
asset class portfolios consisting of globally diversified stock, bond and real estate
investments.

213P.J. Collins et al. / Financial Services Review 24 (2015) 193–216



20 One half of the population will live longer than the average life span; and, in some
cases, the individual’s life span may be many years above the average. Distributions
of life spans approach exponential distributions with long tails. The research of Brown
& McDaid (2003), sponsored by the SOA, reviews 45 research articles that examine
factors, including pre- and post-retirement income, that affect mortality.

21 In place of a simple flat percentage fee, we elect to use the following progressive fee
schedule: 1% on the first million, 65 basis points (bps) on the second million, 35 bps
on each dollar between two and five-million, and finally 25 bps on any amount above
five-million. If the portfolio grows, this laddered fee structure is less detrimental to the
portfolio than a flat 1% fee. A minimum fee of $5,000 is charged to the portfolio if it
ever falls below half a million dollars. The portfolio is passively managed with index
funds and rebalanced annually.

22 These assumptions allow us to bypass the issue of asset location. Gordon Pye (2001)
discusses at length how a portfolio’s asset location affects the withdrawal rate adjust-
ment amount necessary to maintain portfolio sustainability in after tax dollars. Incor-
porating asset location into a risk model adds another dimension of analysis, and
model risk, to retirement planning. Although there are well known rules of “conven-
tional wisdom” that address withdrawal strategies from various accounts, such as
“draw down taxable accounts first, then turn to taxed-deferred accounts,” Coopersmith
and Sumutka (2011) demonstrate the benefits of a tax-efficient optimization approach
over conventional wisdom.

23 Hughen, Laatsch, and Klein (2002) view positive terminal wealth as a potential
“opportunity cost.” The authors develop a concept called “the equivalent payment
value.” This is a way to convert terminal wealth into extra monthly payments
throughout the planning horizon. Blanchett, Kowara, and Chen (2012) optimize based
on a utility function that penalizes portfolio depletion more heavily than it penalizes
excess bequests. See also, Scott, Sharpe, and Watson (2009), which advances the
proposition that investors should not try to generate “unneeded”/low-utility surplus.

24 Thomas J. Sargent’s Noble Prize winning research deals with how investors make
decisions when they doubt the accuracy of their model. When confronted with
ambiguity, they tend to use a family of models and to over-weight bad outcomes as a
mechanism for exercising caution: see Hansen and Sargent (2008) for example. For a
discussion on the wisdom of using multiple models for security valuation, see Collins
(2007).

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