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Introduction to My Annotated Bibliography on the Topic of
‘Longevity Risk and Portfolio Sustainability’

The following document is primarily intended to act as a scholarly reference source. Over the last fifty
years, there has been an accretion of research on this topic from scholars and practitioners in diverse
fields:
Actuaries are interested in the factors that determine pricing of contracts promising lifetime income;
Financial Economists are interested in building models that reflect factors determining the evolution of
retirement portfolios under the stress of expenses and withdrawals, and on using models to optimize
outcomes expressed in both dollar-wealth and utility terms;
Investment Advisors are interested in how best to advise clients on a variety of retirement and
intergenerational wealth management issues;
Trustees charged with providing lifetime income to current beneficiaries and terminal wealth to
remaindermen are interested in how to discharge prudently and impartially their fiduciary duties; and,
Investors are interested in how much money they can safely spend or bequeath from their retirement
portfolio.
Not only is the volume of research vast, but the range of publications reflects pedagogy from academic
fields that traditionally have had little overlap in readership. It is both interesting and beneficial to
examine, from the perspectives of various professions, what questions are raised, how hypotheses are
formulated and tested, and why strategies designed to secure a safe, substantial and sustainable income
are in some cases recommended or, in others, rejected.
From time-to-time, a seemingly “new” idea pops up in a journal aimed towards one audience when, in
fact, the idea has been well developed in a journal written for an entirely different audience. Indeed, in
some cases, the “new” idea was anticipated in journals published a decade or more earlier. It is a rare
idea that, like Botticelli’s Venus, springs forward fully developed ab initio. Indeed, tracing through the
historical scholarship record can be particularly annoying if one encounters individuals with the
effrontery to have published our unique and proprietary ideas before us. Furthermore, important
papers may suffer neglect simply because they appear in publications that are off the beaten track to
many potential readers. Articles on longevity risk management in actuarial journals, for example, are
rarely read by attorneys providing opinions on the prudence of trust portfolio administration despite the
fact that Grantors may direct trustees to provide lifetime income to a trust’s beneficiary.
Also of interest is the chronology of advice, strategies and solutions. Although the annotated
bibliography does not rise to a narrative intellectual history—it was never intended to do so—
nevertheless, if read from beginning to end, it traces the introduction and development of important
ideas and research methods. The bibliography lacks a topical index because it seeks to provide a sense
of how ideas and insights from a variety of professions developed, over time, and, perhaps
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independently from each another. Chronological presentation substitutes for a narrative history of
ideas.
This observation leads straight away to a primary motivation for creating the annotated bibliography. It
is both difficult and time consuming to survey the relevant source materials. Research, however,
demands that the scholar know the work of those who have preceded him. This bibliography should
facilitate what has now become a herculean task of gathering and investigating published studies and
commentary. This said, the research and publication vistas are vast, and no representation can be made
regarding the completeness of this bibliography except that it is incomplete. One is reminded of the
effort needed to survey the relevant literature on, say, Shakespeare or Napoleon—absolute
comprehensiveness would require a lifetime and more. It is hoped that this bibliography will assist those
with the temerity to put pen to paper to understand the nature and scope of previous inquiry and
investigation.
Many of the bibliographical entries summarize complex models in which the solution path aims towards
maximization of a utility function. Summaries of normative articles, sometimes only slightly less
mathematically complex, also appear. These articles provide insights in the areas of asset allocation,
spending, and other financial strategies under a variety of assumptions such as complete markets, log-
normal probability distributions of financial asset returns, constant relative risk aversion, and so forth.
Finally, a generous sampling of practitioner-oriented articles appears.
A bird’s-eye look at the literature reveals a plethora of different research methods, modeling
assumptions, and portfolio allocation/spending preferencing criteria, all of which may produce
significantly different outputs even given the same empirical data. Conclusions are subject to model risk;
and, from time-to-time, practitioners may translate the output from academic model building—an
exercise designed to explore quantitative relationships among variables of interest—into prescriptive
statements for investors. But the mathematical assumptions required for tractable model building often
diverge in both their character and form from common investor utility functions or from the process
underlying the distribution of empirical asset price evolutions. Conversely, from time-to-time,
practitioner-oriented articles may resort to pure empiricism in an attempt to parse historical return
evolutions to find patterns which can be turned into rules for safe and sustainable portfolio withdrawals.
Of course, taken to an extreme, this is mere data mining. Hence, the final purpose of this bibliography.
Conclusions are a function of the research methodology employed by the investigator; and, the
intelligent advisor realizes critical distinctions between the investigative/academic outcomes and the
needs, goals, circumstances, and purposes of an investor. One benefit of reviewing historical research is
to recognize that investors should not be confused with models.
Gathering and filing tasks for this project were formidable. I wish to thank Nicole VanderGeest and
Emma Gavenda for their invaluable assistance.
Patrick Collins

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Longevity Risk and Portfolio Sustainability
Chronological Summary of Articles [1965 – 2014]
DATE TITLE / AUTHOR(S) THESIS COMMENTS
1965 “Uncertain Lifetime, The article is an early analysis of optimization of discounted expected If the choice of a value for λ in a chance-constrained
Life Insurance, and utility under the conditions of an uncertain life span. Investors lacking a programming approach is to be optimized, then the
the Theory of the bequest objective will maximize a utility function for consumption only coefficient of success (success = likelihood of
Consumer,” (c) where the function has a positive first and negative second derivative: portfolio sustainability throughout the remainder of
Menahem E. Yaari life) involves a tradeoff between “safety-first”
The Review of V(c) = (maximize safety by putting all assets into a risk-free
Economic Studies 𝑇𝑇 investment) and opportunity for future growth of
Vol. 32, No.2. pp. ∫0 𝑎𝑎(𝑡𝑡)𝑔𝑔[𝑐𝑐(𝑡𝑡)]𝑑𝑑𝑡𝑡 the portfolio above the risk-free rate. Is maximizing
Where’ a’ is the subjective discount rate—i.e., the investor’s time
137 – 150. expected utility inconsistent with minimizing the
preference rate.
risk of ruin? Prudence is something more than

selecting the lowest failure rate probability. See
For investors with a bequest objective, the function to be maximized is:
later articles reconciling shortfall risk metric with

traditional utility preferencing metric. For example,
U(c) =
“Annuities vs. Safe Withdrawal Rates: Comparing
𝑇𝑇
Floor-with-Upside Approaches Michael Kitces
∫0 𝑎𝑎(𝑡𝑡)𝑔𝑔[𝑐𝑐(𝑡𝑡)]𝑑𝑑𝑡𝑡+ 𝛽𝛽(𝑇𝑇)𝜑𝜑[𝑆𝑆(𝑇𝑇)]
Where Beta is a subjective weighting function for bequest ‘S’ that is itself
[2012].
a function of the time of the bequest.

Under the penalty function approach, penetration
The problem is to find the optimal feasible consumption plan when the
of a floor value equal to funding for a minimum
planning horizon is uncertain.
standard of living may produce disutility

approaching infinity. In such cases a shortfall risk
Yaari notes that the feasibility problem can be solved either through a
metric converges to a utility-based risk metric.
“chance-constrained programming” methodology or through a “penalty

function” procedure. “The chance-constrained programming approach
If an uncertain lifespan results in future cash flows
requires that the constraint (in this case the wealth constraint) be met
being more heavily discounted, a smoothed
with probability λ or more, where λ is some number fixed in advance, say
consumption pattern may not be preferred over a
.95….This approach is, of course very common in statistics where many
front-loaded pattern. Or, a smoothed consumption
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of the standard tests are based on the idea of maximizing some criterion plan (smooth = constant marginal utility of
subject to the constraint that the probability of type I error be less than, consumption) may lead to a front-end loaded
say .05.” However, Yaari points out that the choice of λ is itself a retirement income stream. This is the “Fisher
decision problem: ‘…one might want to choose λ optimally rather than Utility” argument—probability of failure at an
arbitrarily.” advanced age is not as onerous as at an early age.

Note: the decision problem was articulated by I. Fisher who advanced The case of the irrevocable family trust is akin to a
the proposition that income early in retirement had greater utility than case where annuities are available and there exists
the equivalent inflation-adjusted income in late retirement. See “The 7 a positive bequest motive. Is it prudent to fund the
most important equations for your retirement—Milevsky (2012)]. current beneficiary’s income right with an annuity
and the remaindermen’s share with an investment
With the penalty function approach the consumer himself guards against portfolio?
violating the wealth constraint “…by assuming that a violation of the
constraint carries a penalty, i.e., a loss of utility.” Note: The opportunity for an improved future
budget constraint may be offset by a higher
Yaari considers several economies where annuities may or may not be discount rate for future income. This constitutes
available to the investor. A central point is that the optimal utility-of- and important rationale for a portfolio monitoring
consumption plan contains a term for survival probability. The program that reflects the preferences of the
uncertainty of future survival means that the future is discounted more investor rather than performance relative to an
heavily. He demonstrates that absent a bequest objective in a (complete) outside benchmark. This is a variation on the 2-fund
market with fair-valued annuities available to the consumer, all wealth [Tobin] trust solution where consumption is
will be held in an “actuarial note” that is equivalent to an annuity and financed by an annuity and the remainder interest
which pays a rate of return equal to the commercial interest rate plus an by an investment fund. The duty of impartiality is
extra factor reflecting the fact that the investor’s estate must forfeit defined in terms of equalizing marginal utility
further income at death (Terminal Portfolio value = $0). “…positive between beneficiary classes.
assets will always be held in the form of actuarial notes.” [Note: the
“extra factor” is the mortality premium offered by annuity contracts]

When a bequest motive exists, the investor’s problem is to solve for
optimization of two decision values: a feasible consumption plan and a
feasible “saving plan.” Assuming no labor income, the consumption plan
will be funded entirely with annuities while the saving plan is a function
of available investment returns. Ideally, the marginal utility of the
consumption plan will exactly equal that of the saving plan. Yaari makes
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the important observation that when annuities are available, “…the
consumer can separate the consumption decision from the bequest
decision.” In the absence of annuities, such a separation is not possible.
1977 “Savings and The Levhari/Mirman article demonstrates that a (mean-preserving) It is interesting to compare and contrast Levhari and
Consumption with change in the “distribution of lifetime uncertainty” may motivate either Mirman’s analysis of spending from their utility
an Uncertain a decrease in current consumption in the face of a higher probability of a optimization model reflecting a dynamic
Horizon,” David longer life; or, an increase in consumption because sure current programming approach, to prescriptive
Levhari and Leonard consumption is preferred over a more uncertain probability for future recommendation such as the 4% spending rule. The
J. Mirman, Journal of spending. The effect of uncertainty on an individual investor depends on Levhari & Mirman study emphasizes how utility-
Political Economy which motivation is stronger: “…if uncertainty of lifetime is considered optimizing retirement spending decisions are
Vol. 85, No. 2 (1977), as part of a Fisherian lifetime optimization problem, with a risk-averse dynamic and reflect information regarding time
pp. 265 - 281 consumer, consumption in the form of an uncertain lifetime has the horizon, returns, risk preferences, etc. By contrast,
same effect as increasing the rate of discount. This is one of the more many empirical “rules” assume that the myopic
important results found in the contribution of Yaari (1965).” The risk- investor selects a “safe” spending level and
averse investor, confronted by an uncertain life span, may elect to maintains the level under the assumption that past
consume more in the present. results act as a condition precedent for future
The article points out that Champernowne [Uncertainty and Estimation periods (economies).
in Economics Vol. 3 (Holden-Day), 1969—not included in this A choice among approaches—mathematical
bibliography] advances the opposite hypothesis by stressing the modeling approaches include (linear and nonlinear)
precautionary savings aspect of optimization in the face of uncertainty in dynamic programming, maximum likelihood,
the distribution of mortality: “…’the effect of not knowing when’ the optimum (stochastic) control theory, etc.;
consumer ‘will die is to lower the initial value of consumption.’” Hence, empirical/numerical approaches include data
as Levhari & Mirman point out: “The desire to provide for a longer life mining of time series, simulation, etc.—often
together with the desire for more certainty by consuming now pull in depends on (1) tractability, and (2) the structural
opposite directions.” characteristics of the problem—e.g., dimensionality,
Levhari and Mirman compare two individuals each having the same and other characteristics of both the ‘problem’ and
tastes and attributes except with respect to the distribution of lifetime— the model selected to represent the ‘problem.’
i.e., the survival distribution of one investor is more risky than the other. The 2012 article by Huang, Milevsky and Salisbury
Their model assumes constant relative risk aversion: u(x) = x1-γ/(1-γ) with [“Optimal retirement consumption with a stochastic
u(x) = log x for γ = 1. Not surprisingly, it suggests that consumption force of mortality” Insurance: Mathematics and
decisions depend heavily on the utility function, investment returns, and Economics, Vol 51, pp. 282-291]] extends the work
the investor’s subjective discounting (impatience-to-consume). Although of Levhari and Mirman and James Davies
the model can incorporate earned income, its primary focus is on the [“Uncertain Lifetime, Consumption, and Dissaving in
retirement period beginning at time ‘t.’. Investor wealth [W] at time Retirement,” James B. Davies, Journal of Political
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‘t+1’ equals (W – C)r where ‘r’ = return on investments and ‘C’ = Economy Vol. 89, No.3 (1981), pp. 561 – 577] by
t t t
Consumption during period ‘t.’ Each investor’s goal is to maximize a incorporating stochastic morality into the
Von-Neumann-Morgenstern expected (additive and separable) utility consumption utility-based model.
function with respect to lifetime consumption where consumption
during any future period ‘t’ is subject to a discount factor reflective of
each investor’s personal impatience to consume. Adding a term for the
probability of survival [P], the mathematical expression for maximizing
the lifetime stream of expected utility-of-consumption becomes:

𝑇𝑇
𝑡𝑡
𝐸𝐸𝑟𝑟 = �𝑎𝑎 𝑃𝑃𝑡𝑡𝑢𝑢(𝐶𝐶𝑡𝑡)

𝑡𝑡=0
Using dynamic programming methods, the model indicates that
consumption at time 0 is:

C (W) =
0
𝑊𝑊
1
𝑇𝑇 𝛾𝛾 1−𝛾𝛾 𝑖𝑖
If ‘r’ is random, the term ‘ar’ in the∑ 𝑖𝑖a=b0o𝑃𝑃𝑖𝑖ve(𝑎𝑎 e𝑟𝑟qua)𝛾𝛾tion’s denominator
becomes ‘aE(r).’
Holding all else equal, an increase in risk aversion (γ) decreases
consumption in the initial period: “In other words, the individual, being
more sensitive to the possibility of lower consumption in the future,
saves more in the initial period.” However, the primary goal is to
consider an investor who faces an uncertain lifetime. In order to isolate
the influence of uncertainty (the “riskiness of life”), the model assumes a
nonrandom return. The central question is whether consumption is an
increasing or decreasing function of uncertainty in life span.
It turns out, under very restrictive conditions on available return and
subjective discounting, that key factor are the mortality probability
distribution [P] and the consumption risk aversion factor 1/γ. When γ >
1, a riskier distribution of life span reduced current consumption; when 0
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< γ < 1 , current consumption increases. Under more realistic conditions,
“…a straightforward relationship between riskiness and optimal
consumption does not exist….” In some cases, uncertainty elicits greater
consumption; in other cases, greater savings. For example, a high return
on available savings motivates less current consumption; a high utility
discounting factor motivates greater current consumption: “In other
words a small rate of return combined with a riskier horizon will increase
consumption.”
The article concludes by considering an uncertain rate of return. The
model preserves the return’s mean but increases its variance while
keeping lifetime uncertainty unchanged. The effect on consumption
depends on γ. If γ > 1, r1-γ is a convex function of r and hence will reduce
consumption. If 0 < γ < 1, then r1-γ is a concave function of r and
“…increases the proportion of wealth consumed.”
1977 “Mean-Risk Analysis Fishburn reviews a number of preferencing models in the face of An early and important survey of the mathematical
with Risk Associated uncertain outcomes. He contrasts and compares “mean – risk trade off” approaches commonly found in the literature of
with Below-Target and “mean-risk dominance” models to a “mean-risk utility” model. investment decision making. Fishburn provides
Returns,” Peter C. Although there is a substantial body of research using mean-variance or examples of how the form of the utility function can
Fishburn The mean-semivariance models a la Markowitz, Fishburn believes that such reflect both a wide value of risk-aversion
American Economic modeling does not always capture the risk attitudes of investors. The parameters—as opposed to the assumption of
Review, Vol. 67, No. reason for this belief is two-fold: quadratic risk aversion in Markowitz—and can
2 (March, 1977), pp. accommodate a shortfall risk measure.
116 – 126. 1. Mean-variance analysis “…should not be taken very seriously
unless the probability distributions used in the analysis satisfy
certain restrictions.”—i.e., distributions should be IID normal;
and,
2. Investors “…very frequently associate risk with failure to attain a
target return. To the extent that this contention is correct, it
casts serious doubt on variance—or, for that matter, on any
measure of dispersion taken with respect to a parameter(for
example, mean) which changes from distribution to
distribution—as a suitable measure of risk.”

Thus, Fishburn is interested in developing mathematical expressions for
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a variety of models that are based on either mean/variance, or loss
probability/shortfall magnitudes, or expected utility including stochastic
dominance models. The specific model of interest is the α-t model
where α is a risk aversion parameter and t is an investment target wealth
level or rate of return. The Markowitz model is a special case where the
exponent α takes on a value of 2 (quadratic utility) and t is the investor’s
optimal portfolio location on the efficient frontier. Fishburn asserts that
generalized α-t models can accommodate a range of risk aversion
parameters and specific target returns or reference levels. Given a
cumulative probability distribution function where F(x) specifies the
probability of a return not exceeding x (the area under the graph of the
function to the left of x), the mathematical form of such a mean-risk
dominance model is a probability-weighted function of investment
results below the specified target return t:

𝑡𝑡
𝛼𝛼
𝐹𝐹𝛼𝛼(𝑡𝑡) = � (𝑡𝑡−𝑥𝑥) 𝑑𝑑𝐹𝐹(𝑥𝑥)
−∞
Such a flexible α-t model, where α can accommodate many attitudes
towards risk and t can accommodate the investor’s desire to avoid
unacceptable shortfall probabilities and magnitudes, is compatible with
both stochastic dominance models and with von Neumann-Morgenstern
utility-based models. He observes: “The idea of a mean-risk dominance
model in which risk is measured by probability-weighted dispersions
below a target seems rather appealing since it recognizes the desire to
come out well in the long run while avoiding potentially disastrous
setbacks or embarrassing failures to perform up to standard in the short
run.”

The mean-risk dominance model expresses the investor’s preference
criterion as follows: “F dominates G if an only if µ(F) ≥ µ(G) and ρ(F) ≤
ρ(G) with at least one strict inequality. Rho of F is defined by:

𝑡𝑡
𝜌𝜌(𝐹𝐹) = � 𝜑𝜑(𝑡𝑡−𝑥𝑥)𝑑𝑑𝐹𝐹(𝑥𝑥)
−∞
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In which ϕ(y) for y ≥ 0 is a nonnegative nondecreasing function in y with
ϕ(0) = 0 that expresses the ‘riskiness’ of getting a return that is y units
below the target.”
Fishburn develops several theorems including:

• P(α, t) is completely determined by expected returns whenever
all possible returns in both F and G lie at or above the target t.

• If two distributions have the same expected mean and one
distribution is sure to provide a return at or above target while
the other has a positive probability of generating a below-target
return, then a risk-averse investor will prefer the sure thing.

Gambling behavior is revealed whenever all returns from both
distributions are at or below the target. However, for various values of
the risk aversion parameter α, the below-target investor may either
become conservative—lest a bad situation becomes worse--or become
increasingly risk-seeking. “Depending on context and the circumstances
of the decision maker or his firm, t might be formulated as a ruinous
return, as the zero profit return, as the return available from an insured
safe investment, or as a target which reflects a general attitude towards
acceptable performance….” If an investor is primarily concerned about
missing the target but the magnitude of the shortfall is not critical, the
model will incorporate a smaller value for α than if both the shortfall
probability and the magnitude of the shortfall are of importance. The
value of α separates risk-seeking from risk-averse behaviors for below-
target returns.

Fishburn also explores how a mean-risk utility model can express a
decision maker’s preferences. Risk is defined as above, and for two
distributions F and G, F is preferred to G if and only if:

U(µ(F), ρ(F)) > U(µ(G), ρ(G)).

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This results in the following form for the utility function:

U(x) = x for all x ≥ t
U(x) = x – k(t-x)α for all x ≤ t.

After reviewing a number of empirical surveys of how farmers,
businessmen and investors view risk, Fishburn notes: “…most individuals
in investment contexts do indeed exhibit a target return—which can be
above, at, or below the point of no gain and no loss—at which there is a
pronounced change in the shape of their utility functions. A relatively
narrow range of utility functions (mostly linear) holds for above-target
returns; a more wide range of α values are observed for below target
returns.

Generally, the α-t model suggests that investors will avoid distributions
offering a probability of generating returns below the target even if such
distributions have a greater mean. Additionally, “If the α-t utility
model—which presumes the existence of a real valued function U in
mean and risk which increases in mean, decreases in risk , and reflects
the decision maker’s preferences between distributions—is congruent
with the von Neumann-Morgenstern expected utility model with utility
function u, then u can be written as u(x) = x for x ≥ t, u(x) = x – k(t-x)α for
x ≤ t, with k > 0.”
1981 “Uncertain Lifetime, Older investors, lacking bequest motives, either continue to save in Yaari’s model draws on the work of Irving Fisher
Consumption, and retirement or decumulate at rate slower than predicted under standard [The Theory of Interest] published in 1930. Fisher
Dissaving in life-cycle models. Davies argues that lower-than-expected consumption contends that uncertainty in lifespan tends to
Retirement,” James is due to uncertainty in life span: “…in the absence of pensions uncertain increase ‘impatience.’ This, in turn, suggests that
B. Davies, Journal of lifetime will not only depress consumption at all ages but will also have consumers prefer current consumption
Political Economy an increasingly severe proportional impact beyond middle age….On opportunities to future consumption opportunities
Vol. 89, No.3 (1981), conservative assumptions, uncertain lifetime may more than halve the simply because they may not be alive at a future
pp. 561 – 577. mean rate of decumulation among the retired.” date. Technically, an uncertain lifespan produces a
The author cites previous research by Yaari [1965] and Levhari & Mirman higher subjective discount rate—future
[1977]. Yaari’s model suggests that consumption, in the absence of consumption is discounted at a rate greater than
insurance, under an uncertain lifespan grows more slowly than under the prevailing return on savings.
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