Merging Asset Allocation and Longevity Insurance:
An Optimal Perspective on Payout Annuities
Moshe A. Milevsky2
Date: February 20, 2003
The Markowitz mean-variance model is widely accepted as the gold standard for
asset allocation on the way to retirement. Unfortunately, this framework only
considers the risk and return tradeoff in the financial market. It does not consider
the longevity risk people face during retirement. And, while a variety of recent
papers in the Journal of Financial Planning have discussed the mechanics and
importance of payout (also known as lifetime) annuities, the industry literature
currently lacks a coherent and formal model of how much wealth should be
allocated in-and-between asset classes within a payout annuity.
To fill this gap, our paper revisits the importance of longevity insurance – while
discussing the problems with fixed payout annuities -- and then moves on to
address the proper asset allocation between conventional financial assets and
variable payout annuity products. As in the classical Markowitz framework, our
focus is on maximizing a suitably defined objective function in an intuitive,
comprehensible, and practical manner.
In addition to the usual risk and return information from the financial markets, our
modeling framework requires inputs on the relative strength of retiree’s bequest
motives, subjective health status, and liquidity restrictions. To illustrate the model,
we provide some specific case studies and numerical examples to show how a
financial planner can actually apply asset allocation ideas within-and-between
payout annuity products and conventional asset classes.
1 Vice President and Director of Research, Ibbotson Associates, 225 N. Michigan Ave. Suite 700, Chicago,
IL 60601. Phone: (312) 616-1620; Fax: (312) 616-0404; E-mail: firstname.lastname@example.org.
2 Executive Director, The IFID Centre &Associate Professor of Finance, Schulich School of Business,
York University, 4700 Keele street, Toronto, Ontario, M3J 1P3, Canada. Phone: (416) 736-2100 x 66014;
Fax: (416) 736-5487; E-mail: email@example.com.
A number of recent articles in the JFP – for example Ameriks, Veres and
Warshawsky (2001), Duff (2001), Bengen (2001) and Goodman (2002) – have
focused financial planners attention on longevity risk and the probability of
Indeed, the shift in retirement funding from professionally managed defined benefit
plans to defined contribution personal savings vehicles also implies that investors
need to make their own decisions on how to allocate retirement savings, as well as
what product should be used to generate income in retirement. There are two
important risk factors investors must consider when making these decisions: 1)
Financial market risk, i.e., volatility in the capital markets which induces portfolio
values to fluctuate up and down. If the market drops or corrections occur early
during retirement, the portfolio may not be able to cushion the added stress of
systematic withdrawals. This may make the portfolio unable to provide the
necessary income for the desired lifestyle or it may simply run out of money too
soon. 2) Longevity risk, i.e., the risk of living too long or outliving your portfolio.
Life expectancies have been increasing, and retirees should be aware of the
substantial chances for a long retirement, and plan accordingly. This risk is faced by
every investor, especially those taking advantage of early retirement offers or those
who have a family history of a longevity.
Traditionally, asset allocation is determined by constructing efficient portfolios for
various risk levels based on modern portfolio theory (MPT)3. Then, based on the
investor’s risk tolerance, one of the efficient portfolios is chosen. MPT is widely
accepted in the academic and finance industries as the primary tool for developing
asset allocations. Its effectiveness is questionable, however, when dealing with asset
allocations for individual investors in retirement, since longevity risk is not
considered. The purpose of this article is to review the need for longevity insurance
during retirement, and then establish a framework to study the total asset allocation
decision in retirement, which includes both conventional asset classes and
immediate payout annuity products.
2. Why do my clients need longevity insurance?
Americans are living longer on average than ever before. The probability that an
individual retiring at age 65 will reach age 80 is over 70% for females, and over
62% for males. When combined with the life expectancy of a spouse, the odds
reach nearly 90% that at least one spouse will live to 80. And there’s an over 80%
3 Markowitz (1952) and Merton (1971).
chance at least one spouse will live to age 85. For a broader sense of the potential
longevity risk, Table 1 illustrates how long a 65-year-old can expect to live.4
Table 1: The Conditional Probability of Survival at Age 65
At Least One
Member of a
93.8% 92.0% 99.5%
84.4% 79.9% 96.9%
70.9% 62.7% 89.1%
52.8% 41.0% 72.2%
31.6% 19.6% 45.0%
13.4% 5.8% 18.4%
Source: Society of Actuaries RP-2000 Table
For example, the probability that at least one spouse will reach age 75 is computed
as follows: 1 - (1-0.938)*(1-0.920) = 99.5%. As the reader can see from the table,
longevity risk – the risk of outliving one’s resources – is very substantial and is the
main reason that we believe lifetime annuities (alternatively known as payout) will
grow in popularity.
3. Payout annuity and its insurance against longevity risk
Longevity risk can be hedged away with insurance products, namely lifetime payout
annuities. A lifetime payout annuity is an insurance product that exchanges an
accumulated investment into payments that the insurance company pays out over a
specified time or, in this case, over the lifetime of the investor. Payout annuities are
the exact opposite of traditional life (or more aptly named premature death)
There are two basic types of payout annuities: fixed and variable. A fixed payout
annuity pays a fixed dollar amount each period, perhaps with a COLA adjustment,
in real or nominal terms. A variable annuity’s payments fluctuate in value
depending on the investments held and, therefore, disbursements will also fluctuate.
The payment from a lifetime payout annuity is contingent upon the life of the
investor. If the investor dies, he/she will no longer receive any payments, unless a
special guarantee period or estate benefit was purchased at the same time, which is
normally paid for by reducing the benefit stream.
There has been a substantial amount of recent literature on the topic of the costs
and benefits of life annuities, and space constraints prevent us from giving
4 We have chosen age 65 as the standard baseline for retirement, although similar numbers can be
generated for any age.
providing a comprehensive review. Roughly speaking, the relevant literature can be
partitioned into the following categories:
The first category consists of the theoretical insurance economics literature that
investigates the equilibrium supply and demand of life annuities in the context of a
complete market and utility-maximizing investors. This includes the classical work
by Yaari (1965), as well as Richard (1975), Brugiavini (1993), Yagi and Nishigaki
(1993) and Milevsky and Young (2002). Broadly speaking, their main conclusions
are that life annuities should play a substantial role in a retiree’s portfolio.
The empirical annuity literature examines the actual pricing of these products, and
whether consumers are getting their money’s worth. These include a sequence of
papers by Brown, Warshawsky, Mitchell and Poterba (1999, 2000, 2001) in various
A third and final strand attempts to create normative models that help investors
decide how much to annuitize, when to annuitize and the appropriate asset mix
within annuities.These include the work by Milevsky (2000, 2001), Kapur and
Orszag (1999), and Blake, Cairns and Dowd (2000).
3.1 Fixed payout annuity
Chart 1 illustrates the payment stream from a fixed immediate (a.k.a. payout, or
lifetime) annuity. With an initial premium or purchase amount of $100,000, the
annual income payments for a 65 year-old male in today’s environment would be
$706.14 per month, or $8,474 per year.5 The straight line represents the annual
payments before inflation. People who enjoy the security of a steady and
predictable stream of income may find a fixed annuity appealing. The drawback of
a fixed annuity becomes evident over time. Since the payments are the same year
after year, purchasing power is eroded as the annuitant gets older. The second
curved line in the image represents the same payment stream after a hypothetical
3.2% inflation rate is factored in.6 While the annuitant still receives the same
amount, it is no longer able to purchase as much as it used to.
Despite the benefits of longevity insurance and fixed payout amounts, there are
disadvantages with a portfolio that consists solely of fixed annuities. First, because
the nominal value of the payment will remain fixed for the rest of the annuitant’s
life, the value of the payments in real terms (after inflation) will decline over time.
Chart 2 displays the inflation rate during the last 30 years, as measured by changes
in the level of the Consumer Price Index (CPI). Notice that although the inflation
5 This is the average quote obtained by the authors in mid-July, 2002, assuming a 65-year-old male and a
$100,000 premium. The payments from different companies can differ quite substantially from week to
week and from the best to the worst insurance company quotes.
6 The average inflation rate from 1926 to 2001 is 3.2% in the U.S.
rate in the U.S. is currently under 2%, this number is at the low end of the
historical record. In fact, as recently as the early 1990s, the inflation rate was over
6%, and in the early 1980s, it went as high as 13%. The (arithmetic) average during
the last 30 years was approximately 5% per annum.
Secondly, the investor cannot trade-out of the fixed payout annuity once it is
purchased.7 In other words, the lack of liquidity (and reversibility) within a fixed
annuity impedes the optimal asset allocation process and makes the fixed annuity
less desirable, all else being equal. See Browne, Milevsky and Salisbury (2003) for
details on how to quantify this drawback.
Finally, it seems that the current payout rates from fixed payout annuities are at a
historical low, which is consistent with the current interest rate environment. A 65
year-old female might have received as much as $1,150 per month in the early
1980s, in exchange for the same $100,000 initial premium. Today the $100,000
buys closer to $700 per month. In fact, we are currently at historical lows on the
interest rate cycle, and this may be one of the worst times to lock in an interest for
the rest of one’s life. Recall that once the individual has purchased a life annuity
they can no longer cash-in or sell the insurance contract. While we obviously want
to refrain from speculating – and encouraging others to speculate -- on the long-
term direction of interest rates, we want to remind the reader that locking-in a
fixed annuity is implicitly a market timing play. This is why we believe that variable
payout annuities will continue to grow in popularity.
Variable payout annuities
A variable payout annuity is an insurance product that exchanges an accumulated
investment value into annuity units that the insurance company pays out over the
lifetime of the investor. The annuity payments fluctuate in value depending on the
investments held and, therefore, disbursements will also fluctuate. Thus, instead of
getting fixed annuity payments, the annuitant receives the equivalent of a fixed
number of fund units. The insurance company converts these fund units into dollars
at the going net asset value. Therefore, the cash-flow from the variable payout
annuity fluctuates with the underlying investments.
Chart 3 illustrates the annuity payment stream in real terms from a 50% stock/50%
bond portfolio using a life only payment option in an immediate variable annuity.
We generated a Monte Carlo simulation to illustrate the various payment scenarios.
The simulation is generated using historical return statistics of stocks, bonds, and
7 There are payout annuities available that allow the investor to withdraw money from them, but the
investor typically has to pay a surrender or market value adjustment charge. Furthermore, this would only
apply during the certain period of the annuity where payments are guaranteed regardless of life status. In
this paper, we will focus our discussion on the basic type of payout annuities, which does not offer early
withdrawal or death benefits.
inflation from 1926–2001, a $100,000 initial portfolio, and a 3% Assumed
Investment Return (AIR). While the actuarial mechanics behind the AIR are beyond
the scope of this paper, one can think of it as a method of front-loading or back-
loading annuity payments. The initial payment at age 65 is estimated to be $6,615.8
The three lines show the 10th, 25th, and the 50th percentile. In other words, there is a
10% chance that annual inflation-adjusted annuity payments would have fallen
below $5,000, a 25% chance that they would have fallen below $7,027 or higher,
and a 50% chance that they would have grown to over $10,182.
4. Optimal asset allocation mix with payout annuities
Smart asset allocation decisions that take advantage of the diversification benefits
across different asset classes are an effective tool to manage and reduce market risk.
Therefore, to help investors find the appropriate allocation of their savings in
retirement, we must incorporate fixed and variable payout annuities into the
traditional asset allocation models.
4.1 The Rationale
It makes little sense to offer a money market and bond fund in the savings portion
of a personal pension plan, without offering an equity fund to complete the risk
and return spectrum. So, too, it makes little sense to offer fixed payout annuities
without offering variable payout annuities to balance out the risk. Clearly, the latter
is the symmetric extension of the former. And, indeed, since there is a proper asset
allocation involving savings (accumulation) products, the same applies to dissavings
(consumption stage) products.
Classical asset allocation (savings) models used by the popular software vendors and
advisor services input information on the investor’s time horizon and risk aversion
level in order to determine the appropriate asset mix. But, to incorporate payout
annuities and retirement dynamics into asset allocation models, a proper model
requires more information. This would include inputs such as the investor’s
subjective health estimate, the strength of bequest motives and pre-existing pension
We have developed a model for optimally allocating investment assets within and
between two distinct categories. The two categories are annuitized assets and non-
annuitized assets. The annuitized assets include fixed and variable immediate
annuities. The non-annuitized assets include all types of investment instruments,
such as mutual funds, stocks, bonds, and T-bills that do not contain a mortality-
contingent income flow. In addition, our model incorporates the following decision
8 The initial payment is estimated by Ibbotson Associates.
Investor’s subjective probability of survival
Population objective (pricing) probability of survival
Relative weights placed on consumption and bequest
Investor’s utility from “live” consumption and bequest
Risk and return characteristics of risky and risk-free assets
The model is developed based on micro-economic models of consumer behavior.
The appendix provides a more technical discussion about the model. Chart 4
provides a graphical illustration of the tradeoff between the desire for bequest and
liquidity needs and existing pension income. The greater the desire for creating an
estate, or bequest value, the lower the demand (or need for) payout annuities (PA).
This is because life annuities trade-off longevity insurance against the creation of an
4.2 Numerical Results
To understand the normative predictions of the model, let us look at several
different cases so that we can see the effect of changing parameters on the optimal
allocation. We will start with the capital market assumptions that will remain the
same for all four cases. All cases will assume that the individual is a 60 year-old
male who would like to allocate his portfolio across the two investment asset classes
and the two mortality-contingent claim classes. Together, the four ‘allocatable’
products are: 1) risk-free asset; 2) risky asset; 3) immediate fixed annuity; and 4)
immediate variable annuity. We assume that the return from the risk-free (T-bills)
asset class is 5% per annum with no volatility. The return from the risky asset is
log-normally distributed with a mean value of 10% and a standard deviation of
20%. (In other words the investment is expected to earn 10% per annum, but may
actually earn as much as 30% or lose 10% in any given year.) This implies a risk
premium of 5%, which is in line with forward-looking estimates for U.S. equity
markets. As for the mortality parameters, we use a table provided by the U.S. based
Society of Actuaries, called the Individual Annuity Mortality (IAM) 2000 basic table.
These tables are the probabilities of survival for a healthy population of potential
annuitants. Many people might feel they are less (or more) healthy than the
numbers indicated by the IAM 2000; we will therefore allow the subjective
probability of survival to be lower (or higher) than the objective probability of
survival. The utility preferences will be taken from within the Constant Relative
Risk Aversion (CRRA) family, with a CRRA coefficient of ?.
While space constrains us from providing a crash-course on micro-economic theory,
the CRRA can be viewed as measuring a consumer’s aversion to investing in risky
assets. The greater the CRRA value, the lower is their appetite for risk. And, while
we are fully cognizant that few if any investors can identify their personal CRRA
value – and DNA testing has proven elusive so far – we strongly believe this
normative framework can be used to guide a prudent asset mix and to educate the
investor about the risks. Finally, we will employ the 20-year horizon as
representing the one period. In other words, the individual intends to re-allocate
(rebalance) assets after 20 years. 9 In practice, we would recommend investors
rebalance their portfolio much before the 20 year horizon, which require a dynamic
multi-period model. This additional dimension of ‘when to rebalance’ complexity is
beyond the scope of this introductory paper, but is being addressed in a follow-up
report by Peng and Milevsky (2003).
Case #1: Total Altruism and Complete Bequest Motives
In this case we assume the investor’s utility is derived entirely from bequests. In
other words the weight of his utility of bequest is assumed to be one, and the
weight on his utility of consumption is zero, that is, A=0 and D=1. The objective
probability of survival is 65% (roughly equal to the survival probability of a 60-
year-old male in the next 20 years) and the subjective probability is the same 65%.
In other words, we are assuming that the investor does not have any private
information about his or her mortality status that might lead them to believe they
are healthier, or less healthy than average. Using these input parameters in the
model described above, the optimal allocations to the assets across various relative
risk aversion levels are presented in Table 2 and Chart 5.
A few things should be evident from the table. First, immediate annuities get no
allocation, since the investor only cares about bequest. The intuition for this result
can be traced back to a classical paper by M. Yaari (Review of Economics Studies,
1965). Namely, if consumers are 100% altruistic, they will not waste the asset by
annuitizing. Second, the allocation to stocks gradually decreases as the investor’s
risk aversion increases. Thus, without any consumption motive, this becomes the
traditional allocation problem between risk-free and risky assets. This case can be
used as an illustration for extraordinarily wealthy individuals, where the size of
their portfolio far exceeds their consumption needs. In this case, bequest becomes
the dominant factor. Annuities do not get any allocation, as they do not leave any
money for the heirs. For example, for investors with a relative risk aversion level of
2, the optimal allocation is 36% to the risk-free asset and 64% to equity.
Case #2: No Bequest Motives
9 These assumptions can be easily modified to accommodate other utility functions, asset return
distributions, mortality probabilities, and horizons. Note that because we are using a utility function that
has constant relative risk aversion the initial wealth level does not have any impact on the allocations for
the one-period model.
This case maintains the same age (gender), survival probability and time horizon,
but completely eliminates the strength of bequest by replacing A=1 with D=0. In
other words, 100% of the utility weight is placed on “live” consumption. The
optimal allocations to the assets across various risk aversion levels are presented in
Table 3 and Chart 6.
Since the returns on annuities are always higher than the returns on traditional
assets – conditional on the retiree being alive -- the immediate annuities get 100%
of the allocation. The allocation to the immediate variable annuity gradually
decreases, while the allocation to the immediate fixed annuity increases as the risk
aversion of the investor increases. This case can be used as an illustration for
investors who would like to maximize their lifetime consumption and have no
interest in leaving any money behind. (They are alternatively known as the “die
broke” crowd.) All the savings should be used to purchase annuities. Overall, the
optimal allocation between risky and risk-free assets (in this case, they are an
immediate fixed annuity and an immediate variable annuity) are almost identical to
that of Case #1. For investors with a risk aversion level of 2, the optimal allocation
is 36% to immediate fixed annuity and 64% to immediate variable annuity.
Case #3: 20% Bequest Motives and 80% Consumption Motives
This case maintains the same age (gender), survival probability and time horizon,
but changes the strength of bequest from D=0 to a more realistic D=0.2.10 In
other words, 80% of the utility weight is placed on “live” consumption. The
optimal allocations to the assets across various risk aversion levels are presented in
Table 4 and Chart 7.
There are several interesting results in the allocation. First, unlike the previous two
cases, all four of the asset classes are present in the optimal allocations. This is
because immediate annuities are more suitable (relative to traditional assets) for
consumption and traditional investments are more suited for bequest motives in
this one-period framework. When the investor has a more balanced motive between
bequest and consumption, both immediate annuities and traditional asset classes are
selected. In general, the higher the bequest motives, the more the investor should
allocate to traditional investments and the less to immediate annuities.
Second, the allocation between risky (both VIA and equity) and risk-free (cash and
FIA) is almost identical to that in Case #1 and Case #2 at comparable risk aversion
levels. This indicates that the changes in the investor’s bequest vs. consumption
motive do not significantly impact the investor’s behavior regarding risk. The
10 See Bernheim (1991), Hurd (1989), as well as Abel and Warshawsky (1988) for a discussion and
estimates of the ‘strength of bequest’ parameters. We have taken 20% as an approximation.
optimal allocation between risky and risk-free assets is determined by the investor’s
Third, we find the allocation to annuities decreases as the investor’s risk aversion
increases. In other words, more risk averse investors will avoid immediate life
annuities. This makes intuitive sense, since the investor could get little or no utility
from immediate annuity investments if he or she dies shortly after the purchase.
With traditional investments, there will be some left for their heirs. It seems that
higher aversion to risk increases the implicit weight on the utility of bequest. For an
investor with a risk aversion level of 2, the optimal allocation is 22% cash, 38%
equity, 14% FIA, and 26% VIA.