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THE JOURNAL OF WEALTH MANAGEMENT 1
FALL 2011
Portfolios for Investors Who Want to
Reach their Goals While Staying on
the Mean–Variance Efficient Frontier
SANJIV DAS, HARRY MARKOWITZ, JONATHAN SCHEID,
AND MEIR STATMAN
SANJIV DAS
is a professor of finance at
the Leavey School of Busi-
ness, Santa Clara Un iver-
sity, in Santa Clara, CA.
srdas@scu.edu
HARRY MARKOWITZ
is a professor of finance at
the Rady School of Man-
agement, University of
California, San Diego, in
La Jolla, CA.
harryhmm@aol.com
JONATHAN SCH EID
is the president and CIO at
Bellatore Financial, Inc.,
in San Jose, CA.
jscheid@bellatore.com
MEIR STATM AN
is Glenn Klimek Professor
of Finance at the Leavey
School of Business, Santa
Clara University, in Santa
Clara, CA, a nd a visiting
professor at Tilburg Uni-
versit y in the Netherlands.
mstatman@scu.edu
Mental-accounting portfolio
theory is a goal-based port-
folio theory, combining some
of the most appealing features
of Markowitz’s [1952a] mean–variance port-
folio theory and Shefrin and Statman’s [2000]
behavioral portfolio theory.
Mean–variance portfolio theory is a
“production” theory. Investors in that theory
produce portfolios that combine expected
returns and standard deviations of returns at
levels that are best for them. But what do
mean–variance investors want to do with
the money in their portfolios? Do their goals
consist of funding a comfortable retirement
and a modest bequest to children? Do they
consist of a bare-bones retirement and a con-
siderable bequest to charity? The production
of mean–variance efficient portfolios is only
a station on the way to investors’ ultimate
goals, yet mean–variance portfolio theory is
silent about these goals. In contrast, behav-
ioral portfolio theory combines the produc-
tion of optimal portfolios for investors with
the use of the money in investors’ portfolios
to reach their goals.
Mean–variance investors consider their
portfolios as a whole, as depicted in Exhibit 1.
In contrast, behavioral investors begin by
dividing the whole of their portfolios into
mental accounts, each dedicated to a partic-
ular goal and each with its own time horizon.
These are depicted in Exhibit 2. For example,
a $1 million portfolio might be divided into
an $800,000 mental account for a retirement
goal that is 15 years away, a $150,000 mental
account for an education goal that is three
years away, and a $50,000 mental account
for a bequest goal that is 25 years away. Next,
behavioral investors specify threshold levels
they want to reach or exceed for each goal
when t hey arr ive at it s hor i z on. For ex a mp le,
investors might aim to reach or exceed a
$2 million threshold in the mental account
dedicated to the retirement goal, $180,000 in
the mental account dedicated to the educa-
tion goal, and $850,000 in the mental account
dedicated to the bequest goal.
Investors in mean–variance portfolio
theory face a single efficient frontier, con-
sisting of the best feasible combinations of
expected returns and standard deviation of
returns. In contrast, investors in behavioral
portfolio theory face many efficient fron-
tiers, one for each mental account. More-
over, while risk in mean–variance portfolio
theor y is mea s u red by t he st a ndard devia t ion
of return, risk in behavioral portfolio theory
is measured by the probability of failing to
reach the threshold level of a mental account.
Investors in behavioral portfolio theory
choose their best combination on the effi-
cient frontier of expected returns and the
probability of failing to reach the threshold
of each mental account.
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2 PORTFOLIOS FOR INVESTORS WHO WANT TO REACH THEIR GOALS FALL 2011
The road to behavioral portfolio theory started
more than 60 years ago, when Friedman and Savage
[1948] noted that hope for riches and protection from
poverty share roles in our behavior; people who buy
lottery tickets often buy insurance policies as well. So,
people are risk seeking enough to buy lottery tickets
and risk averse enough to buy insurance. Four years
later, Markowitz wrote two articles that ref lect two
very different views of behavior. In one [1952a], he cre-
ated mean–variance theory, and in the other [1952b],
he extended Friedman and Savage’s insurance–lottery
framework. People in mean–variance theor y, unlike
people in the insurance–lottery framework, never buy
lottery tickets; they are always risk averse and never risk
seeking.
Friedman and Savage observed that people buy
lottery tickets because they aspire to reach higher social
classes, whereas they buy insurance as protection against
falling into lower social classes. Markowitz [1952b]
clarified Friedman and Savage’s observation by noting
that people aspire to move up from their current social
class or “customary wealth.” So, people with $10,000
might accept lottery-like odds in the hope of winning
$1 million, and people with $1 million might accept
lottery-like odds in the hope of winning $100 million.
Kahneman and Tversky [1979] extended the work of
Markowitz [1952b] into prospect theory. Prospect theory
describes the behavior of people who accept lottery-like
odds when they are in the domain of losses, such as
when they are below their aspiration levels, but reject
such odd s when they are in the d omain of g a ins, such a s
when they are above their aspiration levels.
Behavioral portfolio theory’s description of port-
folios as collections of mental accounts, or a set of layers
in portfolio pyramids, is part of common investment
advice. The mental account or pyramid structure of
behavioral portfolios is also reflected in the upside-
potential and downside-protection layers of “core and
satellite” portfolios.
One might argue that although the portfolios are
described as layered pyramids, which are consistent with
behavioral portfolio theory, investors actually consider
them as a whole, wh ich i s con sistent with mean–v a r i a n ce
portfolio theory. But the evidence does not support such
an argument. Consider, for example, Question 13 in
the Asset Allocation Planner of Fidelity Investments
[2003]:
If you could increase your chances of improving
your returns by taking more risk, would you:
1. Be willing to take a lot more risk with all your
money?
2. Be w i ll i n g to t a ke a lot more r i s k with some of your
money?
EXHIBIT 1
Mean–Variance Portfolio Theory
Investor s consider the portfolio a s a whole. They want to be on
the mean–variance ef ficient frontier.
EXHIBIT 2
Behavioral Portfolio Theory
Investors divide their portfolios into mental accounting layers
or a portfolio pyram id, whereby each layer is associated with a
particular goal and a particular attitude toward risk.
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THE JOURNAL OF WEALTH MANAGEMENT 3
FALL 2011
3. Be willing to take a little more risk with all your
money.
4. Be willing to take a little more risk with some of
your money?
5. Be unlikely to take much more risk?
Answers 1 and 3 make sense within mean–variance
theory. In that theory, only the risk of all the money
in the overall portfolio matters. But Answers 2 and 4
make no sense within mean–variance theory because
they assume a segmentation of the portfolio into mental
accounts depending on where investors are willing to
take more or less risk with some of their money. Mean–
variance investors have a single attitude toward risk,
not a set of attitudes mental account by mental account.
In contrast, behavioral investors have many attitudes
toward risk, one for each mental account, so they might
be willing to take a lot more risk with some of their
money. Statman [2004] found that the number of inves-
tors who were willing to take a lot mor e risk wit h some of
their money exceeded the number of investors who were
willing to take a little more risk with all their money
by a ratio of approximately 10 to one. Yet taking a lot
more risk with some of our money adds to our overall
portfolio risk about the same as taking a little more risk
with all our money.
Recently, Harry Markowitz joined Das et al.
[2010] in the development of mental accounting port-
folio theory, which combines mean–variance portfolio
theory with several features of behavioral portfolio
theory. These features include the mental accounting
structure of portfolios and the definition of risk as the
probability of failing to reach threshold levels. But mental
accounting portfolio theory does not include two central
features of behavioral portfolio theory. First, investors
in behavioral portfolio theory can be risk seeking in
some of their mental accounts, but risk-seeking pref-
erences are excluded from both mean–variance port-
folio theory and mental accounting portfolio theory.
Second, the optimal securities in behavioral portfolio
theory resemble call options w ith asymmetric distr ibu-
tions of returns rather than securities with symmetric
distributions of returns, such as normal distributions.
We hasten to add that mean–variance portfolio theory
can accommodate many kinds of return distributions,
including asymmetric return distributions.
INVESTOR GOALS AND MENTAL ACCOUNTS
Consider a 50-year-old investor with $1 million in
investable assets. She allocates her money to three mental
account subportfolios, each associated with a goal. She
allocates $800,000 to the retirement account, $150,000
to the education account, and $50,000 to the bequest
account that she hopes to leave for her children. Our
investor has other assets, including a house and future
Social Security payments. Moreover, our investor is still
working and investing her savings. She intends to retire
at 65 and estimates her life expectancy to be 75.
Our investor wants her $800,000 retirement
account to grow to $1,917,247 by the time she is 65,
which implies a compound annualized return of 6%
during the remaining 15 years. Her horizon for the
education account is shorter, only three years, and she
wants that account to grow to $188,957, which implies
a compound annualized return of 8% . Our investor’s
horizon for the bequest account is long—25 years—but
she has an ambitious aspiration; she wants her current
account of $50,000 to grow to $850,003, which implies
a compound annualized return of 12% .
The mean–variance efficient frontier faced by our
investor is determined by the characteristics of avail-
able securities, namely, their expected returns, standard
deviations, and correlations. She creates three mental
account subportfolios associated with her three goals and
will evaluate her progress toward her goals once a year,
adjusting her goals and mental account subportfolios
as necessary. Each mental account subportfolio is opti-
mized by the mean–variance procedure. The optimal
mean–variance subportfolio for the retirement goal is
the one that corresponds to an expected return of 6%.
The 6 % expected ret u r n also deter m ines the ris k of th i s
subportfolio. Mental account subportfolios are created
similarly for the education goal, with its expected return
of 8% , and the bequest goal, with its expected return
of 12% . Exhibit 3 provides a summar y of the mental
account subportfolios and their related return and risk
characteristics.
To keep our example simple, assume there are
only three assets: a bond mutual fund or ETF with an
expected return of 2% a year and a standard deviation
of 5%, a conservative stock mutual fund or ETF with an
expected return of 8% a year and a standard deviation of
20%, and an aggressive stock mutual fund or ETF with
an expected return of 15% a year and a standard deviation
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4 PORTFOLIOS FOR INVESTORS WHO WANT TO R EACH THEIR GOALS FALL 2011
of 40%. The returns of the bond fund are uncorrelated
with the returns of the two stock funds. The returns of
the two stock funds have a 25% correlation.
Our investor calculates three mean–variance effi-
cient subportfolios corresponding to the three goals.
They are displayed in Exhibit 4 along with the overall
portfolio. The standard deviation of the returns of the
re t i rement su bpor t fol io is t he lowe s t at 10.45 % , fo l lowed
by the 15.23% of the education subportfolio and the
25.28% of the bequest subportfolio. The 6.60% expected
return of the overall portfolio is a weighted average of
the returns of the three mental account subportfolios,
but the 11.85% standard deviation of the overall portfolio
is different from the weighted average of the standard
deviations of the three mental account subportfolios.
Asking our investor to state her preferences for
expected return and standard deviations for each goal
is better than asking her to state her preferences for the
expected return and standard deviation of the overall
portfolio. The latter requires that she aggregate the three
goals in her mind, which is difficult. It is easier to match
low risk tolerance with the retirement subportfolio,
medium risk tolerance with the education subportfolio,
and high risk tolerance with the bequest subportfolio
than to match a weighted-average risk attitude toward
the three goals.
Moreover, since our investor has little sense of her
tr ue attitude toward risk in the overall por tfolio, asking
her to state that risk attitude is likely to result in her
choosing a portfolio on the eff icient frontier that does
not correspond well to her true risk attitude. There is a
loss that comes from choosing the wrong portfolio on
the eff icient frontier, and that loss can range between a
few basis points of annual return to several percentage
points. See Exhibit 5 for a schematic representation of
these losses.
Mental account subportfolios and the overall port-
folio are always on the mean–variance eff icient frontier
when there are no constraints on allocation, such as a
preclusion of short or leveraged positions (see Exhibit 6).
Mental account subportfolios may be a few annual basis
points below the mean–variance efficient frontier when
such constraints are imposed. Yet even such small losses
are rarely incurred in usual practice, as financial advi-
sors rarely recommend overall portfolios with short or
leveraged positions.
EXHIBIT 3
Mental Account Subportfolios and Goals
EXHIBIT 4
Mental Account Subportfolios and the Overall Portfolio
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THE JOURNAL OF WEALTH MANAGEMENT 5
FALL 2011
The proportion allocated to the bond fund is
highest in the retirement subportfolio, lower in the edu-
cation subportfolio, and lowest in the bequest portfolio.
Arranging the overall portfolio as a set of three mental
account subportfolios does not imply that we need three
“real” accounts for each fund, or one for the bond fund
in the retirement subportfolio, another for the bond
fund in the education subportfolio, and a third for the
bond fund in the bequest subportfolio. Instead, we have
one real bond fund account and three “virtual” bond
accounts listing the allocation in the bond fund of each
subportfolio. Financial advisors can provide portfolio
reports to their clients in two formats—real account
formats for the overall portfolio and virtual account for-
mats for each of the mental account subportfolios (see
Exhibit 7).
The presentation of the overall portfolio as the sum
of the three mental account subportfolios has an advan-
tage over a sole presentation of the overall portfolio.
The mental account subportfolios’ presentation speaks
the language of investors. Investors want to reach their
goals, not have portfolios only on the mean–variance
eff icient frontier. Goal-based mental account subportfo-
lios let investors articulate each goal, the horizon of each
goal, and the attitude toward risk for each goal. Telling
the investor that the expected return of her overall port-
folio is 6.60 % and its standard deviation is 11.85% tells
her little because the overall portfolio combines three
very different goals. Telling her the expected returns
and standard deviations of each subportfolio highlights
her goals.
MEASURING RISK BY THE PROBABILITY
OF LOSSES AND THEIR AMOUNTS
Our investor is better able to convey her prefer-
ences for expected returns and standard deviations asso-
ciated with each goal than her preferences for expected
returns and standard deviations associated with the
overall portfolio. Yet standard deviations have little intu-
itive meaning to most investors who are not schooled in
statistics. Standard deviations can be made more intui-
tive to understand. For example, we can tell our investor
that the 10.45% standard deviation in her retirement
subportfolio implies that she has approximately a two in
three chance that her realized return would be higher
than –4.45% yet lower than 16.45%. The first is the 6%
expected return minus the 10.45% standard deviation,
and the second is 6% plus 10.45%. This explanation helps
convey the facts of risk somewhat more intuitively but
not much more i ntu itively. ( I ndeed , standard deviations
have little intuitive meaning even to investors schooled
in statistics).
Our investor is likely to prefer to express her atti-
tud e toward risk by sp ecif y i n g a pr obab i l i t y of inc u r r ing
a loss and the amount of loss. Referring to the retirement
mental account subportfolios, she might say that she
EXHIBIT 5
Loss in Units of Expected Return from Misestimated
Risk Attitudes
Misestimation of risk attitudes leads to suboptimal portfolios on
the efficient frontier.
EXHIBIT 6
The Mean–Variance Efficient Frontier with Mental
Account Subportfolios and the Overall Portfolio
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6 PORTFOLIOS FOR INVESTORS WHO WANT TO R EACH THEIR GOALS FALL 2011
does not want too much risk. Specifically, she is willing
to accept no more than a 6.28 % probability of losing
more than 10% of her retirement subportfolio during
the coming year. Our investor is willing to accept con-
siderably more risk in her bequest subportfolio. In that
subportfolio, she is willing to accept a 14.27% prob-
ability of losing more than 15% of her funds during the
coming year.
It turns out that the 10.45% standard deviation
of the retirement subportfolio can be expressed more
intuitively as a probability no higher than 6.28% of
losing more than 10% of her funds in the retirement
subportfolio during the coming year. Similarly, the
25.28% standard deviation of the bequest subportfolio
is expressed more intuitively as a probability no higher
than 14.27% of losing more than 15% of her funds in the
bequest subportfolio during the coming year.
Exhibit 8 compares the mental account subportfo-
lios’ risk with the language of probability.
CONCLUSION
Investors are attracted to Markowitz’s mean–
variance portfolio theory by its logic and practical
application. It seems logical to choose portfolios based
on their overall expected returns and standard devia-
tions of returns. And the mean–variance optimizer is
a practical tool, quick at drawing the efficient frontier
of expected returns and standard deviations of returns.
Yet investors want more than portfolios on the mean–
variance efficient frontier. Ultimately, investors want
their portfolios to satisfy goals such as a secure retire-
ment, college education for their grandchildren, or a
bequest for children.
While Markowitz’s mean–variance portfolio
theor y is si l ent a bou t ultimate portfol io goa ls, t hese g o a l s
are central in Shefrin and Statman’s behavioral port-
folio theory. Among other things, investors in behav-
ioral portfolio theory divide their portfolios into mental
account layers of a portfolio pyramid, whereby each layer
EXHIBIT 7
Real Portfolio and Virtual Portfolios
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THE JOURNAL OF WEALTH MANAGEMENT 7
FALL 2011
is associated with a particular goal and a particular atti-
tude toward risk. Risk in behavioral port folio theory is
the probability of failing to reach the threshold of each
goal, not the standard deviation of the returns of the
overall portfolio. In this article, we combine appealing
features of mean–variance portfolio theory and behav-
ioral portfolio theory in a mental accounting portfolio
theory. Investors divide their portfolios by goals into
mental account layers of a portfolio pyramid, and risk is
the probability of failing to reach the threshold return
of each goal. Yet each mental account is optimized by
the rules of mean–variance portfolio theory.
ENDNOTE
This article is an abbreviated version of “Portfolio
Optimization with Mental Accounts” [Das et al. 2010].
REFERENCES
Das, S., H. Markowitz, J. Scheid, and M. Statman. “Portfolio
Optimization with Mental Accounts.” Journal of Financial and
Quantitative Analysis, Vol. 45, No. 2 (2010), pp. 311-334.
Fidelity Investments, “Asset Allocation Planner Question-
naire.” 2003, Available at: ww w.fidelity.com.
Friedman, M., and L.J. Savage. “The Utility Analysis of
Choices Involving Risk.” Journal of Political Economy, 56
(1948), pp. 279-304.
Kahneman, D., and A. Tversky. “Prospect Theor y: An Anal-
ysis of Decision Making under Risk.” Econometrica, 47 (1979),
pp. 263-291.
Markowitz, H. “Portfolio Selection.” Journal of Finance, 7
(1952a), pp. 77-91.
——. “The Utility of Wealth.” The Journal of Political Economy,
60 (1952b), pp. 151-158.
Shefrin, H., and M. Statman. 2000. “Behavioral Portfolio
Theory.” Journal of Financial and Quantitative Analysis, Vol. 35,
No. 2 (2000), pp. 127-151.
Statman, M., and V. Wood. “Investor Temperament.” Journal
of Investment Consulting, Summer (2004), pp. 1-12.
To order reprints of this article, please contact Dewey Palmieri
at dpalmieri@ iijournals.com or 212-224-3675.
EXHIBIT 8
Probability of Return Lower than Threshold Return
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