111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Portfolio Choiceand PerceivedDiversification
David RodeDepartment of Social and Decision SciencesCarnegie Mellon UniversityPittsburgh, PA 15213
Research Paper Committee:Paul Fischbeck (chair)John H. MillerBaruch Fischhoff
Revision: May 2, 2000
ABSTRACT
The old adage about putting all of one’s eggs in one basket was thought wise long before
the advent of modern portfolio theory and the important results of Markowitz (1959) and others.
However, despite the anecdotal imperative to investors to diversify their portfolios, the empirical
reality is that diversification rarely takes place as recommended by portfolio theory. Institutional
limitations, estimation errors, and transaction costs are often held responsible for the limited
diversification observed in practice. In this paper, however, I explore more psychological reasons
for inconsistent diversifying. I also examine several portfolio selection heuristics commonly used
by investors. In practice, the performance of some very simple heuristics, suggested as more
psychologically appealing or tractable, indicates that boundedly-rational, information-constrained
investors can do nearly as well as the Markowitz optimum. Experimental results also suggest that
subjects fail to recognize core elements of Markowitz diversification. Nevertheless, simple rules
often allow them to mimic nearly-optimal strategies. Thus, advice given to investors should
concentrate on getting them to apply what they know and understand consistently, rather than on the
adoption of new and more complex techniques. Experimental results also suggest that prospect
theory may be better able than utility theory to explain the diversifying behavior of investors.
“It is the part of a wise man to keep himself today for tomorrow,and not venture all his eggs in one basket.”
-Miguel de Cervantes, Don Quixote
“Behold, the fool saith, ‘Put not all thine eggs in the one basket’ -which is but a manner of saying, ‘Scatter your money and yourattention’; but the wise man saith ‘Put all your eggs in the onebasket and - watch that basket!’”
-Mark Twain, The Tragedy of Pudd’nhead Wilson
I. THE EXHORTATION TO “DIVERSIFY”1
The sense that diversifying assets is a good idea was merely well-reasoned speculation until
the appearance of Markowitz’s (1959) mathematical demonstration of its risk-reducing power. In
the Markowitz framework, investors measure the riskiness of investments by their return variance in
the context of a portfolio, or collection of assets. By using the tools of linear and quadratic
programming newly available in the 1950s, Markowitz demonstrated that when risky assets are
aggregated, it is their covariance that often determines the majority of the total risk, rather than only
their variance. Consequently, the total risk of a portfolio, given some careful planning2, could often
be less than the sum of the portfolio’s component pieces. Virtually overnight, financial theory
became concerned with systematic risks, as opposed to the idiosyncratic risks of earlier work in
investment analysis.3
1Thanks go to my committee members, Paul Fischbeck, John Miller, and Baruch Fischhoff, for their helpwith this project and to Carter Butts, Andy Parker, Michele DiPietro, and George Loewenstein for many thoughtfuldiscussions. I would also like to thank the members of the Carmel Hadassah Investment Club, the CCC InvestmentClub, Women Investing Now, and the Ladies of the Court. Special notes of thanks go to Ruth Silverman, SallySleeper, Marilyn Sleeper, Sue Rode, Carol Anderson, and Eileen Boerio for their help in coordinating the investmentclubs’ participation. Of course, the usual disclaimer applies. The material in the first two sections and the appendixcontains background material for readers without a background in finance and capital market theory.
2This “careful planning” is discussed in detail in the appendix.
3In finance, systematic risks are market-level risks, while idiosyncratic risks are company-specific riskfactors. Earlier work used statistics such as debt levels and price-to-earnings ratios to measure risk. These measures,however, are proxies for idiosyncratic risks, which are diversified away in a mean-variance efficient portfolio.
1
The reduction in risk provided by diversification is of such significant value because, if
nothing else, it provides the reduction at minimal cost. In fact, this particular characteristic
(“efficiency”) serves as the foundation for the capital asset pricing models (CAPMs) of Sharpe
(1964), Lintner (1965), and Mossin (1966). Because the reduction in risk provided by
diversification is costless4, investors should not be compensated for bearing risks that could be
diversified away (i.e., firm-specific or idiosyncratic risks).
At least for the last decade, investment advisors have consistently touted the advantages of
diversification to individual and institutional investors alike (e.g., Peavy and Vaughn-Rauscher,
1994; Wasik, 1995; Anderson, 1998; Hays, 1998; Hulbert, 1998; Spragins, 1999). On the surface,
at least, investors have accepted the intuition behind Markowitz diversification.5 Diversification was
perceived as trivial to implement - particularly given the proliferation of mutual funds.6 Computer
programs that perform the required calculations abound and are frequently bundled (in some form)
with personal finance software.7 As for implementation, the widespread availability of mutual funds
makes at least some measure of diversification available to nearly all investors.
4Recall that transaction costs, taxes, market imperfections, and liquidity constraints are assumed away inthe standard asset pricing theories.
5I will use mean-variance efficient , diversified in the Markowitz sense , and Markowitz diversificationinterchangeably. In each case, I am referring only to the normative algorithm for portfolio selection first illustratedby Markowitz (1959). One could easily suppose other measures of diversification, but they would not fall within thenormative framework I am concerned with here. In fact, numerous other criteria for portfolio selection have beenproposed (see Grinold and Kahn (1995) for a survey of information ratio, maximum alpha, multiobjective, and goalprogramming-oriented approaches and Elton and Gruber (1995) for explanations of the maximum geometric meanreturn and safety first models). Mean-variance efficiency is the dominant approach largely because it is utility-maximizing under the assumption of either quadratic utility or normally-distributed asset returns.
6To be clear, I am making the claim that the mean-variance algorithm itself is trivial to implement.Subsequent sections will discuss the difficulties with structuring a portfolio selection problem for use with a mean-variance optimizer. For example, estimation and forecasting of the covariance matrix is a persistently difficult task.Even identifying the universe of possible assets can be a challenging endeavor for many investors.
7Portfolio optimization can be done easily via the Solver in Microsoft’s Excel spreadsheet package. Evenmore simply, there are stand-alone software packages (e.g., Efficient Solutions’ MVO Plus) available and even a freeversion on the WWW (http://finance.wharton.upenn.edu/~stambaug/portopt.html). The WWW version islimited to nine assets, but is full-featured and easy to use.
2
At least since Evans and Archer’s (1968) classic work on diversification, investors have
appeared to accept the advice on holding a “diversified portfolio.” In pursuit of proper
implementation, however, numerous obstacles appeared. There was far less consensus on exactly
what constituted a properly diversified portfolio; the gap between “don’t put all your eggs in one
basket” and Markowitz’ (1959) algorithm was substantial. As Zweig (1998) notes, it’s easier to
talk about building a diversified portfolio than it is to actually build one. Evans and Archer’s (1968)
pioneering work showed that nearly all of the diversifiable risk is eliminated in a portfolio of ten
securities.8 Statman (1987) claimed that the benefits of diversification were essentially exhausted
after 30 securities. Sears and Trennepohl (1993) and Bodie, Kane, and Marcus (1993) appear to
suggest that the correct number is somewhere between the previous limits. Antanasov (1998) claims
that investors can achieve a good level of diversification by holding six to ten mutual funds, but that
more than ten isn’t necessary. Wasik (1995) reports that the National Association of Investment
Clubs (NAIC) advocates the “Rule of 5” - holding “no fewer than 5 stocks.”9 At the very least,
this evidence suggests the potentially dangerous inadequacy of simple heuristics. At the worst, it
suggests that investment “experts” are propagating irrational advice on asset allocation (Canner,
Mankiw, and Weil, 1997). However, as Elton and Gruber (2000) note, correcting Canner et al .
(1997), expert portfolio allocation recommendations cannot necessarily be declared irrational
simply because they disagree with one another. The myriad of different portfolios observed result
from the experts making significantly different assumptions about the inputs and using different
sources of data. This diversity of strategy only exacerbates the complexity of the problem facing
investors.
Even if one assumes agreement on the models and data in question, one problem with such
studies is that they tend only to consider common stocks (and common stock funds) traded in the
United States. In this respect, these authors were speaking of constructing diversified stock
portfolios. The theory, however, was intended to address diversified asset portfolios. Other authors
have addressed this shortcoming (e.g., Grubel (1968) on international investments, Firstenberg,
Ross, and Zisler (1988) on real estate, Blume, Keim, and Patel (1991) on high-yield bonds, etc.), but
the problem remains. There is a tendency for investors to mistake a diversified stock portfolio for a
8Of course, this depends on what is assumed to be the set of all possible securities. Evens and Archer(1968) looked solely at domestic stocks.
9It is worth noting that the NAIC’s justification for this heuristic is that “of the five stocks, one willprobably be a loser, three will produce mediocre results, but the fifth will be a real winner.” (Shefrin, 2000)
3
diversified asset portfolio, and there is a tendency for professionals to advise investors in such a
way as to minimize the importance of including other asset classes in a properly diversified
portfolio (Belec, 1999).
Although investment advisors almost unanimously advocate “diversification,” the advice on
implementation represents a cacophony of disparate rules, many in conflict with the driving
principles of Markowitz efficiency. The lack of clear and consistent advice to investors results in
haphazard response to asset allocation. Investors find it difficult to understand what it is about
diversification that makes it so valuable. As a result, they resort to heuristic approaches that more
often than not address the perception of diversification, rather than the reality.
Mean-Variance Diversification in Practice
In light of rather overwhelming evidence that mean-variance optimization is a conceptually
simple, profitable, and prudent strategy, it is surprising to note that very few investors hold asset
portfolios that are diversified in the Markowitz sense. More fundamentally, there is no real
consensus as to what diversification actually is where implementation is concerned. Hirshleifer and
Riley (p. 72, 1992) note that “unfortunately, we have no handy rule for deciding when a portfolio
may be considered well-diversified.”10 In professional advice frequently given to investors, there
lacks a deep understanding of what gives Markowitz diversification its power. Inherent in this
understanding is the notion of covariance. For most investors this is an ill-defined concept - if it has
any meaning at all. For example, in a survey of 45 investors at a NAIC meeting, there was only
weak evidence that covariance was important in determining portfolio risk (see Table 2). DeBondt
(p. 836, 1998) noted that “[t]he very idea that risk is defined at the level of the portfolio - rather
than at the level of individual assets - and that risk depends on covariation between returns remains
foreign to many investors.”
If we are to take “diversification” to mean simply dividing our capital among several
investments, we are setting an incomplete criterion by which to implement Markowitz
10As will be discussed below, we cannot always say that a portfolio is “almost” well-diversified or that aportfolio “resembles” a diversified portfolio.
4
diversification. It is not sufficient that there be many of them; they must be different.11 Individual
investors frequently invest in a variety of stock mutual funds and believe themselves to hold a
diversified portfolio (NYSE, 1999). Here, they are focusing solely on the quantity of assets in their
portfolios, rather than the class membership of those assets. Many equity mutual funds hold the
same stocks (see Tables 1a and 1b).12 If an investor own shares in three mutual funds and each of
those funds holds shares in Microsoft, Time Warner, and Wal-Mart, is that investor really
diversified (at least to the extent percevied)? Most large mutual funds hold similar investments, if
only because their immense size seriously limits the number of stocks in which it is practical for
them to invest.
11It is a simple statistical fact that it is sufficient for there to be “many” (∞ in the limit) of them so long asthey are not perfectly correlated. So long as all intercorrelations are less than one, as the number of securities in aportfolio approaches the size of the market, the risk of any portfolio will fall to the risk of the market portfolio.These results hold independently of the class membership of the assets themselves.
12The returns of the funds in Table 1a have an average correlation of 0.8189 ± 0.0228 over the period ofJanuary 1, 1997 to December 31, 1998 (using daily returns). The minimum correlation as 0.5731 and the maximumwas 0.9886.
5
Table 1a: Top Holdings of the 10 Largest Mutual FundsDollar amounts are in millions and stocks are listed in alphabetical order
FundF i d e l i t y
M a g e l l a nVanguard 500
IndexW a s h i n g t o n
Mutual Investment Co.
of AmericaFidelity Growth
and Income
Size $93,560 $91,110 $57,150 $53,240 $47,610
America Online AT&T Bank of America AT&T American Express
Cisco Systems Cisco Systems Bristol-Myers Bank of America AT&T
Citigroup Exxon GTE Computer Assoc Citigroup
General Electric General Electric Household Fannie Mae Exxon
Home Depot IBM Monsanto IBM Fannie Mae
MCI WorldCom Intel Sara Lee Microsoft General Electric
Merck Lucent Sprint Philip Morris MCI WorldCom
Microsoft Merck Texaco Sprint Merck
Time Warner Microsoft US West Time Warner Microsoft
Wal-Mart Wal-Mart Wells Fargo Viacom Philip Morris
FundF i d e l i t y
ContrafundAmerican Century
Ultra InvestorsVanguard
Windsor IIJanus Fund
VanguardW e l l i n g t o n
S i z e $41,330 $33,510 $32,400 $31,840 $26,360
Associates First AT&T Anheuser-Busch AIG Alcoa
AT&T AIG Bank of America American Express CIGNA
Cisco Systems America Online Chase Manhattan Charles Schwab Citigroup
CVS Coca-Cola Citigroup Cisco Systems Dow Chemical
McDonald’s EMC Corp GTE Comcast Ford
MCI WorldCom General Electric Honeywell Linear Tech IBM
Microsoft MCI WorldCom SBC Comm Sun Microsystems Motorola
Time Warner Microsoft Wash. Mutual Texas Instrum Pharmacia
Unisys Pfizer Waste Mgmt. Time Warner Union Pacific
Vodafone Time Warner Williams Tyco Intl Xerox
Source: Lipper Analytical Services (1999) and various fund management companies
6
Table 1b: Summary of Top 10 Mutual Fund Holdings
This table includes securities found in the top ten holdings of one fund and heldby another fund in any amount, not just as one of the largest ten holdings.
Stocks Held by 9 of 10 Funds AT&T, Bank of America, Fannie Mae, IBM
Stocks Held by 8 of 10 Funds Time Warner, Citigroup
Stocks Held by 7 of 10 Funds Intel, MCI WorldCom, Cisco Systems
Stocks Held by 6 of 10 FundsMicrosoft, General Electric, Philip Morris,
Merck, Pfizer, Wal-Mart
Source: Lipper Analytical Services (1999)
Shapiro (1999) notes that even companies that seem different on the surface may have
strong performance relationships. He attributes this to the fact that most large companies today
participate in several different businesses and transact in numerous markets. Pepsi owns fast-food
restaurants, General Electric has a substantial financial services subsidiary (in fact, financial services
generates more profit for GE than any other division - nearly 40% of total profit), Westinghouse
owns CBS, just to name a few. The problem is that the managers of these firms tend to see the same
benefits of diversification that potential investors do. An investor hoping to hedge an entertainment
industry exposure by owning Westinghouse is neglecting the fact that Westinghouse’s managers
acted to hedge an industrial exposure by owning CBS. Thus, owning Westinghouse shares does
not hedge the investor’s entertainment industry exposure as intended. The investor has achieved the
perception of diversification without obtaining the intended benefit. One could argue, then, that such
investors are worse off not only because they are not actually diversified, but also because the
perception that they are diversified causes them to ignore very real risks.
It may be the case that investors understand - in general terms - the importance of
diversification, but fail to implement it properly (as a result of transaction costs or capital
constraints, for example). Alternatively, it may be the case that the diversification investors actually
desire is not the diversification that Markowitz (1959) wrote about. Investors may find Markowitz
(1959) diversification “unpalatable” (Fisher and Statman, 1997), a classification that I shall return
to later. Even when it comes to professional investors, the evidence is mixed. Michaud (p. 33, 1989)
provides anecdotal evidence that “a number of experienced investment professionals have
experimented with mean-variance [MV] optimizers only to abandon the effort when they found
their MV-optimized portfolios to be nonintuitive and without obvious investment value . . . . (T)he
optimized portfolios were often found to be unmarketable either internally or externally.”
7
The enduring discrepancy between intent and implementation is striking. Nagy and
Obenberger (1994) surveyed 137 shareholders on factors thought to influence individual investor
behavior. The need for diversification ranked second (43.6% of responses) in a list of 34 variables
(although diversification was left undefined). The past performance of investments ranked seventh
(34.6% of responses) and expected stock market performance ranked twelfth (28.6% of
responses). Interestingly enough, the desire to minimize risk ranked ninth (32.3% of responses).
Perhaps the most interesting result being that the need for diversification is seen as different from
risk minimization. One gets the sense that investors know that diversification is a good thing, but
are not sure exactly why.
Even among experts in the field, there is little evidence of proper diversification. In fact, in
several cases, there is explicit evidence against it. Money (1998) asked six prominent economists
for their TIAA-CREF asset allocations. Not one indicated having anything even close to Markowitz
diversification - not even Harry Markowitz himself. In fact, Markowitz’s comments on the subject
are quite insightful (page 118, Money, 1998):
“I should have computed the historical covariances of the assetclasses and drawn an efficient frontier. Instead, I visualized my griefif the stock market went way up and I wasn’t in it - or if it went waydown and I was completely in it. My intention was to minimize myfuture regret [emphasis added]. So I split my contributions fifty-fifty between bonds and equities.”
Several of the comments include direct acknowledgement of non-Markowitz allocations.13 In each
of the cases, the allocations corresponded to focal points such as 25%, 75% or 100% and involved
“stories” claiming to justify such choices.
DeBondt (p. 832, 1998) analyzed data from a group of individual investors and labeled
“surprising” the “failure of many people to infer basic investment principles from years of
experience.” DeBondt (1998) surveyed 45 investors recruited at a meeting of the National
Association of Investment Clubs (NAIC). Even among this relatively sophisticated group of
investors, it was widely believed that risk could be managed by knowledge and trading skill after
funds had been committed to an investment (Table 2) - a violation of the Efficient Markets
Hypothesis (EMH).
13Of course, caution must be exercised in interpreting these results. Most respondents had significantholdings outside of their TIAA-CREF accounts. Thus, only a fraction of their total wealth was measured.
8
Table 2: Survey of 45 Investors at NAIC Meeting
+2 is Strongly Agree, -2 is Strongly Disagree, 0 is Neither Agree nor DisagreeThe t-statistics test whether the average response score equals zero.
Statement Mean Response t-statistic
I would rather have in my stock portfolio just a fewcompanies that I know well than many companiesthat I know little about.
1.4389% Agree
10.0
If you do not do your homework, I doubt you willachieve much investment success.
0.9270% Agree
5.3
Investing in stocks is like buying lottery tickets.Luck is everything and investment skill plays nomeaningful role.
-1.620% Agree
-20.1
Because most investors do not like risk, risky stockssell at lower market prices.
-0.897% Agree
-5.8
The risk of a stock depends on whether its pricetypically moves with or against the market.
-0.5718% Agree
-3.2
Source: DeBondt (1998)
Trends in Diversifying Practice Over Time
Regardless of investors’ beliefs, their behaviors have changed little: most investors do not
Markowitz diversify. The most commonly used data source for this area is the Federal Reserve
Board’s Survey of Consumer Finances (SCF). Alternatively, many researchers have used records
from brokerage houses. Although there are significant differences14 between the two sources,
together they provide a remarkably complete picture of individual investor behavior. Table 3
summarizes evidence from these sources with regard to portfolio-allocation decisions by the general
population.
14The SCF was originally named the Survey of Financial Characteristics of Consumers (SFCC). It isadministered every three years under the auspices of the Federal Reserve Board of Governors. The granularity of thedata has increased dramatically over the years, as has the proliferation of new financial products. As a result,reporting across years is difficult to compare at best. The SCF/SFCC data is obtained from a randomly-selectedsample of US households. This is in marked contrast to data obtained from brokerage-house records. Particularly inthe early years (e.g., Lease, Lewellen, and Schlarbaum’s (1974) work), this group of individuals constituted a verydifferent demographic group than the SCF data. Each source provides valuable data, but they should not be confusedor directly compared.
9
Table 3: Timeline of Survey Evidence
1 9 6 2
•83% of respondents own tangible assets (e.g., houses, automobiles, etc.)•78% of respondents own liquid assets (bankable deposits)•58% of respondents own life insurance, annuities, or retirement plans•18% of respondents own stock•2% of respondents own bonds
Data from SFCC (Projector, 1963)
1971-1 9 7 2
•Median investor holds no investment assets other than common stock and life insurance•56% hold no mutual funds•70% hold no bonds
Sample from major New York City brokerage house (Lease, Lewellen, and Schlarbaum, 1974)
1 9 7 5•Average number of stocks held by those who hold stock is 3.41
Data based on SFCC information (Blume and Friend, 1975)
1 9 8 2
•19% own stock, but only 40% of those own shares in more than one company and even fewer had abrokerage account (35%) or traded stock at all in the past year (27%)
Data from SCF (Avery et al, 1984a; 1984b)
1 9 8 4
•Limited diversification observed even when assets other than stocks is considered (e.g., real estate,bonds, commodities, etc.)
Data based on SCF information (King and Leape, 1984)
1 9 8 9
•Stock, bond, mutual fund, and retirement plan assets comprise only 13.8% of total household assets•Real estate remains the dominant component of most households’ portfolios
Data from SCF (Kennickell and Starr-McCluer, 1994; Kennickell, Starr-McCluer, and Sundén, 1997)
1 9 9 5•Stock, bond, mutual fund, and retirement plan assets comprise 21% of total household assets
Data from SCF (Kennickell, Starr-McCluer, and Sundén, 1997)
1 9 9 8
•For all shareholders, the average number of stocks held (outside mutual funds) is only 1.5•For those who hold at least some shares directly (most individuals hold shares indirectly throughpensions and retirement savings plans), the average is 3.2
Data from SCF and other sources (NYSE, 1999)
Individual investors increasingly turn to mutual funds to implement portfolio-allocation
decisions. While mutual funds allow smaller, capital-constrained investors to obtain interests
efficiently in numerous assets, we have already demonstrated (Table 1a and 1b) that the
diversification implied by mutual fund investment can often be an illusion. More importantly,
10
investment in mutual funds can itself be costly when taken to extremes (increased monitoring costs,
multiple expense ratios, loads, etc.). The evidence in Figure 1 suggests that investors are investing in
numerous mutual funds - many of which could well be redundant.
Figure 1: Total Number of Mutual Funds Held by Investors Who Report Owning Funds
0%
10%
20%
30%
40%
50%
1 2 3 4 5 6+
Number of Funds Held
Percentage of Investors
1989 1992 1995
Source: NYSE (1999)
These observations suggest that even among the stockholding population, diversification is
haphazardly implemented. Investors hold shares in few companies, but hold more mutual funds
than necessary to achieve a given level of risk reduction.15 The same patterns are still apparent even
among wealthy and, presumably, sophisticated investors (or investors wealthy enough to obtain
sophisticated advisors). Avery and Elliehausen (1986) supplemented the 1983 SCF data with an
oversample of high-income families obtained from IRS tax files and observed many of the same
patterns. Only the very wealthiest group (the 0.5% reporting more than $280,000 in income in 1983
- equivalent to nearly $500,000 today) appears to be pursuing diversification in the Markowitz
sense.
15Diversification is not costless. There must be a balance between the benefits of diversification and thecosts (e.g., transaction costs, monitoring costs, informational costs, etc.) of achieving it. That said, regardless of thecosts involved, diversification past the point of the minimum variance portfolio is worthless from a Markowitzperspective.
11
That only this tiny elite appears diversified is suggestive. One might conclude, for example,
that transaction costs inhibit effective diversification for most investors. If only the wealthiest
investors are in a position to commit capital to a variety of asset classes, some serious questions
about the usefulness of modern portfolio theory are raised. The widespread availability of mutual
funds16 suggests, however, that (lack of) opportunity alone is not to blame. One might also
conclude that individual investors (who must generally make financial decisions for themselves) are
simply not content holding Markowitz-diversified portfolios. An obvious question would be
whether or not investors, knowingly rejecting the Markowitz approach, still considered their own
portfolios to be “diversified.”17 That is, is there a broad consensus that diversification has some
other specific meaning? If so, this would have significant consequences for financial theory,
including the CAPM.
II. COMMONLY SUGGESTED REASONS FOR IMPROPER DIVERSIFICATION
The failure of investors to Markowitz diversify in the presence of near-overwhelming
evidence in its favor has not gone unnoticed by researchers. Several reasons (summarized in Table
4) have been advanced to explain the limited diversification actually achieved in practice. Although
we are primarily concerned with psychological motives in this paper, it is certainly clear that all of
these factors influence actual asset-allocation choices. Moreover, constraints imposed by the
institutional, regulatory, and demographic environments are likely to influence the heuristic decision
processes used by investors.
Hanna and Chen (1995) suggest that demographic trends may be responsible for non-
Markowitz asset-allocation decisions. Investors may face time constraints on their investments that
are ignored under Markowitz diversification. The consumption demands of many older investors
require a higher degree of liquidity, for example, and may limit the ability of those investors to
16There are mutual funds that invest in both domestic and international stocks and bonds, in commodities,in real estate, and in derivatives of various types. Investors could achieve a substantial measure of diversificationsolely through investment in a properly selected mutual fund portfolio.
17One might also consider the potentially large human-capital costs some investors might face inbecoming informed about Markowitz diversification. Many individual investors have probably never heard of theMarkowitz approach.
12
implement a mean-variance efficient strategy. Younger investors may face “cash-in-advance”
constraints limiting their ability to pursue long-term illiquid investments. Thus, even if these
demographic groups are desirous of implementing Markowitz diversification, they may face limits
on their ability to do so.
Insufficient diversification may also be “unintentional.” Many investors ignore asset-
allocation changes resulting from passive rebalancing, or the changes in the weights of the assets
resulting from differential performance of the assets. During long periods of exceptional growth in
the market, actual weights can change substantially. For example, an investor who began the current
bull market (in the early 1990s) with a portfolio equally divided between the S&P 500 and a U.S.
Treasury bond would now have stock holdings worth more than twice the value of the bonds in the
portfolio if the investor took no action during the entire period.
Table 4: Commonly Suggested Reasons for Improper Diversification
Demographics/Life-Cycle ConstraintsAs investors age, they face consumption andinvestment constraints to their portfolio structure notconsidered by mean-variance analysis.
Passive RebalancingInvestors neglect changes to portfolio weights forwhich they are not directly responsible.
Institutional Restrictions
Infinite divisibility, assumed for both linear andquadratic programming models, is frequently violated inpractice. In addition, market depth in certain assets maylimit the ability of investors to take the positionsindicated by Markowitz diversification.
Global ConvergenceAchieving proper diversification is becoming moredifficult because international asset returns arebecoming increasingly intercorrelated.
Transaction Costs
The presence of transaction costs increases the marginalcost of diversification - especially for individualinvestors and less-common financial assets (e.g.,commodities and tangible property investments). Undertransaction costs, we might also include the costs ofbecoming informed about mean-variance efficiency andcollecting the requisite data (historical covarianceinformation, etc.)
Non-Markowitz PreferencesInvestors may have explicit preferences opposingmean-variance efficiency.
13
Institutional restrictions comprise a significant number of the obstacles to diversification
faced by investors - particularly individual investors. Standard asset-pricing models routinely
assume that securities are infinitely divisible and that the entire investment opportunity set is equally
available to all investors. In other words, they assume that there are no constraints on the ownership
and trading of securities other than those imposed by the (rational) preferences of the investors
(who are assumed to care only about the mean and variance of returns). Actual investors, however,
frequently face very real constraints on their ability to participate in securities markets. Restrictions
on the minimum size of investments and on the block size of trades place substantial limitations on
all but the most well-capitalized investors. Even well-capitalized investors, such as mutual funds,
face restrictions on investment, both in terms of market impact costs (e.g., Keim and Madhavan,
1995; 1996) and regulatory restrictions on ownership (e.g., Sutcliffe and Board, 1988).
Structural characteristics of most securities also inhibit implementation of Markowitz
diversification. Infinite divisibility is required to use standard linear and quadratic programming
techniques (Taha, 1997). Unfortunately, the weights on assets in the optimal portfolio tend to be
sensitive to the structure of the problem (Saunders and Woodward, 1979), rendering
“linearization” frequently useless.18 In addition, Bawa, Brown, and Klein (1979) and Michaud
(1989, 1998) refer to the issue of “non-uniqueness” in that, allowing for sampling errors in input
estimates, an optimal region is produced. Within this optimal region of statistically equivalent
portfolios, the actual structure of the statistically optimal portfolios varies widely. As Michaud
(p. 36, 1989) noted, “this means that optimal portfolio structure is fundamentally not well
defined.”
Over time, the returns of various assets around the world have become increasingly
intercorrelated.19 Equity and currency markets in particular have been marked by a growing
18In effect, this is saying that a portfolio manager wishing to implement a MV-efficient strategy requiringthe purchase of 11,742 shares of some stock could not, without consequence, simply purchase an even block of10,000 shares and be considered “approximately efficient.” It may be the case that, within an optimal region - suchas considered by Michaud (1989) - another, entirely different portfolio, would be preferable. In fact, this differentportfolio may not place any weight on the asset initially under consideration. Portfolio rankings are not monotonicin asset weights.
19There is a large macroeconomic literature on global convergence of economies (e.g., Barro and Sala-i-Martin, 1992; Durlauf and Johnson, 1992; Taylor, 1996; Williamson, 1996). The primary concerns of such work arefactor and productivity convergence. However, such research has also addressed issues of capital and labor marketconvergence and issues that have direct bearing on asset prices. The details of such findings, however, are beyond thescope of the present work.
14
convergence in return performance. The most obvious example of this phenomenon in recent
memory has been European monetary unification. This global convergence makes it more difficult
to achieve substantial measures of diversification, putting more emphasis on the number of
securities than on their correlation characteristics and thus, more of a burden on individual
investors. Even looking only within major domestic asset classes, correlations between classes have
risen over time. Table 5 lists the results for several example asset classes of least-squares fits of
monthly correlations over the previous 24 months for the period from 1930 to 1993. Significant
positive slopes are indicative of convergence over time.
Table 5: Domestic Asset Market Convergence, 1930 - 1993
**** denotes significance at the 0.0001 level or better*** denotes significance at the 0.001 level or better
** denotes significance at the 0.01 level or better
The intercept p-values test for difference from zero. The slope p-values test forwhether or not the slope coefficient is strictly greater than zero.
Asset Classes Intercept S lope Adjusted R2
S&P 500 & Government Bonds -0.066*** 0.000578**** 20.5%
S&P 500 & Corporate Bonds 0.048** 0.000512**** 17.9%
Government & Corporate Bonds 0.315**** 0.000891**** 57.2%
Government Bonds & Small Stocks -0.001 0.000248**** 4.9%
Corporate Bonds & T-Bills -0.058*** 0.000265**** 6.0%
Corporate Bonds & Small Stocks 0.126**** 0.000180**** 2.9%
Source: Center for Research in Securities Prices (CRSP), University of Chicago
Although transaction costs are assumed away in theoretical work, they assume a position of
critical importance in implementation. Investors must weigh the costs of making an additional
investment against the reduction in risk provided by such an investment. Because transaction costs
frequently decline as the size of the investment increases, one would expect larger investors, who
enjoy lower trading costs, to have more diversified portfolios. From the empirical evidence
presented earlier (cf. Avery and Elliehausen, 1986), this appears to be the case.
15
There is still the question, however, of how the mean-variance efficient strategy should be
implemented. Investors can either purchase a large quantity of a single asset class or a small
number of different asset classes. Given the transaction-cost structure that most small investors
face, one would expect them to choose the small/different path. In fact, many small investors instead
appear to be pursuing a strategy that maximizes the absolute number of securities that they hold,
without regard to their covariances. Carefully implemented, a mean-variance portfolio strategy could
actually reduce transaction costs for small investors by requiring the purchase of fewer securities.
It has often been suggested that investors do not Markowitz diversify because they have
preferences different than those assumed (implicitly) by Markowitz (1959) and others (e.g., Sharpe,
1964; Samuelson, 1967). These different preferences can be divided into within-framework
preferences and outside-framework preferences. Inside of the traditional mean-variance framework,
it is sometimes assumed that investors react to variables not included in typical utility functions. In
finance, it is often assumed (Huang and Litzenberger, 1988) that investors possess quadratic utility
of the form
U = a r − b σ 2 . [1]
In such formulations, investors can only react to changes in the mean return (r ) and variance (σ 2 ).
Simkowitz and Beedles (1978) suggested that investors take into account the skew of
portfolios returns. Kraus and Litzenberger (1976) and Harlow and Rao (1989) also presented
revisions of the standard CAPM to account for preferences for moments above the first two. In
particular, it has been suggested that investors prefer positive skew (or are negative skewness
averse). Simkowitz and Beedles’ (1978) investors were represented by utility functions that added a
term to [1] to account for the third moment.
U = a r − b σ 2 + c µ 3 á r é [2]
Traditionally, c has been set to zero (as in [1]) or symmetric return distributions have been assumed
(thus eliminating skew). Fama (1965a, 1965b) and Mandelbrot (1963, 1967) have both
demonstrated that the distribution of returns is neither normal nor symmetric. Diversification will
decrease skew so long as the errors are less than perfectly correlated and on average positively
16
skewed. In Simkowitz and Beedles’ (1978) empirical work, residual errors were significantly
positively skewed. In the extreme case in which investors were neutral to dispersion (variance), but
preferred positive skew, no diversification would take place. Simkowitz and Beedles (1978)
suggested that a more realistic model would have b , c > 0 , indicating that some diversification
would take place, though not as much as if the c term were zero.
Outside of the traditional utility-based framework, Baker and Haslem (1973) surveyed
investors to assess preferences for a pre-selected group of factors influencing investment decisions.
They found that investors take far more information into account than merely the mean return and
variance. Subsequently, they reported (Baker and Haslem, 1974) that investors had strong
preferences for stability (of earnings, of dividends, of price, etc.). While diversification would
provide such stability for the overall portfolio, Markowitz diversification frequently involves taking
large positions in assets that may exhibit high degrees of instability on their own.
It may also be the case that investors have “non-economic” preferences - preferences
formed outside the domain of economic reasoning. There are many mutual funds and investment
management companies that advertise as “socially conscious.” That is, they do not invest in firms
that pollute, use nuclear power, are considered anti-labor, are not considered supporters of
“diversity,” or engage in activities involving gambling, weapons, tobacco, alcohol, and the like. Any
limitations on the scope of investments considered violates core assumptions of mean-variance
efficiency. These portfolios may be efficient socially-conscious portfolios, but they are not
Markowitz diversified.
III. THE INFLUENCE OF BEHAVIOR
The reasons discussed in Section II all certainly represent plausible scenarios for why
mean-variance diversification is not observed more frequently in practice. Still, the very meaning of
“well-diversified” is not shared between researchers and practitioners. Even among investors aware
of the Markowitz paradigm, mean-variance efficiency is seen as different from “well-diversified.”
More to the point, a common reaction is to find Markowitz diversification “suspicious” (Green
and Hollifield, 1992), “unmarketable” (Michaud, 1989), or, in Fisher and Statman’s (1997)
particularly apt term, “unpalatable.” The theoretical advantages of Markowitz diversification are
clear; what remains unclear is the source and nature of investor aversion to it. Examining such
17
behavior would seem to be the first step towards either attempting to educate investors on prudent
diversification or constructing a new theory of portfolio choice in greater consonance with empirical
reality and the expressed preferences of investors. We propose to elicit preferences for diversifying
behavior from investors in an attempt to understand the particular attributes of normative portfolio
choice that appear unsatisfying to most investors. In addition, by using well-known results in the
behavioral decision-making literature, we will explore the attractiveness and performance of
alternative portfolio-selection rules. For example, the practical application of different (i.e., non-
Markowitz) heuristics for portfolio choice may produce results sufficiently close to mean-variance
efficiency so as to serve as reasonable, but simpler, substitutes for many investors.
Intent or Implementation (or Both)?
Markowitz diversification has many clear benefits for investors. Unfortunately, the empirical
evidence suggests that it is rarely implemented. Given that investors do not diversify, one might ask
whether it is a question of intent or of implementation. Both approaches depend strongly on the
behavior and preferences of investors (who may not always be rational). With only a handful of
exceptions (Elton, Gruber, and Padberg, 1976, 1977; Michaud, 1989; Green and Hollifield, 1992;
Fisher and Statman, 1997; Shefrin, 2000), the question of investors wanting or even being able to
implement Markowitz diversification has not been considered. For example, although French and
Poterba (1991) find that investors tend to underdiversify internationally beyond what can be
explained by transaction costs and other institutional constraints, they conclude that it “appears to
be the result of investor choices,” without further speculating as to the nature of those choices.
Slovic (1972) noted that even with accurate models and good judgment, experts were unable to
apply consistently what they know. Even though our ability to model and obtain quality data has
grown considerably over the last 30 years, Slovic’s observation remains largely true today
(Michaud, 1989; DeBondt, 1998). Behavior matters.
I have divided the influence of behavior on this problem into two areas: intent and
implementation. By intent, I mean that investors may deliberately and knowingly choose to diversify
in a manner inconsistent with Markowitz (1959). Implicit in such an action may be the recognition
that Markowitz’s algorithm does not minimize the risk as perceived by investors; that variance is an
incomplete measure of investment risk. By implementation, I mean that investors, while intending to
implement the correct algorithm, make mistakes or face cognitive constraints on their ability to
confront complex investment problems requiring the use of heuristic approaches. A note of
18
clarification: I do not mean to imply that investors simply cannot solve quadratic programs in their
heads - of course they can’t. More importantly, with readily available computer programs to
perform the calculations, they do not need to. Now that even the most complicated portfolio-
selection problems can be solved in seconds by computer, the most cognitively demanding portion
is structuring the problem and using the results.
Implementation errors can take numerous forms. Slovic (1972) suggested that even
financial experts are unable to use their models correctly and consistently. Elton, Gruber, and
Padberg (1976, 1977) indicated that multicriterion decision making concerning portfolios was
cognitively taxing and thus, prone to error. Kroll, Levy, and Rapoport (1988) demonstrated that,
even with practice, investors exhibited high rates of requests for useless information, frequent (and
thus overly costly) switching between assets, and sequential dependencies (gambler’s fallacies) in
an experimental investment simulation. DeBondt (1998) surveyed experienced investors and
showed that most failed to use basic principles of good investing despite years of experience.
In this section, I will address possible cognitive and behavioral sources of problems with
both intent and implementation. Support for any of these sources would indicate that investors’
preferences can be manipulated by factors not considered by Markowitz and other normative
portfolio-selection approaches. I present several hypotheses at the end of this section that will be the
subject of experimental tests in Section V.
Intent and the Perception of Risk
“The market is not a weighing machine, on which the value of eachissue is recorded by an exact and impersonal mechanism - rather -the market is a voting machine, whereon countless individualsregister choices that are the product partly of reason and partly ofemotion.” (p. 23, Graham and Dodd, 1934).
Graham and Dodd felt that the investment process was governed, at least in part, by
emotion. Their reference to “weights” and “votes” is particularly apt for our purposes, to the
extent that portfolio allocations that normatively reflect only weights often take on the meaning of
“votes” for some investors. This overattention to individual assets results in underattention to the
portfolio as a whole - a very different perception of risk than Markowitz (1959) intended.
19
Indeed, the question is not so much of finding and using the most efficient selection
algorithm, but in determining what any algorithm is efficient at doing. Markowitz (1959) and others
felt that such an algorithm should minimize portfolio variance because they felt that investment risk
was best (of the feasible measures at the time) represented by statistical variability at the portfolio
level. Statistical variability, however, may not be the measure used, at least implicitly, by most
investors (and, to be fair, Markowitz (1959) and Sharpe (1964) both refer to other measures, such
as semivariance, as possibly being more descriptively accurate). In fact, several researchers have
found evidence to the contrary. Gooding (1975) examined the preferences of investment professors,
portfolio managers, and individual investors and found significant differences between the groups
in the dimensions used to evaluate stocks. In general, portfolio managers and investment professors
used price-to-earnings ratios and downside risk (semivariance), while individual investors used
dividend yields, past returns from holding the stock, and past growth. Although downside risk
comes close, none of the evaluation dimensions involves variance or any portfolio-level risk
measures at all. Watts and Tobin (1967) surveyed 4,000 households attempting to explain a variety
of diverse asset holdings (from furniture to installment debt). Although a number of the variables
were statistically significant, none of the models were able to explain more than 5% of the variance
in any of the asset categories. These findings suggest that there is no clear, simple measure of
investment risk as perceived by investors.
Even if we assume that variance is an appropriate measure, there is evidence that individuals
are poor judges of variability. Lathrop (1967) examined peoples’ tendencies to misjudge estimates
of variability in a simple comparison task. Subjects were presented with cards on which lines of
differing heights were drawn (Figure 2). On the two cards, the mean and standard deviation of the
line heights was constant; only the sequencing of the presentation differed. Subjects reliably used
sequencing information to judge variability, even when they were given specific instructions to
ignore sequencing and that the means were equal.
20
Figure 2: Lathrop’s (1967) Variance Estimation Task
Card 1 Card 2
Perceived as more variable
Although individual investors do not deal with line heights, they do confront a very similar
problem in judging variability. To the extent that investors lack statistical training, the concept of
using distributional moments (e.g., variance) to measure risk may not be meaningful. Many
individual investors, as a heuristic strategy, look at time-series graphs of asset prices to judge how
“risky” a particular asset is (e.g., Gooding (1975) reports this behavior and it was also found
among the investment club members questioned in this paper). In so doing, however, they make
themselves subject to a variety of biases concerning how such information is displayed. For
example, time series’ generally present price data, but risk is measured in terms of return data.
Visually converting a price time series into a return time series is likely to be nontrivial for investors,
even if investors could then calculate variances. In addition, Loewenstein and Prelec (1993) note that
people have preferences for increasing sequences of outcomes (consumption paths in their work)
and against decreasing sequences.
Thus, when given two time series plots, these biases may influence investors’ judgments of
risk, irrespective of the variance of the series. In Figure 3, the increasing series may be perceived as
less volatile (or at least less risky) compared to the decreasing sequence since the decreasing
sequence involves losses to investors. Baker and Haslem (1973, 1974) also noted that investors
have strong preferences for stability (of earnings, of price, of dividends, etc.). Consequently,
investors may judge the smooth series in Figure 4 as less risky than the saw-toothed series.
The series in Figures 3 and 4 were presented as financial time series to 107 students in an
undergraduate decision analysis class. Subjects were asked to indicate which series was risker, or
that they were equally risky. In both cases, students perceived there to be differences between the
riskiness of the graphs.20 For Figure 3, only 19 subjects indicated that the series were equally risky,
20The three responses available to the subjects were A is riskier, B is riskier, and both are equally risky.
21
a figure reliably lower (p < 0.001) then the number who indicated otherwise (44 said the increasing
series was riskier and 44 said the decreasing series was riskier). This suggests that the subjects
were responding to characteristics other than return variances. For Figure 4, only 18 subjects
indicated that the series were equally risky, a figure also reliably lower (p < 0.001) than the number
indicating otherwise. Here, 34 selected the saw-toothed series as riskier and 55 selected the smooth
decreasing series as riskier. This result suggests that subjects were also responding to
characteristics other than price variances. Clearly, the subjects were inferring trends that did not, in
fact, exist, and made judgments of riskiness based on those patterns.
One must be careful, however, in drawing conclusions from the series in Figure 4 since the
smooth series is also a decreasing series. A more extensive testing regimen (in progress) is needed
to explore fully subjects’ risk perceptions from these time series. Nevertheless, it is clear that
subjects tend to be poor assessors of statistical variability.
Figure 3: Increasing versus Decreasing Sequences and Judgments of Riskiness
The increasing series has a greater price variance, but return variances are equal
200 400 600 800 1000
25
50
75
100
125
150
175
200
200 400 600 800 1000
25
50
75
100
125
150
175
200
22
Figure 4: Stable versus Unstable Sequences and Judgments of Riskiness
The saw-toothed series has a greater return variance, but price variances are equal
20 40 60 80 100
80
100
120
140
20 40 60 80 100
80
100
120
140
In general, the statistical variability of the assets may, in fact, have very little bearing on
investors’ perceptions of risk. In this sense, asset variance may be only one element of a larger risk
construct, to use Yates and Stone’s (1992) term. De Bondt (1993) notes that perceived risk
depends on prior performance in forecasting risk and return and that risk premia change “because
risk perceptions change, not because of changes in the public’s willingness to bear risk, or because
objectively the stock became more risky.” As Fischhoff (1985) notes, “people disagree more about
what risk is than about how large it is.”
In practice, and especially for those investors without significant statistical training, the
concept of “risk” may be more of a visceral state than an affectually cold number. Yet, the standard
theories of investment selection completely ignore hedonic responses. If, as investors appear to be
doing, we are to view risk through more than the narrow window of statistical variation, if we are to
consider the “risk experience,” we are led back to our initial question: to minimize risk, what
should investors do (if not minimize portfolio variance)?
If variance is an incomplete and inaccurate measure of perceived investment risk, then are
investors truly better off by minimizing that rather than some other measure? If holding the
Markowitz efficient portfolio still feels risky to investors, does holding that portfolio really make
them better off? On the other hand, if investors are allowed merely to minimize the risk they
perceive, are they naively ignoring very important risks that they may be unaware of? In essence, are
advisors forced to tell investors that, although they may feel uncomfortable holding the (properly
23
diversified) portfolio currently, “it will really be better for them in the long run”? If we are to
resolve these important questions, we must understand what it is about Markowitz portfolios that
investors perceive as riskier (or less desirable) than portfolios structured according to some other
criterion.
Fisher and Statman (1997) referred to this as “palatability” - a reference to an earlier work
by George Stigler. Stigler (1945) used a linear programming model similar to the portfolio-
selection model to identify the lowest cost bundle of food that satisfied the minimum daily
nutritional requirements of the average person. Stigler considered 77 foods, from wheat flour to
sirloin steak to strawberry preserves. Each item had a cost and a set of nutrients. Stigler discovered
that the minimum-cost “food portfolio” (which cost $39.93 per year in 1939 dollars), consisted of
370 pounds of wheat flour, 57 cans of evaporated milk, 111 pounds of cabbage, 23 pounds of
spinach, and 285 pounds of dried navy beans. As Fisher and Statman (1997) noted, this food
portfolio was very efficient, but “unpalatable.”
By the same token, investors may find Markowitz portfolios to be unpalatable. For example,
Saunders and Woodward (1979) presented a Markowitz efficient portfolio that indicated investment
of the entire portfolio in a single asset - the Japanese stock market.21 Investors might find it odd
that the optimally diversified portfolio was not “diversified” at all. Not knowing any better,
investors may perceive the Markowitz weights to be arbitrary, simply because they typically vary
widely. DeBondt’s (1998) NAIC investors had strong beliefs in their investing skill, believing that
they could manage risks after investment through their (apparently) superior abilities (presumably
by knowing when to sell).
Simply put, the Markowitz allocations lack a “story” behind them, and Dawes (1999) has
noted that people “do not appreciate probability [here as probabilistic risk] without the story.” It
may be difficult for investors to understand why the Markowitz algorithm assigns so much weight
to some assets and so little to others when compared to an equal-weighting approach (“they’re all
good stocks, so I put the same in each”). For example, suppose a Markowitz approach assigned
Time Warner a weight of 30% and Microsoft a weight of 5%. Might unsophisticated investors be
tempted to interpret that as saying that Time Warner was six times “better” than Microsoft? The
21The Saunders and Woodward (1979) model also illustrated the sensitivity of optimal portfolio structure todata sources. Had a different time window been used to measure returns and covariances, the results would have beenmarkedly different.
24
weights could become, in a sense, value judgments of the underlying assets themselves, rather than
only measures of their relative risk-to-reward ratios. This story approach fits the advice given by
many professional investment advisors to buy stock in companies whose products they like (e.g., I
like Diet Coke, so I should buy Coca-Cola). Markowitz (1959) envisioned assets as being fully
described by their expected return and variance; people often do not (or cannot).
We must remain open to the possibility that investors simply do not want Markowitz
diversified portfolios and intentionally use other portfolio selection rules. Investors may not believe
that statistical variability is an appropriate measure of risk. Investors may feel that selection rules
which only “weight” rather than “vote” lack a measure of sophistication and investment skill. In
the end, however normatively powerful, Markowitz diversification ultimately may be unpalatable.
Manipulation of Preferences and the Implementation of Portfolio Selection Rules
Even if we assume that investors intentionally pursue minimum variance portfolios in the
name of risk reduction, we still must face the substantial possibility that they make errors in
accomplishing this task. Two frequently-advanced counterarguments to this claim should be
addressed:
(1) The investors that “matter” are experts and they have goodmodels and fast computers, so they don’t make mistakes.
(2) Professional investors are well-trained in their tasks andwould never succumb to such biases in judgment or choice.
These two objections fail to understand fully the nature of the implementation errors that we are
concerned with here. More importantly, despite their face validity, they are frequently and
demonstrably false. Poulton (1994) reviews a number of experiments demonstrating, for a broad
range of activities, that while experts generally perform better than laypersons at various judgment
and choice tasks, they still make mistakes with significant regularity. It is also important to note that
the availability of increasingly more powerful computers does not allow Markowitz diversification
to be implemented with any more accuracy than in years past. In fact, the opposite may be true,
since investment managers can now add additional constraints to the Markowitz model to
accommodate their intuition about what diversification should “look” like - additions that would
25
have been computationally infeasible at the time Markowitz created the model. Thus, we may see a
proliferation of heuristic strategies of unknown value.
Specific to the domain in question, cultural anthropologists O’Barr and Conley (1992)
studied the institutional structure and culture of nearly a dozen large institutional investment groups.
They interviewed institutional researchers and investment managers concerning the decisions that
they made and the actions that they took. O’Barr and Conley (1992) noted that none of the people
that they interviewed described using computer simulations or similarly sophisticated analyses to
select particular investment strategies. Rather, they “heard stories about ‘well, we’ve always done it
this way,’ or ‘how else could you do it?’” (O’Barr, 1998). In response to a question on why
money was invested with a certain manager, one institutional advisor replied “the Chief Financial
Officer likes to play golf with this guy on Saturday mornings, and he made the very strong
suggestion that it would be a good idea if we placed some money with him, so that’s why.” In
summarizing their work, O’Barr (1998) stated that “[t]he more we kept asking questions, the more
we would hear answers . . . which were, from our point of view, anything other than economic in
nature. They were personal; they were historical; they were idiosyncratic; they were cultural; they
were psychological; but they weren’t economic.”
In the remainder of this section, I describe three commonly observed biases from the
behavioral decision-making literature and suggest how these biases, extended to the domain of
investment decision making, could cause errors in the construction of Markowitz diversified
portfolios. In addition, I explore ways in which the “palatability” of Markowitz diversification
might be improved (if investors decide that statistical variance is the conceptualization of risk to
minimize). Table 6 summarizes the biases.
26
Table 6: Consequences for Investor Implementation of Diversification
Common Bias Consequences for Implementation
Illusion of ControlInvestors will tend to underdiversify (relative to Markowitz) throughindividual assets and overdiversify through mutual funds.
Mental AccountingInvestors will tend to overdiversify within asset classes and underdiversifyacross asset classes. That is, they will seek asset class-leveldiversification, rather than portfolio-level diversification.
Sensitivity to FramesThe attractiveness of optimal diversification can be manipulated dependingon whether it is framed as decreasing risk or increasing returns.
Implementation Biases in Detail
DeBondt (1998) observed that the (relatively sophisticated) investors in his study believed
that investment risk could be “managed” after the commitment of funds through “knowledge and
trading skill.” This belief, he felt, caused investors to place wealth in only a few assets. It would
stand to reason that, if investors believed risk could be managed in this fashion, maintaining a small
portfolio might be all that most investors could cognitively handle. This is similar to Langer’s
(1975) “illusion of control.” In her experiments, subjects were found to assign higher values to
risky (and random) situations in which they had some personal involvement. For example, subjects
were willing to bet more on the toss of a coin they threw themselves, rather than one which someone
else threw for them.
In an investing context, investors appear to believe that the riskiness of an investment is not
attached to the security itself, but rather is a function of each investor’s skill and luck. Investors do
not maintain large (and thus more diversified) portfolios because they cannot actively “manage”
the risk for so many different assets. At the same time, they are eager to hold numerous mutual
funds. The number of investors holding six or more funds quadrupled between 1989 and 1995 (see
Figure 1). In effect, mutual funds allow these investors to achieve (via proxy) the control over
investments they desire while still holding large portfolios. The problem with this approach,
however, is that most large mutual funds have largely identical holdings (see Tables 1a and 1b).
Further, most large mutual funds are all common stock funds, thus other asset classes are excluded.
Because of the overlap between fund holdings and the exclusion of non-equity assets, mutual funds
tend to be expensive and relatively ineffective methods of achieving Markowitz diversification.
27
Another possible reason for the popularity of mutual funds could be regret aversion. If a
fund goes up, the investor feels good for picking a “winner.” However, if the fund goes down, the
investor can blame the loss on the fund manager. This explanation would imply that investors feel
they are better at picking fund managers than at picking investments. Consequently, it is interesting
that the addition of a layer of uncertainty (the quality of the manager) might make investors more
comfortable.
All of this is not to say that mutual funds themselves are bad instruments for implementing
Markowitz diversification.22 Rather, that mutual funds, regardless of their focus, should not be
considered the only action taken in furtherance of diversification. Investors cannot simply say, “I
have several mutual funds, therefore I am diversified.” This illusion of control would lead investors
to underdiversify through individual assets (stocks, commodities, real estate, etc.) and overdiversify
through (relatively expensive) mutual funds.
Thaler (1985) demonstrated that individuals create “mental accounts” for different
quantities of money and behave differently in their management. For example, individuals
simultaneously maintain a large credit card debt at a high interest rate and hold money in a low-
yielding savings account earmarked as “college savings.” Such violations of the assumption of
asset integration conflict with rational theories of choice. In the context of investments, it has often
been noted that investors segregate their investment assets (e.g., retirement savings, children’s
college fund, vacation savings, home down-payment savings, etc.). Although the labels have no
bearing on normative financial behavior, individuals often do behave differently depending on which
investment pool they are dealing with.23
22Mutual funds are certainly quite useful at achieving cost-effective within-asset diversification - especiallyfor capital-constrained investors and for investors interested in relatively “exotic” asset classes. The key is thatinvestors must then remember to build a diversified mutual-fund portfolio. Investors should also keep in mind anadditional “cost” of mutual funds: with very few exceptions, short selling of mutual fund shares is prohibited. Notonly are there limited facilities available for short-selling fund shares, such transactions are generally discouraged bythe fund-management companies.
23In some cases, there are legitimate reasons for differential treatment across accounts. For example, it ispointless to hold tax-exempt securities (such as municipal bonds) in tax-deferred accounts (such as IRAs).
28
In the case of diversification, mental accounting will cause investors to overdiversify within
asset classes and underdiversify across asset classes.24 Investors will fail to see their entire
investment (net asset) position as an integrated whole and instead act to implement diversification in
each asset subcategory (e.g., a diversified stock portfolio, a diversified bond portfolio, etc.). This
bias has two implications for investing: (1) investors are engaging in excessive (and thus inefficient)
trading within asset classes, and (2) investors are dissuaded from venturing outside of current asset-
class choices and into different classes entirely.
Consider an investor who currently owns shares in ten different companies. This investor,
when considering a new investment, would do better in terms of portfolio risk reduction to choose
an asset other than common stock (say, for example, gold bullion).25 However, the mental
accounting hypothesis suggests that adding an eleventh common stock to such a portfolio would be
preferred to adding a single gold (commodity) investment. All else being equal, although the entire
portfolio is better off under the commodity investment, this choice results in the investor holding an
undiversified commodity portfolio. The prospect of holding this undiversified subcategory portfolio
is more salient to the investor than the integrated asset position.
The mathematical programming models of diversification in the appendix are all stated as
minimization problems: minimize variance while holding returns fixed. Alternatively, however, we
could have stated these problems as their duals: maximize returns while holding variance fixed. For
the same inputs, the vector of allocations for each problem will be identical. Here, we may more
generally state the following as a result of this duality: for any suboptimally (Markowitz) diversified
portfolio (we will discuss exactly what this means momentarily), there is another feasible allocation
of assets that will either increase returns or decrease risks. This feasible allocation can be achieved
by either the minimization or maximization procedures.
24The terms overdiversify and underdiversify are used relative to Markowitz diversification. A portfolio isunderdiversified (relative to Markowitz) if it still contains idiosyncratic risk. On the other hand, a portfolio isoverdiversified if it contains more assets than are necessary to remove idiosyncratic risks once transaction costs aretaken into account.
25Jacob (1974) considers a similar problem and observes that careful selection of the securities in smallportfolios can have impacts similar to the large-portfolio effect of diversification. However, while her basic premise(small portfolios can be efficient too) is valid, her work does not generalize well and ignores the importantcomputational and sampling problems suggested by Michaud (1989).
29
In practice, however, people are profoundly influenced by how choices are framed, or
described. The framing of decisions is a key component of Kahneman and Tversky’s (1979)
prospect theory. In prospect theory, the framing of outcomes as gains or losses from a reference
level influences the risk preferences of the decision maker.26 Tversky and Kahneman (1981)
further investigated the bias by frames by manipulating the reference level. For example, a medical
decision (whether or not to have surgery) can be framed in terms of either the probability of survival
or the probability of death. Because there is a very straightforward relationship between the two
(they are complementary: Pr á Survival é = 1 − Pr á Death é ), the decisions made under both frames
should be identical. However, in experiments (e.g., Kahneman and Tversky, 1979; Tversky and
Kahneman, 1981), people are strongly influenced by the selection of the reference level and the
classification as gains or losses.
Diversification can be thought of in terms of maximizing the ratio of (excess) return to
standard deviation in a portfolio. This ratio [3] is known as the Sharpe ratio (Sharpe, 1966) and is
the slope of the steepest line (known as the ex ante capital market line) passing through the riskless
rate and still tangent to the efficient frontier.
S i =
r i − r
f
σ i
[3]
It is trivial to show that the portfolio with the maximum Sharpe ratio is the market portfolio (e.g.,
Grinold and Kahn, 1995). Since we have previously stated that the market portfolio is, by definition,
the maximally diversified portfolio, the Sharpe ratio can also be used to measure diversification.
Any non-Markowitz diversified portfolio will have a Sharpe ratio less than that of the market
portfolio. Let S p and S
m be, respectively, the Sharpe ratios of a non-Markowitz diversified portfolio
and of the market portfolio. For any portfolio p Ö m there exists a budget feasible reallocation p N
which is preferred to p. There are multiple paths by which this reallocation can be implemented.
Clearly, a dominating portfolio can be selected that either increases return, holding variance
constant, decreases variance, holding return constant, or some combination thereof.
In a portfolio context, investors face two possible frames in which diversification could be
presented. In one case, an investor with a suboptimally diversified portfolio could be given the
26Prospect theory holds that people are risk averse in the domain of gains and risk seeking in the domain oflosses.
30
opportunity, through implementation of Markowitz diversification, to increase portfolio return
without taking any more risk. Alternatively, an investor could be given the opportunity to reduce
portfolio risk without diminishing the portfolio’s return. Although both of these options are
identical at one level (they both return the investor to an optimal risk-reward ratio), they are likely to
evoke different judgments as to their attractiveness. Thus, the portfolio seeming attractive to a
utility-maximizing investor might not be attractive to an investor better characterized by prospect
theory (and vice versa). This will be discussed in greater detail in Section V.
Stochastic Dominance Investors and Prospect Theory Investors
We will use stochastic dominance as the decision rule followed by expected utility-
maximizing decision makers. For the sake of clarity in the following analysis, we shall restrict our
examples to those dealing with normally distributed returns. However, the generalization to any
symmetric distribution is straightforward. The comparison in question involves three portfolios,
which we shall term A, B, and C. Let A represent an inefficient portfolio. Let B and C represent more
efficient, but not necessarily frontier portfolios, such that the Sharpe ratios have the following
relationship: S A < S
B = S
C # S
M . That is, on the basis of Sharpe ratio maximization, an investor would
be indifferent between B and C. We shall assume that portfolios can be completely described by
their mean and variance: 7 r , σ 2 ? . Let B = 7 r + ρ , σ 2 ? represent the “increasing return” portfolio
and C = 7 r , á σ − υ é 2 ? represent the “decreasing variance” portfolio. ρ > 0 and υ > 0 represent,
respectively, the return and risk premiums.
Clearly, B first-order stochastically dominates (FSD) A. This was first illustrated by
Markowitz (1952). Further, C second-order stochastically dominates (SSD) A. This has been
demonstrated by Huang and Litzenberger (1988). Here, we are interested, however, in the
comparison between B and C: which would an expected utility-maximizing investor choose? We
can easily see the answer to this question in Figure 5. The increasing-return portfolio is FSD-
preferred to the inefficient portfolio and SSD-preferred to the decreasing-variance portfolio. Thus,
expected utility-maximizing investors will implement Markowitz diversification via the increasing-
return portfolio.
31
Figure 5: Stochastic Dominance Relationships Among Three Portfolios
Return
0.2
0.4
0.6
0.8
1
Probability
Decreasing
Variance
Increasing
Return
Inefficient
Portfolio
r
r +
ρ
For those investors who are not expected utility maximizers, we might assume that they are
better described by Kahneman and Tversky’s prospect theory (1979; Tversky and Kahneman,
1992).27 Prospect theory provides a descriptive model of risky decision making. A prospect is a
collection of payoff-probability pairs (x, p) constituting a discrete probability density function for a
risky choice.
For our present purposes, we shall capture the two primary parts of prospect theory: the
decision weighting function [4] and the value function [5]. Just as expected utility theory measures
the utility of a set of outcomes as the product of probabilities and final wealth utilities, prospect
theory measures the prospect value V [6] of a set of outcomes as the product of their decision
weights and gain or loss values. The decision weights adjust probabilities to account for the
observed tendency of individuals to overweight small probabilities and underweight large
probabilities. Accordingly, Kahneman and Tversky (1979) proposed a decision weighting function
π á p é that was subadditive. That is, for probabilities p and q where p + q = 1 , the corresponding
decision weighting function would indicate that π á p é + π á q é < 1 . In addition, prospect theory’s
value function takes into account the tendency for individuals to care more about losses than gains.
27And, in fact, the subjects in Section V exhibit behavior consistent with prospect theory when confrontedwith the tasks used by Kahneman and Tversky (1979).
32
Thus, the value function disproportionately weights losses more heavily than gains. A significant
difference between expected utility theory and prospect theory is that prospect values are defined on
gains and losses (from some reference point) and expected utilities are defined on final wealth
levels.
Although Kahneman and Tversky’s (1979) original work did not specify functional forms
for the components of prospect theory, their subsequent work (Tversky and Kahneman, 1992)
estimated both the functional forms and the parameters28 and it is those characteristics that we shall
use in this paper. Figure 6 illustrates the decision weighting function and the value function.
π á p é = p γ
á p γ + á 1 − p é γ é
1 / γ [4]
υ á x é = : ; <
= =
= = = x α x$ 0
− λ á − x é β x < 0
[5]
V = 3 i
π á p i é υ á x
i é [6]
28The parameters used are α = β = 0 . 88, and λ = 2 . 25. In the decision-weighting function, γ is assumed to
be 0.61 for gains and 0.69 for losses. α and β control the curvature of the value function, λ controls the degree ofloss aversion present (λ = 1 would indicate symmetric treatment of losses), and γ controls the curvature of thedecision-weighting function (γ = 1 represents a standard probability).
33
Figure 6: Prospect Theory’s Decision Weighting Function and Value Function
The dashed line in the decision weighting function represents a perfectlycalibrated decision weight, linear between the endpoints of 0 and 1.
0.2 0.4 0.6 0.8 1Probability
0.2
0.4
0.6
0.8
1Decision Weight
-20 -10 10 20Gain or Loss
-30
-20
-10
10
Value
Under prospect thory, a portfolio can be represented as a discretized version of its PDF. For
example, we might approximate a normally distributed return as the prospect {{-0.2, 0.25},
{0.1, 0.5}, {0.4, 0.25}}. Obviously, the approximation can be made arbitrarily close to the
continuous form. Standard prospect theory cannot accommodate continuous prospects, but such a
level of detail is not needed for the current analysis.29
It is possible, under prospect theory, that investors, contrary to expected utility theory, would
prefer the variance-reduction option over the increasing-return option. Although under some
conditions even prospect-theoretic decision makers would prefer the return-increasing option, the
opportunity to eliminate extreme losses through diversification can make the variance-reduction
option the preferred choice. For example, consider the securities illustrated in Table 7.
29Prospect theory has been extended by Tversky and Kahneman (1992) to accommodate continuousprospects, but the additional complexity required for doing so is not justified by the minimal improvement found inthe present application.
34
Table 7: Inefficient and Efficient Portfolios as Prospects
Type Portfolio Portfolios as Prospects Return Std. Dev.
InefficientStatus
Quo/Reference{{-0.50, 0.05}, {-0.10, 0.10}, {0.00, 0.20},{0.10, 0.40}, {0.15, 0.15}, {0.20, 0.10}}
4.75% 0.1503
“Efficient”Increasing Return
{{-0.48, 0.05}, {-0.08, 0.10}, {0.02, 0.20},{0.12, 0.40}, {0.17, 0.15}, {0.22, 0.10}}
6.75% 0.1503
Decreasing Variance {{-0.0525, 0.25}, {0.0475, 0.50}, {0.1475, 0.25}} 4.75% 0.0707
These “portfolio prospects” are illustrated graphically in Figure 7. These structures are consistent
with historical patterns of asset returns, particularly the stock market. For example, although the
average return on the S&P 500 index is positive, the negative returns tend to be larger than the
positive ones (though there are fewer of them). Stated another way, the tails of the return
distributions tend to be heavier on the side of losses than on the side of gains.
Prospect theory assigns values of -0.0500 to the status quo portfolio, -0.0104 to the
increasing return portfolio, and 0.0334 to the variance reduction portfolio. Because prospect theory
only produces ordinal rankings, we can see that for this example, the variance reduction option is
preferred (it has the largest value). Naturally, these results are sensitive to the parameter values
assumed in the decision-weighting and value functions. However, as Figure 8a clearly indicates, the
variance-reduction option remains preferred over a broad range of value-function parameters (λ).
Figure 8a graphs the difference between the values of the variance-reduction option and the
increasing-return option. The λ-parameter determines the steepness of the value curve for losses.
The larger the λ, the steeper the curve. An investor would be indifferent, in this example, at λ = 1 . 45.
As λ 6 1 , the treatment of gains and losses becomes more symmetric. As this happens, prospect
theory begins to indicate that the increasing-return option is preferred, in accordance with utility
theory. Figure 8b illustrates the regions for which the decreasing-variance option is preferred as α
and λ change (with α = β ).
35
Figure 7: Portfolios as ProspectsThe solid, thin line is the status quo portfolio, the dashed line is the increasing-return portfolio,
and the thick line is the variance-reduction portfolio
-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2Return
0.2
0.4
0.6
0.8
1
Cumulative Probability
Figure 8a: Sensitivity of Variance-Reduction Preference to Changes in λ
1
2
3
4
5
l
Parameter
-
0.2
-
0.15
-
0.1
-
0.05
0.05
0.1
0.15
0.2
VH
incrret L
-
VH
decrvar L
111111111111
000000000000
111111111111111111
000000000000000000
Increasing-returnpreferred
111111111111111111
000000000000000000
Decreasing-variancepreferred
36
Figure 8b: Sensitivity of Decreasing-Variance Preference Regions to α and λ (α = β )
Combinations for which the increasing-return portfolio is preferred are indicatedby darker shading; decreasing-variance preference is indicated by no shading. The
parameters estimated by Kahneman and Tversky are highlighted (K).
Increasing Loss Aversion (λ parameter)
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 K
1.0
Incr
easi
ng R
isk
Ave
rsio
n (α
= β
par
amet
ers)
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Increasing-Return Preferred
Decreasing-Variance Preferred
Clearly, other parameter values are possible and the results will change accordingly. The
purpose of this example was to demonstrate that, for some decision makers, the frame of the
problem (increase return or decrease variance) may cause decisions to deviate from the normative
expected utility maximizing choice. Thus, it is a question worth asking: do preferences for or
against (Markowitz) diversification depend on how the benefits are framed by investors? Expected
utility maximizers would be more motivated by the increasing return frame. In this most recent
example, prospect-theoretic decision makers would have been more motivated by the variance-
reduction frame.
37
In this section, I have suggested how three commonly observed results in the behavioral
decision theory literature could impact investor behavior toward diversification. I have suggested
that the lack of evidence for diversifying practices observed among investors could be a function of
intent or implementation. That is, investors may understand Markowitz diversification, but choose
not to implement it, finding it “unpalatable” (intent). Or, investors may simply be subject to biases
during the process of implementing heuristic portfolio selection rules intended to emulate mean-
variance efficiency (implementation).
IV. ALTERNATIVE HEURISTICS FOR PORTFOLIO CHOICE
Investors frequently use heuristic rules to pursue diversification in response to the
computational and cognitive complexity of normative optimization. Section III examined many of
the possible biases facing investors in the implementation of portfolio diversification. Consequently,
the use of such non-optimal rules almost always results in a different asset allocation than would be
achieved under the normative paradigm. An important question concerns their performance relative
to the theoretical optimum: how much return (if statistically any) are investors sacrificing in order to
get simplicity? In other cases, investors explicitly desire the use of alternative algorithms to pursue
diversification. As such, they may or may not recognize that Markowitz diversification is
numerically superior, but feel that it results in a distribution of assets which is psychologically
unappealing.
Whatever rules investors are using, they clearly do not resemble the optimal ones. However,
it remains to be explored whether they are using these (sub-optimal) heuristic rules in order to (1)
be explicitly different than Markowitz, or (2) satisfice by having these rules serve as proxy for the
Markowitz portfolio selection process. The first reason concerns the question of whether or not
investors’ attempts to act differently than Markowitz would result in systematic deviations from
optimal behavior. That is, do investors generally agree on how to “act differently”? The second
reason concerns the question of just how good (or bad) the heuristic rules are at controlling
portfolio normatively-defined risk. Additionally, even if investors choose to “act differently,” are
the outcomes substantially different from those stemming from the normative approach?
38
In this section, I examine in detail two alternative methods of “diversifying” a portfolio that
have been suggested in the extant literature. Fisher and Statman (1997) have suggested that
investors simply ignore covariation among returns altogether, and thus, seek to minimize the
weighted variance of the portfolio. Alternatively, most professional investment managers seem to
advocate simply increasing the number of securities held in relatively equivalent increments (Green
and Hollifield, 1992). Below, we shall examine these portfolio-selection rules and compare their
performance with the Markowitz algorithm.
This first alternative was suggested by Fisher and Statman (1997). They suggested that
portfolios should be framed as a whole, but often are not in practice. Rather, investors build
portfolios as pyramids, layer by layer, and covariances are overlooked in the process. Thus,
investors minimize the diagonal elements of the variance-covariance matrix and ignore the off-
diagonal elements. This amounts to minimizing the weighted variance of the assets. In contrast to
the Markowitz portfolio-selection problem presented in the appendix, investors would instead face
[7, 8, 9]. The weights are given by ω i , the returns by r
i , r í is the target minimum return, and the
variance of each asset is given by σ 2
i . Thus, this problem simply excludes the covariance terms,
represented by σ i , j
in the standard problem as outlined in the appendix.
minω
i
n
3 i = 1
ω 2
i σ 2
i [7]
such that:n
3 i = 1
ω i r
i $ r í [8]
n
3 i = 1
ω i = 1 [9]
This problem results in the following statement [10] of the Lagrangian, which can be solved for the
optimal vector of portfolios weights:
ã = n
3 i = 1
ω 2
i σ 2
i − λ
ä
ã å å å
n
3 i = 1
ω i r
i − r í
ë
í ì ì ì − µ
ä
ã å å å
n
3 i = 1
ω i − 1
ë
í ì ì ì [10]
39
The difference between this “naive” formulation and the Markowitz formulation is obvious in the
absence of any covariance or correlation terms. The resulting portfolios, while minimizing the
fluctuations in individual assets, often do nothing (or exacerbate) fluctuations in the portfolio as a
whole.
The second alternative technique, which simply uses the absolute quantity of securities in a
portfolio, has been suggested by Green and Hollifield (1992) among others. They take the term
“well-diversified” to mean that weight in the portfolio is spread “reasonably evenly” across many
assets (without regard to their correlation).30 This method is perhaps the easiest to think about and,
to an extent, is guaranteed to work. As I demonstrate in the appendix, all of the diversifiable variance
in a portfolio can be eliminated merely by increasing the number of assets to approach the size of
the market. The implementation of this strategy, however, is likely to revolve around transaction
costs. There are diminishing marginal (and eventually absolute) returns to increasing portfolio size,
since the transaction costs remain relatively constant while the incremental reductions in portfolio
variance decrease in size.31
Assuming for the moment that there are no transaction costs and all assets have equal
variance and are held in equal proportion, Figure 9 illustrates the reduction in diversifiable risk
provided simply by increasing the number of assets in a portfolio. Recall, however, that even those
investors owning some shares directly own stock in an average of 3.2 firms (NYSE, 1999)
30More recently, Benartzi and Thaler (1998) studied asset allocation choices made in retirement plans (suchas 401k plans). They termed this equal division approach the “1/n approach” or “naive diversification.” In this paper,we have referred to the Fisher and Statman’s (1997) minimum-weighted-variance heuristic as “naive diversification” -the terminology has not been standardized. Interestingly, in Benartzi and Thaler’s (1998) work, the rule leads toallocation choices that are dependent on the choice set provided to the investors. For example, if a retirement plancontains two stock funds and one bond fund, investors will end up with 66.7% stock and 33.3% bonds. If, however,the retirement plan offers one stock fund and two bond funds, the allocations will be reversed. Although this isconceptually similar (in outcomes) to the regret strategy followed by Markowitz (Money, 1998), the Benartzi andThaler (1998) theory concerns cognitive limitations. Markowitz (Money, 1998) explicitly acknowledged the correctprocedure, but chose another allocation to satisfy preferences outside of the mean-variance framework.
31Brennan (p. 484, 1975) considers exactly this issue. Brennan noted that “it is no longer necessarilyoptimal for [investors] to hold a perfectly [Markowitz] diversified portfolio” in the presence of transaction costs.Further, he noted that practical application of normative diversification was severely hampered by the significantincrease in problem complexity (e.g., the shift from linear to quadratic, dynamic, and mixed integer-nonlinearprogramming (MINLP) techniques) that resulted from the incorporation of transaction costs, taxes, marketsegmentation, and the like. His work approached the issue in a somewhat different manner than I do below, but thegeneral results hold.
40
(although this does not include mutual funds and the figure refers to assets in general, not just
stock). Clearly, however, investors are nowhere close to achieving substantial risk reduction simply
as a result of increasing the total number of assets held.
Figure 9: The Effect of Portfolio Size on Diversifiable Risk in a Portfolio
20 40 60 80 100Size
20
40
60
80
100
Percent of Diversifiable Risk Remaining
At the limit, portfolio risk can be reduced to the level of the nondiversifiable or systematic
risk present (market risk). In a world of transaction costs, however, at some point the marginal
benefit from risk reduction is exceeded by the marginal cost of transacting in an additional security.
This is easily demonstrated. Consider a model, in the spirit of Brennan (1975), based on the
quadratic utility [11] of Huang and Litzenberger (1988).
u = γ r p − δ σ 2
p [11]
We will assume, for the sake of simplicity, that all assets are held in equal proportion, and that the
correlation between all assets is 0.5.32 Also for simplicity, assume a constant unit variance among
all assets. Finally, let the return on all assets (or, at least their average return) be 0.1. We will admit
transaction costs by reducing the expected portfolio return by some amount proportional to the total
number of assets in the portfolio. Set this amount to c = 0 . 00125 (or any small fraction of the
average asset return). Thus, the portfolio expected return is given by [12] and the portfolio variance
by [13].
32Obviously, the results will hold in general for any correlations less than 1.
41
r p = ä
ã å å å
n
3 i = 1
r i
n
ë
í ì ì ì − cn [12]
σ 2
p =
n
3 i = 1
n
3 j = 1
ρ i , j
n 2 [13]
Then, for some coefficients33 of the utility function, we obtain the relationship between portfolio
size, transaction costs, and utility illustrated in Figure 10.
Figure 10: The Relationship Between Diversification and Transaction Costs
This figure uses Equation 11 and the parameters defined above
10 20 30 40 50
- 0.7
- 0.6
- 0.5
- 0.4
Utility
Number of Securities
More generally, given our assumptions, we have
r p = ä
ã å å å
n
3 i = 1
r i
n
ë
í ì ì ì − cn = 1
n
ä
ã å å å
n
3 i = 1
r i
ë
í ì ì ì − cn [14]
Since we have assumed r i to be constant across assets, without loss of generality let r
i = r ú i .
Therefore, r p = r − cn . Similarly,
33For Figure 3, the coefficients are γ = 2 and δ = 1 . Only the ratio (representing the coefficient of risk
aversion) actually matters. In practice, γ is usually set to 1 and in general any γ , δ > 0 are admissible. For example,
in the case of quadratic utility, we can divide [11] through by γ . Now, let δ í = δ / γ . Utility is now given by
u í = u / γ = r p − δ í σ 2
p , but u í is an order-preserving transformation of u ú γ > 0 .
42
σ 2
p =
n
3 i = 1
n
3 j = 1
ρ i , j
n 2 = 1
n 2
n
3 i = 1
n
3 j = 1
ρ i , j
[15]
This is represented by the matrix in [16], which has the closed form, for a constant correlation, of
[17].
σ 2
p = 1
n 2 A
1
ρ i , j
!
ρ i , j
"
˛
1
[16]
σ 2
p = 1
n 2 n + á n 2 − n é ρ
i , j[17]
Again, without loss of generality, let ρ i , j
= ρ ú i Ö j . Obviously, ρ = 1 if i = j . Therefore,
σ 2
p = n + á n 2 − n é ρ
n 2 [18]
= 1 + á n − 1 é ρ n
[19]
Combining [14] and [19], we can reexpress utility as [20].
u = γ á r − cn é − δ ä ã å å 1 + á n − 1 é ρ
n ë
í ì ì [20]
Differentiating with respect to n and setting equal to zero, we obtain [21] as the first-order
condition.
M u M n
= − γ c − δ ρ n
+ δ 1 + á n − 1 é ρ n 2
= 0 [21]
The solution to [21] is
n í = " i δ á ρ − 1 é c γ
with i = − 1 [22]
43
Clearly, a complex answer is not useful. We can, however, transform [22] in such a way as to
eliminate the complex value. Therefore, the optimal34 number of assets to hold in the presence of
fixed transaction costs is [23].
n í = − i δ á ρ − 1 é c γ
= − i á − 1 é δ á 1 − ρ é
c γ = δ á 1 − ρ é
c γ [23]
Thus, depending the nature of transaction costs and the utility functions of investors, the maximum
naively-optimal number of securities in a portfolio is finite (n í ñ 14 in Figure 10), even though the
reduction in variance can continue indefinitely.35 In effect, investors continue to purchase additional
assets until the cost of doing so becomes prohibitively expensive (relative to their benefit).
34To be thorough, we should state that the second-order condition guaranteeing u is at a maximum ispresent as well: M 2 u / M 2 n = 2 δ á ρ − 1 é n − 3 < 0 . This condition holds for all ρ < 1 as we have previously asserted.
35For n í to be finite, δ á 1 − ρ é á c γ é
− 1 < 4 . This implies that δ á 1 − ρ é á c γ é − 1 < 4 . Consider both extremes:
ρ = − 1 and ρ = 1 . At ρ = 1 , this value is zero. At ρ = − 1 , this value is 2 δ á c γ é − 1 . Given the presence of transaction
costs (c > 0 ) and that γ $ 0 implies that investors are indifferent to return or averse to it (which is unreasonable), it is
then sufficient to state that any finite value of δ is enough to prove that n í is also finite.
44
A Numerical Comparison of Markowitz and Heuristic Measures of Diversification36
We shall now consider a numerical comparison of the Markowitz and naive portfolio-
selection algorithms.37 We can compare the Markowitz and Fisher and Statman (1997) models
directly in terms of percentage of variance reduced. We shall consider an investment-opportunity
set of 42 different assets. Although these assets are by no means exhaustive, they do represent a
broad cross-section of different investment types (domestic common stock, international common
stock, domestic bonds, real estate, precious metals, commodities, and currencies). To analyze the
Green and Hollifield (1992) algorithm, we shall make a different set of tradeoffs. Table 8
summarizes the various asset classes (daily returns and standard deviations are annualized).
36The data used in this section were compiled from several sources. Most index data were obtained fromCSI Data. Currency and precious metals data were obtained from Olsen and Associates. CRB index data wereobtained from CRB/Bridge Associates. Real estate investment trust (REIT) index data were obtained from theNational Association of Real Estate Investment Trusts (NAREIT). The data represent daily total returns (capitalappreciation and dividend or interest income) to U.S.-domiciled investors during the period January 5, 1994 toDecember 31, 1998 (1,259 observations). When dates did not overlap for foreign investments (because of holidaysnot recognized by U.S. financial markets), the asset value for the closest U.S. business day was used. Returns werecalculated in the usual manner: U.S. investment, commodity, and real estate returns were calculated asr
t = ln á Π
t é − ln á Π t − 1 é . Foreign currency returns used European convention and were calculated as r FC
t = ln á S
t − 1 é − ln á S t é .
Foreign index returns were repatriated daily to U.S. dollars and calculated as r FM
t = ln á Π
t é − ln á Π t − 1 é + ln á S
t − 1 é − ln á S t é .
The asset’s price is given by Π except for foreign currencies which are valued at their spot rate S.
37As we have stated before, the Markowitz algorithm refers to the portfolio-selection process developed byMarkowitz (1959). When we are referring to naive portfolio-selection rules, we are referring to any non-normativeselection procedure. In the context of this paper, we are referring to the procedure that minimizes the weightedvariance of asset returns as the Fisher and Statman (1997) method and the procedure that maximizes n as the Greenand Hollifield (1992) method. These authors speculated that some investors might use these approaches, but neitherpair explored them in detail or advocated them as anything but suggestive of investor behavior.
45
Table 8: Asset Class Performance Summary, 1994-1998
ID Asset ClassAverageReturn
StandardDeviation
ID Asset ClassAverageReturn
StandardDeviation
1 S&P 500 Index 19.24% 14.39% 22 KLSE Composite -23.81% 38.08%
2 NASDAQ/NMS Index 20.34% 17.84% 23 Malaysian Ringgit -7.38% 12.34%
3 Bridge/CRB Index 1.75% 8.13% 24 NZSE 40 Index -2.03% 17.54%
4 NAREIT Total Return 9.11% 8.77% 25 New Zealand Dollar -1.22% 7.52%
5 Silver (NY spot) -0.69% 20.96% 26 Swiss Market Index 18.79% 17.71%
6 Gold (NY spot) -6.29% 8.34% 27 Swiss Franc 1.57% 10.21%
7 Platinum (NY spot) -1.93% 15.24% 28 ATX Index -0.03% 17.34%
8 FTSE-100 Index 13.37% 15.32% 29 Austrian Schilling 0.76% 9.24%
9 British Pound 2.37% 6.54% 30 Madrid General Index 19.83% 20.06%
10 All Ordinaries Share 2.46% 16.46% 31 Spanish Peseta 0.28% 9.62%
11 Australian Dollar -2.28% 8.22% 32 Johannesburg All-Share -9.52% 22.29%
12 TSE 300 Index 4.29% 13.56% 33 South African Rand -10.86% 10.04%
13 Canadian Dollar -3.24% 4.34% 34 IPC Mexico Index -15.16% 41.32%
14 BEL-20 Index 18.03% 14.17% 35 Mexican Peso -23.08% 22.43%
15 Belgian Franc 0.92% 8.83% 36 DAX General 16.78% 18.66%
16 KFX Index 14.86% 15.56% 37 German Mark 0.77% 8.84%
17 Danish Krone 1.19% 8.30% 38 CAC-40 Index 12.16% 19.05%
18 Straits Times Index -11.72% 25.25% 39 French Franc 1.02% 8.22%
19 Singapore Dollar -0.68% 5.82% 40 Egyptian Pound -0.28% 1.90%
20 Hang Seng Index -3.72% 31.56% 41 Dutch Guilder 0.62% 8.89%
21 Hong Kong Dollar -0.06% 0.39% 42 Israeli Shekel -6.71% 5.13%
For these 42 assets, the optimal portfolios are calculated using the Markowitz algorithm and
the Fisher-Statman algorithm. Table 9 lists the annualized return and standard deviation of the
optimal portfolio, as well as the weights of the constituent assets. There are two possible approaches
to the portfolio-selection problem. Investors can either select a desired return and minimize variance
or select a desired variance and maximize return. Since investors are likely to have more intuition
with regard to selecting a desired level of return, we will fix return and minimize variance. For the
sake of comparison, the desired portfolio return has been set to the level of the S&P 500 average
46
return (19.2345% annually, 0.07694% daily). To obtain this level of return by investing only in the
S&P 500 requires an annualized standard deviation of 14.39%. Keep in mind that the S&P 500
itself represents a diversified portfolio, and this must be taken into account when assessing the
reductions in variance obtained through the portfolio-selection rules under consideration.38
Table 9: Numerical Results Using Different Portfolio-Selection Rules
Portfolio Selection Method
S&P 500 StandardMarkowitz
StandardFisher-
Statman
Pos-WtdMarkowitz
Pos-WtdFisher-
Statman
Required Return 19.2% 1 9 . 2 % 19.2% 19.2% 19.2%
Standard Deviation 14.4% 5 . 2 % 7.2% 12.1% 12.3%
Percent of OptimalStandard Deviation 276% 1 0 0 % 139% 232% 236%
Asset ID Asset Weights (columns may not add to 1 due to rounding)
1 1 0.07725 0.10242 0.31249 0.26189
2 0 0.01253 0.07045 0.22572 0.23064
3 0 0.05761 0.03216 0 0
4 0 0.07845 0.13217 0 0
5 0 0.04627 -0.00124 0 0
6 0 -0.26475 -0.09558 0 0
7 0 0.02535 -0.00815 0 0
8 0 0.07826 0.06305 0 0
9 0 0.05104 0.06535 0 0
10 0 0.01944 0.01069 0 0
11 0 0.00859 -0.03373 0 0
12 0 0.03406 0.02660 0 0
38For comparison, the Dow Jones Industrial Average had an annualized return of 17.510% over the sameperiod, with a standard deviation of 14.689%. However, the average standard deviation of the individual Dowcomponent stocks was 27.372% (median of 27.252%). The standard deviation of most individual domestic stocks isapproximately 30% per annum , so the results for the S&P 500 are likely to be similar. Any differences are likely toarise from two facts: (1) the Dow is price-weighted (meaning higher-priced stocks receive more weight in the index)and the S&P 500 is capitalization-weighted (meaning stocks are weighted according to their price per sharemultiplied by their number of shares outstanding), and (2) the S&P 500 includes a greater number of smaller firmsthan does the Dow (meaning that the average standard deviation of S&P 500 component stocks is likely to behigher).
47
13 0 -0.28942 -0.17592 0 0
14 0 0.14675 0.09899 0.24420 0.16585
15 0 0.22350 0.01556 0 0
16 0 0.06107 0.06783 0 0
17 0 0.14151 0.02187 0 0
18 0 -0.06102 -0.01975 0 0
19 0 0.19841 -0.01574 0 0
20 0 0.00642 -0.00386 0 0
21 0 1.49866 0.94233 0 0
22 0 -0.01137 -0.01778 0 0
23 0 -0.06276 -0.05153 0 0
24 0 -0.02637 -0.00654 0 0
25 0 0.05072 -0.01997 0 0
26 0 0.01243 0.06607 0.07952 0.14824
27 0 0.04894 0.01848 0 0
28 0 -0.09834 0.00060 0 0
29 0 -0.03312 0.01222 0 0
30 0 0.06229 0.05428 0.13807 0.15998
31 0 -0.05649 0.00551 0 0
32 0 0.05170 -0.02050 0 0
33 0 -0.25647 -0.11533 0 0
34 0 -0.01604 -0.00957 0 0
35 0 -0.05903 -0.04967 0 0
36 0 0.01962 0.05321 0 0.03339
37 0 -0.12422 0.01341 0 0
38 0 -0.06792 0.03716 0 0
39 0 0.12937 0.01960 0 0
40 0 0.03840 -0.02653 0 0
41 0 -0.31058 0.01117 0 0
42 0 -0.44075 -0.26979 0 0
As can be seen from Table 9, use of the weight-unconstrained, non-normative algorithms
requires investors to take substantially more risk than is necessary. At the same time, the positively-
48
weighted results are very similar. We can also observe that portfolio composition varies widely.
That is, the portfolio considered to be “optimally diversified” under one measure of diversification
bears little resemblance to the portfolio considered optimal under another measure. This is
important because investors may tend to think of these heuristic approaches as allowing them to be
“almost” optimal. In fact, as Michaud (1989) noted, composition of optimal portfolios varies
widely for even small parameter changes. Thus, portfolios similar to the Markowitz optimal
allocation (those close in weight-space) need not have similar performance (be close in risk-return
space). Whether or not the different heuristics produce portfolio returns and variances comparable
to the optimal procedure is another question. As Table 9 indicates, in the weight-constrained case,
the answer would appear to be that they do; very little is sacrificed by using the (simpler) heuristic
approach.
To compare the Green-Hollifield algorithm, we must take a different approach. The Green-
Hollifield approach requires the relative weights to be fixed; the only possible adjustment being
increasing or decreasing the total number of assets in the portfolio without regard to their individual
covariances. In effect, Green-Hollifield implies that inattention to individual securities can be
compensated by having “a lot” of them. This claim can be examined without reference to a specific
return by using the Sharpe ratio. Using the same data as in Table 9, Table 10 compares the Sharpe
ratios of the various portfolios, with higher Sharpe ratios indicating greater efficiency.39
39The equal weight portfolios only select from asset classes with positive expected returns. The risklessrate used to calculate the Sharpe ratios was the average constant-maturity one-year US Treasury bond rate (5.495%)over the same period as the asset class returns were calculated.
49
Table 10: Sharpe Ratio Comparison of Normative and Heuristic Portfolio Selection Rules
Portfolio Sharpe Ratio
Standard Markowitz 2.640
Standard Fisher-Statman 1.901
Positively-Weighted Markowitz 1.140
Positively-Weighted Fisher-Statman 1.119
S&P 500 0.955
Green-Hollifield (Equal Weights, 22 assets) 0.412
Mean of 500 Randomly-Selected 10-asset Portfolios 0.356 ± 0.0209
Mean of 500 Randomly-Selected 5-asset Portfolios 0.320 ± 0.0313
As can clearly be seen, some heuristics (such as Fisher-Statman) perform better than others (Green-
Hollifield). In general, however, they perform better than randomly selected portfolios.
To illustrate more vividly the relative inefficiency of the equal-weights approach, we will
examine the decrease in risk possible by comparing the additional number of both high-correlation
and low-correlation securities required to reduce variance by a fixed amount. The Green-Hollifield
approach is more likely to add high-correlation securities since it pays no attention to individual
covariances. In contrast, the Markowitz algorithm specifically targets low-correlation securities for
addition to risky portfolios. The end result is that Markowitz portfolios require far fewer securities
to achieve the same level of reduction in portfolio risk (which is why Markowitz portfolios are
known as efficient portfolios).
Consider the following numerical example. The Green-Hollifield algorithm suggests
holding assets in roughly equal proportions. Thus, we shall let ω i = n − 1 ú i . Also, for the sake of
convenience, we shall assume all assets have a unit variance. Finally, we will assume that all assets
belong to one of two correlation classes: high and low. These are, of course, relative terms. In
practical terms, one could think of them as representing within-asset class correlations (the high
correlations) and across-asset class correlations (the low correlations). Green and Hollifield (1992)
50
describe investors approaching the diversification process as believing that simply adding more of
the same assets will properly diversify their holdings. This, naturally, is true. However, as can be
seen below, this can be a remarkably expensive process when lower-correlation assets are available.
As we have noted above, the variance of a portfolio can be represented as [24].
σ 2
p =
n
3 i = 1
n
3 j = 1
ω i ω
j σ
i σ
j ρ
i , j[24]
However, we have made the assumptions that the weights are all equal and that all assets have unit
variances. Thus, we are left with a much simpler equation [25].
σ 2
p = 1
n 2
n
3 i = 1
n
3 j = 1
ρ i , j
, ρ i , j
= : ; <
= =
= = 1 i = j ρ
H iÖ j
[25]
If we consider a five asset portfolio and let ρ H
= 0 . 75, we have σ 2
p = 0 . 80. Now, if we add one lower-
correlation asset (ρ i , 6
= 0 . 65 ú i ), the portfolio variance drops to [26].
σ ˆ 2
p = 1
á n + 1 é 2
n + 1
3 i = 1
n + 1
3 j = 1
ρ ˆ i , j
, ρ ˆ i , j
=
:
;
<
= = = = = = = = = = = =
= = = = = = = = = = = = =
1 i = j ρ
L iÖ j , i = 6
ρ L iÖ j , j = 6
ρ H iÖ j Ö 6
[26]
If we let ρ L = 0 . 65, then the addition of a sixth less-correlated asset has caused the portfolio
variance to drop from 0.80 to 0.761. However, if we instead add only same-correlation assets, the
important question is how many must we add to provide the same reduction in variance as a single
less-correlated asset? Figure 11 plots the high-correlation portfolio variance for different values of n
against the variance of the less-correlated-additional-asset portfolio.40 The dashed line represents
the asymptotic portfolio variance. The thin line is the variance of the low-correlation portfolio after
the addition of a sixth (less-correlated) asset.
40Note that because of the assumptions that we have made in this problem (namely, constant non-negativeweights), the portfolio variance asymptotically approaches the average correlation (0.75 in this example). Ifnegatively-weighted asset positions were allowed, we could achieve further reductions in portfolio variance.
51
Figure 11: High-Correlation vs. Low-Correlation Additions to a Portfolio
5 10 15 20 25 30Number of Assets
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Variance
As Figure 4 shows, to achieve the same reduction in variance from adding a single less-correlated
asset, investors must add approximately 13 additional high-correlation assets! This additional
burden is critically important in a world of transaction costs.
More generally, let ρ H and ρ
L represent the high and low correlation values. The variance of
the original portfolio is [29].
σ 2
p = 1
n 2
n
3 i = 1
n
3 j = 1
ρ i , j
, ρ i , j
= : ; <
= =
= = 1 i = j ρ
H iÖ j
[27]
= 1
n 2 n + ρ
H á n 2 − n é [28]
σ 2
p = 1
n á 1 − ρ
H é + ρ H
[29]
The variance of the portfolio adding a single less-correlated asset is given by [32].
52
σ ˆ 2
p = 1
á n + 1 é 2
n + 1
3 i = 1
n + 1
3 j = 1
ρ ˆ i , j
, ρ ˆ i , j
=
:
;
<
= = = = = = = = = = = =
= = = = = = = = = = = = =
1 i = j ρ
L iÖ j , i = n + 1
ρ L iÖ j , j = n + 1
ρ H iÖ j Ö n + 1
[30]
= 1
á n + 1 é 2
n + ρ H á n 2 − n é + 1 + 2 n ρ
L [31]
σ ˆ 2
p =
2 n ρ L + á n − 1 é n ρ
H + n + 1
á n + 1 é 2
[32]
We are interested in the number of securities which the basic portfolio must contain to match the
variance of the portfolio that added a single less-correlated asset. Let N represent this number. We
can set these two variance measures equal to each other [33] and solve for N [34].41
σ 2
p = σ ˆ 2
p = 1
N á 1 − ρ
H é + ρ H
= 2 n ρ
L + á n − 1 é n ρ
H + n + 1
á n + 1 é 2
[33]
N = á n + 1 é
2 á ρ
H − 1 é
3 n ρ H
− 2 n ρ L + ρ
H − n − 1
[34]
Additionally, we can examine the sensitivity of N to changes in the correlation values of the different
classes of assets. Figure 12 illustrates the number of high-correlation securities required to equalize
the variance between the five-asset high-only and mixed-correlation portfolios. The line presented
represents the minimum number of securities required for a small (0.1) difference in correlations
41For the previous numerical example, the exact number is N = 18:
N = á 5 + 1 é 2 ( 0 . 75− 1 )
3 A 5 A 0 . 75− 2 A 5 A 0 . 65+ 0 . 75− 5 − 1
= 36 A á − 0 . 25 é 15 A 0 . 75− 10 A 0 . 65− 5 . 25
= − 9
− 0 . 5 = 18
53
between high and low-correlation assets. For larger correlation differences, more assets are
required. For example, the equalizing number for the high and low correlations of -0.2 and -0.9 is
216.
Figure 12: Equalizing Number of Securities for Small (0.1)Correlation Differences Between Two Asset Types for
Five-Asset Portfolios
0
5
10
15
20
25
30
35
40
Low-to-High Correlation Change
Equivalent Number of High Correlation
Assets
In this section, we have demonstrated why Markowitz diversification is so effective as a
portfolio selection rule, minimizing both transaction costs and normatively-defined risk. More
importantly, we have demonstrated two additional facts: (1) under practical implementation
constraints, heuristic approaches to portfolio selection do not always cause investors to take more
risk than is necessary, and (2) portfolio-selection rules chosen because they “feel” diversified or
appear to select diversified portfolios produce portfolios that bear little resemblance to one another
(or to Markowitz portfolios). This suggests that investors’ concepts of diversification vary widely
and, at the same time, show little recognition of the optimal method. Further, this suggests that not
all heuristic approaches are necessarily bad; many can produce performance comparable to the
theoretical optimum. If this is true, one consequence for investors is that, under short-sale
54
constraints, covariation is just not something investors need to be concerned about; it’s impact on
portfolio risk is negligible. Naturally, there are dangers in over-generalizing any such result.
However, one could argue that an imperfect understanding of covariation could be more costly than
no understanding at all.
V. EXPERIMENTAL EVIDENCE
Based on the results in previous sections, some of the questions that we are interested in
addressing are:
(1) Do investors recognize the effect of diversification on risk?(2) Is the desire for, or likelihood of implementing, diversification
dependent on the amount of money at stake?(3) How is diversification perceived by investors?(4) Can the implementation of diversification be manipulated
through framing? If so, is the result consistent with prospecttheory?
Some of these questions are specific and involve direct claims. Others are more open-ended and are
designed to explore the diversifying behavior of individual investors. Despite its central role in
financial theory, the literature has largely ignored individual investor risk perception in the context
of diversifying behavior.
The subjects in the following experiments were drawn from several sources. Many of the
subjects were third- and fourth-year undergraduates in a decision analysis class. The remaining
subjects were members of amateur investment clubs. The investment-club members were drawn
from three clubs in the Pittsburgh region and one in San Diego. Although some results will present
aggregated data, the most interesting will involve differences between the subject groups.42 In
general, the decision analysis students are more quantitatively inclined, but have less (if any) actual
investment experience. By comparison, the investment-club members are generally not
quantitatively inclined, but have years, and in many cases, decades of actual investment experience.
42The differences being between students and investment club members. The Kruskal-Wallis rank sum andMood median tests were used to show that there were no significant differences between clubs in the mean or medianof questions involving numerical responses.
55
The investment club members were all women, although this was not part of the
experimental design. The average club has existed for 4 years and has 10 members. Each of the
clubs examined had between $15,000 and $150,000 invested at the time of the survey. When asked
how diversified they felt their personal portfolios (i.e., their investors both inside and outside of
their share in the club’s holdings) were, the club members surveyed provided a median response of
5 (µ = 4.8, n = 35) on a scale of 1 (not very diversified) to 7 (extremely diversified). When asked
how important, on a scale of 1 (not very important) to 7 (extremely important) holding a diversified
portfolio was (to them), the median response was 5 (µ = 5.26, n = 35). Responses to these two
questions were correlated at the 0.57 level (p < 0.001 by the Kendall rank correlation test). Table 11
summarizes the survey results for the investment club members.
Table 11: Summary of Survey Questions and Results (n = 35)
How diversified do you consider the assets you currently own?
Rank Frequency
Not very diversified 1 -
2 1
3 5
4 8
5 9
6 10
Extremely Diversified 7 2
How important to you is holding a diversified portfolio?
Not very important 1 -
2 -
3 4
4 8
5 7
6 7
Extremely important 7 9
56
Hypothesis 1: Subjects Recognize and Respond to Diversification Opportunities
Nagy and Obenberger’s (1994) results suggest that investors perceive diversification as a
separate activity from portfolio risk management. I use survey results and a coin-toss experiment to
analyze this claim. This hypothesis has two parts: recognition and response. Thirty-five members
of amateur investment clubs were judged on recognition ability for (Markowitz) diversification
based on their answers to the following survey questions:
(1) Which of the following choices best describes your definition of diversification?
Holding a large number of securities 2Holding securities in equally-sized amounts 0Holding securities of firms in different industries 29Holding securities of firms that have performed differently in the past 2Other (free response): real estate, firms of different sizes 2
(2) Indicate whether your agree or disagree with the following statements:
The risk of a stock depends on whether its price typically moves with or against themarket.
[8 Agree 25 Disagree p < 0.05]
In my stock portfolio, I would rather have just a few companies that I know well,rather than many companies I know little about.
[28 Agree 7 Disagree p < 0.05]
Their responses indicate that most subjects do not recognize the core elements of Markowitz
diversification (either covariation of returns or large numbers of securities). Instead, they tend to
focus on company-specific factors. This claim is supported by responses to the following question:
(3) Buying stock in local companies or in companies whose products I like and use is agood idea.
[ 24 Agree 11 Disagree p < 0.05]
Familiarity with firm operations and the “type” of company were important influences on
investors. In fact, two subjects indicated that they were unable to complete another investment-
choice experiment (the build/eliminate experiment discussed below) because the experiment “only
gives [them] numbers” (a distribution of returns) and “doesn’t have any information about what
the company does.”
57
It may be the case, however, that even though subjects fundamentally don’t understand
portfolio theory, their heuristic strategies may allow them to behave in a manner consistent with
normative diversification. Subjects’ responses to diversification opportunities were measured by
choices in a coin-toss experiment. One hundred and two students in an undergraduate decision
analysis class and 35 members of investment clubs were randomly given one of two questions. The
questions asked how many coins the subject would prefer to flip to obtain a given chance of
winning some dollar amount43 (or nothing): 1 coin to win $16 (or $1,600), 2 coins to win $8 (or
$800), 4 coins to win $4 (or $400), 8 coins to win $2 (or $200), or 16 coins to win $1 (or $100)
on each coin.
E á R é = 3 # of coins
EV á Coin Toss é [35]
E á R é = n A p
H
2 = n A á 16n − 1 é
2 = 16
2 = 8 [36]
Because the coin tosses are uncorrelated, their total variance is just the sum of the variances of the
individual coin tosses.
V = V á x 1 + ̨ + x
n é = n
3 i = 1
V á x i é [37]
Further, since the payoff for tails is always 0, we can simplify and state that for n coin tosses, the
variance of all tosses together is [39]. Adding the fact that p H
= 16n − 1 , we can simplify further [40].
V = 1 2
ä
ã å å å −
p H
2
ë
í ì ì ì
2
+ ä
ã å å å p
H −
p H
2
ë
í ì ì ì
2
[38]
V = np2
H
4 [39]
V = n á 16n − 1 é 2
4 = 64
n [40]
43The dollar amounts in parentheses represent the payoffs for the large dollar amount version of thegamble.
58
Clearly, the expected payoff [36] is constant, but the variance [40] of the gambles is reduced as
more coins are tossed. This is exactly what adding securities (with correlations less than 1) does to
a portfolio. Table 12 and Figure 13 report the results.
Table 12: Results of the Coin-Toss Experiment
Choiceand
Condition
Investment ClubMembers
Decision AnalysisStudents
N = 19 N = 16 N = 55 N = 47
Payoff Amount Large Small Large Small
1 or 2 coins 52.6% 25.0% 18.2% 44.7%
4 coins 10.5% 25.0% 12.7% 8.5%
8 or 16 coins 36.8% 50.0% 69.1% 46.8%
If subjects were responding to diversification opportunities, they should choose to toss
more coins rather than fewer.44 This claim is supported in only two of the four cases. The decision
analysis students facing large dollar gambles (p < 0.0001) and the investment club members facing
small dollar gambles (p = 0.0186) all significantly preferred to toss more coins to fewer. The other
two groups did not attain significance (p = 0.4375 for the students in the small dollar condition and
p = 0.6015 for the club members in the large dollar condition).
Although this provides some evidence that diversification opportunities are responded to,
this behavior is also consistent with simple heuristics such as “buy a lot of things” (toss as many
coins as possible). However, only 2 of the 35 subjects indicated that diversification meant “hold a
large number of securities.” Alternatively, some of this behavior could be seen as consistent with
certain risk-seeking utility functions (e.g., subjects preferring to increase the probability of
obtaining the maximum payoff). However, subsequent responses to lottery comparison questions
suggest that the subjects were moderately risk-averse, if anything.
44The four-coin-toss responses were discarded for the purposes of this analysis and the one-tailed binomialtest was used.
59
Figure 13: Frequency of Condition Choices by Subjects in Both Groups
Decision Analysis Students
18.2%12.7%
69.1%
44.7%
8.5%
46.8%
0%
25%
50%
75%
1 or 2 Coins 4 Coins 8 or 16 CoinsLarge Small
Investment Club Members
52.6%
10.5%
36.8%
25.0% 25.0%
50.0%
0%
25%
50%
75%
1 or 2 Coins 4 Coins 8 or 16 Coins
Large Small
Hypothesis 2: Subjects are more likely to toss more coins as the payoffs increase.
The results for this hypothesis are also split between the groups. The students significantly
(p = 0.0024) increased the number of coins tossed for the large dollar gambles. However, this was
not supported for the club members (p = 0.9120), who actually appeared to reduce the number of
coins tossed when faced with larger gambles.
One possible explanation for the contrasting behavior of the students and club members
could be differential treatment of the monetary amounts involved. As the payoffs increased, the
students became more risk-averse, but the club members became more risk-seeking. This result
highlights a particular challenge of doing experimental research in finance: subjects respond to
experimental stimuli with different schemata than they respond to real stimuli. In this case, the very
same investment club members who responded that diversification was very important turned the
page and chose to stake the entire payoff of a gamble on the toss of a single coin. In fact, the rank
correlation between the stated importance of diversification and the number of coins tossed was
insignificantly different from zero (r = 0.037, p = 0.444) in the small condition and significantly
negative in the large condition (r = -0.359, p = 0.066). It is plausible to conclude that the investment
club members see no correspondence at all between the abstracted experimental situations (coin
toss gambles) and real investment experience.
60
Hypothesis 3: Portfolio Selection Behavior is Better Explained by Prospect Theory thanby Expected Utility Theory
Earlier in this paper, I demonstrated that, while expected utility maximizers (Markowitz
diversifiers) would choose to optimize by increasing the return on their portfolios, prospect-
theoretic decision makers would prefer the decreasing variance option. Simply stated, the two
decision rules would produce different rankings of the portfolios. To test this Hypothesis 3, an
experiment involving 3 hypothetical securities was developed. Subjects were asked told either (a)
that they held all three and had to eliminate one, or (b) held none and could add as many of the three
(in equal proportion) as they wished. If prospect theory is a more accurate description of human
decision making than expected utility theory, here in the context of investing, I would expect to see
subjects’ choices more closely aligned with the prospect theory rankings.45
Recall that Markowitz diversification can be implemented as the maximization of the Sharpe
ratio of a portfolio. This measure of excess reward per unit of risk is calculated as the return of an
asset in excess of risk-free rate divided by the standard deviation of returns: S i = á r
i − r
f é σ − 1
i .
Clearly, the value S i can be increased by increasing the numerator (return) or by decreasing the
denominator (risk). Thus, for any Markowitz-suboptimal investor, there are two paths to obtaining a
more diversified portfolio: (1) increasing the return without taking more risk, or (2) reducing risk
without sacrificing any return. In Section III we illustrated how prospect-theoretic decision makers
could prefer either option depending on the λ-parameter in the value function. Using Tversky and
Kahneman’s (1992) estimated parameters, the prospect theory decision maker would prefer the
45The functional forms and parameters used in the prospect theory examples in the previous section remainthe same throughout this section as well.
61
variance reduction alternative.46 Expected utility-maximizing investors will always prefer the
increasing return alternative.
Thus, exploring such preferences for risk provides an indication as to which model (utility
or prospect theory) gives a better explanation of investor portfolio selection behavior. In the
experiment, three assets were defined according to the convention used in Section III. Table 13
presents the three assets.47
Table 13: Three Assets in Prospect Theory Notation
Asset 1 Asset 2 Asset 3
Probability Return Probability Return Probability Return
10% chance 5% return 10% chance 16% return 10% chance 40% return
5% chance 4% return 80% chance 3% return 50% chance 3% return
85% chance 3.3% return 10% chance -5% return 40% chance -5% return
In each case the expected return was identical: 3.5%. However, each asset had a different standard
deviation and investors could also consider combinations of the various assets:
{{12},{13},{23},{123}}. Naturally, each combination portfolio had the same expected return as
well, although standard deviation (and thus the Sharpe ratio) varied for each asset. The three assets
and their combinations could also be evaluated for their “prospective value” using the equations
46The investment club members were given four questions taken from the original prospect theory paper(Kahneman and Tversky, 1979) to illustrate their consistency with prospect-theoretic decision making. Theirresponses (CLUB) were consistent with those obtained from Kahneman and Tversky’s (K&T) subjects. For thefollowing four prospects, the percentage of subjects choosing A for each group is given:
A: (4,000, 0.80) B: 3,000 CLUB: 26% K&T: 20%A: (4,000, 0.20) B: (3,000, 0.25) CLUB: 58% K&T: 65%A: (6,000, 0.45) B: (3,000, 0.90) CLUB: 21% K&T: 14%A: (6,000, 0.001) B: (3,000, 0.002) CLUB: 63% K&T: 73%
In none of the cases could the null hypothesis of equal proportions be rejected.
47The rankings presented below are relatively robust to changes in the returns assumed in Table 13. Mostimportantly, to elimination of the negative returns (by increasing all of the payoffs by a fixed amount). The Sharperatio rankings remain unchanged and the prospect theory rankings only move slightly, usually by switching onlyone rank.
62
from Section III. Table 14 displays the various statistics for each possible portfolio and their rank
under both expected utility theory (maximizing the Sharpe ratio) and prospect theory (maximizing
the prospective value).
Table 14: Three-Asset Summary Statistics for Expected Utility and Prospect Theory
PortfolioExpectedReturn
StandardDeviation
SharpeRatio
SharpeRank
ProspectiveValue
ProspectiveValue Rank
1 3.50% 0.52% 6.73 1 0.0536 5
2 3.50% 4.80% 0.73 4 0.0375 7
3 3.50% 12.70% 0.28 7 0.0393 6
1 & 2 3.50% 2.40% 1.45 2 0.0695 4
1 & 3 3.50% 6.40% 0.55 5 0.0869 2
2 & 3 3.50% 6.80% 0.51 6 0.0824 3
1 & 2 & 3 3.50% 4.50% 0.77 3 0.1308 1
Investors were faced with either of two decision problems: (1) to start with no assets and
select which one, two, or three assets they wished to hold (in equal amounts) in a portfolio, or (2) to
start with all three assets and choose one asset to eliminate (with money held in that asset
reallocated evenly to the remaining two assets).
For the portfolio construction task, by examining Table 14, we can see that utility
maximizers would most prefer to hold their entire portfolio in a single asset (Asset 1).48 Asset 1’s
Sharpe ratio is higher than any of the other assets available, so an investor’s entire portfolio would
be concentrated in that single asset and any adjustment to the risk level desired made by taking a
position in the riskless asset.49 On the contrary, prospect theory investors would most prefer to add
48Consequently, it is implied that investors were not given the entire universe of possible investmentassets and not all assets were members of the efficient set (otherwise one asset would not dominate all othercombination of assets). Rather, they were given a subset of all possible feasible assets. Alternatively, we canconceive of the assets themselves as portfolios. Since Asset 1 has the highest Sharpe ratio, it could be considered anindex fund representative of the market portfolio. The short-sale restriction could also explain the maximal efficiencyof a single asset.
49As noted previously, however, no riskless asset was available to investors in the experiment.
63
all three assets to their portfolio, as the portfolio of all three assets results in the highest prospective
value. The correlation between the expected utility and prospect theory rankings was 0.071.
In the asset elimination task, utility maximizing investors could actually reach a more
optimal position by eliminating Asset 3 (i.e., moving from {123} to {12}). By eliminating Asset 3,
the Sharpe ratio of their portfolio would increase from 0.77 to 1.45. Prospect theory investors
would prefer to eliminate Asset 2 (i.e., moving from {123} to {13}). However, they would actually
prefer not to change at all, since their prospective value drops from 0.1308 to 0.0869. In the
experiment, however, subjects were not given the option of remaining at the status quo. The context
of the experiment was that their company’s retirement savings plan was eliminating one of the
available assets to cut costs. Subjects were told to select the asset they would least mind being
eliminated. The results of the two groups are presented in Tables 15 and 16. Table 15 presents the
data for the elimination condition and Table 16 presents the data from the building condition.
Table 15: Choice Frequencies in the Elimination Condition
EliminationStudents(n = 45)
Investment Clubs(n = 17)
ExpectedUtility Ranking
Prospect TheoryRanking
1 8 3 3 2
2 12 6 2 1
3 25 8 1 3
Table 16: Choice Frequencies in the Building Condition
BuildingStudents(n = 51)
Investment Clubs(n = 16)
Expected UtilityRanking
Prospect TheoryRanking
1 13 1 1 5
2 7 0 4 7
3 4 1 7 6
12 9 2 2 4
13 8 2 5 2
23 1 0 6 3
123 9 10 3 1
64
Although the small number of data points in the elimination condition makes it difficult to
analyze, it is clear that both the student and club members’ rankings are identical. In addition, they
both clearly correspond to the expected utility rankings. Thus, we can conclude that people’s
behavior is largely consistent with expected utility theory for decisions involving the removal of
assets from portfolios. In this experiment, the unattractiveness of Asset 3 stemmed from the fact
that it had a much larger probability of earning a negative return than either of the other two assets.
Similarly, the popularity of Asset 1 is likely a consequence of its strictly positive returns.
Different results for the club members are observed in the build condition. Using the ranked
data, we can evaluate the alignment of subjects’ choices with the two decision rules by examining
their rank correlations. Table 17 presents such information. The three shaded cells are particularly
important. First, we can see that the two subject groups have begun to diverge in their responses.
Although they responded identically in the elimination condition, their rankings only correlated at
the 0.581 level in the building condition. In addition, the decision analysis students’ rankings were
far more consistent with expected utility theory, while the investment club members’ rankings were
more consistent with prospect theory.
Table 17: Correlations Between Subject Group and Normative Rankings
Decision AnalysisStudents
Investment ClubMembers
Expected UtilityRanking
Prospect TheoryRanking
Decision AnalysisStudents 1.000
Investment ClubMembers 0 . 5 8 1 1.000
Expected UtilityRanking 0 . 9 2 0 0.344 1.000
Prospect TheoryRanking
0.1982 0 . 7 0 7 0.071 1.000
However, the fact that there is but a single ranking produced for each group precludes us
from testing the significance of those correlations. Instead, we shall explore several other analyses
and show that Hypothesis 3 is supported, at least in part, by each of them. Figures 14 and 15 show
the frequency of responses ordered with expected utility theory’s ranking (Figure 14) and with
prospect theory’s ranking (Figure 15).
65
Figure 14: Frequency of Responses Ordered by Expected Utility Theory’s Ranking
0%
10%
20%
30%
40%
50%
60%
70%
1 1 & 2 1&2&3 2 1 & 3 2 & 3 3
Expected Utility Ranking (More Preferred to Less Preferred)
Percent of Respondents
Club Members Students
Figure 15: Frequency of Responses Ordered by Prospect Theory’s Ranking
0%
10%
20%
30%
40%
50%
60%
70%
1&2&3 1 & 3 2 & 3 1 & 2 1 3 2
Prospect Theory Ranking (More Preferred to Less Preferred)
Percent of Respondents
Club Members Students
It is clear simply from examining these frequency distributions that the students’ choices are better
explained by the expected utility ranking and that the club members’ choices are better explained by
the prospect theory ranking.
66
We can take an additional step to verify the difference between the groups’ alignment with
the two decision rules. For both groups and both rules, we will replace the indicated choice with the
rank of that choice under either expected utility theory or prospect theory. To the extent that a
group’s decisions are more consistent with one rule than another, its mean rank should be lower
when coded according to that rule’s ranked values. Mean ranks closer to one indicate increasing
alignment with the coding rule used (expected utility theory or prospect theory). Thus, by
comparing the mean ranks of the two groups, we can ascertain which rule provides a better model of
the subjects’ portfolio selection behavior. The students’ choices had an average rank of 3.14 when
ranked by expected utility and 3.96 when ranked by prospect theory. The investment club
members’ choices had an average rank of 3.25 when ranked by expected utility and 2.06 when
ranked by prospect theory. These differences were significant in the hypothesized direction. The
students used expected utility theory (3.14 < 3.96, p = 0.0178 by the Mann-Whitney-Wilcoxon
test) and the club members used prospect theory (2.06 < 3.25, p = 0.0065 by the Mann-Whitney-
Wilcoxon test).
One possible explanation for this result involves the relatively abstracted nature of the
experiment and the more quantitative nature of the decision analysis students. At least in theory, the
decision analysis students are familiar with expected utility theory and should be able to “solve”
the problem correctly. The investment club members, although having more “practical” experience,
are less “problem-solving” inclined and thus more likely to fall prey to the biases behind prospect
theory.
One possible question on implementation concerns the use of ranks imputed from the
frequency data. Since ranks are not actually elicited from the subjects, there is some cause for
concern that each subject’s own ranking could be different than the imputed consensus rank. To
address this concern, assume that each subject has an internal ranking of the seven possible choices
in accordance with a decision rule. Although subjects intend to implement their internal rankings
(by selecting as their most preferred alternative the highest internally-ranked alternative), suppose
that there exists some probability of error (as a result of miscalculation, for example). I will assume
that any such errors could extend only three levels deep. This means that any differences between
indicated ranks and the theoretical rankings greater than three levels would not be attributed to
implementation error. In addition, the greater the difference, the less likely it is attributable to
implementation error (and the more likely it is because some alternative rule is used).
67
There are 5,040 possible ways to rank 7 choices (ties excluded). By assigning these
weighted ranking scores to all possible choices, we can examine the number of different rankings
(decision rules) which could be more preferred to the hypothesized choices. Using this method, no
decision rule results in a higher rank score for the decision analysis students than expected utility
theory (i.e., the three highest frequency choices were those ranked 1, 2, and 3 by expected utility
theory). By comparison, prospect theory’s rank score for the students placed it 4,594th. Similarly,
for the investment club members, no decision rule results in a higher rank score than prospect
theory. By comparison, expected utility’s rank score for the club members placed it 1,321st.
To clarify this point, although there are several different rankings which have equal scores
(expected utility tied with 23 others for the students and prospect theory tied with 119 others for the
club members), none exceed these values. Thus, although we may be reasonably certain of the
rankings elicited, attributing them to a particular theory is less certain. For example, the prospect
theory rankings are identical to the rankings of a person who wished only to rank more highly
those portfolios containing larger numbers of securities. This problem in judging the
“reasonableness” of portfolio choices is well known (Elton and Gruber, 2000).
Summary
The experimental evidence suggests the following about the four questions raised at the
beginning of this section:
(1) In general, subjects preferred to toss more coins to fewer. However, this did notappear to be related to their knowledge of how Markowitz diversification actuallyworked. Rather, it appeared to be attributable to the use of simple heuristics (“moreis better”).
(2) The desire for diversification does depend on the amount of money at stake,although in two different ways. The decision analysis students chose to toss morecoins as the payoffs increase, as one might expect. However, the investment clubmembers tended to prefer tossing fewer coins in the large payoff condition. Moreinterestingly, the number of coins tossed was uncorrelated with the statedimportance of diversification for the investment club members.
(3) Diversification is almost universally perceived as “looks different” by investors,rather than the Markowitz version, which might be characterized as “performsdifferently.” The majority of respondents, in fact, disagreed that the risk of a stockdepends on its covariation with the market.
68
(4) Asset allocation decisions can be significantly affected by framing issues. There aresignificant differences in the rankings of portfolios produced by expected utility andprospect theory. The decision analysis students tended to behave in accordance withexpected utility theory while the investment club members tended to behave inaccordance with prospect theory.
In addition to these specific findings of interest, there were several other observations that are
relevant to the subject. These general findings fall into three areas: the impact of experimental
abstraction, the lack of portfolio thinking, and the application of prospect theory to portfolio
management questions.
One possible result emerging as a consequence of the second finding is the warning to
experimenters to treat abstraction with caution. Although it is clearly advantageous to have simple,
clear experiments, some people exhibit different behavior in naturalistic versus contrived settings.
The lack of any correlation between the subjects’ stated desire for diversification and the counter-
intuitive result in the coin toss experiment seems to suggest either that subjects’ perceptions of their
own behavior are grossly inaccurate, or that they are simply treating the two theoretically-equivalent
situations are functionally different. Of course, similar results have long been demonstrated with
regard to the purchase of insurance and the participation in gambling. If anything, this would be
suggestive of incorporating a richer experience into the modeling of investment decision making.
We use the phrase “lack of portfolio thinking” to suggest a similarity to the “lack of
systems thinking” suggested by Fischhoff et al ., (1978) and others. Investors, as with people in
general, often fail to see the “big picture” with regard to the complexity of problems and the risks
they are facing. In an investing context, the potentially counterintuitive result of reducing risk by
combining two risky assets is often overlooked in favor of selecting what appear to be low-risk
assets in isolation. Too often for investors, this is either an inefficient strategy, or one fraught with
hidden exposures at the portfolio level. Even when investors use proxies for investment risk, those
proxies are solely on a single-investment basis (e.g., the debt-to-equity ratio, whether or not the
company makes a good product, etc.). Although still incorrect, the use of proxies might be more
useful to investors if they took those proxy values in the context of their entire portfolio.
Finally, the application of prospect theory to portfolio management seems promising. If, as
we have found in this paper, the behavior of investors is more accurately modeled by prospect
theory (at least for certain tasks), the next logical step would be to reformulate asset pricing theory
69
under prospect theoretic assumptions. Shefrin and Statman’s (1994) Behavioral Capital Asset
Pricing Theory is in the same spirit as this effort, but does not explicitly use prospect theory.
VI. CONCLUSION
Investors are told constantly that “diversification” is a good thing. Unfortunately, they are
rarely consistently told what diversification means. Not only is some of the advice given to investors
on diversification wrong, but the sheer “diversity” of the advice, which is often conflicting, can
often be confusing to investors. This confusion, together with the complexity of implementing
Markowitz diversification leads many investors to make poor asset allocation decisions.
These poor (and costly) decisions are particularly unfortunate because careful and
consistent use of many of the heuristics investors find appealing can lead to nearly optimal
performance. The greatest benefits of Markowitz diversification generally result from taking
extreme, and often negative, positions in various assets. Due to institutional and regulatory
limitations, as well as capital constraints for most investors, these portfolios are infeasible. The
added benefit of implementing Markowitz diversification on portfolios facing these constraints is
minimal and not worth the added complexity and potential for error of the normative diversification
algorithms.
Thus, the important information which must be provided to investors is the reason why
Markowitz diversification works: covariation. More generally, that risks must be calculated and
managed at the portfolio level, rather than at the individual asset level. The experiments carried out
in this paper demonstrated that investors do not make a clear connection between risk reduction and
diversification; that they pursue diversification for its own sake, rather than to achieve a reduction in
portfolio risk. This is a costly strategy. Further, our experimental evidence suggests that portfolio
decisions can be influenced by how those decisions are presented.
All of this evidence suggests that investors are simply unable to reconcile the numerous
pieces of advice they receive on portfolio management in time sufficient to make effective decisions.
Investors must often move quickly to implement portfolio changes, and often in an environment
characterized by substantial uncertainty. Here, the pressure is on getting investors the information
70
they need in a form they can quickly, correctly, and consistently interpret. As the numerical results
illustrated, in a realistically-constrained trading environment, the consistent application of very
simple strategies produces results nearly as good as the (constrained) Markowitz optimum. Advice
given to investors should be built around this fact, not on the pursuit of more complicated
investment management strategies.
71
APPENDIX: HOW AND WHY DIVERSIFICATION MATTERS
The Markowitz Problem
It was Markowitz (1959) who first showed that the riskiness of a portfolio of assets
depends on the covariance of its assets, rather than just their variance. Intuitively, this makes sense:
by investing in assets with different patterns of returns, the overall pattern of (portfolio) returns
would be more even (i.e., fewer extreme composite returns).50 In Markowitz’ (1959) model, risk-
averse investors sought to achieve a given expected return with the minimum amount of risk
possible. The question posed was: “what is the minimum amount of risk which must be accepted to
produce a given return with a given set of assets?” Risk, in the context of the investor’s portfolio,
was represented by the variance of the portfolio. In turn, the variance of the portfolio depended on
the covariation of the assets in the portfolio (because the sum of two correlated random variables
depends on their covariance).
Thus, the basic minimum variance portfolio problem was created [A1, A2, A3]. Note that ω i
represents the fraction of wealth invested in security i, σ 2
i represents the variance of security i, and
σ i , j
represents the covariance of security i with security j. A portfolio p contains n assets, each with
expected return given by r i . r í represents the target portfolio return desired by the investor.
minω
i
var á r p é = σ 2
p =
n
3 i = 1
ω 2
i σ 2
i +
n
3 i = 1
n
3 j = 1
i Ö j
ω i ω
j σ
i , j [A1]
such that: E á r p é =
n
3 i = 1
ω i r
i $ r í [A2]
n
3 i = 1
ω i = 1 [A3]
50A simple example is provided by the following two securities and a coin toss. The first security pays out$10 on heads and nothing on tails. The second security pays out $10 on tails and nothing on heads. Each of thesesecurities in isolation has the potential to leave investors with nothing. However, by investing both of them,investors can be guaranteed a $10 return in each period. That is, each individual security is risky, but thecombination is riskless.
72
Let ρ i , j
= σ i , j á σ
i σ
j é − 1 be the correlation of security i’s returns with security j’s returns. In addition,
we shall make use of the following facts [A4, A5, A6].
(i) σ i , j
= σ j , i
(Covariance is symmetric) [A4]
(ii) σ i , i
= σ 2
i (The covariance of i with itself is the variance) [A5]
(iii) ρ i , i
= 1 (The correlation of i with itself is 1) [A6]
We may now simplify the minimum variance problem objective function [A1] as [A 1 N ].
minω
i
var á r p é = σ 2
p =
n
3 i = 1
n
3 j = 1
ω i ω
j ρ
i , j σ
i σ
j [A 1 N ]
The benefit of diversification can now be seen clearly as a consequence of the Markowitz
formulation of the portfolio selection or minimum variance problem. By including assets which are
less than perfectly correlated (assets for which ρ i , j
< 1 ) - and especially which are negatively
correlated - the variance of the overall portfolio is reduced.
Markowitz’ (1959) formulation contributed directly to the subsequent Capital Asset Pricing
Models (CAPM) of Sharpe (1964), Linter (1965), and Mossin (1966). This body of work
demonstrated that portfolio risk could be split into diversifiable (idiosyncratic) and nondiversifiable
(systematic) risks. Investors should only be compensated for bearing nondiversifiable risk. Thus
Markowitz diversification plays a central role in the hypothesized behavior of CAPM investors.
Investors, theoretically, should diversify away as much risk as possible. The amount of risk that can
be eliminated by diversification is E á σ 2
i é − E á σ i , j é : the average variance minus the average
covariance. To see this, assume for simplicity that all securities are held in equal proportion:
ω i = n − 1 ú i . Substituting in [A1], the variance of this portfolio is given by [A7].
σ 2
p =
n
3 i = 1 á
1
n é 2 σ 2
i +
n
3 i = 1
n
3 j = 1
i Ö j
á 1
n é á 1
n é σ i , j
[A7]
73
Note that the first term in [A7] now represents the average. Let E á σ 2
i é = n
3 i = 1 á
1
n é σ 2
i . The average in
[A7] can now be written more clearly as [A8].
n
3 i = 1 á
1
n é 2 σ 2
i = 1
n E á σ 2
i é [A8]
Similarly, the second term in the equation represents the average covariance between all the
securities in the portfolio. Because there are n 2 − n covariance terms, we have [A9]. First, however,
use the simplification in [A8].
E á σ i , j é =
n
3 i = 1
n
3 j = 1
σ i , j
n á n − 1 é = 1
n á n − 1 é
n
3 i = 1
n
3 j = 1
σ i , j
[A9]
n
3 i = 1
n
3 j = 1
i Ö j
á 1
n é á 1
n é σ i , j
= n − 1 n
E á σ i , j é [A10]
Back in the form of [A7], however, we now have [A11].
E á σ 2
p é = 1 n
E á σ 2
i é + n − 1 n
E á σ i , j é [A11]
If [A11] is written in a different way we can exactly split the portfolio variance into diversifiable and
nondiversifiable risk [A12].
E á σ 2
p é = 1 n
E á σ 2
i é − E á σ i , j é + E á σ
i , j é [A12]
As the size of the portfolio grows, the first term (the diversifiable risk) gets smaller and smaller.
Finally, at the limit where the size of the portfolio reaches the size of the market, we have [A13].
limn 6 4
E á σ 2
p é = E á σ i , j é [A13]
74
Even this example, however, only captures part of the effect of diversification. It is true, of
course, that simply adding less-than-perfectly-correlated securities to a portfolio will reduce the
overall variability of that portfolio, but the real power lies in the covariance of the assets one chooses
to include. Including assets which are less correlated with existing ones makes a greater difference
than including assets which are more strongly correlated. This is precisely the condition which is so
commonly ignored in practice.
The solution to the optimization problem in [A1, A2, A3] can be obtained through straight-
forward calculus. Incorporating the two constraints by adding the variables λ and µ , we form the
Lagrangian [A14]51:
ã = 1 2
n
3 i , j = 1
ω i ω
j σ
i , j − λ
ä
ã å å å
n
3 i = 1
ω i r
i − r í
ë
í ì ì ì − µ
ä
ã å å å
n
3 i = 1
ω i − 1
ë
í ì ì ì [A14]
From [A14] we obtain a system of n + 2 linear equations and n + 2 variables in ω 1 , ..., ω
n , λ , µ .
M ã M ω
1
= 0
!
M ã M ω
n
= 0
M ã M λ
= 0
M ã M µ
= 0
B
C
D
E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E
E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E n + 2 equations [A15]
The solution to the system in [A15] is the vector of weights (ω 1 , ..., ω
n ) constituting the minimum
variance portfolio.
51The 1/2 in front of the function is used as a convenience to simplify the resulting equations (by using thesymmetry of the covariance matrix). Obviously, it has no impact on the outcome of the optimization problem.
75
Samuelson (1967) showed that, under certain conditions52, the optimal portfolio for each
risk-averter would include some positive amount of each asset. That is, ω i > 0 ú i , for all risk-averse
investors. However, Hadar, Russell, and Seo (1977) demonstrated that, without other restrictive
conditions, there was no specific diversified portfolio which all risk-averters unanimously preferred.
They were only able to demonstrate that (a) given two identically distributed assets, any mixture is
preferred to a specialized portfolio and (b) given a symmetric joint distribution of individual assets,
every risk-averter should prefer an equal allocation of each asset to any other weighting. Extending
such results, Hadar, Russell, and Seo (1977) were able to prove that even when risk averters
unanimously judge one asset superior by second-order stochastic dominance (SSD) to all others,
they will still prefer certain diversified portfolios (the diversified portfolios are SSD-preferred to all
specialized portfolios). This is strong evidence of the power of diversification.
Adjustments to the Markowitz Portfolio Selection Model
The Markowitz model allows - in fact, requires - unlimited short selling and unlimited
borrowing and lending. In many cases, especially those involving individual investors, this is not
practical. The government regulates the extent to which individuals may engage in short-selling
transactions and other leveraged investments. As a result, a model which restricts short sales was
seen as more realistic. The addition of short-sale constraints is important because in portfolios
which are optimal given such constraints, many assets typically have zero weights (Elton and
Gruber, 1995).
The short-selling restriction is made by incorporating an additional series of constraints
[A16] into the standard problem [A1, A2, A3].
ω i $ θ
i ú i = 1 , ..., n [A16]
θ is typically set to zero, but could be set to any number to represent regulatory or preferential
limitations on investment in specific assets. Unfortunately, the addition of such constraints
transforms the simple linear form of [A1, A2, A3] into a quadratic form and thus a more difficult
optimization problem (Wagner, 1975; Hillier and Liebermann, 1980). Despite being inherently
52The conditions were that the assets had to be independently distributed, have equal means, and havepositive finite variances.
76
more difficult than linear programming problems, today there are many different algorithms
available for solving such problems. The earliest of these was Wolfe’s (1959) generalized simplex
method. Wolfe’s algorithm solves the Kuhn-Tucker equations by using the standard linear
programming simplex algorithm with additional checks to determine whether complimentary
slackness is maintained at each pivot. If a normal pivot would violate the complementary slackness
conditions, the next best point is brought in and the process is iterated. Numerous faster methods
(e.g., active set methods, adaptive grid search, simulated annealing, etc.) have also been developed
and large-scale quadratic programs (and even most MINLP problems) are no longer a challenge.
Still, short-sale constraints are not the only restrictions frequently added to portfolio
selection problems. In practice, numerous other constraints are used, making the problems far more
complex. These constraints often include ranges for the asset weights and restrictions on the lot
sizes of assets (e.g., consider single-family homes as an asset class: an investor must purchase an
integer quantity as fractional homes are infeasible). Green and Hollifield (1992) note that, in
practice, mean-variance methods are commonly implemented with extensive sets of constraints that
enforce what portfolio managers feel “diversification” should look like. Often, constraints are put
into place in order to satisfy investor demands not considered in the Markowitz framework. For
example, preference for a constant income stream from a portfolio may place a lower bound on the
expected yield of the portfolio assets. Markowitz optimization only takes total returns into account.
When short-sale constraints are considered, the important question of whether or not mean-
variance efficient portfolios can exist arises. Sharpe (1964) claimed that the market portfolio (which
is always positively-weighted) would be maximally mean-variance efficient. However, whenever
mean-variance efficient portfolios are generated from historical data, they are almost never
positively-weighted. Best and Grauer (1992) address this important paradox. They are able to show
that when positively-weighted minimum variance portfolios exist, they occupy a very small segment
of the (weight-unconstrained) efficient frontier. More importantly, as the size of the universe of
possible assets increases, the positively-weighted segment shrinks to a single point. Stated another
way, the more assets an investor has to consider, the less likely it is that the investor arrives at a
portfolio which is both positively weighted and mean-variance efficient. Further, Best and Grauer
(1992) demonstrate that the existence of the positively-weighted minimum variance portfolio is
extremely sensitive to small perturbations in mean returns. Their conclusion (ibid., p. 535) is that it
77
is “highly unlikely that the market portfolio, or any other positively weighted portfolio . . . could be
mean-variance efficient unless the security market line relationship holds as a virtual identity.”53
The point is that what started out as a simple optimization problem frequently becomes very
complex in practice. Further, this complexity is often added in order to achieve the appearance of
diversification - or, more correctly, what investors perceive as diversification. Not only does this
result54 in the selection of sub-optimal portfolios (in the Markowitz sense), it makes for a vastly
more complicated process of portfolio selection. In many cases, the process loses the strong
intuition of the traditional (but basic) formulation.
53The security market line represents the hypothesized CAPM relationship for equilibrium asset returns inmean return-beta space: r = r
f + β á r
m − r
f é . β measures the relative marginal contribution of an asset to the risk of the
market portfolio m. Intuitively then, the security market line illustrates the contribution of an individual asset to therisk-return tradeoff for an efficient portfolio (the market portfolio is guaranteed to be efficient as a condition of assetmarket equilibrium).
54Or, most probably result, in the Best and Grauer (1992) sense.
78
REFERENCES
Anderson, J. The Power of Diversification. Black Enterprise (December, 1998).
Antanasov, M. Why Owning More Funds Won’t Make You Merrier. Fortune (April 13, 1998).
Avery, R., G. Elliehausen, G. Canner, and T. Gustafson. Survey of Consumer Finances, 1983. Federal Reserve Bulletin (September, 1984a).
Avery, R., G. Elliehausen, G. Canner, and T. Gustafson. Survey of Consumer Finances, 1983: ASecond Report. Federal Reserve Bulletin (December, 1984b).
Avery, R., and G. Elliehausen. Financial Characteristics of High-Income Families. Federal ReserveBulletin (March, 1986).
Baker, H., and J. Haslem. Information Needs of Individual Investors. Journal of Accountancy(November, 1973).
Baker, H., and J. Haslem. Toward the Development of Client-Specified Valuation Models. Journalof Finance 29 (1974).
Barro, R., and X. Sala-i-Martin. Convergence. Journal of Political Economy 100 (1992).
Bawa, V., S. Brown, and R. Klein. Estimation Risk and Optimal Portfolio Choice . (New York, NY:North-Holland, 1979).
Belec, P. Time to Stuff Money into Bed Mattresses? Reuters Online Headlines. April 10, 1999.
Benartzi, S., and R. Thaler. (1998). Illusionary Diversification and its Implications for the U.S. andChilean Retirement Systems. Working Paper, University of California, Los Angeles.
Best, M., and R. Grauer. Positively Weighted Minimum-Variance Portfolios and the Structure ofAsset Expected Returns. Journal of Financial and Quantitative Analysis 27 (1992).
Black, F. Capital Market Equilibrium with Restricted Borrowing. Journal of Business 45 (1972).
Blume, M., and I. Friend. The Asset Structure of Individual Portfolios and Some Implications forUtility Functions. Journal of Finance 30 (1975).
Blume, M., D. Keim, and S. Patel. Returns and Volatility of Low-Grade Bonds: 1977-1989.Journal of Finance 46 (1991).
Bodie, Z., A. Kane, and A. Marcus. Investments. (Burr Ridge, IL: Irwin, 1993).
Brennan, M. The Optimal Number of Securities in a Risky Portfolio When There are Fixed Costsof Transacting: Theory and Some Empirical Results. Journal of Financial and QuantitativeAnalysis 10 (1975).
Bromiley, P., and S. Curley. Individual Differences in Risk Taking. In J. F. Yates, ed., Risk-TakingBehavior (New York, NY: Wiley, 1992).
79
Canner, N., N. Mankiw, and D. Weil. An Asset Allocation Puzzle. American Economic Review 87(1997).
Dawes, R. A Message from Psychologists to Economists: Mere Predictability Doesn’t Matter Likeit Should (Without a Good Story Appended to it). Journal of Economic Behavior andOrganization 39 (1999).
De Bondt, W. Betting on Trends: Intuitive Forecasts of Financial Risk and Return. InternationalJournal of Forecasting 9 (1993).
De Bondt, W. A Portrait of the Individual Investor. European Economic Review 42 (1998).
Durlauf, S., and P. Johnson. (1992). Local Versus Global Convergence Across NationalEconomies. NBER Working Paper 3996.
Elton, E., and M. Gruber. Modern Portfolio Theory and Investment Analysis, Fifth Edition . (NewYork, NY: John Wiley, 1995.
Elton, E., and M. Gruber. The Rationality of Asset Allocation Recommendations. Journal ofFinancial and Quantitative Analysis 35 (2000).
Elton, E., M. Gruber, and M. Padberg. Simple Criteria for Optimal Portfolio Selection. Journal ofFinance 31 (1976).
Elton, E., M. Gruber, and M. Padberg. Simple Rules for Optimal Portfolio Selection: The Multi-Group Case. Journal of Financial and Quantitative Analysis 12 (1977).
Evans, J., and S. Archer. Diversification and the Reduction of Dispersion: An Empirical Analysis.Journal of Finance 23 (1968).
Fama, E. The Behavior of Stock Market Prices. Journal of Business 38 (1965a).
Fama, E. Portfolio Analysis in a Stable Paretian Market. Management Science 11 (1965b).
Firstenberg, P., S. Ross, and R. Zisler. Real Estate: The Whole Story. Journal of PortfolioManagement (Spring, 1988).
Fisher, K., and M. Statman. The Mean-Variance-Optimization Puzzle: Security Portfolios and FoodPortfolios. Financial Analysts Journal (July/August, 1997).
Fischhoff, B., P. Slovic, and S. Lichtenstein. Fault Trees: Sensitivity of Estimated FailureProbabilities to Problem Representation. Journal of Experimental Psychology: HumanPerceptions and Performance 4 (1978).
Fischhoff, B. Managing Risk Perceptions. Issues in Science and Technology 2 (1985).
French, K., and J. Poterba. Investor Diversification and International Equity Markets. AmericanEconomic Review 81 (1991).
Gooding, A. Quantification of Investors’ Perceptions of Common Stocks: Risk and ReturnDimensions. Journal of Finance 30 (1975).
80
Graham, B., and D. Dodd. Security Analysis. (New York, NY: McGraw-Hill, 1934).
Green, R., and B. Hollifield. When Will Mean-Variance Efficient Portfolios Be Well Diversified?Journal of Finance 47 (1992).
Grinold, R., and R. Kahn. Active Portfolio Management. (Chicago, IL: Irwin, 1995).
Grubel, H. Internationally Diversified Portfolios: Welfare Gains and Capital Flows. AmericanEconomic Review 58 (1968).
Hadar, J., and W. Russell. Rules for Ordering Uncertain Prospects. American Economic Review 59(1969).
Hadar, J., W. Russell, and T. Seo. Gains From Diversification. Review of Economic Studies 44(2)(1977).
Hanna, S., and P. Chen. Optimal Portfolios: An Expected Utility/Simulation Approach.Proceedings of the Academy of Financial Services (1995).
Harlow, W., and R. Rao. Asset Pricing in a Generalized Mean-Lower Partial Moment Framework:Theory and Evidence. Journal of Financial and Quantitative Analysis 24 (1989).
Hays, K. Are You Overstocked? Working Woman (June, 1998).
Hillier, F., and G. Liebermann. Introduction to Operations Research, Third Edition . (SanFrancisco, CA: Holden-Day, 1980).
Hirshleifer, J., and J. Riley. The Analytics of Uncertainty and Information. (New York, NY:Cambridge University Press, 1992).
Huang, C., and R. Litzenberger. Foundations for Financial Economics . (Englewood Cliffs, NJ:Prentice-Hall, 1988).
Hulbert, M. Hedge Your Risks: Allocate. Forbes (June 15, 1998).
Jacob, N. A Limited-Diversification Portfolio Selection Model for the Small Investor. Journal ofFinance 29 (1974).
Kahneman, D., and A. Tversky. Prospect Theory: An Analysis of Decision Under Risk.Econometrica 47 (1979).
Keim, D., and A. Madhavan. Anatomy of the Trading Process: Empirical Evidence on the Behaviorof Institutional Traders. Journal of Financial Economics 37 (1995).
Keim, D., and A. Madhavan. The Upstairs Market for Large-Block Transactions: Analysis andMeasurement of Price Effects. Review of Financial Studies 9 (1996).
Kennickell, A., and J. Shack-Marquez. Changes in Family Finances from 1983 to 1989: Evidencefrom the Survey of Consumer Finances. Federal Reserve Bulletin (January, 1992).
81
Kennickell, A., and M. Starr-McCluer. Changes in Family Finances from 1989 to 1992: Evidencefrom the Survey of Consumer Finances. Federal Reserve Bulletin (October, 1994).
Kennickell, A., M. Starr-McCluer, and A. Sundén. Family Finances in the U.S.: Recent Evidencefrom the Survey of Consumer Finances. Federal Reserve Bulletin (January, 1997).
King, M., and J. Leape. (1984). Wealth and Portfolio Composition: Theory and Evidence. WorkingPaper 1468, National Bureau of Economic Research.
Kraus, A., and R. Litzenberger. Skewness Preference and the Valuation of Risk Assets. Journal ofFinance 31 (1976).
Kraus, A., and H. Stoll. Price Impacts of Block Trading on the New York Stock Exchange. Journalof Finance 27 (1972).
Kroll, Y., H. Levy, and A. Rapoport. Experimental Tests of the Mean-Variance Model for PortfolioSelection. Organizational Behavior and Human Decision Processes 42 (1988).
Langer, E. The Illusion of Control. Journal of Personality and Social Psychology 32 (1975).
Lathrop, R. Perceived Variability. Journal of Experimental Psychology 73 (1967).
Lease, R., W. Lewellen, and G. Schlarbaum. The Individual Investor: Attributes and Attitudes.Journal of Finance: Papers and Proceedings 29 (1974).
Lintner, J. Security Prices, Risk, and Maximal Gains From Diversification. Journal of Finance 20(1965): 79-96.
Loewenstein, G., and D. Prelec. Preferences for Sequences of Outcomes. Psychological Review100 (1993).
Mandelbrot, B. The Variation of Certain Speculative Prices. Journal of Business 36 (1963).
Mandelbrot, B. The Variation of Some Other Speculative Prices. Journal of Business 40 (1967).
Markowitz, H. Portfolio Selection. Journal of Finance 7 (1952).
Markowitz, H. Portfolio Selection. (Cambridge, MA: Basil Blackwell, 1959).
Michaud, R. The Markowitz Optimization Enigma: Is ‘Optimized’ Optimal? Financial AnalystsJournal (January-February, 1989).
Michaud, R. Efficient Asset Management. (Boston, MA: Harvard Business School Press, 1998).
Money. How the Big Brains Invest at TIAA-CREF. Page 118 (January 1998).
Mossin, J. Equilibrium in a Capital Asset Market. Econometrica 34 (1966): 768-783.
Mossin, J. Theory of Financial Markets. (Englewood Cliffs, NJ: Prentice Hall, 1973).
82
Nagy, R., and R. Obenberger. Factors Influencing Individual Investor Behavior. Financial AnalystsJournal (July-August, 1994).
NYSE. (1999). Shareownership 1998. New York Stock Exchange document.
O’Barr, W., and J. Conley. Fortune and Folly: The Wealth and Power of Institutional Investing.(Homewood, IL: Business One Irwin, 1992).
O’Barr, W. (1998). The Cultural World of Economic Behavior. Comments at the Aspen Meetingof the Institute of Psychology and Markets. December, 1998.
Peavy, J., and M. Vaughn-Rauscher. Risk Management Through Diversification. Trusts andEstates 133 (September, 1994).
Poulton, E. Behavioral Decision Theory. (New York, NY: Cambridge University Press, 1994).
Projector, D. (1963) 1962 Survey of Financial Characteristics of Consumers. Federal Reserve BankBoard of Governors.
Samuelson, P. General Proof that Diversification Pays. Journal of Financial and QuantitativeAnalysis 2 (1967).
Saunders, A., and R. Woodward. Gains From International Portfolio Diversification: UK Evidence1971-1975. Journal of Business Finance and Accounting 4 (1979).
Sears, R., and G. Trennepohl. Investment Management. (Fort Worth, TX: Dryden Press, 1993).
Shapiro, H. The Rewards of Risk. Hemispheres (February, 1999).
Sharpe, W. Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.Journal of Finance 19 (1964): 425-442.
Sharpe, W. Mutual Fund Performance. Journal of Business 39 (1966).
Shefrin, H. Beyond Greed and Fear: Understanding Behavioral Finance and the Psychology ofInvesting. (Boston, MA: Harvard Business School Press, 2000).
Shefrin, H., and M. Statman. Behavioral Capital Asset Pricing Theory. Journal of Financial andQuantitative Analysis 29 (1994).
Simkowitz, M., and W. Beedles. Diversification in a Three-Moment World. Journal of Financialand Quantitative Analysis 13 (1978).
Slovic, P. Psychological Study of Human Judgment: Implications for Investment Decision Making.Journal of Finance 27 (1972).
Spragins, E. Investing Secrets of Idiots. Newsweek (April 5, 1999).
Statman, M. How Many Stocks Make a Diversified Portfolio? Journal of Financial andQuantitative Analysis 22 (1987).
83
Stigler, G. The Cost of Subsistence. Journal of Farm Economics 27 (1945).
Sutcliffe, C., and J. Board. Forced Diversification. Quarterly Review of Economics and Business28 (1988).
Taha, H. Operations Research: An Introduction, Sixth Edition . (Upper Saddle River, NJ: Prentice-Hall, 1997).
Taylor, A. (1996). Sources of Convergence in the Late Nineteenth Century. NBER Working Paper5806.
Thaler, R. Mental Accounting and Consumer Choice. Marketing Science 4 (1985).
Tversky, A., and D. Kahneman. The Framing of Decisions and the Psychology of Choice. Science211 (1981).
Tversky, A., and D. Kahneman. Advances in Prospect Theory: Cumulative Representation ofUncertainty. Journal of Risk and Uncertainty 5 (1992).
Wagner, H. Principles of Operations Research, Second Edition. (Englewood Cliffs, NJ: Prentice-Hall, 1975).
Wasik, J. The Investment Club Book. (New York, NY: Warner Books, 1995).
Watts, H., and J. Tobin. Consumer Expenditures and the Capital Account. In D. Hester and J.Tobin, eds., Studies of Portfolio Behavior. (New York, NY: Wiley, 1967).
Williamson, J. Globalization, Convergence, and History. Journal of Economic History 56 (1996).
Wolfe, P. The Simplex Method for Quadratic Programming. Econometrica 27 (1959).
Yates, J. F., and E. Stone. The Risk Construct. In J. F. Yates, ed., Risk-Taking Behavior . (NewYork, NY: Wiley, 1992).
Zweig, J. Diversification Pitfalls. Money (October, 1998).
84