PERSPECTIVES

The Early History of Portfolio Theory:

1600–1960
Harry M. Markowitz

D
iversification of investments was a selection, both as a possible hypothesis about actual
well-established practice long before I behavior and as a maxim for how investors ought to
published my paper on portfolio selection act. The article assumed that “beliefs” or projections
in 1952. For example, A. Wiesenberger’s about securities follow the same probability rules
annual reports in Investment Companies prior to that random variables obey. From this assumption,
1952 (beginning 1941) showed that these firms held it follows that (1) the expected return on the portfolio
large numbers of securities. They were neither the is a weighted average of the expected returns on
first to provide diversification for their customers individual securities and (2) the variance of return
(they were modeled on the investment trusts of on the portfolio is a particular function of the vari-
Scotland and England, which began in the middle ances of, and the covariances between, securities
of the 19th century), nor was diversification new and their weights in the portfolio.
then. In the Merchant of Venice, Shakespeare has the Markowitz (1952) distinguished between effi-
merchant Antonio say: cient and inefficient portfolios. Subsequently,
My ventures are not in one bottom trusted, someone aptly coined the phrase “efficient fron-
Nor to one place; nor is my whole estate tier” for what I referred to as the “set of efficient
Upon the fortune of this present year; mean–variance combinations.” I had proposed that
Therefore, my merchandise makes me not sad. means, variances, and covariances of securities be
Act I, Scene 1 estimated by a combination of statistical analysis
Clearly, Shakespeare not only knew about diversi- and security analyst judgment. From these esti-
fication but, at an intuitive level, understood cova- mates, the set of efficient mean–variance combina-
riance. tions can be derived and presented to the investor
What was lacking prior to 1952 was an ade- for choice of the desired risk–return combination. I
quate theory of investment that covered the effects used geometrical analyses of three- and
of diversification when risks are correlated, distin- four-security examples to illustrate properties of
guished between efficient and inefficient portfo- efficient sets, assuming nonnegative investments
lios, and analyzed risk–return trade-offs on the subject to a budget constraint. In particular, I
portfolio as a whole. This article traces the develop- showed in the 1952 article that the set of efficient
ment of portfolio theory in the 1950s (including the portfolios is piecewise linear (made up of con-
contributions of A.D. Roy, James Tobin, and me) nected straight lines) and the set of efficient mean–
and compares it with theory prior to 1950 (includ- variance combinations is piecewise parabolic.
ing the contributions of J.R. Hicks, J. Marschak, J.B. Roy also proposed making choices on the basis
Williams, and D.H. Leavens). of mean and variance of the portfolio as a whole.
Specifically, he proposed choosing the portfolio
Portfolio Theory: 1952 that maximizes portfolio (E – d)/σ, where d is a
On the basis of Markowitz (1952), I am often called fixed (for the analysis) disastrous return and σ is
standard deviation of return. Roy’s formula for the
the father of modern portfolio theory (MPT), but
variance of the portfolio, like the one I presented,
Roy (1952) can claim an equal share of this honor.
included the covariances of returns among securi-
This section summarizes the contributions of both.
ties. The chief differences between the Roy analysis
My 1952 article on portfolio selection proposed
and my analysis were that (1) mine required non-
expected (mean) return, E, and variance of return, V,
negative investments whereas Roy’s allowed the
of the portfolio as a whole as criteria for portfolio amount invested in any security to be positive or
negative and (2) I proposed allowing the investor
to choose a desired portfolio from the efficient fron-
Harry M. Markowitz is president of Harry Markowitz tier whereas Roy recommended choice of a specific
Company. portfolio.

Jully/August 1999 5
Financial Analysts Journal

Comparing the two articles, one might wonder The “General” Portfolio Selection Problem.
why I got a Nobel Prize for mine and Roy did not For the case in which one and only one feasible
for his. Perhaps the reason had to do with the portfolio minimizes variance among portfolios
differences described in the preceding paragraph, with any given feasible expected return, Marko-
but the more likely reason was visibility to the witz (1952) illustrated that the set of efficient port-
Nobel Committee in 1990. Roy’s 1952 article was folios is piecewise linear. It may be traced out by
his first and last article in finance. He made this one starting with the unique point (portfolio) with min-
tremendous contribution and then disappeared imum feasible variance, moving in a straight line
from the field, whereas I wrote two books from this point, then perhaps, after some distance,
(Markowitz 1959; Markowitz 1987) and an assort- moving along a different straight line, and so on,
ment of articles in the field. Thus, by 1990, I was still until the efficient portfolio with maximum
active and Roy may have vanished from the Nobel expected return is reached.1 Note that we are not
Committee’s radar screen. discussing here the shape of efficient mean–
variance combinations or the shape of efficient
Problems with Markowitz (1952). I am temp- mean–standard deviation combinations. Rather,
ted to include a disclaimer when I send requested we are discussing the shape of the set of efficient
copies of Markowitz (1952) that warns the reader portfolios in “portfolio space.”
that the 1952 piece should be considered only a
The set of portfolios described in the preceding
historical document—not a reflection of my current
paragraph is not a piecewise linear approximation to
views about portfolio theory. There are at least four
the problem; rather, the exact solution is itself
reasons for such a warning. The first two are two
technical errors described in this section. A third is piecewise linear. The points (portfolios) at which
that, although the article noted that the same port- the successive linear pieces meet are called “corner
folios that minimize standard deviation for given E portfolios” because the efficient set turns a corner
also minimize variance for given E, it failed to point and heads in a new direction at each such point.
out that standard deviation (rather than variance) The starting and ending points (with, respectively,
is the intuitively meaningful measure of disper- minimum variance and maximum mean) are also
sion. For example, “Tchebychev’s inequality” says called corner portfolios.
that 75 percent of any probability distribution lies Markowitz (1952) did not present the formulas
between the mean and ±2 standard deviations—not for the straight lines that make up the set of efficient
two variances. Finally, the most serious differences portfolios. These formulas were supplied in
between Markowitz (1952) and the views I now Markowitz (1956), but Markowitz (1956) solved a
hold concern questions about “why mean and vari- much more general problem than discussed in
ance?” and “mean and variance of what?”. The Markowitz (1952). A portfolio in Markowitz (1952)
views expressed in Markowitz (1952) were held by was considered feasible (“legitimate”) if it satisfied
me very briefly. Those expressed in Markowitz one equation (the budget constraint) and its values
(1959) have been held by me virtually unchanged (investments) were not negative. Markowitz (1956),
since about 1955. I will discuss these views in the however, solved the (single-period mean–variance)
section on Markowitz (1959). portfolio selection problem for a wide variety of
As for the technical errors: First, it has been possible feasible sets, including the Markowitz
known since Markowitz (1956) that variance, V, is (1952) and Roy feasible sets as special cases.
a strictly convex function of expected return among Specifically, Markowitz (1956) allowed the
efficient EV combinations. Markowitz (1952) portfolio analyst to designate none, some, or all
explained, correctly, that the curve is piecewise variables to be subject to nonnegativity constraints
parabolic. Figure 6 of Markowitz (1952) showed (as in Markowitz 1952) and the remaining variables
two such parabola segments meeting at a point. The to not be thus constrained (as in Roy). In addition
problem with the figure is that its parabolas meet to (or instead of) the budget constraint, the portfo-
in such a way that the resulting curve is not convex. lio analyst could specify zero, one, or more linear
This cannot happen. Second, Figure 1 in Markowitz equality constraints (sums or weighted sums of
(1952) was supposed to portray the set of all feasible variables required to equal some constant) and/or
EV combinations. In particular, it showed the “inef- linear inequality constraints (sums or weighted
ficient border,” with maximum V for a given E, as a sums of variables required to be no greater or no
concave function. This is also an error. Since my less than some constant). A portfolio analyst can set
1956 article, we know that the curve relating maxi- down a system of constraints of these kinds such
mum V for a given E is neither concave nor convex that no portfolio can meet all constraints. In this
(see Markowitz 1987, Chapter 10, for a description case, we say that the model is “infeasible.” Other-
of possibilities). wise, it is a “feasible model.”

6 Association for Investment Management and Research

 The Early History of Portfolio Theory

In addition to permitting any system of con- for more than three assets, the general approach has
straints, Markowitz (1956) made an assumption2 been to display qualitative results in terms of
that assured that if the model was feasible, then (as graphs” (p. 1851). I assume that at the time, Merton
in Markowitz 1952) there was a unique feasible had not read Markowitz (1956) or Appendix A of
portfolio that minimized variance among portfo- Markowitz (1959).
lios with any given feasible E.
Markowitz (1956) showed that the set of effi- Markowitz Portfolio Theory circa
cient portfolios is piecewise linear in the general
model, as in the special case of Markowitz (1952). 1959
Depending on the constraints imposed by the port- Markowitz (1959) was primarily written during the
folio analyst, one of the linear pieces of the efficient 1955–56 academic year while I was at the Cowles
set could extend without end in the direction of Foundation for Research in Economics at Yale at the
increasing E, as in the case of the Roy model. (Note invitation of Tobin. At the time, Tobin was already
that if the analysis contains 1,000 securities, the working on what was to become Tobin (1958),
lines we are discussing here are straight lines in which is discussed in the next section.
1,000-dimensional “portfolio space.” These lines I had left the University of Chicago for the
may be hard to visualize and impossible to draw, RAND Corporation in 1951; my coursework was
but they are not hard to work with algebraically.) finished, but my dissertation (on portfolio theory)
Markowitz (1956) presented a computing pro- was still to be written. My RAND work had nothing
cedure, the “critical line algorithm,” that computes to do with portfolio theory. So, my stay at the
each corner portfolio in turn and the efficient line Cowles Foundation on leave from RAND provided
segment between them, perhaps ending with an an extended period when I could work exclusively
efficient line “segment” on which feasible E on, as well as write about, portfolio theory. The
increases without end. The formulas for the effi- following subsections summarize the principal
cient line segments are all of the same pattern. ways in which my views on portfolio theory
Along a given “critical line,” some of the variables evolved during this period, as expressed in
that are required to be nonnegative are said to be Markowitz (1959).
OUT and are set to zero; the others are said to be
IN and are free to vary. Variables not constrained A Still More General Mean–Variance
to be nonnegative are always IN. On the critical Analysis. The central focus of Markowitz (1959)
line, some inequalities are called SLACK and are was to explain portfolio theory to a reader who
ignored; the others are BINDING and are treated lacked advanced mathematics. The first four chap-
(in the formula for the particular critical line) as if ters introduced and illustrated mean–variance
they were equalities. With its particular combina- analysis, defined the concepts of mean, variance,
tion of BINDING constraints and IN variables, the and covariance, and derived the formulas for the
formula for the critical line is the same as if the mean and variance of a portfolio. Chapter 7 defined
problem were to minimize V for various E subject mean–variance efficiency and presented a geomet-
to only equality constraints. In effect, OUT vari- ric analysis of efficient sets, much like Markowitz
ables and SLACK constraints are deleted from the (1952) but without the two errors noted previously.
problem. Chapter 8 introduced the reader to some matrix
At each step, the algorithm uses the formula notation and illustrated the critical line algorithm
for the current critical line for easy determination in terms of a numerical example.
of the next corner portfolio. The next critical line, The proof that the critical line algorithm pro-
which the current critical line meets at the corner, duces the desired result was presented in Appen-
has the same IN variables and BINDING con- dix A of Markowitz (1959). Here, the result was
straints as the current line except for one of the more general than that in Markowitz (1956). The
following—one variable moves from OUT to IN or result in Markowitz (1956) made an assumption
moves from IN to OUT or one constraint moves sufficient to assure that a unique feasible portfolio
from BINDING to SLACK or from SLACK to would minimize variance for any given E.
BINDING. This similarity between successive crit- Markowitz (1959) made no such assumption;
ical lines greatly facilitates the solution of one line rather, it demonstrated that the critical line algo-
when given the solution of the preceding critical rithm will work for any covariance matrix. The
line.3 reason it works is as follows: Recall that the equa-
Merton (1972) said, “The characteristics of the tions for a critical line depend on which variables
mean–variance efficient portfolio frontier have are IN and which are OUT. Appendix A showed
been discussed at length in the literature. However, that each IN set encountered in tracing out the

July/August 1999 7
Financial Analysts Journal

efficient frontier is such that the associated equa- tive difference in the efficacy of diversification
tions for the critical line are solvable.4 depending on whether one assumes correlated or
uncorrelated returns.
Models of Covariance. Markowitz (1959, pp.
96–101) argued that analysis of a large portfolio Semideviation. Semivariance is defined like
consisting of many different assets has too many variance (as an expected squared deviation from
covariances for a security analysis team to carefully something) except that it counts only deviations
consider them individually, but such a team can below some value. This value may be the mean of
carefully consider and estimate the parameters of a the distribution or some fixed value, such as zero
model of covariance. This point was illustrated in return. Semideviation is the square root of semi-
terms of what is now called a single-index or variance. Chapter 9 of Markowitz (1959) defined
one-factor (linear) model. The 1959 discussion semivariance and presented a three-security geo-
briefly noted the possibility of a more complex metric analysis illustrating how the critical line
model—perhaps involving multiple indexes, non- algorithm can be modified to trace out mean–
linear relationships, or distributions that vary semideviation-efficient sets. Appendix A pre-
through time. sented the formal description of this modification
Markowitz (1959) presented no empirical anal- for any number of securities and a proof that it
ysis of the ability of particular models to represent works.
the real covariance matrix (as in Sharpe 1963,
Cohen and Pogue 1967, Elton and Gruber 1973, or Mean and Variance of What? Why Mean and
Rosenberg 1974), and I did not yet realize how a Variance? The basic ideas of Markowitz (1952)
(linear) factor model could be used to simplify the came to me sometime in 1950 while I was reading
computation of critical lines, as would be done in Williams (1938) in the Business School library at the
Sharpe (1963) and in Cohen and Pogue. University of Chicago. I was considering applying
mathematical or econometric techniques to the
The Law of the Average Covariance. Chap- stock market for my Ph.D. dissertation for the Eco-
ter 5 of Markowitz (1959) considered, among other nomics Department. I had not taken any finance
things, what happens to the variance of an equally courses, nor did I own any securities, but I had
weighted portfolio as the number of investments recently read Graham and Dodd (1934), had exam-
increases. It showed that the existence of correlated ined Wiesenberger (circa 1950), and was now read-
returns has major implications for the efficacy of ing Williams.
diversification. With uncorrelated returns, portfo- Williams asserted that the value of a stock is
lio risk approaches zero as diversification the expected present value of its future dividends.
increases. With correlated returns, even with My thought process went as follows: If an investor
unlimited diversification, risk can remain substan- is only interested in some kind of expected value
tial. Specifically, as the number of stocks increases, for securities, he/she must be only interested in
the variance of an equally weighted portfolio that expected value for the portfolio, but the maxi-
approaches the “average covariance” (i.e., portfolio mization of an expected value of a portfolio (subject
variance approaches the number you get by adding to a budget constraint in nonnegative investments)
up all covariances and then dividing by the number does not imply the desirability of diversification.
of them). I now refer to this as the “law of the Diversification makes sense as well as being com-
average covariance.” mon practice. What was missing from the analysis,
For example, if all securities had the same vari- I thought, was a measure of risk. Standard devia-
ance Vs and every pair of securities (other than the tion or variance came to mind. On examining the
security with itself) had the same correlation coef- formula for the variance of a weighted sum of
ficient ρ, the average covariance would be ρVs and random variables (found in Uspensky 1937 on the
portfolio variance would approach ρVs; therefore, library shelf), I was elated to see the way covari-
portfolio standard deviation would approach ances entered. Clearly, effective diversification
ρσ s . If the correlation coefficient that all pairs required avoiding securities with high covariance.
shared was, for example, 0.25, then the standard Dealing with two quantities—mean and
deviation of the portfolio would approach 0.5 times variance—and being an economics student, I natu-
the standard deviation of a single security. In this rally drew a trade-off curve. Being, more specifi-
case, investing in an unlimited number of securities cally, a student of T.C. Koopmans (see Koopmans
would result in a portfolio whose standard devia- 1951), I labeled dominated EV combinations “inef-
tion was 50 percent as great as that of a completely ficient” and undominated ones “efficient.”
undiversified portfolio. Clearly, there is a qualita- The Markowitz (1952) position on the ques-

8 Association for Investment Management and Research

 The Early History of Portfolio Theory

tions used as the heading for this subsection dif- Chapter 13 applied the theory of rational
fered little from my initial thoughts while reading behavior—which was developed by John von Neu-
Williams. Markowitz (1952) started by rejecting the mann and Oskar Morgenstern (1944), Leonard J.
rule that the “investor does (or should) maximize Savage (1954), Richard Bellman (1957), and others,
the discounted . . . [expected] value of future and was reviewed in Chapters 10 through 12—to
returns,” both as a hypothesis about actual behav- the problem of how to invest. It began with a
ior and as a maxim for recommended behavior, many-period consumption–investment game and
because it “fails to imply diversification no matter made enough assumptions to assure that the
how the anticipated returns are formed.” Before dynamic programming solution to the game as a
presenting the mean–variance rule, Markowitz whole would consist of maximizing a sequence of
(1952) said: single-period “derived” utility functions that
It will be convenient at this point to consider a depended only on end-of-period wealth. Chapter
static model. Instead of speaking of the time 13 then asked whether knowledge of the mean and
series of returns on the ith security (ri,1, ri,2, . . ., variance of a return distribution allows one to esti-
ri,t, . . .) we will speak of “the flow of returns” mate fairly well the distribution’s expected utility.
(ri) from the ith security. The flow of returns The analysis here did not assume either normally
distributed returns or a quadratic utility function
from the portfolio as a whole is R = ∑ Xi ri . (as in Tobin 1958). It did consider the robustness of
(pp. 45–46)
quadratic approximations to utility functions. In
The flow of returns concept is not heard from after other words, if you know the mean and variance of
this point. Shortly, Markowitz (1952) introduced a distribution, can you approximate its expected
“elementary concepts and results of mathematical utility? See also Levy and Markowitz (1979). Fur-
statistics,” including the mean and variance of a thermore, Chapter 13 considered what kinds of
sum of random variables. “The return (R) on the approximations to expected utility are implied by
portfolio as a whole is a weighted sum of random other measures of risk.
variables (where the investor can choose the The last six pages of the chapter sketched how
weights).” From this point forward, Markowitz one could or might (“could” in the easy cases,
(1952) was primarily concerned with how to choose “might” in the hard cases) incorporate into a formal
the weights Xi so that portfolios will be mean– portfolio analysis considerations such as (1) con-
variance efficient. sumer durables, (2) nonportfolio sources of income,
Markowitz (1952) stated that its “chief limita- (3) changing probability distributions, (4) illiquidi-
tions” are that “(1) we do not derive our results ties, and (5) taxes. As compared with later analyses,
analytically for the n-security case; . . . (2) we the Chapter 13 consumption–investment game was
assume static probability beliefs.” This work in discrete time rather than continuous time (as in
expressed the intention of removing these limita- Merton 1969), did not reflect the discovery of
tions in the future. Markowitz (1956) and Appendix myopic utility functions (as did Mossin 1968 and
A of Markowitz (1959) addressed the first issue, Samuelson 1969), and did not consider the behavior
and Chapter 13 of Markowitz (1959) addressed the of a market populated by consumers/investors
second issue. playing this game. Its objective was to provide a
Chapters 10–12 of Markowitz (1959) reviewed theoretical foundation for portfolio analysis as a
the theory of rational decision making under risk practical way to approximately maximize the
and uncertainty. Chapter 10 was concerned with derived utility function of a rational investor.
rational decision making in a single period with
known odds; Chapter 11 reviewed many-period Tobin (1958)
optimizing behavior (again, with known odds);
Tobin was concerned with the demand for money
Chapter 12 considered single- or many-period
as distinguished from other “monetary assets.”
rational behavior when the odds might be
Monetary assets, including cash, were defined by
unknown. The introduction in Chapter 10 empha-
Tobin as “marketable, fixed in money value, free of
sized that the theory reviewed there applies to an
default risk.” Tobin stated:
idealized rational decision maker with limited
Liquidity preference theory takes as given the
information but unlimited computing powers and choices determining how much wealth is to be
is not necessarily a hypothesis about actual human invested in monetary assets and concerns itself
behavior. This position contrasts with Markowitz with the allocation of these amounts among
(1952), which offered the mean–variance rule both cash and alternative monetary assets. (p. 66)
as a hypothesis about actual behavior and as a Tobin assumed that the investor seeks a mean–
maxim for recommended behavior. variance-efficient combination of monetary assets.

July/August 1999 9
Financial Analysts Journal

He justified the use of expected return and stan- for economic theory, mainly comparative stat-
dard deviation as criteria on either of two bases: ics, that can be derived from assuming that
Utility functions are quadratic, or probability dis- investors do in fact follow such rules. (p. 85,
tributions are from some two-parameter family of Note 1)
return distributions. To this extent, at least, the focus of Sharpe (1964) is
Much of Tobin’s article analyzed the demand the same as that of Tobin. Tobin and Sharpe are also
for money when “consols”5 are the only other mon- similar in postulating a model with n risky and one
etary asset. The next-to-last section of the article riskless security. The principal differences between
was on “multiple alternatives to cash.” Here, Tobin the two are (1) a difference in assumption between
presented his seminal result now known as the their mathematical models and (2) the economic
Tobin Separation Theorem. Tobin assumed a port- phenomena concerning which the respective mod-
folio selection model with n risky assets and one els are asserted.
riskless asset, cash. Because these assets were mon- As for assumptions, Tobin assumed that one
etary assets, the risk was market risk, not default can invest (i.e., lend) at the risk-free rate. Sharpe
risk. Holdings had to be nonnegative. Borrowing assumed that the investor can either borrow or lend
was not permitted. Implicitly, Tobin assumed that at the same rate. (Tobin usually assumed that the
the covariance matrix for risky assets is nonsingu- rate is zero, but he noted that this assumption is not
lar (or he could have made the slightly more gen- essential.) This, perhaps seemingly small, difference
eral assumption of Markowitz 1956). Tobin showed between the two models makes for a substantial
that these premises imply that for a given set of difference in their conclusions. First, if investors can
means, variances, and covariances among efficient borrow or lend all they want at the risk-free rate
portfolios containing any cash at all, the propor- (and the covariance matrix among the n risky stocks
tions among risky stocks are always the same: is nonsingular), then all efficient portfolios consist
of one particular combination of risky assets, per-
. . . the proportionate composition of the
non-cash assets is independent of their aggre- haps plus borrowing or lending. The implication is
gate share of the investment balance. This fact that, in equilibrium, the market portfolio (plus bor-
makes it possible to describe the investor’s rowing or lending) is the only efficient portfolio. In
decisions as if there were a single non-cash the Tobin model, in contrast, if investors have het-
asset, a composite formed by combining the erogeneous risk tolerances—so some hold cash and
multitude of actual non-cash assets in fixed others do not—the market portfolio can be quite
proportions. (p. 84) inefficient, even when all investors have the same
The primary purpose of Tobin’s analysis was beliefs and all hold mean–variance-efficient portfo-
to provide an improved theory of the holding of lios (see Markowitz 1987, Chapter 12).
cash. He concluded that the preceding analysis Probably the most remarkable conclusion
. . . is a logically more satisfactory foundation Sharpe drew from his premises was that in CAPM
for liquidity preference than the Keynesian equilibrium, the expected return of each security is
theory. . . . Moreover, it has the empirical advan- linearly related to its beta and only its beta. This
tage of explaining diversification—the same conclusion is not necessarily true in the Tobin
individual holds both cash and “consols”—
model (see Markowitz 1987, Chapter 12).
while the Keynesian theory implies that each
investor will hold only one asset. (p. 85)
The second major difference between the two
works is that Sharpe postulated that his model
At a meeting with Tobin in attendance, I once
applied to all securities, indeed all “capital assets,”
referred to his 1958 article as the first capital asset
whereas Tobin postulated only that his model
pricing model (CAPM). Tobin declined the honor.
applied to “monetary assets.” In fact, Tobin
It is beyond the scope of this article, which has a
expressed doubts that cash should be considered
1960 cutoff, to detail the contributions of William
risk free:
Sharpe (1964), John Lintner (1965), Jan Mossin
It is for this reason that the present analysis has
(1966), and others in the development of capital
been deliberately limited . . . to choices among
asset pricing models. A comparison of the assump- monetary assets. Among these assets cash is
tions and conclusions of Tobin with those of Sharpe relatively riskless, even though in the wider
may, however, help locate Tobin in the develop- context of portfolio selection, the risk of
ment of today’s financial economics. changes in purchasing power, which all mone-
Tobin contrasted his interest to mine as fol- tary assets share, may be relevant to many
lows: investors.
Markowitz’s main interest is prescription of Between them, Tobin’s assumptions were more
rules of rational behavior for investors: the cautious; Sharpe’s revolutionized financial eco-
main concern of this paper is the implications nomics.

10 Association for Investment Management and Research

 The Early History of Portfolio Theory

Hicks (1935, 1962) frontier. In other words, Hicks presented what we

call the Tobin Separation Theorem.
The Hicks (1962) article on liquidity included the
Hicks also analyzed the efficient frontier
following paragraph:
beyond the point where the holding of cash goes to
It would obviously be convenient if we could
zero. In particular, he noted that as we go out along
take just one measure of “certainty”; the mea-
sure which would suggest itself, when thinking the frontier in the direction of increasing risk and
on these lines, is the standard deviation. The return, securities leave the efficient portfolio and
chooser would then be supposed to be making do not return. (This last point is not necessarily true
his choice between different total outcomes on with correlated returns.6)
the basis of mean value (or “expectation”) and Returning to the portion of the frontier that
standard deviation only. A quite simple theory contains cash, if the Hicks (1962) results are, in fact,
can be built up on that basis, and it yields few a formalization of those in Hicks (1935)—in the
conclusions that do not make perfectly good sense of transcribing into mathematics results that
sense. It may indeed be regarded as a straight- were previously described verbally—then the
forward generalisation of Keynesian Liquidity
Tobin Separation Theorem should properly be
Preference. We would be interpreting Liquidity
called the Hicks or Hicks–Tobin Separation Theo-
Preference as a willingness to sacrifice some-
thing in terms of mean value in order to dimin- rem. Let us examine Hicks (1935) to see if it did
ish the expected variance (of the whole anticipate Tobin as described in the appendix to
portfolio). Instead of looking simply at the sin- Hicks (1962).
gle choice between money and bonds, we could Within Hicks (1935), the topic of Section V is
introduce many sorts of securities and show the closest to that of “The Pure Theory of Portfolio
distribution between them determined on the Investment” in the appendix of Hicks (1962). Pre-
same principle. It all works out very nicely, ceding sections of Hicks (1935) discussed, among
being indeed no more than a formalisation of an other things, the need for an improved theory of
approach with which economists have been money and the desirability of building a theory of
familiar since 1936 (or perhaps I may say 1935). money along the same lines as the existing theory
[A footnote to the last sentence of this paragraph of value. They also discussed, among other things,
explained as follows:] Referring to my article, the relationship between the Hicks (1935) analysis
“A Suggestion for Simplifying the Theory of and that of Keynes as well as the existence of “fric-
Money.” Economica (February 1935). (p. 792)
tions,” such as “the cost of transferring assets from
The formalization was spelled out in a mathemati- one form to another.” In Section IV, Hicks (1935)
cal appendix to Hicks (1962) titled “The Pure The- introduced risk into his analysis. Specifically, he
ory of Portfolio Investment” and in a footnote on p. noted, “The risk-factor comes into our problem in
796 that presents an Eσ − efficient set diagram. two ways: First, as affecting the expected period of
The appendix presented a mathematical investment, and second, as affecting the expected
model that is almost exactly the Tobin model with net yield of investment” (p. 7). In a statement appli-
no reference to Tobin. The difference between the cable to both sources of risk, Hicks continued:
Hicks and Tobin models is that Hicks assumed that Where risk is present, the particular expectation
all correlations are zero whereas Tobin permitted of a riskless situation is replaced by a band of
any nonsingular covariance matrix. Specifically, possibilities, each of which is considered more
Hicks presented the general formula for portfolio or less probable. It is convenient to represent
these probabilities to oneself, in statistical fash-
variance written in terms of correlations, rather
ion, by a mean value, and some appropriate
than covariances, and then stated: measure of dispersion. (No single measure will
It can, I believe, be shown that the main prop- be wholly satisfactory, but here this difficulty
erties which I hope to demonstrate, remain may be overlooked.) (p. 8)
valid whatever the r’s; but I shall not attempt Hicks (1935) never designated standard deviation
to offer a general proof in this place. I shall
or any other specific measure as the measure he
simplify by assuming that the prospects of the
various investments are uncorrelated (rjk = 0
meant when speaking of risk. After discussing
uncertainty of the period of the investment, he
when k ≠ j): an assumption with which, in any
case, it is natural to begin. concluded Section IV thus:
To turn now to the other uncertainty—
In the discussion that followed, Hicks (1962)
uncertainty of the yield of investment. Here
derived the Tobin conclusion that among portfolios again we have a penumbra. . . . Indeed, without
that include cash, there is a linear relationship assuming this to be the normal case, it would
between portfolio mean and standard deviation be impossible to explain some of the most obvi-
and that the proportions among risky assets remain ous of the observed facts of the capital market.
constant along this linear portion of the efficient (p. 8)

July/August 1999 11
Financial Analysts Journal

The theory of investment that Hicks (1935) mulation to the analysis of portfolio selection
presented in Section V may be summarized as fol- in general is the shortest of steps, but one not
lows: taken by Marschak. (p. 14)
It is one of the peculiarities of risk that the G.M. Constantinides and A.G. Malliaris (1995)
total risk incurred when more than one risky described the role of Marschak (1938) as follows.
investment is undertaken does not bear any sim- The asset allocation decision was not ade-
ple relation to the risk involved in each of the quately addressed by neoclassical economists.
particular investments taken separately. In most . . . The methodology of deterministic calculus
cases, the “law of large numbers” comes into is adequate for the decision of maximizing a
play (quite how, cannot be discussed here). . . . consumer’s utility subject to a budget con-
Now, in a world where cost of investment straint. Portfolio selection involves making a
was negligible, everyone would be able to take decision under uncertainty. The probabilistic
considerable advantage of this sort of risk notions of expected return and risk become
reduction. By dividing up his capital into small very important. Neoclassical economists did
portions, and spreading his risks, he would be not have such a methodology available to
able to insure himself against any large total them. . . . An early and important attempt to do
risk on the whole amount. But in actuality, the that was made by Marschak (1938) who
cost of investment, making it definitely unprof- expressed preferences for investments by indif-
itable to invest less than a certain minimum ference curves in the mean–variance space. (pp.
amount in any particular direction, closes the 1–2)
possibility of risk reduction along these lines to An account of Marschak is, therefore, manda-
all those who do not possess the command over tory in a history of portfolio theory through 1960,
considerable quantities of capital. . . . if for no other reason than that these scholars
By investing only a proportion of total assets judged it to be important. On the other hand, I
in risky enterprises, and investing the remain- know of one authority who apparently did not
der in ways which are considered more safe, it think the article to be important for the develop-
will be possible for the individual to adjust his
ment of portfolio theory. My thesis supervisor was
whole risk situation to that which he most pre-
Marschak himself, and he never mentioned Mar-
fers, more closely than he could do by investing
schak (1938). When I expressed interest in applying
in any single enterprise. (pp. 9–10)
mathematical or econometric techniques to the
Hicks (1935) was a forerunner of Tobin in seek-
stock market, Marschak told me of Alfred Cowles
ing to explain the demand for money as a conse-
own interest in financial applications, resulting, for
quence of the investor’s desire for low risk as well
example, in Cowles 1939 work.7 Then, Marschak
as high return. Beyond that, there is little similarity sent me to Marshall Ketchum in the Business
between the two authors. Hicks (1935), unlike School at the University of Chicago for a reading
Tobin or the appendix to Hicks (1962), did not list in finance. This list included Williams (1938)
designate standard deviation or any other specific and, as I described, led to the day in the library
measure of dispersion as representing risk for the when my version of portfolio theory was born.
analysis; therefore, he could not show a formula Marschak kept track of my work, read my disser-
relating risk on the portfolio to risk on individual tation, but never mentioned his 1938 article.
assets. Hicks (1935) did not distinguish between So, which authority is correct concerning the
efficient and inefficient portfolios, contained no place of Marschak in the development of portfolio
drawing of an efficient frontier, and had no hint of theory? Like Hicks, Marschak sought to achieve a
any kind of theorem to the effect that all efficient better theory of money by integrating it with the
portfolios that include cash have the same propor- General Theory of Prices. In the introductory sec-
tions among risky assets. tion of the article, Marschak explained that
Thus, there is no reason why the theorem that to treat monetary problems and indeed, more
currently bears Tobin’s name should include any generally, problems of investment with the
other name. tools of a properly generalized Economic
Theory . . . requires, first, an extension of the
concept of human tastes: by taking into account
Marschak (1938) not only men’s aversion for waiting but also
Kenneth Arrow (1991) said of Marschak (1938): their desire for safety, and other traits of behav-
Jacob Marschak . . . made some efforts to con- iour not present in the world of perfect cer-
struct an ordinal theory of choice under uncer- tainty as postulated in the classical static
tainty. He assumed a preference ordering in the economics. Second, the production conditions,
space of parameters of probability dis- assumed hereto to be objectively given,
tributions—in the simplest case, the space of become, more realistically, mere subjective
the mean and the variance. . . . From this for- expectations of the investors—and all individ-

12 Association for Investment Management and Research

 The Early History of Portfolio Theory

uals are investors (in any but a timeless econ- ing Williams occurred in my reading early parts of
omy) just as all market transactions are the book. Later in the book, Williams observed that
investments. The problem is: to explain the the future dividends of a stock or the interest and
objective quantities of goods and claims held at principal of a bond may be uncertain. He said that,
any point of time, and the objective market
in this case, probabilities should be assigned to
prices at which they are exchanged, given the
various possible values of the security and the
subjective tastes and expectations of the indi-
viduals at this point of time. (p. 312) mean of these values used as the value of the secu-
In the next five sections, Marschak presented rity. Finally, he assured readers that by investing in
the equations of the standard economic analysis of sufficiently many securities, risk can be virtually
production, consumption, and price formation. eliminated. In particular, in the section titled
Section 7 dealt with choice when outcomes are “Uncertainly and the Premium for Risk” (starting
random. No new equations were introduced in this on p. 67 in the chapter on “Evaluation by the Rule
section. Rather, Marschak used the prior equations of Present Worth”), he used as an example an inves-
with new meanings: tor appraising a risky 20-year bond “bearing a 4 per
We may, then, use the previous formal setup if
cent coupon and selling at 40 to yield 12 per cent to
we reinterpret the notation: x, y . . . shall mean, maturity, even though the pure interest seems to be
not future yields, but parameters (e.g., only 4 per cent.” His remarks apply to any investor
moments and joint moments) of the who “cannot tell for sure” what the present worth
joint-frequency distribution of future yields. is of the dividends or of the interest and principal
Thus, x may be interpreted as the mathematical to be received:
expectation of first year’s meat consumption, y Whenever the value of a security is uncertain
may be its standard deviation, z may be the and has to be expressed in terms of probabil-
correlation coefficient between meat and salt ity, the correct value to choose is the mean
consumption in a given year, t may be the third value. . . . The customary way to find the value
moment of milk consumption in second year, of a risky security has always been to add a
etc. (p. 320) “premium for risk” to the pure interest rate,
Marschak noted that people usually like high mean and then use the sum as the interest rate for
and low standard deviation; also, “they like meat discounting future receipts. In the case of the
consumption to be accompanied by salt consump- bond under discussion, which at 40 would
tion” (i.e., z as well as x in the preceding quotation yield 12 per cent to maturity, the “premium
“are positive utilities” as opposed to standard devi- for risk” is 8 per cent when the pure interest
ation, y, which is “a disutility”). He noted that peo- rate is 4 per cent.
ple “like ‘long odds’ (i.e., high positive skewness of Strictly speaking, however, there is no risk
yields.” However, it “is sufficiently realistic . . . to in buying the bond in question if its price is
confine ourselves, for each yield, to two parameters right. Given adequate diversification, gains on
such purchases will offset losses, and a return
only: the mathematical expectation (‘lucrativity’)
at the pure interest rate will be obtained. Thus
and the coefficient of variation (‘risk’).”
the net risk turns out to be nil. To say that a
So, is Marschak’s article a forerunner of port- “premium for risk” is needed is really an ellip-
folio theory or not? Yes and no. It is not a step (say, tical way of saying that payment of the full face
beyond Hicks 1935) toward portfolio theory because value of interest and principal is not to be
it does not consider portfolios. The means, standard expected on the average.
deviations, and correlations of the analysis, includ- In my 1952 article, I said that Williams’s prescrip-
ing the means (and so on) of end products con- tion has the investor
sumed, appear directly in the utility and diversify his funds among all those securities
transformation functions with no analysis of how which give maximum expected return. The law
they combine to form moments of the investor’s of large numbers will insure that the actual yield
portfolio as a whole. On the other hand, Marschak’s of the portfolio will be almost the same as the
1938 work is a landmark on the road to a theory of expected yield. This rule is a special case of the
markets whose participants act under risk and expected returns–variance of returns rule. . . . It
uncertainty, as later developed in Tobin and the assumes that there is a portfolio which gives
CAPMs. It is the farthest advance of economics both maximum expected return and minimum
under risk and uncertainty prior to the publication variance, and it commends this portfolio to the
of von Neumann and Morgenstern (1944). investor.
This presumption, that the law of large
numbers applies to a portfolio of securities,
Williams (1938) cannot be accepted. The returns from securities
The episode reported previously in which I discov- are too intercorrelated. Diversification cannot
ered the rudiments of portfolio theory while read- eliminate all variance.

July/August 1999 13
Financial Analysts Journal

That is still my view. It should be noted, however, more conscientious student of this literature
that Williams’s “dividend discount model” remains informs us otherwise, that Leavens was correct that
one of the standard ways to estimate the security the majority discussed diversification in general
means needed for a mean–variance analyses (see terms and did “not clearly indicate why it is desir-
Farrell 1985). able.” Let us further assume that the financial ana-
lysts who did indicate why it is desirable did not
Leavens (1945) include covariance in their formal analyses and had
not developed the notion of an efficient frontier.
Lawrence Klein called my attention to an article on
Thus, we conclude our survey with Leavens as
the diversification of investments by Leavens, a
representative of finance theory’s analysis of risk as
former member of the Cowles Commission. Leav-
of 1945 and, presumably, until Roy and Markowitz
ens (1945) said:
in the 1950s.
An examination of some fifty books and articles
on investment that have appeared during the
last quarter of a century shows that most of The End of the Beginning
them refer to the desirability of diversification.
The majority, however, discuss it in general One day in 1960, having said what I had to say
terms and do not clearly indicate why it is about portfolio theory in my 1959 book, I was sit-
desirable. ting in my office at the RAND Corporation in Santa
Leavens illustrated the benefits of diversification Monica, California, working on something quite
on the assumption that risks are independent. different, when a young man presented himself at
However, the last paragraph of Leavens cautioned: my door, introduced himself as Bill Sharpe, and
The assumption, mentioned earlier, that each said that he also was employed at RAND and was
security is acted upon by independent causes, working toward a Ph.D. degree at UCLA. He was
is important, although it cannot always be fully looking for a thesis topic. His professor, Fred
met in practice. Diversification among compa- Weston, had reminded Sharpe of my 1952 article,
nies in one industry cannot protect against which they had covered in class, and suggested that
unfavorable factors that may affect the whole
he ask me for suggestions on a thesis topic. We
industry; additional diversification among
talked about the need for models of covariance.
industries is needed for that purpose. Nor can
diversification among industries protect This conversation started Sharpe out on the first of
against cyclical factors that may depress all his (ultimately many) lines of research, which
industries at the same time. resulted in Sharpe (1963).
Thus, Leavens understood intuitively, as did For all we know, the day Sharpe introduced
Shakespeare 350 years earlier, that some kind of himself to me at RAND could have been exactly 10
model of covariance is at work and that it is relevant years after the day I read Williams. On that day in
to the investment process. But he did not incorpo- 1960, there was no talk about the possibility of using
rate it into his formal analysis. portfolio theory to revolutionize the theory of
Leavens did not provide us with his reading financial markets, as done in Sharpe (1964), nor was
list of “some fifty books and articles.” This omission there any inkling of the flood of discoveries and
is fortunate because I am probably not prepared to applications, many by Sharpe himself, that were to
read them all and the reader is surely not ready to occur in investment theory and financial economics
read accounts of them. Let us assume, until some during the next four decades.

14 Association for Investment Management and Research

 The Early History of Portfolio Theory

Notes
1. Given the assumptions of Markowitz (1952), if more than ducing “slack variables” as in linear programming. The
one portfolio has maximum feasible E, only one of these critical line algorithm works even if the constraint matrix,
portfolios will be efficient, namely, the one with the smallest A, as well as the covariance matrix, C, is rank deficient. The
V. This one will be reached by the “tracing out” process critical line algorithm begins with George Dantzig’s (1963)
described. simplex algorithm to maximize E or determine that E is
2. The assumption was that V is strictly convex over the set of unbounded. The simplex algorithm introduces “dummy
feasible portfolios. This assumption is weaker than requir- slacks,” some of which remain in the critical line algorithm
ing the covariance matrix to be nonsingular. if A is rank deficient (see Markowitz 1987, Chapters 8 and
3. In the text, I am discussing the shape of efficient sets in 9). Historically, not only did I have great teachers at the
portfolio space. As observed in Markowitz (1952), the set of University of Chicago, including Jacob Marschak, T.C.
efficient EV combinations is piecewise parabolic, with each Koopmans, Milton Friedman, and L.J. Savage, but I was
line segment in portfolio space corresponding to a parabolic especially fortunate to have Dantzig as a mentor when I
segment in EV space. As discussed previously, Markowitz worked at RAND.
(1956) understood that successive parabolas meet in such a 5. Government bonds in Great Britain, originally issued in
way that efficient V as a function of E is strictly convex. 1751, that (similarly to an annuity) pay perpetual interest
Markowitz (1956) noted that typically there is no kink
and have no date of maturity.
where two successive efficient parabola segments meet: The
slope of the one parabola equals that of the other at the 6. See Chapter 11 in Markowitz (1987) for a three-security
corner portfolio where they meet. Markowitz (1956) did, example of risky assets in which an asset leaves and later
however, note the possibility of a kink in the efficient EV set reenters the efficient portfolio. Add cash to the analysis in
if a certain condition occurred, but the 1956 work did not such a way that the coming and going of the security
provide a numerical example of a problem containing such happens above the tangency of the line from (0, r0) to the
a kink. For numerical examples of problems with kinks in frontier. Perhaps, if you wish, add a constant to all expected
the efficient EV set, see Dybvig (1984) and Chapter 10 of returns, including r0, to assure that r0 ≥ 0.
Markowitz (1987). 7. The Cowles Commission for Research in Economics,
4. The equations in Markowitz (1956) also depended on which endowed by Alfred Cowles, was affiliated with the Univer-
inequalities were BINDING. Markowitz (1959) wrote ine- sity of Chicago at the time. Marschak was formerly its
qualities as equalities, without loss of generality, by intro- director. I was a student member.

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16 Association for Investment Management and Research