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A Portfolio of Nobel Laureates: Markowitz, Miller and Sharpe
Author(s): Hal Varian
Source: The Journal of Economic Perspectives, Vol. 7, No. 1 (Winter, 1993), pp. 159-169
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Journal of EconomicPerspectives- Volume7, Number1-Winter 1993-Pages 159-169
A Portfolio of Nobel Laureates:
Markowitz, Miller and Sharpe
Hal Varian
inance is one of the great success stories of quantitative economics. A
recent ad in The Economist for a “mathematical economist” described an
“excellent opportunity for numerate individual with background in
capital markets.” In today’s market, numeracy pays!
But it was not always so. According to Robert Merton (1990):
As recently as a generation ago, finance theory was still little more than a
collection of anecdotes, rules of thumb, and manipulations of accounting
data. The most sophisticated tool of analysis was discounted value and the
central intellectual controversy centered on whether to use present value
or internal rate of return to rank corporate investments. The subsequent
evolution from this conceptual potpourri to a rigorous economic theory
subjected to scientific empirical examination was, of course, the work of
many, but most observers would agree that Arrow, Debreu, Lintner,
Markowitz, Miller, Modigliani, Samuelson, Sharpe, and Tobin were the
early pioneers in this transformation.
Three of these pioneers of quantitative finance have now been justly honored:
Harry Markowitz, Merton Miller and William Sharpe received the Nobel Prize
in Economic Science in 1990.
From today’s perspective it is hard to understand what finance was like
before portfolio theory. Risk and return are such fundamental concepts of
finance courses that it is hard to realize that these were once a novelty. But
* Hal Varian is Reuben Kempf Professor of Economics and Professor of Finance,
Universityof Michigan, Ann Arbor, Michigan.
Journal of EconomicPerspectives
these esoteric theories of the last generation form the basic content of MBA
courses today.
The history of the quantitative revolution in finance has recently been
summarized in Bernstein (1992). Here I attempt to provide a very brief history
of this enterprise, drawing upon the work of Bernstein and the accounts of the
Nobel laureates in Markowitz (1991), Miller (1991) and Sharpe (1991). Readers
interested in more detailed accounts of the development of modern financial
theory should consult these works.
Harry Markowitz
Harry Markowitz was born in 1927 in Chicago. He attended the University
of Chicago and majored in economics. He found the subject appealing enough
to go on to graduate school and eventually arrived at the thesis stage. While
waiting to see Jacob Marschak he struck up a conversation with a stockbroker
who suggested that he might write a thesis about the stock market. Markowitz
was excited by this idea and started to read in the area.
One of his first books was The Theory of Investment Value by John Burr
Williams, (1938). Williams argued that the value of a stock should be the
present value of its dividends-which
was then a novel theory. Markowitz
quickly recognized the problem with this theory: future dividends are not
known for certain-they are random variables. This observation led Markowitz
to make the natural extension of the Williams’ theory: the value of a stock
should be the expectedpresent value of its dividend stream.
But if an investor wants to maximize the expected value of portfolio of
stocks he owns, then it is obvious that he should buy only one stock-the one
that has the highest expected return. To Markowitz, this was patently unrealistic. It was clear to him that investors must care not only about the expected
return of their wealth, but also about the risk. He was then naturally led to
examine the problem of finding the portfolio with the maximum expected
return for a given level of risk.
The fact that investors should care about both the risk and the return of
their investments is so commonplace today that it is hard to believe that this
view was not appreciated in 1952. Even Keynes (1939) said, “To suppose that
safety-first consists in having a small gamble in a large number of different
[companies] … strikes me as a travesty of investment policy.” Luckily, Keynes
was not held in high repute in Chicago, even in those days, and Markowitz was
not deterred from his investigations.
Markowitz posed the problem of minimizing the variance of a portfolio
taking as a constraint a required expected return. This way of posing the
problem contained two significant insights. First, Markowitz realized that the
mathematics could not pick out a single optimal portfolio, but rather could only
identify a set of efficient portfolios-the
set of portfolios that had the lowest
Hal Varian 161
Harry Markowitz
possible risk for each possible expected return. Secondly, Markowitz recognized
that the appropriate risk facing an investor was portfolio risk-how much his
entire portfolio of risky assets would fluctuate.
Today, we pose the problem of portfolio selection as a quadratic programming problem. The choice variables are the fractions of wealth invested in each
of the available risky assets, the quadratic objective function is the variance of
return on the resulting portfolio, and the linear constraint is that the expected
return of the portfolio achieve some target value. Variables may be subjected to
nonnegativity constraints or not, depending on whether short sales are feasible.
The first-order conditions for this quadratic programming problem require that the marginal increase in variance from investing a bit more in a
given asset should be proportional to the expected return of that asset. The key
insight that arises from this first-order condition is that the marginal increase in
variance depends on both the variance of a given asset’s return plus the
covariance of the asset return with all other asset returns in the portfolio.
Markowitz’s formulation of portfolio optimization leads quickly to the
fundamental point that the riskiness of a stock should not be measured just by
the variance of the stock, but also by the covariance. In fact, if a portfolio is
highly diversified, so that the amount invested in any given asset is “small,”
and the returns on the stocks are highly correlated, then most of the marginal
risk from increasing the fraction of a given asset in a portfolio is due to
this covariance effect.
This was, perhaps, the central insight of Markowitz’s contribution to
finance. But it is far from the end of the story. As every graduate student
knows, the first-order conditions are only the first step in solving an optimization problem. In 1952, linear programming was in its infancy and quadratic
programming was not widely known. Nevertheless, Markowitz succeeded in
developing practical methods to determine the “critical line” describing
Journal of EconomicPerspectives
mean-variance efficient portfolios. The initial work in his thesis was described in
two papers Markowitz (1952, 1956) and culminated in his classic book
(Markowitz, 1959).
When Markowitz defended his dissertation at the University of Chicago,
Milton Friedman gave him a hard time, arguing that portfolio theory was not a
part of economics, and therefore that Markowitz should not receive a Ph.D. in
economics. Markowitz (1991) says, ” . . . this point I am now willing to concede:
at the time I defended my dissertation, portfolio theory was not part of
Economics. But now it is.”
William Sharpe
Markowitz’s model of portfolio selection focused only on the choice of risky
assets. Tobin (1958), motivated by Keynes’ theory of liquidity preference,
extended the model to include a riskless asset. In doing so, he discovered a
surprising fact. The set of efficient risk-return combinations turned out to be a
straight line!
The logic of Tobin’s discovery can be seen with simple geometry. The
hyperbola in Figure 1 depicts the combination of mean returns and standard
deviation of returns that can be achieved by the various portfolios of risky
assets. Each set of risky assets will generate some such hyperbola depicting the
feasible combinations of risk and return.
The risk-free return has a standard deviation of zero, so it can be represented by a point on the vertical axis, (0, RO). Now make the following
geometric construction: draw a line through the point (0, RO) and rotate it
clockwise until it just touches the set of efficient portfolios. Call the point where
Figure 1
of risky
riskyand risk-freeassets
A _ E i:
f iE; iEi;i. ^
… T:X4…
0r .
Standard Deviation
A Portfolioof NobelLaureates:Markowitz,Millerand Sharpe 163
William Sharpe
it touches this line (o,,,
Now observe that every efficient portfolio consistof
asset can be achieved by combining only two
portfolios-one portfolio consisting only of the risk free asset, and one consisting of the portfolio that yields the risk-return combination (a.,,, Rm).
For example, if you want an expected return and standard deviation that is
halfway between (0, R0) and GYm,,,,
R.), just put half of your wealth in the
risk-free asset and half in the risky portfolio. Points to the right of the risky
portfolio can be achieved by leverage: borrowmoney at the rate R0 and invest it
in the risky portfolio.
Tobin’s discovery dramatically simplified portfolio selection: his analysis
showed the same portfolio of risky assets is appropriate for everyone. All that
varies is how much money you choose to put in risky assets and how much you
choose to put in the riskless asset. Each investor can limit his investment choices
to two “mutual funds:” a money market fund that invests only in the riskless
asset (e.g., Treasury bills) and another fund that invests only in the magical
portfolio that yields (o,,,,,R.).
But one still needs to determine just which stocks, and which proportions
of stocks, comprise the magic portfolio rn-and that is a difficult and costly
computation. The next contribution to portfolio theory was a simplified way to
perform this computation. William Sharpe was a doctoral student at UCLA,one
of the first students there to take courses in both economics and finance. When
it came time to write a thesis, Fred Weston suggested
that he talk with Harry
Markowitz, who was then at RAND. Markowitz became Sharpe’s unofficial
thesis advisor and put him to work trying to simplify the computational aspects
of portfolio theory.
Journal of EconomicPerspectives
Sharpe explored an approach now known as the “market model” or the
“single factor” model. It assumes that the return on each security is linearly
related to a single index, usually taken to be the return on some stock market
index such as the S & P500. Thus the (random) return on asset a at time t can
be written as
= C + bRmt + Eat
where Rmt is the return on the S&P 500, say, and Eat is an error term with
expected value of zero. In this equation c is the expected return of the asset if
the market is expected to have a zero return, while the parameter b measures
the sensitivity of the asset to “market conditions.” A stock that has b = 1 is just
as risky as the market index: if the S&P index increases by 10 percent in a
given year, we would expect this stock to increase by c + 10 percent. A stock
that has b < 1 is less volatile than the market index, while one with b > 1 is
more volatile. Sharpe’s motivation in formulating this model was empirical:
most stocks move together, most of the time. Hence, it is natural to think that a
single factor (or small number of factors) determines most of the cross-sectional
variation in returns.
This linear relationship can easily be estimated by ordinary least squares,
and the estimated coefficients can be used to construct covariances, which, in
turn, can be used to construct optimal portfolios. Sharpe’s approach reduced
the dimensionality of the portfolio problem dramatically and made it much
simpler to compute efficient portfolios. Problems that took 33 minutes of
computer time using the Markowitz model took only 30 seconds with Sharpe’s
model. This work led to Sharpe (1963) and a Ph.D. thesis.
Later, while teaching at the University of Washington, Sharpe turned his
attention to equilibrium theory in capital markets. Up until this point portfolio
theory was a theory of individual behavior-how an individual might choose
his investments given the set of available assets.
What would happen, Sharpe asked, if everyone behaved like Markowitz
portfolio optimizers? Tobin had shown that everyone would hold the same
portfolio of risky assets. If Mr. A had 5 percent of his stock market wealth
invested in IBM, then Ms. B should invest 5 percent of her stock portfolio in
IBM. Of course, they might have different amounts of money invested in the
stock market, but each would choose the same portfolio of risky assets. But
Sharpe then realized that if everyone held the same portfolio of risky assets,
then it would be easy to measure that portfolio: you just need to look at the
total wealth invested in IBM, say, and divide that by the total wealth in the stock
market. The portfolio of risky assets that was optimal for each individual would
just be the portfolio of risky assets held by the market.
This insight gave Sharpe an empirical proxy for the risky portfolio in the
Tobin analysis: in equilibrium it would simply be the market portfolio. This
observation has the important implication that the market portfolio is
Hal Varian
mean-variance efficient-that is, it lies on the frontier of the efficient set, and
therefore satisfies the first-order conditions for efficiency.
Some simple’ manipulations of those first-order conditions then yield the
celebrated Capital Asset Pricing Model (CAPM):
In words, the expected return on any asset a is the risk-free rate plus the risk
premium. The risk premium is the “beta” of the asset a times the expected
excess return on the market portfolio.
The “beta” of an asset turns out to be the covariance of that asset’s return
with the market return divided by the variance of the market return. This is
simply the theoretical regression coefficient between the return on asset a and
the market return, a result remarkably consistent with the single-factor model
proposed in Sharpe’s thesis.
Meanwhile, back on the east coast, Jack Treynor and John Lintner were
independently discovering the same fundamental pricing equation of the CAPM.
Treynor’s work was never published; Sharpe (1964) and Lintner (1965) remain
the classical citations for the CAPM.
The Capital Asset Pricing Model was truly a revolutionary discovery for
financial economics. It is a prime example of how to take a theory of individual
optimizing behavior and aggregate it to determine equilibrium pricing relationships. Furthermore, since the demand for an asset inevitably depends on the
prices of all assets, due to the nature of the portfolio optimization problem, it is
inherently a general equilibrium theory.
Sharpe’s two major contributions, the single factor model and the CAPM,
are often confused. The first is a “supply side” model of how returns are
generated; the second is a “demand side” model. The models can hold
independently, or separately, and both are used in practice.
Subsequent research has relaxed many of the conditions of the original
CAPM (like unlimited short sales) and provided some qualifications about the
empirical observables of the model. Sharpe (1991) provides a brief review of
these points. Despite these qualifications, the CAPM still reigns as one of the
fundamental achievements of financial economics, taught in every finance
textbook and intermediate microeconomics texts.2
Merton Miller
In 1990 Merton Miller was named a Distinguished Fellow of the American
Economic Association in honor of his many contributions. He has worked on a
ISharpe’s proof of the CAPM was given in a footnote.
2Or at least, the good ones.
Journal of EconomicPerspectives
variety of topics in economics and finance, but the idea singled out by the
Nobel Committee was one of his early papers on corporate finance. Portfolio
theory and the CAPM focus on the behavior of the demanders of securities-the
individual investors. Corporate finance focusses on the suppliers of the
securities-the corporations that issue stocks and bonds.
Merton Miller joined Carnegie Tech in 1952 to teach economic history and
public finance. In 1956, the dean asked Miller to teach corporate finance in the
business school. At first Miller wasn’t interested, since finance was then viewed
as being a bit too grubby for an economist to dabble in. But after appropriate
inducements, Miller sat in on the corporate finance class in the fall and started
to teach it the following term.
One of the major issues in corporate finance, then and now, was how to
raise capital in the best way. Broadly speaking a firm can issue new equity or
new debt to raise money. Each has its advantages and disadvantages: issuing
debt increases the fixed costs of the firm, while issuing equity dilutes the shares
of the existing shareholders. There were lots of rules of thumb about when to
do one and when to do the other. Miller started to look at some data to see if he
could determine how corporate financial structure affected firms’ values.
He found, much to his surprise, that there was no particular relationship
between financial structure and firm value. Some firms had a lot of debt; some
had a lot of equity, but there didn’t appear to be much of a pattern in terms of
how the debt-equity ratio affected market value.
It has been seriously suggested that there should be a Journal of Negative
Results which could contain reports of all those regressions with insignificant
regression coefficients and abysmal R-squares. If such a journal had existed,
Miller might well have published his findings there. But there was no such
journal, so Miller had to think about why there might be no relationship
between capital structure and firm value.
Franco Modigliani, whose office was next to Miller’s, had been working on
some of the same issues from the theoretical side. He was concerned with
providing microeconomic foundations for Keynesian models of investment.
Building on previous work by Durand (1952), Modigliani had sketched out
some models of financial structure that seemed to imply that there was no
preferred capital structure. Miller and Modigliani joined forces, and the world
of corporate finance has never been the same.
Miller and Modigliani considered a simple world without taxes or transactions costs and showed that in such a world, the value of a firm would be
independent of its capital structure. Their argument was a novel application of
the arbitrage principle, or the law of one price. Since the MM theorem has been
described at least twice in this journal (Miller, 1988; Varian, 1987), I will give
only a very brief outline of the theorem.
The easiest way to think about the MM theorem, in my view, is that it is a
consequence of value additivity. Consider any portfolio of assets. Then value
additivity says that the value of the portfolio must be the sum of the values of
A Portfolioof NobelLaureates:Markowitz,Millerand Sharpe 167
Merton Miller
the assets that make it up. At first, this principal seems to contradict the insights
of Markowitz about portfolio diversification: certainly an asset should be worth
more combined in a portfolio with other assets than it is standing alone due to
the benefits of diversification.
But the point is that asset values in a well-functioning securities market
already reflect the value achievable by portfolio optimization. This is the chief
insight of the Capital Asset Pricing Model: the equilibrium value of an asset
depends on how it covaries with other assets, not on its risk as a stand-alone
In any event, the principal of value additivity is even more fundamental
than the Capital Asset Pricing Model, since it rests solely on arbitrage considerations. If a slice of bread and a piece of ham were worth more together as a
sandwich than separately, everyone would buy bread and ham and make
sandwiches-for a free lunch! The excess demand for bread and ham would
push up the price of each, restoring the equilibrium relationship that the value
of the whole has to be the sum of the value of the parts.
From this observation, the MM theorem follows quickly. The value of the
firm is defined to be the sum of the values of its debt and its equity. If the firm
could increase its value by changing how much of its cash flow is paid to
bondholders and how much to stockholders, any individual investor could
construct a free lunch. The investor would buy a fraction f of the outstanding
stocks and the same fraction f of the outstanding bonds, which would give him
a fraction f of the total cash flow. He could then repackage this cash flow in the
same way as the firm could, thereby increasing the value of the total
portfolio-and violating value additivity.
This sort of “home-made leverage” argument is one way to prove the MM
theorem. But it is a particularly powerful way since it doesn’t appeal to any
Journal of Economic Perspectives
particular model of consumer or firm behavior. It rests solely on the principle
of arbitrage-there can be no free lunches in equilibrium.
The theory of the MM proposition is solidly established. The controversies
all arise from the assumption of a frictionless world: in particular, no costs to
bankruptcy, no asymmetric information, and no taxes. The latter is probably
more important than the former. In the United States, at least, interest
payments on debt are tax deductible while dividends to shareholders are taxed
at both the corporate and individual level.
Since the MM proposition showed that debt and equity were perfect
substitutes in the absence of taxes, the favorable tax treatment given to debt
should imply that all firms are 100 percent debt-financed. This is contrary to
fact-although for a while in the 1980s it looked as though it might come true.
Miller (1988), and the comments on this article by Bhattacharya, Modigliani,
Ross, and Stiglitz, describe the current state of research on the MM theorem.
Suffice it to say that there is still doubt about exactly which frictions are the
most relevant ones.
I happened to have lunch with Merton Miller in October 1990, the
weekend before the Nobel prize winners were announced. Part of the lunchtime
conversation was devoted to speculation about who might win the Nobel Prize
in Economics that year. Mert thought that someone from Chicago might well
receive the prize that year, and he suggested a few worthy possibilities-his
own name not among them, of course. He was awarded the Nobel prize two
days later. His 1990 forecast was a bit like the MM theorem itself-it was right
in principle, but the details were a little off!
In reviewing the work of these three economists, we see a common thread
of theory and empiricism running through their research. It isn’t enough just
to formulate a theory of portfolio choice you’ve got to find a feasible way to
compute optimal portfolios as well. It isn’t enough to formulate a theory of
capital market equilibrium-the theory should be estimated and tested. It isn’t
enough just to look at a scatterplot of firm values and debt-equity ratios-we
need a theory for why there should or should not be a relationship among
these variables.
Financial economics has been so successful because of this fruitful relationship between theory and data. Many of the same people who formulated the
theories also collected and analyzed the data. This is a model that the rest of the
economics profession would do well to emulate.
Hal Varian 169
Bernstein, P., Capital Ideas. New York: The
Free Press, 1992.
Durand, D., “Costs of debt and equity funds
for business: Trends and problems of measurement,” Conference on Research in Business
Finance, 215-47. New York: National Bureau
of Economic Research, 1952.
Keynes, J. M., “Memorandum for the estates committee,” Kings College, Cambridge,
May 8, 1939. In Moggridge, D., CollectedWritings of John Maynard Keynes, Vol. XII, 66-68.
New York: Cambridge University Press, 1983.
Lintner, J., “The valuation of risky assets
and the selection of risky investments in stock
portfolios and capital budgets,” Review of Economicsand Statistics,February 1965, 47, 13-37.
Markowitz, H., “Portfolio selection,” The
Journal of Finance, March 1952, 7, 77-91.
Markowitz, H., “The Optimization of a
Quadratic Function Subject to Linear Constraints,” Naval Research Logistics Quarterly,
1956, 3.
Markowitz, H., Portfolio Selection: Efficient
Diversification of Investment. New York: Wiley,
Markowitz, H., “Foundations of portfolio
theory,” Journal of Finance, June 1991, 46,
Merton, R., Continuous Time Finance. Oxford: Basil Blackwell, 1990.
Miller, M., “The Modigliani-Miller propositions after thirty years,” Journal of Economic
Perspective, Fall 1988, 2:4, 99-120.
Miller, M., “Leverage,” Journal of Finance,
June 1991, 46, 479-88.
Sharpe, W., “A simplified model for portfolio analysis,” Management Science, 1963, 9,
Sharpe, W., “Capital asset prices: A theory
of market equilibrium under conditions of
risk,” Journal of Finance, September 1964, 19,
Sharpe, W., “Capital asset prices with and
without negative holdings,” Journal of Finance,
June 1991, 46, 489-509.
Tobin, J., “Liquidity preference as behavior
towards risk,” Review of Economic Studies,
February 1958, 25, 65-86.
Varian, H., “The arbitrage principle in financial economics.” Journal of Economic Perspectives, Fall 1987, 1:2, 55-72.
Williams, J. B., The Theory of Investment
Value, Cambridge: Harvard University Press,

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