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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):

F

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.

160

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

162

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

Expected

Return

Efficient

portfolios

of risky

assets

Efficientportfolioswith

riskyand risk-freeassets

/

A _ E i:

.

;

f iE; iEi;i. ^

0

f!422;iEd~~~~~~~..

… T:X4…

;~~~~

~~~~~~.

~

f

ErRE

………

Rm0

_

0r .

3.

E–E

Standard Deviation

A Portfolioof NobelLaureates:Markowitz,Millerand Sharpe 163

William Sharpe

it touches this line (o,,,

Now observe that every efficient portfolio consistof

assets

and

the

riskless

asset can be achieved by combining only two

ing

risky

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.

164

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

Rat

= 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

165

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):

Ra

=Ro

+

3a(Rm-

Ro).

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.

166

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

investment.

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

168

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!

Summary

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

References

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,

1959.

Markowitz, H., “Foundations of portfolio

theory,” Journal of Finance, June 1991, 46,

469-77.

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,

277-93.

Sharpe, W., “Capital asset prices: A theory

of market equilibrium under conditions of

risk,” Journal of Finance, September 1964, 19,

425-42.

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,

1938.

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