Portfolio Theory :
The First Half-Century
(plus a little)
Harry M. Markowitz
“Portfolio Theory”
Before 1950
Markowitz, Harry M. (1999)
The Early History of Portfolio Theory:
1600 – 1960 Financial Analyst Journal
July/August, 5-16
It is sometimes said that investors did not diversify much before 1952.
Not correct. See:
Graham and Dodd [1934], Wiesenberger [1941-],
Shakespeare "Merchant of Venice"
(quoted by P. Bernstein):
My ventures are not in one bottom trusted,
Nor to one place; nor is my whole estate
Upon the fortune of this present year;
Therefore, my merchandise makes me not sad.
- Act I, Scene 1
What was lacking in 1952 was an adequate theory covering:
• The effect of diversification when risks are correlated
• Risk/return tradeoff on the portfolio as a whole.
See: Hicks [1935], Leavens [1945], Williams [1938]
First, some terminology:
Standard Deviation ()
A measure of “dispersion”.
Variance (V)
=
(std Dev)2
Correlation ()
A measure of how closely two random
variables go up and down together.
Covariance (cov)
Cov (i,j) =  (i,j) *  (i) * (j)
"A Suggestion for Simplifying the Theory of Money",
Economics, February, pp. 1-19. Hicks, J. R. [1935],
Assumes uncorrelated returns.
Big difference between correlated and uncorrelated returns:
With uncorrelated returns
as diversification increases, portfolio risk approaches zero.
With correlated returns
even with unlimited diversification portfolio risk remains substantial.
See the Law of the Average Covariance
Hicks assumed that indivisibility limited the amount of diversification, therefore limited risk reduction:
... since most people do not possess sufficient resources to enable them to take much advantage of the law
of large numbers, and since even the large capitalist cannot annihilate his risks altogether in this manner,
there will be a tendency to spread capital over a number of investments, not for this purpose, but for another.
By investing only a proportion of total assets in risky enterprises, and investing the remainder in ways which
are considered more safe, it will be possible for the individual to adjust his whole risk-situation to that which
he most prefers, more closely than he could do by investing in any single enterprise.
Hicks [1962] includes an appendix showing some details along these lines,
assuming independent returns:
It can, I believe, be shown that the main properties which I hope to demonstrate
remain valid whatever the r's; but I shall not offer a general proof in this place. I shall simplify
by assuming that the prospects of the various investments are uncorrelated rjk = 0 when k  j):
an assumption with which, in any case, it is natural to begin.
Thus, for a different reason (indivisibility rather than correlation) Hicks arrived at one
of the conclusions of Tobin [1958], Sharpe [1964] and Lintner [1965]: the cautious
investor will hold cash for portfolio reasons.
Importance of Covariance
An extreme example:
A security is likely to have a high return but has a small chance of going broke.
Is a small investment in this security a reasonably safe bet?
Not if the remainder of the portfolio consists of similar bets all of which will go
broke at the same time.
Law of the Average Covariance
See Markowitz [1959] Chapter 5
For an equal weighted portfolio as the number of securities held increases
Portfolio Variance  Average Covariance
E.g. 1 For uncorrelated securities
Portfolio Variance  0
E.g. 2 If all securities have the same variance Vs, and all distinct pairs of securities have
the same correlation coefficient , then
Portfolio Variance Vp   Vs
Portfolio Std Dev p 
E.g. 3 If  =
.25
.10

s
Then p /s =
.500
.316
even with unlimited diversification!
Williams, John B. [1938], The Theory of Investment Value,
Harvard University Press, Cambridge, Mass.
Most of the ideas in Markowitz [1952] popped up one day in 1950 when Markowitz read
Williams [1938].
Williams asserted:
The value of a stock should be the present value of its future dividends.
Markowitz thought:
Future is uncertain; must mean expected present value.
If one is only interested in the expected values of securities, then one must only be
interested in the expected value of the portfolio.
One maximizes expected value by putting all resources into the single security
with greatest expected value.
Diversification exists and is good.
See Wiesenberger.
Clearly investors (and investment companies) seek return, avoid risk. Let's
measure these by expected value and variance, like the statisticians do.
Two criteria: let's draw a tradeoff curve.
Markowitz (Ph.D. candidate, Economics, University of Chicago; zero investment
experience) was currently taking a course by Koopmans on efficient and
inefficient allocation of resources. Hence efficient versus inefficient portfolios.
Portfolio return is a weighted sum of security returns. What is the formula for the
variance of a weighted sum?
Formula supplied by Uspensky [1937]. Wow! Covariance's!
Later in the book Williams addresses the issue of risk and uncertainty:
“Whenever the value of a security is uncertain and has to be expressed
in terms of probability, the correct value to choose is the mean value….The
customary way to find the value of a risky security has always been to add a
“premium for risk” to the pure interest rate, and then use the sum as the interest
rate for discounting future receipts. In the case of the bond under discussion,
which at 40 would yield 12 per cent to maturity, the “premium for risk” is 8 per cent
when the pure interest rate is 4 per cent.
Strictly speaking, however, there is no risk in buying the bond in question
if its price is right. Given adequate diversification, gains on such purchases will
offset losses, and a return at the pure interest rate will be obtained. Thus the net
risk turns out to be nil. To say that a “premium for risk” is needed is really an
elliptical way of saying that payment of the full face value of interest and principal
is not to be expected on the average”.
pp.67,68.
Leavens, D. H. [1945], “Diversification of Planning”,
Trusts and Estates, 80, May, pp. 469-73.
An examination of some fifty books and articles on investment that have
appeared during the last quarter of a century shows that most of them refer to the
desirability of diversification. The majority, however, discuss it in general terms and
do not clearly indicate why it is desirable.
Leavens illustrated the benefits of diversification on the assumption that
risks are independent. However, the last paragraph of Leavens cautioned:
The assumption, mentioned earlier, that each security is acted upon by
independent causes, is important, although it cannot always be fully met in practice.
Diversification among companies in one industry cannot protect against
unfavorable factors that may affect the whole industry; additional diversification
among industries is needed for that purpose. Nor can diversification among
industries protect against cyclical factors that may depress all industries at the
same time.
Portfolio Theory
Circa 1952-1959
Markowitz, Harry M. (1952)
Portfolio Selection, The Journal of Finance,
7(1), March, 77-91
Roy, A. D. (1952), Safety First and the Holding of Assets”
Econometrica, 20, pp.431-49
Markowitz, Harry M. (1959)
Portfolio Selection: Efficient Diversification
of Investments, Yale University Press,
1970; 2nd edition, Basis Blackwell, 1991.
Will discuss Tobin (1958) later
Markowitz, H. [1952], "Portfolio Selection",
The Journal of Finance, Vol. 7, No. 1, pp. 77-91.
Proposes mean (a.k.a. expected) return and variance of return on portfolio as a
whole as criteria for portfolio selection, as
• maxim
• hypothesis
Assumes that "beliefs" about securities follow probability rules for random
variables. Relates formulas for
• expected return on portfolio as a weighted average of the expected returns
on individual securities
• variance of return as a function of variances of, and covariances between,
securities. .
Distinguishes between efficient and inefficient portfolios.
Proposes that means, variances and covariances of securities be estimated by a
combination of statistic analysis and security analyst judgment (no details supplied)
and the set of mean-variance, or mean-standard deviation combinations implied by
these be presented to investor for choice of desired risk-return combination.
Uses geometric analysis of 3- and 4-security examples to illustrate properties of
efficient sets, assuming non-negative investments subject to budget constraint.
• Set of efficient portfolios is piecewise linear (connected straight lines);
• Set of efficient mean-variance combinations is piecewise parabolic.
Roy, A.D. [1952], "Safety First and the Holding of Assets",
Econometrica, 20, pp. 431-49.
Proposes portfolio choice based on mean and variance of portfolio as a whole.
Specifically proposes choosing portfolio which maximizes
m-d
σ
Where m is mean return, σ is standard deviation and d is a fixed (for the analysis) disastrous
return.
Why did Markowitz get a Nobel Prize and not Roy?
Differences between Markowitz [1952] and Roy [1952]??
More likely: visibility to Nobel Committee in 1990
Roy [1952] is Roy’s first and last paper in finance
Markowitz also wrote Markowitz [1959, 1987] and assorted articles.
Markowitz, H. [1959], Portfolio Selection:
Efficient Diversification of Investments,
2nd ed. Basil Blackwell, 1991.
Written mostly during the academic year 1954-55 at Yale University at the invitation of Jim Tobin.
Its contents can be classified into two categories:
I.
Introductions for the non-mathematician to
A. "Portfolio analysis": What is it? Examples.
Chapters 1 and 2.
B. Definitions of expected value, variance, covariance. Formulas for expected value and
variance of a portfolio.
Chapters 3 and 4.
Models of covariance; in particular the one factor model
(pp. 96-100), but not the possibility of "diagonalizing“
the covariance matrix as in Sharpe [1963].
C. The Law of the Average Covariance. Chapter 5.
D. The definition and geometry of efficient sets. Chapter 7.
(Like Markowitz [1952] minus 2 bugs.)
E. Introductory expositions of needed topics in utility analysis, dynamic programming,
and Bayesian analysis.
II. Technical results to date
A. Computation of mean-variance efficient sets
1. The "critical line algorithm". See Chapters 7, 8 and Appendix A. Originally
presented in Markowitz [1956]. See also Perold [1984], Markowitz [1987]
2. Seeks mean-variance efficient sets subject to any system of linear
equality or inequality constraints in variables which may or may not be
required to be non-negative.
3. The critical line algorithm traces out efficient portfolios for all efficient EV
combinations in the time it takes for a quadratic programming algorithm
to minimize V for a few values of E.
4. Markowitz [1956] assumes a "non-singularity" requirement. Markowitz
[1959] Appendix A shows this to not be required. See especially
Markowitz [1987].
MPT Investment Process
return
Expected return model
risk
Volatility and
correlation estimates
Constraints on portfolio
choice e.g. turnover
constraints
PORTFOLIO OPTIMIZATION
Risk – Return
Efficient Frontier
Choice of Portfolio
B. Choice of criteria
1.
Markowitz [1959] does not justify mean-variance by assuming either
normal distributions or quadratic utility (unlike Tobin [1958])!
2.
Conjecture: for common utility functions and for distributions like those of
portfolio returns, if you know mean and variance you can well estimate
expected utility.
A few examples in Markowitz [1959]
Chapters 6 and 13.
More in Young and Trent [1969]
Levy and Markowitz [1979]
Literature cited in Markowitz [1987]
Pp.63-68, Hlawitschka [1994]
Distinguish between having a quadratic utility function and using a
quadratic approximation to a given utility function.
3.
Relationship between single period and many period analysis. Derived
utility function, in the manner of Bellman [1957]. See Markowitz [1959]
Chapters 11 and 13.
4.
If the derived utility function includes state variables besides wealth, and a
quadratic approximation is good enough, then the mean-variance analysis
should include, as criteria, covariance between portfolio return and other
state variables. See Chapter 13 for use of "exogenous assets“ for this and
other purposes.
C. Semivariance, a.k.a. downside risk.
Definition; geometric analysis.
Markowitz [1959] Chapter 9.
Solution by critical line algorithm; see Appendix A.
As criteria: see Chapter 13.
Variations on a Theme
1. Risk can be
-total variance
-tracking error
tracking what?
2. Risk can be
-variance
-semivariance
3. Analysis can include “exogenous assets: -e.g., income from other sources
4. Constraints can include bounds on individual holdings sector holdings turnover
CAN
vs
SHOULD
Then what happened?
Models of Covariance
1959-1974
Markowitz, Harry M., (1959)
Portfolio Selection: Efficient
Diversification of Investments
John Wiley & Sons, New York:
And 1991, 2nd ed., Basil Blackwell,
Cambridge Ma
Sharpe, William F., (1963), A Simplified
Model For Portfolio Analysis,
Management Science 9,
Pp 277-293
Cohen, K.J., and J.A. Pogue, (1967),
An Empirical Evaluation of Alternative
Portfolio Selection Models,
Journal of Business 40, 166-193
Rosenberg, B. (1974), Extra-Market
Components Of Covariance in
Security Returns, Journal of Finance
And Quantitative Analysis, 9 (2) 263-273
Models of Covariance
Rather than estimating individual covariances, a model of covariance
can be supplied, e.g.,
- One Factor model
ri   i  i F  ui
- Two Factor model
ri   i   i1 F1   i 2 F2  ui
- etc.
- BARRA model
- Scenario models
In case of the two-factor model, for example,
V
2V ( F ) V (U )
( Rp )  B12V ( F1)  B2
2
where
B1   i1 X i
B2   i 2 X i
V
(U )   X i2V (Ui )
Idiosyncratic Risk
Only the last term “diversifies”:
1 1 
2
2
( 1 ) 2V  ( 1 )2V  1 V
2
2
2
MPT

CAPM
Tobin, James(1958)
Liquidity Preference as Behavior
Towards Risk, The Review of
Economic Studies 67, pp65-68
Sharpe, William F., (1963)
A Simplified Model for
Portfolio Analysis
Management Science 9, pp 277-293
Lintner, John (1965)
Security Prices, Risk, and
Maximal Gains from Diversification
The Journal of Finance, Vol.20, No.4
December 1965
MPT  CAPM
-Standard CAPMs assume that everyone seeks mean-variance efficiency
 investor can use mean-variance analysis even if no one else does
 maybe better if the market is not efficient
-Standard CAPMs assume that everyone has the same beliefs
 mean-variance makes no such assumption
-Standard CAPMs make special assumptions concerning constraint set
 investors can borrow all they want at the risk-free rate; or
 can short and use the proceeds without limit
 mean-variance analysis can model actual constraints
-CAPM conclusions depend heavily on special assumptions about constraint sets
Therefore, if an empirical tests rejects CAPM, that doesn’t even prove that
investors are not using mean-variance; certainly doesn’t prove you shouldn’t.
Portfolio Theory in Practice
A Half-Century Later
“In theory, there is no difference between
theory and practice. In practice there is.”
(source forgotten)
Why is using portfolio theory
not
like being pregnant?
Because you can use portfolio theory a little bit
Uses of Portfolio Theory in Investment Management
1. Full use (by some quantitative managers)
Expected returns
Usually by a proprietary expected return model
Covariances
Historical or factor model
Constraints
Upper bounds
Turnover
Liquidity
Some rule for picking portfolio from efficient frontier.
2.
Top down, asset class, analysis
a. Select asset classes
e.g., large cap stocks, small cap stocks,
long term bonds, etc.
b. Review history (e.g., Ibbotson)
c. Make estimates
e.g., will equity premium be as high in the
next few years as it has been “recently”?
d. Generate frontier.
e. Pick portfolio - perhaps use Monte Carlo to help see consequences
f. Implement
Index funds
Active managers
Widely used by financial advisors and some institutional investors.
Gary Brinson, et. al.(various).
3.
Risk Control Only
a. Arrive at current or proposed portfolio in any manner
b. Determine exposures to factors using model of covariance
-only look at B1, B2…
-let model evaluate
V(Rp) using V(F1), V(F2)…
V(u1), V(u2),…
c. Merits and problems of making more of the model explicit
-Merit: the model can figure the tradeoffs
-Problem: someone has to estimate - E(ri) and/or V(Fi), V(ui) etc.
4.
Combined with simulation analysis
(a.k.a. Monte Carlo analysis)
Simulation is used to answer questions like
if the investor saves at a certain rate
and
reinvests each period in a particular point
on the efficient frontier
what is the probability distribution of
retirement income?
Decisions, Decisions
I
Full risk-return analysis
-criteria: variance, semivariance, tracking error.
-model of covariance
-estimation of means, and inputs to covariance model.
-constraints: liquidity, turnover, max holdings
-backtest or Monte Carlo to evaluate choices
II
Top down
-asset classes
-history
-estimates
-implementation of asset class allocations.
III
Risk control only
-which model of covariance?
-how to recognize if overexposed to a risk factor; how to adjust if so.
IV
Combined with simulation
-model?
-parameters?
MPT at 51?
Still feeling well and going strong!