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 JournalJuly/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 estateUpon 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 CovarianceSee 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 s

 E.g. 3 If = Then p /s =

.25 .500 .10 .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 Diversificationof 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].

Constraints on portfolio choice e.g. turnover

constraints

MPT Investment Process

Expected return model

Volatility and correlation estimates

PORTFOLIO OPTIMIZATIONRisk – Return

Efficient Frontier

Choice of Portfolio

risk

retu

rn

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 InvestmentsJohn 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 FinanceAnd 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

- Two Factor model

- etc. - BARRA model - Scenario models

iiii uFr

iiiii uFFr 2211

2 21 1 1 1 1 ( ) ( )2 2 2 2 2

V V V

In case of the two-factor model, for example,

2 21 1 2 2)( ( ) ( ) ( )pR B V F B V F V U V

1 1 iiB X

2 2 iiB X

2( ) ( )i iU X V UV

where

Idiosyncratic Risk

Only the last term “diversifies”:

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 DiversificationThe Journal of Finance, Vol.20, No.4December 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 betweentheory 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 returnsUsually by a proprietary expected return model

  Covariances

Historical or factor model 

ConstraintsUpper boundsTurnoverLiquidity

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 estimatese.g., will equity premium be as high in thenext few years as it has been “recently”?

 d. Generate frontier.

 e. Pick portfolio - perhaps use Monte Carlo to help see consequences

f. Implement

Index fundsActive 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 pointon 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!