using modern portfolio theory to maintain an efficiently diversified portfolio

CFA Institute

Using Modern Portfolio Theory to Maintain an Efficiently Diversified PortfolioAuthor(s): Lawrence FisherSource: Financial Analysts Journal, Vol. 31, No. 3 (May - Jun., 1975), pp. 73-85Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4477827 .Accessed: 17/06/2014 20:01

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by Lawrence Fisher

Using Modern Portfolio Theory

To Maintain an Efficiently

Diversified Portfolio *

In the twenty years since Harry Markowitz (1952) described the concept of efficient diversification of investments, theory seems to have changed much more than practice. I suspect that one of the main impediments to the application of modern portfolio theory is that described by the following example:

Suppose you had provided the necessary estimates of risk and return and had used a computer program to help select a portfolio that was efficient according to the market ("diagonal") model (Markowitz, 1959, pp. 98-101; Sharpe, 1963). Now, some time has passed. The prices of some of the securities in the portfolio have gone up more than others. Hence the composition of the portfolio has changed. Moreover, your own estimates of return are likely to have changed; and statistically derived estimates of risk have been revised. If you used the computer program with the revised data, you would probably find that your current portfolio bore little resem- blance to any members of the newly computed set of "efficient" portfolios. If you made a general practice of changing portfolios under such circumstances, you would soon find that brokerage commissions and income taxes had reduced your capital to a small fraction of its original value. For this reason you have not adopted the Markowitz-Sharpe methods of portfolio selection.'

There are at least two modifications of the diagonal model that tend to reduce this kind of instability. One of them modifies the procedure so that the number of securities in the portfolio is at least equal to an arbitrary minimum. The other requires the analyst to modify his forecasts so that whatever portfolio is held will strongly resemble the market portfolio unless the analyst can show that he now has access to tomorrow's closing quotations.

The first modification was added-I believe by Sharpe-to the portfolio-selection computer program at an early stage. It was placed there because of legal requirements faced by diversified investment companies. The modification takes the form

1. Footnotes appear at end of article.

Lawrence Fisher is Professor of Finance, Graduate School of Business, the University of Chicago and Associate Director of The Center for Research in Security Prices.

*This article develops an idea that I have had in rudi- mentary form for ten years. That the expected returns implied by the assumption that a portfolio is efficient could be found up to a linear transformation was the reaction of Harry Markowitz when I discussed the idea of a "dual" with him in May 1964. The idea was later pursued for a short time by David Duvel in connection with my workshop in business finance at the University of Chicago. In 1964, Benjamin F. King had not yet com- pleted his study of market and industry factors (1966); neither had anyone made a really serious attempt to select a portfolio with the aid of Sharpe's simplified model (1963). When Duvel spent his time working with the notion of implicit return, the fact that systematic risk could be estimated from past data (although not without difficulty) was only suspected. (Cf. Blume, 1970; Treynor, Priest, Fisher and Higgins, 1968; Jensen, 1968, 1969.) The renewed work on the development of the idea has been encouraged by the reaction of the students in my course on investments and by the support of Jas. H. Oliphant & Co., Inc. in preparing early drafts of this article.

FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975 LG 73


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of requiring that no more than, say, five per cent of the portfolio be invested in any one security. Hence all portfolios printed out by the computer will contain at least, say, 20 securities.2

Portfolios consisting of 20 securities will tend to have more in common with one another than portfolios of only three, four, or five. Thus the constraint that makes the program applicable to mutual funds also makes composition more nearly stable. But, when used for investors who are not subject to the constraint, the modified computer algorithm seems paradoxical at best. The instructions for its use say: Tell us your beliefs about the future; we will tell you the behavior that they imply. How- ever, because the program tells one to buy more stocks than his beliefs imply it implicitly treats the input data it uses as false.

The second method, due to Treynor and Black (1973), is to use an application of modern statistical theory: Bayesian analysis. Treynor and Black say, in effect, recognize that your own estimates of return and risk are imperfect. Collect data to see whether you can give your estimates any credence. If you can't, hold the market portfolio. If your estimates have some merit, hold an average of the market portfolio and the portfolio you would choose if your estimates could be accepted at face value. This average of "passive" and "active" portfolios will contain a rather substantial number of securities.

The first method is unsatisfactory and hasn't been applied. The Treynor and Black method is theoretically sound but was still unpublished hence not widely known at the time this article was written. Neither method, however, takes transaction costs into explicit account.

In this article I sketch out a method of applying modern portfolio theory in a manner that recognizes that a new set of estimates of the expected returns and risks of securities may differ from the old primarily because of statistical sam- pling error and may, therefore, imply little actual revision of the composition of the portfolio. To apply the suggested procedure the analyst asks, in a systematic way, whether expected return or the estimated risk incurred by holding a security has changed enough to make it worth the cost of buying more or selling off all or part of his holdings. In making the decision, transaction costs are explicitly taken into account.

To carry out the procedure, the portfolio analyst begins by combining data on the composition of the portfolio with estimates of risk. From the combination he obtains a set of estimates of expected returns that is consistent with the proposition that the currently held portfolio is an efficient portfolio.

He then examines each of these implicitly expected returns to see whether it is reasonable. If in the analyst's view the implicit estimates are all reasonable, the analysis is complete. If not, he alters one estimate at a time and considers the characteristics of the successively modified portfolios that result from changing the estimates. In considering the modifications, he looks at both the incremental costs and benefits associated with each modification until he has either considered all reasonable revisions of the implicitly expected returns or until he has found that a further potential modification is not worthwhile. In the process of seeing whether the implicit estimates are reasonable ones, he can easily include (in somewhat modified form) the principal considerations suggested by Treynor and Black.

In Section I, I describe the procedure for finding the implicit estimates of expected return for all stocks now included in a portfolio and also show how to find upper limits of implicit estimates for each security not held. The mathematical derivation of the procedures is given in detail in the appendix. Sections II and III outline my sug- gestions for using the differences between the estimates implicit in the composition of the portfolio and estimates that the analyst prefers to guide the alteration of the portfolio. The procedures suggested will permit nearly optimum adjustments-adjustments that are about as good as can be obtained without using more computing power than exists in the world (and getting it free of charge). In Section II, I assume that the analyst's strategy is a "passive" one. He is trying to approximate the market portfolio with a manageable number of securities. In Section III, I assume that the analyst has at least some estimates of risk or return that differ markedly from those implicit in either his own portfolio as it currently stands or in the market portfolio. Section IV takes up two empirical problems, the appropriate way to relate transaction costs to expected return and the need for reliable estimates of risk. Section V is a summary.

I. Finding the Rates of Return Implied by the Assumption that the Currently

Held Portfolio is Efficient

Background for the Diagonal Model Modern capital theory begins with the idea

presented by Markowitz (1952) "that the investor does (or should) consider expected return [E] a desirable thing and variance of return [V] an undesirable thing....

"Suppose that the set of all obtainable (E, V) combinations were [known]. The E-V rule states

74 E FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975



that the investor would (or should) want to select one of those portfolios which give rise to the (E,V) combinations that [are] efficient ... i.e., those with minimum V for given E or more and maximum E for given V or less." To find out what set of combinations of ex-

pected return and variance is obtainable one must form beliefs about the probability distributions of return for all assets that might be included in the portfofio. It is not sufficient to describe each security's distribution separately, for security prices tend to move together. In order for the probability distributions to be specified completely, a separate distribution must be specified for the security in question for each combination of possible outcomes for all other securities. In general that task is so large as to be inconceivable. If Markowitz's E-V rule is applied, however, complete specifica- tion of the probability distributions is not even desirable.3 What is desirable for each security is a statement of its expected return, i.e., the mean of the probability distribution; its variance of return; and the covariances of its return with the returns of all other securities. These parameter estimates are necessary because the expected return of the portfolio is a weighted average of the expected returns of the securities included and the variance of the portfolio's return depends on both variances of returns and covariances.4 However, the number of covariances is slightly more than proportional to the square of the number of securities under consideration (cf. Markowitz, 1959, or Sharpe, 1963). Thus, if only the 1,400 common stocks listed on the New York Stock Exchange are considered, beliefs need to be formed about 1,400 expected returns; 1,400 variances; and 989,300 covariances. Mere processing of data about beliefs that were well formed would be a substantial task. However, meaningful estimates can be made only with the aid of statistical inference. Unfortunately, brute force techniques of taking the 1,400 time series of returns two at a time in order to estimate each of the million covariances from past data will provide unsatisfactory results. The E-V method tends to include in the portfolio those securities whose covariances (with one another) are small or-better yet-negative. If brute force estimates of covariance were used, over 20,000 of them would be low by the equivalent of more than two standard deviations, even if it could be assumed that the procedure was otherwise valid. Hence, instead of being really efficient, the portfolio selected would more likely be the result of estimation error. If one investor alone used the technique, over time he would be likely to hold a series of portfolios that was not much different from any other set of ran-

domly selected portfolios. However, if many investors were selecting on the basis of the same erroneous data, they would all tend to lose money.

In order to form reasonable beliefs about the relevant parameters of the probability distributions, one needs to have some theory about the behavior of the stock market and then must estimate the parameters in a manner that is consistent with that theory. After that, he can compare the behavior pattern of real or hypothetical portfolios with the pattern predicted by the theory. Only if the theory passes the tests by providing a reason- ably good description of the actual pattern, should he be willing to act on the basis of the parameter estimates consistent with the theory.

One theory is that each security behaves independently from other securities. This theory is false because there is, and has been for centuries, overwhelming evidence that stock prices tend to move together (cf. Markowitz, 1952). It is also wrong as a description of the beliefs investors act as though they have. If investors thought that prices of individual stocks moved independently of one another they would think that well-diversified portfolios were essentially riskless. Then, on the average, stocks, long-term bonds and short-term Treasury bills would all have the same returns.5 There is, however, sufficient co-movement among stocks that indexes of stock market prices are widely disseminated by the news media. Moreover, the most widely disseminated index in the United States, the Dow Jones Industrial Average, is con- structed so as to use computational procedures that have been indefensible since the hand-cranked desk calculator became available about 60 years ago. However, it has withstood the onslaught of newer and statistically valid indexes because of the extent of the co-movement.

The mathematically simplest theory that in- cludes co-movement is the well known market or "diagonal" model described briefly by Markowitz (1959, pp. 97-101) and developed by Sharpe in his landmark paper (1963).6 As Sharpe put it:

"The major characteristic of the diagonal model is the assumption that the returns of the various securities are related only through common rela- tionships with some basic underlying factor. The return from any security is determined solely by random factors and this single outside element." To apply the diagonal model, one must supply

estimates of the mean (or "expectation") and variance of return for each of the S securities that one is selecting from. However, instead of the S(S-1)/2 estimates of covariance needed to cover each possible pair of securities, one need supply only the variance of the "market factor" and the

FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975 O 75



covariance between the return of each stock and the return of the market factor. The whole ana- lytical process is essentially equivalent if the regression coefficient of each security on the market factor ("beta") is supplied instead of an explicit estimate of the covariance.

The computer then can grind away on the input data and produce the implied set of efficient portfolios. In other words, the computer program takes the input data and suggests sets of fractions of the total value of the portfolio to be invested in each security. The main problem in applying the results is that, even if only a small percentage of the input data is changed, the set of "efficient" portfolios is likely to be changed completely.

Toward a Solution: The Use of the Dual of Programming Solutions

In attempting to solve the problem of instability, we begin by noting that, in mathematical programming, one may often obtain a "dual" of the solution. For example, linear programming is fre- quently used to find the minimum cost combination of, say, feed ingredients that will provide a balanced diet. One can write the linear programming code so as to provide as part of the output the maximum price (which is lower than the current market price) at which each potential ingredient that is not being used would enter the mixture and the range for each of those ingredients that are used within which its price could fluctuate without causing a different recipe to become optimal.7 By checking current prices of ingredients with the "shadow" prices that form the dual of the solution for the most recent recipe, the food chemist can see whether it is necessary to rerun the computer program.

To this observer it seems that an analogous use of the "dual" of the portfolio allocation results might be worthwhile. Instead of taking estimates of return and risk and using them to find an "optimum" recipe for the portfolio, I propose taking the risk estimates and the current composition of the portfolio and seeing what expected rates of return are consistent with the notion that the currently held portfolio is efficient.8 The analyst can then see whether these implicit rates of return are reasonable. If they are, no change in the portfolio will be called for. If some of them are unrea- sonable, the analyst's estimate can be substituted for the implicit estimate and the portfolio-selection program run in order to select the modified portfolio.

Let us see how the "dual" of the result of finding the set of efficient portfolios can be used to find estimates of expected return.

Form of Implicit Estimates of Expected Return In the appendix, I derive the relationship be-

tween expected return and risk factors that exists in any mean-variance (E-V) efficient portfolio, which is the kind of portfolio suggested by Markowitz and Sharpe.

In Markowitz's general case, it is shown that, for any of the N securities included in an efficient portfolio,

N

E.= El+ (2/i) IXJ C.- (1) j=l

where, following Sharpe's notation (1970),

Ei= The expected rate of return on the ith security

= slope of the "efficient frontier" at the point in the E-V (Expected return- variance of return) plane where this portfolio lies

Xi = fraction of total market value of the portfolio represented by the jth security,

Cij = covariance of return between the ill and jth securities (note that the variance of return for the ith security, =Cii),and

El = the lending rate or rate of return on a riskless security, for which

VI = C 1,2 = C1, ...= C 1N = -000-

If the security in question is not in the portfolio (and if short sales are not allowed), i.e., for i > N,

N

Ei ' EE+ (2/i) Xi Ci. (2)9 j=1

In practice, direct estimates of the covariance of each security in the portfolio with every security that is a candidate for inclusion are unlikely to be available- primarily because direct and reliable statistical estimation of all relevant covariances is impossible. Fortunately, one may use the market (or "diagonal") model to derive usable covariance estimates from statistically or otherwise estimated beta coefficients in the manner suggested by Markowitz (1959, pp. 98-101). When the derived estimates are substituted into Equation 1, one obtains-after a fair amount of algebraic manipula- tion-the following:

Ei = (2/i) ( bibpVM+ Xi Si/ ) + El (5)101*1

[*] Equations 3 and 4 appear in footnote 9 at the end of the article.

76 O FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975



where

bi = the best available estimate of the ith

b = security's beta coefficient, p the best available estimate of the

current portfolio's beta coefficient, VM = the variance of return of the market

index used in the estimation of the beta coefficients, and

s2 = residual variance of return of the ith stock (=M-b'VM).

To apply Equation 5 one must have available an estimate of beta and of residual variance for each security that is included in the portfolio or is a candidate for inclusion, an estimate of variance for the market index (or market factor), the riskless rate of return ("lending rate"), and either an assumed value of slope parameter A or a value of expected rate of return for one risky asset that is in the portfolio." This estimated return must be consistent with the point estimates of beta and residual variance for that security. From the composition of the portfolio and the estimates of beta for the securities, one can calculate the beta value for the portfolio. If the expected rate of return for a security has been supplied, he can rearrange Equation 5 and solve for X. Then, he substitutes bi, xi, and s? for each security in the portfolio into Equation 5 and calculates and writes down the expected rate of return consistent with the portfolio's being efficient. Next, he converts Equation 5 to the market-model analogue of Inequality 2 by substituting ' for = and dropping the Xis? term since failure to hold a security means that its Xi = 0.00, thus the value of Xis, must also be zero. Hence, the upper bound of E, for each excluded security is found by substituting bi into the first term in parentheses in Equation 5 and 0.00 for the product of Xs,.

When one obtains the estimates of expected return implicit in the current composition of the portfolio, several possible procedures seem to me to be good alternatives.

Perhaps the simplest procedure would apply when the investment strategy was essentially passive-i.e., when the actual portfolio held was meant to be approximately the market portfolio but contained a smaller number of securities. This case will be taken up in Section II.

When the portfolio analyst has available still other estimates of individual returns, taking account of the considerations raised by Treynor and Black makes the analysis somewhat more com- plicated. Application of my method to that problem is discussed in Section III.

Portfolio selection based on the market model

necessarily relies on the quality of beta estimates; the estimation of beta is not a simple task. Some considerations that will be treated at still greater length elsewhere are discussed in Section IV, which also considers transaction costs.

II. Application of the Dual Method to a Passive Portfolio

Suppose one holds a passive portfolio (in the Treynor-Black sense). Suppose further that he has chosen to hold a relatively small number of securities in order to reduce brokerage commissions, safety-deposit box rentals or other managerial ex- penses.

In this case, he would first obtain a set of "portfolio-implicit" returns from Equation 5. Then he would use Equation 5 to calculate a second set of expected returns. The second set would be the returns implied by the Capital Assets Pricing Model. To calculate the second set of expected returns, he would take X, as a fraction of the total market value of all wealth instead of a fraction of the market value of the portfolio under consideration. He would compare the two sets of estimates. For a given value of X the set for the portfolio would probably be slightly larger than the set for the market. If the portfolio were optimal, the apparent difference would, in fact, represent the savings in the cost of managing the smaller portfolio. 2 2

To adjust the portfolio, we would first alter X for one of the sets of estimates in order to make the weighted averages of E agree. Then we would begin repeated application of the following procedure.

1) Substitute the "market" (or "best") estimate for the security whose portfolio estimate differed from its market estimate by the greatest amount.

2) The quadratic programming code would then be used to find a set of efficient portfolios from the securities included and, perhaps, a large excluded security.

3) Compare the combinations of expectation and variance of return from the two portfolios to see whether the potential benefit from changing the composition of the portfolio is worth the cost. To compute expectation and variance we would apply Equations 6 and 7:

N

E =ZXiE (6) i=1

where Ei is the "best" estimate of expected return for the ith asset, and

N

VP=VMb 2 +Xi2S2 (7)13 i=




In Equation 6, the "best" estimates rather than the estimates implicit in the composite portfolio resulting from the mixture of "best" and implicit estimates used in the quadratic program are used in order to see what the potential improvement is.'4

Actually making the comparison is likely to be a two-step process because transaction costs must be taken into account. One promising way to do so is to note which securities must be sold (entirely or in part) and which bought in moving from the original portfolio to the new one. In making the final comparison the best estimates of return for these securities must be altered to take account of the income that will not be received if commissions and taxes are paid. What seems to be relevant is the expected return on the money to be received by selling or to be used in buying each security. Thus the alterations must reduce expected returns for securities that are candidates for sale. The amount of alteration of return to make depends on several factors. It is discussed at length in Section IV.'5

4) If the comparison indicates that the incremental benefit from the new portfolio exceeds the incremental cost of switching to it, continue the analysis by changing the estimate for the security with the next largest difference between implicit (based on the new tentative portfolio) and "best" estimates of expected return and going back to step 2.

5) When one finds that a switch is not worthwhile, he actually exchanges the original portfolio for the next-to-last tentative portfolio.'6

III. Adjusting an Active Portfolio

If one has available estimates of expected return that were found by some means other than applying Equation 5 to the composition of his portfolio or to the market portfolio-if he has made forecasts on the basis of information not generally known (or not properly interpreted by others)-he can try to get superior performance. In this case, it is quite likely that his estimates will differ substantially from the estimates implied by the composition of the portfolio. This will be true if the market quickly recognizes that he was right and prices become consistent with his valuations and also if, instead, he soon learns that the market was correct and he was wrong!

However, before one tries to apply my method of adjustment to an active portfolio, it is important to take into account two of the considerations raised by Treynor and Black. The most obviously important one is that, if he is sufficiently con- cerned with the problem of efficient diversification to have read to this point, he must have discovered that his (or his firm's) forecasts of return are less

than perfect. Moreover, it is quite likely that, if he used a weighted average of the "market" forecast and his own forecast, the one that would work best would give at least positive weight to the market forecast. If he has any forecasting ability, his own forecast will also receive positive weight.

If one diversifies efficiently, given his own forecasts, it is important that he use adjusted (weighted average) forecasts rather than his analyst's raw forecasts. Moreover, if he generally makes forecasts for a security and if his forecasts are more reliable than those implied by the Capital Assets Pricing Model, he will have available estimates of sS that are smaller than the corresponding value of sS found from proper application of that model. In such a happy case, the former rather than the latter should be used in finding the portfolio-implicit estimates of expected return.

The first alterations of procedure to apply my method to an active portfolio rather than a passive one are (1) to use s, estimated from forecasts used for the active portfolio rather than from the market model alone and (2) to compare portfolio estimates with adjusted forecasts. As with the passive portfolio, in final comparisons best estimates of return are used for the original portfolio and estimates altered to take account of transaction costs are used for the new portfolio. The process may be simpler than in the passive portfolio case, however, because it may be more readily apparent which securities are likely to be bought and which are likely to be sold.

The second consideration raised by Treynor and Black in their analysis is that, in selecting an active portfolio, one should worry not about forecasting return itself but rather about forecasting the excess of return over that implied by the Capital Assets Pricing Model. In applying my method to an active portfolio this consideration is also important. In the Capital Assets Pricing Model the only deter- minant of expected return is the security's beta coefficient. Hence as the statistical estimate of beta changes-as it must if betas are good estimates-the estimate of expected return implicit in the Capital Assets Pricing Model also changes. In comparing one's own with portfolio estimates of expected return, it is important that the assumptions on which they are based be consistent. It is consequently necessary that one takes changes in estimates of beta into account when he revises his own estimates. They will automatically be taken into account if he follows the Treynor-Black procedure of trying to forecast what the market model regards as residual return. Indeed, if the forecasts are in terms of expected "residual" return, Equa- tion 5 can be modified to

78 Z FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975



Ui= (2/i) Xi si (8)

where

Ui = implicit expectation of "residual" return

Then, in seeing whether the expectations implicit in the assumption that the present portfolio is efficient are reasonable, he need only compare the values of Ui with his current (adjusted) forecasts of "residual" return.

IV. Other Problems

Transaction Costs In previous sections, I have asserted that my

method makes it easy to take transaction costs into account. This is precisely true in the formal sense that the E-V criteria are formulated as one-period models and, morever, that one period is infinitesi- mal in length. Hence the cost of changing can be compared directly with the expected benefits.

When transaction costs can be neglected, it has been shown that the one-period model also applies to the case where the investor holds a portfolio or series of portfolios over a long period (Fama, 1972). However, when transaction costs must be considered, the adjustment problem becomes much more complex. This subsection suggests what to do to take transaction costs into account in the more realistic multi-period case.

When there is only one period to go, I can compare the expected utility of holding my present portfolio with the expected utility of an altered portfolio (which has a smaller initial market value but is better diversified). Although the number of comparisons may be substantial, they can be made one by one in the manner described in Section II.

However, when there are many periods to go, even the formal solution of the problem of adjusting given transaction costs becomes rather in- tractable. To see that the problem is difficult, consider the following argument:

As time passes, an inefficiently diversified portfolio will tend to become more and more poorly diversified. Therefore, proper adjustment of the portfolio now will not only provide the benefits of diversification in the coming period but also will reduce the need for future adjustment. However, even if we diversified as much as possible now, we would still find that the portfolio was imperfectly diversified in the future. Hence the costs incurred in adjusting the composition of the portfolio now must be amortized

over a period that is shorter than the investor's time horizon. In a formal sense, the adjustmnent problem with

transaction costs is a stochastic dynamic programming problem (Chen, Jen, and Zionts, 1970).17 But it is not feasible to obtain exact solutions to realistic problems with the present generation of digital computers.'8

Some casual empiricism suggests, however, that assuming a one-year holding period would not be very far from the mark-perhaps longer for passive portfolios and shorter for active portfolios. The assumption of a single one-year period implies that transaction costs would not be incurred unless the benefits expected within one year in the form of higher return or decreased variance would be equal to or greater than the immediate cost. A shorter assumed period would require amortization of costs over a shorter period which would tend to reduce turnover; a longer assumed holding period would tend to increase it. Simulation studies should make it feasible to find the period in which the portfolio would on the average turn over once and thus cause transaction costs to be amortized over the actual period to which they apply.

For example, suppose switching from one stock to another cost an amount equal to two per cent of the value of the stock sold-if one sold a thousand dollars' worth of stock A he could get 980 dollars' worth of Stock B. Then, the cost of switching should be amortized over the expected length of time he will hold the newly acquired stock. If, with an assumed transaction cost of two per cent per annum used to reduce the expected return from any newly acquired shares, the average turnover rate for the portfolio were 100 per cent, then the assumed and actual average holding periods would be the same. However, if the turnover rate were less than 100 per cent, the assumed amortization period should be increased somewhat. Then the adjusted return on potential new stocks will be raised and the actual turnover rate will increase. A quickly conVerging algorithm for finding the appropriate amortization period is given by Equa- tion 9:

L(a)j l fL(a),j L(h)j 9 where

L(a)i= Length of amortization period assumed on the ith iteration

L(h)i= Average length of holding period that is consistent with L(a) .

This algorithm will converge under many but not under all conditions. However, it is important to




note that what I have proposed here is only a rule of thunab and one that has not been thoroughly evaluated. '9

Effect of the Quality of Estimates of Risk The model presented in Equation 5 states that

the expected return implied by the assumption that the currently held portfolio is efficient depends on the estimate of the stock's systematic risk (or beta coefficient). If the stock makes up a substantial part of the portfolio, this implicit portfolio return also depends on the residual variance of the stock's return. Since the accuracy with which beta is estimated affects the residual variance, the useful- ness of Equation 5 depends critically on the reliability of the estimate of beta.

Consider first the "passive" portfolio. Since both the portfolio and the market estimates of return depend on the same beta, the problem is somewhat less critical than if one were getting estimates of return that did not depend directly on beta. However, the reported residual variance must be computed properly or one will hold the wrong number of securities in his portfolio.

The actual value of s2 that must be used in Equa- tion 5 is the value of the mean squared forecast error (MSFE) given the estimate of beta. This MSFE will be the sum of (1) the actual residual variance about the true regression line and (2) the product of (a) the expected squared error of b- as an estimator of beta in the near future and (b) the variance of return of the market factor. In many cases, the service that provides the estimates of beta provides the standard error of beta as an estimate of the average beta over the time period covered by the data used in the estimate. Kamin and I show elsewhere (Fisher, 1970; Fisher and Kamin, 1971 and 1972) that beta changes over time in a manner resembling a random walk. Thus any estimate of beta made from past data will have a much larger standard error as a forecaster of future beta than is implied by the usual methods of estimation. If inappropriate methods have been used to estimate beta, therefore, it is quite likely that si will be reported as being substantially smaller than it really is. If one applies my method with bad data about betas, he will be likely to hold too few securities. He will be disappointed with the results. If values of s. are computed properly but betas are estimated by inefficient methods, residual variance will be correct, given the quality of the betas being used. However, one would have to hold more securities than he would if more reliable estimates of beta were available and he would therefore incur higher managerial costs.

For an active portfolio it is even more important

that accurate estimates of beta be available. Recall that the appropriate variable in Treynor and Black's theoretical framework is the forecast not of total return, not of return in excess of some pure interest rate, but rather return in excess of that implied by the market model-the forecast of "residual return" in the sense introduced by Fama, Fisher, Jensen and Roll (1969). Kamin and I have found that, taken at face value, some kinds of estimates of beta are worse than the simple assumption that all stocks have the same beta. That is, although the b's they give are correlated with actual betas, poor methods-which some people regard as "standard"-will, on the average, yield estimates whose mean squared error is larger than the variance among the actual betas. If one's only choice were to use such estimates at face value or to use the same beta for all stocks, the latter would be preferable. However, it is possible to adjust even these substandard estimates of beta so they can be of some use. It is much better, however, to use the most reliable estimates of beta that can be obtained; otherwise the old data-processing rule of GIGO (garbage in, garbage out) is likely to apply.

If estimates of beta are reliable, then it is feasible to make meaningful forecasts of "re- siduals," rather than forecasts merely of excess return or total return. If estimates of beta are poor, it will be difficult for the analyst to decide whether he should change his forecast of total return merely because somebody's report about beta has changed. Then it will be harder for the analyst to think in terms that are relevant for the use of Equations 5 or 8. With reliable estimates of beta and appropriate estimates of sS at hand, using my method for adjusting either active or passive portfolios should be a feasible exercise of the analyst's talent.

V. Summary If one attempts straightforward application of the idea of E-V efficient portfolios to the actual management of portfolios through time, he is likely to find that the computer algorithms generate succes- sive sets of "efficient" portfolios that have little in common with each other. Because holding these portfolios successively would generate huge transaction costs, the concept of E-V efficiency has seen little practical application.

In this article, I have proposed reversing the order of analysis. Instead of providing data on expected returns and risk to the computer program and having it report the "efficient" set of portfolios, I suggest providing the computer with data on risk and the composition of the present portfolio and then asking the computer to tell us what

80 O FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975



expected returns are implied by the data on risk, the present composition of the portfolio, and the assumption that the present portfolio is efficient. Then the portfolio may be adjusted by comparing these implicit portfolio estimates with estimates that the analyst thinks are better.

If the analyst thinks all of the "portfolio" estimates are reasonable, his task is complete. If he thinks that some of them are wrong, he re- estimates the expected return and risk of the present portfolio. Then he substitutes a "reasonable" estimate of return for the security with the worst portfolio estimate and allows the computer to find a new but only tentative set of efficient portfolios. He then looks at the E and V of the new but tentative set (using the reasonable estimates for the computations) and sees whether it would be worthwhile to incur the incremental cost of switching to the tentative portfolio. If it is worth the cost of switching, he repeats the procedure substituting one estimate at a time until he has made as much adjustment as it pays to make.

My procedure may be applied to either "passive" or "active" portfolios (in the sense defined by Treynor and Black).

In applying my method, one takes up changes in the composition of the portfolio a step at a time and goes only as far as it pays to go. Inherent in this step-by-step procedure is the consideration of transaction costs. Unfortunately, no theoretically sound empirical method of taking transaction costs into account has been found for the case where the portfolio will be held for many periods, although the problem is being actively pursued. Therefore, it is necessary to suggest rules of thumb for trans- lating commission and other transaction costs into quantitative adjustments of expected return.

It is also necessary to be careful in selecting the statistically derived estimates of either systematic risk (beta) or specific risk (si).

If this care is taken, the analyst may find application of my method rewarding. m

APPENDIX Derivation of the Expression for

Implicit Estimate of Expected Return

In this appendix, I shall derive the equations and inequalities for finding the estimate of expected return for each security under consideration that is consistent with the assumption that the currently held portfolio is efficient. The notation and the first part of the analysis follow that in William F. Sharpe, Portfolio Theory and Capital Markets (New York: McGraw-Hill, 1970), Sec. B, pp. 244- 247.

The "basic" portfolio selection problem is stated as:

Minimize

Z = -REp + Vp (Al)

where

N

Ep =EXiE (A2)

and

N N

VP= XiXjCij , (A3)

subject to

N

Ex =1 . (A4) i=1

In these equations

Z = the function defined,

Ep = expected return of the portfolio,

Ei = expected return of the 'th security,

Xi = fraction of portfolio invested in the ith security,

VP = variance of return of the portfolio,

Cii = covariance of return between the ith and jth securities (n.b., by definition, the variance of return for the ith security, V, = CH), and

A = the slope, i.e., dV/dE, of the "efficient frontier" in the E-V plane.*

* In the process of actually solving the problem of describing the entire set of E-V efficient portfolios, the critical-line method, for example, requires that one begin by making A infinite. In this case, the only efficient portfolio is the one made up entirely of the security that has the greatest expected return. Then A is successively reduced and the changes in the composition of the efficient portfolio are noted. As A is reduced, the nature of the changes depends on the constraints placed on the solution. In the "basic" problem, the only constraints are state of nature-in the form of expected returns, variances, and covariances-and the requirement that the net amount in the portfolio be equal to the resources available, i.e., that Xi = 1. In the "standard" problem, there are constraints on the maximum and/or minimum value of an XN, such as 0 < X < .05. See Markowitz (1959, cc. 7 and 8), or Sharpe (1970, ch. 4 and supp., Secs. B and C).

FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975 C 81



The process of solving the basic problem requires consideration of a new objective function:

N

Z'=Z + f(1-ZX ) (A5)

where

a Lagrangian multiplier that guarantees satisfaction of the constraint in equation A4.

Equation A5 may be restated as

N

Z'= -)Ep + VP + ;f 1I . E Xi) .(A 6) i=l

For the value of . in question, the problem is solved when X (the vector X,, X2, ..., XN) is chosen so that

Z'/8X = 8Z'/aX2 = ... - 8Z'/aXN

=aZ'/a8'f = O (A7)

But

dZ' 8X, = aZ 8Xj -'f

=-; 8E p / ax? + avp /,Xi- f .(A8)

Moreover,

Ep /Xi= E*; (A9)

and

N

aVp/Xi= 2 7CcXj . (AIO)

Thus N

z'/x =-;,Ei-Af+2ECijXj . t(Al1) j~i

Since, by equation A7,

N

iEi + .f = 2 CiJXJ . (A12) j=l

Suppose there were a riskless security, for which all covariances of return-including the covariance with own return, i.e., including the variance of return-were zero. Assign this security subscript

number 1. For such a security, the right hand side (r.h.s.) of Equation A12 would be zero. Then, if any of the riskless security were actually held,

El=-lf/;, . (A13)

If no such security exists or, even if it does exist, it is not held-and not sold short-in the portfolio in question, then El will represent a constant term in what follows. If the "first security" is held, El is both the constant term of what follows and the riskless rate of interest per time unit equal to whatever time unit applies to the statement of other returns. We may now state the equation that gives the difference between the expected return on a security that is held in an efficient portfolio and the riskless rate (or EI) up to a constant of propor- tionality:

N

Ej::= El+(2/i XXCj . (1)

If a security is not held in the efficient portfolio in question, then we know that its expected return must be less than the r.h.s. of Equation 1. There- fore, we may replace the equal sign of Equation 1 and write Inequality 2 as shown in the text.

To obtain Inequality 3, we note that, when Xi may be equal to zero but all other values of Xi that are less than Xmmi are excluded, observation that a particular security is not represented in an efficient portfolio merely tells us that, if there were no excluded region for the security, we would have set Xi < Xmin. If the desirable amount, in the absence of the forbidden region, were infinitesimally less than Xmin, then, to find the upper limit on the possible range of Ei consistent with the idea that the portfolio is efficient, we must set Xi equal to Xmi and thereby reduce the values of fractions of securities actually held (X) by multiplying them by (1-Xmin). Since, for a security not held the value of the subscript i lies outside the range 1, .. ., N, Xmin Vi appears as an explicit term in Inequality 3 of the text.

Simplification The diagonal model suggested by Markowitz

(1959, pp. 97-101) and developed by Sharpe (1963) assumes that there is some market factor RM that is a random variable. It also assumes that each security's return R, is a random variable and that the random variables are related according to the equation

RJi k+SIRM+ ii (A14) t Thus far, the analysis has followed that given by Sharpe (1970).

82 0 FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1975



where

ui = a random "residual" return,which has a mean of zero, variance a2, and is uncorrelated with iij for all i #j.

Note that if it is valid, Equation A14 merely states a relationship as of the moment at which the portfolio is chosen. The econometric problems of the estimation of ai, ,3;, and o-J-as well as VM -have a growing literature of their own.

From Equation A 14 and the assumed in- dependence among the distributions of 'u, Mark- owitz (1959, p. 100) obtained (but expressed in different notation)

\K= Cji3,VM+ , (A15)

and

Ci- = fiSj fli vm(A16)

Hence, if the diagonal model is valid, Equation 1 may be restated as

N

Ei = El + (2/i)(Xi Si2+ XS:j (Al17) j-1

or

N

Ei= El+ (2/,.) (Xij+ST 2

Xj3jVM ) . (A18) j=l

But N

fp = XJjXJ . A19) j=1

Therefore, we may write

Ei= El + (2/i) (Xi ai + /j fPVM) . (A20)

In this article we are interested in finding an estimate of expected return under the assumption that the portfolio now held is efficient. In fact, we have no way of knowing what the actual values of the vectors o.2 and ,B are or what the actual value of VM is. We must rely on estimates. To make the distinction clear, equation A20 is restated in the text as

Ei = E, + (2/i) (bi bPVM + XiS2) , (5)

where the symbols are as defined in the text. i

Footnotes 1. Of course, you might hold the "market portfolio," in

which all possible securities are held in the same proportions as they bear to the total. If one accepts a number of assumptions, e.g., that all potential investors agree on the risk and return of all securities, that transaction costs can be neglected, such a portfolio must be efficient. (See Sharpe, 1964; Lintner, 1965; Fama and Miller, 1972; etc.) However, such a market portfolio can never be achieved exactly for a variety of reasons-primarily because for most investors the income that they expect to receive from selling their own services in the future constitutes at least a substantial fraction of their "assets." Since unconditional lump-sum sales of lifetime personal services are not permitted in free societies, no actual portfolio can be a market portfolio. Moreover, it appears that through time compensation for services is less than perfectly correlated with income to property. Hence no actual portfolio can be perfectly correlated with the idealized market portfolio. Thus, the holding of a market portfolio, while an extremely useful concept for theoretical analysis, can only be approximated to a greater or lesser extent by actual investors.

2. As applied experimentally, this modification seemed also to invariably produce portfolios that contained only the minimum number of securities. I suspect that this is the result that led Sharpe to develop still further simplifications (e.g., Sharpe, 1967).

3. Or one of a set of rules that use only a limited number of parameters, see Fama (1965).

4. See, for example, almost any of the works cited thus far or equations A2 and A3 of the appendix.

5. However, the framework for analysis given in this paper would be usable even by an investor who really thought that security prices changed independently of one another.

6. Note, however, that although Markowitz's text discusses a relationship between return and the change in the level of an index, his mathematical summary description (fn., p. 100) and Sharpe's article are in terms of the level of the index. Those ex- pository errors have led to a substantial amount of invalid empirical work by other authors.

7. Provided other prices were unchanged. 8. The idea of a minimum-cost balanced diet for

people was the reason for one of the original formu- lations of linear programming-Stigler's (1945).

In developing the Capital Assets Pricing Model, Sharpe's own analysis of the equilibrium prices of securities was one in which the market was treated as a single efficient portfolio. Then the "dual" suggested that there was an implicit set of equilibrium prices (Sharpe, 1964-however, Sharpe had made oral presentations of the argument at least as early as February 1962). Jack Treynor's unpublished paper (1961) that also arrived at the Capital Assets Pricing Model used a method of argument more nearly analogous to much earlier theories that distinguished between insurable "risk" and uncertainty, e.g., t Sharpe (1963, sec. V).




Knight (1921), and the notion of equilibrium in the capital markets as, for example, in Modigliani and Miller (1958).

9. However, we note that if the rule being followed requires that any security that enters the portfolio at all be held in a substantial amount, e.g., making up at least one or two per cent of the portfolio, the true upper bound on the implicit return of the stock is between that of Inequality 2 and

Ei < E I + (2 /i) [Xminvi N

+ (1 -Xmjn) E XjCij ] (3) j=1

where

Xtnin= the least positive fraction of the portfolio that will be invested in a security (Rule - Hold none or a fraction at least as large as Xminl)-

Finding that Inequality 2 is satisfied means that the security will not be held. If Inequality 3 is not satisfied, then the security should be added to the portfolio (provided Equation 1 holds for all securities in the portfolio and Inequality 2 holds for all other securities not in the portfolio). However, if In- equality 3 holds and Inequality 2 does not hold, it may be desirable to include the security. My hunch is that a finer screen is provided by modifying In- equality 3 to replace Xmnn by

X min = Xmin 2 O 2 0.7 Xmin . (4)

It should also be noted, though, that if Xmin is about 0.02, the difference between the right-hand sides of Inequalities 2 and 3 is likely to be only about 0.0003 per month or 0.0036 (36 basis points) per annum.

10. Note that Equation 5 is equivalent to one of Lint- ner's (1965). If VM is interpreted as referring to a "market factor" rather than a "market index," it is also equivalent to Fama's Equation 27 (Fama, 1968). Note, however, that Fama's and Lintner's Xi and X. refer to fractions of securities in "market" portfolios, while my Xi and Xi refer to the fractions in the portfolio actually held. If all investors agreed exactly on their estimates of risk and expected return for all securities and if there were no taxes or other transaction costs, then, of course, the assumption that a portfolio was efficient could never hold unless the only risky assets included could be described as a portion of the market portfolio.

11. The slope parameter may be available if the portfolio was chosen or previously modified with the aid of a portfolio-selection computer program.

12. If the difference exceeded the apparent savings, then it would indicate that too small a portfolio was being held. If the difference were less than half the apparent savings, then we might infer that too many securities were held.

13. Equation 7 applies precisely if the index used represents the market factor. However, if the index used represents the market "portfolio," the variance

of the index should be reduced to arrive at an estimate of the variance of the market factor. See Fama (1968). The net effect of such an adjustment would be to increase bp by about the same percentage that VM was reduced. Hence, the adjustment would tend to increase all estimates of VP.

14. The quadratic program is likely to report a large set of corner portfolios (see Sharpe, 1963). The portfolios to be compared could then either have the same value of bp or be most nearly similar in overall composition. Similarity may be compared by finding the mean squared difference between corresponding values of X. If the portfolios to be compared are lending portfolios, there will be little difficulty since the "new" efficient set will contain a single combination of risky assets. If borrowing is permitted and desired, the situation is similar. In these two cases the comparisons should probably hold bp, or the amount lent, or the amount borrowed constant. If the holding of riskless assets is exactly zero rather than positive (with lending) or negative (with borrowing) then either hold bp constant or compare the old portfolio with the new one that could be bought for the smallest payment of taxes and commissions.

15. With appropriate knowledge about the amount to alter estimates of return, the formal process could be reduced to having the quadratic programming code use a pair of "securities" for each real security that appears in the original portfolio. The variance and covariance for two members of the pair would reflect the fact that their outcomes are perfectly correlated. The altered expected return on the first member of each pair would be the raised one. The altered expected return on the second member of the pair would be the lowered one. The portfolio would be constrained so that the maximum amount of any first security in a pair would be the amount in the original portfolio and so that the minimum holding of any security including those not currently held (except, perhaps, riskless borrowed money) would be zero. The basic difference between this procedure and the one discussed in the main body of the text is that this procedure would require simultaneous sub- stitution of a set of pairs of best estimates for the estimates derived by assuming that the original portfolios were efficiently diversified.

16. When the diagonal model is used, as we are doing here, it seems unlikely that this stepwise procedure will cause us to stop just short of the point where a major improvement would have been obtained if only a few more steps were taken.

17. Samuelson's similar conclusion applies in the absence of transaction costs.

18. Jules Kamin (1973) has shown that it is feasible to reduce the problem of periodically adjusting the composition of a portfolio containing two risky assets to a one-state-variable dynamic programming problem when transaction costs are proportional to the size of the transaction and the investor's utility function is of the type known as myopic or iso- elastic (cf. Samuelson, 1969; Mossin, 1968).

He has also written a computer program for per- forming the dynamic program and has analyzed the nature of optimum solutions for a number of



simulated cases. However, even if the basic nature of his solution applies to many-stock portfolios (a sur- mise which, at the moment, is unproved), exact solution appears to require analysis of an N-minus- one-state-variable dynamic program.

Unfortunately, in dynamic programming problems, the number of machine instructions that must be executed in a k-state-variable analysis is approximately equal to the number of instructions for a one-state variable problem raised to the kth power. Since computers execute several million instructions per dollar and Kamin's two-security program costs a dollar or so to run, it appears likely that the exact solutions for three or more securities will not be economically feasible with this or the next generation of computers-unless a better form of algorithm is found.

19. It is certainly conceivable that there would be some securities that one would expect to hold for many years and others that would be held only for very short periods. In that case, the rule of amortizing transaction costs over the same period for all securities or even for all lots of a given security can be only a rough approximation. We can quickly think of other 'rules of thumb." For example: We might suppose that the average holding period for all the shares of an issue in the portfolio would be inversely proportional to the product of X. and s2, etc.

References Blume, Marshall E., "Portfolio Theory: A Step Towards

Its Practical Application," Journal of Business, Vol. 43, No. 2 (April 1970), pp. 152-173.

Chen, A.H., Jen, F.C. and Zionts, S., "The Optimal Portfolio Revision Policy," Journal of Business, Vol. 44, No. 1 (January 1971), pp. 51-61.

Fama, Eugene F., "Portfolio Analysis in a Stable Paretian Market," Management Science, Vol. 11, No. 3 (January 1965), pp. 404-419.

, "Risk, Return, and Equilibrium: Some Clarifying Comments," Journal of Finance, Vol. 23, No. 1 (March 1968).

, "Risk and Rate of Return in the Long Run," Proceedings of the Seminar on the Analysis of Security Prices (November 1972), Center for Research in Security Prices, University of Chicago.

Fama, Eugene F., Fisher, Lawrence, Jensen, Michael C. and Roll, Richard, "Adjustment of Stock Prices to New Information," International Economic Review, Vol. 10, No. 1 (February 1969), pp. 1-21.

Fama, Eugene F. and Miller, Merton, Theory of Fi- nance (New York: Holt, Rinehart & Winston, Inc., 1972).

Fisher, Lawrence, "The Estimation of Systematic Risk: Some New Findings," Proceedings of the Seminar on the Analysis of Security Prices (May 1970), Cen- ter for Research in Security Prices, University of Chicago.

Fisher, Lawrence and Kamin, Jules, "Good Betas and Bad Betas: How To Tell the Difference," Proceed- ings of the Seminar on the Analysis of Security Prices (November 1971), Center for Research in Security Prices, University of Chicago.

, "On the Estimation of Systematic Risk," Journal of the Midwest Finance Association, 1 (1972).

Jensen, Michael C., "The Performance of Mutual Funds in the Period 1945 through 1965," Journal of Finance, Vol. 23, No. 2 (May 1968), pp. 389-416.

, "Risk, The Pricing of Capital Assets and the Evaluation of Investment Portfolios," Journal of Business, Vol. 42, No. 2 (April 1969), pp. 167-247.

Kamin, Jules, "Effects of Transaction Costs on the Ad- justment through Time of the Composition of Port- folios Held by Risk-Averse Investors: Implications for Capital Market Theory and the Optimum Strategy for Two Risky Assets," unpublished Ph.D. dissertation (December 1973), Graduate School of Business, University of Chicago.

King, Benjamin F., "Market and Industry Factors in Stock Price Behavior," Journal of Business, Vol. 39, No. 1, Part 2 (January 1966), pp. 139-190.

Knight, Frank H., Risk, Uncertainty, and Profit (New York: Hart, Schaffner & Marx, 1921).

Lintner, John, "Security Prices, Risk, and Maximal Gains from Diversification," Journal of Finance, Vol. 20 (December 1965), pp. 587-616.

Markowitz, Harry, "Portfolio Selection," Journal of Finance, Vol. 7 (March 1952), pp. 77-91.

, Portfolio Selection: Efficient Diver- sification of Investments (New York: John Wiley & Sons, Inc., 1959). Page references are to the second printing (New Haven: Yale, 1970).

Mossin, Jan, "Optimal Multiperiod Portfolio Policy," Journal of Business, Vol. 41, No. 2 (April 1968), pp. 215-229.

Samuelson, Paul A., "Lifetime Portfolio Selection by Dynamic Stochastic Programming," Review of Economics and Statistics, Vol. LI, No. 3 (April 1969), pp. 239-246.

Sharpe, William F., "A Simplified Model for Portfolio Analysis," Management Science, Vol. 9, No. 3 (January 1963), pp. 277-293.

, "Capital Asset Prices: A Theory of Market Equilibrium, Under Conditions of Risk," Journal of Finance, Vol. 19, No. 3 (September 1964), pp. 425- 442.

, "A Linear Programming Algorithm for Mutual Funds Portfolio Selection," Management Science, Vol. 13 (March 1967), pp. 499-510.

, Portfolio Theory and Capital Markets (New York: McGraw-Hill Book Company, 1970).

Stigler, G.J., "The Cost of Subsistence," Journal of Farm Economics, Vol. 27 (1945), pp. 303-314.

Treynor, Jack L., "'Toward a Theory of the Market Value of Risky Assets," unpublished, 1961.

Treynor, Jack L., Priest, William W., Fisher, Lawrence and Higgins, Catherine A., "Using Portfolio Com- position to Estimate Risk," Financial Analysts Jour- nal (September/October 1968).

Treynor, Jack L. and Black, Fischer, "How to Use Security Analysis to Improve Portfolio Selection," Journal of Business, Vol. 46. No. 1 (January 1973), pp. 66-86.



Article Contentsp. 73p. 74p. 75p. 76p. 77p. 78p. 79p. 80p. 81p. 82p. 83p. [84]p. [85]

Issue Table of ContentsFinancial Analysts Journal, Vol. 31, No. 3 (May - Jun., 1975), pp. 1-96Front Matter [pp. 1-57]Editor's Comment: If You're so Smart, Why Ain't You Rich? [p. 6]Letters to the EditorMoxibustion Explained [p. 8]Defense Unnecessarily Weak [p. 8]Kerrigan Responds to Critic [pp. 8-9]

Securities Law and Regulation: Fiduciary Responsibility under the 1974 Pension Act [pp. 10+12+14+16+89-92]Editorial Viewpoint: On the Monetary Role of Gold [p. 18]Unemployment and Inflation [pp. 21-23+26-28]Inflation Accounting: Public Utilities [pp. 30-34+62]Musings of a Portfolio Manager [pp. 37-40]Interpreting Analysts' Recommendations [pp. 42-46+48+62]Homogeneous Stock Groupings: Implications for Portfolio Management [pp. 50-56+58-62]Active Portfolio Management: How to Beat the Index Funds [pp. 63-72]Using Modern Portfolio Theory to Maintain an Efficiently Diversified Portfolio [pp. 73-85]Portfolio Diversification Strategies [pp. 86-88]Book ReviewsReview: untitled [pp. 93-94]Review: untitled [p. 94]

FAF Newsletter [pp. 95-96]Back Matter

using modern portfolio theory to maintain an efficiently diversified portfolio

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