INFORMS Open Forum

Discover more about Harry, his accomplishments, and his life through this heartfelt tribute of appreciation by INFORMS member, John Guerard. 

Harry Markowitz: An Appreciation

John Guerard

Scientific Advisory Board, McKinley Capital Management, LLC

Anchorage, AK

jguerard@mckinleycapital.com

Professor Harry Markowitz passed on June 22, 2023; some four years short of reaching 100 years old. Dr. Markowitz was not a traditional economist. That fact was well-established and documented from his thesis defense at the University of Chicago. When Milton Friedman uttered lines to the effect that Dr. Markowitz's thesis has nothing wrong with it, but is not an economics dissertation, Dr. Friedman applied a very narrow definition of economics. Harry, as I will hereafter refer to him, is acknowledged as a (the) creator of Portfolio Theory. His dissertation was its genesis. Harry was not a traditional person at all. Harry was included in a Springer monograph, Profiles in Operations Research (2011), as were several economists, several of whom won, or were finalists, for the Nobel Prize. Yes, for his work in portfolio theory, sparce matrices and simulation, Harry was awarded the 1989 John von Neumann Theory Prize from the Operations Research Society of America. Using ideas from sparce matrices, Harry developed an approach and a pivot selection rule, now called Markowitz rule,  which applied to a large-scale problem, reduced the generation of the transformed matrix's non-zero elements. The sparsity of the transformed matrix is the key to solve large scale linear equations efficiently.

In 1990, Harry, William F. Sharpe, hereafter "Bill", and  Merton Miller shared the Nobel Prize in Economic Sciences. In two years, Harry had reached the pinnacle of two disciplines, economics and operations research.  During these two years, Harry was the chaired Professor of Finance at Baruch College, City University of New York, and a co-head of research at Daiwa Securities Trust Company, in Jersey City, New Jersey.  This is who Harry was in 1990. Harry belonged to a very small subset of mankind, who were great thought-thinkers and even smaller subset who were great thought-thinkers and accomplished the objectives of their thoughts.

Harry Markowitz was born in Chicago on August 24, 1927. It has been reported for years that Harry's father was a grocer. Harry told many of his followers and audiences that he was not aware of The Depression growing up because his family always had food. Moreover, he told his audiences that he enjoyed comic books and reading, although he was not a great student, but passed an entrance exam to gain admission to The University of Chicago. There he read David Hume and studied liberal arts, graduating in 1947. Harry entered the graduate program in Economics where he studied under economists, including  Milton Friedman, Tjalling Koopmans, Jacob Marschak, and Leonard Savage. Professor Tjalling Koopmans was the Director of the Cowles Foundation. Harry did well at Chicago enough to be invited to become a Research Fellow at the Cowles Foundation in 1950-1951. Harry would earn an MA in Economics in 1952 and Ph.D. in 1954.

Why should forecasters care about Harry Markowitz? Harry's portfolio construction process required three inputs; models of expected returns, covariances, and portfolio constraints.  Analysts' earnings forecasts, revisions, and the direction of revisions (breadth) are key inputs to expected returns modeling.  Time series models have been used to forecast expected returns, often combining analysts' and time series models, and models of covariance. Furthermore, the Markowitz portfolio returns were subjected to his Data Mining Corrections test to statistically "verify" that the expected returns model's portfolio return did not occur by chance. Did we use only data available to "the little man inside the computer" was Harry's question. Did we get luck, is Harry's second question. How much future outperformance could be expected, as Harry's third question. His fourth question was what have you done for me lately. Model parameters should be re-estimated over time and models enhanced. Does the original model still work, after 20-30-years, and have the enhancements maintained their statistical advantages over previous model estimations?

1.     The Cowles Foundation: Behavioral Finance and Portfolio Selection

Why was the Cowles Foundation so important to Harry? The Cowles Foundation for Research in Economics at Yale University has as its purpose the conduct and encouragement of research in economics. The Cowles Foundation seeks to foster the development and application of rigorous logical, mathematical, and statistical methods of analysis.[1] The Cowles Foundation continues the work of the Cowles Commission for Research in Economics in publishing papers and monographs. The Cowles Foundation was founded in 1932 by Alfred Cowles at Colorado Springs. Mr. Cowles authored several papers on forecasting stock prices during the Depression. The Cowles Commission moved to Chicago in 1939 and was affiliated with the University of Chicago until 1955.  Cowles Foundation Director Tjalling Koopmans invited Harry to become a Research Associate in 1950-1951. Other Research Associates in 1950-1951 included Carl Christ, Gerard Debreu, Jacob Marschak, and Leonid Hurwicz.[2] The Cowles Foundation adopted its motto "Theory and Measurement," in 1952. It was during the 1950 -1951 period that Harry researched three Cowles Foundation papers that would be published in 1952 while he was at the Rand Corporation. The first paper was Cowles Foundation Paper No. 57, "The Utility of Wealth", better known to the world as "The Utility of Wealth", Journal of Political Economy 60 (1952), pp. 151-158.  The JPE, as the Journal of Political Economy is popularly known, has been a top-four economics journal since its founding under James Laurence Laughlin in the early 1890s. The Markowitz paper on utility theory was a contradiction to the Friedman and Savage classic paper, which sought to explain the existence of insurance and lotteries.[3] Instead of different social classes, as Friedman and Savage hypothesized, Harry based his utility discussion on income and economic observation. Harry stated that lower income persons would not accept a fair (lottery) bet below a certain level, nor would higher income persons accept a fair lottery bet above a certain level.  People with wealth levels between the lower wealth and the higher wealth level would accept a fair bet. [4] As Harry stated on p. 152:

"We do not observe persons of middle income taking large symmetric bets. We expect people to be repelled by such bets. If such a bet were made, it would certainly be considered unusual and probably irrational."

The bet which this person would like most, according to the Friedman and Savage (F&S) hypothesis, is one which if won would raise him to rich level of income, if lost would lower him to the lower level of income. The investor would be willing to take a small chance of a large loss for a large chance of a small gain. Contrarily,  he would be anxious to underwrite insurance. He would even be willing to extend insurance at an expected loss to himself.

Harry, on p. 153 stated:

"Again such behavior is not observed. On the contrary we find that the insurance business is done by companies of such great wealth that they can diversify to the point of almost eliminating risk. In general, it seems to me that circumstances in which a moderately wealthy person is willing to risk a large fraction of his wealth at actuarially unfair odds will arise very rarely."

Harry believed that the Friedman and Savage utility function was inconsistent with economic behavior.  The Markowitz utility function was shaped like a snake, changing second derivatives as wealth increased. Harry noted on page 155:

"To summarize my hypothesis: the utility function has three inflection points. The middle inflection point is defined to be at the "customary" level of wealth. Except in cases of recent windfall gains and losses, customary wealth equals present wealth. The first inflection point is below, the third inflection point is above, customary wealth. The distance between the inflection points is a nondecreasing function of wealth. The curve is monotonically increasing but bounded; it is first concave, then convex, then concave, and finally convex. "

The Markowitz utility theory paper became one of the great foundations of behavioral economics and has had a profound impact on researchers in behavioral finance.

The second paper,  "Portfolio Selection", Cowles Foundation Paper No. 60, is better known to the world as "Portfolio Selection", Journal of Finance 7 (1952), pp. 77-91. Harry developed the concept of the Efficient Frontier, where efficient portfolios offered the greatest returns for a given risk, or the minimum risk for a given level of return. Investment managers should not buy low price-to-earnings stocks, as advocated by Graham and Dodd (1934) and John Burr Williams (1938) without assessing their risk. It was risk and return that must be considered for an optimal portfolio. The Efficient Frontier required statistical inputs of stocks' expected returns and  the measurement of risk, the stocks' variances and the covariances among the stocks. Furthermore, risk could be reduced  by investing in assets with lower covariances.  Harry emphasized the role of covariances. Investment managers should systematically create portfolios to maximize return for a given level of risk.

The concept of the efficient frontier can now be found in most textbooks on financial management or investment management and analysis. The efficient frontier is used to discuss why many investors prefer stock-intensive portfolios to cash or bond investments, why smaller-capitalized stocks may be preferred to larger-capitalized stocks, and why portfolios composed of global stocks are very often preferred to portfolios composed of domestic-only stocks. Before Harry's work, investors described in words what was meant by the risk–return trade-off; after Harry, investors were able to precisely quantify, in mathematical terms, the risk–return trade-off (Rubinstein 1991). Investors tended to disregard the risk of assets. The risk–aversion of the investor identified the portfolios to choose among the portfolios on the efficient frontier. If one seeks to maximize the long-run rate of return of the portfolio and maximize the expected utility of (final) terminal wealth, then one selects the portfolio that maximizes the excess return-to-risk measure, known as the Sharpe Ratio. This portfolio maximizes the geometric mean of the strategy, as well (Bloch et al. 1993; Markowitz 1959: Levy 2012).

The third paper of 1952 was Cowles Foundation Paper No. 67, co-authored with Leo Goodman, on individual social welfare functions published in the American Journal of Sociology (1952).

The Cowles Foundation Research Consultants in 1950 -1951 included Kenneth Arrow, Harold Davis, William Hood, Trygve Haavelmo, Lawrence Klein, Franco Modigliani, and Herbert Simon. Why do we stress the numbers of the papers? Harry did, when he talked to friends and researchers. He referred to his 1952 papers by their Cowles Foundation numbers. Herbert Simon published Cowles papers 47 and 70; Kenneth Arrow published papers Numbers 54 and 62; Lawrence Klein authored papers 55 and 69; and Paul Samuelson authored Cowles Paper Number 61. All these Cowles papers were published in 1951-1952! Of this Cowles Foundation group, Harry, Debreu, Arrow, Klein, Modigliani, and Simon won Nobel Prizes in Economics. Harry received some of the greatest training available to American economists.

In 1955, the professional research staff of the Commission accepted appointments at Yale and, along with other members of the Yale Department of Economics, and Cowles moved to yale. Harry took a leave from Rand and went to Yale, at the invitation of Professor and the new Cowles Foundation Director James Tobin. From August 1955 to May 1956, Harry was at the Cowles Foundation at Yale, supported by a grant from the Merrill Foundation for Advancement of Financial Knowledge. It was at Yale that Harry penned his Portfolio Selection: Efficient Diversification of Investments (New York: John Wiley & Sons 1959). [5] Harry expected educated persons to be familiar with his book. He expected readers to remember Chapter 5 was on diversification and how risk could be altered by covariances and the number of stocks. Chapter 6 introduced the Geometric Mean to its readers, and its promise to produce the greatest return in the long run. Chapter 7 was the geometric analysis of the Efficient Frontier, one seen in his 1952 Journal of Finance (JF) paper. Chapter 8 crunched the numbers on the Efficient Frontier. Chapter 9 was on the semi-variance, or downside risk.  Chapter 10 was the maximization of expected utility and Chapter 12 was applications to portfolio selection. In some 350 pages, Harry wrote the Cowles Foundation monograph 16, Portfolio Selection, that changed the world of investments. Harry always referred to his book to the author as Cowles Foundation 16. Of the first 21 Cowles Foundation monographs, 11 were authored or co-authored by Nobel Prize winners. Harry always enjoyed showing younger researchers the list of Cowles Foundation monographs and telling them of he, Koopmans, Klein, Debreu, and Tobin and the Cowles Foundation researchers brought "Theory and Measurement" to Economics.

2.     RAND, Simulation, and Arbitrage

Harry had not completed his dissertation despite these three major publications. Harry's early career was both extremely productive and he was very ambitious. The author worked with Harry on the Markowitz chapter in Profiles of Operations Research (Guerard, 2011), to completely understand his employment between 1952 to 1971. This section draws heavily on that chapter. In 1952, he felt that it was time to find a job. While attending a meeting of the American Economic Association, he met members of the RAND Corporation's economics department. At the time, RAND was a research organization in Santa Monica, California, supported by the U.S. Air Force. RAND expressed interest in Harry's research and offered him a job at 50% more than an offer he already had from a university. Harry left the University of Chicago for RAND in early 1952. Harry's initial RAND research was applying linear programming (LP) to economic problems. Harry's first exposure to LP occurred at RAND when he was asked to read George Dantzig's paper on the simplex method and to supervise the computer programming and the running of RAND's first simplex code on an IBM Card Programmed Calculator (CPC). George Dantzig, the developer of LP and the simplex method, joined the RAND staff in June 1952, and over time, enhanced RAND's ability to solve LP problems.

In the 1950s, RAND was using computer-based simulation procedures to evaluate military situations, with special emphasis on war games. To that end, RAND created a logistics laboratory within its Economics Department. Although Harry was familiar with simulation ideas and techniques as applied to industrial operations and warfare research, his first hands-on exposure to such simulations was when he was assigned to the laboratory to coordinate the development of its computer-based simulation models (Markowitz 2002). The laboratory's first simulation model, called LP1, ''was a man-machine simulation in which actual air force logistics officers played the role of air force logistics officers. Some time after LP1 was finished, Harry received a job offer from the General Electric (GE) Company and accepted a position within the Manufacturing Services Department located in New York City. Alan Rowe was now with GE and was the supervisor of the programming of a large, detailed job-shop simulator He applied his ideas to the building of the GE Transformer Department's shop simulator, the General Electric Manufacturing Simulator (GEMS). Although GEMS was well received at GE, it was not as flexible as Harry hoped it would be. Rather than enhance GEMS at GE,  where it would then be proprietary, Harry returned to RAND to put his ideas to work. At RAND, Harry teamed up with Bernard Hausner and, with the help of Herbert Karr, developed SIMSCRIPT, a very powerful, influential, and long-lived computer-based simulation system (Markowitz et al. 1963). Hausner was a RAND computer programmer and was responsible for the development of the computer-based simulation language using Harry's novel simulation concepts of entities, sets, and events. The initial name for the language was Simulation Programming System One (SPS1).  SIMSCRIPT introduced some novel concepts. It was ''designed to facilitate the programming of 'discrete event' simulation models, especially 'asynchronous' discrete event simulators, as compared to continuous time or difference equation models'' (Markowitz 2002, p. 157).

Sparse matrix methods are now widely used to solve very large systems of simultaneous equations whose coefficients are mostly zero. SIMSCRIPT has been widely used to program computer simulations of manufacturing, transportation, and computer systems as well as war games. The company that would become  CACI International was founded by Herb Karr and Harry Markowitz on July 17, 1962 as California Analysis Center, Inc. They helped develop SIMSCRIPT, the first simulation programming language, at RAND and after it was released to the public domain, CACI was founded to provide support and training for SIMSCRIPT. Under contract to IBM, Harry, based on SIMSCRIPT II, helped to develop an experimental programming language integrated with a database management system, the EAS-E system (Markowitz 2002). EAS-E (pronounced EASY) was built around the entity, attribute, and set (EAS) view of application development. IBM used EAS-E for an internal application, but it was never released as a product. CACI developed and marketed a proprietary version of SIMSCRIPT II, SIMSCRIPT II.5. In 1968, Harry left CACI and, for the first time, joined academia as a professor at the business school of the University of California-Los Angeles (UCLA). In 1968, Markowitz joined Arbitrage Management company founded by  Michael Goodkin. Working with Paul Samuelson and Robert Merton, he created a hedge fund that represents the first known attempt at computerized arbitrage trading. He took over as chief executive in 1970. After a successful run as a private hedge fund, AMC was sold to Stuart & Co. in 1971. A year later, Markowitz left the company. During his academic career, Harry taught MBA and Ph.D. level courses in investments and portfolio management. He was a professor at UCLA (1968–1969), University of Pennsylvania's Wharton School (1972–1974), and Rutgers University (1980–1982). From 1974 to 1983, he was a staff member at IBM's T. J. Watson Research Center, Yorktown Heights, New York. In 1982, Harry was President of the American Finance Association, one of two presidents of the association from industry in the past 40 years and was appointed the Marvin Speiser Distinguished Professor of Finance and Economics at Baruch College, City University of New York. In 1993, he retired from Baruch College as Distinguished Professor Emeritus. Harry moved to San Diego in 1993, where he lived with his wife Barbara. He was an adjunct professor at the Rady School of Management, University of California, San Diego.

3.     The Nobel Prize, Daiwa Securities, and Applied Investment Research at McKinley Capital Management

Harry continued to consult during his academic tenure, serving, from 1984, as President of the Harry Markowitz Company, and, from 1990 to 2000, as Director of Research, Global Portfolio Research Department (GPRD), for the Daiwa Securities Trust Company, the U.S. affiliate of Japan's Daiwa Securities. The GPRD published several journal articles in addition to managing institutional assets. 

In 1990, Harry Markowitz developed an equity research group, the Daiwa Portfolio Optimization System (DPOS), at Daiwa Securities Trust Company in Jersey City, NJ. Financial modeling used traditional fundamental variables, such as earnings-to-price, book value-to-price, cash flow-to-price, sales-to-price, cash flow-to-price, small size, institutional holdings, earnings forecasts, revisions, recommendations, and breadth, earnings surprises, and dividend yield variables, identified in Jacobs and Levy (1988), and on-going conversations with William (Bill) Ziemba as anomalies. The DPOS research was reported in Bloch, Guerard, Markowitz, Todd, and Xu (1993), which included testing of 368 models of Japanese and US stocks. The geometric mean-maximizing model was identified in November 1990 and funded as Fund Academy, after Harry was awarded the Nobel Prize in January 1991. Professor Joshua Livnat of New York University hosted a joint NYU /Daiwa research conference in 1991 to celebrate Harry, and Paul Samuelson was the luncheon speaker. The conference papers were peer-reviewed and published in Japan and the World Economy, an NYU economics journal. Fund Academy substantially outperformed in its first year, as noted in the Bloch et al (1993) paper, and came to national attention, being featured by Jason Zweig in Forbes (1993) on Harry and DPOS. DPOS used robust regression to address outlier issues in the fundamental data, and latent root regression to address multicollinearity problems. Furthermore, Guerard and Takano (1992), Guerard, Takano, and Yamane (1993) reported weighted latent root regression results in the US and Japan.  The Fund Academy was ranked as the second highest-returning Japanese Equity Fund of 65 managers, after 3.5 years, as reported in Guerard, Takano, and Yamane (1993).

In June 2008, The International Symposium on Forecasting (ISF) met in Nice, France. Nice, is of course, often ranked in the top three among beautiful beaches in the world. Harry Markowitz was invited by the Program Committee to give a keynote address but was unable to attend the meeting. Harry gave, via a professional recording studio, the address to a packed conference audience in Nice. His topic was "Three Decades of Portfolio Management". McKinley Capital Management (MCM) sponsored the Markowitz video. Martin Gruber, the Nomura Professor of Finance at New York University, gave the portfolio seminar to the ISF audience in Nice. Several of the conference participants had published in the authored tribute to Harry, Handbook  of Portfolio Construction: Current Applications of Markowitz Techniques (New York: Springer, 2010), as a "thank you gift" for his 2008 ISF presentation. Marty Gruber and Ed Elton had produced an outstanding tribute to Harry in 1979, entitled Portfolio Theory: 25 Years After (Amsterdam: North-Holland).  In his role as the Head of the MCM Scientific Advisor Board, Harry published with John and Ganlin Xu, as they had at Daiwa, some twenty years earlier, including Guerard, Markowitz, and Xu (2014), Guerard, Markowitz, and Xu  (2015), and Markowitz, Guerard, Xu, and Beheshti (2021). The papers updated the testing of the DPOS models, which were enhanced by the addition of the I/B/E/S proprietary earnings  forecasting variables. Earnings forecasting analysis was well established in the IJF with Brown (1993). The research of Harry Markowitz on earnings forecasting shifted out the efficient frontier, compared to the use of traditional, fundamental models of stock selection. Robust regression is a highly (appropriate) statistically significant technique to use in recreating expected returns.[6] Harry Markowitz joined McKinley Capital's Scientific Advisory Board in 2014 where the author was serving as the Director of Quantitative Research. Harry's contributions to portfolios selection are so great that normally at least one-quarter to one-third of MBA investments texts are based on his mean-variance analysis and efficient frontiers. The reader is referred to standard investment textbooks and research monographs such as Sharpe (1970), Elton, Gruber, Brown and Goetzmann (2007), Conner, Goldberg, and Korajczyk  (2010) ,and Levy (2012).

Harry was brilliant, and yet human. In 2003, Richard (Dick) Michaud challenged Harry and his optimizer to a Horse Race, as one saw in a Pensions & Investments (P&I) article in 2003. Michaud won. However, Dick Michaud was not only gracious, but his firm, New Frontiers Advisors, LLC, established an annual award in conjunction with the Journal of Investment Management (JOIM), in 2010 to honor Harry. The Annual award serves to recognize the impact of Markowitz on modern finance and to encourage future research and innovation. The 2015 Special Distinction Awards were given to Clifford S. Asness, Antti Ilmanen, Ronen Israel, and Tobias J. Moskowitz for their paper "Investing with Style," and Harry M. Markowitz for "Consumption, Investment and Insurance in the Game of Life." You could disagree with Harry, but you had better bring your "A+" game to the table if you expect to be competitive. As Harry often said when he encountered persons not familiar with his Portfolio Selection Cowles Foundation 16 monograph, "I have forgotten more than you will ever know". 

On October 17, 2015, the University of California at San Diego, UCSD hosted a great party to celebrate the 25th anniversary of Dr. Harry Markowitz's Nobel Prize in Economics for Modern Portfolio Theory, a construct he actually discovered almost 40 years prior. The Rady School of Management at UC San Diego hosted a celebration the last night of conference to honor Markowitz, who was then 88. Guest speakers included Martin Gruber, Stephen Horan, Jack Rivkin, Bruce Jacobs, and Robert Arnott. Harry taught at Rady as an Adjunct for years and left UCSD his Nobel Prize Medal.

4.     Harry Worked Well with Others, Particularly Jacobs and Levy

            In the early 2000s, Harry worked in joint research with Bruce Jacobs and Ken Levy of Jacobs Levy Equity Management, a provider of quantitative equity strategies for institutional clients, where he helped to construct the JLM Market Simulator (Jacobs et al., 2004, 2010). The JLM simulator is an asynchronous simulation that investors can use to create a model of the market using their own inputs. The investor's portfolio selection choice comes from the risk–aversion coefficient parameter that helps the client choose from a desired portfolio on the efficient frontier. Together, they examined optimization of portfolios with short positions,(Jacobs et al. 2005, 2006). Later, Jacobs and Levy incorporated  a term for leverage aversion within the investor utility function. They extended the mean-variance optimization model to become a mean-variance-leverage optimization model. The leverage term captures the unique risk inherent in margin calls, see Jacobs and Levy (2013), Markowitz (2013).

            Bruce Jacobs and Ken Levy have been great friends of Harry for decades. As Wharton alumni, they have been exceedingly generous in creating and endowing the Wharton-Jacobs Levy Prize for Quantitative Financial Innovation and the Jacobs Levy Equity Management Center for Quantitative Financial Research at The Wharton School.  Harry won the Wharton-Jacobs Levy Prize in 2013 for his groundbreaking innovations in individual retirement planning.

5.     What Readers Should Know

Harry Markowitz was a brilliant man who forever changed the world of investments in 1952 through 1959. However, much of Wall Street did not understand or implement much of Harry's until the early 1970s. As technology improved and computer power became far cheaper, quantitative management firms' assets under management grew exponentially. The Quant models that Harry Markowitz's Daiwa Team, DPOS, built in 1991 have continued to produce highly statistically significant stock selection and Active Returns. Has the world changed? Yes. Is everything that Harry taught us from 1952 -1959 to 2019 obsolete? No! Can they be enhanced? Yes, more predictive data sources may become available. (Legal) insider trading enhances returns. KLD data for Socially Responsible Investing (SRI) can help reduce portfolio risk and increase Sharpe, Treynor, and Information Ratios for institutional institutions. Text recognition and Natural Language Processing (NLP) can enhance returns, as was in the Guerard and Lahiri IJF Special Issue of  2015. What should we be doing? Harry would want to update the DPOS models and tests, particularly during COVID, answering his often-asked question, "What have you done for me lately?".

Acknowledgements

The author is grateful to Professors William F. Sharpe, Martin J. Gruber, and Haim Levy for their comments and helpful edits. Practitioners, Ph.D.'s in industry, Richard Michaud, Bruce Jacobs, and Ganlin Xu, provided comments and edits. The author appreciates the helpful comments of the Editor, Pierre Pinson, and an Associate Editor. Any errors remaining are the sole responsibility of the author.

References

Bloch, M., Guerard Jr., J. B., Markowitz, H. M., Todd, P., and Xu, G.-L. (1993). A comparison of some aspects of the U.S. and Japanese equity markets. Japan and the World Economy, 5, 3-26.

Brown. L.D. (1993). Earnings forecasting research: its implications for capital markets research. International Journal of Forecasting 9, 295-320.

Brown, L.D. and Zhou, L. (2015). Interactions between analysts' and managers' earnings forecasts. International Journal of Forecasting, 31, 501-514.

Chernoff, J. (2003). "Markowitz says that Michaud Built a Better Mousetrap", Pensions & Investments, December 22, 2003.

Connor, G., Goldberg, L. and Korajczyk, R.A. (2010). Portfolio Risk Analysis. Princeton: Princeton University Press.

Elton, E. J., Gruber, M. J., and Gultekin, M. (1981). Expectations and share prices. Management Science, 27, 975-987.

Elton, E.J., M.J. Gruber, S.J. Brown, and W.N. Goetzman. (2007). Modern Portfolio Theory and Investment Analysis. John Wiley and Sons, Inc., Seventh Edition.

Guerard, J.B. (2011). "Harry Markowitz". In: Assad, A., Gass, S. (eds) Profiles in Operations Research. International Series in Operations Research & Management Science, vol 147. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-6281-2_35.

Guerard, J.B., Jr., H. M. Markowitz, and Xu, G. 2015. Earnings forecasting in a global stock selection model and efficient portfolio construction and management. International Journal of Forecasting, 31, 550-560.

Guerard, J.B., Jr., Xu, G. and Markowitz, H.M. (2021). A further analysis of robust regression modeling and data mining corrections testing in global stocks. Annals of Operations Research (2021) 303:175–195.

Jacobs, B. I. and Levy, K. (1988). Disentangling equity return regularities: New insights and investment opportunities. Financial Analysts Journal 44, 18-43.

Jacobs, B.I., K. N. Levy, and Markowitz, H. M. (2005). Portfolio optimization with factor, scenarios, and realistic short positions, Operations Research 53, 586 -599.

Jacobs, B.I., K. N. Levy, and Markowitz., H. M. (2006). Trimability and fast optimization of long-short portfolios, Financial Analysts Journal, 62, 36-46.

Jacobs, B., and Levy, K. 2013. Leverage aversion, efficient frontiers, and the efficient region. The Journal of Portfolio Management 39, 54-64.

Levy, H. (2012). The Capital Asset Pricing Model in the 21^st Century. Cambridge: Cambridge University Press.

Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7, 77-91.

Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investment. Cowles Foundation Monograph No.16. New York, John Wiley and Sons.

Markowitz, H. M. (1976). Investment in the long run: new evidence for an old rule. Journal of Finance, 31, 1273-1286.

Markowitz, H. M., and Xu, G. L. (1994.) Data mining corrections. Journal of Portfolio Management 21, 60-69.

Markowitz, H. M. (2002.) Efficient portfolios, sparse matrices, and entities: a retrospective. Operations Research 50, 154–160.

Markowitz, H. M. (2013). How to represent mark-to-market possibilities with the general portfolio selection model. The Journal of Portfolio Management 39, 1-3.

Markowitz, H.M. J.B. Guerard, Jr., G.  Xu, and B. Beheshti. (2021). Financial anomalies in portfolio construction and management." The Journal of Portfolio Management 47, 51-64.

Michaud, R. O. (1989). The Markowitz optimization enigma: Is 'optimized' optimal? Financial Analysts Journal 45, 31-42.

Rudd, A, and B. Rosenberg. (1979). "Realistic Portfolio Optimization." In Portfolio Theory, 25 Years After. E. Elton and M. Gruber, Editors. Amsterdam: North-Holland.

Rubinstein, M. (1991). Portfolio selection: A fifty-year retrospective. Journal of Finance 57, 1041-1045.

Sharpe. W. F. (1970). Portfolio Theory and Capital Markets. New York: McGraw-Hill.

Timmermann, A. and Granger, C.W.J. (2004). Efficient market hypothesis and forecasting, International Journal of Forecasting 20, 14-27.

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