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Markowitz Mean-Variance Diagram

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Markowitz Mean-Variance DiagramMarkowitz Mean-Variance DiagramThe Markowitz mean-variance diagram plays a central role in the development of theoreticalfinance. In setting the foundation for the capital asset pricing model, it represents the beginningof modern portfolio theory. Prior to Harry Markowitz’s contribution, the field of finance reliedmuch less on mathematical technique. Contributions to the literature tended to be descriptive, orinvolved only simple operations applied to accounting data. The principle of diversification,while accepted as a rule of thumb, was not well understood. Markowitz’s mean-varianceparadigm, summed up succinctly in his famous diagram, set finance on the path to becoming atechnical scientific discipline, more a branch of economics than of business administration.The Markowitz diagram is based on the idea that all the information about a portfolio of riskyassets that is relevant to a risk averse investor can be summed up in the values of two parameters:the standard deviation and the expected value of the portfolio’s return, briefly stated as the riskand return. The diagram, presented in Figure 1, contains four essential features: i) the set ofparameter pairs of feasible portfolios represented by the shaded area, ii) the efficient frontieralong the upper edge of the feasible set, iii) the linear asset allocation line running from the pointon the vertical axis at the rate of return on the risk-free asset and tangent to the efficient frontier,and iv) the super-efficient portfolio parameter pair located at the point of tangency. The feasibleportfolios are constructed by considering an initial endowment that can be spread across differentrisky assets in a multitude of ways. The efficient frontier represents those portfolios for whichthe expected return is the highest for any level of risk, and for which the risk is the lowest for anylevel of expected return. Its curvature, in bending towards the vertical axis, is the result of thebenefit of diversifying among assets whose returns are not perfectly correlated.Figure 1. The shaded area is the feasible set and the upper boundary is the efficient frontier. P*marks the super-efficient portfolio located at the point of tangency between the asset allocationline and the efficient frontier. RRFrepresents the risk-free rate of return.Points on the asset allocation line represent the parameter pairs associated with portfolios madeup of a combination of the super-efficient portfolio and the risk-free asset. Points on the line tothe left of the efficient frontier are associated with portfolios that combine long positions in boththe super-efficient portfolio and the risk-free asset, while points to the right are associated with aleveraged long position in risky assets created through borrowing at the risk-free rate. Ifinvestors have access to risk-free borrowing and lending, the optimal portfolio will lie along theasset allocation line, since portfolios below and to the right will be inefficient and portfoliosabove and to the left will not be feasible.The origin of the mean-variance paradigm is recounted in vivid detail by Markowitz (1999)1. Asa Ph.D. student in economics at the University of Chicago, he went to see Jacob Marschak foradvice on a topic for his dissertation. While waiting in the ante room, he happened to meet astockbroker, also waiting to see Marschak, who suggested that he write his dissertation on thestockmarket. Markowitz relayed the suggestion to Marschak and found him receptive to the idea.Both men were members of the Cowles Commission, as it was called at the time, andeconometric investigations of the stockmarket was of special interest to their benefactor, AlfredCowles. Marschak sent Markowitz to a professor in the business school for advice onbackground reading in the current literature on investments.Markowitz recalls that it was sometime in 1950, while going through the books on investmentssuggested by the business school professor, that he hit on his seminal idea regardingdiversification. The presumption of previous writers had been that, through the law of largenumbers, diversification could eliminate all risk as long as one had a large enough number ofstocks in one’s portfolio. The advice that followed was to invest in those assets with the highestexpected return. Markowitz recognized the error in the implicit assumption that stock returnswere uncorrelated and sought to discover a measure of risk for a portfolio of assets withimperfectly correlated returns. The idea of using variance as a measure of risk came to himindependently, although Irving Fisher and others had previously thought of it. Next, he found theformula for the variance of the returns of a portfolio of risky assets in a book on probability.On examining the formula for the variance of a weighted sum of random variables (foundin Uspensky 1937 on the library shelf), I was elated to see the way covariances entered.… Dealing with two quantities—mean and variance—and being an economics student, Inaturally drew a trade-off curve. Being, more specifically, a student of T. C. Koopmans…, I labeled dominated EV combinations “inefficient” and the undominated ones“efficient.” (Markowitz 1999, p. 8)By a curious coincidence, in April of the same year that Markowitz made his original sketchin the library at the University of Chicago, an article by Marschak appeared in Econometrica, that while not actually showing the diagram, sketches out some of the essential details. Insummarizing, the state of the art in the “theory of assets”, Marschak writes:“[I]f more than one commodity or time-point is considered, say γ(τ + 1) = ε quantities,each prospect may be characterized by ε means, ε variances, ε(ε – 1)/ 2 correlations, etc.Each distribution parameter was regarded as being a “good” (or possibly a “bad”) in theeyes of the man. … Tastes were described by indifference surfaces drawn in parameterspace P, and the empirical properties of these surfaces were discussed. For example, therisk-aversion (risk-discount), i.e., the rate of substitution between the mean and thevariance of income, expressed by the relevant slope of an indifference surface, was statedto be positive.” (Marschak 1950, p. 118-19)Markowitz’s diagram of the efficient frontier first appeared in print in the Journal of Finance in1952. Unlike the now standard depiction presented in Figure 1, Markowitz’s construction hadthe axes


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