Harry Markowitz
Encyclopedia
Harry Max Markowitz is an American economist and a recipient of the John von Neumann Theory Prize
and the Nobel Memorial Prize in Economic Sciences
.
Markowitz is a professor of finance at the Rady School of Management
at the University of California, San Diego
(UCSD). He is best known for his pioneering work in Modern Portfolio Theory
, studying the effects of asset risk
, return
, correlation
and diversification
on probable investment portfolio returns.
, an interest he continued to follow during his undergraduate years at the University of Chicago
. After receiving his B.A., Markowitz decided to continue his studies at the University of Chicago, choosing to specialize in economics. There he had the opportunity to study under important economists, including Milton Friedman
, Tjalling Koopmans
, Jacob Marschak
and Leonard Savage. While still a student, he was invited to become a member of the Cowles Commission for Research in Economics, which was in Chicago at the time.
Markowitz chose to apply mathematics to the analysis of the stock market
as the topic for his dissertation. Jacob Marschak, who was the thesis advisor, encouraged him to pursue the topic, noting that it had also been a favorite interest of Alfred Cowles
, the founder of the Cowles Commission. While researching the then current understanding of stock prices, which at the time consisted in the present value
model of John Burr Williams
, Markowitz realized that the theory lacks an analysis of the impact of risk. This insight led to the development of his seminal theory of portfolio
allocation under uncertainty, published in 1952 by the Journal of Finance
.
In 1952, Harry Markowitz went to work for the RAND Corporation, where he met George Dantzig
. With Dantzig's help, Markowitz continued to research optimization
techniques, further developing the critical line algorithm for the identification of the optimal mean-variance portfolios, lying on what was later named the Markowitz frontier
. In 1955, he received a PhD from the University of Chicago with a thesis on the portfolio theory. The topic was so novel that, while Markowitz was defending his dissertation, Milton Friedman argued his contribution was not economics. During 1955–1956 Markowitz spent a year at the Cowles Foundation, which had moved to Yale University
, at the invitation of James Tobin
. He published the critical line algorithm in a 1956 paper and used this time at the foundation to write a book on portfolio allocation which was published in 1959.
Markowitz won the Nobel Memorial Prize in Economic Sciences
in 1990 while a professor of finance at Baruch College
of the City University of New York
. In the preceding year, he received the John von Neumann Theory Prize from the Operations Research Society of America (now Institute for Operations Research and the Management Sciences
, INFORMS) for his contributions in the theory of three fields: portfolio theory; sparse matrix methods; and simulation language programming (SIMSCRIPT
). 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. SIMSCRIPT (I) included the Buddy memory allocation
method, which was also developed by Markowitz.
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.
Markowitz now divides his time between teaching (he is an adjunct professor at the Rady School of Management at the University of California at San Diego, UCSD); video casting lectures; and consulting (out of his Harry Markowitz Company offices). He currently serves on the Advisory Board of SkyView Investment Advisors, an alternative investment advisory firm and fund of hedge funds. Markowitz also serves on the Investment Committee of LWI Financial Inc. ("Loring Ward"), a San Jose, California-based investment advisor; on the advisory panel of Robert D. Arnott
's Newport Beach, California
based investment management firm, Research Affiliates; on the Advisory Board of Mark Hebner's Irvine, California and internet based investment advisory firm, Index Funds Advisors; and as an advisor to the Investment Committee of 1st Global, a Dallas, Texas-based wealth management and investment advisory firm.
Dr. Markowitz is co-founder and Chief Architect of GuidedChoice, a 401(k) managed accounts provider and investment advisor. Dr. Markowitz’s more recent work has included designing the backbone software analytics for the GuidedChoice investment solution and heading the GuidedChoice Investment Committee. He is actively involved in designing the next step in the retirement process: assisting retirees with wealth distribution through GuidedSpending.
can lower the portfolio's risk
for a given return expectation (alternately, no additional expected return can be gained without increasing the risk of the portfolio). The Markowitz Efficient Frontier
is the set of all portfolios that will give the highest expected return for each given level of risk. These concepts of efficiency were essential to the development of the Capital Asset Pricing Model
.
Markowitz also co-edited the textbook The Theory and Practice of Investment Management with Frank J. Fabozzi
of Yale School of Management
.
-Variance
Model due to the fact that it is based on expected returns (mean) and the standard deviation
(variance) of the various portfolios.
Harry Markowitz made the following assumption while developing the HM model, which were :
1. Risk of a portfolio
is based on the variability of returns from the said portfolio.
2. An investor is risk averse.
3. An investor prefers to increase consumption
.
4. The investor's utility function is concave and increasing, due to his risk aversion and consumption preference.
5. Analysis is based on single period model of investment
.
6. An investor either maximizes his portfolio return for a given level of risk or maximum return for minimum risk.
7. An investor is rational in nature.
To choose the best portfolio from a number of possible portfolios, each with different return and risk, two separate decisions are to be made :
1. Determination a set of efficient portfolios.
2. Selection of best portfolio out of the efficient set.
(a) From the portfolios that have the same return, the investor will prefer the portfolio with lower risk, and
(b) From the portfolios that have the same risk level, an investor will prefer the portfolio with higher rate of return.
As the investor is rational, they would like to have higher return. And as he is risk averse, he wants to have lower risk. In Figure 1, the shaded area PVWP includes all the possible securities an investor can invest in. The efficient portfolios are the ones that lie on the boundary of PQVW. For example, at risk level x2, there are three portfolios S, T, U. But portfolio S is called the efficient portfolio as it has the highest return, y2, compared to T and U. All the portfolios that lie on the boundary of PQVW are efficient portfolios for a given risk level.
The boundary PQVW is called the Efficient Frontier. All portfolios that lie below the Efficient Frontier are not good enough because the return would be lower for the given risk. Portfolios that lie to the right of the Efficient Frontier would not be good enough, as there is higher risk for a given rate of return. All portfolios lying on the boundary of PQVW are called Efficient Portfolios. The Efficient Frontier is the same for all investors, as all investors want maximum return with the lowest possible risk and they are risk averse.
Figure 2 shows the risk-return indifference curve
for the investors. Indifference curves C1, C2 and C3 are shown. Each of the different points on a particular indifference curve shows a different combination of risk and return, which provide the same satisfaction to the investors. Each curve to the left represents higher utility
or satisfaction. The goal of the investor would be to maximize his satisfaction by moving to a curve that is higher. An investor might have satisfaction represented by C2, but if his satisfaction/utility increases, he/she then moves to curve C3 Thus, at any point of time, an investor will be indifferent between combinations S1 and S2, or S5 and S6.
The investor's optimal portfolio is found at the point of tangency of the efficient frontier with the indifference curve
. This point marks the highest level of satisfaction the investor can obtain. This is shown in Figure 3. R is the point where the efficient frontier is tangent to indifference curve C3, and is also an efficient portfolio. With this portfolio, the investor will get highest satisfaction as well as best risk-return combination. Any other portfolio, say X, isn't the optimal portfolio even though it lies on the same indifference curve as it is outside the efficient frontier. Portfolio Y is also not optimal as it does not lie on the indifference curve, even though it lies in the portfolio region. Another investor having other sets of indifference curves might have some different portfolio as his best/optimal portfolio.
All portfolios so far have been evaluated in terms of risky securities only, and it is possible to inclued risk-free securities in a portfolio as well. A portfolio with risk-free securities will enable an investor to achieve a higher level of satisfaction. This has been explained in Figure 4.
R1 is the risk-free return, or the return from government
securities, as government securities have no risk. R1PX is drawn so that it is tangent to the efficient frontier. Any point on the line R1PX shows a combination of different proportions of risk-free securities and efficient portfolios. The satisfaction an investor obtains from portfolios on the line R1PX is more than the satisfaction obtained from the portfolio P. All portfolio combinations to the left of P show combinations of risky and risk-free assets, and all those to the right of P represent purchases of risky assets made with funds borrowed at the risk-free rate.
In the case that an investor has invested all his funds, additional funds can be borrowed at risk-free rate and a portfolio combination that lies on R1PX can be obtained. R1PX is known as the Capital Market Line (CML). this line represents the risk-return trade off in the capital market
. The CML is an upward sloping curve, which means that the investor will take higher risk if the return of the portfolio is also higher. The portfolio P is the most efficient portfolio, as it lies on both the CML and Efficient Frontier, and every investor would prefer to attain this portfolio, P. The P portfolio is known as the Market Portfolio and is also the most diversified portfolio. It consists of all shares and other securities in the capital market.
In the market for portfolios that consists of risky and risk-free securities, the CML represents the equilibrium condition. The Capital Market Line says that the return from a portfolio is the risk-free rate plus risk premium. Risk premium is the product of the market price of risk and the quantity of risk, and the risk is the standard deviation of the portfolio.
The CML equation is :
RP = IRF + (RM - IRF)σP/σM
Where,
RP = Expected Return of Portfolio
RM = Return on the Market Portfolio
IRF = Risk-Free rate of interest
σM = Standard Deviation
of the market portfolio
σP = Standard Deviation of portfolio
(RM - IRF)/σM is the slope of CML. (RM - IRF) is a measure of the risk premium, or the reward for holding risky portfolio instead of risk-free portfolio. σM is the risk of the market portfolio. Therefore, the slope measures the reward per unit of market risk.
The characteristic features of CML are:
1. At the tangent point, i.e. Portfolio P, is the optimum combination of risky investments and the market
portfolio.
2. Only efficient portfolios that consist of risk free investments and the market portfolio P lie on the CML.
3. CML is always upward sloping as the price of risk has to be positive. A rational investor will not invest unless he knows he will be compensated for that risk.
Figure 5 shows that an investor will choose a portfolio on the efficient frontier, in the absence of risk-free investments. But when risk-free investments are introduced, the investor can can choose the portfolio on the CML (which represents the combination of risky and risk-free investments). This can be done with borrowing or lending at the risk-free rate of interest (IRF) and the purchase of efficient portfolio P. The portfolio an investor will choose depends on his preference of risk. The portion from IRF to P, is investment in risk-free assets and is called Lending Portfolio. In this portion, the investor will lend a portion at risk-free rate. The portion beyond P is called Borrowing Portfolio, where the investor borrows some funds at risk-free rate to buy more of portfolio P.
2. There are numerous calculations involved that are complicated because from a given set of securities, a very large number of portfolio combinations can be made.
3. The expected return and variance will also have to computed for each securities.
http://people.maths.ox.ac.uk/~zhouxy/download/mvjump_part2.pdf
http://www.cpdwise.com/tutorial/The-Markowitz-Model.html
http://home.dacor.net/norton/finance-math/problems_w_Markowitz.pdf
John von Neumann Theory Prize
The John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciencesis awarded annually to an individual who has made fundamental and sustained contributions to theory in operations research and the management sciences.The Prize named after mathematician John von...
and the Nobel Memorial Prize in Economic Sciences
Nobel Memorial Prize in Economic Sciences
The Nobel Memorial Prize in Economic Sciences, commonly referred to as the Nobel Prize in Economics, but officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel , is an award for outstanding contributions to the field of economics, generally regarded as one of the...
.
Markowitz is a professor of finance at the Rady School of Management
Rady School of Management
The Rady School of Management at the University of California, San Diego is a graduate-level business school offering full-time and part-time Master of Business Administration degree programs in addition to non-degree executive development programs, Ph.D.s and undergraduate courses including a...
at the University of California, San Diego
University of California, San Diego
The University of California, San Diego, commonly known as UCSD or UC San Diego, is a public research university located in the La Jolla neighborhood of San Diego, California, United States...
(UCSD). He is best known for his pioneering work in Modern Portfolio Theory
Modern portfolio theory
Modern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
, studying the effects of asset risk
Risk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...
, return
Return
-In business, economics, and finance:* Rate of return, the financial term for the profit or loss derived from an investment* Tax return , various meanings relating to taxation...
, correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
and diversification
Diversification (finance)
In finance, diversification means reducing risk by investing in a variety of assets. If the asset values do not move up and down in perfect synchrony, a diversified portfolio will have less risk than the weighted average risk of its constituent assets, and often less risk than the least risky of...
on probable investment portfolio returns.
Autobiography
"Autobiography of Harry M. Marchoviz, The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1990" can be viewed on the official website of The Nobel Foundation.Biography
Harry Markowitz was born on August 24, 1927 in Chicago, to his Jewish parents Morris and Mildred Markowitz. During high school, Markowitz developed an interest in physics and philosophy, in particular the ideas of David HumeDavid Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
, an interest he continued to follow during his undergraduate years at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
. After receiving his B.A., Markowitz decided to continue his studies at the University of Chicago, choosing to specialize in economics. There he had the opportunity to study under important economists, including Milton Friedman
Milton Friedman
Milton Friedman was an American economist, statistician, academic, and author who taught at the University of Chicago for more than three decades...
, Tjalling Koopmans
Tjalling Koopmans
Tjalling Charles Koopmans was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences....
, Jacob Marschak
Jacob Marschak
Jacob Marschak was an American economist of Ukrainian Jewish origin.- Life :...
and Leonard Savage. While still a student, he was invited to become a member of the Cowles Commission for Research in Economics, which was in Chicago at the time.
Markowitz chose to apply mathematics to the analysis of the stock market
Stock market
A stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...
as the topic for his dissertation. Jacob Marschak, who was the thesis advisor, encouraged him to pursue the topic, noting that it had also been a favorite interest of Alfred Cowles
Alfred Cowles
Alfred Cowles, 3rd was an American economist, businessman and founder of the Cowles Commission. He graduated from Yale in 1913, where he was a member of Skull and Bones....
, the founder of the Cowles Commission. While researching the then current understanding of stock prices, which at the time consisted in the present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
model of John Burr Williams
John Burr Williams
John Burr Williams , one of the first economists to view stock prices as determined by “intrinsic value”, is recognised as a founder and developer of fundamental analysis. He is best known for his 1938 text "The Theory of Investment Value", based on his Ph.D...
, Markowitz realized that the theory lacks an analysis of the impact of risk. This insight led to the development of his seminal theory of portfolio
Portfolio (finance)
Portfolio is a financial term denoting a collection of investments held by an investment company, hedge fund, financial institution or individual.-Definition:The term portfolio refers to any collection of financial assets such as stocks, bonds and cash...
allocation under uncertainty, published in 1952 by the Journal of Finance
Journal of Finance
The Journal of Finance is a peer-reviewed academic journal published by Wiley-Blackwell on behalf of the American Finance Association. It was established in 1946. Its current editors are Campbell R. Harvey and John R. Graham...
.
In 1952, Harry Markowitz went to work for the RAND Corporation, where he met George Dantzig
George Dantzig
George Bernard Dantzig was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics....
. With Dantzig's help, Markowitz continued to research optimization
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
techniques, further developing the critical line algorithm for the identification of the optimal mean-variance portfolios, lying on what was later named the Markowitz frontier
Modern portfolio theory
Modern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
. In 1955, he received a PhD from the University of Chicago with a thesis on the portfolio theory. The topic was so novel that, while Markowitz was defending his dissertation, Milton Friedman argued his contribution was not economics. During 1955–1956 Markowitz spent a year at the Cowles Foundation, which had moved to Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
, at the invitation of James Tobin
James Tobin
James Tobin was an American economist who, in his lifetime, served on the Council of Economic Advisors and the Board of Governors of the Federal Reserve System, and taught at Harvard and Yale Universities. He developed the ideas of Keynesian economics, and advocated government intervention to...
. He published the critical line algorithm in a 1956 paper and used this time at the foundation to write a book on portfolio allocation which was published in 1959.
Markowitz won the Nobel Memorial Prize in Economic Sciences
Nobel Memorial Prize in Economic Sciences
The Nobel Memorial Prize in Economic Sciences, commonly referred to as the Nobel Prize in Economics, but officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel , is an award for outstanding contributions to the field of economics, generally regarded as one of the...
in 1990 while a professor of finance at Baruch College
Baruch College
Bernard M. Baruch College, more commonly known as Baruch College, is a constituent college of the City University of New York, located in the Flatiron district of Manhattan, New York City. With an acceptance rate of just 23%, Baruch is among the most competitive and diverse colleges in the nation...
of the City University of New York
City University of New York
The City University of New York is the public university system of New York City, with its administrative offices in Yorkville in Manhattan. It is the largest urban university in the United States, consisting of 23 institutions: 11 senior colleges, six community colleges, the William E...
. In the preceding year, he received the John von Neumann Theory Prize from the Operations Research Society of America (now Institute for Operations Research and the Management Sciences
Institute for Operations Research and the Management Sciences
The Institute for Operations Research and the Management Sciences is an international society for practitioners in the fields of operations research and management science...
, INFORMS) for his contributions in the theory of three fields: portfolio theory; sparse matrix methods; and simulation language programming (SIMSCRIPT
SIMSCRIPT
SIMSCRIPT is a free-form, English-like general-purpose simulation language conceived by Harry Markowitz and Bernard Hausner at the RAND Corporation in 1963. It was implemented as a Fortran preprocessor on the IBM 7090 and was designed for large discrete event simulations...
). 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. SIMSCRIPT (I) included the Buddy memory allocation
Buddy memory allocation
The buddy memory allocation technique is a memory allocation algorithm that divides memory into partitions to try to satisfy a memory request as suitably as possible. This system makes use of splitting memory into halves to try to give a best-fit...
method, which was also developed by Markowitz.
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.
Markowitz now divides his time between teaching (he is an adjunct professor at the Rady School of Management at the University of California at San Diego, UCSD); video casting lectures; and consulting (out of his Harry Markowitz Company offices). He currently serves on the Advisory Board of SkyView Investment Advisors, an alternative investment advisory firm and fund of hedge funds. Markowitz also serves on the Investment Committee of LWI Financial Inc. ("Loring Ward"), a San Jose, California-based investment advisor; on the advisory panel of Robert D. Arnott
Robert D. Arnott
Robert D. Arnott is an American entrepreneur, investor, editor and writer who focuses on articles about quantitative investing. He is the father of Richard Wiles-Arnott, Sydney Arnott, and Robin Arnott. He edited the CFA Institute's Financial Analysts Journal from 2002–2006, and has edited three...
's Newport Beach, California
Newport Beach, California
Newport Beach, incorporated in 1906, is a city in Orange County, California, south of downtown Santa Ana. The population was 85,186 at the 2010 census.The city's median family income and property values consistently place high in national rankings...
based investment management firm, Research Affiliates; on the Advisory Board of Mark Hebner's Irvine, California and internet based investment advisory firm, Index Funds Advisors; and as an advisor to the Investment Committee of 1st Global, a Dallas, Texas-based wealth management and investment advisory firm.
Dr. Markowitz is co-founder and Chief Architect of GuidedChoice, a 401(k) managed accounts provider and investment advisor. Dr. Markowitz’s more recent work has included designing the backbone software analytics for the GuidedChoice investment solution and heading the GuidedChoice Investment Committee. He is actively involved in designing the next step in the retirement process: assisting retirees with wealth distribution through GuidedSpending.
Research
A Markowitz Efficient Portfolio is one where no added diversificationDiversification
Diversification may refer to:* Diversification involves spreading investments* Diversification is a corporate strategy to increase market penetration...
can lower the portfolio's risk
Risk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...
for a given return expectation (alternately, no additional expected return can be gained without increasing the risk of the portfolio). The Markowitz Efficient Frontier
Modern portfolio theory
Modern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
is the set of all portfolios that will give the highest expected return for each given level of risk. These concepts of efficiency were essential to the development of the Capital Asset Pricing Model
Capital asset pricing model
In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
.
Markowitz also co-edited the textbook The Theory and Practice of Investment Management with Frank J. Fabozzi
Frank J. Fabozzi
Frank J. Fabozzi is Professor of Finance at EDHEC Business School, one of Europe’s leading business schools, and a Member of . He was previously a Professor in the Practice of Finance and Becton Fellow in the Yale School of Management...
of Yale School of Management
Yale School of Management
The Yale School of Management is the graduate business school of Yale University and is located on Hillhouse Avenue in New Haven, Connecticut, United States. The School offers Master of Business Administration and Ph.D. degree programs. As of January 2011, 454 students were enrolled in its MBA...
.
Introduction
Harry Markowitz put forward this model in 1952. It assists in the selection of the most efficient by analyzing various possible portfolios of the given securities. By choosing securities that do not 'move' exactly together, the HM model shows investors how to reduce their risk. The HM model is also called MeanMean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
-Variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
Model due to the fact that it is based on expected returns (mean) and the standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
(variance) of the various portfolios.
Harry Markowitz made the following assumption while developing the HM model, which were :
1. Risk of a portfolio
Portfolio
Portfolio literally means "a case for carrying loose papers," ....
is based on the variability of returns from the said portfolio.
2. An investor is risk averse.
3. An investor prefers to increase consumption
Consumption
Consumption may refer to:Economics* Use of final goods by a consumer until disposal* Consumption * Consumption function, an economic formula* Consumption Sociology* Consumption Other...
.
4. The investor's utility function is concave and increasing, due to his risk aversion and consumption preference.
5. Analysis is based on single period model of investment
Investment
Investment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...
.
6. An investor either maximizes his portfolio return for a given level of risk or maximum return for minimum risk.
7. An investor is rational in nature.
To choose the best portfolio from a number of possible portfolios, each with different return and risk, two separate decisions are to be made :
1. Determination a set of efficient portfolios.
2. Selection of best portfolio out of the efficient set.
Determining the Efficient Set
A portfolio that gives maximum return for a given risk, or minimum risk for given return is an efficient portfolio. Thus, portfolios are selected as follows:(a) From the portfolios that have the same return, the investor will prefer the portfolio with lower risk, and
(b) From the portfolios that have the same risk level, an investor will prefer the portfolio with higher rate of return.
As the investor is rational, they would like to have higher return. And as he is risk averse, he wants to have lower risk. In Figure 1, the shaded area PVWP includes all the possible securities an investor can invest in. The efficient portfolios are the ones that lie on the boundary of PQVW. For example, at risk level x2, there are three portfolios S, T, U. But portfolio S is called the efficient portfolio as it has the highest return, y2, compared to T and U. All the portfolios that lie on the boundary of PQVW are efficient portfolios for a given risk level.
The boundary PQVW is called the Efficient Frontier. All portfolios that lie below the Efficient Frontier are not good enough because the return would be lower for the given risk. Portfolios that lie to the right of the Efficient Frontier would not be good enough, as there is higher risk for a given rate of return. All portfolios lying on the boundary of PQVW are called Efficient Portfolios. The Efficient Frontier is the same for all investors, as all investors want maximum return with the lowest possible risk and they are risk averse.
Choosing the best Portfolio
For selection of the optimal portfolio or the best portfolio, the risk-return preferences are analyzed. An investor who is highly risk averse will hold a portfolio on the lower left hand of the portfolio, and an investor who isn’t too risk averse will choose a portfolio on the upper portion of the frontier.Figure 2 shows the risk-return indifference curve
Indifference curve
In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the indifference curve...
for the investors. Indifference curves C1, C2 and C3 are shown. Each of the different points on a particular indifference curve shows a different combination of risk and return, which provide the same satisfaction to the investors. Each curve to the left represents higher utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
or satisfaction. The goal of the investor would be to maximize his satisfaction by moving to a curve that is higher. An investor might have satisfaction represented by C2, but if his satisfaction/utility increases, he/she then moves to curve C3 Thus, at any point of time, an investor will be indifferent between combinations S1 and S2, or S5 and S6.
The investor's optimal portfolio is found at the point of tangency of the efficient frontier with the indifference curve
Indifference curve
In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the indifference curve...
. This point marks the highest level of satisfaction the investor can obtain. This is shown in Figure 3. R is the point where the efficient frontier is tangent to indifference curve C3, and is also an efficient portfolio. With this portfolio, the investor will get highest satisfaction as well as best risk-return combination. Any other portfolio, say X, isn't the optimal portfolio even though it lies on the same indifference curve as it is outside the efficient frontier. Portfolio Y is also not optimal as it does not lie on the indifference curve, even though it lies in the portfolio region. Another investor having other sets of indifference curves might have some different portfolio as his best/optimal portfolio.
All portfolios so far have been evaluated in terms of risky securities only, and it is possible to inclued risk-free securities in a portfolio as well. A portfolio with risk-free securities will enable an investor to achieve a higher level of satisfaction. This has been explained in Figure 4.
R1 is the risk-free return, or the return from government
Government
Government refers to the legislators, administrators, and arbitrators in the administrative bureaucracy who control a state at a given time, and to the system of government by which they are organized...
securities, as government securities have no risk. R1PX is drawn so that it is tangent to the efficient frontier. Any point on the line R1PX shows a combination of different proportions of risk-free securities and efficient portfolios. The satisfaction an investor obtains from portfolios on the line R1PX is more than the satisfaction obtained from the portfolio P. All portfolio combinations to the left of P show combinations of risky and risk-free assets, and all those to the right of P represent purchases of risky assets made with funds borrowed at the risk-free rate.
In the case that an investor has invested all his funds, additional funds can be borrowed at risk-free rate and a portfolio combination that lies on R1PX can be obtained. R1PX is known as the Capital Market Line (CML). this line represents the risk-return trade off in the capital market
Capital market
A capital market is a market for securities , where business enterprises and governments can raise long-term funds. It is defined as a market in which money is provided for periods longer than a year, as the raising of short-term funds takes place on other markets...
. The CML is an upward sloping curve, which means that the investor will take higher risk if the return of the portfolio is also higher. The portfolio P is the most efficient portfolio, as it lies on both the CML and Efficient Frontier, and every investor would prefer to attain this portfolio, P. The P portfolio is known as the Market Portfolio and is also the most diversified portfolio. It consists of all shares and other securities in the capital market.
In the market for portfolios that consists of risky and risk-free securities, the CML represents the equilibrium condition. The Capital Market Line says that the return from a portfolio is the risk-free rate plus risk premium. Risk premium is the product of the market price of risk and the quantity of risk, and the risk is the standard deviation of the portfolio.
The CML equation is :
RP = IRF + (RM - IRF)σP/σM
Where,
RP = Expected Return of Portfolio
RM = Return on the Market Portfolio
IRF = Risk-Free rate of interest
Interest
Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....
σM = Standard Deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of the market portfolio
σP = Standard Deviation of portfolio
(RM - IRF)/σM is the slope of CML. (RM - IRF) is a measure of the risk premium, or the reward for holding risky portfolio instead of risk-free portfolio. σM is the risk of the market portfolio. Therefore, the slope measures the reward per unit of market risk.
The characteristic features of CML are:
1. At the tangent point, i.e. Portfolio P, is the optimum combination of risky investments and the market
Market
A market is one of many varieties of systems, institutions, procedures, social relations and infrastructures whereby parties engage in exchange. While parties may exchange goods and services by barter, most markets rely on sellers offering their goods or services in exchange for money from buyers...
portfolio.
2. Only efficient portfolios that consist of risk free investments and the market portfolio P lie on the CML.
3. CML is always upward sloping as the price of risk has to be positive. A rational investor will not invest unless he knows he will be compensated for that risk.
Figure 5 shows that an investor will choose a portfolio on the efficient frontier, in the absence of risk-free investments. But when risk-free investments are introduced, the investor can can choose the portfolio on the CML (which represents the combination of risky and risk-free investments). This can be done with borrowing or lending at the risk-free rate of interest (IRF) and the purchase of efficient portfolio P. The portfolio an investor will choose depends on his preference of risk. The portion from IRF to P, is investment in risk-free assets and is called Lending Portfolio. In this portion, the investor will lend a portion at risk-free rate. The portion beyond P is called Borrowing Portfolio, where the investor borrows some funds at risk-free rate to buy more of portfolio P.
Demerits of the HM Model
1. It requires lots of data to be included. An investor must obtain variances of return, covariance of returns and estimates of return for all the securities in a portfolio.2. There are numerous calculations involved that are complicated because from a given set of securities, a very large number of portfolio combinations can be made.
3. The expected return and variance will also have to computed for each securities.
Further Reading
http://people.orie.cornell.edu/~leventhal/Markowitz.pdfhttp://people.maths.ox.ac.uk/~zhouxy/download/mvjump_part2.pdf
http://www.cpdwise.com/tutorial/The-Markowitz-Model.html
http://home.dacor.net/norton/finance-math/problems_w_Markowitz.pdf
Selected publications
(reprinted by Yale University Press, 1970, ISBN 978-0300013726; 2nd ed. Basil Blackwell, 1991, ISBN 978-1557861085)See also
- List of economists
- List of Jewish Nobel laureates
- Modern Portfolio TheoryModern portfolio theoryModern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
- Capital asset pricing modelCapital asset pricing modelIn finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
- Risk management toolsRisk management toolsRisk Management is a non-intuitive field of study, where the most simple of models consist of a probability multiplied by an impact. Even understanding individual risks is difficult as multiple probabilities can contribute to Risk total probability, and impacts can be "units" of cost, time, events...
- RiskRiskRisk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...
External links
- The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, 1990
- Autobiography, The Nobel Prizes 1990, Editor Tore Frängsmyr, [Nobel Foundation], Stockholm, 1991
- Banquet Speech, December 10, 1990
- Nobel Prize Lecture: Foundations of Portfolio Theory, December 7, 1990 ( PDF format)
- Oral history interview with Harry M. Markowitz, Charles Babbage InstituteCharles Babbage InstituteThe Charles Babbage Institute is a research center at the University of Minnesota specializing in the history of information technology, particularly the history since 1935 of digital computing, programming/software, and computer networking....
, University of Minnesota – Markowitz discusses his development of portfolio theory, sparse matricesSparse matrixIn the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....
, and his work at the RAND Corporation and elsewhere on simulation software development (including computer language SIMSCRIPTSIMSCRIPTSIMSCRIPT is a free-form, English-like general-purpose simulation language conceived by Harry Markowitz and Bernard Hausner at the RAND Corporation in 1963. It was implemented as a Fortran preprocessor on the IBM 7090 and was designed for large discrete event simulations...
), modeling, and operations research. - History of Finance, interviews, The American Finance Association
- Adjunct Professor of Finance, bio, Rady School of Management, University of California at San Diego
- 1st Global Engages Dr. Harry M. Markowitz, 1st Global