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INVESTMENT ANALYSIS ACT 4221. Optimal Portfolios and Efficient Frontier. Lecture 05A. Harry M. Markowitz, 1927-. Founder of Modern Portfolio Theory. Harry Markowitz shared the Nobel memorial prize in 1990 with William F. Sharpe and Merton H. Miller . Major Works of Harry M. Markowitz

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Optimal Portfolios and Efficient Frontier

Lecture 05A


Harry M. Markowitz, 1927-

Founder of Modern Portfolio Theory


Harry Markowitz shared the Nobel memorial prize in 1990 with William F. Sharpe and Merton H. Miller.

  • Major Works of Harry M. Markowitz
        • "The Utility of Wealth", 1952, JPE
        • "Portfolio Selection",1952, J of Finance
        • "Social Welfare Functions Based on Individual Rankings" with L.A. Goodman, 1952, AJS
        • "The Optimization of a Quadratic Function Subject to Linear Constraints", 1956, Naval Research Logistics Quarterly
        • Portfolio Selection: Efficient diversification of investment. 1958
        • "Approximating Expected Utility by a Function of Mean and Variance", 1979, with H. Levy, AER
        • Mean-Variance Analysis in Portfolio Choice and Capital Markets, 1987
        • "Foundations of Portfolio Theory", 1991, J of Finance

Resources on Harry M. Markowitz

  • HET Pages: Risk Aversion
  • Autobiography of Markowitz at Nobel site
  • Press release of Nobel award (1990).
  • Markowitz entry at
  • Markowitz Page at Britannica Guide to the Nobel Prizes
  • Markowitz Page at Nobel Prize Internet Archive
  • Markowitz entry at Bartleby
  • Citation from 1989 John von Neumann Theory Prize from INFORMS
  • "Diversification pitfalls", 1998, Money
  • Markowitz Page at Laura Forgette
  • Investor risk attitudes
  • Markowitz portfolio theory
    • Expected return and risk for individual risky asset or a portfolio of assets
    • Covariance and correlation
    • Hows and whys of diversification
  • Efficient frontier of risky assets
  • Risky asset portfolio selection
background assumptions
Background Assumptions
  • As an investor you want to maximize the returns for a given level of risk
  • Your portfolio includes all of your assets, not just financial assets
  • The relationship between the returns for assets in the portfolio is important
  • A good portfolio is not simply a collection of individually good investments
risk aversion
Risk Aversion

Portfolio theory assumes that investors are averse to risk

  • Given a choice between two assets with equal expected rates of return, risk averse investors will select the asset with the lower level of risk
  • It also means that a riskier investment has to offer a higher expected return or else nobody will buy it
definition of risk
Definition of Risk
  • One definition: Uncertainty of future outcomes relative to expectations
  • Alternative definitions:
    • Probability of an adverse outcome (losing money)
    • Range of returns
    • Returns below expectations
      • Semivariance – measures deviations only below the mean
markowitz portfolio theory
Markowitz Portfolio Theory
  • Derives the expected rate of return for a portfolio of assets and an expected risk measure
  • Markowitz demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk under reasonable assumptions
  • The portfolio variance formula shows how to effectively diversify a portfolio
markowitz portfolio theory assumptions
Markowitz Portfolio Theory Assumptions
  • Investors consider probability distribution of expected returns over some holding period
  • Investors minimize one-period expected utility
    • Utility exhibits diminishing marginal utility of wealth
  • Investors estimate portfolio risk on the basis of the variability of expected returns
  • Investors base decisions solely on expected return and risk
  • Given risk, investors prefer higher returns to lower returns
  • Given expected returns, investors prefer less risk to more risk
efficient portfolios
Efficient Portfolios
  • Under these assumptions, a portfolio of assets is efficient if no other asset or portfolio of assets offers:
    • Higher expected return with the same (or lower) risk, or
    • Lower risk with the same (or higher) expected return
efficient portfolios1
Efficient Portfolios
  • All other portfolios in attainable set are dominated by efficient set
  • Global minimum variance portfolio
    • Smallest risk of the efficient set of portfolios
  • Efficient set
    • The efficient frontier with risk greater than or equal to the global minimum variance portfolio
expected rates of return
Expected Rates of Return
  • Individual risky asset
    • Weighted average of all possible returns
    • Probabilities serve as the weights
  • Portfolio
    • Weighted average of expected returns (Ri) for the individual investments in the portfolio
    • Percentages invested in each asset (wi) serve as the weights
portfolio risk
Portfolio Risk
  • Measured by the variance or standard deviation of the portfolio’s return
    • Portfolio risk is not a weighted average of the risk of the individual securities in the portfolio
risk reduction in portfolios
Risk Reduction in Portfolios
  • Assume all risk sources for a portfolio of securities are independent
    • The larger the number of securities the smaller the exposure to any particular risk
    • “Insurance principle”
  • Only decision: How many securities to hold?
risk reduction in portfolios1
Risk Reduction in Portfolios
  • Random diversification
    • Diversifying without looking at relevant investment characteristics
    • Marginal risk reduction gets smaller and smaller as more securities are added
  • A large number of securities is not required for significant risk reduction
  • International diversification benefits

Portfolio Risk and Diversification

sport %




Portfolio risk

Market Risk

10 20 30 40 ...... 100+

Number of securities in portfolio

markowitz diversification
Markowitz Diversification
  • Non-random diversification
    • Active measurement and management of portfolio risk
    • Investigate relationships between portfolio securities before making a decision to invest
    • Takes advantage of expected return and risk for individual securities and how security returns move together
covariance of returns
Covariance of Returns
  • Before calculating the portfolio risk, several other measures need to be understood
  • Covariance
    • Measures the extent to which two variables move together
    • For two assets, i and j, the covariance of rates of return is defined as:
correlation coefficient
Correlation Coefficient
  • Scaled statistical measure of association
  • rij = correlation coefficient between securities i and j
    • rij = +1.0 = perfect positive correlation
    • rij = -1.0 = perfect negative (inverse) correlation
    • rij = 0.0 = zero correlation
portfolio standard deviation
Portfolio Standard Deviation


sport=standard deviation of the portfolio returns

wi=proportion of asset i in value of portfolio

si=standard deviation of asset i’s returns

Covij=the covariance between the returns on assets i and j

portfolio standard deviation1
Portfolio Standard Deviation
  • Portfolio standard deviation is a function of:
    • The variances of the individual assets that make up the portfolio
    • The covariances between all of the assets in the portfolio
  • The larger the portfolio, the more the impact of covariance and the lower the impact of the individual security variance
implications for portfolio formation
Implications for Portfolio Formation
  • Combining assets together with low correlations reduces portfolio risk more
    • The lower the correlation, the lower the portfolio standard deviation
    • Combining two assets with perfect negative correlation reduces the portfolio standard deviation to nearly zero
  • Even for assets that are positively correlated, the portfolio risk tends to fall as assets are added to the portfolio
implications for portfolio formation1
Implications for Portfolio Formation
  • Assets differ in terms of expected rates of return, standard deviations, and correlations with one another
  • Decision: select weights to determine the minimum variance combination for a given level of expected return
    • Non-random diversification
estimation issues
Estimation Issues
  • Diversification results depend on accurate statistical inputs
  • Estimates of
    • Expected returns
    • Standard deviations of returns
    • Correlation coefficients between returns
      • With 100 assets, 4,950 correlation estimates
  • Estimation risk refers to potential errors
the single index model
The Single Index Model
  • Relates returns on each security to the returns on a common index, such as the S&P 500 Stock Index
  • Expressed by the following equation
  • Divides return into two components
    • a unique part, ai
    • a market-related part, biRM
the single index model1
The Single Index Model
  • b measures the sensitivity of a stock to stock market movements
  • If securities are only related in their common response to the market
    • Securities covary together only because of their common relationship to the market index
    • Security covariances depend only on market risk and can be written as:
the single index model2
The Single Index Model
  • Single index model helps split a security’s total risk into
    • Total risk = market risk + unique risk
  • Multi-Index models as an alternative
    • Between the full variance-covariance method of Markowitz and the single-index model
the efficient frontier
The Efficient Frontier
  • The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return
  • Frontier will be portfolios of investments rather than individual securities
    • Exceptions being the asset with the highest return and the asset with the lowest risk
efficient frontier and alternative portfolios
Efficient Frontier and Alternative Portfolios

Efficient Frontier





Standard Deviation of Return

the efficient frontier and portfolio selection
The Efficient Frontier and Portfolio Selection
  • Any portfolio that plots “inside” the efficient frontier (such as point C) is dominated by other portfolios
    • For example, Portfolio A gives the same expected return with lower risk, and Portfolio B gives greater expected return with the same risk
  • Would we expect all investors to choose the same efficient portfolio?
    • No, individual choices would depend on relative appetites for return as opposed to risk
investor utility
Investor Utility
  • An individual investor’s utility curve specifies the trade-offs she is willing to make between expected return and risk
  • Each utility curve represent equal utility
    • Curves higher and to the left represent greater utility (more return with lower risk)
  • The interaction of the individual’s utility and the efficient frontier should jointly determine portfolio selection
the efficient frontier and investor utility
The Efficient Frontier and Investor Utility
  • The optimal portfolio has the highest utility for a given investor
  • It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility
  • Greater slope of utility curve implies greater risk aversion
selecting an optimal risky portfolio
Selecting an Optimal Risky Portfolio





Efficient Frontier






Standard Deviation of Return

investor differences and portfolio selection
Investor Differences and Portfolio Selection
  • A relatively more conservative investor would perhaps choose Portfolio X
    • On the efficient frontier and on the highest attainable utility curve
  • A relatively more aggressive investor would perhaps choose Portfolio Y
    • On the efficient frontier and on the highest attainable utility curve
selecting optimal asset classes
Selecting Optimal Asset Classes
  • Another way to use Markowitz model is with asset classes
    • Allocation of portfolio assets to broad asset categories
      • Asset class rather than individual security decisions most important for investors
    • Different asset classes offers various returns and levels of risk
      • Correlation coefficients may be quite low