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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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INVESTMENT ANALYSISACT 4221

Optimal Portfolios and Efficient Frontier

Lecture 05A

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Harry M. Markowitz, 1927-

Founder of Modern Portfolio Theory

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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
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Resources on Harry M. Markowitz

  • HET Pages: Risk Aversion
  • Autobiography of Markowitz at Nobel site
  • Press release of Nobel award (1990).
  • Markowitz entry at Britannica.com
  • 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
agenda
Agenda
  • 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
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Portfolio Risk and Diversification

sport %

35

20

0

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

where:

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

E(R)

B

A

C

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

E(R)

U3’

U2’

U1’

Efficient Frontier

Y

U3

X

U2

U1

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