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## Optimal Portfolios and Efficient Frontier

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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 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

- 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

- 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

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

- 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

- 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

- 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

- 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 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

- 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

- 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

- 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 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 %

35

20

0

Portfolio risk

Market Risk

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

Number of securities in portfolio

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

- 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

- 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

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 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

- 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 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

- 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

- 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 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 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 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

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

- 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 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

E(R)

U3’

U2’

U1’

Efficient Frontier

Y

U3

X

U2

U1

Standard Deviation of Return

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

- 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

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