Optimal Portfolios and Efficient Frontier

1. INVESTMENT ANALYSISACT 4221 Optimal Portfolios and Efficient Frontier Lecture 05A

3. Harry M. Markowitz, 1927- Founder of Modern Portfolio Theory

4. 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

5. Resources on Harry M. Markowitz • HET Pages: Risk Aversion • Autobiography ofMarkowitz 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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?

17. 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

18. Portfolio Risk and Diversification sport % 35 20 0 Portfolio risk Market Risk 10 20 30 40 ...... 100+ Number of securities in portfolio

19. 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

20. 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:

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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:

28. 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

29. 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

30. Efficient Frontier and Alternative Portfolios Efficient Frontier E(R) B A C Standard Deviation of Return

31. 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

32. 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

33. 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

34. Selecting an Optimal Risky Portfolio E(R) U3’ U2’ U1’ Efficient Frontier Y U3 X U2 U1 Standard Deviation of Return

35. 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

36. 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