Chapter 8

1. Chapter 8 Portfolio Selection

2. Learning Objectives • State three steps involved in building a portfolio. • Apply the Markowitz efficient portfolio selection model. • Describe the effect of risk-free borrowing and lending on the efficient frontier. • Discuss the separation theorem and its importance to modern investment theory. • Separate total risk into systematic and non-systematic risk.

3. Portfolio Selection • Diversification is key to optimal risk management • Analysis required because of the infinite number of portfolios of risky assets • How should investors select the best risky portfolio? • How could riskless assets be used? • Investors can invest in both risky and riskless assets and buy assets on margin or with borrowed funds

4. Building a Portfolio • Step 1: Use the Markowitz portfolio selection model to identify optimal combinations • Step 2: Consider borrowing and lending possibilities • Step 3: Choose the final portfolio based on your preferences for return relative to risk

5. Portfolio Theory • Optimal diversification takes into account all available information (as opposed to random diversification) • Assumptions in portfolio theory • A single investment period (e.g., one year) • Liquid position (e.g., no transaction costs) • Preferences based only on a portfolio’s expected return and risk

6. An Efficient Portfolio • Smallest portfolio risk for a given level of expected return • Largest expected return for a given level of portfolio risk • From the set of all possible portfolios • Only locate and analyze the subset known as the efficient set • Lowest risk for given level of return

7. An Efficient Portfolio • Fig. 8.1 pg 219 • All other portfolios in attainable set are dominated byefficient set • Global minimum variance portfolio • Smallest risk of the efficient set of portfolios • Efficient set (frontier) • Segment of the minimum variance frontier above the global minimum variance portfolio • The set of efficient portfolios composed entirely of risky securities generated by the Markowitz portfolio model

8. Efficient Portfolios • Efficient frontier or Efficient set (curved line from A to B) • Global minimum variance portfolio (represented by point A) B x E(R) A y C Risk = 

9. Efficient Portfolios • The basic Markowitz model is solved by a complex technique called quadratic programming • The expected returns, standard deviations, and correlation coefficients for the securities being considered are inputs in the Markowitz analysis • The portfolio weights are the only variables that can be manipulated to solve the portfolio problem of determining efficient portfolios

10. 1- Selecting an Optimal Portfolio of Risky Assets • In finance we assume investors are risk averse (i.e., they require additional expected return for assuming additional risk) • Indifference curves describe investor preferences for risk and return (Fig. 8.2 pg 221) • Indifference curves • Cannot intersect since they represent different levels of desirability • Are upward sloping for risk-averse investors • Greater slope implies greater risk aversion • Investors have an infinite number of indifference curves • Higher indifference curves are more desirable

11. Selecting an Optimal Portfolio of Risky Assets • The optimal portfolio for a risk-averse investor occurs at the point of tangency between the investor’s highest indifference curve and the efficient set of portfolios (Fig. 8.3 pg 221) • This portfolio maximizes investor utility because the indifference curves reflect investor preferences, while the efficient set represents portfolio possibilities

12. Selecting an Optimal Portfolio of Risky Assets • Markowitz portfolio selection model • Generates a frontier of efficient portfolios which are equally good • Does not address the issue of riskless borrowing or lending (investors are not allowed to use leverage) • Different investors will estimate the efficient frontier differently (this results from estimating the inputs to the Markowitz model differently) • Element of uncertainty in application (i.e., uncertainty is inherent in security analysis)

13. Alternative Methods of Obtaining the Efficient Frontier • For a portfolio of n securities: • The full variance-covariance model of Markowitz requires [n (n + 3)] / 2 estimates • The single-index model requires (3n + 2) estimates • Example: calculate the required number of estimates needed by both models for a portfolio of 250 securities

14. Selecting Optimal Asset Classes • Another way to use the Markowitz model is with asset classes • Allocation of portfolio assets to broad asset categories (i.e., how much of the portfolio’s assets are to be invested in stocks, bonds, money market securities, etc.) • Asset class rather than individual security decisions most important for investors • The rationale behind the asset allocation approach is that different asset classes offer various returns and levels of risk • Correlation coefficients may be quite low

15. Example: Selecting Optimal Asset Classes • (Pg 223) Consider the performance of two Canadian portfolio managers, A and B, between 1999 and 2003. • Manager A maintained an equally weighted portfolio with respect to T-bills, long-term government bonds, and common stocks. • Manager B was more conservative and allocated 20% of funds to each stocks and bonds, with the remaining 60% to T-bills. • Assume each manager matched the risk-return performance on the relevant asset class benchmark index for the proportion of their portfolio invested in each of the three asset classes.

16. Example: Selecting Optimal Asset Classes • Over the period, the average annual return and standard deviation for the asset classes were: • T-bills: 4.01% & 1.29% • Government bonds: 5.6% & 7.96% • Common stocks: 7.68% & 20.42% • Calculate the annual return earned by managers A & B and the standard deviations of their portfolios.

17. Optimal Risky Portfolios Investor Utility Function E (R) Efficient Frontier * 

18. 2- Borrowing and Lending Possibilities • Risk-free assets • Certain-to-be-earned expected return(this is nominal return and not real return which is uncertain since inflation is uncertain) • Zero variance • No covariance or correlation with risky assets (ρ_RF = 0 since the risk-free rate is a constant which by nature has no correlation with the changing returns on risky securities) • Usually proxied by a Treasury Bill • Amount to be received at maturity is free of default risk, known with certainty

19. Borrowing and Lending Possibilities • Adding a risk-free asset extends and changes the efficient frontier • Investors can now invest part of their wealth in the risk-free asset and the remainder in any of the risky portfolios in the Markowitz efficient set • It allows the Markowitz portfolio theory to be extended in such a way that the efficient frontier is completely changed

20. Risk-Free Lending • Riskless assets can be combined with any portfolio in the efficient set AB (comprised only of risky assets) • Z implies lending • Set of portfolios on line RF to T dominates all portfolios below it L B E(R) T Z X RF A Risk

21. Impact of Risk-Free Lending • If wRF placed in a risk-free asset and (1- wRF) in risky portfolio X: • Expected portfolio return • Risk of the portfolio (correlation and covariance for the risk-free asset is zero) • Expected return and risk of the portfolio with lending is a weighted average

22. Example: Impact of Risk-Free Lending • (Pg 226) Assume that portfolio X has an expected return of 15% with a standard deviation of 30%, and that the risk-free security has an expected return of 3%. • If 60% of investable funds is placed in RF and 40% in portfolio X, calculate the expected return and standard deviation of the resulting portfolio.

23. Impact of Risk-Free Lending • An investor could change positions on the line RF-X by varying wRF. As more of the investable funds are placed in the risk-free asset, both the expected return and the risk of the portfolio decline. • It is apparent that the segment of the efficient frontier below X (i.e., A to X) is now dominated by the line RF-X. • Therefore, the ability to invest in RF provides investors with a more efficient set of portfolios from which to choose which lies along line RF-T.

24. Borrowing Possibilities • Investor no longer restricted to own wealth • One way to accomplish this borrowing is to buy stocks on margin • Interest paid on borrowed money • Higher returns sought to cover expense • Assume borrowing at risk-free rate (RF) • Risk will increase as the amount of borrowing increases • Financial leverage

25. Borrowing Possibilities • Borrowing additional investable funds and investing them together with the investor’s own wealth allows investors to seek higher expected returns while assuming greater risk • These borrowed funds can be used to leverage the portfolio position beyond the tangency point T, which represents 100% of an investor’s wealth in Risky asset portfolio T • The straight line RF-T is now extended upward, and can be designated RF-T-L

26. The New Efficient Set • Risk-free investing and borrowing creates a new set of expected return-risk possibilities • Addition of risk-free asset results in • A change in the efficient set from an arc to a straight line tangent to the feasible set without the riskless asset • Chosen portfolio depends on investor’s risk-return preferences (i.e., investors can be anywhere they choose on line RF-T-L, depending on their risk-return preference)

27. The New Efficient Set • In real life, it is unlikely that the typical investor can borrow at the same rate as that offered by riskless securities because borrowing rates generally exceed lending rates • The straight line RF-T-L will be transformed into a line with a “kink” at point T (Fig. 8.6 pg 230)

28. 3- Portfolio Choice • The more conservative the investor, the more that is placed in risk-free lending and the less in borrowing (i.e., closer to the risk-free rate RF) • The more aggressive the investor, the less that is placed in risk-free lending and the more in borrowing (i.e., closer to, or on, point T, representing full investment in a portfolio of risky assets) • Even more aggressive investors could go beyond point T by using leverage to move up the line RF-T-L

29. The Separation Theorem • Investors use their preferences (reflected in an indifference curve) to determine their optimal portfolio along the new efficient frontier RF-T-L • Separation Theorem • The investment decision (which portfolio of risky assets to hold) is separate from the financing decision (how to allocate investable funds between the risk-free asset and the risky asset)

30. Separation Theorem • All investors • Invest in the same portfolio of risky assets T • Attain any point on the straight line RF-T-L by either borrowing or lending at the rate RF, depending on their preferences • Risky portfolios are not tailored to each individual’s taste • The separation theorem argues that the tailoring process is inappropriate

31. Implications of Portfolio Selection • Investors should focus on risk that cannot be managed by diversification • Total risk = • Systematic (non-diversifiable) risk + • Non-systematic (diversifiable) risk

32. Systematic risk • Systematic risk (unavoidable) • Variability in a security’s total returns directly associated with economy-wide events • Common to virtually all securities • E.g., interest rate risk, market risk, and inflation risk

33. Non-Systematic Risk • Non-Systematic Risk • Variability of a security’s total return not related to general market variability • Diversification decreases this risk • The relevant risk of an individual stock is its contribution to the riskiness of a well-diversified portfolio • Portfolios rather than individual assets most important Recent Canadian research suggests that 70 or more stocks are required to obtain a well diversified portfolio

34. Portfolio Risk and Diversification p % 35 20 0 Total risk Diversifiable Risk Systematic Risk 10 20 30 40 ...... 100+ Number of securities in portfolio

35. Appendix 8-A: Modern Portfolio Theory and the Portfolio Management Process • Demonstrated by the impact on regulations governing the investment behaviour of professional money managers • Require them to adhere to “prudence, loyalty, reasonable administrative cost, and diversification” • Portfolio management process: • Designing an investment policy • Developing and implementing an asset mix • Monitoring the economy, the markets, and the client • Adjusting the portfolio and measuring performance