Chapter 8 Portfolio Selection

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. 2

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 2

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 3

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 4

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 5

An Efficient Portfolio Fig. 8.1 pg 219 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 (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 6

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 =  7

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

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 8

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

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) 9

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

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 10

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.

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.

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

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 11

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

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 12

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 13

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.

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.

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 14

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

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) 15

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)

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 16

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) 17

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 18

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 19

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 19

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 20

Portfolio Risk and Diversification 35 20 Total risk Diversifiable Risk Systematic Risk 10 20 30 40 ...... 100+ Number of securities in portfolio 21

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