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Chapter 7 Why Diversification Is a Good Idea. The most important lesson learned is an old truth ratified. - General Maxwell R. Thurman. Outline. Introduction Carrying your eggs in more than one basket Role of uncorrelated securities Lessons from Evans and Archer

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## Chapter 7 Why Diversification Is a Good Idea

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**The most important lesson learned**is an old truth ratified. - General Maxwell R. Thurman**Outline**• Introduction • Carrying your eggs in more than one basket • Role of uncorrelated securities • Lessons from Evans and Archer • Diversification and beta • Capital asset pricing model • Equity risk premium • Using a scatter diagram to measure beta • Arbitrage pricing theory**Introduction**• Diversification of a portfolio is logically a good idea • Virtually all stock portfolios seek to diversify in one respect or another**Carrying Your Eggs in More Than One Basket**• Investments in your own ego • The concept of risk aversion revisited • Multiple investment objectives**Investments in Your Own Ego**• Never put a large percentage of investment funds into a single security • If the security appreciates, the ego is stroked and this may plant a speculative seed • If the security never moves, the ego views this as neutral rather than an opportunity cost • If the security declines, your ego has a very difficult time letting go**The Concept of Risk Aversion Revisited**• Diversification is logical • If you drop the basket, all eggs break • Diversification is mathematically sound • Most people are risk averse • People take risks only if they believe they will be rewarded for taking them**The Concept of Risk Aversion Revisited (cont’d)**• Diversification is more important now • Journal of Finance article shows that volatility of individual firms has increased • Investors need more stocks to adequately diversify**Multiple Investment Objectives**• Multiple objectives justify carrying your eggs in more than one basket • Some people find mutual funds “unexciting” • Many investors hold their investment funds in more than one account so that they can “play with” part of the total • E.g., a retirement account and a separate brokerage account for trading individual securities**Role of Uncorrelated Securities**• Variance of a linear combination: the practical meaning • Portfolio programming in a nutshell • Concept of dominance • Harry Markowitz: the founder of portfolio theory**Variance of A Linear Combination**• One measure of risk is the variance of return • The variance of an n-security portfolio is:**Variance of A Linear Combination (cont’d)**• The variance of a two-security portfolio is:**Variance of A Linear Combination (cont’d)**• Return variance is a security’s total risk • Most investors want portfolio variance to be as low as possible without having to give up any return Total Risk Risk from A Risk from B Interactive Risk**Variance of A Linear Combination (cont’d)**• If two securities have low correlation, the interactive risk will be small • If two securities are uncorrelated, the interactive risk drops out • If two securities are negatively correlated, interactive risk would be negative and would reduce total risk**Portfolio Programming in A Nutshell**• Various portfolio combinations may result in a given return • The investor wants to choose the portfolio combination that provides the least amount of variance**Portfolio Programming in A Nutshell (cont’d)**Example Assume the following statistics for Stocks A, B, and C:**Portfolio Programming in A Nutshell (cont’d)**Example (cont’d) The correlation coefficients between the three stocks are:**Portfolio Programming in A Nutshell (cont’d)**Example (cont’d) An investor seeks a portfolio return of 12%. Which combinations of the three stocks accomplish this objective? Which of those combinations achieves the least amount of risk?**Portfolio Programming in A Nutshell (cont’d)**Example (cont’d) Solution: Two combinations achieve a 12% return: • 50% in B, 50% in C: (.5)(14%) + (.5)(10%) = 12% • 20% in A, 80% in C: (.2)(20%) + (.8)(10%) = 12%**Portfolio Programming in A Nutshell (cont’d)**Example (cont’d) Solution (cont’d): Calculate the variance of the B/C combination:**Portfolio Programming in A Nutshell (cont’d)**Example (cont’d) Solution (cont’d): Calculate the variance of the A/C combination:**Portfolio Programming in A Nutshell (cont’d)**Example (cont’d) Solution (cont’d): Investing 50% in Stock B and 50% in Stock C achieves an expected return of 12% with the lower portfolio variance. Thus, the investor will likely prefer this combination to the alternative of investing 20% in Stock A and 80% in Stock C.**Concept of Dominance**• Dominance is a situation in which investors universally prefer one alternative over another • All rational investors will clearly prefer one alternative**Concept of Dominance (cont’d)**• A portfolio dominates all others if: • For its level of expected return, there is no other portfolio with less risk • For its level of risk, there is no other portfolio with a higher expected return**Concept of Dominance (cont’d)**Example (cont’d) In the previous example, the B/C combination dominates the A/C combination: B/C combination dominates A/C Expected Return Risk**Harry Markowitz: Founder of Portfolio Theory**• Introduction • Terminology • Quadratic programming**Introduction**• Harry Markowitz’s “Portfolio Selection” Journal of Finance article (1952) set the stage for modern portfolio theory • The first major publication indicating the important of security return correlation in the construction of stock portfolios • Markowitz showed that for a given level of expected return and for a given security universe, knowledge of the covariance and correlation matrices are required**Terminology**• Security Universe • Efficient frontier • Capital market line and the market portfolio • Security market line • Expansion of the SML to four quadrants • Corner portfolio**Security Universe**• The security universe is the collection of all possible investments • For some institutions, only certain investments may be eligible • E.g., the manager of a small cap stock mutual fund would not include large cap stocks**Efficient Frontier**• Construct a risk/return plot of all possible portfolios • Those portfolios that are not dominated constitute the efficient frontier**Efficient Frontier (cont’d)**Expected Return 100% investment in security with highest E(R) No points plot above the line Points below the efficient frontier are dominated All portfolios on the line are efficient 100% investment in minimum variance portfolio Standard Deviation**Efficient Frontier (cont’d)**• The farther you move to the left on the efficient frontier, the greater the number of securities in the portfolio**Efficient Frontier (cont’d)**• When a risk-free investment is available, the shape of the efficient frontier changes • The expected return and variance of a risk-free rate/stock return combination are simply a weighted average of the two expected returns and variance • The risk-free rate has a variance of zero**Efficient Frontier (cont’d)**Expected Return C B Rf A Standard Deviation**Efficient Frontier (cont’d)**• The efficient frontier with a risk-free rate: • Extends from the risk-free rate to point B • The line is tangent to the risky securities efficient frontier • Follows the curve from point B to point C**Capital Market Line and the Market Portfolio**• The tangent line passing from the risk-free rate through point B is the capital market line (CML) • When the security universe includes all possible investments, point B is the market portfolio • It contains every risky assets in the proportion of its market value to the aggregate market value of all assets • It is the only risky assets risk-averse investors will hold**Capital Market Line and the Market Portfolio (cont’d)**• Implication for investors: • Regardless of the level of risk-aversion, all investors should hold only two securities: • The market portfolio • The risk-free rate • Conservative investors will choose a point near the lower left of the CML • Growth-oriented investors will stay near the market portfolio**Capital Market Line and the Market Portfolio (cont’d)**• Any risky portfolio that is partially invested in the risk-free asset is a lending portfolio • Investors can achieve portfolio returns greater than the market portfolio by constructing a borrowing portfolio**Capital Market Line and the Market Portfolio (cont’d)**Expected Return C B Rf A Standard Deviation**Security Market Line**• The graphical relationship between expected return and beta is the security market line (SML) • The slope of the SML is the market price of risk • The slope of the SML changes periodically as the risk-free rate and the market’s expected return change**Security Market Line (cont’d)**Expected Return E(R) Market Portfolio Rf 1.0 Beta**Expansion of the SML to Four Quadrants**• There are securities with negative betas and negative expected returns • A reason for purchasing these securities is their risk-reduction potential • E.g., buy car insurance without expecting an accident • E.g., buy fire insurance without expecting a fire**Security Market Line (cont’d)**Expected Return Securities with Negative Expected Returns Beta**Corner Portfolio**• A corner portfolio occurs every time a new security enters an efficient portfolio or an old security leaves • Moving along the risky efficient frontier from right to left, securities are added and deleted until you arrive at the minimum variance portfolio**Quadratic Programming**• The Markowitz algorithm is an application of quadratic programming • The objective function involves portfolio variance • Quadratic programming is very similar to linear programming**Lessons from Evans and Archer**• Introduction • Methodology • Results • Implications • Words of caution**Introduction**• Evans and Archer’s 1968 Journal of Finance article • Very consequential research regarding portfolio construction • Shows how naïve diversification reduces the dispersion of returns in a stock portfolio • Naïve diversification refers to the selection of portfolio components randomly**Methodology**• Used computer simulations: • Measured the average variance of portfolios of different sizes, up to portfolios with dozens of components • Purpose was to investigate the effects of portfolio size on portfolio risk when securities are randomly selected**Results**• Definitions • General results • Strength in numbers • Biggest benefits come first • Superfluous diversification

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