CHAPTER FOURTEEN WHY DIVERSIFY? © 2001 South-Western College Publishing

2 Outline  Use More Than One Basket for Your Eggs  The Axiom  The Concept of Risk Aversion Revisited  Preliminary Steps in Forming a Portfolio  The Reduced Security Universe  Security Statistics  Interpreting the Statistics  The Role of Uncorrelated Securities  The Variance of a Linear Combination  Diversification and Utility  The Concept of Dominance

3 Outline  The Efficient Frontier  Optimum Diversification of Risky Assets  The Minimum Variance Portfolio  The Effect of a Risk-free Rate  The Efficient Frontier with Borrowing  Different Borrowing and Lending Rates  Naive Diversification  The Single Index Model

4 Use More Than One Basket for Your Eggs  Failure to diversify may violate the terms of fiduciary trust.  Risk aversion seems to be an instinctive trait in human beings.  Don’t put all your eggs in one basket.

5 Preliminary Steps in Forming a Portfolio  Identify a collection of eligible investments known as the security universe.  Compute statistics for the chosen securities. e.g. mean of return variance / standard deviation of return matrix of correlation coefficients

6 Preliminary Steps in Forming a Portfolio  Interpret the statistics. 1. Do the values seem reasonable? 2. Is any unusual price behavior expected to recur? 3. Are any of the results unsustainable? 4. Low correlations: Fact or fantasy?

7 The Role of Uncorrelated Securities  The expected return of a portfolio is a weighted average of the component expected returns. where x i = the proportion invested in security i

The Role of Uncorrelated Securities two-security portfolio risk = risk A + risk B + interactive risk  The total risk of a portfolio comes from the variance of the components and from the relationships among the components. 8

9 The Role of Uncorrelated Securities expected return risk better performance  A portfolio dominates all others if no other equally risky portfolio has a higher expected return, or if no portfolio with the same expected return has less risk.  The point of diversification is to achieve a given level of expected return while bearing the least possible risk.

10 The Efficient Frontier : Optimum Diversification of Risky Assets expected return risk (standard deviation of returns) impossible portfolios dominated portfolios efficient frontier  The efficient frontier contains portfolios that are not dominated.

11 The Efficient Frontier : The Minimum Variance Portfolio expected return risk (standard deviation of returns) single security with the highest expected return minimum variance portfolio  The right extreme of the efficient frontier is a single security; the left extreme is the minimum variance portfolio.

12 The Efficient Frontier : The Effect of a Risk-Free Rate expected return risk (standard deviation of returns) dominated portfolios impossible portfolios M RfRf C efficient frontier: R f to M to C  When a risk-free investment complements the set of risky securities, the shape of the efficient frontier changes markedly.

13 The Efficient Frontier : The Effect of a Risk-Free Rate  In capital market theory, point M is called the market portfolio.  The straight portion of the line is tangent to the risky securities efficient frontier at point M and is called the capital market line.  Since buying a Treasury bill amounts to lending money to the U.S. Treasury, a portfolio partially invested in the risk-free rate is often called a lending portfolio.

14 The Efficient Frontier with Borrowing expected return risk (standard deviation of returns) dominated portfolios impossible portfolios M RfRf C efficient frontier: the ray from R f through M lending borrowing  Buying on margin involves financial leverage, thereby magnifying the risk and expected return characteristics of the portfolio. Such a portfolio is called a borrowing portfolio.

15 The Efficient Frontier : Different Borrowing and Lending Rates expected return dominated portfolios impossible portfolios M RLRL N efficient frontier : R L to M, the curve to N, then the ray from N risk (standard deviation of returns) RBRB  Most of us cannot borrow and lend at the same interest rate.

16 The Efficient Frontier : Naive Diversification  As portfolio size increases, total portfolio risk, on average, declines. After a certain point, however, the marginal reduction in risk from the addition of another security is modest. total risk nondiversifiable risk number of securities  Naive diversification is the random selection of portfolio components without conducting any serious security analysis.

17 The Efficient Frontier : Naive Diversification  The remaining risk, when no further diversification occurs, is pure market risk.  Market risk is also called systematic risk and is measured by beta.  A security with average market risk has a beta equal to 1.0. Riskier securities have a beta greater than one, and vice versa.

18 The Efficient Frontier : The Single Index Model  A pair-wise comparison of the thousands of stocks in existence would be an unwieldy task. To get around this problem, the single index model compares all securities to a benchmark measure.  The single index model relates security returns to their betas, thereby measuring how each security varies with the overall market.

The Efficient Frontier : The Single Index Model  Beta is the statistic relating an individual security’s returns to those of the market index. 19

The Efficient Frontier : The Single Index Model  The relationship between beta and expected return is the essence of the capital asset pricing model (CAPM), which states that a security’s expected return is a linear function of its beta. 20

21 The Efficient Frontier : The Single Index Model beta E(R i ) - R f security market line

22 Review  Use More Than One Basket for Your Eggs  The Axiom  The Concept of Risk Aversion Revisited  Preliminary Steps in Forming a Portfolio  The Reduced Security Universe  Security Statistics  Interpreting the Statistics  The Role of Uncorrelated Securities  The Variance of a Linear Combination  Diversification and Utility  The Concept of Dominance

23 Review  The Efficient Frontier  Optimum Diversification of Risky Assets  The Minimum Variance Portfolio  The Effect of a Risk-free Rate  The Efficient Frontier with Borrowing  Different Borrowing and Lending Rates  Naive Diversification  The Single Index Model