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1 Chapter 6 The Mathematics of Diversification

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2 O! This learning, what a thing it is! - William Shakespeare

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3 Outline u Introduction u Linear combinations u Single-index model u Multi-index model

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4 Introduction u The reason for portfolio theory mathematics: To show why diversification is a good idea To show why diversification makes sense logically

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5 Introduction (cont’d) u Harry Markowitz’s efficient portfolios: Those portfolios providing the maximum return for their level of risk Those portfolios providing the minimum risk for a certain level of return

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6 Linear Combinations u Introduction u Return u Variance

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7 Introduction u A portfolio’s performance is the result of the performance of its components The return realized on a portfolio is a linear combination of the returns on the individual investments The variance of the portfolio is not a linear combination of component variances

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8 Return u The expected return of a portfolio is a weighted average of the expected returns of the components:

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9 Variance u Introduction u Two-security case u Minimum variance portfolio u Correlation and risk reduction u The n-security case

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10 Introduction u Understanding portfolio variance is the essence of understanding the mathematics of diversification The variance of a linear combination of random variables is not a weighted average of the component variances

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11 Introduction (cont’d) u For an n-security portfolio, the portfolio variance is:

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12 Two-Security Case u For a two-security portfolio containing Stock A and Stock B, the variance is:

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13 Two Security Case (cont’d) Example Assume the following statistics for Stock A and Stock B: Stock AStock B Expected return Variance Standard deviation Weight40%60% Correlation coefficient.50

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14 Two Security Case (cont’d) Example (cont’d) What is the expected return and variance of this two- security portfolio?

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15 Two Security Case (cont’d) Example (cont’d) Solution: The expected return of this two-security portfolio is:

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16 Two Security Case (cont’d) Example (cont’d) Solution (cont’d): The variance of this two-security portfolio is:

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17 Minimum Variance Portfolio u The minimum variance portfolio is the particular combination of securities that will result in the least possible variance u Solving for the minimum variance portfolio requires basic calculus

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18 Minimum Variance Portfolio (cont’d) u For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:

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19 Minimum Variance Portfolio (cont’d) Example (cont’d) Assume the same statistics for Stocks A and B as in the previous example. What are the weights of the minimum variance portfolio in this case?

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20 Minimum Variance Portfolio (cont’d) Example (cont’d) Solution: The weights of the minimum variance portfolios in this case are:

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21 Minimum Variance Portfolio (cont’d) Example (cont’d) Weight A Portfolio Variance

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22 Correlation and Risk Reduction u Portfolio risk decreases as the correlation coefficient in the returns of two securities decreases u Risk reduction is greatest when the securities are perfectly negatively correlated u If the securities are perfectly positively correlated, there is no risk reduction

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23 The n-Security Case u For an n-security portfolio, the variance is:

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24 The n-Security Case (cont’d) u The equation includes the correlation coefficient (or covariance) between all pairs of securities in the portfolio

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25 The n-Security Case (cont’d) u A covariance matrix is a tabular presentation of the pairwise combinations of all portfolio components The required number of covariances to compute a portfolio variance is (n 2 – n)/2 Any portfolio construction technique using the full covariance matrix is called a Markowitz model

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26 Single-Index Model u Computational advantages u Portfolio statistics with the single-index model

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27 Computational Advantages u The single-index model compares all securities to a single benchmark An alternative to comparing a security to each of the others By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other

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28 Computational Advantages (cont’d) u A single index drastically reduces the number of computations needed to determine portfolio variance A security’s beta is an example:

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29 Portfolio Statistics With the Single-Index Model u Beta of a portfolio: u Variance of a portfolio:

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30 Portfolio Statistics With the Single-Index Model (cont’d) u Variance of a portfolio component: u Covariance of two portfolio components:

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31 Multi-Index Model u A multi-index model considers independent variables other than the performance of an overall market index Of particular interest are industry effects –Factors associated with a particular line of business –E.g., the performance of grocery stores vs. steel companies in a recession

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32 Multi-Index Model (cont’d) u The general form of a multi-index model:

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