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1 Chapter 5 The Mathematics of Diversification

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2 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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3 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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4 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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5 Return u The expected return of a portfolio is a weighted average of the expected returns of the components:

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6 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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7 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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8 Introduction (cont’d) u For an n-security portfolio, the portfolio variance is:

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

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10 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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11 Two Security Case (cont’d) Example (cont’d) Solution: The expected return of this two-security portfolio is:

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

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13 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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14 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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15 Minimum Variance Portfolio (cont’d) Example (cont’d) Solution: The weights of the minimum variance portfolios in the previous case are:

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

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17 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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18 The n-Security Case u For an n-security portfolio, the variance is:

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19 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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20 Example of Variance-Covariance Matrix Computation in Excel

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23 Portfolio Mathematics (Matrix Form) u Define w as the (vertical) vector of weights on the different assets. u Define the (vertical) vector of expected returns u Let V be their variance-covariance matrix u The variance of the portfolio is thus: Portfolio optimization consists of minimizing this variance subject to the constraint of achieving a given expected return.

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24 Portfolio Variance in the 2-asset case We have: Hence:

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25 Covariance Between Two Portfolios (Matrix Form) u Define w 1 as the (vertical) vector of weights on the different assets in portfolio P 1. u Define w 2 as the (vertical) vector of weights on the different assets in portfolio P 2. u Define the (vertical) vector of expected returns u Let V be their variance-covariance matrix u The covariance between the two portfolios is:

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26 The Optimization Problem u Minimize Subject to: where E(R p ) is the desired (target) expected return on the portfolio and is a vector of ones and the vector is defined as:

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27 Lagrangian Method Min Or: Min Thus: Min

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28 Taking Derivatives

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30 u The last equation solves the mean-variance portfolio problem. The equation gives us the optimal weights achieving the lowest portfolio variance given a desired expected portfolio return. u Finally, plugging the optimal portfolio weights back into the variance gives us the efficient portfolio frontier:

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31 Global Minimum Variance Portfolio u In a similar fashion, we can solve for the global minimum variance portfolio: The global minimum variance portfolio is the efficient frontier portfolio that displays the absolute minimum variance.

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32 Another Way to Derive the Mean- Variance Efficient Portfolio Frontier u Make use of the following property: if two portfolios lie on the efficient frontier, any linear combination of these portfolios will also lie on the frontier. Therefore, just find two mean-variance efficient portfolios, and compute/plot the mean and standard deviation of various linear combinations of these portfolios.

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35 Some Excel Tips u To give a name to an array (i.e., to name a matrix or a vector): Highlight the array (the numbers defining the matrix) Click on ‘Insert’, then ‘Name’, and finally ‘Define’ and type in the desired name.

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36 Excel Tips (Cont’d) u To compute the inverse of a matrix previously named (as an example) “V”: Type the following formula: ‘=minverse(V)’ and click ENTER. Re-select the cell where you just entered the formula, and highlight a larger area/array of the size that you predict the inverse matrix will take. Press F2, then CTRL + SHIFT + ENTER

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37 Excel Tips (end) u To multiply two matrices named “V” and “W”: Type the following formula: ‘=mmult(V,W)’ and click ENTER. Re-select the cell where you just entered the formula, and highlight a larger area/array of the size that you predict the product matrix will take. Press F2, then CTRL + SHIFT + ENTER

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38 Single-Index Model 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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39 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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40 Portfolio Statistics With the Single-Index Model u Beta of a portfolio: u Variance of a portfolio:

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41 Proof

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42 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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43 Proof

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44 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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45 Multi-Index Model (cont’d) u The general form of a multi-index model:

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