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Ch10 - Chapter 10 Portfolio Theory Road Map Part A...

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Chapter 10 Portfolio Theory Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of discount rates. Historic asset returns. Time value of money. Risk. Portfolio theory. Capital Asset Pricing Model (CAPM). Arbitrage Pricing Theory (APT). Part D Introduction to corporate finance. Main Issues Returns of Portfolios Diversification Optimal Portfolio Selection and Frontier Portfolios Frontier Portfolios with a Risk-free Asset Individual Assets’ Contribution to Portfolio Risk
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10-2 Portfolio Theory Chapter 10 Contents 1 Introduction and Overview . . . . . . . . . . . . . . . . . . . 10-3 2 Returns of Portfolios . . . . . . . . . . . . . . . . . . . . . . 10-4 2.1 Portfolio of Two Assets . . . . . . . . . . . . . . . . . . . . . . . 10-4 2.2 Portfolio of Multiple Assets . . . . . . . . . . . . . . . . . . . . . 10-9 3 Diversification . . . . . . . . . . . . . . . . . . . . . . . . . 10-12 4 Optimal Portfolio Selection . . . . . . . . . . . . . . . . . . . 10-16 4.1 Portfolio Frontier with Two Assets . . . . . . . . . . . . . . . . . 10-18 4.2 Portfolio Frontier with Multiple Assets . . . . . . . . . . . . . . . 10-22 5 Portfolio Frontier with A Safe Asset . . . . . . . . . . . . . . 10-23 6 Individual Assets and Portfolios . . . . . . . . . . . . . . . . 10-25 6.1 Contribution of An Asset to A Portfolio . . . . . . . . . . . . . . 10-26 6.2 Individual Asset and Frontier Portfolios . . . . . . . . . . . . . . . 10-29 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-31 8 Appendix A: Solve Frontier Portfolios . . . . . . . . . . . . . 10-32 9 Appendix B: Portfolios Analytics . . . . . . . . . . . . . . . . 10-35 9.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-35 9.2 Frontier Portfolios without Risk-free Asset . . . . . . . . . . . . . 10-37 9.3 Frontier Portfolios with A Risk-Free Asset . . . . . . . . . . . . . 10-44 9.4 Properties of Frontier Portfolios . . . . . . . . . . . . . . . . . . . 10-46 10 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-48 15.407 Lecture Notes Fall 2003 c Jiang Wang
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Chapter 10 Portfolio Theory 10-3 1 Introduction and Overview In order to understand risk-return trade-off, we observe: 1. Risks in individual asset returns have two components: (a) Systematic risks—common to many assets (b) Non-systematic risks—specific to individual assets. 2. Systematic risks and non-systematic risks are different: (a) Systematic risks are non-diversifiable (b) Non-systematic risks are diversifiable. 3. Forming portfolios can eliminate non-systematic risks. 4. Investors hold diversified portfolios instead of single assets. 5. Investors care only about portfolio risks—systematic risks. 6. Return on an asset compensates only for systematic risks. c Jiang Wang Fall 2003 15.407 Lecture Notes
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10-4 Portfolio Theory Chapter 10 2 Returns of Portfolios 2.1 Portfolio of Two Assets We start with two assets, 1 and 2, whose returns, { ˜ r 1 , ˜ r 2 } , are characterized by their mean, variance and covariances. Mean returns: Asset 1 2 Mean Return ¯ r 1 ¯ r 2 Variances and covariances (given by the covariance matrix): ˜ r 1 ˜ r 2 ˜ r 1 σ 2 1 σ 12 ˜ r 2 σ 21 σ 2 2 Covariance of an asset with itself is its variance: σ 11 = σ 2 1 and σ 22 = σ 2 2 . Example. Monthly stock returns on IBM ( ˜ r 1 ) and Merck ( ˜ r 2 ): Mean returns ¯ r 1 ¯ r 2 0.0149 0.0100 Covariance matrix ˜ r 1 ˜ r 2 ˜ r 1 0.007770 0.002095 ˜ r 2 0.002095 0.003587 15.407 Lecture Notes Fall 2003 c Jiang Wang
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Chapter 10 Portfolio Theory 10-5 A portfolio of these two assets is characterized by the value invested in each asset.
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