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CHAPTER THREE: Portfolio Theory, Fund Separation and CAPM

CHAPTER THREE: Portfolio Theory, Fund Separation and CAPM. Markowitz Portfolio Selection. There is no single portfolio that is best for everyone. The Life Cycle — different consumption preference Time Horizons — different terms preference Risk Tolerance — different risk aversion.

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CHAPTER THREE: Portfolio Theory, Fund Separation and CAPM

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  1. CHAPTER THREE: Portfolio Theory, Fund Separation and CAPM

  2. Markowitz Portfolio Selection There is no single portfolio that is best for everyone. • The Life Cycle — different consumption preference • Time Horizons — different terms preference • Risk Tolerance — different risk aversion • Limited Variety of Portfolio — Limited “finished products” in markets

  3. Expected Return Risk Weight Asset 1 Asset 2 iscorrelation coefficient: The Trade-Off Between Expected Return and Risk Markowitz’s contribution 1:The measurement of return and risk Portfolio of two assets

  4. Suppose , how to achieve a target expected return ? Mini Case 1: Portfolio of the Riskless Asset and a Single Risky Asset Is the portfolio efficient ?

  5. Asset 1 Asset 2 Expected Return0.14 0.08 Standard Deviation0.20 0.15 Correlation Coefficient0.6 The Diversification Principle Mini Case 2: Portfolio of Two Risky Assets The Diversification Principle — The standard deviation of the combination is less than the combination of the standard deviations.

  6. Symbol Proportion in Asset 1 Proportion in Asset 2 Portfolio Expected Return Portfolio Standard Deviation R 0 100% 8% 0.15 C 10% 90% 8.6% 0.1479 Minimum Variance Portfolio 17% 83% 9.02% 0.1474 .1400 S .1100 D 50% 50% 11% 0.1569 D S 100% 0 14% 0.20 .0902 .0860 C R .0800 0 .2000 .1479 .1569 .1500 The Optimal Combination of Two Risky Assets Hyperbola Frontier of Two Risky Assets Combination Minimum Variance Portfolio

  7. Suppose , Then Let , Let , — Diversification 0 Systematic Exposure Markowitz’s contribution 2:Diversification.

  8. Expected return: : Covariance: : Mini Case 3: Portfolio of Many Risky Assets ? Resolving the quadratic programming, get the minimum variance frontier

  9. Indifference Curve of Utility Optimal Portfolio of Risky Assets 0 The Mean-Variance Frontier Efficient Frontier of Risky Assets

  10. Proposition! The variance of a diversified portfolio is irrelevant to the variance of individual assets. It is relevant to the covariance between them and equals the average of all the covariance.

  11. Systematic risk cannot be diversified

  12. Proposition! Only unsystematic risks can be diversified. Systematic risks cannot be diversified. They can be hedged and transferred only. Markowitz’s contribution 3:Distinguishing systematic and unsystematic risks.

  13. Only systematic risk premium contained, no unsystematic risk premium contained. Both systematic and unsystematic volatilities contained Proposition! There is systematic risk premium contained in the expected return. Unsystematic risk premium cannot be got through transaction in competitive markets.

  14. 0 Two Fund Separation Theorem: Practice: The portfolio frontier can be generated by any two distinct frontier portfolios. If individuals prefer frontier portfolios, they can simply hold a linear combination of two frontier portfolios or mutual funds.

  15. Orthogonal Characterization of the Mean-Variance Frontier

  16. Orthogonal Characterization of the Mean-Variance Frontier

  17. Proposition: Every return rican be represented as R* P(x)=1 1 0 Re* E=0 E=1 P(x)=0

  18. The Portfolio Frontier: where is R*? Efficient Frontier of Risky Assets w3 w2 w1 R* 0

  19. Some Properties of the Orthogonal Characterization

  20. Indifference Curve 2 CAL 2 Indifference Curve 1 CAL 1 Pcan be the linear combination of M and 0 Capital Market Line (CML) CAL — Capital Allocation Line CML

  21. — The weight invested inportfolio M — The weight invested inrisk-free security Combination of M and Risk-free Security

  22. Substitute: Market Index Security Market Value Composition Stock A $66 billion 66% Stock B $22 billion 22% Treasury $12 billion 12% Total $100 billion 100% Market Portfolio • Definition: • A portfolio that holds all assets in proportion to their observed market values is called theMarket Portfolio. M is a market portfolio of risky assets ! • Two fund separation • Market clearing

  23. Capital Asset Pricing Model (CAPM) • Assumptions: • 1. Many investors, they are price – takers. The market is perfectly competitive. • 2. All investors plan for one identical holding period. • 3. Investments to publicly traded financial assets. Financing at a fixed risk – free rate is unlimited. • 4. The market is frictionless, no tax, no transaction costs. • 5. All investors are rational mean – variance optimizers. • 6. No information asymmetry. All investors have their homogeneous expectations.

  24. Portfolio of risky assets The weights If (market portfolio), Derivation of CAPM The exposure of the market portfolio of risky assets is only related to the correlation between individual assets and the portfolio.

  25. Derivation of CAPM: Security Market Line SML E(rM)-rF 0 1.0

  26. are additive Security Market Line (SML) • Model

  27. Understanding Risk in CAPM • In CAPM, we can decompose an asset’s return into three pieces: where • Three characteristic of an asset: • Beta • Sigma • Aplha

  28. 0 The market becomes more conservative SML The market becomes more aggressive Risk neutral 1.0

  29. Summary of Chapter Three • The Key of Investments  Trade-Off Between Expected Return and Risk • Diversification  Only Systematic Risk Can Get Premium • Two Fund Separation  Any Trade in the Market can be Considered as a Trade Between Two Mutual Funds • CAPM — Individual Asset Pricing

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