7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions Asset Pricing Chapter VII. The Capital Asset Pricing Model: Another View About Risk June 20, 2006 Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions Equilibrium theory (in search of appropriate risk premium) Exchange economy I � Supply = Demand: for all asset j, w ij Y 0 i = p j Q j i Implications for returns Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proof of the CAPM relationship Appendix 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions Traditional Approach All agents are mean-variance maximizers Same beliefs (expected returns and covariance matrix) There exists a risk free asset Common linear efficient frontier Separation/Two fund theorem T=M Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proof of the CAPM relationship Appendix 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions a. The market portfolio is efficient since it is on the efficient frontier. b. All individual optimal portfolios are located on the half line originating at point ( 0 , r f ) The slope of the CML r M − r f σ M r p = r f + r M − r f σ p (1) σ M Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proof of the CAPM relationship Appendix 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions E ( r ) CML E ( r M ) M r f j s M s Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proof of the CAPM relationship Appendix 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions Refer to Figure 7.1. Consider a portfolio with a fraction 1- a of wealth invested in an arbitrary security j and a fraction a in the market portfolio ¯ r p = α ¯ r M + ( 1 − α )¯ r j σ 2 p = α 2 σ 2 M + ( 1 − α ) 2 σ 2 j + 2 α ( 1 − α ) σ jM As α varies we trace a locus that - passes through M (- and through j ) - cannot cross the CML (why?) - hence must be tangent to the CML at M Tangency = d ¯ d σ p | α = 1 = slope of the locus at M = slope of CML = ¯ r p r M − r f σ M Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proof of the CAPM relationship Appendix 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions r M − r f ) σ jM ¯ r j = r f + (¯ (2) σ 2 M Define: β j = σ jM σ 2 M � r M − r f � � r M − r f � r j = r f + β j σ M = r f + ρ jM σ j (3) σ M σ M Only a portion of total risk is remunerated = Systematic Risk Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proof of the CAPM relationship Appendix 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions ˜ r j = α + β j ˜ r M + ε j (4) σ 2 j = β 2 j σ 2 M + σ 2 ε j , (5) β j = ˆ σ jM ˆ . σ 2 ˆ M ¯ r j − r f = (¯ r M − r f ) β j (6) β is the only factor; SML is linear Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proof of the CAPM relationship Appendix 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions E ( r ) SML E ( r i ) E ( r M ) r f Slope SML = Er M – r f = ( E ( r i ) – r f ) / b i b M b M =1 b i b Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions Ft + 1 C ˜ j With ˜ r j = − 1, the CAPM implies pj , t ˜ ! CFj , t + 1 − 1 , ˜ cov r M ˜ pj , t ! CF j , t + 1 = r f + β j ( E ˜ ( E ˜ E − 1 r M − r f ) = r f + r M − r f ) , σ 2 p j , t M or ˜ ! CF j , t + 1 E (˜ 1 r M ) − r f cov ( ˜ E − 1 = r f + CF j , t + 1 , ˜ r M )[ ] . σ 2 p j , t p j , t M Solving for p j , t yields r M )[ E ˜ rM − rf “ ” ˜ − cov ( ˜ E CF j , t + 1 CF j , t + 1 , ˜ ] σ 2 M p j , t = , 1 + r f which one may also write “ ” ˜ − p j , t β j [ E ˜ r M − r f ] E CF j , t + 1 p j , t = . 1 + r f Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proposition 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio Proposition 7.2 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions Mathematics of the Portfolio Frontier Goal : Understand better what the CAPm is really about - In the process: generalize. No risk free asset Vector of expected returns e Returns are linearly independent V ij = σ ij Asset Pricing
7.1 Introduction 7.2 The Traditional Approach to the CAPM 7.3 Valuing Risky Cash Flows with the CAPM 7.4 The Mathematics of the Portfolio Frontier: Many Risky Assets and No Risk-Free Asset 7.5 Characterizing Efficient Portfolios (No Risk-Free Assets) Proposition 7.1 7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio Proposition 7.2 7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium) 7.8 The Standard CAPM 7.9 What have we accomplish? 7.10 Conclusions � � σ 2 � � w 1 � σ 12 � � w T Vw = 1 w 1 σ 2 w 1 w 2 = 1 + w 2 σ 21 w 1 σ 12 σ 2 σ 21 w 2 2 = w 2 1 σ 2 1 + w 1 w 2 σ 21 + w 1 w 2 σ 12 + w 2 2 σ 2 2 = w 2 1 σ 2 1 + w 2 2 σ 2 2 + 2 w 1 w 2 σ 12 ≥ 0 since σ 12 = ρ 12 σ 1 σ 2 ≥ − σ 1 σ 2 . Definition 7.1 A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same E (˜ r p ) Asset Pricing
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