Asset Pricing - Chapter VII. The Capital Asset Pricing ... for... · 7.7 The Zero-Beta Capital Asset

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  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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 PricingChapter VII. The Capital Asset Pricing Model: Another View

    About Risk

    June 20, 2006

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Supply = Demand: for all asset j,Ii

    wijY0i = pjQj

    Implications for returns

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proof of the CAPM relationship Appendix 7.1

    Traditional Approach

    All agents are mean-variance maximizersSame beliefs (expected returns and covariance matrix)There exists a risk free asset

    Common linear efficient frontierSeparation/Two fund theoremT=M

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proof of the CAPM relationship Appendix 7.1

    a. The market portfolio is efficient since it is on the efficientfrontier.b. All individual optimal portfolios are located on the halfline originating at point (0, rf )

    The slope of the CML rMrfM

    rp = rf +rM rf

    Mp (1)

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proof of the CAPM relationship Appendix 7.1

    s

    E (r)

    rf

    M

    sM

    CML

    E (rM)

    j

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proof of the CAPM relationship Appendix 7.1

    Refer to Figure 7.1. Consider a portfolio with a fraction 1- a of wealthinvested in an arbitrary security j and a fraction a in the marketportfolio

    rp = rM + (1 )rj2p =

    22M + (1 )22j + 2(1 )jMAs 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 MTangency = drpdp |=1 = slope of the locus at M = slope of CML =

    rMrfM

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proof of the CAPM relationship Appendix 7.1

    rj = rf + (rM rf )jM

    2M(2)

    Define:j =jM2M

    r j = rf +(

    rM rfM

    )jM = rf +

    (rM rf

    M

    )jMj (3)

    Only a portion of total risk is remunerated = Systematic Risk

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proof of the CAPM relationship Appendix 7.1

    rj = + j rM + j (4)

    2j = 2j

    2M +

    2j, (5)

    j =jM

    2M.

    rj rf = (rM rf ) j (6)

    is the only factor; SML is linear

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proof of the CAPM relationship Appendix 7.1

    b

    E(r)

    rf

    bM

    SML

    E(rM)

    bM =1

    E(ri)

    Slope SML = ErM rf = (E(ri) rf) /bi

    bi

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    With rj =CF t+1j

    pj,t 1, the CAPM implies

    E

    CF j,t+1

    pj,t 1!

    = rf + j (ErM rf ) = rf +cov

    CF j,t+1

    pj,t 1, rM

    !2M

    (ErM rf ),

    or

    E

    CF j,t+1

    pj,t 1!

    = rf +1

    pj,tcov(CF j,t+1, rM )[

    E (rM ) rf2M

    ].

    Solving for pj,t yields

    pj,t =

    E

    CF j,t+1 cov(CF j,t+1, rM )[

    ErMrf2M

    ]

    1 + rf,

    which one may also write

    pj,t =E

    CF j,t+1 pj,t j [ErM rf ]

    1 + rf.

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proposition 7.1Proposition 7.2

    Mathematics of the Portfolio Frontier

    Goal: Understand better what the CAPm is really about - In theprocess: generalize.

    No risk free assetVector of expected returns eReturns are linearly independentVij = ij

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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

    Proposition 7.1Proposition 7.2

    wT Vw =(

    w1 w2) ( 21 12

    21 22

    ) (w1w2

    )=

    (w121 + w221 w112 + w2

    22

    ) ( w1w2

    )= w21

    21 + w1w221 + w1w212 + w

    22

    22

    = w21 21 + w

    22

    22 + 2w1w212 0

    since 12 = 1212 12.

    Definition 7.1 A frontier portfolio is one which displaysminimum variance among all feasible portfolioswith the same E(rp)

    Asset Pricing

  • 7.1 Introduction7.2 The Traditional Approach to the CAPM

    7.3 Valuing Risky Cash Flows with the CAPM7.4 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 Stand