Asset Pricing Chapter VII. The Capital Asset Pricing Model: Another - - PowerPoint PPT Presentation

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 Supply = Demand: for all asset j,

I

• i

wijY0i = pjQj 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) 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 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) 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 efficient

frontier.

• b. All individual optimal portfolios are located on the half

line originating at point (0, rf) The slope of the CML r M−rf

σM

r p = rf + r M − rf σM σp (1)

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 Proof of the CAPM relationship Appendix 7.1 s E (r) rf M sM CML E (rM) j 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 Proof of the CAPM relationship Appendix 7.1

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 ¯ rp = α¯ rM + (1 − α)¯ rj σ2

p = α2σ2 M + (1 − α)2σ2 j + 2α(1 − α)σjM

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¯

rp dσp |α=1 = slope of the locus at M = slope of CML = ¯ rM−rf σM

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 Proof of the CAPM relationship Appendix 7.1

¯ rj = rf + (¯ rM − rf)σjM σ2

M

(2) Define:βj = σjM

σ2

M

r j = rf + r M − rf σM

• βjσM = rf +

r M − rf σM

• ρjMσj

(3) 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) 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 = α + βj˜ rM + εj (4) σ2

j = β2 j σ2 M + σ2 εj,

(5) ˆ βj = ˆ σjM ˆ σ2

M

. ¯ rj − rf = (¯ r M − rf) βj (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) 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 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 With ˜ rj =

C ˜ Ft+1 j pj,t

− 1, the CAPM implies E ˜ CF j,t+1 pj,t − 1 ! = rf + βj (E˜ rM − rf ) = rf + cov ˜

CFj,t+1 pj,t

− 1, ˜ rM ! σ2

M

(E˜ rM − rf ),

• r

E ˜ CF j,t+1 pj,t − 1 ! = rf + 1 pj,t cov( ˜ CF j,t+1, ˜ rM )[ E (˜ rM ) − rf σ2

M

]. Solving for pj,t yields pj,t = E “ ˜ CF j,t+1 ” − cov( ˜ CFj,t+1, ˜ rM )[ E˜

rM −rf σ2 M

] 1 + rf , which one may also write pj,t = E “ ˜ CFj,t+1 ” − pj,t βj [E˜ rM − rf ] 1 + rf . 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 Proposition 7.1 Proposition 7.2

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 Vij = σij

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 Proposition 7.1 Proposition 7.2

wTVw =

• w1

w2 σ2

1

σ12 σ21 σ2

2

w1 w2

• =
• w1σ2

1 + w2σ21

w1σ12 = w2

1σ2 1 + w1w2σ21 + w1w2σ12 + w2 2σ2 2

= w2

1σ2 1 + w2 2σ2 2 + 2w1w2σ12 ≥ 0

since σ12 = ρ12σ1σ2 ≥ −σ1σ2. Definition 7.1 A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same E(˜ rp)

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 Proposition 7.1 Proposition 7.2

min

w

1 2wTVw (λ) (γ) s.t. wTe = E wT1 = 1 N

• i=1

wiE(˜ r i) = E (˜ rp) = E

• N
• i=1

wi = 1

• 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 Proposition 7.1 Proposition 7.2

wp = CE − A D

• λ
• scalar

V −1e

vector

+ B − AE D

• γ
• scalar

V −11

vector

= 1 D

• B
• V −11
• − A
• V −1e
• + 1

D

• C
• V −1e
• − A
• V −11
• E

wp = g

• vector

+ h

• vector

E

• scalar

(7) If E=0, wp = g If E=1, wp = g + h

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 Proposition 7.1 Proposition 7.2

Proposition 7.1 The entire set of frontier portfolios can be generated by (are affine combinations of) g and g + h. Proof To see this, let q be an arbitrary frontier portfolio with E (˜ rq) as its expected return. Consider portfolio weights (proportions) πg = 1 − E (˜ rq) and πg+h = E (˜ rq); then, as asserted, [1 − E (˜ rq)] g+E (˜ rq) (g + h) = g+hE (˜ rq) = wq .

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 Proposition 7.1 Proposition 7.2

Proposition 7.2 The portfolio frontier can be described as affine combinations of any two frontier portfolios, not just the frontier portfolios g and g + h. Proof To confirm this assertion, let p1 and p2 be any two distinct frontier portfolios; since the frontier portfolios are different, E (˜ rp1) = E (˜ rp2). Let q be an arbitrary frontier portfolio, with expected return equal to E (˜ rq). Since E (˜ rp1) = E (˜ rp2), there must exist a unique number α such that E (˜ rq) = αE (˜ rp1) + (1 − α) E (˜ rp2) (8) Now consider a portfolio of p1 and p2 with weights α, 1 − α, respectively, as determined by Equation (8). We must show that wq = αwp1 + (1 − α) wp2.

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 Proposition 7.1 Proposition 7.2

αwp1 + (1 − α) wp2 = α [g + hE (˜ rp1)] + (1 − α) [g + hE (˜ rp2)] = g + h [αE (˜ rp1) + (1 − α) E (˜ rp2)] = g + hE (˜ rq) = wq, since q is a frontier portfolio.

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 Proposition 7.1 Proposition 7.2

For any portfolio on the frontier, σ2 (˜ rp) = [g + hE (˜ rp)]T V [g + hE (˜ rp)] , with g and h as defined

• earlier. Multiplying all this out yields;

σ2 (˜ rp) = C D

• E (˜

rp) − A C 2 + 1 C , (9)

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 Proposition 7.1 Proposition 7.2

(i) the expected return of the minimum variance portfolio is A/C; (ii) the variance of the minimum variance portfolio is given by 1

C ;

(iii) Equation (9) is the equation of a parabola with vertex 1

C , A C

• in the expected return/variance space and of a

hyperbola in the expected return/standard deviation space. See Figures 7.3 and 7.4.

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 Proposition 7.1 Proposition 7.2

The Set of Frontier Portfolios: Mean/Variance Space

Var(r) E(r) 1/C A/C 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 Proposition 7.1 Proposition 7.2

The Set of Frontier Portfolios: Mean/Variance Space

SD(r) E(r) sqr(1/C ) A/C Minimum variance portfolio w = g + h w = g 1 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 Proposition 7.1 Proposition 7.2

The Set of Frontier Portfolios: Short Selling Allowed

SD(r) E(r) MVP A B Corresponds to short selling A to buy more of B Corresponds to short selling B to buy more of A 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 Definition 7.2 Proposition 7.3 Proposition 7.4

Efficient Portfolios: Characteristics

Definition 7.2: Efficient portfolios are those frontier portfolios for which the expected return exceeds A/C, the expected return of the minimum variance portfolio.

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 Definition 7.2 Proposition 7.3 Proposition 7.4 Proposition 7.3 Any convex combination of frontier portfolios is also a frontier portfolio Proof Let (¯ w1...¯ wN), define N frontier portfolios (¯ wi represents the vector defining the composition

• f the ith portfolio) and αi , t =, ..., N be real numbers such that PN

i=1 αi = 1. Lastly, let

E (˜ ri ) denote the expected return of the portfolio with weights ¯ wi . The weights corresponding to a linear combination of the above N portfolios are:

N

X

i=1

αi ¯ wi =

N

X

i=1

αi (g + hE (˜ ri )) =

N

X

i=1

αi g + h

N

X

i=1

αi E (˜ ri ) = g + h 2 4

N

X

i=1

αi E (˜ ri ) 3 5 Thus

N

P

i=1

αi ¯ wi is a frontier portfolio with E (r) =

N

P

i=1

αi E (˜ ri ). 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 Definition 7.2 Proposition 7.3 Proposition 7.4

Proposition 7.4 The set of efficient portfolios is a convex set.

This does not mean, however, that the frontier of this set is convex-shaped in the risk-return space. Proof Suppose each of the N portfolios under consideration was efficient; then E (˜ ri) ≥ A

C , for every portfolio i.

However,

N

• i=1

αiE (˜ ri) ≥

N

• i=1

αi A

C = A C ; thus, the convex

combination is efficient as well. So the set of efficient portfolios, as characterized by their portfolio weights, is a convex set.

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 Proposition 7.5

Proposition 7.5 For any frontier portfolio p, except the minimum variance portfolio, there exists a unique frontier portfolio with which p has zero covariance. We will call this portfolio the zero covariance portfolio relative to p, and denote its vector of portfolio weights by ZC (p). Proof by construction

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 Proposition 7.5

The Set of Frontier Portfolios: Location of the Zero-Covariance Portfolio

Var(r) E(r) MVP p (1/C ) A/C ZC( p) E[rZC( p)]

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 Proposition 7.5

Let q be any portfolio (not necessary on the frontier) and let p be any frontier portfolio. E ˜ rj

• = E

˜ rZC(M)

• + βMj
• E (˜

rM) − E ˜ rZC(M)

• (10)

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

The Zero-Beta CAPM

(i) agents maximize expected utility with increasing and strictly concave utility of money functions and asset returns are multivariate normally distributed, or (ii) each agent chooses a portfolio with the objective of maximizing a derived utility function of the form W(e, σ2), W1 > 0, W2 < 0, W concave. (iii) common time horizon, (iv) homogeneous beliefs about e and V

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

(-) All investors hold mean-variance efficient portfolios (-) The market portfolio is convex combination of efficient portfolios (is efficient) E (˜ rq) = E ˜ rZC(M)

• + βMq
• E (˜

rM) − E ˜ rZC(M)

• (11)

E ˜ rj

• = E

˜ rZC(M)

• + βMj
• E (˜

rM) − E ˜ rZC(M)

• (12)

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

The Standard CAPM

min

w

1 2wTVw s.t.wTe + (1 − wT1)rf = E Solving this problem gives wp = V −1

• nxn

(e − rf1)

• nx1
• nx1

E (˜ rp) − rf H

• a number

(13)

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

σ2 (˜ rp) = wT

p Vwp = [E (˜

rp) − rf]2 H , and (14) cov (˜ rq,˜ rp) = wT

q Vwp = [E (˜

rq) − rf] [E (˜ rp) − rf] H (15) E (˜ rq) − rf = Hcov (˜ rq,˜ rp) E (˜ rp) − rf (16)

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

E (˜ rq) − rf = cov (˜ rq,˜ rp) E (˜ rp) − rf [E (˜ rp) − rf]2 σ2 (˜ rp)

• r

E (˜ rq) − rf = cov (˜ rq,˜ rp) σ2 (˜ rp) [E (˜ rp) − rf] (17) E (˜ rq) − rf = cov (˜ rq,˜ rM) σ2 (˜ rM) [E (˜ rM) − rf] ,

• r

E (˜ rq) = rf + βqM [E (˜ rM) − rf] (18)

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

What have we accomplish?

The pure mathematics of the mean-variance portfolio frontier goes a long way. In particular in producing a SML - like relationship where any frontier portfolio and its zero -covariance kin are the heroes The CAPM = a set of hypothesis guaranteeing that the efficient frontier is relevant (mean-variance optimizing) and the same for everyone (homogeneous expectations and identical horizons)

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

The implication: every investor holds a mean-variance efficient portfolio Since the efficient frontier is a convex set, this implies that the market portfolio is efficient. This is the key lesson of the

• CAPM. It does not rely on the existence of a risk free asset.

The mathematics of the efficient frontier then produces the SML. In the process, we have obtained easily workable formulas permitting to compute efficient portfolio weights with or without risk-free asset.

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

Conclusions

The asset management implications of the CAPM The testability of the CAPM: what is M? the fragility of betas to this definition ( The Roll Critique) The market may not be the only factor (Fama-French) remains: do not bear diversifiable risk

Asset Pricing