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
P
wij Y0i = pj Qj
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, rf )
The slope of the CML
r M −rf
σM
r p = rf +
r M − rf
σp
σM
Asset Pricing
(1)
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
M
E (rM)
rf
j
sM
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
r̄p = αr̄M + (1 − α)r̄j
2
σp2 = α2 σM
+ (1 − α)2 σj2 + 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
d r̄
Tangency = dσpp |α=1 = slope of the locus at M = slope of CML =
Asset Pricing
r̄M −rf
σM
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
r̄j = rf + (r̄M − rf )
Define:βj =
σjM
2
σM
(2)
σjM
2
σM
r j = rf +
r M − rf
σM
βj σM = rf +
r M − rf
σM
ρjM σj
Only a portion of total risk is remunerated = Systematic Risk
Asset Pricing
(3)
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
r̃j = α + βj r̃M + εj
(4)
2
σj2 = βj2 σM
+ σε2j ,
(5)
β̂j =
σ̂jM
.
2
σ̂M
r̄j − rf = (r̄ M − rf ) βj
β is the only factor; SML is linear
Asset Pricing
(6)
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(ri)
E(rM)
rf
Slope SML = ErM – rf = (E(ri) – rf) /bi
bM
bM =1
bi
Asset Pricing
b
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 r̃j =
C F̃ t+1
j
pj,t
E
− 1, the CAPM implies
˜
CF
j,t+1
pj,t
or
E
cov
!
−1
˜
CF
j,t+1
pj,t
= rf + βj (E r̃M − rf ) = rf +
˜
CF
j,t+1
pj,t
!
−1
= rf +
1
pj,t
!
− 1, r̃M
(E r̃M − rf ),
2
σM
˜
cov (CF
j,t+1 , r̃M )[
E (r̃M ) − rf
2
σM
Solving for pj,t yields
E
“
˜
CF
”
j,t+1
˜
− cov (CF
j,t+1 , r̃M )[
pj,t =
Er̃M −rf
σ2
M
1 + rf
which one may also write
E
pj,t =
“
˜
CF
”
j,t+1
− pj,t βj [E r̃M − 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)
Proposition 7.1
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance
Proposition
Portfolio
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
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)
Proposition 7.1
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance
Proposition
Portfolio
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
T
w Vw =
w1 w2
σ12 σ12
σ21 σ22
w1
w2
=
w1 σ12 + w2 σ21 w1 σ1
= w12 σ12 + w1 w2 σ21 + w1 w2 σ12 + w22 σ22
= w12 σ12 + w22 σ22 + 2w1 w2 σ12 ≥ 0
since σ12 = ρ12 σ1 σ2 ≥ −σ1 σ2 .
Definition 7.1 A frontier portfolio is one which displays
minimum variance among all feasible portfolios
with the same E(r̃p )
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
Proposition
Portfolio
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
(λ)
(γ)
1
min w T Vw
w 2
N
P
T
wi E(r̃ i ) = E (r̃p ) = E
s.t. w e = E
i=1
N
P
T
w 1=1
wi = 1
i=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)
Proposition 7.1
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance
Proposition
Portfolio
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
wp =
CE − A −1
B − AE −1
V
e} +
V 1
{z
|
D }
D }| {z }
| {z
|
{z
vector
vector
γ
λ
|{z}
|{z}
scalar
=
scalar
i 1 h i
1 h −1 B V 1 − A V −1 e +
C V −1 e − A V −1 1 E
D
D
wp = g + |{z}
h |{z}
E
|{z}
vector
vector scalar
If E=0, wp = g
If E=1, wp = g + h
Asset Pricing
(7)
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
Proposition
Portfolio
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
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 (r̃q ) as its expected return. Consider
portfolio weights (proportions) πg = 1 − E (r̃q ) and
πg+h = E (r̃q ); then, as asserted,
[1 − E (r̃q )] g +E (r̃q ) (g + h) = g +hE (r̃q ) = 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)
Proposition 7.1
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance
Proposition
Portfolio
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
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 (r̃p1 ) 6= E (r̃p2 ). Let q be an arbitrary
frontier portfolio, with expected return equal to E (r̃q ).
Since E (r̃p1 ) 6= E (r̃p2 ), there must exist a unique
number α such that
E (r̃q ) = αE (r̃p1 ) + (1 − α) E (r̃p2 )
(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)
Proposition 7.1
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance
Proposition
Portfolio
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
αwp1 + (1 − α) wp2
= α [g + hE (r̃p1 )] + (1 − α) [g + hE (r̃p2 )]
= g + h [αE (r̃p1 ) + (1 − α) E (r̃p2 )]
= g + hE (r̃q )
= 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)
Proposition 7.1
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance
Proposition
Portfolio
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
For any portfolio on the frontier,
σ 2 (r̃p ) = [g + hE (r̃p )]T V [g + hE (r̃p )] , with g and h as defined
earlier. Multiplying all this out yields;
C
σ (r̃p ) =
D
2
A
E (r̃p ) −
C
2
Asset Pricing
+
1
,
C
(9)
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
Proposition
Portfolio
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
(i) the expected return of the minimum variance portfolio is
A/C;
(ii) the variance of the minimum variance portfolio is given by C1 ;
(iii) Equation
(9) is the equation of a parabola with vertex
1 A
C , 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)
Proposition 7.1
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance
Proposition
Portfolio
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
The Set of Frontier Portfolios: Mean/Variance Space
E(r)
A/C
1/C
Var(r)
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
Proposition
Portfolio
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
The Set of Frontier Portfolios: Mean/Variance Space
E(r)
Minimum variance portfolio
A/C
w=g+h
1
w=g
sqr(1/C )
SD(r)
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
Proposition
Portfolio
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
The Set of Frontier Portfolios: Short Selling Allowed
E(r)
B
Corresponds to short selling
A to buy more of B
MVP
Corresponds to short selling
B to buy more of A
A
SD(r)
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
Definition 7.2
7.5 Characterizing Efficient Portfolios (No Risk-Free Assets)
Proposition 7.3
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio
Proposition 7.4
7.7 The Zero-Beta Capital Asset Pricing Model (Equilibrium)
7.8 The Standard CAPM
7.9 What have we accomplish?
7.10 Conclusions
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
Definition 7.2
7.5 Characterizing Efficient Portfolios (No Risk-Free Assets)
Proposition 7.3
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio
Proposition 7.4
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 7.3 Any convex combination of frontier portfolios is also a frontier portfolio
Proof Let (w̄1 ...w̄N ), define N frontier portfolios (w̄i represents the vector defining the composition
P
of the ith portfolio) and αi , t =, ..., N be real numbers such that N
i=1 αi = 1. Lastly, let
E (r̃i ) denote the expected return of the portfolio with weights w̄i . The weights
corresponding to a linear combination of the above N portfolios are:
N
X
αi w̄i
=
i=1
N
X
αi (g + hE (r̃i ))
i=1
=
N
X
αi g + h
i=1
=
N
X
αi E (r̃i )
i=1
2
3
N
X
g +h4
αi E (r̃i )5
i=1
Thus
N
P
i=1
αi w̄i is a frontier portfolio with E (r ) =
N
P
i=1
Asset Pricing
αi E (r̃i ). 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
Definition 7.2
7.5 Characterizing Efficient Portfolios (No Risk-Free Assets)
Proposition 7.3
7.6 Background for Deriving the Zero-Beta CAPM: Notion of a Zero Covariance Portfolio
Proposition 7.4
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 7.4 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 (r̃i ) ≥ CA , for every portfolio i.
N
N
P
P
However,
αi E (r̃i ) ≥
αi CA = CA ; thus, the convex
i=1
i=1
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)
Proposition 7.5
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 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)
Proposition 7.5
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 Set of Frontier Portfolios: Location of the Zero-Covariance Portfolio
E(r)
p
A/C
MVP
ZC( p)
E[rZC( p)]
(1/C )
Var(r)
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 7.5
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
Let q be any portfolio (not necessary on the frontier) and let p
be any frontier portfolio.
E r̃j = E r̃ZC(M) + βMj E (r̃M ) − E r̃ZC(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, σ 2 ), 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 (r̃q ) = E r̃ZC(M) + βMq E (r̃M ) − E r̃ZC(M)
(11)
E r̃j = E r̃ZC(M) + βMj E (r̃M ) − E r̃ZC(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
1
min w T Vw
w 2
s.t.w T e + (1 − w T 1)rf = E
Solving this problem gives
E (r̃p ) − rf
−1
wp = V
rf 1)
|{z} |(e −
{z } | {z
H
}
nxn
nx1
|
{z
} a number
nx1
Asset Pricing
(13)
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 (r̃p ) − rf ]2
, and
H
[E (r̃q ) − rf ] [E (r̃p ) − rf ]
cov (r̃q , r̃p ) = wqT Vwp =
H
σ 2 (r̃p ) = wpT Vwp =
E (r̃q ) − rf =
Hcov (r̃q , r̃p )
E (r̃p ) − rf
Asset Pricing
(14)
(15)
(16)
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 (r̃q ) − rf =
cov (r̃q , r̃p ) [E (r̃p ) − rf ]2
E (r̃p ) − rf
σ 2 (r̃p )
E (r̃q ) − rf =
cov (r̃q , r̃p )
[E (r̃p ) − rf ]
σ 2 (r̃p )
or
E (r̃q ) − rf =
(17)
cov (r̃q , r̃M )
[E (r̃M ) − rf ] ,
σ 2 (r̃M )
or
E (r̃q ) = rf + βqM [E (r̃M ) − rf ]
Asset Pricing
(18)
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