Lecture
Lecture 07: 07: Mean Mean--Variance Analysis & Variance Analysis &
Capital Asset Pricing Model Capital Asset Pricing Model ((CAPM CAPM))
((CAPM CAPM))
Prof. Markus K. Brunnermeier
Slide Slide 0707--11 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
O i
O i
Overview Overview
1. Simple CAPM with quadratic utility functions
(derived from state
(derived from state--price beta model) price beta model)
2.
2. Mean Mean--variance preferences variance preferences
–– Portfolio Theory Portfolio Theoryyy –– CAPM CAPM (intuition) (intuition)
33 CAPM CAPM 3.
3. CAPM CAPM
–– Projections Projections
Pricing Kernel and Expectation Kernel Pricing Kernel and Expectation Kernel
Slide Slide 0707--22 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
–– Pricing Kernel and Expectation Kernel Pricing Kernel and Expectation Kernel
R ll St t
R ll St t i i B t B t d l d l Recall State
Recall State--price Beta model price Beta model
Recall:
Recall:
E[
E[R R hh ]] -- R R ff = = ββ hh E[R E[R ** -- R R ff ]]
E[
E[R R ] ] R R ββ E[R E[R R R ]]
where β h := Cov[R * ,R h ] / Var[R * ]
very general
very general –– but what is R but what is R ** in reality in reality ??
Slide Slide 0707--33 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Simple CAPM with Quadratic Expected Utility Simple CAPM with Quadratic Expected Utility Simple CAPM with Quadratic Expected Utility Simple CAPM with Quadratic Expected Utility
1. All agents are identical
• Expected utility U(x x ) = ∑ π u(x x ) ⇒ m= ∂ u / E[∂ u]
• Expected utility U(x 0 , x 1 ) = ∑ s π s u(x 0 , x s ) ⇒ m= ∂ 1 u / E[∂ 0 u]
• Quadratic u(x 0 ,x 1 )=v 0 (x 0 ) - (x 1 - α) 2
⇒ ∂ 1 1 u = [-2(x [ ( 1 1 1,1 - α),…, -2(x ), , ( S 1 S,1 - α)] )]
• E[R h ] – R f = - Cov[m,R h ] / E[m]
= -R f Cov[∂ 1 u, R h ] / E[∂ 0 u]
= -R f Cov[-2(x - α) R h ] / E[∂ u]
= -R Cov[-2(x 1 - α), R ] / E[∂ 0 u]
= R f 2Cov[x 1 ,R h ] / E[∂ 0 u]
• Also holds for market portfolio
• E[R m ] – R f = R f 2Cov[x 1 ,R m ]/E[∂ 0 u]
Slide Slide 0707--44 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
⇒
Simple CAPM with Quadratic Expected Utility Simple CAPM with Quadratic Expected Utility Simple CAPM with Quadratic Expected Utility Simple CAPM with Quadratic Expected Utility
2. Homogenous agents + Exchange economy
d d i f l l d i h R
⇒ x 1 = agg. endowment and is perfectly correlated with R m
E[
E[R R hh ]= ]=R R ff + + ββ hh {E[ {E[R R m m ]]--R R ff }} Market Security Line Market Security Line
Slide Slide 0707--55 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
N.B.: R * =R f (a+b 1 R M )/(a+b 1 R f ) in this case (where b 1 <0)!
O i
O i
Overview Overview
1.
1. Simple CAPM with quadratic utility functions Simple CAPM with quadratic utility functions
(derived from state
(derived from state--price beta model) price beta model)
2.
2. Mean Mean--variance analysis variance analysis
–– Portfolio Theory Portfolio Theory
((P P f li f f li f i i ffi i ffi i f f i i ) ) ((Portfolio frontier, efficient frontier, …) Portfolio frontier, efficient frontier, …) –– CAPM CAPM (Intuition) (Intuition)
33 CAPM CAPM 3.
3. CAPM CAPM
–– Projections Projections
P i i K l d E i K l
P i i K l d E i K l
Slide Slide 0707--66 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
–– Pricing Kernel and Expectation Kernel Pricing Kernel and Expectation Kernel
Definition: Mean
Definition: Mean--Variance Dominance e e o : o : e e Variance Dominance V V ce o ce o ce ce
& Efficient Frontier
& Efficient Frontier
• Asset (portfolio) A mean-variance dominates asset (portfolio) B
asset (portfolio) B
if μ A ≥ μ B and σ A < σ Β or if μ A >μ B while σ A ≤σ B . Effi i t f ti l i f ll d i t d
• Efficient frontier: loci of all non-dominated
portfolios in the mean-standard deviation space.
B d fi iti (“ ti l”) i
By definition, no (“rational”) mean-variance investor would choose to hold a portfolio not l t d th ffi i t f ti
Slide Slide 0707--77 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
located on the efficient frontier.
E d P f li R & V i E d P f li R & V i
Expected Portfolio Returns & Variance Expected Portfolio Returns & Variance
• Expected returns (linear)
μ p := E[r p ] = w j μ j , where each w j = P h
jh
j• Variance
2
V [ ]
0V ( )
µ σ
12σ
12¶ µ
w
1¶
μ p [ p ] j μ j , j P
j
h
jσ
p2:= V ar[r
p] = w
0V w = (w
1w
2) µ
1 12
σ
21σ
22¶ µ
1
w
2¶
= (w
1σ
12+ w
2σ
21w
1σ
12+ w
2σ
22)
µ w
1w
2¶
recall that correlation µ
w
2¶
= w
21σ
12+ w
22σ
22+ 2w
1w
2σ
12≥ 0 since σ
12≤ −σ
1σ
2.
Slide Slide 0707--88 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
coefficient ∈ [-1,1]
Ill t ti f 2 A t C Ill t ti f 2 A t C
Illustration of 2 Asset Case Illustration of 2 Asset Case
• For certain weights: w 1 and (1-w 1 ) μ p = w 1 E[r 1 ]+ (1-w 1 ) E[r 2 ]
μ p 1 [ 1 ] ( 1 ) [ 2 ]
σ 2 p = w 1 2 σ 1 2 + (1-w 1 ) 2 σ 2 2 + 2 w 1 (1-w 1 )σ 1 σ 2 ρ 1,2 ( Specify σ 2 p and one gets weights and μ p ’s)
( p y p g g μ p )
• Special cases [w
1to obtain certain σ
R]
– ρ = 1 ⇒ w = (+/-σ – σ ) / (σ – σ ) – ρ 1,2 1 ⇒ w 1 (+/-σ p – σ 2 ) / (σ 1 – σ 2 ) – ρ 1,2 = -1 ⇒ w 1 = (+/- σ p + σ 2 ) / (σ 1 + σ 2 )
Slide Slide 0707--99 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
1,2 w
1=
1,2
μ
p= w
1μ
1+ (1 − w
1)μ
21 σ1−σ2
μ p
E[r
2]
E[r ]
σ 1 σ p σ 2
E[r
1]
The Efficient Frontier: Two Perfectly Correlated Risky Assets Lower part with … is irrelevant
Slide Slide 0707--1010 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
The Efficient Frontier: Two Perfectly Correlated Risky Assets
1,2 w
1=
+1,2
μ
p= w
1μ
1+ (1 − w
1)μ
21 σ1+σ2
E[ ] E[r
2]
E[r ]
σ2
σ1+σ2
μ
1+
σ σ11+σ2
μ
2slope:
μσ2−μ11+σ2
σ
pslope: −
μσ2−μ11+σ2
σ
pσ 1 σ 2
E[r
1]
σ1+σ2 pEfficient Frontier: Two Perfectly Negative Correlated Risky Assets
Slide Slide 0707--1111 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Efficient Frontier: Two Perfectly Negative Correlated Risky Assets
1,2 1,2
E[r
2]
E[r
1]
σ 1 σ 2
Efficient Frontier: Two Imperfectly Correlated Risky Assets
Slide Slide 0707--1212 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
For σ = 0 For σ 1 = 0
E[r ]
μ p
E[r
2]
E[r
1]
σ 1 σ p σ 2
E[r
1]
The Efficient Frontier: One Risky and One Risk Free Asset
Slide Slide 0707--1313 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Effi i f i i h i k Effi i f i i h i k
Efficient frontier with n risky assets Efficient frontier with n risky assets
• A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same expected portfolio
return E[r]
return.
Vw 2 w
min 1 T
w
( ) λ s . .t w
Te = E ⎟
⎠ ⎞
⎜ ⎝
⎛ ∑
Nw
iE ( r~
i) = E
2
w
σ
( ) γ w
T1 = 1 ⎜ ⎝ ⎛ ∑ = ⎟ ⎠ ⎞
⎠
⎝
=
=
1
N
w
1
i i
1 i
Slide Slide 0707--1414
⎠
⎝
i=1∂ L
∂ L
∂w = V w − λe − γ1 = 0
∂ L = E w T e = 0
∂ L
∂λ = E − w T e = 0
∂ L
∂γ = 1 − w T 1 = 0
The first FOC can be written as:
∂γ
V w p = λe + γ1 or
λV −1 + V −1 1 w p = λV 1 e + γV 1 1
e T w p = λ(e T V −1 e) + γ(e T V −1 1)
Slide Slide 0707--1515
p ( ) + γ( )
N i h T T i h fi f h d f Noting that e T w p = w T p e, using the first foc, the second foc can be written as
E[˜ r ] = e T w = λ (e T V −1 e) +γ (e T V −1 1)
pre-multiplying first foc with 1 (instead of e T ) yields
E[r p ] = e w p = λ (e V e)
| {z }
:=B
+γ (e V 1)
| {z }
=:A pre-multiplying first foc with 1 (instead of e ) yields
1 T w p = w p T 1 = λ(1 T V −1 e) + γ(1 T V −1 1) = 1 ( T 1 ) ( T 1 )
1 = λ (1 T V −1 e)
| {z }
=:A
+γ (1 T V −1 1)
| {z }
=:C Solving both equations for λ and γ
λ = CE D −A and γ = B −AE D
Slide Slide 0707--1616
where D = BC − A 2 .
Hence, w p = λ V -1 e + γ V -1 1 becomes
1
1
1 −
−
−
− +
= V
D AE e B
D V A w
pCE
(vector) (vector)
( ) ( )
[ B V 1 A V e ] 1 [ C ( ) ( ) V e A V 1 ] E
1
−1 −1 −1 −1− +
−
=
λ (scalar) γ (scalar)
( ) ( )
[ ] [ C ( ) ( ) V e V ]
e D V D V
• Result: Portfolio weights are linear in expected portfolio return w
p= g + h E
Slide Slide 0707--1717
p
If E = 0, w
p= g
If E = 1, w
p= g + h Hence, g and g+h are portfolios
on the frontier.
Ch t i ti f F ti P tf li Ch t i ti f F ti P tf li
• Proposition 6.1: The entire set of frontier
f li b d b ("
Characterization of Frontier Portfolios Characterization of Frontier Portfolios
portfolios can be generated by ("are convex combinations" of) g and g+h.
• Proposition 6.2. The portfolio frontier can be described as convex combinations of any two f y frontier portfolios, not just the frontier portfolios g and g+h.
• Proposition 6.3 : Any convex combination of frontier portfolios is also a frontier portfolio
Slide Slide 0707--1818 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
frontier portfolios is also a frontier portfolio.
…Characterization of Frontier Portfolios…
…Characterization of Frontier Portfolios…
• For any portfolio on the frontier, with g and h as defined earlier.
( ) [ ( ) ] [ ( )
p]
T p
p
g hE r V g hE r
r
E [ ~ ] ~ ~
2
= + +
σ
Multiplying all this out yields:
Slide Slide 0707--1919
10:13 Lecture
MeanMean--Variance Analysis and CAPMVariance Analysis and CAPM…Characterization of Frontier Portfolios…
…Characterization of Frontier Portfolios…
• (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; y
• (iii) equation (6.17) is the equation of a parabola with
(1/C A/C) i h d / i d
vertex (1/C, A/C) in the expected return/variance space and of a hyperbola in the expected return/standard deviation space. See Figures 6.3 and 6.4.
Slide Slide 0707--2020 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
p g
Slide Slide 0707--2121 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space
Slide Slide 0707--2222 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space
Slide Slide 0707--2323 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed
Efficient Frontier with risk
Efficient Frontier with risk--free asset free asset Efficient Frontier with risk
Efficient Frontier with risk free asset free asset
Slide Slide 0707--2424
The Efficient Frontier: One Risk Free and n Risky Assets
Efficient Frontier with risk
Efficient Frontier with risk--free asset free asset Efficient Frontier with risk
Efficient Frontier with risk free asset free asset
1 T
min w 1 2 w T V w
s.t. w T e + (1 − w T 1)r f = E[r p ]
FOC: w p = λV −1 (e − r f 1)
f
Multiplying by (e–r
f1)
Tand solving for λ yields λ = (e E[r p ] −r f
−r f 1) T V −1 (e −r f 1)
w p = V −1 (e − r f 1)
| {z }
1
E[r
p] −r
fH
2Slide Slide 0707--2525
where is a n
n ×1
H = q
B − 2Ar
f+ Cr
f2Effi i f i i h i k Effi i f i i h i k ff Efficient frontier with risk
Efficient frontier with risk--free asset free asset
T
• Result 1: Excess return in frontier excess return
Cov[r q , r p ] = w q T V w p
= w q T (e − r f 1) E[r p ] − r f H 2
q f
| {z }
E[r
q] −r
fH 2
= (E[r q ] − r f )([E[r p ] − r f )
= H 2
V ar[r p , r p ] = (E[r p ] − r f ) 2 H 2
E[r q ] − r f = Cov[r q , r p ]
V ar[r p ] (E[r p ] − r f )
H 2
Slide Slide 0707--2626
V ar[r p ]
| {z }
:=β q,p
Holds for any frontier portfolio p, in particular the market portfol
Efficient Frontier with risk
Efficient Frontier with risk--free asset free asset Efficient Frontier with risk
Efficient Frontier with risk free asset free asset
• Result 2: Frontier is linear in (E[r], σ)-space
V ar[r r ] (E[r p ] − r f ) 2 V ar[r p , r p ] = p f
H 2 E[r p ] = r f + Hσ p
H = E[r p σ ] −r f
p p
where H is the Sharpe ratio
Slide Slide 0707--2727
T F d S ti
T F d S ti
Two Fund Separation Two Fund Separation
• Doing it in two steps:
– First solve frontier for n risky asset First solve frontier for n risky asset – Then solve tangency point
• Advantage:
• Advantage:
– Same portfolio of n risky asset for different agents with different risk aversion
with different risk aversion
– Useful for applying equilibrium argument (later)
Slide Slide 0707--2828
Two
Two Fund Separation Fund Separation Two
Two Fund Separation Fund Separation
Price of Risk =
= highest Sharpe ratio
Slide Slide 0707--2929
Optimal Portfolios of Two Investors with Different Risk Aversion
M
M V i V i P f P f Mean
Mean--Variance Preferences Variance Preferences
∂U
∂μ > 0, ∂σ ∂U
2< 0
• U(μ p , σ p ) with
– Example:
∂μ
p∂σ
pE[W ] −
γ2V ar[W ] p
• Also in expected utility framework
– quadratic utility function (with portfolio return R) U(R) = a + b R + c R ( )
2vNM: E[U(R)] = a + b E[R] + c E[R
2]
= a + b μ
p+ c μ
p2+ c σ
p2= g(μ
pp, σ
pp)
– asset returns normally distributed ⇒ R=∑
jw
jr
jnormal
• if U(.) is CARA ⇒ certainty equivalent = μ
p- ρ
A/2σ
2p(Use moment generating function)
Slide Slide 0707--3030
(Use moment generating function)
Equilibrium leads to CAPM Equilibrium leads to CAPM
• Portfolio theory: only analysis of demand
– price/returns are taken as given
i i f i k f li i f ll i
– composition of risky portfolio is same for all investors
• Equilibrium Demand = Supply (market portfolio)
• Equilibrium Demand = Supply (market portfolio)
• CAPM allows to derive CAPM allows to derive
– equilibrium prices/ returns.
– risk-premium
Slide Slide 0707--3131 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
The CAPM with a risk
The CAPM with a risk free bond free bond The CAPM with a risk
The CAPM with a risk--free bond free bond
• The market portfolio is efficient since it is on the The market portfolio is efficient since it is on the efficient frontier.
• All individual optimal portfolios are located on
• All individual optimal portfolios are located on the half-line originating at point (0,r f ).
R R
E [ ] −
• The slope of Capital Market Line (CML): .
M f
M
R
R E
σ
− ] [
p M
f M
f p
R R
R E R
E σ
σ + −
= [ ]
] [
Slide Slide 0707--3232 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
The Capital Market Line The Capital Market Line The Capital Market Line The Capital Market Line
CML
r
MM
j
r
fSlide Slide 0707--3333 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
σ
pσ
MThe Security Market Line The Security Market Line
E(r)
The Security Market Line The Security Market Line
E(ri)
SML
E(rM)
rf
slope SML = (E(ri)-rf) /βi
β βi
βM=1
Slide Slide 0707--3434 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Overview Overview
1.
1. Simple CAPM with quadratic utility functions Simple CAPM with quadratic utility functions (derived from state
(derived from state--price beta model) price beta model) 2.
2. Mean Mean--variance preferences variance preferences
–– Portfolio Theory Portfolio Theory –– CAPM CAPM (Intuition) (Intuition)
3.
3. CAPM CAPM (modern derivation) (modern derivation)
–– Projections Projections
–– Pricing Kernel and Expectation Kernel Pricing Kernel and Expectation Kernel
Slide Slide 0707--3535 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Projections Projections
• States s=1,…,S with π s >0
• Probability inner product Probability inner product
( f l h)
• π-norm (measure of length)
Slide Slide 0707--3636 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
⇒
⇒ shrink
y axes y
x x
x and y are π-orthogonal iff [x,y] π = 0, I.e. E[xy]=0
Slide Slide 0707--3737 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
…Projections…
…Projections…
• Z space of all linear combinations of vectors z 1 , …,z n
• Given a vector y ∈ R S solve
• [smallest distance between vector y and Z space]
Slide Slide 0707--3838 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• [smallest distance between vector y and Z space]
…Projections
…Projections
y ε
y Z
E[ε z j ]=0 for each j=1,…,n (from FOC) ε⊥ z
Slide Slide 0707--3939 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
y Z is the (orthogonal) projection on Z
y = y Z + ε’ , y Z ∈ Z, ε ⊥ z
Expected Value and Co
Expected Value and Co--Variance… Variance…
squeeze axis by
(1,1)
x [x,y]=E[xy]=Cov[x,y] + E[x]E[y]
x [x,y] E[xy] Cov[x,y] E[x]E[y]
[x,x]=E[x 2 ]=Var[x]+E[x] 2
||x||= E[x 2 ] ½
Slide Slide 0707--4040 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
E t d V l d C
E t d V l d C V i V i
…Expected Value and Co
…Expected Value and Co--Variance Variance
E[x] = [x, 1]=
Slide Slide 0707--4141 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Overview Overview
1.
1. Simple CAPM with quadratic utility functions Simple CAPM with quadratic utility functions
(derived from state
(derived from state--price beta model) price beta model)
2.
2. Mean Mean--variance preferences variance preferences
–– Portfolio Theory Portfolio Theory –– CAPM CAPM (Intuition) (Intuition)
3.
3. CAPM CAPM (modern derivation) (modern derivation)
P j i P j i –– Projections Projections
–– Pricing Kernel and Expectation Kernel Pricing Kernel and Expectation Kernel
Slide Slide 0707--4242 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
New (LeRoy & Werner) Notation New (LeRoy & Werner) Notation
• Main changes (new versus old)
– gross return: gross return: r = R r R
– SDF: μ = m
– pricing kernel: pricing kernel: k = m k q m * – Asset span: M = <X>
– income/endowment: w t = e t
Slide Slide 0707--4343 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Pricing Kernel k
Pricing Kernel k qq … …
M f f ibl ff
• M space of feasible payoffs.
• If no arbitrage and π >>0 there exists g
SDF μ ∈ R S , μ >>0, such that q(z)=E(μ z).
∈ M SDF d t b i t
• μ ∈ M – SDF need not be in asset span.
• A pricing kernel is a k q ∈ M such that for each z ∈ M, q(z)=E(k q z).
• (k q = m * in our old notation.)
Slide Slide 0707--4444 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
( q )
P i i K l
P i i K l E E l l
…Pricing Kernel
…Pricing Kernel -- Examples… Examples…
• Example 1:
– S=3,π S 3,π 1/3 for s 1,2,3, s =1/3 for s=1,2,3,
– x 1 =(1,0,0), x 2 =(0,1,1), p=(1/3,2/3).
– Then k=(1 1 1) is the unique pricing kernel Then k (1,1,1) is the unique pricing kernel.
• Example 2:
S 3 s 1/3 f 1 2 3 – S=3,π s =1/3 for s=1,2,3,
– x 1 =(1,0,0), x 2 =(0,1,0), p=(1/3,2/3).
Th k (1 2 0) i th i i i k l
Slide Slide 0707--4545 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
– Then k=(1,2,0) is the unique pricing kernel.
P i i K l
P i i K l U i U i
…Pricing Kernel
…Pricing Kernel –– Uniqueness Uniqueness
• If a state price density exists, there exists a If a state price density exists, there exists a unique pricing kernel.
– If dim(M) = m (markets are complete), If dim(M) m (markets are complete),
there are exactly m equations and m unknowns If dim(M) ≤ m (markets may be incomplete) – If dim(M) ≤ m, (markets may be incomplete)
For any state price density (=SDF) μ and any z ∈ M E[(μ k )z]=0
E[(μ-k q )z]=0
μ=(μ-k q )+k q ⇒ k q is the “projection'' of μ on M.
• Complete markets ⇒ k =μ (SDF=state price density)
Slide Slide 0707--4646 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• Complete markets ⇒, k q =μ (SDF=state price density)
E t ti K l k
E t ti K l k
Expectations Kernel k Expectations Kernel k ee
• An expectations kernel is a vector k ∈ M An expectations kernel is a vector k e ∈ M
– Such that E(z)=E(k e z) for each z ∈ M.
• Example p
– S=3, π s =1/3, for s=1,2,3, x 1 =(1,0,0), x 2 =(0,1,0).
– Then the unique k e =(1,1,0).
• If π >>0, there exists a unique expectations kernel.
• Let e=(1,…, 1) then for any z∈ M E[( k ) ] 0
• E[(e-k e )z]=0
• k e is the “projection” of e on M
k if b d b li t d ( if k t l t )
Slide Slide 0707--4747 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• k e = e if bond can be replicated (e.g. if markets are complete)
Mean Variance Frontier Mean Variance Frontier Mean Variance Frontier Mean Variance Frontier
• Definition 1: z ∈ M is in the mean variance frontier if there exists no z’ ∈ M such that
E[z’]= E[z], q(z')= q(z) and var[z’] < var[z].
• Definition 2: Let E the space generated by k q and k e .
• Decompose z=z E +ε, with z E ∈ E and ε ⊥ E.
• Hence, E[ε]= E[ε k e ]=0, q(ε)= E[ε k q ]=0 Cov[ε,z E ]=E[ε z E ]=0, since ε ⊥ E.
• var[z] = var[z E ]+var[ε] (price of ε is zero, but positive variance)
• If z in mean variance frontier ⇒ z ∈ E.
• Every z ∈ E is in mean variance frontier
Slide Slide 0707--4848 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• Every z ∈ E is in mean variance frontier.
F ti R t F ti R t
Frontier Returns…
Frontier Returns…
• Frontier returns are the returns of frontier payoffs with
• Frontier returns are the returns of frontier payoffs with non-zero prices.
• x
Slide Slide 0707--4949 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• graphically: payoffs with price of p=1.
M R S R 3 M = R S = R 3
Mean-Variance Payoff Frontier Mean-Variance Payoff Frontier
ε
k q
Mean-Variance Return Frontier
p=1-line = return-line (orthogonal to k q )
Slide Slide 0707--5050 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
p ( g q )
M V i (P ff) F ti Mean-Variance (Payoff) Frontier
(1 1 1) standard deviation
0 k q
(1,1,1)
expected return standard deviation
q
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
NB: graphical illustrated of expected returns and standard deviation
changes if bond is not in payoff span.
M V i (P ff) F ti
efficient (return) frontier Mean-Variance (Payoff) Frontier
(1 1 1) standard deviation
0 k q
(1,1,1)
expected return standard deviation
q
inefficient (return) frontier
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
…Frontier Returns
…Frontier Returns
(if agent is risk-neutral)
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Mi i V i P tf li
Mi i V i P tf li
Minimum Variance Portfolio Minimum Variance Portfolio
• Take FOC w.r.t. λ of
H MVP h f
• Hence, MVP has return of
Slide Slide 0707--5454 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Illustration of MVP Illustration of MVP Illustration of MVP Illustration of MVP
E t d t M = R 2 and S=3
Mi i d d
Expected return of MVP
(1,1,1)
Minimum standard deviation
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
M
M V i V i Effi i t R t Effi i t R t Mean
Mean--Variance Efficient Returns Variance Efficient Returns
• Definition: A return is mean-variance efficient if there is no other return with same variance but greater
expectation expectation.
• Mean variance efficient returns are frontier returns with E[r ] ≥ E[r ]
E[r λ ] ≥ E[r λ0 ].
• If risk-free asset can be replicated
M i ffi i t t d t λ ≤ 0
– Mean variance efficient returns correspond to λ ≤ 0.
– Pricing kernel (portfolio) is not mean-variance efficient, since
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Z
Z C C i i F F ti R t ti R t Zero
Zero--Covariance Frontier Returns Covariance Frontier Returns
• Take two frontier portfolios with returns
• Take two frontier portfolios with returns and
• C C
• The portfolios have zero co-variance if p
• For all λ ≠ λ 0 μ exists
• μ=0 if risk-free bond can be replicated
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Illustration of ZC Portfolio…
Illustration of ZC Portfolio…
Illustration of ZC Portfolio…
Illustration of ZC Portfolio…
M = R 2 and S=3
arbitrary portfolio p
(1,1,1) y p p
Recall:
Slide Slide 0707--5858 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
…Illustration of ZC Portfolio
…Illustration of ZC Portfolio
…Illustration of ZC Portfolio
…Illustration of ZC Portfolio
Green lines do not necessarily cross.
arbitrary portfolio p
(1,1,1) y p p
ZC of pp
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
B t P i i B t P i i
Beta Pricing…
Beta Pricing…
• Frontier Returns (are on linear subspace). Hence
• Consider any asset with payoff x y p y j j
– It can be decomposed in x j = x j E + ε j
– q(x q(x j j )=q(x ) q(x j j E ) and E[x ) and E[x j j ]=E[x ] E[x j j E ], since ], since ε ⊥ E. ε ⊥ E.
– Let r j E be the return of x j E – x
– x
– Using above and assuming λ ≠ λ 0 and μ is ZC-portfolio of λ
Slide Slide 0707--6060 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
ZC portfolio of λ,
B t P i i B t P i i
…Beta Pricing
…Beta Pricing
• Taking expectations and deriving covariance
• _
• If risk-free asset can be replicated, beta-pricing equation simplifies to
equation simplifies to
• Problem: How to identify frontier returns
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• Problem: How to identify frontier returns
C it l A t P i i M d l C it l A t P i i M d l
Capital Asset Pricing Model…
Capital Asset Pricing Model…
• CAPM = market return is frontier return
– Derive conditions under which market return is frontier return – Two periods: 0,1.
– Endowment: individual w i 1 at time 1, aggregate
where the orthogonal projection of on M is.
– The market payoff:
– Assume q(m) ≠ 0, let r m =m / q(m), and
assume that r m is not the minimum variance return.
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
C it l A t P i i M d l C it l A t P i i M d l
…Capital Asset Pricing Model
…Capital Asset Pricing Model
• If r m0 is the frontier return that has zero
covariance with r m then, for every security j,
• E[r j ]=E[r m0 ] + β j (E[r m ]-E[r m0 ]), with β j j =cov[r j j ,r m ] / var[r m ].
• If a risk free asset exists, equation becomes,
• E[r [ j j ]= r ] f f + β β j j (E[r ( [ m m ]- r ] f f ) )
• N.B. first equation always hold if there are only two assets.
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
q y y
O td t d t i l f ll O td t d t i l f ll
Outdated material follows Outdated material follows
• Traditional derivation of CAPM is less elegant
• Not relevant for exams Not relevant for exams
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Ch t i ti f F ti P tf li Ch t i ti f F ti P tf li
• Proposition 6.1: The entire set of frontier
f li b d b ("
Characterization of Frontier Portfolios Characterization of Frontier Portfolios
portfolios can be generated by ("are convex combinations" of) g and g+h.
• Proposition 6.2. The portfolio frontier can be described as convex combinations of any two f y frontier portfolios, not just the frontier portfolios g and g+h.
• Proposition 6.3 : Any convex combination of frontier portfolios is also a frontier portfolio
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
frontier portfolios is also a frontier portfolio.
Ch t i ti f F ti P tf li Ch t i ti f F ti P tf li
• Proposition 6.1: The entire set of frontier
Characterization of Frontier Portfolios Characterization of Frontier Portfolios
p f f
portfolios can be generated by ("are convex combinations" of) g and g+h. f) g g
– Proof: To see this let q be an arbitrary frontier portfolio with as its expected return.
Consider portfolio eights (proportions)
( ) r~
qE
Consider portfolio weights (proportions)
then, as asserted,
( )
q g h( )
qg
= 1 − E r~ and π = E r~
π
+[ 1 E ( ) r~ ] g + E ( ) r~ ( g + h ) g + hE ( ) r~ w ( )
[ 1 − E r
q] g + E ( ) r
q( g + h ) = g + hE ( ) r
q= w
q.
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• Proposition 6.2. The portfolio frontier can be described as convex combinations of any two frontier portfolios, not just the frontier portfolios g and g+h.
• Proof: To see this, let p 1 and p 2 be any two distinct frontier portfolios; since the frontier portfolios are E[r p1 ] ≠ E[r p2 ] different. Let q be an arbitrary frontier portfolio, with d e e t. et q be a a b t a y o t e po t o o, w t
expected return equal to E[r q ]. Since E[r p1 ] ≠ E[r p2 ] , there must exist a unique number such that
E[r ] = α E[r ] + (1 α) E[r ] (6 16) E[r q ] = α E[r p1 ] + (1 - α) E[r p2 ] (6.16)
Now consider a portfolio of p 1 and p 2 with weights α, 1- α, respectively, as determined by (6.16). We must show that
w q = α w p1 + (1 - α) w p2 .
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Proof of Proposition 6.2 (continued)
( )
2[ ( )
1] ( ) [ ( )
2]
1 p p p
p
1 w g hE r~ 1 g hE r~
w + − α = α + + − α +
α
( ) ( ) ( )
[ ( ) ( ) ( ) ]
[ ]
( ) r~ , by constructi on
hE g
r~
E 1
r~
E h g
q
p
p1 2
+
=
α
− +
α +
=
portfolio.
frontier a
is q since ,
w
q=
Slide Slide 0707--6868 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
f li i l f i f li portfolios is also a frontier portfolio.
• Proof : Let , define N frontier portfolios ( represents the vector defining the composition of the i th
( w
1... w
N)
w
i( represents the vector defining the composition of the i portfolios) and let , i = 1, ..., N be real numbers such that . Lastly, let denote the expected return of portfolio with weights
w
iα
i N1
1
i i
=
∑ α
=E ( ) r
iof portfolio with weights .
The weights corresponding to a linear combination of the above N portfolios are :
w
i( )
( )
∑ α +
∑α =
=
=
N 1
i i i
N 1
i i
w
ig hE r~
( )
+ ∑α
= ∑α ∑α
Ng + h ∑α
NE ( ) r~
=
= i=1 i i 1i i
g h E r
( ) ⎥⎦ ⎤
⎢⎣ ⎡∑α +
=
=N 1
i i
E r~
ih g
N
( )
N( )
Slide Slide 0707--6969 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Thus ∑α is a frontier portfolio with .
= N
1
i i
w
i( )
N( )
i1
i i
E r~
r
E = ∑α
=
…Characterization of Frontier Portfolios…
…Characterization of Frontier Portfolios…
• For any portfolio on the frontier, with g and h as defined earlier.
( ) [ ( ) ] [ ( )
p]
T p
p
g hE r V g hE r
r
E [ ~ ] ~ ~
2
= + +
σ
Multiplying all this out yields:
Slide Slide 0707--7070 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
…Characterization of Frontier Portfolios…
…Characterization of Frontier Portfolios…
• (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; y
• (iii) equation (6.17) is the equation of a parabola with
(1/C A/C) i h d / i d
vertex (1/C, A/C) in the expected return/variance space and of a hyperbola in the expected return/standard deviation space. See Figures 6.3 and 6.4.
Slide Slide 0707--7171 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
p g
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space
Slide Slide 0707--7373 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed
Characterization of Efficient Portfolios Characterization of Efficient Portfolios Characterization of Efficient Portfolios Characterization of Efficient Portfolios
(No Risk
(No Risk--Free Assets) Free Assets)
• Definition 6.2: Efficient portfolios are those frontier
tf li hi h t i d i t d
portfolios which are not mean-variance dominated.
• Lemma: Efficient portfolios are those frontier
portfolios for which the expected return exceeds A/C portfolios for which the expected return exceeds A/C, the expected return of the minimum variance
portfolio portfolio.
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
• Proposition 6.4 : 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 considered above was
• Proof : Suppose each of the N portfolios considered above was efficient; then E ( ) r~
i≥ A / C , for every portfolio i.
A A
N
However
N; thus, the convex
combination is efficient as well. So the set of efficient
f li h i d b h i f li i h i
( ) C
A C
r~ A
E
N1
i i
N 1
i i i
≥ ∑ α =
∑α
= =portfolios, as characterized by their portfolio weights, is a convex set.
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Zero Covariance Portfolio Zero Covariance Portfolio
• Zero-Cov Portfolio is useful for Zero-Beta CAPM
• Proposition 6.5: For any frontier portfolio p, except the Proposition 6.5: 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 We will call this portfolio the zero covariance portfolio relative to p", and denote its vector of portfolio weights by
( ) p . ZC
• Proof: by construction.
Slide Slide 0707--7777 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
collect all expected returns terms, add and subtract A 2 C/DC 2 collect all expected returns terms, add and subtract A C/DC and note that the remaining term (1/C)[(BC/D)-(A 2 /D)]=1/C, since D=BC-A 2
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For zero co-variance portfolio ZC(p) For zero co variance portfolio ZC(p)
For graphical illustration, let’s draw this line:
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Graphical Representation:
line through
p (Var[r ] E[r ]) AND p (Var[r p ], E[r p ]) AND
MVP (1/C, A/C) (use ) for σ 2 (r) = 0 you get E[r ZC(p) ]
( ) ( )
C 1 C
r~ A D E
r~ C
2 p
p
2
⎟ +
⎠
⎜ ⎞
⎝
⎛ −
= σ
for σ (r) 0 you get E[r ZC(p) ]
Slide Slide 0707--8080 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
E ( r )
A/C
p
A/C
MVP
ZC(p)
E[r ( ZC(p)]
ZC (p) on frontier
(1/C) Var ( r )
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM
Figure 6-6 The Set of Frontier Portfolios: Location of the Zero-Covariance Portfolio
Zero
Zero Beta CAPM Beta CAPM Zero
Zero--Beta CAPM Beta CAPM
(no risk
(no risk--free asset) free asset)
(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
maximizing a derived utility function of the form , U concave.
(iii) common time horizon,
( ) e, , U 0, U 0
U σ
2 1>
2<
( ) ,
(iv) homogeneous beliefs about e and σ 2
Slide Slide 0707--8282 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
– All investors hold mean-variance efficient portfolios
th k t tf li i bi ti f ffi i t tf li
– the market portfolio is convex combination of efficient portfolios
⇒ is efficient.
– Cov[r
p,r
q] = λ E[r
q]+ γ (q need not be on the frontier) (6.22) – Cov[r
p,r
ZC(p)] = λ E[r
ZC(p)] + γ=0
– Cov[r
pp q,r
q] = λ {E[r
qq]-E[r
ZC(p)(p)]}
– Var[r
p] = λ {E[r
p]-E[r
ZC(p)]} .
Divide third by fourth equation:
( ) r~
qE ( r~
ZC( )
M)
Mq[ E ( ) r~
ME ( r~
ZC( )
M) ]
E = + β − (6.28)
( ) ( r~ E r~ ) [ E ( ) r~ E ( r~ ) ]
E = + β (6 29)
Slide Slide 0707--8383 Mean
Mean--Variance Analysis and CAPMVariance Analysis and CAPM
( ) r
jE ( r
ZC( )
M)
Mj[ E ( ) r
ME ( r
ZC( )
M) ]
E = + β − (6.29)
Z
Z B t CAPM B t CAPM Zero
Zero--Beta CAPM Beta CAPM
• mean variance framework (quadratic utility or normal returns)
• In equilibrium market portfolio which is a In equilibrium, market portfolio, which is a convex combination of individual portfolios E[r q ] = E[r ZC(M) ]+ β Mq [E[r M ]-E[r ZC(M) ]]
E[r j ] = E[r ZC(M) ]+ β Mj [E[r M ]-E[r ZC(M) ]]
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Mean--Variance Analysis and CAPMVariance Analysis and CAPM