lecture lecture 07: 07: meanmean--variance analysis …markus/teaching/fin501/07lecture.pdf ·...

Fin 501: Asset PricingFin 501: Asset Pricing

Lecture Lecture 07: 07: MeanMean--Variance Analysis &Variance Analysis &Capital Asset Pricing ModelCapital Asset Pricing Model((CAPMCAPM))((CAPMCAPM))

Prof. Markus K. Brunnermeier

Slide Slide 0707--11MeanMean--Variance Analysis and CAPMVariance Analysis and CAPM


O iO iOverviewOverview

1. Simple CAPM with quadratic utility functions(derived from state(derived from state--price beta model)price beta model)

2.2. MeanMean--variance preferencesvariance preferences–– Portfolio TheoryPortfolio Theoryyy–– CAPM CAPM (intuition)(intuition)

33 CAPMCAPM3.3. CAPMCAPM–– ProjectionsProjections

Pricing Kernel and Expectation KernelPricing Kernel and Expectation KernelSlide Slide 0707--22MeanMean--Variance Analysis and CAPMVariance Analysis and CAPM

–– Pricing Kernel and Expectation KernelPricing Kernel and Expectation Kernel


R ll St tR ll St t i B t d li B t d lRecall StateRecall State--price Beta modelprice Beta model

Recall:Recall:E[E[RRhh]] -- RRff == ββhh E[RE[R**-- RRff]]E[E[RR ] ] RR ββ E[RE[R RR ]]

where βh := Cov[R*,Rh] / Var[R*]

very general very general –– but what is Rbut what is R** in realityin reality??



Simple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected Utility1. All agents are identical

• Expected utility U(x x ) = ∑ π u(x x ) ⇒ m= ∂ u / E[∂ u]• Expected utility U(x0, x1) = ∑s πs u(x0, xs) ⇒ m= ∂1u / E[∂0u]• Quadratic u(x0,x1)=v0(x0) - (x1- α)2

⇒ ∂1u = [-2(x1 1- α),…, -2(xS 1- α)]1 [ ( 1,1 ), , ( S,1 )]• E[Rh] – Rf = - Cov[m,Rh] / E[m]

= -Rf Cov[∂1u, Rh] / E[∂0u]= -Rf Cov[-2(x - α) Rh] / E[∂ u]= -R Cov[-2(x1- α), R ] / E[∂0u]= Rf 2Cov[x1,Rh] / E[∂0u]

• Also holds for market portfolio• E[Rm] – Rf = Rf 2Cov[x1,Rm]/E[∂0u]


⇒


Simple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected Utility

2. Homogenous agents + Exchange economy d d i f l l d i h R⇒ x1 = agg. endowment and is perfectly correlated with Rm

E[E[RRhh]=]=RRff + + ββhh {E[{E[RRmm]]--RRff}} Market Security LineMarket Security Line

Slide Slide 0707--55MeanMean--Variance Analysis and CAPMVariance Analysis and CAPMN.B.: R*=Rf (a+b1RM)/(a+b1Rf) in this case (where b1<0)!


O iO iOverviewOverview

1.1. Simple CAPM with quadratic utility functionsSimple CAPM with quadratic utility functions(derived from state(derived from state--price beta model)price beta model)

2.2. MeanMean--variance analysisvariance analysis–– Portfolio Theory Portfolio Theory

((P f li f i ffi i f i )P f li f i ffi i f i )((Portfolio frontier, efficient frontier, …)Portfolio frontier, efficient frontier, …)–– CAPM CAPM (Intuition)(Intuition)

33 CAPMCAPM3.3. CAPMCAPM–– ProjectionsProjections

P i i K l d E i K lP i i K l d E i K lSlide Slide 0707--66MeanMean--Variance Analysis and CAPMVariance Analysis and CAPM

–– Pricing Kernel and Expectation KernelPricing Kernel and Expectation Kernel


Definition: MeanDefinition: Mean--Variance DominanceVariance Dominancee o : ee o : e V ce o ceV ce o ce& Efficient Frontier& Efficient Frontier

• Asset (portfolio) A mean-variance dominates asset (portfolio) Basset (portfolio) B if μA ≥ μB and σA < σΒ or if μA>μBwhile σA ≤σB.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”) iBy definition, no (“rational”) mean-variance investor would choose to hold a portfolio not l t d th ffi i t f ti


located on the efficient frontier.


E d P f li R & V iE d P f li R & V iExpected Portfolio Returns & VarianceExpected Portfolio Returns & Variance

• Expected returns (linear)μp := E[rp] = wjμj , where each wj =

hjPhj

• Variance2 V [ ] 0V ( )

µσ21 σ12

¶µw1

¶μp [ p] jμj , j P

jhj

σ2p := V ar[rp] = w0V w = (w1 w2)µ

1 12

σ21 σ22

¶µ1

w2

¶= (w1σ

21 + w2σ21 w1σ12 + w2σ

22)

µw1w2

¶

recall that correlation

µw2

¶= w21σ

21 + w

22σ

22 + 2w1w2σ12 ≥ 0

since σ12 ≤ −σ1σ2.


coefficient ∈ [-1,1]


Ill t ti f 2 A t CIll t ti f 2 A t CIllustration of 2 Asset CaseIllustration of 2 Asset Case

• For certain weights: w1 and (1-w1)μp = w1 E[r1]+ (1-w1) E[r2] μp 1 [ 1] ( 1) [ 2]σ2

p = w12 σ1

2 + (1-w1)2 σ22 + 2 w1(1-w1)σ1 σ2 ρ1,2

(Specify σ2p and one gets weights and μp’s)( p y p g g μp )

• Special cases [w1 to obtain certain σR]

– ρ = 1 ⇒ w = (+/-σ – σ ) / (σ – σ )– ρ1,2 1 ⇒ w1 (+/-σp – σ2) / (σ1 – σ2)– ρ1,2 = -1 ⇒ w1 = (+/- σp + σ2) / (σ1 + σ2)



For ρ1,2 = 1: Hence, σp = |w1σ1 + (1− w1)σ2| w1 =±σp−σ2ρ1,2 ,

μp = w1μ1 + (1− w1)μ21 σ1−σ2

μp

E[r2]

E[r ]

σ1 σp σ2

E[r1]

The Efficient Frontier: Two Perfectly Correlated Risky Assets

Lower part with … is irrelevant


The Efficient Frontier: Two Perfectly Correlated Risky Assets


For ρ1,2 = -1: Hence, σp = |w1σ1 − (1− w1)σ2| w1 =±σp+σ2

+ρ1,2 ,

μp = w1μ1 + (1− w1)μ21 σ1+σ2

E[ ]E[r2]

E[r ]

σ2σ1+σ2

μ1 +σ1

σ1+σ2μ2

slope: μ2−μ1σ1+σ2

σp

slope: −μ2−μ1σ1+σ2

σp

σ1 σ2

E[r1] σ1+σ2 p

Efficient Frontier: Two Perfectly Negative Correlated Risky Assets


Efficient Frontier: Two Perfectly Negative Correlated Risky Assets


For -1 < ρ1,2 < 1:ρ1,2

E[r2]

E[r1]

σ1 σ2

Efficient Frontier: Two Imperfectly Correlated Risky Assets



For σ = 0For σ1 = 0

E[r ]

μp

E[r2]

E[r1]

σ1 σp σ2

E[r1]

The Efficient Frontier: One Risky and One Risk Free Asset



Effi i f i i h i kEffi i f i i h i kEfficient frontier with n risky assetsEfficient 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.

Vww21min T

w

( )λ Eew .t.s T = ⎟⎠⎞⎜

⎝⎛ =∑ E)r~(Ew

N

ii

2w

σ

( )γ 11w T = ⎟⎠⎞⎜

⎝⎛ =∑

⎠⎝

=

=

1wN

1ii

1i

Slide Slide 0707--1414

⎠⎝ =1i


∂L∂L∂w = V w − λe− γ1 = 0∂L = E wTe = 0∂L∂λ = E − wTe = 0∂L∂γ = 1− wT1 = 0

The first FOC can be written as:

∂γ

V wp = λe+ γ1 or

λV −1 + V −11wp = λV 1e+ γV 11

eTwp = λ(eTV −1e) + γ(eTV −11)Slide Slide 0707--1515

p ( ) + γ( )


N i h T T i h fi f h d fNoting that eT wp = wTpe, using the first foc, the second foc

can be written as

E[r̃ ] = eTw = λ (eTV −1e)+γ (eTV −11)

pre-multiplying first foc with 1 (instead of eT) yields

E[rp] = e wp = λ (e V e)| {z }:=B

+γ (e V 1)| {z }=:A

pre-multiplying first foc with 1 (instead of e ) yields

1Twp = wTp 1 = λ(1TV −1e) + γ(1TV −11) = 1

( T 1 ) ( T 1 )1 = λ (1TV −1e)| {z }=:A

+γ (1TV −11)| {z }=:C

Solving both equations for λ and γ

λ= CE−AD and γ = B−AE

D


where D = BC −A2.


Hence, wp = λ V-1e + γ V-11 becomes

111 −− −+

−= V

DAEBeV

DACEwp

(vector) (vector)

( ) ( )[ ] ( ) ( )[ ]E1VAeVC1eVA1VB1 1111 −−−− −+−=

λ (scalar) γ (scalar)

( ) ( )[ ] ( ) ( )[ ]VeVCD

eVVD

• Result: Portfolio weights are linear in expected portfolio return E h g wp +=


pIf E = 0, wp = gIf E = 1, wp = g + h

Hence, g and g+h are portfolios on the frontier.


Ch t i ti f F ti P tf liCh 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 convexcombinations" of) g and g+h.

• Proposition 6.2. The portfolio frontier can bedescribed as convex combinations of any twof yfrontier portfolios, not just the frontier portfolios gand g+h.

• Proposition 6.3 : Any convex combination offrontier portfolios is also a frontier portfolio


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.

( ) ( )[ ] ( )[ ]pT

pp rhEgVrhEgrE ~~]~[2 ++=σ

Multiplying all this out yields:

Slide Slide 0707--191910:13 Lecture MeanMean--Variance Analysis and CAPMVariance Analysis and CAPM



• (i) the expected return of the minimum variance portfoliois A/C;

• (ii) the variance of the minimum variance portfolio is givenby 1/C;y

• (iii) equation (6.17) is the equation of a parabola with(1/C A/C) i h d / i dvertex (1/C, A/C) in the expected return/variance space and

of a hyperbola in the expected return/standard deviationspace. See Figures 6.3 and 6.4.


p g



Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space



Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space



Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed


Efficient Frontier with riskEfficient Frontier with risk--free assetfree assetEfficient Frontier with riskEfficient Frontier with risk free assetfree asset


The Efficient Frontier: One Risk Free and n Risky Assets



1 Tminw12w

TV w

s.t. wTe+ (1− wT1)rf = E[rp]

FOC: wp = λV −1(e− rf1)f

Multiplying by (e–rf 1)T and solving for λ yields λ =E[rp]−rf

(e−rf1)TV −1(e−rf1)

wp = V−1(e− rf1)| {z }

1

E[rp]−rfH2

Slide Slide 0707--2525where is a n

n×1H =

qB − 2Arf + Cr2f


Effi i f i i h i kEffi i f i i h i k ffEfficient frontier with riskEfficient frontier with risk--free assetfree assetT

• Result 1: Excess return in frontier excess returnCov[rq, rp] = wTq V wp

= wTq (e− rf1)E[rp]− rf

H2q f| {z }E[rq]−rf

H2

=(E[rq]− rf )([E[rp]− rf )

=H2

V ar[rp, rp] =(E[rp]− rf )2

H2

E[rq]− rf =Cov[rq, rp]

V ar[rp](E[rp]− rf)

H2


V ar[rp]| {z }:=βq,p

Holds for any frontier portfolio p, in particular the market portfol



• Result 2: Frontier is linear in (E[r], σ)-spaceV ar[r r ]

(E[rp]− rf )2V ar[rp, rp] =

p f

H2

E[rp] = rf +Hσp

H =E[rp]−rf

σpp

where H is the Sharpe ratio



T F d S tiT F d S tiTwo Fund SeparationTwo Fund Separation

• Doing it in two steps:– First solve frontier for n risky assetFirst solve frontier for n risky asset– Then solve tangency point

• Advantage:• Advantage:– Same portfolio of n risky asset for different agents

with different risk aversionwith different risk aversion– Useful for applying equilibrium argument (later)



TwoTwo Fund SeparationFund SeparationTwo Two Fund SeparationFund Separation

Price of Risk == highest

Sharpe ratio

Slide Slide 0707--2929Optimal Portfolios of Two Investors with Different Risk Aversion


MM V i P fV i P fMeanMean--Variance PreferencesVariance Preferences∂U∂μ > 0, ∂U

∂σ2 < 0• U(μp, σp) with

– Example:

∂μp ∂σp

E[W ]− γ2V ar[W ]

p• Also in expected utility framework

– quadratic utility function (with portfolio return R)U(R) = a + b R + c R2( )

vNM: E[U(R)] = a + b E[R] + c E[R2]= a + b μp + c μp

2 + c σp2

= g(μp, σp)p p

– asset returns normally distributed ⇒ R=∑j wj rj normal• if U(.) is CARA ⇒ certainty equivalent = μp - ρA/2σ2

p(Use moment generating function)


(Use moment generating function)


Equilibrium leads to CAPMEquilibrium 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 deriveCAPM allows to derive– equilibrium prices/ returns.– risk-premium



The CAPM with a riskThe CAPM with a risk free bondfree bondThe CAPM with a riskThe CAPM with a risk--free bondfree bond

• The market portfolio is efficient since it is on theThe market portfolio is efficient since it is on theefficient frontier.

• All individual optimal portfolios are located on• All individual optimal portfolios are located onthe half-line originating at point (0,rf).

RRE −][• The slope of Capital Market Line (CML): .

M

fM RREσ

−][

pM

fMfp

RRERRE σ

σ−

+=][

][



The Capital Market LineThe Capital Market LineThe Capital Market LineThe Capital Market Line

CML

MrM

jrf


σ pσM


The Security Market LineThe Security Market LineE(r)

The Security Market LineThe Security Market Line

E(ri)SML

E(rM)

rfslope SML = (E(ri)-rf) /β i

ββ iβ M= 1



OverviewOverview


2.2. MeanMean--variance preferencesvariance preferences–– Portfolio TheoryPortfolio Theory–– CAPM CAPM (Intuition)(Intuition)

3.3. CAPM CAPM (modern derivation)(modern derivation)

–– ProjectionsProjections–– Pricing Kernel and Expectation KernelPricing Kernel and Expectation Kernel



ProjectionsProjections

• States s=1,…,S with πs >0 • Probability inner productProbability inner product

( f l h)• π-norm (measure of length)



⇒⇒shrinkaxesy y

x x

x and y are π-orthogonal iff [x,y]π = 0, I.e. E[xy]=0



…Projections……Projections…

• Z space of all linear combinations of vectors z1, …,zn• Given a vector y ∈ RS solve

• [smallest distance between vector y and Z space]


• [smallest distance between vector y and Z space]


…Projections…Projections

yε

yZ

E[ε zj]=0 for each j=1,…,n (from FOC)ε⊥ z


yZ is the (orthogonal) projection on Zy = yZ + ε’ , yZ ∈ Z, ε ⊥ z


Expected Value and CoExpected 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[x2]=Var[x]+E[x]2

||x||= E[x2]½



E t d V l d CE t d V l d C V iV i…Expected Value and Co…Expected Value and Co--Variance Variance

E[x] = [x, 1]=



OverviewOverview


2.2. MeanMean--variance preferencesvariance preferences–– Portfolio TheoryPortfolio Theory–– CAPM CAPM (Intuition)(Intuition)

3.3. CAPM CAPM (modern derivation)(modern derivation)

P j iP j i–– ProjectionsProjections–– Pricing Kernel and Expectation KernelPricing Kernel and Expectation Kernel



New (LeRoy & Werner) NotationNew (LeRoy & Werner) Notation

• Main changes (new versus old)– gross return: r = Rgross return: r R– SDF: μ = m– pricing kernel: k = m*pricing kernel: kq m

– Asset span: M = <X>– income/endowment: wt = et



Pricing Kernel kPricing Kernel kqq……M f f ibl ff• M space of feasible payoffs.

• If no arbitrage and π >>0 there exists gSDF μ ∈ RS, 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 kq ∈M such that for each z ∈M, q(z)=E(kq z).

• (kq = m* in our old notation.)


( q )


P i i K lP i i K l E lE l…Pricing Kernel …Pricing Kernel -- Examples…Examples…

• Example 1:– S=3,πs=1/3 for s=1,2,3,S 3,π 1/3 for s 1,2,3, – x1=(1,0,0), x2=(0,1,1), p=(1/3,2/3).– Then k=(1 1 1) is the unique pricing kernelThen k (1,1,1) is the unique pricing kernel.

• Example 2:S 3 s 1/3 f 1 2 3– S=3,πs=1/3 for s=1,2,3,

– x1=(1,0,0), x2=(0,1,0), p=(1/3,2/3).Th k (1 2 0) i th i i i k l


– Then k=(1,2,0) is the unique pricing kernel.


P i i K lP i i K l U iU i…Pricing Kernel …Pricing Kernel –– UniquenessUniqueness• If a state price density exists, there exists aIf 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 ∈ME[(μ k )z]=0E[(μ-kq)z]=0μ=(μ-kq)+kq ⇒ kq is the “projection'' of μ on M.• Complete markets ⇒ k =μ (SDF=state price density)


• Complete markets ⇒, kq=μ (SDF=state price density)


E t ti K l kE t ti K l kExpectations Kernel kExpectations Kernel kee• An expectations kernel is a vector k ∈MAn expectations kernel is a vector ke∈M

– Such that E(z)=E(ke z) for each z ∈M.• Example p

– S=3, πs=1/3, for s=1,2,3, x1=(1,0,0), x2=(0,1,0). – Then the unique ke=(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-ke)z]=0• ke is the “projection” of e on M

k if b d b li t d ( if k t l t )


• ke = e if bond can be replicated (e.g. if markets are complete)


Mean Variance FrontierMean Variance FrontierMean Variance FrontierMean 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 kq and ke.• Decompose z=zE+ε, with zE∈ E and ε ⊥ E. • Hence, E[ε]= E[ε ke]=0, q(ε)= E[ε kq]=0

Cov[ε,zE]=E[ε zE]=0, since ε ⊥ E.• var[z] = var[zE]+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


• Every z ∈ E is in mean variance frontier.


F ti R tF ti R tFrontier 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


• graphically: payoffs with price of p=1.


M RS R3M = RS = R3

Mean-Variance Payoff FrontierMean-Variance Payoff Frontier

ε

kq

Mean-Variance Return Frontierp=1-line = return-line (orthogonal to kq)


p ( g q)


M V i (P ff) F tiMean-Variance (Payoff) Frontier

(1 1 1) standard deviation

0kq

(1,1,1)

expected return

standard deviation

q


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

0kq

(1,1,1)

expected return

standard deviation

q

inefficient (return) frontier



…Frontier Returns…Frontier Returns(if agent is risk-neutral)



Mi i V i P tf liMi i V i P tf liMinimum Variance PortfolioMinimum Variance Portfolio

• Take FOC w.r.t. λ of

H MVP h f• Hence, MVP has return of



Illustration of MVPIllustration of MVPIllustration of MVPIllustration of MVP

E t d tM = R2 and S=3

Mi i d d

Expected return of MVP

(1,1,1)

Minimum standard deviation



MM V i Effi i t R tV i Effi i t R tMeanMean--Variance Efficient ReturnsVariance Efficient Returns• Definition: A return is mean-variance efficient if there

is no other return with same variance but greater expectationexpectation.

• Mean variance efficient returns are frontier returns withE[r ] ≥ E[r ]E[rλ] ≥ E[rλ0].

• If risk-free asset can be replicatedM 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



ZZ C i F ti R tC i F ti R tZeroZero--Covariance Frontier ReturnsCovariance Frontier Returns• Take two frontier portfolios with returns• Take two frontier portfolios with returns

and• CC

• The portfolios have zero co-variance ifp

• For all λ ≠ λ0 μ exists• μ=0 if risk-free bond can be replicated



Illustration of ZC Portfolio…Illustration of ZC Portfolio…Illustration of ZC Portfolio…Illustration of ZC Portfolio…M = R2 and S=3

arbitrary portfolio p(1,1,1)

y p p

Recall:



…Illustration of ZC Portfolio…Illustration of ZC Portfolio…Illustration of ZC Portfolio…Illustration of ZC PortfolioGreen lines do not necessarily cross.

arbitrary portfolio p(1,1,1)

y p p

ZC of pp



B t P i iB t P i iBeta Pricing…Beta Pricing…• Frontier Returns (are on linear subspace). Hence

• Consider any asset with payoff xjy p y j– It can be decomposed in xj = xj

E + εj

– q(xj)=q(xjE) and E[xj]=E[xj

E], since ε ⊥ E.q(xj) q(xj ) and E[xj] E[xj ], since ε ⊥ E. – Let rj

E be the return of xjE

– x– x– Using above and assuming λ ≠ λ0 and μ is

ZC-portfolio of λ


ZC portfolio of λ,


B t P i iB t P i i…Beta Pricing…Beta Pricing

• Taking expectations and deriving covariance• _

• If risk-free asset can be replicated, beta-pricing equation simplifies toequation simplifies to

• Problem: How to identify frontier returnsSlide Slide 0707--6161MeanMean--Variance Analysis and CAPMVariance Analysis and CAPM

• Problem: How to identify frontier returns


C it l A t P i i M d lC it l A t P i i M d lCapital 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 wi1 at time 1, aggregate

where the orthogonal projection of on M is. – The market payoff:– Assume q(m) ≠ 0, let rm=m / q(m), and

assume that rm is not the minimum variance return.



C it l A t P i i M d lC it l A t P i i M d l…Capital Asset Pricing Model…Capital Asset Pricing Model

• If rm0 is the frontier return that has zero covariance with rm then, for every security j,

• E[rj]=E[rm0] + βj (E[rm]-E[rm0]), with βj=cov[rj,rm] / var[rm].j j

• If a risk free asset exists, equation becomes,• E[rj]= rf + βj (E[rm]- rf) [ j] f βj ( [ m] f)

• N.B. first equation always hold if there are only two assets.


q y y


O td t d t i l f llO td t d t i l f llOutdated material followsOutdated material follows

• Traditional derivation of CAPM is less elegant• Not relevant for examsNot relevant for exams




f li b d b ("

Characterization of Frontier Portfolios Characterization of Frontier Portfolios

portfolios can be generated by ("are convexcombinations" of) g and g+h.

• Proposition 6.2. The portfolio frontier can bedescribed as convex combinations of any twof yfrontier portfolios, not just the frontier portfolios gand g+h.

• Proposition 6.3 : Any convex combination offrontier portfolios is also a frontier portfolio


frontier portfolios is also a frontier portfolio.



Characterization of Frontier Portfolios Characterization of Frontier Portfolios p f f

portfolios can be generated by ("are convexcombinations" 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)( )qr~E

Consider portfolio weights (proportions) then, as asserted,( ) ( )qhgqg r~E and r~E1 =π−=π +

( )[ ] ( )( ) ( ) wr~hEghgr~Egr~E1 +++( )[ ] ( )( ) ( ) .wrhEghgrEgrE1 qqqq =+=++−



• Proposition 6.2. The portfolio frontier can be described asconvex combinations of any two frontier portfolios, not justthe frontier portfolios g and g+h.

• Proof: To see this, let p1 and p2 be any two distinct frontier portfolios; since the frontier portfolios are E[rp1] ≠ E[rp2] 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 texpected return equal to E[rq]. Since E[rp1] ≠ E[rp2] , there must exist a unique number such that

E[r ] = α E[r ] + (1 α) E[r ] (6 16)E[rq] = α E[rp1] + (1 - α) E[rp2] (6.16)Now consider a portfolio of p1 and p2 with weights α, 1- α,respectively, as determined by (6.16). We must show that

wq = α wp1 + (1 - α) wp2.



Proof of Proposition 6.2 (continued)

( ) ( )[ ] ( ) ( )[ ]2121 pppp r~hEg1r~hEgw1w +α−++α=α−+α

( ) ( ) ( )[ ]( ) ( ) ( )[ ]( ) onconstructiby ,r~hEg

r~E1r~Ehg

q

pp 21

+=

α−+α+=

portfolio.frontier a is q since ,wq=



• Proposition 6.3 : Any convex combination of frontierf li i l f i f liportfolios is also a frontier portfolio.

• Proof : Let , define N frontier portfolios( represents the vector defining the composition of the ith

( )N1 w...w

iw( represents the vector defining the composition of the iportfolios) and let , i = 1, ..., N be real numbers suchthat . Lastly, let denote the expected returnof portfolio with weights

iwiα

1N1i i =∑ α= ( )irE

of portfolio with weights .The weights corresponding to a linear combination of theabove N portfolios are :

iw

( )( )∑ +α=∑α==

N

1iii

N

1iii r~hEgw

( )∑α+∑α=NN

r~Ehg ( )∑α+∑α=== 1i

ii1i

i rEhg

( )⎥⎦⎤

⎢⎣⎡∑α+=

=

N

1iii r~Ehg

N( ) ( )

N


Thus is a frontier portfolio with .∑α=

N

1iiiw ( ) ( )i

N

1ii r~ErE ∑α=

=



• For any portfolio on the frontier,with g and h as defined earlier.

( ) ( )[ ] ( )[ ]pT

pp rhEgVrhEgrE ~~]~[2 ++=σ

Multiplying all this out yields:




• (i) the expected return of the minimum variance portfoliois A/C;

• (ii) the variance of the minimum variance portfolio is givenby 1/C;y

• (iii) equation (6.17) is the equation of a parabola with(1/C A/C) i h d / i dvertex (1/C, A/C) in the expected return/variance space and

of a hyperbola in the expected return/standard deviationspace. See Figures 6.3 and 6.4.


p g



Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space



Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space



Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed


Characterization of Efficient PortfoliosCharacterization of Efficient PortfoliosCharacterization 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 dportfolios which are not mean-variance dominated.

• Lemma: Efficient portfolios are those frontier portfolios for which the expected return exceeds A/Cportfolios for which the expected return exceeds A/C, the expected return of the minimum variance portfolioportfolio.



• 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 wasefficient; then , for every portfolio i.( ) C/Ar~E i ≥

AANN

However ; thus, the convexcombination is efficient as well. So the set of efficient

f li h i d b h i f li i h i

( )CA

CAr~E

N

1ii

N

1iii ∑ =α≥∑α

==

portfolios, as characterized by their portfolio weights, is aconvex set.



Zero Covariance PortfolioZero Covariance Portfolio• Zero-Cov Portfolio is useful for Zero-Beta CAPM

• Proposition 6.5: For any frontier portfolio p, except theProposition 6.5: For any frontier portfolio p, except theminimum variance portfolio, there exists a unique frontierportfolio with which p has zero covariance.We will call this portfolio the "zero covariance portfolioWe will call this portfolio the zero covariance portfolio relative to p", and denote its vector of portfolio weights by

.( )pZC

• Proof: by construction.



collect all expected returns terms, add and subtract A2C/DC2collect all expected returns terms, add and subtract A C/DCand note that the remaining term (1/C)[(BC/D)-(A2/D)]=1/C, since D=BC-A2



For zero co-variance portfolio ZC(p)For zero co variance portfolio ZC(p)

For graphical illustration, let’s draw this line:



Graphical Representation:

line through p (Var[r ] E[r ]) ANDp (Var[rp], E[rp]) ANDMVP (1/C, A/C) (use )

for σ2(r) = 0 you get E[rZC(p)]

( ) ( )C1

CAr~E

DCr~

2

pp2 +⎟

⎠⎞

⎜⎝⎛ −=σ

for σ (r) 0 you get E[rZC(p)]



E ( r )

A/C

p

A/CMVP

ZC(p)E[r ( ZC(p)] ZC (p) on frontier

(1/C) Var ( r )


Figure 6-6 The Set of Frontier Portfolios: Location of the Zero-Covariance Portfolio


ZeroZero Beta CAPMBeta CAPMZeroZero--Beta CAPM Beta CAPM (no risk(no risk--free asset)free asset)

(i) agents maximize expected utility with increasing andstrictly concave utility of money functions and assetreturns are multivariate normally distributed, or

(ii) each agent chooses a portfolio with the objective ofmaximizing a derived utility function of the formmaximizing a derived utility function of the form

, U concave.(iii) common time horizon,

( ) 0 U0, U,e,U 212 <>σ

( ) ,(iv) homogeneous beliefs about e and σ2



– 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[rp,rq] = λ E[rq]+ γ (q need not be on the frontier) (6.22)– Cov[rp,rZC(p)] = λ E[rZC(p)] + γ=0

– Cov[rp,rq] = λ {E[rq]-E[rZC(p)]}p q q (p)

– Var[rp] = λ {E[rp]-E[rZC(p)]} .

Divide third by fourth equation:

( ) ( )( ) ( ) ( )( )[ ]MZCMMqMZCq r~Er~Er~Er~E −β+= (6.28)

( ) ( ) ( ) ( )[ ]r~Er~Er~Er~E β+= (6 29)


( ) ( )( ) ( ) ( )( )[ ]MZCMMjMZCj rErErErE −β+= (6.29)


ZZ B t CAPMB t CAPMZeroZero--Beta CAPMBeta CAPM

• mean variance framework (quadratic utility or normal returns)

• In equilibrium market portfolio which is aIn equilibrium, market portfolio, which is a convex combination of individual portfolios

E[rq] = E[rZC(M)]+ βMq[E[rM]-E[rZC(M)]]

E[rj] = E[rZC(M)]+ βMj[E[rM]-E[rZC(M)]]


lecture lecture 07: 07: meanmean--variance analysis …markus/teaching/fin501/07lecture.pdf ·...

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