Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections)...

05:2005:20 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)

Fin 501: Asset PricingFin 501: Asset Pricing

OverviewOverview

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

• Mean-variance preferences– Portfolio Theory– CAPM (traditional derivation)

• With risk-free bond• Zero-beta CAPM

• CAPM (modern derivation)

– Projections– Pricing Kernel and Expectation Kernel



ProjectionsProjections

• States s=1,…,S with s >0

• Probability inner product

• -norm (measure of length)



)shrinkaxes

x x

y y

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



……Projections…Projections…

• Z space of all linear combinations of vectors z1, …,zn

• Given a vector y 2 RS solve

• [smallest distance between vector y and Z space]



y

yZ

E[ zj]=0 for each j=1,…,n (from FOC)? z yZ is the (orthogonal) projection on Zy = yZ + ’ , yZ 2 Z, ? z

……ProjectionsProjections



Expected Value and Co-Variance…Expected Value and Co-Variance…squeeze axis by

x

(1,1)

[x,y]=E[xy]=Cov[x,y] + E[x]E[y][x,x]=E[x2]=Var[x]+E[x]2

||x||= E[x2]½



……Expected Value and Co-Variance Expected Value and Co-Variance

E[x] = [x, 1]=



OverviewOverview

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

• Mean-variance preferences– Portfolio Theory– CAPM (traditional derivation)

• With risk-free bond• Zero-beta CAPM

• CAPM (modern derivation)

– Projections– Pricing Kernel and Expectation Kernel



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

• Main changes (new versus old)– gross return: r = R– SDF: = m

– pricing kernel: kq = m*

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



Pricing Kernel kPricing Kernel kqq……

• M space of feasible payoffs.

• If no arbitrage and >>0 there exists

SDF 2 RS, >>0, such that q(z)=E( z).

• 2 M – SDF need not be in asset span.

• A pricing kernel is a kq 2 M such that for

each z 2 M, q(z)=E(kq z).

• (kq = m* in our old notation.)



……Pricing Kernel - Examples…Pricing Kernel - Examples…

• Example 1:– S=3,s=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 kernel.

• Example 2:– S=3,s=1/3 for s=1,2,3,

– x1=(1,0,0), x2=(0,1,0), p=(1/3,2/3).

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



……Pricing Kernel – UniquenessPricing Kernel – Uniqueness• If a state price density exists, there exists a unique pricing kernel.– If dim(M) = m (markets are complete),

there are exactly m equations and m unknowns

– If dim(M) · m, (markets may be incomplete)

For any state price density (=SDF) and any z 2 M

E[(-kq)z]=0

=(-kq)+kq ) kq is the ``projection'' of on M.• Complete markets ), kq= (SDF=state price density)



Expectations Kernel kExpectations Kernel kee

• An expectations kernel is a vector ke2 M – Such that E(z)=E(ke z) for each z 2 M.

• Example – S=3, s=1/3, for s=1,2,3, x1=(1,0,0), x2=(0,1,0). – Then the unique $ke=(1,1,0).$

• If >>0, there exists a unique expectations kernel.• Let e=(1,…, 1) then for any z2 M• E[(e-ke)z]=0• ke is the “projection” of e on M• ke = e if bond can be replicated (e.g. if markets are complete)



Mean Variance FrontierMean Variance Frontier• Definition 1: z 2 M is in the mean variance frontier if

there exists no z’ 2 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+, with zE2 E and ? E.

• Hence, E[]= E[ ke]=0, q()= E[ kq]=0

Cov[,zE]=E[ zE]=0, since ? E.• var[z] = var[zE]+var[] (price of is zero, but positive variance)

• If z in mean variance frontier ) z 2 E.• Every z 2 E is in mean variance frontier.



Frontier Returns…Frontier Returns…• Frontier returns are the returns of frontier payoffs with non-

zero prices.

• x

• graphically: payoffs with price of p=1.



kq

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

M = RS = R3

Mean-Variance Payoff Frontier



0kq

(1,1,1)

expected return

standard deviation

Mean-Variance (Payoff) Frontier



0kq

(1,1,1)

inefficient (return) frontier

efficient (return) frontier

expected return

standard deviation

Mean-Variance (Payoff) Frontier



……Frontier ReturnsFrontier Returns



Minimum Variance PortfolioMinimum Variance Portfolio

• Take FOC w.r.t. of

• Hence, MVP has return of



Mean-Variance Efficient ReturnsMean-Variance Efficient Returns

• Definition: A return is mean-variance efficient if there is no other return with same variance but greater expectation.

• Mean variance efficient returns are frontier returns with E[r] ¸ E[r0].

• If risk-free asset can be replicated– Mean variance efficient returns correspond to · 0.

– Pricing kernel (portfolio) is not mean-variance efficient, since



Zero-Covariance Frontier ReturnsZero-Covariance Frontier Returns• Take two frontier portfolios with returns

and• C

• The portfolios have zero co-variance if

• For all 0 exists• =0 if risk-free bond can be replicated



(1,1,1)

Illustration of MVPIllustration of MVP

Minimum standard deviation

Expected return of MVP

M = R2 and S=3



(1,1,1)

Illustration of ZC Portfolio…Illustration of ZC Portfolio…

arbitrary portfolio p

Recall:

M = R2 and S=3



(1,1,1)

……Illustration of ZC PortfolioIllustration of ZC Portfolio

arbitrary portfolio p

ZC of p



Beta Pricing…Beta Pricing…• Frontier Returns (are on linear subspace). Hence

• Consider any asset with payoff xj

– It can be decomposed in xj = xjE + j

– q(xj)=q(xjE) and E[xj]=E[xj

E], since ? E.

– Let rjE be the return of xj

E

– Rdddf

– Using above and assuming lambda0 and is ZC-portfolio of ,



……Beta PricingBeta Pricing

• Taking expectations and deriving covariance• _

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

• Problem: How to identify frontier returns



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 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.



……Capital Asset Pricing ModelCapital Asset Pricing Model

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

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

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

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

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections)...

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