fin 501: asset pricing 11:41 lecture 07mean-variance analysis and capm (derivation with projections)...
TRANSCRIPT
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
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
ProjectionsProjections
• States s=1,…,S with s >0
• Probability inner product
• -norm (measure of length)
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
)shrinkaxes
x x
y y
x and y are -orthogonal iff [x,y] = 0, I.e. E[xy]=0
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
……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]
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
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
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
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
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
……Expected Value and Co-Variance Expected Value and Co-Variance
E[x] = [x, 1]=
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
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
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
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
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.)
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
……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.
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
……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)
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
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)
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
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.
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
Frontier Returns…Frontier Returns…• Frontier returns are the returns of frontier payoffs with non-
zero prices.
• x
• graphically: payoffs with price of p=1.
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
kq
Mean-Variance Return Frontierp=1-line = return-line (orthogonal to kq)
M = RS = R3
Mean-Variance Payoff Frontier
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
0kq
(1,1,1)
expected return
standard deviation
Mean-Variance (Payoff) Frontier
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
0kq
(1,1,1)
inefficient (return) frontier
efficient (return) frontier
expected return
standard deviation
Mean-Variance (Payoff) Frontier
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
……Frontier ReturnsFrontier Returns
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
Minimum Variance PortfolioMinimum Variance Portfolio
• Take FOC w.r.t. of
• Hence, MVP has return of
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
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
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
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
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
(1,1,1)
Illustration of MVPIllustration of MVP
Minimum standard deviation
Expected return of MVP
M = R2 and S=3
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
(1,1,1)
Illustration of ZC Portfolio…Illustration of ZC Portfolio…
arbitrary portfolio p
Recall:
M = R2 and S=3
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
(1,1,1)
……Illustration of ZC PortfolioIllustration of ZC Portfolio
arbitrary portfolio p
ZC of p
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
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 ,
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
……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
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
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.
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
……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.