### 07Lecture_a_CAPM(projections)

```Fin 501: Asset Pricing
Overview
• Simple CAPM with quadratic utility functions
(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
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Projections
• States s=1,…,S with ps >0
• Probability inner product
• p-norm
16:27 Lecture 07
(measure of length)
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
)
shrink
axes
y
x
y
x
x and y are p-orthogonal iff [x,y]p = 0, I.e. E[xy]=0
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…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]
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…Projections
y
e
yZ
E[e zj]=0 for each j=1,…,n (from FOC)
e? z
yZ is the (orthogonal) projection on Z
Z + e’ , yZ 2 Z, e ? z
y
=
y
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Expected Value and Co-Variance…
squeeze axis by
(1,1)
x
16:27 Lecture 07
[x,y]=E[xy]=Cov[x,y] + E[x]E[y]
[x,x]=E[x2]=Var[x]+E[x]2
||x||= E[x2]½
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…Expected Value and Co-Variance
E[x] = [x, 1]=
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Overview
• Simple CAPM with quadratic utility functions
(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
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
New Notation (LeRoy & Werner)
• Main changes (new versus old)
– gross return:
– SDF:
– pricing kernel:
r=R
m=m
kq = m*
– Asset span:
– income/endowment:
M = <X>
wt = et
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Pricing Kernel kq…
• M space of feasible payoffs.
• If no arbitrage and p >>0 there exists
SDF m 2 RS, m >>0, such that q(z)=E(m z).
• m 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.)
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…Pricing Kernel - Examples…
• Example 1:
– S=3,ps=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,ps=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.
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…Pricing 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) m and any z 2 M
E[(m-kq)z]=0
m=(m-kq)+kq ) kq is the ``projection'' of m on M.
• Complete markets ), kq=m (SDF=state price density)
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Expectations Kernel ke
• An expectations kernel is a vector ke2 M
– Such that E(z)=E(ke z) for each z 2 M.
• Example
– S=3, ps=1/3, for s=1,2,3, x1=(1,0,0), x2=(0,1,0).
– Then the unique \$ke=(1,1,0).\$
•
•
•
•
•
If p >>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)
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Mean 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+e, with zE2 E and e ? E.
• Hence, E[e]= E[e ke]=0, q(e)= E[e kq]=0
Cov[e,zE]=E[e zE]=0, since e ? E.
• var[z] = var[zE]+var[e] (price of e is zero, but positive variance)
• If z in mean variance frontier ) z 2 E.
• Every z 2 E is in mean variance frontier.
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Frontier Returns…
• Frontier returns are the returns of frontier payoffs with
non-zero prices.
• x
• graphically: payoffs with price of p=1.
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
M = RS = R3
Mean-Variance Payoff Frontier
e
kq
Mean-Variance Return Frontier
p=1-line = return-line (orthogonal to kq)
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Mean-Variance (Payoff) Frontier
(1,1,1)
0
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kq
Mean-Variance Analysis and CAPM
(Derivation with Projections)
standard deviation
expected return
Fin 501: Asset Pricing
Mean-Variance (Payoff) Frontier
efficient (return) frontier
(1,1,1)
0
standard deviation
expected return
kq
inefficient (return) frontier
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
…Frontier Returns
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Fin 501: Asset Pricing
Minimum Variance Portfolio
• Take FOC w.r.t. l of
• Hence, MVP has return of
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Mean-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[rl] ¸ E[rl0].
• If risk-free asset can be replicated
– Mean variance efficient returns correspond to l · 0.
– Pricing kernel (portfolio) is not mean-variance efficient, since
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Zero-Covariance Frontier Returns
• Take two frontier portfolios with returns
and
• C
• The portfolios have zero co-variance if
• For all l  l0 m exists
• m=0 if risk-free bond can be replicated
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Illustration of MVP
M = R2 and S=3
Expected return
of MVP
Minimum standard
deviation
(1,1,1)
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Illustration of ZC Portfolio…
M = R2 and S=3
arbitrary portfolio p
(1,1,1)
16:27 Lecture 07
Recall:
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…Illustration of ZC Portfolio
arbitrary portfolio p
(1,1,1)
ZC of p
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
Beta Pricing…
• Frontier Returns (are on linear subspace). Hence
• Consider any asset with payoff xj
–
–
–
–
–
It can be decomposed in xj = xjE + ej
q(xj)=q(xjE) and E[xj]=E[xjE], since e ? E.
Let rjE be the return of xjE
Rdddf
Using above and assuming l  lambda0 and m is
ZC-portfolio of l,
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…Beta Pricing
• Taking expectations and deriving covariance
• _
• If risk-free asset can be replicated, beta-pricing
equation simplifies to
• Problem: How to identify frontier returns
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
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.
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
Fin 501: Asset Pricing
…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] + bj (E[rm]-E[rm0]), with
bj=cov[rj,rm] / var[rm].
• If a risk free asset exists, equation becomes,
•
E[rj]= rf + bj (E[rm]- rf)
• N.B. first equation always hold if there are only two assets.
16:27 Lecture 07
Mean-Variance Analysis and CAPM
(Derivation with Projections)
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