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

05:03 Lecture 07 Mean-Variance Analysis and CAPM

(Derivation with Projections)


Projections

• States s=1,…,S with p s

• Probability inner product

>0

• p

-norm (measure of length)



Fin 501: Asset Pricing y x

) shrink axes y x x and y are p

-orthogonal iff [x,y] p

= 0, I.e. E[xy]=0




…Projections…

•

Z space of all linear combinations of vectors z

1

, …,z n

• Given a vector y 2 R S solve

• [smallest distance between vector y and Z space]



…Projections y e

Fin 501: Asset Pricing y Z

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

?

z y Z is the (orthogonal) projection on Z y = y Z +

05:03 Lecture 07 e ’ , y Z 2 Z , e

?

z

Mean-Variance Analysis and CAPM



Expected Value and Co-Variance… squeeze axis by

(1,1) 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




…Expected Value and Co-Variance

E[x] = [x, 1 ]=




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




New Notation (LeRoy & Werner)

• Main changes (new versus old)

– gross return:

– SDF: r = R m

= m

– pricing kernel: k q

= m *

– Asset span:

– income/endowment:

M = <X> w t

= e t




Pricing Kernel k q

…

• M space of feasible payoffs.

• If no arbitrage and p

>>0 there exists

SDF m

2 R S , m

>>0, such that q(z)=E( m z).

• m

2 M – SDF need not be in asset span.

• A pricing kernel is a k q each z 2 M , q(z)=E(k q

2 M z).

such that for

• (k q

= m * in our old notation.)




…Pricing Kernel - Examples…

• Example 1:

– S=3, p 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.

• Example 2:

– S=3, p s =1/3 for s=1,2,3,

– x

1

=(1,0,0), x

2

=(0,1,0), p=(1/3,2/3).

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




…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

-k q

)z]=0 m

=( m

-k q

)+k q

) k q is the ``projection'' of m on M

• Complete markets ) , k q

= m

(SDF=state price density)

.




Expectations Kernel k e

• An expectations kernel is a vector k e

– Such that E(z)=E(k e z) for each z 2 M .

2 M

• Example

– S=3, p 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 p

>>0, there exists a unique expectations kernel.

• Let e=(1,…, 1) then for any z 2 M

•

E[(e-k e

)z]=0

• k

• k e e is the “projection” of e on M

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




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 k q

• Decompose z=z E + e

, with z E 2 E and e

? E . and k e

.

• Hence, E[ e

]= E[ e k e

]=0, q( e

)= E[ e k q

]=0

Cov[ e

,z E ]=E[ e z E ]=0, since e

? E .

• var[z] = var[z E ]+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.




Frontier Returns…

• Frontier returns are the returns of frontier payoffs with non-zero prices.

• x

• graphically: payoffs with price of p=1.



M = R S = R 3


Mean-Variance Payoff Frontier e

05:03 Lecture 07 k q

Mean-Variance Return Frontier p=1-line = return-line (orthogonal to k q

)

Mean-Variance Analysis and CAPM


Mean-Variance (Payoff) Frontier

(1,1,1)

0 k q



Fin 501: Asset Pricing standard deviation expected return

Mean-Variance (Payoff) Frontier

Fin 501: Asset Pricing efficient (return) frontier

(1,1,1)

0 k q standard deviation expected return inefficient (return) frontier



…Frontier Returns





Minimum Variance Portfolio

• Take FOC w.r.t. l of

• Hence, MVP has return of




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[r l

]

¸

E[r l

0

].

• If risk-free asset can be replicated

– Mean variance efficient returns correspond to l ·

0.

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




Zero-Covariance Frontier Returns

• Take two frontier portfolios with returns and

• C

• The portfolios have zero co-variance if

• For all l  l

0 m exists

• m

=0 if risk-free bond can be replicated



Expected return of MVP

Illustration of MVP


M = R 2 and S=3

Minimum standard deviation

(1,1,1)




Illustration of ZC Portfolio…

M = R 2 and S=3

(1,1,1) arbitrary portfolio p

Recall:




…Illustration of ZC Portfolio arbitrary portfolio p

(1,1,1)

ZC of p




Beta Pricing…

• Frontier Returns (are on linear subspace). Hence

• Consider any asset with payoff x

– It can be decomposed in x j

= x j

E j

+ e j

– q(x j

)=q(x j

E ) and E[x j

]=E[x j

E ], since e

? E .

– Let r j

E be the return of x j

E

– Rdddf

– Using above and assuming l  lambda

ZC-portfolio of l

,

0 and m is




…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




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.




…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

] + b b j j

(E[r

=cov[r j

,r m

]-E[r m0

]), with m

] / var[r m

].

• If a risk free asset exists, equation becomes,

E[r j

]= r f

+ b j

(E[r m

]- r f

)

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



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