Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Overview Simple CAPM with quadratic utility functions (derived from state-price beta model)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

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Projections States s=1,…,S with  s >0 Probability inner product  -norm (measure of length)

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) ) shrink axes xx y y x and y are  -orthogonal iff [x,y]  = 0, I.e. E[xy]=0

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

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) y yZyZ  E[  z j ]=0 for each j=1,…,n (from FOC)  ? z y Z is the (orthogonal) projection on Z y = y Z +  ’, y Z 2 Z,  ? z …Projections

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) 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[x 2 ]=Var[x]+E[x] 2 ||x||= E[x 2 ] ½

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) …Expected Value and Co-Variance E[x] = [x, 1]=

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Overview Simple CAPM with quadratic utility functions (derived from state-price beta model)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

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) New Notation (LeRoy & Werner) Main changes (new versus old) – gross return: r = R – SDF:  = m – pricing kernel:k q = m * – Asset span: M = – income/endowment:w t = e t

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Pricing Kernel k q … M space of feasible payoffs. If no arbitrage and  >>0 there exists SDF  2 R S,  >>0, such that q(z)=E(  z).  2 M – SDF need not be in asset span. A pricing kernel is a k q 2 M such that for each z 2 M, q(z)=E(k q z). (k q = m * in our old notation.)

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) …Pricing Kernel - Examples… Example 1: –S=3,  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,  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.

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) …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)  and any z 2 M E[(  -k q )z]=0  =(  -k q )+k q ) k q is the ``projection'' of  on M. Complete markets ), k q =  (SDF=state price density)

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Expectations Kernel k e An expectations kernel is a vector k e 2 M –Such that E(z)=E(k e z) for each z 2 M. Example –S=3,  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  >>0, there exists a unique expectations kernel. Let e=(1,…, 1) then for any z 2 M E[(e-k e )z]=0 k e is the “projection” of e on M k e = e if bond can be replicated (e.g. if markets are complete)

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) 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 and k e. Decompose z=z E + , with z E 2 E and  ? E. Hence, E[  ]= E[  k e ]=0, q(  )= E[  k q ]=0 Cov[ ,z E ]=E[  z E ]=0, since  ? E. var[z] = var[z E ]+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.

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Frontier Returns… Frontier returns are the returns of frontier payoffs with non-zero prices. x graphically: payoffs with price of p=1.

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) kqkq Mean-Variance Return Frontier p=1-line = return-line (orthogonal to k q ) M = R S = R 3 Mean-Variance Payoff Frontier 

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) 0 kqkq (1,1,1) expected return standard deviation Mean-Variance (Payoff) Frontier

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) 0 kqkq (1,1,1) inefficient (return) frontier efficient (return) frontier expected return standard deviation Mean-Variance (Payoff) Frontier

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

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Minimum Variance Portfolio Take FOC w.r.t. of Hence, MVP has return of

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) 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 ] ¸ E[r 0 ]. If risk-free asset can be replicated –Mean variance efficient returns correspond to · 0. –Pricing kernel (portfolio) is not mean-variance efficient, since

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

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) (1,1,1) Illustration of MVP Minimum standard deviation Expected return of MVP M = R 2 and S=3

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) (1,1,1) Illustration of ZC Portfolio… arbitrary portfolio p Recall: M = R 2 and S=3

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) (1,1,1) …Illustration of ZC Portfolio arbitrary portfolio p ZC of p

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) Beta Pricing… Frontier Returns (are on linear subspace). Hence Consider any asset with payoff x j –It can be decomposed in x j = x j E +  j –q(x j )=q(x j E ) and E[x j ]=E[x j E ], since  ? E. –Let r j E be the return of x j E –Rdddf –Using above and assuming  lambda 0 and  is ZC-portfolio of,

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

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

Fin 501: Asset Pricing 11:41 Lecture 07Mean-Variance Analysis and CAPM (Derivation with Projections) …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 ] +  j (E[r m ]-E[r m0 ]), with  j =cov[r j,r m ] / var[r m ]. If a risk free asset exists, equation becomes, E[r j ]= r f +  j (E[r m ]- r f ) N.B. first equation always hold if there are only two assets.