#### Transcript 07Lecture_a_CAPM(projections)

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### 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 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### Projections

• States s=1,…,S with p s • Probability inner product >0 • p -norm (measure of length) 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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y x ) shrink axes y x x and y are p -orthogonal iff [x,y] p = 0, I.e. E[xy]=0 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### …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] 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

### …Projections

y e

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y Z E[ e e ?

z z j ]=0 for each j=1,…,n (from FOC) y Z is the (orthogonal) projection on Z y = y Z + e ’ , y Z 08:49 Lecture 07 2 Z , e ?

z Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### …Expected Value and Co-Variance

E[x] = [x,

**1**

]= 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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 =

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### 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.) 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### …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. 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### …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''**

• Complete markets ) , k q = m of m on M (SDF=state price density) .

08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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, 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) 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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•

### 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.

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### Frontier Returns…

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

• x • graphically: payoffs with price of p=1.

08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

M = R S = R 3

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Mean-Variance Payoff Frontier e 08:49 Lecture 07 k q Mean-Variance Return Frontier p=1-line = return-line (orthogonal to k q ) Mean-Variance Analysis and CAPM (Derivation with Projections)

Mean-Variance (Payoff) Frontier (1,1,1) 0 k q 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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standard deviation expected return

Mean-Variance (Payoff) Frontier

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efficient (return) frontier (1,1,1) 0 k q standard deviation expected return inefficient (return) frontier 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

### …Frontier Returns

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08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### Minimum Variance Portfolio

• Take FOC w.r.t. l of • Hence, MVP has return of 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

Expected return of MVP

### Illustration of MVP

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M = R 2 and S=3 Minimum standard deviation (1,1,1) 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### Illustration of ZC Portfolio…

M = R 2 and S=3 (1,1,1) arbitrary portfolio p Recall: 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### …Illustration of ZC Portfolio

arbitrary portfolio p (1,1,1) ZC of p 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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 + 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 l ZC-portfolio of l , ? E . lambda 0 and m is 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### …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 08:49 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)

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### 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) assume that r m 0, let r m =m / q(m), and is not the minimum variance return.

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### …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 j (E[r m ]-E[r m0 ]), with b j =cov[r j ,r 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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