Lecture 07: Mean-Variance Analysis & Variance Analysis & Capital Asset

Lecture 07: MeanMean--VarianceVariance Analysis & Capital Asset Pricing Model (CAPM)

Prof. Markus K. Brunnermeier

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--11 Fin 501: Asset Pricing OiOverview

1. Simple CAPM with quadratic utility functions (derived from statestate--priceprice beta model) 2. MeanMean--variancevariance preferences – Portfolio Theory – CAPM (intuition) 3. CAPM – Projections – Pricing Kernel and Expectation Kernel

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--22 Fin 501: Asset Pricing RllSttRecall Statee--prpriBtice Beta mod dlel

Recall: E[Rh] - Rf = βh E[R*- Rf] where βh := Cov[R*,Rh] / Var[R*]

very general – but what is R* in reality?

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--33 Fin 501: Asset Pricing Simple CAPM with Quadratic Expected Utility 1. All agents are identical

• Expected utility U(x 0, x1)=) = ∑s πs u(x0, xs) ⇒ m= ∂1u/E[u / E[∂0u] 2 • Quadratic u(x0,x1)=v0(x0) - (x1- α) ⇒ ∂1u = [-2(x111,1- α),… , -2(xS1S,1- α)] •E[Rh] – Rf = - Cov[m,Rh] / E[m] f h = -R Cov[∂1u, R ] / E[∂0u] f h = -R Cov[-2(x1- α), R ]/E[] / E[∂0u] f h = R 2Cov[x1,R ] / E[∂0u] • Also holds for market portfolio m f f m •E[R] – R = R 2Cov[x1,R ]/E[∂0u]

⇒

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--44 Fin 501: Asset Pricing Simple CAPM with Quadratic Expected Utility

2. Homogenous agents + Exchange economy m ⇒ x1 = agg. enddiflldihdowment and is perfectly correlated with R

E[E[RRh]=]=RRf + βh {E[{E[RRm]]--RRf} Market Security Line

* f M f N.B.: R =R (a+b1R )/(a+b1R ) in this case (where b1<0)! MeanMean--VarianceVariance Analysis and CAPM Slide 0707--55 Fin 501: Asset Pricing OiOverview

1. Simple CAPM with quadratic utility functions (derived from state-state-priceprice beta model) 2. MeanMean--variancevariance analysis – Portfolio Theory (Port fo lio f ronti er, effi ci ent f ronti er, …) – CAPM (Intuition) 3. CAPM – Projections – PPiiricing K ernel and dE Expectati on K ernel

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--66 Fin 501: Asset Pricing Deeo:efinition: Meanan--VVceoceariance Dominance & Efficient Frontier

• Asset (portfolio) A mean-variance dominates asset (portfolio) B

if μA ≥ μB and σA < σΒ or if μA>μBwhile σA ≤σB. • Effici ent f ronti er: lifllloci of all non-ditddominated portfolios in the mean-standard deviation space. BBy de finiti on, no (“ rati onal”) mean-variance investor would choose to hold a portfolio not locat ed on th e effi ci ent ftifrontier.

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--77 Fin 501: Asset Pricing EdPfliR&ViExpected Portfolio Returns & Variance

• Expected returns (linear) h j μ p := E [rp ] = w j μ j , where each w j = h j j •Variance 2 P 2 σ 1 σ 12 w 1 σ p := V ar [rp ] = w 0 V w = (w 1 w 2 ) 2 σ 21 σ 2 w 2 µ ¶ µ ¶ 2 2 w 1 =(w 1 σ + w 2 σ 21 w 1 σ 12 + w 2 σ ) 1 2 w 2 2 2 2 2 µ ¶ = w 1 σ 1 + w 2 σ 2 +2w 1 w 2 σ 12 ≥ 0 since12 σ≤−σ 1 σ 2 . recall that correlation coefficient ∈ [-1,1]

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--88 Fin 501: Asset Pricing Illust rati on of 2 A sse t C ase

• For certain weights: w1 and (1-w1) μp = w1 E[r1]](+ (1-w1))[ E[r2] 2 2 2 2 2 σ p = w1 σ1 + (1-w1) σ2 + 2 w1(1-w1)σ1 σ2 ρ1,2 2 (Sppyecify σ p and one gggets weights and μp’s)

• Special cases [w1 to obtain certain σR]

– ρ1,2 = 1 ⇒ w1 = (+/-σp – σ2)/() / (σ1 – σ2)

– ρ1,2 = -1 ⇒ w1 = (+/- σp + σ2) / (σ1 + σ2)

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--99 Fin 501: Asset Pricing

σ p σ 2 For ρ = 1: σ p = |w 1 σ 1 +(1− w 1 )σ 2 | Hence, ± − 1,2 w 1 = σ 1 σ 2 μ p = w 1 μ 1 +(1 − w 1 )μ 2 −

E[r2]

μp

E[r1]

σ1 σp σ2 Lower part with … is irrelevant

The Efficient Frontier: Two Perfectly Correlated Risky Assets

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1010 Fin 501: Asset Pricing

σ p +σ 2 For ρ = -1:σ p = |w 1 σ 1 − (1 − w 1 )σ 2 | Hence, ± 1,2 w 1 = σ 1 + σ 2 μ p = w 1 μ 1 +(1 − w 1 )μ 2

E[E[r2] μ 2 μ 1 slope: σ 1 −+σ 2 σ p σ2 σ1 μ1 + μ2 σ1+σ2 σ1+σ2 μ 2 μ 1 slope: − σ 1 −+ σ 2 σ p E[r1]

σ1 σ2

Efficient Frontier: Two Perfectly Negative Correlated Risky Assets

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1111 Fin 501: Asset Pricing

For -1 < ρ1,2 < 1:

E[r2]

E[r1]

σ1 σ2

Efficient Frontier: Two Imperfectly Correlated Risky Assets

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1212 Fin 501: Asset Pricing

For σ1 =0= 0

E[r2]

μp

E[r1]

σ1 σp σ2

The Efficient Frontier: One Risky and One Risk Free Asset

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1313 Fin 501: Asset Pricing Effici ent f ronti er wi th n ri sk y assets

• A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same expected portfolio E[r] return. 1 min wTVw w 2

T N σ ()λ s.t. w e = E ⎛ ~ ⎞ ⎜ ∑ wiE( ri ) = E⎟ ⎝ i=1 ⎠ N T ⎛ ⎞ ()γ w 1 = 1 ⎜ ∑ wi = 1⎟ ⎝ i=1 ⎠

Slide 0707--1414 Fin 501: Asset Pricing

∂ = Vw λe γ1=0 ∂wL − − ∂ = E wT e = 0 ∂λL − ∂ =1 wT 1=0 ∂Lγ − The first FOC can be written as: Vwp = λe + γ1or 1 1 wp = λV − e + γV − 1 T T 1 T 1 e wp = λ(e V − e) + γ(e V − 1)

Slide 0707--1515 Fin 501: Asset Pricing

T T NiNoting th at e wp = w pe, usihfifhing the first foc, the second df foc can be written as T T 1 T 1 E[r˜p] = e wp = λ (e V − e) +γ (e V − 1) :=B =:A pre-multiplying first foc with | 1 { (instead z } of e |T) yields {z } T T T 1 T 1 1 wp = wp 1=λ(1 V − e)+γ(1 V − 1) = 1 T 1 T 1 1=λ (1 V − e) +γ (1 V − 1) =:A =:C Solving| both{z equations} for| λ and{z γ } CE A B AE λ = D− and γ = −D where D = BC A2. − Slide 0707--1616 Fin 501: Asset Pricing

-1 -1 Hence, wp = λ V e + γ V 1 becomes CE − A B − AE w = V−1e + V−11 p D D (vector) (vector) λ (scalar) γ (scalar)

1 1 = [B(V−11)− A(V−1e)]+ [C(V−1e)− A(V−11)]E D D

• Result: Portfolio weights are linear in expected w = g + h E portfolio return p

If E = 0, wp = g Hence, g and g+h are portfolios Slide 0707--1717 If E = 1, wp = g + h on the frontier. Fin 501: Asset Pricing Charac ter iza tion of F ronti er P ortf oli os • Proposition 6.1: The entire set of frontier portflifolios can begenerated by ("are convex combinations" of) g and g+h.

• Proposition 6.2. The portfolio frontier can be described as convex combinations of any two frontier portfolios, not just the frontier portfolios g and g+h.

• Proposition 6.3 :Any convex combination of frontier portfolios is also a frontier portfolio.

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1818 Fin 501: Asset Pricing

…Characterization of Frontier Portfolios…

2 ~ ~ T ~ • For any portfolio on the frontier, σ ()E[rp ] = [g + hE(rp )]V [g + hE(rp )] with g and h as defined earlier.

Multiplying all this out yields:

10:13 Lecture MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1919 Fin 501: Asset Pricing

…Characterization of Frontier Portfolios…

• (i) the expected return of the minimum variance portfolio is A/C;

• (ii) the variance of the minimum variance portfolio is given by 1/C;

• (iii) equation (6.17) is the equation of a parabola with vertex (1/C, A/C) inthe expected return/i/variance space and of a hyperbola in the expected return/standard deviation space. See Figures 6.3 and 6.4.

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2020 Fin 501: Asset Pricing

Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2121 Fin 501: Asset Pricing

Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2222 Fin 501: Asset Pricing

Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2323 Fin 501: Asset Pricing Efficient Frontier with risk-free asset

The Efficient Frontier: One Risk Free and n Risky Assets Slide 0707--2424 Fin 501: Asset Pricing Efficient Frontier with risk -free asset

1 T minw 2w Vw T T s.t. w e +(1 w 1)r = E[rp] − f 1 FOC: wp = λV (e r 1) − − f T E[rp] rf Multiplying by (e–rf 1) and solving for λ yields λ = − (e r 1)T V 1(e r 1) − f − − f

1 E[rp] rf w = V − (e r 1) − p − f H2 n 1 × 2 where H = B 2 Ar f + Cr f Slide 0707- -25is25 a n | {z } − q Fin 501: Asset Pricing Effici ent f ronti er wi th ri sk -free asset • Result 1: Excess return in frontier excess return T Cov[rq,rp]=wq Vwp E[r ] r = wT (e r 1) p − f q − f H2 E[r ] r q − f (E[r ] r )([E[r ] r ) = | q{z− f } p − f H2 (E[r ] r )2 Var[r ,r ]= p − f p p H2 Cov[rq,rp] E[rq] rf = (E[rp] rf) − V ar[rp] − :=βq,p Holds for any frontier portfolio p, in particular the marketSlide portfo 0707--2626 l | {z } Fin 501: Asset Pricing Efficient Frontier with risk -free asset

• Result 2: Frontier is linear in (E[r], σ)-space (E[r ] r )2 V ar[r , r ] = p − f p p H2 E[rp]=rf + Hσp

E[r ] r H = p − f σp where H is the Sharpe ratio

Slide 0707--2727 Fin 501: Asset Pricing TFdStiTwo Fund Separation

• Doing it in two steps: – First solve frontier for n risky asset – Then solve tangency point • Advantage: – Same portfolio of n risky asset for different agents with different risk aversion – Useful for applying equilibrium argument (later)

Slide 0707--2828 Fin 501: Asset Pricing Two Fund Separation

Price of Risk = = highest Sharpe ratio

Optimal Portfolios of Two Investors with Different Risk Aversion Slide 0707--2929 Fin 501: Asset Pricing Meanean--ViVariance P Pfreferences ∂U ∂U > 0, 2 < 0 ∂ μ p ∂ σ p •U(μp, σp) with E[W ] γ Var[W] − 2 –Example: • Also in expected utility framework – quadratic utility function (with portfolio return R) U(()R) = a + b R + c R2 vNM: E[U(R)] = a + b E[R] + c E[R2] 2 2 = a + b μp + c μp + c σp = g(μp, σp)

j j – asset returns normally distributed ⇒ R=∑j w r normal 2 • if U(.) is CARA ⇒ certainty equivalent = μp - ρA/2σ p (Use moment generating function)

Slide 0707--3030 Fin 501: Asset Pricing

Equilibrium leads to CAPM

• Portfolio theory: only analysis of demand – price/returns are taken as given – compositi on of ri sk y portf oli o i s same f or all i nvestors

• Equilibrium Demand = Supply (market portfolio)

• CAPM allows to derive – equilibrium prices/ returns. – risk-premium

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3131 Fin 501: Asset Pricing The CAPM with a risk-free bond

• The market portfolio is efficient since it is on the efficient frontier. • All individual optimal portfolios are located on

the half-line originating at point (0,rf). E[R ]− R • The slope of Capital Market Line (CML): M f . σ M

E[R ]− R Mσ f E[Rp ] = R f + σ p M

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3232 Fin 501: Asset Pricing The Capital Market Line

CML

M rM

rf j

σ M σ p

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3333 Fin 501: Asset Pricing The Security Market Line

E(r)

SML

E(ri)

E(rM)

rf slope SML = (E(ri)-rf) /βi

βM=1 β i β

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3434 Fin 501: Asset Pricing Overview

1. Simple CAPM with quadratic utility functions (derived from state-state-priceprice beta model) 2. MeanMean--variancevariance preferences – Portfolio Theory ––CAPMCAPM (Intuition) 3. CAPM (modern derivation) – Projections – Pricing Kernel and Expectation Kernel

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3535 Fin 501: Asset Pricing Projections

• States s=1,…,S with πs >0 • Probability inner product

• π-norm (flh)(measure of length)

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3636 Fin 501: Asset Pricing

⇒ shrink y axes y xx

x and y are π-orthogonal iff [x,y]π = 0, I.e. E[xy]=0

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3737 Fin 501: Asset Pricing …Projections…

• Z space of all linear combinations of vectors z1, …,zn • Given a vector y ∈ RS solve

• [smallest distance between vector y and Z space]

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--3838 Fin 501: Asset Pricing …Projections

y ε

E[ε zj]=0 for each j=1,…,n (from FOC) ε⊥ z yZ is the (orthogonal) projection on Z y = yZ + ε’ , yZ , ε z ∈ ZMeanMean-⊥-VarianceVariance Analysis and CAPM Slide 0707--3939 Fin 501: Asset Pricing Expected Value and CoCo--Variance…Variance… squeeze axis by (1,1)

x [x,y]=E[xy] =Cov[x,y] + E[x]E[y] [x,x]=E[x2]=Var[x]+E[x]2 ||x||= E[x2]½

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4040 Fin 501: Asset Pricing …EtdVldCExpected Value and Coo--ViVariance

E[x] = [x, 1]=

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4141 Fin 501: Asset Pricing Overview

1. Simple CAPM with quadratic utility functions (derived from statestate--priceprice beta model) 2. MeanMean--variancevariance preferences – Portfolio Theory – CAPM (Intuition) 3. CAPM (modern derivation) – PjProjecti ons – Pricing Kernel and Expectation Kernel

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4242 Fin 501: Asset Pricing New (LeRoy & Werner) Notation

• Main changes (new versus old) – gross return: r = R –SDF: μ = m * – pricing kernel: kq = m

– Asset span: M =

– income/endowment: wt = et

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4343 Fin 501: Asset Pricing

Pricing Kernel kq… • M space offf feas ible payoff s. • If no arbitrage and π >>0 there exists SDF μ ∈ RS, μ >>0, such that q(z)=E(μ z). • μ ∈ M – SDF nee d no t be in asse t span.

• A pricing kernel is a kq ∈ M such that for each z ∈ M, q(z)=E(kq z). * • (kq = m in our old notation.)

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4444 Fin 501: Asset Pricing …PiiPricing K ernel - ElExamples…

• Example 1: – SS3,=3,πs=1/3 for s = 1,2,3,

–x1=(1,0,0), x2=(0,1,1), p=(1/3,2/3). – Then kk(1=(1, 1, 1) is the unique pricing kernel. • Example 2: – S3S=3,πs=1/3f1/3 for s=1 123,2,3,

–x1=(1,0,0), x2=(0,1,0), p=(1/3,2/3). – Then k =(1 (120)i,2,0) is th e uni que pri ci ng k ernel .

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4545 Fin 501: Asset Pricing …PiiPricing K ernel – UiUniqueness • 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) μ and any z ∈ M

E[(μ-kq)z]=0

μ=(μ-kq)+kq ⇒ kq is the “projection'' of μ on M.

• Complete markets ⇒, kq=μ (SDF=state price density)

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4646 Fin 501: Asset Pricing

EttiKlkExpectations Kernel ke

• An expectations kernel is a vector k e∈ M – Such that E(z)=E(ke z) for each z ∈ M. •Example s – S=3, π =1/3, for s=1,2,3, x1=(1,0,0), x2=(0,1,0). – Then the unique ke=(1,1,0). • If π >>0, there exists a unique expectations kernel. • Let e=(1,…, 1) then for any z∈ M

• E[(e-ke)]0)z]=0 •ke is the “projection” of e on M

• ke = e if b ond can b e repli cat ed (ifkt(e.g. if markets are compl lt)ete)

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4747 Fin 501: Asset Pricing Mean Variance Frontier • Definition 1: z ∈ M is in the mean variance frontier if there exists no z’ ∈ 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+ε, with zE∈ E and ε ⊥ E.

• Hence, E[ε]= E[ε ke]=0, q(ε)= E[ε kq]=0 Cov[ε,zE]=E[ε zE]=0, since ε ⊥ E. • var[z] = var[zE]+var[ε] (price of ε is zero, but positive variance) • If z in mean variance frontier ⇒ z ∈ E. • Every z ∈ E is in mean variance frontier .

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4848 Fin 501: Asset Pricing FtiRtFrontier Returns…

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

•x

• graphically: payoffs with price of p=1. MeanMean--VarianceVariance Analysis and CAPM Slide 0707--4949 Fin 501: Asset Pricing

M = RS = R3

Mean-Variance Payoff Frontier ε

Mean-Variance Return Frontier

p=1-line = return-line ((gorthogonal to kq)

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5050 Fin 501: Asset Pricing

Mean-ViVariance (Payo ff)Fff) Fron tier

(111)(1,1,1) standard deviation

0 expected return kq

NB: graphical illustrated of expected returns and standard deviation changes if bond is not in payoff span. MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5151 Fin 501: Asset Pricing

Mean-ViVariance (Payo ff)Fff) Fron tier

efficient (return) frontier

(111)(1,1,1) standard deviation

0 expected return kq

inefficient (return) frontier

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5252 Fin 501: Asset Pricing …Frontier Returns (if agent is risk-neutral)

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5353 Fin 501: Asset Pricing Min imum Var iance Por tfo lio

• Take FOC w.r.t. λ of

• HMVPhfHence, MVP has return of

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5454 Fin 501: Asset Pricing Illustration of MVP

M = R2 and S=3 EtdtExpected return of MVP

Min imum stand ard (1,1,1) deviation

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5555 Fin 501: Asset Pricing Meanean--Var iance Effi ci ent R et urns

• 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[rλ] ≥ E[rλ0]. • If risk-free asset can be replicated – Mean var iance e ffici ent ret urns correspond t o λ ≤ 0. – Pricing kernel (portfolio) is not mean-variance efficient, since

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5656 Fin 501: Asset Pricing Zeroero--CiCovariance F FtiRtrontier Returns

• Take two frontier portfolios with returns and • C

•The portfolios have zero co-variance if

• For all λ≠λ0 μ exists • μ=0 if risk-free bond can be replicated

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5757 Fin 501: Asset Pricing Illustration of ZC Portfolio…

M = R2 and S=3

arbitraryyp portfolio p (1,1,1) Recall:

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5858 Fin 501: Asset Pricing …Illustration of ZC Portfolio

Green lines do not necessarily cross.

arbitraryyp portfolio p (1,1,1)

ZC of p

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--5959 Fin 501: Asset Pricing BtBeta P Piiricing… • Frontier Returns (are on linear subspace). Hence

• Consider anyypy asset with payoff xj E – It can be decomposed in xj = xj + εj E E – q(xj))q(=q(xj )andE[) and E[xj]]E[=E[xj ], since ε ⊥ E. E E – Let rj be the return of xj – x

– Using above and assuming λ≠λ0 and μ is ZC-portfolio of λ,

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--6060 Fin 501: Asset Pricing …BtBeta P Piiricing

• Taking expectations and deriving covariance • _

• If risk-free asset can be replicated, beta-pricing equation simplifies to

• Problem: How to identify frontier returns

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--6161 Fin 501: Asset Pricing CitlAtPiiMdlCapital Asset Pricing Model…

• CAPM = market return is frontier return – Derive conditions under which market return is frontier return – Two periods: 0,1.

i – Endowment: individual w 1 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.

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--6262 Fin 501: Asset Pricing …CitlAtPiiMdlCapital Asset Pricing Model

• If rm0 is the frontier return that has zero covariance with rm then, for every security j, •E[rj]=E[rm0] + βj (E[rm]-E[rm0]), with βj=cov[rj,rm] / var[rm]. • If a risk free asset exists, equation becomes,

•E[rj]= rf + βj (([E[rm]-rf)

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

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--6363 Fin 501: Asset Pricing OtdtdOutdated mat eri ilfllal follows

• Traditional derivation of CAPM is less elegant • Not relevant for exams

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--6464 Fin 501: Asset Pricing Charac ter iza tion of F ronti er P ortf oli os • Proposition 6.1: The entire set of frontier portflifolios can begenerated by ("are convex combinations" of) g and g+h.

• Proposition 6.2. The portfolio frontier can be described as convex combinations of any two frontier portfolios, not just the frontier portfolios g and g+h.

• Proposition 6.3 :Any convex combination of frontier portfolios is also a frontier portfolio.

MeanMean--VarianceVariance Analysis and CAPM Slide 0707--6565 Fin 501: Asset Pricing Charac ter iza tion of F ronti er P ortf oli os •Proposition 6.1: The entire set of frontier portfolios can be generated by ("are convex combinations" of) g and g+h. – Proof: To see this let q be an arbitrary frontier portfolio with ~ as its expected return. E(rq ) Consider portfolio weights (proportions) ~ ~ πg = 1− E(rq ) and πg+h = E(rq ) then, as asserted, ~ ~ ~ [1− E(rq )]g + E(rq )(g + h)= g + hE(rq )= wq.

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-6666 Fin 501: Asset Pricing

• Proposition 6.2. The portfolio frontier can be described as convex combinations of any two frontier portfolios, not just the frontier portfolios g and g+h.

• Proof: To see this, let p1 and p2 be any two distinct frontier portfolios; since the frontier portfolios are E[rp1] ≠ E[rp2] ddeet.etqifferent. Let q be a n a rb it ra ry fro nt ie r po rt fo lio, w it h expected return equal to E[rq]. Since E[rp1] ≠ E[rp2] , there must exist a unique number such that

E[rq]=] = α E[rp1]+(1] + (1 - α)E[r) E[rp2] (6.16) Now consider a portfolio of p1 and p2 with weights α,1-α, respectively, as determined by (6.16). We must show that

wq = α wp1 +(1-α)wp2.

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-6767 Fin 501: Asset Pricing

Proof of Proposition 6.2 (continued)

αw + ()1− α w = α[g + hE(~r )]+ (1− α)[g + hE(~r )] p1 p2 p1 p2

= g + h[αE(~r )+ (1− α)E(~r )] p1 p2 ~ = g + hE()rq , by construction

= wq , since q is a frontier portfolio.

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-6868 Fin 501: Asset Pricing • Proposition 6.3 :Any convex combination of frontier portflifolios isalso a fifrontier portflifolio. • Proof : Let()w1 ... wN , define N frontier portfolios th ( wi represents the vector defining the composition of the i portfolios) and let , i = 1, ..., N be real numbers such αi thatN . Lastly, let denote the expected return ∑i=1 αi = 1 E(ri ) of portfolio with weights . wi The weights corresponding to a linear combination of the above N portfolios are : N N ~ ∑αiwi = ∑()αi g + hE()ri i=1 i=1 N N ~ = ∑αig + h∑αiE(ri ) i=1 i=1 ⎡ N ~ ⎤ = g + h ∑αiE()ri ⎣⎢i=1 ⎦⎥ N N ~ Thus∑αiwi is a frontier portfolio withE(r)= ∑αiE(ri ) . i=1 i=1 MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-6969 Fin 501: Asset Pricing

…Characterization of Frontier Portfolios…

2 ~ ~ T ~ • For any portfolio on the frontier, σ ()E[rp ] = [g + hE(rp )]V [g + hE(rp )] with g and h as defined earlier.

Multiplying all this out yields:

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7070 Fin 501: Asset Pricing

…Characterization of Frontier Portfolios…

• (i) the expected return of the minimum variance portfolio is A/C;

• (ii) the variance of the minimum variance portfolio is given by 1/C;

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7171 Fin 501: Asset Pricing

Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7272 Fin 501: Asset Pricing

Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7373 Fin 501: Asset Pricing

Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7474 Fin 501: Asset Pricing Characterization of Efficient Portfolios (No RiskRisk--FreeFree Assets)

• Definition 6.2: Efficient portfolios are those frontier portfo lios whi ch are not mean-variditdiance dominated. • Lemma: Efficient portfolios are those frontier portfolios for which the expected return exceeds A/C, the expected return of the minimum variance portfolio.

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7575 Fin 501: Asset Pricing

• Proposition 6.4 : The set of efficient portfolios is a convex set. – This does not mean,,, however, that the frontier of this set is convex- shaped in the risk-return space.

• Proof : Suppose each of the N portfolios considered above was ~ efficient; thenE(ri )≥ A / C , for every portfolio i.

N ~ N A A However∑αiE(ri )≥ ∑αi = ; thus, the convex i=1 i=1 C C combination is efficient as well. So the set of efficient portflifolios, as chidharacterized bythiheir portflifolioweihights, isa convex set.

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7676 Fin 501: Asset Pricing Zero Covariance Portfolio

• Zero-Cov Portfolio is useful for Zero-Beta CAPM

• Proposition 6.5: For any frontier portfolio p, except the minimum variance portfolio, there exists a unique frontier portfolio with which p has zero covariance. We will call this portfolio the "zero covariance portfolio relative to p", and denote its vector of portfolio weights by ZC()p .

• Proof: by construction.

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7777 Fin 501: Asset Pricing

collect all expected returns terms, add and subtract A 2C/DC2 and note that the remaining term (1/C)[(BC/D)-(A2/D)]=1/C, since D=BC-A2

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7878 Fin 501: Asset Pricing

For zero co-variance portfolio ZC(p)

For graphical illustration, let’s draw this line:

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-7979 Fin 501: Asset Pricing

Graphical Representation:

line through

p (Var[rp]E[r], E[rp]) AND 2 2 ~ C ⎛ ~ A ⎞ 1 MVP (1/C, A/C) (use σ () rp = ⎜ E () rp − ⎟ + ) D ⎝ C ⎠ C 2 for σ (r) = 0 you get E[rZC(p)]

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-8080 Fin 501: Asset Pricing

E ( r )

A/C MVP

ZC(p) E[r ( ZC(p)] ZC (p) on frontier

(1/C) Var ( r )

Figure 6-6 The Set of Frontier Portfolios: Location of the Zero-Covariance Portfolio

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-8181 Fin 501: Asset Pricing Zero-Beta CAPM (no riskrisk--freefree asset)

(i) agents maximize expected utility with increasing and strictly concave utility of money functions and asset returns are multivariate normally distributed, or (ii) each agent chooses a portfolio with the objective of maximizing a derived utility function of the form 2 U(e,σ ), U1 > 0, U2 < 0 , U concave. (()iii) common time horizon, (iv) homogeneous beliefs about e and σ2

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-8282 Fin 501: Asset Pricing

– All investors hold mean-variance efficient portfolios

– the mar ke t por tfo lio is convex com bina tion o f e ffic ien t por tfoli os ⇒ is efficient.

– Cov[rp,rq] = λ E[rq]+ γ (q need not be on the frontier) (6.22)

– Cov[rp,rZC(p)] = λ E[rZC(p)] + γ=0

– Cov[rp,rq] = λ {E[rq]-E[rZC(p)]}

– Var[rp] = λ {E[rp]-E[rZC(p)]} .

Divide third by fourth equation: ~ ~ ~ ~ E(rq )= E(rZC()M )+ βMq [E(rM )− E(rZC()M )] (6.28)

~ ~ ~ ~ E(rj )= E(rZC()M )+ βMj[E(rM )− E(rZC(M ))] (6.29)

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-8383 Fin 501: Asset Pricing Zeroero--BBteta CAPM

• mean variance framework (quadratic utility or normal returns) • In equilibrium , market portfolio , which is a convex combination of individual portfolios

E[rq] = E[rZC(M)]+ βMq[E[rM]-E[rZC(M)]]

E[rj] = E[rZC(M)]+ βMj[E[rM]-E[rZC(M)]]

MeanMean--VarianceVariance Analysis and CAPM Slide 07-07-8484