Lecture 05: Mean-Variance Analysis & Capital Asset Pricing ...

1 16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--11 Lecture 05: MeanLecture 05: Mean--Variance Analysis &Variance Analysis & Capital Asset Pricing ModelCapital Asset Pricing Model((CAPMCAPM))Prof. Markus K. Brunnermeier16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--22 OverviewOverview Simple CAPM with quadratic utility functionsSimple CAPM with quadratic utility functions(derived from state(derived from state--price beta model)price beta model) Mean-Variance preferences Portfolio Theory CAPM (intuition) CAPM Projections Pricing Kernel and Expectation Kernel16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--33 Recall StateRecall State--price Beta modelprice Beta modelRecall:Recall.

2 E[RE[Rhh] ] --RRff= = hhE[RE[R**--RRff]]where h:= Cov[R*,Rh] / Var[R*]very general very general but what is Rbut what is R**in realityin reality??16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--44 Simple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected Utility1. All agents are identical Expected utility U(x0, x1) = s su(x0, xs) m= 1u / E[ 0u] Quadratic u(x0,x1)=v0(x0) - (x1- )2 1u = [-2(x1,1- ),.., -2(xS,1- )] E[Rh] Rf= - Cov[m,Rh] / E[m]= -RfCov[ 1u, Rh] / E[ 0u]= -RfCov[-2(x1- ), Rh] / E[ 0u]= Rf2 Cov[x1,Rh] / E[ 0u] Also holds for market portfolio E[Rm] Rf= Rf2 Cov[x1,Rm]/E[ 0u] 16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--55 Simple CAPM with Quadratic Expected UtilitySimple CAPM with Quadratic Expected Utility2.

3 Homogenous agents + Exchange economy x1= agg. endowment and is perfectly correlated with RmE[RE[Rhh]=]=RRff+ + hh{{E[RE[Rmm]]--RRff}}Market Security LineMarket Security :R*=Rf(a+b1RM)/(a+b1Rf) in this case (where b1<0)!16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--66 OverviewOverview Simple CAPM with quadratic utility functionsSimple CAPM with quadratic utility functions(derived from state(derived from state--price beta model)price beta model) Mean-Variance Analysis Portfolio Theory (Portfolio frontier, efficient frontier, ..) CAPM (Intuition) CAPM Projections Pricing Kernel and Expectation Kernel16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--77 Asset (portfolio) A Mean-Variance dominatesasset (portfolio) B if A Band A< or if A> Bwhile A B.

4 Efficient frontier: loci of all non-dominated portfolios in the mean-standard deviation space. By definition, no ( rational ) Mean-Variance investor would choose to hold a portfolio not located on the efficient : MeanDefinition: Mean--Variance DominanceVariance Dominance& Efficient Frontier& Efficient Frontier16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--88 Expected Portfolio Returns & VarianceExpected Portfolio Returns & Variance Expected returns (linear) Variancerecall that correlation coefficient [-1,1] p:=E[rp]=wj j,whereeach j=hjPjhj 2p:=Var[rp]=w0Vw=(w1w2) 21 12 21 22 w1w2 =(w1 21+w2 21w1 12+w2 22) w1w2 =w21 21+w22 22+2w1w2 12 0since 12 1 :14 Lecture 0516.

5 14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--99 For certain weights: w1and (1-w1) p= w1E[r1]+ (1-w1) E[r2] 2p= w12 12+ (1-w1)2 22+ 2 w1(1-w1) 1 2 1,2(Specify 2pand one gets weights and p s) Special cases [w1to obtain certain R] 1,2= 1 w1= (+/- p 2) / ( 1 2) 1,2= -1 w1= (+/- p+ 2) / ( 1+ 2)Illustration of 2 Asset CaseIllustration of 2 Asset Case16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1010 The Efficient Frontier: Two Perfectly Correlated Risky AssetsFor 1,2= 1:Hence, 1 p 2 pLower part with .. is irrelevantE[r2]E[r1] p=|w1 1+(1 w1) 2| p=w1 1+(1 w1) 2w1= p 2 1 216:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1111 Efficient Frontier: Two Perfectly Negative Correlated Risky AssetsFor 1,2= -1:Hence, 1 2E[r2]E[r1] p=|w1 1 (1 w1) 2| p=w1 1+(1 w1) 2 2 1+ 2 1+ 1 1+ 2 2slope: 2 1 1+ 2 pslope: 2 1 1+ 2 pw1= p+ 2 1+ 216:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1212 1 2E[r2]E[r1]For -1 < 1,2<1:Efficient Frontier: Two Imperfectly Correlated Risky Assets16:14 Lecture 0516.

6 14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1313 The Efficient Frontier: One Risky and One Risk Free AssetFor 1 = 0 1 p 2 pE[r2]E[r1]16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1414 The Efficient Frontier: One Risk Free and n Risky AssetsEfficient Frontier with Efficient Frontier with n risky assets and one riskn risky assets and one risk--free assetfree asset16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1515 MeanMean--Variance PreferencesVariance Preferences U( p, p) with quadratic utility function (with portfolio return R)U(R) = a + b R + c R2vNM: E[U(R)] = a + b E[R] + c E[R2]= a + b p+ c p2+ c p2= g( p, p) Asset returns normally distributed R= jwjrjnormal if U(.)

7 Is CARA certainty equivalent = p- A/2 2p(Use moment generating function) U p>0, U 2p<016:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1616 Optimal Portfolios of Two Investors with Different Risk AversionOptimal Portfolio: Two Fund SeparationOptimal Portfolio: Two Fund SeparationPrice of Risk == highest Sharperatio16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1717 Equilibrium leads to CAPME quilibrium leads to CAPM Portfolio theory: only Analysis of demand price/returns are taken as given composition of risky portfolio is same for all investors Equilibrium Demand = Supply (market portfolio) CAPM allows to derive equilibrium prices/ returns.

8 Risk-premium16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1818 The CAPM with a riskThe CAPM with a risk--free bondfree bond The market portfolio is efficient since it is on theefficient frontier. All individual optimal portfolios are located on the half-line originating at point (0,rf). The slope of Capital Market Line(CML): .MfMRRE ][pMfMfpRRERRE +=][][16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--1919 pjMrM MrfCMLThe Capital Market LineThe Capital Market Line16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--2020 Proof of the CAPM relationship[old traditional derivation] Refer to previous figure.

9 Consider a portfolio with a fraction 1 - of wealth invested in an arbitrary security j and a fraction in the market portfolioAs varies we trace a locus which- passes through M(- and through j)- cannot cross the CML (why?)- hence must be tangent to the CML at MTangency = = slope of the locus at M = = slope of CML =jMjMpjMp )1(2)1()1(22222 + += +=d pd p| =1 M rf M16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--2121Do you see the connection to earlier state-price beta model? R*= RMslope of tangent!slope of locus==at =1 p= Md pd p| =1=( M j) M 2M jM= M rf ME[rj]= j=rf+ jM 2M( M rf)16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--2222 i M=1rfE(rM)E(ri)E(r)slope SML = (E(ri)-rf) / iSMLThe Security Market LineThe Security Market Line16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525.

10 Financial Economics ISlide 05 Slide 05--2323 OverviewOverview Simple CAPM with quadratic utility functionsSimple CAPM with quadratic utility functions(derived from state(derived from state--price beta model)price beta model) Mean-Variance preferences Portfolio Theory CAPM (Intuition) CAPM (modern derivation) Projections Pricing Kernel and Expectation Kernel16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--2424 ProjectionsProjections States s=1,..,S with s>0 Probability inner product -norm (measure of length)16:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide 05--2525 shrinkaxesxxyyx and y are -orthogonal iff [x,y] = 0, E[xy]=016:14 Lecture 0516:14 Lecture 05 MeanMean--Variance Analysis and CAPMV ariance Analysis and CAPMEco 525: Financial Economics IEco 525: Financial Economics ISlide 05 Slide Zspace of all linear combinations of vectors z1.