Eco 525 Financial Economics I Lecture 05 Mean Variance Analysis Capital Asset Pricing Model CAPM Prof Markus K Brunnermeier 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 1 Eco 525 Financial Economics I Overview Simple CAPM with quadratic utility functions derived from state price beta model Mean variance preferences Portfolio Theory CAPM intuition CAPM Projections Pricing Kernel and Expectation Kernel 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 2 Eco 525 Financial Economics I Recall State price Beta model Recall E Rh Rf h E R Rf where h Cov R Rh Var R very general but what is R in reality 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 3 Eco 525 Financial Economics I Simple CAPM with Quadratic Expected Utility 1 All agents are identical Expected utility U x0 x1 s s u 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 Rf Cov 1u Rh E 0u Rf Cov 2 x1 Rh E 0u Rf 2Cov x1 Rh E 0u Also holds for market portfolio E Rm Rf Rf 2Cov x1 Rm E 0u 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 4 Eco 525 Financial Economics I Simple CAPM with Quadratic Expected Utility 2 Homogenous agents Exchange economy x1 agg endowment and is perfectly correlated with Rm E Rh Rf h E Rm Rf Market Security Line N B R Rf a b1RM a b1Rf in this case where b1 0 16 14 Lecture 05 Mean Variance Analysis and CAPM 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 5 Eco 525 Financial Economics I Overview Simple CAPM with quadratic utility functions derived from state price beta model Mean variance analysis Portfolio Theory Portfolio frontier efficient frontier CAPM Intuition CAPM Projections Pricing Kernel and Expectation Kernel 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 6 Eco 525 Financial Economics I Definition Mean Variance Dominance Efficient Frontier Asset portfolio A mean variance dominates asset portfolio B if A B and A or if A Bwhile A B 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 frontier 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 7 Eco 525 Financial Economics I Expected Portfolio Returns Variance Expected returns linear p E rp wj j where each j Ph j j Variance p2 V ar rp w 0 V w w1 w2 21 12 22 w1 12 w2 21 w1 12 w2 22 w12 12 w22 22 2w1 w2 12 0 since 12 1 2 16 14 Lecture 05 12 hj Mean Variance Analysis and CAPM w1 w2 w1 w2 recall that correlation coefficient 1 1 Slide 05 8 Eco 525 Financial Economics I Illustration of 2 Asset Case For certain weights w1 and 1 w1 p w1 E r1 1 w1 E r2 2p w12 12 1 w1 2 22 2 w1 1 w1 1 2 1 2 Specify 2p and one gets weights and p s Special cases w1 to obtain certain R 1 2 1 w1 p 2 1 2 1 2 1 w1 p 2 1 2 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 9 Eco 525 Financial Economics I For 1 2 1 p w1 1 1 w1 2 p w1 1 1 w1 2 Hence w1 p 2 1 2 E r2 p E r1 1 p 2 Lower part with is irrelevant The Efficient Frontier Two Perfectly Correlated Risky Assets 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 10 Eco 525 Financial Economics I For 1 2 1 p p w1 1 1 w1 2 w1 1 1 w1 2 E r2 2 1 2 1 Hence w1 slope 1 1 2 2 p 2 1 2 2 1 1 2 p 2 1 slope 1 2 p E r1 1 2 Efficient Frontier Two Perfectly Negative Correlated Risky Assets 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 11 Eco 525 Financial Economics I For 1 1 2 1 E r2 E r1 1 2 Efficient Frontier Two Imperfectly Correlated Risky Assets 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 12 Eco 525 Financial Economics I For 1 0 E r2 p E r1 1 p 2 The Efficient Frontier One Risky and One Risk Free Asset 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 13 Eco 525 Financial Economics I Efficient Frontier with n risky assets and one risk free asset The Efficient Frontier One Risk Free and n Risky Assets 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 14 Eco 525 Financial Economics I Mean Variance Preferences U p p with U p 0 U 2 p 0 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 a b p c p2 c p2 g p p asset returns normally distributed R j wj rj normal if U is CARA certainty equivalent p A 2 2p Use moment generating function 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 15 Eco 525 Financial Economics I Optimal Portfolio Two Fund Separation Price of Risk highest Sharpe ratio Optimal Portfolios of Two Investors with Different Risk Aversion 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 16 Eco 525 Financial Economics I Equilibrium 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 risk premium 16 14 Lecture 05 Mean Variance Analysis and CAPM Slide 05 17 Eco 525 Financial Economics I 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 M E R p R f 16 14 Lecture 05 E RM R f M Mean Variance Analysis and CAPM p Slide 05 18 Eco 525 Financial Economics I The Capital Market Line CML M rM rf j M 16 14 Lecture 05 Mean Variance Analysis and CAPM p Slide 05 19 Eco 525 Financial Economics I Proof of the CAPM relationship old traditional derivation Refer to previous figure Consider a portfolio with a fraction 1 of wealth invested in an arbitrary security j and a fraction in the market portfolio p M 1 j p2 2 M2 1 2 2j 2 1 jM As 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 M Tangency d p slope of the locus at M d p 1 slope of CML …