Microfoundations of Financial Economics Professor André Farber Solvay Business School Université Libre de Bruxelles

2 June 2015 PhD 03 |2 CAPM: the real stuff Today we will look at various classical derivations of the CAPM. 1. Mossin Equilibrium of an exchange economy Based on quadratic utility functions 2. Mathematics of the efficient frontier

2 June 2015 PhD 03 |3 William Forsyth Sharpe From Wikipedia, the free encyclopedia. William Forsyth Sharpe (born June 16, 1934) is Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel.June Stanford University1990Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel Dr. Sharpe taught at the University of Washington and the University of California at Irvine. In 1970 he joined the Stanford University. He was one of the originators of the Capital Asset Pricing Model, created the Sharpe ratio for risk-adjusted investment performance analysis, contributed to the development of the binomial method for the valuation of options, the gradient method for asset allocation optimization, and returns-based style analysis for evaluating the style and performance of investment funds.University of WashingtonUniversity of California at Irvine1970Capital Asset Pricing ModelSharpe ratio binomial methodoptions He served as a President of the American Finance Association.American Finance Association He received his Ph.D., M.A., and B.A. in Economics from the University of California at Los Angeles. He is also the recipient of a Doctor of Humane Letters, Honoris Causa from DePaul University, a Doctor Honoris Causa from the University of Alicante (Spain), a Doctor Honoris Causa from the University of Vienna and the UCLA Medal, UCLA's highest honor.EconomicsUniversity of California at Los AngelesDePaul UniversityUniversity of AlicanteSpainUniversity of Vienna –Bibliography  Portfolio Theory and Capital Markets (McGraw-Hill, 1970 and 2000)  Asset Allocation Tools (Scientific Press, 1987)  Fundamentals of Investments (with Gordon J. Alexander and Jeffrey Bailey, Prentice-Hall, 2000)  Investments (with Gordon J. Alexander and Jeffrey Bailey, Prentice-Hall, 1999)

2 June 2015 PhD 03 |4 John Lintner Wikipedia does not yet have a page called John Lintner.  To start the page, begin typing in the box below. When you're done, press the "Save page" button. Your changes should be visible immediately.  If you have created this page in the past few minutes and it has not yet appeared, it may not be visible due to a delay in updating the database. Please wait and check again later before attempting to recreate the page.  Please do not create an article to promote yourself, a website, a product, or a business (see Wikipedia:What Wikipedia is not).Wikipedia:What Wikipedia is not  If you are new to Wikipedia, please read the tutorial before creating your first article, and only use the sandbox for editing experiments.tutorial your first articlesandbox  Search for John Lintner in Wikipedia Search for John Lintner in Wikipedia

2 June 2015 PhD 03 |5 Jan Mossin From Wikipedia, the free encyclopedia. Jan Mossin (b in Oslo – d. 1987) was a Norwegian economist. He graduated with a siviløkonom degree from the Norwegian School of Economics and Business Administration (NHH) in After a couple of years in business, he started his PhD studies in the Spring semester of 1962 at Carnegie Mellon University (then Carnegie Institute of Technology).1936Oslo1987Norwegian School of Economics and Business Administration (NHH) Carnegie Mellon University One of the papers in his doctoral dissertation was a very important contribution to the Capital Asset Pricing Model (CAPM). At Carnegie Mellon he was, among others, awarded the Alexander Henderson Award for 1968 for this contribution. If Jan Mossin had lived longer he would most likely had been a candidate for the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1990 together with Professors Sharpe and Lintner.Capital Asset Pricing Model (CAPM)Alexander Henderson Award1968Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel1990 After he had finished his PhD he returned to NHH where he in 1968 was tenured professor.NHH1968

2 June 2015 PhD 03 |6 CAPM à la Mossin 1 period model Investor i has quadratic utility function over future wealth Y n firms issue shares 1 share outstanding per firm p j price per share x j (s)payoff of firm j in state s σ jk covariance of payoffs of firm j and k

2 June 2015 PhD 03 |7 Investor’s problem

2 June 2015 PhD 03 |8 FOC Note: with z j * solution of: j=1,…,n

2 June 2015 PhD 03 |9 Market clearing conditions j=1,…,n

2 June 2015 PhD 03 |10 Equilibrium

2 June 2015 PhD 03 |11 Equilibrium (2) FOC (in equilibrium) can be written as: Solving for p j :

2 June 2015 PhD 03 |12 Equilibrium (3) Define: λ is a measure of the market risk aversion, the same for all companies b j is the contribution of company j to the market’s total variance we can write the equilibrium value of the firm as:

2 June 2015 PhD 03 |13 Beta formulation The equilibrium price can be written as: Define:

2 June 2015 PhD 03 |14 Mean-Variance Frontier Calculation: brute force Mean variance portfolio: s.t. Matrix notations:

2 June 2015 PhD 03 |15 Some math… Lagrange: FOC: Define:

2 June 2015 PhD 03 |16 Interpretation 1 g+h 0 H E The frontier can be spanned by two frontier returns Minimum variance portfolio MVP A/C

2 June 2015 PhD 03 |17 Zero covariance portfolio The covariance between any two frontier portfolios p and q is: For any two frontier portfolios p (except the MVP), there exists a unique frontier portfolio with which p has zero covariance:

2 June 2015 PhD 03 |18 Zero Covariance Portfolio in the σ, E space p zc(p) E[R zc(p) ] E(Rp)E(Rp) E(R)E(R) σ(R)σ(R)

2 June 2015 PhD 03 |19 Toward a Zero-Beta Capital Asset Pricing Model p q zc(p) E(Rp)E(Rp) E[R zc(p) ] σ(R)σ(R)

2 June 2015 PhD 03 |20 Some math Proof on demand – see DD Chap 7 Apply to ZC portfolio: Apply to p: Divide: Rearrange:

2 June 2015 PhD 03 |21 Another proof (more intuitive??) Consider a fraction a invested in stock q and (1-a) in p The slope is equal to the slope of tangent As:

2 June 2015 PhD 03 |22 Zero-Beta CAPM In equilibrium, the market portfolio is on the efficient frontier If there exist a risk free asset: E(R zc(M) ) = R f Empirical test: Roll critique If proxy used for the market portfolio, linear relationship doesn’t hold

2 June 2015 PhD 03 |23 Next session Efficient frontier in Hilbert space (wooow..) Where are the SDF in the CAPM?