Empirical Financial Economics Asset pricing and Mean Variance Efficiency

Eigenvalues and Eigenvectors  Eigenvalues and eigenvectors satisfy  Eigenvectors diagonalize covariance matrix

Normal Distribution results  Basic result used in univariate tests:

Multivariate Normal results  Direct extension to multivariate case:

Mean variance facts

The geometry of mean variance Note: returns are in excess of the risk free rate

Tests of Mean Variance Efficiency  Mean variance efficiency implies CAPM  For Normal with mean and covariance matrix, is distributed as noncentral Chi Square with degrees of freedom and noncentrality

MacBeth T 2 test  Regress excess return on market excess return  Define orthogonal return  Market efficiency implies, estimate.

MacBeth T 2 test (continued)  The T 2 test statistic is distributed as noncentral Chi Square with m degrees of freedom and noncentrality parameter  The quadratic form is interpreted as the Sharpe ratio of the optimal orthogonal portfolio  This is interpreted as a test of Mean Variance Efficiency  Gibbons Ross and Shanken adjust for unknown Gibbons, M, S. Ross and J. Shanken, 1989 A test of the efficiency of a given portfolio Econometrica 57,

The geometry of mean variance Note: returns are in excess of the risk free rate

Multiple period consumption- investment problem  Multiperiod problem:  First order conditions:  Stochastic discount factor interpretation:

Stochastic discount factor and the asset pricing model  If there is a risk free asset:  which yields the basic pricing relationship

Stochastic discount factor and mean variance efficiency  Consider the regression model  The coefficients are proportional to the negative of minimum variance portfolio weights, so

The geometry of mean variance Note: returns are in excess of the risk free rate

Hansen Jagannathan Bounds  Risk aversion times standard deviation of consumption is given by:  “Equity premium puzzle”: Sharpe ratio of market implies a risk aversion coefficient of about 50  Consider

Non negative discount factors  Negative discount rates possible when market returns are high  Consider a positive discount rate constraint:

Stochastic discount factor and the asset pricing model  If there is a risk free asset:  which yields the basic pricing relationship

Where does m come from?  Stein’s lemma  If the vector f t+1 and r t+1 are jointly Normal  Taylor series expansion  Linear term: CAPM, higher order terms?  Put option payoff

Multivariate Asset Pricing  Consider  Unconditional means are given by  Model for observations is

Principal Factors  Single factor case  Define factor in terms of returns  What factor maximizes explained variance?  Satisfied by with criterion equal to

Principal Factors  Multiple factor case  Covariance matrix  Define and the first columns  Then  This is the “principal factor” solution  Factor analysis seeks to diagonalize  Satisfied by with criterion equal to

Importance of the largest eigenvalue

The Economy What does it mean to randomly select security i? Restrictive? Harding, M., 2008 Explaining the single factor bias of arbitrage pricing models in finite samples Economics Letters 99,

k Equally important factors  Each factor is priced and contributes equally (on average) to variance:  Eigenvalues are given by

Important result  The larger the number of equally important factors, the more certain would a casual empirical investigator be there was only one factor!

Numerical example

What are the factors?  Where W is the Helmert rotation: The average is one and the remaining average to zero

Implications for pricing  Regress returns on factor loadings  Suppose k factors are priced:  Only one factor will appear to be priced!

Application of Principal Components Yield curve factors: level, slope and curvature

A more interesting example Yield curve factors: level, slope and curvature

Application of Principal Components Procedure: 1.Estimate B* using principal components 2.Choose an orthogonal rotation to minimize a function that penalizes departures from

Conclusion  Mean variance efficiency and asset pricing  Important role of Sharpe ratio  Implicit assumption of Multivariate Normality  Limitations of data driven approach