asset pricing and mean variance efficiency
TRANSCRIPT
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 T2 test
Regress excess return on market excess return
Define orthogonal return Market efficiency implies , estimate .
MacBeth T2 test (continued)
The T2 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, 1121-1152
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 lemmaIf the vector ft+1 and rt+1 are jointly Normal
Taylor series expansionLinear 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 caseDefine factor in terms of returnsWhat factor maximizes explained variance?
Satisfied by with criterion equal to
Principal Factors
Multiple factor caseCovariance matrix Define and the first columnsThen This is the “principal factor” solutionFactor analysis seeks to diagonalize
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 finitesamples Economics Letters 99, 85-88.
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 andthe 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 ratioImplicit assumption of Multivariate NormalityLimitations of data driven approach