Download - Asset pricing and Mean Variance Efficiency
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Empirical Financial Economics
Asset pricing and Mean Variance Efficiency
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Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors satisfy
Eigenvectors diagonalize covariance matrix
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Normal Distribution results
Basic result used in univariate tests:
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Multivariate Normal results
Direct extension to multivariate case:
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Mean variance facts
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The geometry of mean variance
Note: returns are in excess of the risk free rate
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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
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MacBeth T2 test
Regress excess return on market excess return
Define orthogonal return Market efficiency implies , estimate .
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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
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The geometry of mean variance
Note: returns are in excess of the risk free rate
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Multiple period consumption-investment problem
Multiperiod problem:
First order conditions:
Stochastic discount factor interpretation:
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Stochastic discount factor and the asset pricing model
If there is a risk free asset:
which yields the basic pricing relationship
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Stochastic discount factor and mean variance efficiency
Consider the regression model
The coefficients are proportional to the negative of minimum variance portfolio weights, so
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The geometry of mean variance
Note: returns are in excess of the risk free rate
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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
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Non negative discount factors
Negative discount rates possible when market returns are high
Consider a positive discount rate constraint:
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Stochastic discount factor and the asset pricing model
If there is a risk free asset:
which yields the basic pricing relationship
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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
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Multivariate Asset Pricing
Consider
Unconditional means are given by
Model for observations is
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Principal Factors
Single factor caseDefine factor in terms of returnsWhat factor maximizes explained variance?
Satisfied by with criterion equal to
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Principal Factors
Multiple factor caseCovariance matrix Define and the first columnsThen This is the “principal factor” solutionFactor analysis seeks to diagonalize
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Importance of the largest eigenvalue
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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.
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k Equally important factors
Each factor is priced and contributes equally (on average) to variance:
Eigenvalues are given by
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Important result
The larger the number of equally important factors, the more certain would a casual empirical investigator be there was only one factor!
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Numerical example
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What are the factors?
Where W is the Helmert rotation:
The average is one andthe remaining average to zero
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Implications for pricing
Regress returns on factor loadings
Suppose k factors are priced:
Only one factor will appear to be priced!
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Application of Principal Components
Yield curve factors: level, slope and curvature
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A more interesting example
Yield curve factors: level, slope and curvature
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Application of Principal Components
Procedure:
1. Estimate B* using principal components
2. Choose an orthogonal rotation to minimize a function
that penalizes departures from
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Conclusion
Mean variance efficiency and asset pricing
Important role of Sharpe ratioImplicit assumption of Multivariate NormalityLimitations of data driven approach