R/Finance 2014
Portfolio Construction and Systematic Trading
with Factor Entropy Pooling
Meucci, Ardia, Colasante
Presented by Marcello Colasante
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Factor Entropy Pooling: purpose
What is the optimal investment strategy if we believe that,
qualitatively, higher price on earnings imply higher returns, but
we do not know precisely?
Inequality views of
Sharpe-ratios
Cumulative PnL
Factor Entropy Pooling
Standard approach
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Factor Entropy Pooling: purpose
What is the set of expected returns and covariances that are
consistent with CAPM equilibrium and thus can be used
effectively as a starting point of mean variance optimization?
Equality views consistent with
equilibrium
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Reference model
• Set of risk drivers represented by probability density function
is the number of risk drivers
Approach
1. Non-parametric
2. Parametric
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Entropy pooling
• Framework:
1. Prior distribution
2. Views
3. Posterior distribution
Relative entropy (target function)
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Case study
• Normal assumption:
1. Prior distribution
2. Views on expectations and covariances
3. Posterior distribution (analytical solution)
Reletive entropy (explicit form)
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Problem
• General views are not addressed by analytical solution
• Numerical approach is computationally expensive:
1. Large number
of parameters
2. Constrained specification
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Solution
• Covariance matrix of low-rank-diagonal type
• Consistence with a systematic-idiosyncratic linear factor
model
uncorrelated
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• Numerical approach with general views is possible:
1. Small number
of parameters (
)
2. Unconstrained specification
3. Analytical expression of the gradient and the Hessian of the
entropy
4. The high-dimensional inverses
that appear in the
gradient and in the Hessian are obtained analytically by
means the binomial inverse theorem
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Views on ranking
• We back-test a standard reversal strategy processing ranking
(inequality) trading signals:
Step 1. Momentum/reversal indicator
Step 2. Reorder the stocks in such a way that
Step 3. Lower ranking gives rise to a lower Sharpe ratio
is a buffer that induces stronger inequalities.
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Step 4. Standard approach
Problem
1. Sharpe ratios never change through time
2. Volatilities are not updated
Solution
Step 4’. Compute the optimal parameters that satisfy the signal
inequalities and are closest to the estimated covariances and
expected returns
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Fan plot
Factor Entropy Pooling
Standard approach
Cumulative P&L generated by the reversal strategy back-test for various parametrizations.
The plot reports the median (solid line), the 50% percentile range (dim shading) and the
90% percentile range (dimmer shading).
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Views on equilibrium
Step 1. Target optimal portfolio
Step 2. Equilibrium constraints
Step 3. BL-equilibrium parameters
Step 3’. Generalized FEP-equilibrium parameters
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Factor Entropy Pooling
Historical
Black-Litterman
Historical means and covariances (blue) for various pairs of stocks versus respective
implied expected returns and covariances: Black-Litterman (black) and Factor Entropy
Pooling (red).
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