1. R/Finance 2014
Portfolio Construction and Systematic Trading with Factor Entropy Pooling
Meucci, Ardia, Colasante
Presented by Marcello Colasante
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DO IT: Advanced Risk and Portfolio Management® Bootcamp
2014/05/16

2. 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
2014/05/16

3. 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
2014/05/16

4. Set of risk drivers represented by probability density function
Reference model
Set of risk drivers represented by probability density function
is the number of risk drivers
Approach
Non-parametric
Parametric
2014/05/16

5. Posterior distribution
Entropy pooling
Framework:
Prior distribution
Views
Posterior distribution
Relative entropy (target function)
2014/05/16

6. Views on expectations and covariances
Case study
Normal assumption:
Prior distribution
Views on expectations and covariances
Posterior distribution (analytical solution)
Reletive entropy (explicit form)
2014/05/16

7. General views are not addressed by analytical solution
Problem
General views are not addressed by analytical solution
Numerical approach is computationally expensive:
Large number of parameters
Constrained specification
2014/05/16

8. Covariance matrix of low-rank-diagonal type
Solution
Covariance matrix of low-rank-diagonal type
Consistence with a systematic-idiosyncratic linear factor model
uncorrelated
2014/05/16

9. Numerical approach with general views is possible:
Small number of parameters ( )
Unconstrained specification
Analytical expression of the gradient and the Hessian of the entropy
The high-dimensional inverses that appear in the gradient and in the Hessian are obtained analytically by means the binomial inverse theorem
2014/05/16

10. Step 1. Momentum/reversal indicator
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.
2014/05/16

11. Step 4. Standard approach Problem
Sharpe ratios never change through time
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
2014/05/16

12. 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).
2014/05/16

13. Step 1. Target optimal portfolio Step 2. Equilibrium constraints
Views on equilibrium
Step 1. Target optimal portfolio
Step 2. Equilibrium constraints
Step 3. BL-equilibrium parameters
Step 3’. Generalized FEP-equilibrium parameters
2014/05/16

14. 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).
2014/05/16