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Portfolio Construction and Systematic Trading with Factor

Entropy Pooling

Meucci, Ardia, Colasante

Presented by Marcello Colasante

R/Finance 2014

2014/05/16

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Factor Entropy Pooling: purposeWhat is the optimal investment strategy if we believe that, qualitatively, higher price on earnings imply higher returns, but we do not know precisely?

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Inequality views of Sharpe-ratios

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Factor Entropy Pooling: purposeWhat 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?

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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 driversApproach1. 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 distribution2. 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

Problem1. Sharpe ratios never change through time2. Volatilities are not updated

SolutionStep 4’. Compute the optimal parameters that satisfy the signal inequalities and are closest to the estimated covariances and expected returns

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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

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Views on equilibrium

Step 1. Target optimal portfolioStep 2. Equilibrium constraints

Step 3. BL-equilibrium parameters

Step 3’. Generalized FEP-equilibrium parameters

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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