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R/Finance 2014 Portfolio Construction and Systematic Trading with Factor Entropy Pooling Meucci, Ardia, Colasante Presented by Marcello Colasante STUDY IT: (white papers and code) DO IT: Advanced Risk and Portfolio Management® Bootcamp 2014/05/16

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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 2014/05/16

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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 2014/05/16

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**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

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**Posterior distribution**

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

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**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

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**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

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**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

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**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

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**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

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**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

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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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**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

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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

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