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Optimizing Expected Shortfall under an ℓ1 Constraint-An Analytic Approach.

Gábor Papp1, Imre Kondor2,3,4, Fabio Caccioli3,5,6

  • 1Institute for Physics, Eötvös Loránd University, 1117 Budapest, Hungary.

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Regularizers stabilize financial risk estimations like Expected Shortfall (ES), overcoming limitations in portfolio optimization. Applying an ℓ1 regularizer analytically extends the feasible optimization range by managing asset volatility.

Keywords:
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Area of Science:

  • Quantitative Finance
  • Statistical Physics
  • Risk Management

Background:

  • Expected Shortfall (ES) is the current financial regulatory market risk measure.
  • ES estimation and optimization are unstable against sample fluctuations, especially with high N/T ratios.
  • The critical ratio (r=N/T) for ES feasibility depends on the confidence level α.

Purpose of the Study:

  • To analytically calculate ES under an ℓ1 regularizer using methods from statistical physics.
  • To investigate the impact of regularizers, including the no-short selling constraint, on ES estimation and portfolio optimization.
  • To extend the feasible range of portfolio optimization by mitigating estimation instability.

Main Methods:

  • Analytical calculation of ES using the method of replicas from statistical physics.
  • Application of an ℓ1 regularizer to attenuate fluctuations in ES estimation.
  • Analysis of the no-short selling constraint as a special case of an asymmetric ℓ1 regularizer.

Main Results:

  • The ℓ1 regularizer stabilizes ES estimation and extends the feasible optimization range (r=N/T).
  • The no-short constraint acts as a volatility cutoff, zeroing out high-volatility asset weights.
  • A nontrivial mapping between regularized and unregularized problems was found, involving renormalization of order parameters.

Conclusions:

  • Regularization, particularly ℓ1, is crucial for stable and feasible ES estimation and portfolio optimization.
  • The no-short selling constraint effectively manages portfolio risk by filtering high-volatility assets.
  • The study provides a theoretical framework for understanding the impact of regularization on financial risk measures.