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.
Entropy (Basel, Switzerland)
|April 30, 2021
Summary
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.
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.
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