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Optimizing Expected Shortfall under an ℓ_{1} Constraint—An Analytic Approach

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## Abstract

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## 1. Introduction

## 2. Method and Preliminaries

## 3. Results

#### 3.1. The Limit of Complete Information

#### 3.2. Without Regularization

#### 3.3. No Short Selling

#### 3.4. No-Short Mapping

#### 3.5. Mapping for Generic ${\ell}_{1}$ Constraint

#### 3.6. Solutions for the Order Parameters

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The boundary of the region where the optimization of ES is feasible in the unregularized case (nr); its image under the map for a finite ${\eta}^{-}=0.05$, ${\eta}^{+}=0$ regularizer; and the same under the no-short map (ns).

**Figure 2.**Dependence of ${q}_{0}$ at ${r}_{c}$ (

**left**), critical point (

**middle**), and proportion of zero weights at ${r}_{c}$ (

**right**) as a function of the regularization strength, ${\eta}^{-}=\eta $ (${\eta}^{+}=0$). Note the logarithmic scale in the left panel.

**Figure 3.**Dependence of ${q}_{0}$(

**left**), $\Delta $ (

**middle**) and “chemical potential” $\lambda $ (

**right**) on $r=N/T$, for the unregularized (blue), ${\eta}^{-}=0.05,{\eta}^{+}=0$ regularized (green), and no-short (yellow) cases.

**Figure 4.**Dependence of the out-of-sample estimation error (

**left**), proportion of zero weights (

**center**), and in-sample ES (

**right**) on $r=N/T$, for the non-regularized (blue), ${\eta}^{-}=\eta $ (${\eta}^{+}=0$) regularized (green), and no-short (orange) cases.

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**MDPI and ACS Style**

Papp, G.; Kondor, I.; Caccioli, F. Optimizing Expected Shortfall under an *ℓ*_{1} Constraint—An Analytic Approach. *Entropy* **2021**, *23*, 523.
https://doi.org/10.3390/e23050523

**AMA Style**

Papp G, Kondor I, Caccioli F. Optimizing Expected Shortfall under an *ℓ*_{1} Constraint—An Analytic Approach. *Entropy*. 2021; 23(5):523.
https://doi.org/10.3390/e23050523

**Chicago/Turabian Style**

Papp, Gábor, Imre Kondor, and Fabio Caccioli. 2021. "Optimizing Expected Shortfall under an *ℓ*_{1} Constraint—An Analytic Approach" *Entropy* 23, no. 5: 523.
https://doi.org/10.3390/e23050523