# Forecasting Efficient Risk/Return Frontier for Equity Risk with a KTAP Approach—A Case Study in Milan Stock Exchange

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## Abstract

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## 1. Introduction

## 2. The Risk/Return Efficient Frontier in Portfolio Management

#### Efficient Frontier

## 3. Statement of the Problem and Derivation of the Mathematical Structure

- ${\eta}_{hk}$ - the interaction rate, that is, the frequency of interactions of the test a-particle with a field a-particle;
- ${\mathcal{B}}_{hk}^{i}$ - the probability that the test a-particle in the h-th risk state shifts to the i-th risk state due to an encounter with a field a-particle in the k-th risk state.

#### Modelling the Dynamics of Risky Assets in a Portfolio

- Interactions between uncorrelated a-particles (${\rho}_{hk}=0$), or of an a particle with itself ($h=k$):$$\left\{\begin{array}{c}{\mathcal{B}}_{hk}^{i=h}=1\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\forall i,h,k=1,\cdots ,n\hfill \\ {\mathcal{B}}_{hk}^{i\ne h}=0\hfill \end{array}\right.$$
- Interactions between positively correlated a-particles (${\rho}_{hk}>0$):
- if $\phantom{\rule{1.em}{0ex}}h=n\phantom{\rule{1.em}{0ex}}$ then$$\left\{\begin{array}{c}{\mathcal{B}}_{hk}^{i=n}=1\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\forall \phantom{\rule{0.222222em}{0ex}}i,h,k=1,\cdots ,n\hfill \\ {\mathcal{B}}_{hk}^{i\ne n}=0\hfill \end{array}\right.$$
- if $\phantom{\rule{1.em}{0ex}}h\ne n\phantom{\rule{1.em}{0ex}}$ then$$\phantom{\rule{1.em}{0ex}}\left\{\begin{array}{c}{\mathcal{B}}_{hk}^{i=h+1}={\rho}_{hk}\hfill \\ {\mathcal{B}}_{hk}^{i=h}=1-{\rho}_{hk},\hfill & \forall \phantom{\rule{0.222222em}{0ex}}i,h,k=1,\cdots ,n\hfill \\ {\mathcal{B}}_{hk}^{i\ne h,h+1}=0\hfill \end{array}\right.$$

- Interactions between negatively correlated a-particles (${\rho}_{hk}<0$):
- if $\phantom{\rule{1.em}{0ex}}h=1\phantom{\rule{1.em}{0ex}}$ then$$\phantom{\rule{1.em}{0ex}}\left\{\begin{array}{c}{\mathcal{B}}_{hk}^{i=1}=1\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\forall \phantom{\rule{0.222222em}{0ex}}i,h,k=1,\cdots ,n\hfill \\ {\mathcal{B}}_{hk}^{i\ne 1}=0\hfill \end{array}\right.$$
- if $\phantom{\rule{1.em}{0ex}}h\ne 1\phantom{\rule{1.em}{0ex}}$ then$$\phantom{\rule{1.em}{0ex}}\left\{\begin{array}{c}{\mathcal{B}}_{hk}^{i=h-1}=-{\rho}_{hk}\hfill \\ {\mathcal{B}}_{hk}^{i=h}=1+{\rho}_{hk},\hfill & \forall \phantom{\rule{0.222222em}{0ex}}i,h,k=1,\cdots ,n\hfill \\ {\mathcal{B}}_{hk}^{i\ne h-1,h}=0\hfill \end{array}\right.$$

## 4. A Case Study on Principal Components of Milan Stock Exchange

## 5. Results and Discussion

#### Basel Committee Capital Accords and the CVaR as Risk Measure

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. CVaR as a Coherent Measure of Risk

## Appendix B. Materials and Methods

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**Figure 1.**Logarithmic return distributions for each stock of the panel data (logarithmic return on the abscissa and frequency on the ordinate).

**Figure 3.**Risk distribution for the 13 efficient portfolios of the risk-return frontier (on the abscissa) of the case study with the mean-variance optimization at time ${t}_{0}$ (

**left**) and its evolution at the time horizon $T=10$ (months) (

**right**).

**Figure 4.**Conditional Value-at-Risk (CVaR) vs mean-variance optimization for the given panel of data.

**Figure 6.**Risk distribution for the 13 efficient portfolios of the risk-return frontier (on the abscissa) of the case study with the CVaR optimization at time ${t}_{0}$ (

**left**) and its evolution at the time horizon $T\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}10$ (months) (

**right**).

Ticker | Stock |
---|---|

CPR.MI | Davide Campari-Milano S.p.A. |

SRG.MI | Snam S.p.A. |

UNI.MI | Unipol Gruppo S.p.A. |

AZM.MI | Azimut Holding S.p.A. |

FCA.MI | Fiat Chrysler Automibiles NV |

REC.MI | Recordati Industria Chimica e Farmaceutica S.p.A. |

SFER.MI | Salvatore Ferragamo S.p.A. |

TRN.MI | Tenaris S.A. |

MONC.MI | Moncler S.p.A. |

UBI.MI | Unione di Banche Italiane S.p.A. |

JUVE.MI | Juventus Football Club S.p.A. |

BPE.MI | BPER Banca S.p.A. |

ENI.MI | Eni S.p.A. |

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**MDPI and ACS Style**

Dolfin, M.; Leonida, L.; Muzzupappa, E.
Forecasting Efficient Risk/Return Frontier for Equity Risk with a KTAP Approach—A Case Study in Milan Stock Exchange. *Symmetry* **2019**, *11*, 1055.
https://doi.org/10.3390/sym11081055

**AMA Style**

Dolfin M, Leonida L, Muzzupappa E.
Forecasting Efficient Risk/Return Frontier for Equity Risk with a KTAP Approach—A Case Study in Milan Stock Exchange. *Symmetry*. 2019; 11(8):1055.
https://doi.org/10.3390/sym11081055

**Chicago/Turabian Style**

Dolfin, Marina, Leone Leonida, and Eleonora Muzzupappa.
2019. "Forecasting Efficient Risk/Return Frontier for Equity Risk with a KTAP Approach—A Case Study in Milan Stock Exchange" *Symmetry* 11, no. 8: 1055.
https://doi.org/10.3390/sym11081055