# Advanced Expected Tail Loss Measurement and Quantification for the Moroccan All Shares Index Portfolio

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## Abstract

**:**

## 1. Introduction

## 2. Value at Risk and Its Complements

#### 2.1. Value at Risk Parameters

- -
- The level of confidence;
- -
- The horizon;
- -
- The distribution of risk factors.

#### 2.2. Risk Measurement Calculation Methods

#### 2.2.1. Historical Value at Risk

#### 2.2.2. Parametric Value at Risk

#### 2.2.3. The Value at Risk Monte Carlo

**Step 1**: Generation of random vectors resulting from the chosen law.

**Step 2**: Decomposition of Cholesky.

- ${\sigma}_{ii}:$ Variance;
- ${\sigma}_{ij}:$ Covariance between the security i and the security j.

**Step 3**: Calculate the return scenarios.

- $\mathrm{A}:$ The Cholesky factorization matrix of the variance–covariance matrix $\mathrm{\Sigma}$;
- ${\mathrm{Z}}^{\mathrm{t}}:$ Transposed random vectors;
- $\mathsf{\mu}:$ Vector of simulated scenario averages.

**Step 4**: Calculate the VaR.

#### 2.3. Value at Risk Complementary Risk Measures

#### 2.3.1. Subadditivity Remedied by PVaR

- $\mathrm{VaR}\left(\omega \right):\mathrm{the}\text{}\mathrm{Global}\text{}\mathrm{portfolio}\text{}\mathrm{VaR};$
- ${\omega}_{i}:\mathrm{the}\text{}\mathrm{weight}\text{}\mathrm{of}\text{}\mathrm{security}\text{}\mathrm{i}\text{}\mathrm{making}\text{}\mathrm{up}\text{}\mathrm{the}\text{}\mathrm{portfolio}.$

**Demonstration of the PVaR:**To try to extract the formula of the PVaR, we will refer to Euler’s law (or Euler’s theorem).

**Theorem**

**1.**

- $y=\left({y}_{1},{y}_{2}\dots \dots \dots {y}_{N}\right);$
- ${y}_{n}=k{\omega}_{n}.$

**Demonstration of Equation (34):**According to the diversification theory supported by Markowitz, the variance of a portfolio composed of three securities is as follows:

#### 2.3.2. The Expected Tail Loss

## 3. Results of Loss Measurement and Quantification for the Moroccan All Shares Index Portfolio: Modeling by VaR and ETL

#### 3.1. Modelling by VaR

#### 3.1.1. Application of Historical VaR to the MASI Portfolio

- i : the index of the securities making up the MASI,
- ${\omega}_{i}$ and ${R}_{i}$: the weight and return, respectively, on asset i at time t.

#### 3.1.2. Application of Parametric VaR to the MASI Portfolio

- ${\mathrm{H}}_{0}:$ the Sample follows a normal law;
- ${\mathrm{H}}_{1}:$ the Sample doesn^′ t follow a normal law.

- n: number of observations;
- S: asymmetry coefficient;
- K: flattening coefficient.

- $\Sigma \text{}:\text{}\mathrm{variance}\u2013\mathrm{covariance}$ matrix;
- $\Omega \text{}:\text{}\mathrm{weight}\text{}\mathrm{vector}$;
- ${\omega}^{t}:\text{}\mathrm{the}\text{}\mathrm{transposition}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{weight}\text{}\mathrm{vector}$.

#### 3.1.3. Cornish–Fischer Expansion or Adjustment

- $S=-0.3438$ (asymmetry coefficient);
- $K=8.92318$ (flattening coefficient).

#### 3.1.4. Application of Monte Carlo VaR to the MASI Portfolio

#### 3.1.5. Application of PVaR to the MASI Portfolio

#### 3.2. Modelling by ETL

#### 3.2.1. Application of Historical ETL to the MASI Portfolio

#### 3.2.2. Application of Parametric ETL to the MASI Portfolio

#### 3.2.3. Application of Monte Carlo ETL to the MASI Portfolio

## 4. Discussion

## 5. Conclusions

- (i)
- The loss to be faced depends on the narrowness of the Moroccan stock market (a small number of sectors and quoted securities). In other words, few stocks create intense volatility across the entire portfolio. Indeed, the first five values in terms of weight alone account for more than 80% of the total loss calculated on the basis of the two approaches (VaR and ETL).
- (ii)
- The downward trend in the stock market from 2009 to June 2016 sets out the risk to be undertaken in any investment in the MASI portfolio. This is consolidated by our VaR and ETL calculations, which show significant levels of losses compared with the mixed returns on the various subfunds of the Moroccan financial market.

## Author Contributions

## Conflicts of Interest

## References

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Date | Afric Industries | Afriquia GAZ | … | Alliances | Wafa Assurance | Portfolio Performance |
---|---|---|---|---|---|---|

2 January 2014 | 0.00% | −4.88% | … | −1.17% | 0.00% | 0.07% |

3 January 2014 | 0.00% | 0.00% | … | 1.17% | 0.00% | −0.07% |

6 January 2014 | 0.00% | 0.00% | … | −0.15% | 0.00% | −0.04% |

… | … | … | … | … | … | … |

28 June 2017 | 0.00% | 0.00% | … | 0.00% | 0.00% | 0.13% |

29 June 2017 | 0.00% | 0.00% | … | −0.83% | 0.59% | 0.05% |

30 June 2017 | 1.22% | 0.00% | … | 0.83% | 0.00% | 0.26% |

VaR Parameters | Historical Value at Risk | |
---|---|---|

% | MAD | |

VaR _{(95%,1-day)} | −0.89% | 1,001,162,895.63 MAD |

VaR _{(99%,1-day)} | −1.41% | 1,596,808,209.94 MAD |

VaR _{(95%,10-days)} | −2.80% | 3,165,955,059.03 MAD |

VaR _{(99%,10-days)} | −4.47% | 5,049,550,929.87 MAD |

Stocks | Afriquia GAZ | Afriqia Industrie | … | Alliance | Wafa Assur |
---|---|---|---|---|---|

Afriquia GAZ | 0.0003123 | 0.0000004 | … | 0.0000110 | 0.0000153 |

Afriqia industrie | 0.0000004 | 0.0002066 | … | 0.0000144 | −0.0000151 |

… | … | … | … | … | … |

Alliance | 0.0000110 | 0.0000144 | … | 0.0002081 | 0.0000224 |

Wafa assur | 0.0000153 | −0.0000151 | … | 0.0000224 | 0.0004371 |

VaR Parameters | Parametric Value at Risk | |||
---|---|---|---|---|

Relative | Absolute | |||

% | MAD | % | MAD | |

VaR _{(95%,1-day)} | −0.92% | −1,041,208,988.18 MAD | −0.91% | −1,024,691,792.13 MAD |

VaR _{(99%,1-day)} | −1.31% | −1,472,601,741.82 MAD | −1.29% | −1,456,084,545.77 MAD |

VaR _{(95%,10-days)} | −2.91% | −3,292,591,922.90 MAD | −2.87% | −3,240,359,962.80 MAD |

VaR _{(99%,10-days)} | −4.12% | −4,656,775,590.49 MAD | −4.07% | −4,604,543,630.40 MAD |

VaR Parameters | Adjusted Parametric Value at Risk | |||
---|---|---|---|---|

Relative | Absolute | |||

% | MAD | % | MAD | |

VaR _{(95%,1-day)} | −0.94% | −1,063,909,251.02 MAD | −0.93% | −1,047,392,054.96 MAD |

VaR _{(99%,1-day)} | −1.81% | −2,043,346,992.54 MAD | −1.79% | −2,026,829,796.48 MAD |

VaR _{(95%,10-days)} | −2.98% | −3,364,376,456.94 MAD | −2.93% | −3,312,144,496.84 MAD |

VaR _{(99%,10-days)} | −5.72% | −6,461,630,546.48 MAD | −5.67% | −6,409,398,586.38 MAD |

VaR Parameters | Monte Carlo Value at Risk | |
---|---|---|

% | MAD | |

VaR _{(95%,1-day)} | −1.04% | 1,178,476.625.61 MAD |

VaR _{(99%,1-day)} | −1.48% | 1,669,542,616.39 MAD |

VaR _{(95%,10-days)} | −3.30% | 3,726,670,306.19 MAD |

VaR _{(99%,10-days)} | −4.67% | 5,279,557,318.51 MAD |

Stocks | PVaR Contribution |
---|---|

Attijariwafa bank | −0.271% |

Itissalat al maghrib | −0.241% |

Lafarge ciments | −0.180% |

Cosumar | −0.130% |

Holcim maroc | −0.070% |

Bmce bank | −0.070% |

BCP | −0.060% |

Douja prom addoha | −0.050% |

Ciments du maroc | −0.050% |

Wafa assurance | −0.040% |

CGI | −0.020% |

Managem | −0.020% |

VaR Parameters | Monte Carlo Value at Risk | |
---|---|---|

% | MAD | |

VaR _{(95%,1-day)} | −1.2282% | −1,388,012,915.66 MAD |

VaR _{(99%,1-day)} | −1.9215% | −2,171,479,166.29 MAD |

VaR _{(95%,10-days)} | −3.8840% | −4,389,282,235.23 MAD |

VaR _{(99%,10-days)} | −6.0763% | −6,866,820,057.09 MAD |

VaR Parameters | Monte Carlo Value at Risk | |
---|---|---|

% | MAD | |

VaR _{(95%,1-day)} | −1.16% | −1,305,718,077.30 MAD |

VaR _{(99%,1-day)} | −1.49% | −1,687,107,567.62 MAD |

VaR _{(95%,10-days)} | −3.65% | −4,129,043,106.32 MAD |

VaR _{(99%,10-days)} | −4.72% | −5,335,102,571.39 MAD |

VaR Parameters | Monte Carlo Value at Risk | |
---|---|---|

% | MAD | |

VaR _{(95%,1-day)} | −1.23277% | 1,393,158,385.82 MAD |

VaR _{(99%,1-day)} | −1.47733% | 1,669,542,616.39 MAD |

VaR _{(95%,10-days)} | −3.89835% | 4,405,553,640.55 MAD |

VaR _{(99%,10-days)} | −4.67170% | 5,279,557,318.51 MAD |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Airouss, M.; Tahiri, M.; Lahlou, A.; Hassouni, A. Advanced Expected Tail Loss Measurement and Quantification for the Moroccan All Shares Index Portfolio. *Mathematics* **2018**, *6*, 38.
https://doi.org/10.3390/math6030038

**AMA Style**

Airouss M, Tahiri M, Lahlou A, Hassouni A. Advanced Expected Tail Loss Measurement and Quantification for the Moroccan All Shares Index Portfolio. *Mathematics*. 2018; 6(3):38.
https://doi.org/10.3390/math6030038

**Chicago/Turabian Style**

Airouss, Marouane, Mohamed Tahiri, Amale Lahlou, and Abdelhak Hassouni. 2018. "Advanced Expected Tail Loss Measurement and Quantification for the Moroccan All Shares Index Portfolio" *Mathematics* 6, no. 3: 38.
https://doi.org/10.3390/math6030038