# Managing Meteorological Risk through Expected Shortfall

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## Abstract

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## 1. Introduction

#### Contributions in the Literature

## 2. Weather Derivatives and Risk Measures

- Heating degree days (HDD) contracts defined, over some time interval $\left[{t}_{1};{t}_{2}\right]$, as$$\mathrm{HDD}=\sum _{i={t}_{1}}^{{t}_{2}}max(K-T\left(i\right),0)$$

- Cooling degree days (CDD) contracts defined as$$\mathrm{CDD}=\sum _{i={t}_{1}}^{{t}_{2}}max(T\left(i\right)-K,0),$$
- Cumulative average temperature (CAT) is defined as$$\mathrm{CAT}=\sum _{i={t}_{1}}^{{t}_{2}}T\left(i\right).$$

## 3. Data Collection

#### Estimation of VaR and ES Using Historical and Parametric Approaches

## 4. Temperature-Based Model

## 5. Worst Case Approach Based on VaR and ES

## 6. Pricing a Temperature-Based Weather Derivative

## 7. Hedging Strategies with Hybrid Instruments Based on Value-at-Risk and Expected Shortfall

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

- the Value-at-Risk (VaR) for a risk $\mathbf{X}$ with $1-\alpha $ confidence level is defined as$${\mathrm{VaR}}_{\alpha}\left(\mathbf{X}\right)=-\underset{z}{inf}\left\{{F}_{\mathbf{X}}\left(z\right)>\alpha \right\},$$If $\mathbf{X}\in {\mathcal{A}}^{C}$$\left(\mathbf{X}\in \mathcal{A}\right)$, then ${\mathrm{VaR}}_{\alpha}\left(\mathbf{X}\right)>0$ $\left({\mathrm{VaR}}_{\alpha}\left(\mathbf{X}\right)<0\right)$
- the Expected Shortfall (ES) for a risk $\mathbf{X}$ with $1-\alpha $ confidence level is expressed as$${\mathrm{ES}}_{\alpha}\left(\mathbf{X}\right)=-\mathbb{E}\left[\mathbf{X}|\mathbf{X}\le -{\mathrm{VaR}}_{\alpha}\left(\mathbf{X}\right)\right]$$

## Appendix B

## Appendix C

**Table A1.**Descriptive Statistics for seasonal HDDs for the period of time 1970–2016 and 10% percentile threshold.

Temp. (${}^{\xb0}$C) | Mean | Var | Std Dev | Min | Max | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

VaR ($K=4.25{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 40.43 | 1503.19 | 38.77 | 3.25 | 236.65 | 3.05 | 13.68 |

ES ($K=2.27{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 13.06 | 231.48 | 15.21 | 0 | 65.86 | 1.99 | 4.28 |

**Table A2.**Descriptive Statistics for seasonal CDDs for the period of time 1970–2016 and 90% percentile threshold.

Temp. (${}^{\xb0}$C) | Mean | Var | Std Dev | Min | Max | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

VaR ($K=26{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 19.38 | 316.61 | 17.79 | 0 | 83.5 | 1.44 | 2.53 |

ES ($K=20.26{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 303.41 | 8112.51 | 90.07 | 102.51 | 565.11 | 0.27 | 0.47 |

**Table A3.**Descriptive Statistics for monthly HDDs for the period of time 1970–2016 and 10% percentile threshold.

Temp. (${}^{\xb0}$C) | Mean | Var | Std Dev | Min | Max | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

VaR ($K=16.5{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 40.43 | 3.44 | 17.46 | 4.18 | 0 | 1.77 | 2.78 |

ES ($K=15.24{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 1.08 | 4.33 | 2.08 | 0 | 9.04 | 2.44 | 5.62 |

**Table A4.**Descriptive Statistics for monthly CDDs for the period of time 1970–2016 and 90% percentile threshold.

Temp. (${}^{\xb0}$C) | Mean | Var | Std Dev | Min | Max | Skewness | Kurtosis |
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VaR ($K=24.4{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 3.94 | 43.47 | 6.59 | 0 | 27.5 | 2.07 | 3.64 |

ES ($K=19.97{\phantom{\rule{3.33333pt}{0ex}}}^{\xb0}$C) | 45.54 | 744.39 | 27.28 | 12.26 | 144.75 | 1.39 | 2.56 |

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**Figure 1.**Thresholds ${K}_{\alpha}^{\mathrm{VaR}}$ and ${K}_{\alpha}^{\mathrm{ES}}$ for heating degree days (HDD) and cooling degree days (CDD) for different confidence levels and in different months (HDD corresponds to percentiles 0% to 50% while CDD corresponds to percentiles 50% to 100%). Note that the abbreviations ${J}_{1}$, ${A}_{1}$, ${J}_{2}$, ${J}_{3}$, and ${A}_{2}$ are used to identify the months of January, April, June, July, and August, respectively.

**Figure 5.**The daily average temperature, Value-at-Risk (VaR), and Expected Shortfall (ES) estimated with the historical method.

**Figure 9.**Daily errors of simulated data when compared to observed 2017 temperatures (mean absolute percentage error (MAPE), mean absolute error (MAE), and root mean square error (RMSE)).

**Figure 10.**Simulated VaR and ES for 2017 obtained by means of 1000 trajectories using the historical method.

**Figure 11.**Simulated VaR and ES for 2017 applying in 1000 different trajectories using the normal method.

Mean | Var | Std Dev | Min | Max | Skewness | Kurtosis | |
---|---|---|---|---|---|---|---|

Temp. (${}^{\xb0}$C) | $14.26$ | $51.34$ | $7.17$ | $-8.1$ | 34 | $0.028$ | $2.01$ |

Jarque-Bera | Lillefors (Kolmogorov-Smirnov) | |
---|---|---|

Statistic value | $693.99$ | $0.0564$ |

p-value | $0.001$ | $0.001$ |

**Table 3.**Pure financial derivative values for heating degree days (HDD) and CDD computed from data simulated for 2017—thresholds $5\%$ for HDD and $95\%$ for CDD—tick size $\lambda =20$ EUR—tail values have not been removed.

Month | HDD (VaR/ES) | CDD (VaR/ES) |
---|---|---|

January | $48.92/19.15$ EUR | $55.09/846.03$ EUR |

February | $35.49/11.84$ EUR | $116.39/805.06$ EUR |

March | $50.65/17.18$ EUR | $43.59/958.18$ EUR |

April | $36.43/12.71$ EUR | $83.09/845.55$ EUR |

May | $33.81/11.39$ EUR | $46.08/849.19$ EUR |

June | $25.90/11.66$ EUR | $73.05/833.31$ EUR |

July | $35.28/10.16$ EUR | $34.13/741.01$ EUR |

August | $38.17/12.88$ EUR | $30.76/816.59$ EUR |

September | $34.63/11.83$ EUR | $69.72/834.55$ EUR |

October | $45.56/14.59$ EUR | $28.51/938.44$ EUR |

November | $33.52/10.98$ EUR | $79.63/954.93$ EUR |

December | $51.28/17.91$ EUR | $38.86/924.04$ EUR |

**Table 4.**Pure financial derivative values for HDD and CDD computed from data simulated for 2017—thresholds $5\%$ for HDD and $95\%$ for CDD—tick size $\lambda =20$ EUR—tail values have been removed.

Month | HDD (VaR/ES) | CDD (VaR/ES) |
---|---|---|

January | $48.90/19.14$ EUR | $55.08/846.02$ EUR |

February | $35.48/11.83$ EUR | $116.38/805.05$ EUR |

March | $50.64/17.17$ EUR | $43.59/958.18$ EUR |

April | $36.42/12.70$ EUR | $83.07/845.54$ EUR |

May | $33.80/11.38$ EUR | $46.09/849.18$ EUR |

June | $25.90/11.66$ EUR | $73.04/833.30$ EUR |

July | $35.27/10.15$ EUR | $34.12/741.00$ EUR |

August | $38.16/12.87$ EUR | $30.75/816.58$ EUR |

September | $34.62/11.82$ EUR | $69.70/834.53$ EUR |

October | $45.55/14.58$ EUR | $28.50/938.43$ EUR |

November | $33.51/10.97$ EUR | $79.62/954.92$ EUR |

December | $51.27/17.90$ EUR | $38.85/924.03$ EUR |

**Table 5.**Final prices (i.e., financial value + risk loadings) for HDD and CDD—simulations for 2017—tick size $\lambda =20$ Euro—thresholds $5\%$ for HDD and $95\%$ for CDD—tail values have not been removed.

Months | HDD (VaR/ES) | CDD (VaR/ES) |
---|---|---|

January | $61.09/24.09$ EUR | $65.08/926.09$ EUR |

February | $41.95/16.70$ EUR | $170.62/889.76$ EUR |

March | $67.43/25.57$ EUR | $57.31/1084.86$ EUR |

April | $46.99/17.57$ EUR | $96.4/928.79$ EUR |

May | $41.03/13.98$ EUR | $52.64/935.06$ EUR |

June | $33.39/15.85$ EUR | $84.54/920.02$ EUR |

July | $44.59/13.76$ EUR | $40.28/822.81$ EUR |

August | $47.43/18.22$ EUR | $38.48/922.24$ EUR |

September | $44.32/14.27$ EUR | $79.82/929.98$ EUR |

October | $56.18/18.39$ EUR | $36.13/1048.33$ EUR |

November | $43.19/14.08$ EUR | $92.49/1029.81$ EUR |

December | $63.59/22.69$ EUR | $49.41/1006.49$ EUR |

**Table 6.**HDD and CDD Profit/Losses for 2017—Prices in Table 5 have been compared with actual 2017 pay-offs.

Months | HDD (VaR/ES) | CDD (VaR/ES) |
---|---|---|

January | $85.51/20.92$ EUR | −65.08/−744.92 EUR |

February | −41.95/−16.70 EUR | −120.62/354.61 EUR |

March | −67.43/−25.56 EUR | −12.71/408.03 EUR |

June | −33.39/−15.85 EUR | −52.54/828.07 EUR |

July | −44.59/−13.76 EUR | −21.28/188.90 EUR |

August | −38.43/−18.22 EUR | $147.52/816.34$ EUR |

**Table 7.**Comparison between HDDs pure financial prices, financial prices + risk loadings (RL) and Profit and Loss (P/L) − winter season 2017.

Months | Pure Fin. Prices (VaR/ES) | Fin. Prices + RL (VaR/ES) | P/L (VaR/ES) |
---|---|---|---|

Winter season | $138.12/48.92$ EUR | $162.64/58.4$ EUR | $111.36/28.61$ EUR |

Jan-Feb-Mar | $135.06/48.17$ EUR | $170.47/66.36$ EUR | −23.87/−21.34 EUR |

**Table 8.**Comparison between CDDs pure financial prices, financial prices + RL and P/L − summer season 2017.

Months | Pure Fin. Prices (VaR/ES) | Fin. Prices + RL (VaR/ES) | P/L (VaR/ES) |
---|---|---|---|

Summer season | $88.60/2638.38$ EUR | $107.97/2877.86$ EUR | $242.03/1754.69$ EUR |

Jun-Jul-Aug | $137.94/2390.91$ EUR | $163.30/2665.07$ EUR | $73.70/1833.31$ EUR |

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**MDPI and ACS Style**

Stefani, S.; Kutrolli, G.; Moretto, E.; Kulakov, S. Managing Meteorological Risk through Expected Shortfall. *Risks* **2020**, *8*, 118.
https://doi.org/10.3390/risks8040118

**AMA Style**

Stefani S, Kutrolli G, Moretto E, Kulakov S. Managing Meteorological Risk through Expected Shortfall. *Risks*. 2020; 8(4):118.
https://doi.org/10.3390/risks8040118

**Chicago/Turabian Style**

Stefani, Silvana, Gleda Kutrolli, Enrico Moretto, and Sergei Kulakov. 2020. "Managing Meteorological Risk through Expected Shortfall" *Risks* 8, no. 4: 118.
https://doi.org/10.3390/risks8040118