# CARL and His POT: Measuring Risks in Commodity Markets

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. POT Methods

#### 2.2. CARL Models

## 3. Empirical Application

#### 3.1. Data Description

#### 3.2. ES Testing Procedures

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Evolution over time of 5%-VaR and 5%-ES calculated using the ABS model combined with the POT method.

Statistic | WTI | Natural Gas | Gold | Corn |
---|---|---|---|---|

Mean | 0.017 | −0.019 | 0.033 | 0.009 |

Min | −13.065 | −14.893 | −9.821 | −26.862 |

Max | 16.410 | 26.771 | 8.625 | 12.757 |

Standard Deviation | 2.363 | 3.201 | 1.188 | 1.948 |

Kurtosis | 7.134 | 7.536 | 8.186 | 15.195 |

Skewness | 0.115 | 0.685 | −0.371 | −0.691 |

Jarque-Bera | 2500.4 | 3274.2 | 4002.4 | 21,964.9 |

J-B p-value | 0.000 | 0.000 | 0.000 | 0.000 |

ADF p-value | 0.000 | 0.000 | 0.000 | 0.000 |

N. of observations | 3500 | 3500 | 3500 | 3500 |

Correlation | WTI | Natural Gas | Gold | Corn |
---|---|---|---|---|

WTI | 1 | |||

Natural gas | 0.244 | 1 | ||

Gold | 0.242 | 0.074 | 1 | |

Corn | 0.241 | 0.107 | 0.174 | 1 |

**Table 3.**p-values for the Testing Procedures for $5\%$-ES based on $K=4$. The threshold Q is equal to the 85% unconditional quantile of the losses ${y}_{t}$.

WTI | Gold | Natural Gas | Corn | |
---|---|---|---|---|

Pearson Test; TVPOT-CARL-IND Model | 0.0716 | 0.0696 | 0.0493 | 0.0883 |

Pearson Test; TVPOT-CARL-IND-X Model | 0.3282 | 0.1630 | 0.2558 | 0.0412 |

Pearson Test; TVPOT-CARL-ABS Model | 0.5879 | 0.0129 | 0.0897 | 0.0061 |

Pearson Test; TVPOT-CARL-ABS-X Model | 0.7000 | 0.0541 | 0.2124 | 0.0564 |

Pearson Test; GARCH Model | 0.2583 | 0.0787 | 0.0600 | 0.0000 |

Pearson Test; HS | 0.0000 | 0.0102 | 0.1954 | 0.0878 |

Pearson Test; FHS | 0.0307 | 0.0303 | 0.0152 | 0.0735 |

Pearson Test; QAV Model | 0.0000 | 0.0019 | 0.2236 | 0.0108 |

Nass Test; TVPOT-CARL-IND Model | 0.0726 | 0.0705 | 0.0502 | 0.0893 |

Nass Test; TVPOT-CARL-IND-X Model | 0.3281 | 0.1639 | 0.2562 | 0.0420 |

Nass Test; TVPOT-CARL-ABS Model | 0.5859 | 0.0133 | 0.0908 | 0.0064 |

Nass Test; TVPOT-CARL-ABS-X Model | 0.6975 | 0.0550 | 0.2131 | 0.0574 |

Nass Test; GARCH Model | 0.2587 | 0.0797 | 0.0609 | 0.0000 |

Nass Test; HS | 0.0000 | 0.0106 | 0.1956 | 0.0659 |

Nass Test; FHS | 0.0314 | 0.0310 | 0.0157 | 0.0810 |

Nass Test; QAV Model | 0.0000 | 0.0020 | 0.2113 | 0.0112 |

LRT Test; TVPOT-CARL-IND Model | 0.0479 | 0.0465 | 0.0284 | 0.0552 |

LRT Test; TVPOT-CARL-IND-X Model | 0.2791 | 0.1212 | 0.2180 | 0.0193 |

LRT Test; TVPOT-CARL-ABS Model | 0.5583 | 0.0051 | 0.0647 | 0.0017 |

LRT Test; TVPOT-CARL-ABS-X Model | 0.6912 | 0.0294 | 0.1736 | 0.0329 |

LRT Test; GARCH Model | 0.3190 | 0.0786 | 0.0296 | 0.0000 |

LRT Test; HS | 0.0000 | 0.0049 | 0.1780 | 0.0628 |

LRT Test; FHS | 0.0178 | 0.0102 | 0.0196 | 0.0722 |

LRT Test; QAV Model | 0.0000 | 0.0004 | 0.1877 | 0.0284 |

**Table 4.**p-values for the Testing Procedures for $5\%$-ES based on $K=8$. The threshold Q is equal to the 85% unconditional quantile of the losses ${y}_{t}$.

WTI | Gold | Natural Gas | Corn | |
---|---|---|---|---|

Pearson Test; TVPOT-CARL-IND Model | 0.2318 | 0.2374 | 0.1576 | 0.2739 |

Pearson Test; TVPOT-CARL-IND-X Model | 0.1231 | 0.3822 | 0.2025 | 0.2738 |

Pearson Test; TVPOT-CARL-ABS Model | 0.2986 | 0.0605 | 0.1253 | 0.0377 |

Pearson Test; TVPOT-CARL-ABS-X Model | 0.3281 | 0.1279 | 0.1489 | 0.2723 |

Pearson Test; GARCH Model | 0.0153 | 0.0000 | 0.1265 | 0.0000 |

Pearson Test; HS | 0.0000 | 0.0375 | 0.0221 | 0.2518 |

Pearson Test; FHS | 0.0356 | 0.0596 | 0.0444 | 0.2658 |

Pearson Test; QAV Model | 0.0000 | 0.0003 | 0.1213 | 0.0142 |

Nass Test; TVPOT-CARL-IND Model | 0.2336 | 0.2391 | 0.1600 | 0.2754 |

Nass Test; TVPOT-CARL-IND-X Model | 0.1256 | 0.3820 | 0.2045 | 0.2757 |

Nass Test; TVPOT-CARL-ABS Model | 0.2996 | 0.0627 | 0.1278 | 0.0395 |

Nass Test; TVPOT-CARL-ABS-X Model | 0.3287 | 0.1304 | 0.1477 | 0.2736 |

Nass Test; GARCH Model | 0.0163 | 0.0000 | 0.1290 | 0.0000 |

Nass Test; HS | 0.0000 | 0.0393 | 0.0235 | 0.2529 |

Nass Test; FHS | 0.0373 | 0.0617 | 0.0463 | 0.2629 |

Nass Test; QAV Model | 0.0000 | 0.0004 | 0.1167 | 0.0152 |

LRT Test; TVPOT-CARL-IND Model | 0.1421 | 0.1507 | 0.0925 | 0.2749 |

LRT Test; TVPOT-CARL-IND-X Model | 0.0456 | 0.2799 | 0.1314 | 0.2754 |

LRT Test; TVPOT-CARL-ABS Model | 0.2064 | 0.0183 | 0.0622 | 0.0107 |

LRT Test; TVPOT-CARL-ABS-X Model | 0.2274 | 0.0695 | 0.1735 | 0.2733 |

LRT Test; GARCH Model | 0.0496 | 0.0003 | 0.0466 | 0.0000 |

LRT Test; HS | 0.0000 | 0.0125 | 0.0398 | 0.2483 |

LRT Test; FHS | 0.0092 | 0.0139 | 0.0466 | 0.2652 |

LRT Test; QAV Model | 0.0000 | 0.0000 | 0.0769 | 0.0357 |

**Table 5.**p-values for the Testing Procedures for $5\%$-ES based on $K=4$ and $K=8$. The threshold Q is equal to the 83% unconditional quantile of the losses ${y}_{t}$.

WTI | Gold | Natural Gas | Corn | ||
---|---|---|---|---|---|

Pearson Test; TVPOT-CARL-IND Model | 0.2394 | 0.0612 | 0.0603 | 0.0562 | |

Pearson Test; TVPOT-CARL-IND-X Model | 0.5092 | 0.0964 | 0.1462 | 0.0606 | |

Pearson Test; TVPOT-CARL-ABS Model | 0.7089 | 0.0131 | 0.0474 | 0.0084 | |

Pearson Test; TVPOT-CARL-ABS-X Model | 0.7518 | 0.0804 | 0.0839 | 0.1046 | |

Nass Test; TVPOT-CARL-IND Model | 0.2399 | 0.0622 | 0.0613 | 0.0571 | |

$K=4$ | Nass Test; TVPOT-CARL-IND-X Model | 0.5078 | 0.0974 | 0.1472 | 0.0615 |

Nass Test; TVPOT-CARL-ABS Model | 0.7063 | 0.0135 | 0.0482 | 0.0087 | |

Nass Test; TVPOT-CARL-ABS-X Model | 0.7508 | 0.0814 | 0.0849 | 0.1056 | |

LRT Test; TVPOT-CARL-IND Model | 0.2022 | 0.0365 | 0.0408 | 0.0351 | |

LRT Test; TVPOT-CARL-IND-X Model | 0.4673 | 0.0635 | 0.1125 | 0.0387 | |

LRT Test; TVPOT-CARL-ABS Model | 0.6808 | 0.0055 | 0.0307 | 0.0028 | |

LRT Test; TVPOT-CARL-ABS-X Model | 0.6110 | 0.0510 | 0.0566 | 0.0796 | |

Pearson Test; TVPOT-CARL-IND Model | 0.2200 | 0.3348 | 0.2215 | 0.2693 | |

Pearson Test; TVPOT-CARL-IND-X Model | 0.1260 | 0.4058 | 0.4691 | 0.2488 | |

Pearson Test; TVPOT-CARL-ABS Model | 0.3967 | 0.0307 | 0.2348 | 0.0630 | |

Pearson Test; TVPOT-CARL-ABS-X Model | 0.4063 | 0.3113 | 0.3827 | 0.3606 | |

Nass Test; TVPOT-CARL-IND Model | 0.2219 | 0.3352 | 0.2234 | 0.2707 | |

$K=8$ | Nass Test; TVPOT-CARL-IND-X Model | 0.1285 | 0.4053 | 0.4676 | 0.2504 |

Nass Test; TVPOT-CARL-ABS Model | 0.3963 | 0.0323 | 0.2366 | 0.0651 | |

Nass Test; TVPOT-CARL-ABS-X Model | 0.4058 | 0.3121 | 0.3825 | 0.3607 | |

LRT Test; TVPOT-CARL-IND Model | 0.1522 | 0.2396 | 0.1328 | 0.1818 | |

LRT Test; TVPOT-CARL-IND-X Model | 0.0547 | 0.3092 | 0.3878 | 0.1621 | |

LRT Test; TVPOT-CARL-ABS Model | 0.3227 | 0.0069 | 0.1629 | 0.0229 | |

LRT Test; TVPOT-CARL-ABS-X Model | 0.3278 | 0.2071 | 0.2921 | 0.2799 |

**Table 6.**p-values for the Testing Procedures for $5\%$-ES based on $K=4$ and $K=8$. The threshold Q is equal to the 87% unconditional quantile of the losses ${y}_{t}$.

WTI | Gold | Natural Gas | Corn | ||
---|---|---|---|---|---|

Pearson Test; TVPOT-CARL-IND Model | 0.3600 | 0.0551 | 0.0090 | 0.0259 | |

Pearson Test; TVPOT-CARL-IND-X Model | 0.1534 | 0.0937 | 0.1236 | 0.0831 | |

Pearson Test; TVPOT-CARL-ABS Model | 0.6923 | 0.0144 | 0.0397 | 0.0112 | |

Pearson Test; TVPOT-CARL-ABS-X Model | 0.6961 | 0.0600 | 0.0553 | 0.0303 | |

Nass Test; TVPOT-CARL-IND Model | 0.3597 | 0.0561 | 0.0094 | 0.0266 | |

$K=4$ | Nass Test; TVPOT-CARL-IND-X Model | 0.1543 | 0.0947 | 0.1246 | 0.0841 |

Nass Test; TVPOT-CARL-ABS Model | 0.6898 | 0.0149 | 0.0405 | 0.0116 | |

Nass Test; TVPOT-CARL-ABS-X Model | 0.6971 | 0.0609 | 0.0562 | 0.0311 | |

LRT Test; TVPOT-CARL-IND Model | 0.3238 | 0.0348 | 0.0034 | 0.0111 | |

LRT Test; TVPOT-CARL-IND-X Model | 0.1564 | 0.0597 | 0.0858 | 0.0578 | |

LRT Test; TVPOT-CARL-ABS Model | 0.6521 | 0.0058 | 0.0220 | 0.0042 | |

LRT Test; TVPOT-CARL-ABS-X Model | 0.6970 | 0.0356 | 0.0313 | 0.0156 | |

Pearson Test; TVPOT-CARL-IND Model | 0.6558 | 0.1940 | 0.0228 | 0.0776 | |

Pearson Test; TVPOT-CARL-IND-X Model | 0.1213 | 0.2612 | 0.3367 | 0.1966 | |

Pearson Test; TVPOT-CARL-ABS Model | 0.0371 | 0.0430 | 0.1741 | 0.0649 | |

Pearson Test; TVPOT-CARL-ABS-X Model | 0.7857 | 0.0784 | 0.2262 | 0.1238 | |

Nass Test; TVPOT-CARL-IND Model | 0.6518 | 0.1961 | 0.0241 | 0.0800 | |

$K=8$ | Nass Test; TVPOT-CARL-IND-X Model | 0.1238 | 0.2627 | 0.3372 | 0.1987 |

Nass Test; TVPOT-CARL-ABS Model | 0.0389 | 0.0449 | 0.1764 | 0.0671 | |

Nass Test; TVPOT-CARL-ABS-X Model | 0.7809 | 0.0807 | 0.2281 | 0.1262 | |

LRT Test; TVPOT-CARL-IND Model | 0.5764 | 0.1160 | 0.0065 | 0.0288 | |

LRT Test; TVPOT-CARL-IND-X Model | 0.1112 | 0.1625 | 0.2079 | 0.1200 | |

LRT Test; TVPOT-CARL-ABS Model | 0.0249 | 0.0139 | 0.0919 | 0.0192 | |

LRT Test; TVPOT-CARL-ABS-X Model | 0.7339 | 0.0447 | 0.1262 | 0.0654 |

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**MDPI and ACS Style**

Algieri, B.; Leccadito, A.
CARL and His POT: Measuring Risks in Commodity Markets. *Risks* **2020**, *8*, 27.
https://doi.org/10.3390/risks8010027

**AMA Style**

Algieri B, Leccadito A.
CARL and His POT: Measuring Risks in Commodity Markets. *Risks*. 2020; 8(1):27.
https://doi.org/10.3390/risks8010027

**Chicago/Turabian Style**

Algieri, Bernardina, and Arturo Leccadito.
2020. "CARL and His POT: Measuring Risks in Commodity Markets" *Risks* 8, no. 1: 27.
https://doi.org/10.3390/risks8010027