# Lambda Value at Risk and Regulatory Capital: A Dynamic Approach to Tail Risk

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Current Risk Measures and $\mathsf{\Lambda}VaR$

#### 2.2. The Proposal of $\mathsf{\Lambda}VaR$ Estimation: A Dynamic Benchmark Approach

#### 2.3. Backtesting Method

## 3. Empirical Analysis

#### Risk Measures Computation, Backtesting, and Comparison

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Descriptive Statistics of the Dataset

**Table A1.**Annual descriptive statistics for the equities and indexes in each year under analysis. The dataset includes 12 stocks belonging to the S&P500, the FTSE 100, and the EURO STOXX 50. The dataset contains six 1-year windows from January 2006 to December 2011. For each stock and index, we report the minimum daily return, the maximum daily return, the annual mean (the average daily return is annualized), the annual standard deviation (the daily standard deviation is annualized), skewness, kurtosis, the Jarque–Bera (JB) test statistic, and its null hypothesis h ($h=A$ if ${H}_{0}$ is accepted and $h=R$ otherwise).

2006 | 2007 | |||||||||||||||||

Min Daily Return | Max Daily Return | Annual Mean | Annual std | Skewness | Kurtosis | JB | H | p-Value | Min Daily Return | Max Daily Return | Annual Mean | Annual std | Skewness | Kurtosis | JB | H | p-Value | |

FP FP | $-0.0388$ | 0.0334 | 0.0326 | 0.1807 | $-0.3998$ | 3.6305 | 10.8004 | R | 0.0119 | $-0.0434$ | 0.0458 | 0.0131 | 0.2053 | 0.1085 | 3.6228 | 4.5312 | A | 0.0828 |

SAN SQ | $-0.0354$ | 0.0340 | 0.2265 | 0.1764 | $-0.3483$ | 3.6016 | 8.8246 | R | 0.0194 | $-0.0450$ | 0.0407 | 0.0157 | 0.2112 | $-0.0848$ | 3.7085 | 5.5288 | A | 0.0540 |

VOW3 GY | $-0.0664$ | 0.0875 | 0.5331 | 0.2884 | 0.6285 | 6.8101 | 167.6764 | R | 0.0010 | $-0.0929$ | 0.0775 | 0.6116 | 0.3010 | $-0.0793$ | 6.7672 | 148.0897 | R | 0.0010 |

BNP FP | $-0.0441$ | 0.0411 | 0.1838 | 0.2157 | $-0.0998$ | 3.2898 | 1.2895 | A | 0.4873 | $-0.0522$ | 0.0501 | $-0.1345$ | 0.2649 | 0.0604 | 3.7158 | 5.4897 | A | 0.0548 |

DBK GY | $-0.0482$ | 0.0416 | 0.2132 | 0.2055 | $-0.3727$ | 3.7980 | 12.4206 | R | 0.0084 | $-0.0528$ | 0.0419 | $-0.1578$ | 0.2347 | 0.0817 | 3.8776 | 8.3019 | R | 0.0222 |

TEF SQ | $-0.0404$ | 0.0362 | 0.2387 | 0.1532 | $-0.2186$ | 4.6203 | 29.3396 | R | 0.0010 | $-0.0374$ | 0.0675 | 0.3186 | 0.2018 | 0.3926 | 5.8767 | 92.6257 | R | 0.0010 |

ISP IM | $-0.0550$ | 0.0733 | 0.1883 | 0.2139 | 0.4339 | 6.5060 | 135.8837 | R | 0.0010 | $-0.0594$ | 0.0328 | $-0.0882$ | 0.1891 | $-0.2301$ | 5.0088 | 44.2408 | R | 0.0010 |

ENEL IM | $-0.0575$ | 0.0323 | 0.1554 | 0.1322 | $-1.1327$ | 12.4250 | 978.7849 | R | 0.0010 | $-0.0394$ | 0.0243 | 0.0257 | 0.1501 | $-0.5195$ | 4.3647 | 30.6449 | R | 0.0010 |

C UN | $-0.0471$ | 0.0300 | 0.0146 | 0.1619 | $-0.4009$ | 5.0078 | 48.6896 | R | 0.0010 | $-0.0826$ | 0.0680 | $-0.6955$ | 0.3012 | $-0.1538$ | 6.1061 | 101.4809 | R | 0.0010 |

MSFT UW | $-0.1274$ | 0.0560 | 0.0305 | 0.2274 | $-2.4345$ | 27.9181 | 6714.7750 | R | 0.0010 | $-0.0481$ | 0.0849 | 0.1039 | 0.2299 | 0.6769 | 7.4973 | 229.7779 | R | 0.0010 |

RBS LN | $-0.0334$ | 0.0400 | 0.1195 | 0.1571 | 0.3047 | 4.3665 | 23.3198 | R | 0.0014 | $-0.4096$ | 0.0879 | -0.8447 | 0.5197 | $-7.4942$ | 94.7552 | 90038.2373 | R | 0.0010 |

ULVR LN | $-0.0599$ | 0.0453 | 0.1291 | 0.1714 | $-0.5139$ | 7.4596 | 218.1719 | R | 0.0010 | $-0.0347$ | 0.0568 | 0.2977 | 0.2086 | 0.5671 | 4.4899 | 36.5219 | R | 0.0010 |

Indexes’ Statistics | Indexes’ Statistics | |||||||||||||||||

SPX Index | $-0.0290$ | 0.0220 | 0.0215 | 0.1234 | $-0.2404$ | 3.5449 | 5.5002 | A | 0.0545 | $-0.0413$ | 0.0313 | $-0.0391$ | 0.1644 | $-0.5905$ | 4.9352 | 53.5407 | R | 0.0010 |

SX5E Index | $-0.0341$ | 0.0264 | 0.1389 | 0.1461 | $-0.4683$ | 4.2074 | 24.3246 | R | 0.0013 | $-0.0293$ | 0.0289 | 0.0622 | 0.1591 | $-0.2096$ | 3.4765 | 4.1959 | A | 0.0965 |

UKX Index | $-0.0244$ | 0.0237 | 0.1210 | 0.1225 | $-0.1826$ | 4.0389 | 12.6325 | R | 0.0080 | $-0.0438$ | 0.0326 | $-0.0387$ | 0.1830 | $-0.4139$ | 4.3609 | 26.6163 | R | 0.0010 |

2008 | 2009 | |||||||||||||||||

Min Daily Return | Max Daily Return | Annual Mean | Annual std | Skewness | Kurtosis | JB | H | p-Value | Min Daily Return | Max Daily Return | Annual Mean | Annual std | Skewness | Kurtosis | JB | H | p-Value | |

FP FP | $-0.0964$ | 0.1279 | $-0.3778$ | 0.4800 | 0.5332 | 6.8048 | 162.6412 | R | 0.0010 | $-0.0592$ | 0.0854 | 0.0904 | 0.2942 | 0.1161 | 5.1276 | 47.7145 | R | 0.0010 |

SAN SQ | $-0.1272$ | 0.1339 | $-0.7142$ | 0.5410 | 0.1200 | 6.0451 | 97.1911 | R | 0.0010 | $-0.0852$ | 0.1257 | 0.5008 | 0.4566 | 0.2267 | 5.6305 | 74.2226 | R | 0.0010 |

VOW3 GY | $-0.2086$ | 0.1797 | $-0.9460$ | 0.6583 | $-0.8299$ | 10.4683 | 609.7016 | R | 0.0010 | $-0.1717$ | 0.1397 | 0.4458 | 0.5869 | $-0.2108$ | 5.3791 | 60.8111 | R | 0.0010 |

BNP FP | $-0.1893$ | 0.1613 | $-0.9074$ | 0.6240 | $-0.1873$ | 6.3064 | 115.3370 | R | 0.0010 | $-0.1430$ | 0.1887 | 0.5602 | 0.6176 | 0.8090 | 8.0523 | 293.1566 | R | 0.0010 |

DBK GY | $-0.1754$ | 0.2125 | $-1.2575$ | 0.7341 | 0.2452 | 7.3222 | 197.1055 | R | 0.0010 | $-0.1269$ | 0.1986 | 0.5447 | 0.6598 | 0.4548 | 5.8678 | 94.2840 | R | 0.0010 |

TEF SQ | $-0.0954$ | 0.1022 | $-0.3257$ | 0.3750 | 0.0187 | 6.6780 | 140.9284 | R | 0.0010 | $-0.0377$ | 0.0570 | 0.1916 | 0.2015 | 0.1586 | 4.1838 | 15.6458 | R | 0.0045 |

ISP IM | $-0.1846$ | 0.1614 | $-0.7518$ | 0.5905 | $-0.1282$ | 8.4487 | 309.9411 | R | 0.0010 | $-0.1665$ | 0.1460 | 0.2049 | 0.5107 | $-0.3946$ | 7.3550 | 204.0519 | R | 0.0010 |

ENEL IM | $-0.1007$ | 0.1682 | $-0.6123$ | 0.4222 | 0.5062 | 11.0113 | 679.2366 | R | 0.0010 | $-0.1203$ | 0.0743 | $-0.0059$ | 0.3460 | $-0.8385$ | 7.6000 | 249.7127 | R | 0.0010 |

C UN | $-0.3049$ | 0.4290 | $-1.3997$ | 1.1321 | 0.4480 | 10.2519 | 556.1805 | R | 0.0010 | $-0.4917$ | 0.3188 | $-0.8075$ | 1.2630 | $-0.5920$ | 11.0411 | 688.1444 | R | 0.0010 |

MSFT UW | $-0.0861$ | 0.1665 | $-0.5525$ | 0.4995 | 0.7136 | 6.7986 | 171.5222 | R | 0.0010 | $-0.1324$ | 0.0887 | 0.3860 | 0.3617 | $-0.5212$ | 9.2412 | 417.0772 | R | 0.0010 |

RBS LN | $-0.4981$ | 0.2773 | $-2.1988$ | 1.0237 | $-1.6843$ | 18.3559 | 2574.4868 | R | 0.0010 | $-1.0957$ | 0.3050 | $-0.6141$ | 1.4588 | $-6.4827$ | 80.8619 | 64901.8805 | R | 0.0010 |

ULVR LN | $-0.0842$ | 0.0717 | $-0.1675$ | 0.3922 | $-0.0967$ | 4.0791 | 12.5204 | R | 0.0082 | $-0.0605$ | 0.0936 | 0.2483 | 0.2625 | 0.5203 | 6.9602 | 174.6435 | R | 0.0010 |

Indexes’ Statistics | Indexes’ Statistics | |||||||||||||||||

SPX Index | $-0.0872$ | 0.1104 | $-0.4517$ | 0.4304 | 0.1654 | 5.6222 | 72.7629 | R | 0.0010 | $-0.0517$ | 0.0656 | 0.1518 | 0.2494 | 0.2158 | 5.2018 | 52.4405 | R | 0.0010 |

SX5E Index | $-0.0821$ | 0.1044 | $-0.5932$ | 0.3899 | 0.3188 | 6.5992 | 139.1742 | R | 0.0010 | $-0.0524$ | 0.0588 | 0.1536 | 0.2799 | $-0.1597$ | 3.9723 | 10.9097 | R | 0.0116 |

UKX Index | $-0.0923$ | 0.0962 | $-0.6465$ | 0.4000 | 0.3106 | 6.6233 | 140.7746 | R | 0.0010 | $-0.0668$ | 0.0443 | 0.2303 | 0.2622 | $-0.2272$ | 3.9021 | 10.6280 | R | 0.0124 |

2010 | 2011 | |||||||||||||||||

Min Daily Return | Max Daily Return | Annual Mean | Annual std | Skewness | Kurtosis | JB | H | p-Value | Min Daily Return | Max Daily Return | Annual Mean | Annual std | Skewness | Kurtosis | JB | H | p-Value | |

FP FP | $-0.0427$ | 0.0737 | $-0.1226$ | 0.2254 | 0.1622 | 5.7403 | 79.3164 | R | 0.0010 | $-0.0585$ | 0.0476 | $-0.0037$ | 0.2457 | $-0.2340$ | 3.7736 | 8.7549 | R | 0.0197 |

SAN SQ | $-0.0987$ | 0.2088 | $-0.3645$ | 0.4453 | 1.4085 | 15.3195 | 1663.5904 | R | 0.0010 | $-0.0870$ | 0.0913 | $-0.2924$ | 0.3823 | 0.1693 | 4.0687 | 13.4584 | R | 0.0068 |

VOW3 GY | $-0.0629$ | 0.0745 | 0.7057 | 0.3498 | $-0.0003$ | 3.2736 | 0.7800 | A | 0.5000 | $-0.0736$ | 0.0994 | $-0.0464$ | 0.4342 | 0.1226 | 3.5902 | 4.3738 | A | 0.0893 |

BNP FP | $-0.0770$ | 0.1898 | $-0.1405$ | 0.4181 | 1.1895 | 13.1948 | 1141.6031 | R | 0.0010 | $-0.1399$ | 0.1564 | $-0.4380$ | 0.5922 | 0.2772 | 5.8701 | 91.5016 | R | 0.0010 |

DBK GY | $-0.0755$ | 0.1210 | $-0.1624$ | 0.3301 | 0.4727 | 7.2044 | 193.4445 | R | 0.0010 | $-0.0927$ | 0.1428 | $-0.2762$ | 0.4869 | 0.3426 | 6.0348 | 103.6490 | R | 0.0010 |

TEF SQ | $-0.0761$ | 0.1131 | $-0.1291$ | 0.2619 | 0.5956 | 13.1055 | 1078.5520 | R | 0.0010 | $-0.0590$ | 0.0497 | $-0.2306$ | 0.2695 | $-0.1712$ | 4.2022 | 16.7312 | R | 0.0037 |

ISP IM | $-0.0808$ | 0.1796 | $-0.3890$ | 0.4205 | 1.0616 | 11.3222 | 768.4168 | R | 0.0010 | $-0.1720$ | 0.0980 | $-0.3756$ | 0.6431 | $-0.5304$ | 4.4557 | 34.7431 | R | 0.0010 |

ENEL IM | $-0.0574$ | 0.0767 | $-0.0574$ | 0.2226 | 0.1074 | 6.9702 | 164.6747 | R | 0.0010 | $-0.0816$ | 0.0703 | $-0.1689$ | 0.3213 | $-0.4818$ | 4.7037 | 41.0263 | R | 0.0010 |

C UN | $-0.0676$ | 0.0731 | 0.4268 | 0.3605 | $-0.0866$ | 3.7236 | 5.7664 | R | 0.0494 | $-0.1774$ | 0.1290 | $-0.5406$ | 0.4876 | $-0.5330$ | 8.0922 | 289.8362 | R | 0.0010 |

MSFT UW | $-0.0407$ | 0.0388 | 0.0029 | 0.2103 | -0.2385 | 3.7083 | 7.5948 | R | 0.0272 | $-0.0529$ | 0.0433 | $-0.0408$ | 0.2129 | $-0.2159$ | 4.5799 | 28.7242 | R | 0.0010 |

RBS LN | $-0.1955$ | 0.1200 | 0.2371 | 0.5011 | $-0.4706$ | 9.3214 | 425.4745 | R | 0.0010 | $-0.1315$ | 0.0958 | $-0.6427$ | 0.5344 | $-0.1668$ | 3.7965 | 7.9866 | R | 0.0241 |

ULVR LN | $-0.0828$ | 0.0611 | 0.0314 | 0.2176 | $-0.6901$ | 9.8276 | 505.4270 | R | 0.0010 | $-0.0272$ | 0.0366 | 0.1324 | 0.1687 | 0.1154 | 3.3345 | 1.7682 | A | 0.3672 |

Indexes’ Statistics | Indexes’ Statistics | |||||||||||||||||

SPX Index | $-0.0400$ | 0.0348 | 0.2020 | 0.1642 | $-0.2917$ | 4.4570 | 25.6590 | R | 0.0011 | $-0.0670$ | 0.0457 | 0.0300 | 0.1969 | $-0.6397$ | 7.5329 | 237.5537 | R | 0.0010 |

SX5E Index | $-0.0482$ | 0.0985 | $-0.0503$ | 0.2363 | 0.7062 | 10.4500 | 598.9229 | R | 0.0010 | $-0.0654$ | 0.0590 | $-0.1819$ | 0.2885 | $-0.2408$ | 4.5356 | 27.7334 | R | 0.0010 |

UKX Index | $-0.0358$ | 0.0471 | 0.1329 | 0.1748 | $-0.0519$ | 4.6341 | 27.9268 | R | 0.0010 | $-0.0466$ | 0.0379 | $-0.0307$ | 0.2004 | $-0.3651$ | 4.3841 | 26.2260 | R | 0.0010 |

## Appendix B. Violations and Kupiec’s Portion of Failure Test 2008

**Table A2.**Violations and Kupiec’s POF test for each equity and risk model. The table displays the violations, the POF test statistic (log-likelihood ratio, LR), and the outcome (${H}_{0}$) for all the stocks and risk models during the 2008 crisis year (A = accepted and R = rejected).

Violations and Kupiec’s POF-Test 2008 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

FP FP | SAN SQ | VOW3 GY | BNP FP | DBK GY | TEF SQ | ISP IM | ENEL IM | C UN | MSFT UW | RBS LN | ULVR LN | |||

VaR | 1% | LR | 9.711 | 15.210 | 15.210 | 15.210 | 24.852 | 7.297 | 18.253 | 7.297 | 43.847 | 28.383 | 18.253 | 9.711 |

H0 | R | R | R | R | R | R | R | R | R | R | R | R | ||

VIOL | 9 | 11 | 11 | 11 | 14 | 8 | 12 | 8 | 19 | 15 | 12 | 9 | ||

2% | LR | 10.439 | 17.230 | 17.230 | 17.230 | 22.401 | 5.016 | 19.756 | 14.830 | 30.998 | 22.401 | 14.830 | 6.653 | |

H0 | R | R | R | R | R | R | R | R | R | R | R | R | ||

VIOL | 14 | 17 | 17 | 17 | 19 | 11 | 18 | 16 | 22 | 19 | 16 | 12 | ||

3% | LR | 13.864 | 35.598 | 22.612 | 13.864 | 25.038 | 6.860 | 35.598 | 18.039 | 27.553 | 27.553 | 13.864 | 4.132 | |

H0 | R | R | R | R | R | R | R | R | R | R | R | R | ||

VIOL | 20 | 29 | 24 | 20 | 25 | 16 | 29 | 22 | 26 | 26 | 20 | 14 | ||

$\mathsf{\Lambda}$VaR 1% (decreasing) | linear (VaR 5%) | LR | 3.280 | 7.297 | 0.654 | 7.297 | 12.356 | 0.654 | 3.280 | 5.141 | 7.297 | 5.141 | 12.356 | 3.280 |

H0 | A | R | A | R | R | A | A | R | R | R | R | A | ||

VIOL | 6 | 8 | 4 | 8 | 10 | 4 | 6 | 7 | 8 | 7 | 10 | 6 | ||

linear (VaR 1%) | LR | 0.654 | 3.280 | 0.059 | 7.297 | 9.711 | 0.152 | 0.654 | 3.280 | 7.297 | 1.762 | 12.356 | 0.654 | |

H0 | A | A | A | R | R | A | A | A | R | A | R | A | ||

VIOL | 4 | 6 | 3 | 8 | 9 | 2 | 4 | 6 | 8 | 5 | 10 | 4.000 | ||

$\mathsf{\Lambda}$VaR 1% (increasing) | linear (VaR 5%) | LR | 0.059 | 0.152 | 0.059 | 0.654 | 1.762 | 0.152 | 0.654 | 3.280 | 1.762 | 0.654 | 1.762 | 0.654 |

H0 | A | A | A | A | A | A | A | A | A | A | A | A | ||

VIOL | 3 | 2 | 3 | 4 | 5 | 2 | 4 | 6 | 5 | 4 | 5 | 4 | ||

linear (VaR 1%) | LR | 0.059 | 0.152 | 0.059 | 0.654 | 1.762 | 0.152 | 0.654 | 3.280 | 1.762 | 0.654 | 1.762 | 0.654 | |

H0 | A | A | A | A | A | A | A | A | A | A | A | A | ||

VIOL | 3 | 2 | 3 | 4 | 5 | 2 | 4 | 6 | 5 | 4 | 5 | 4 | ||

$\mathsf{\Lambda}$VaR 1.5% (decreasing) | linear (VaR 5%) | LR | 3.361 | 8.811 | 4.955 | 8.811 | 15.990 | 2.027 | 8.811 | 2.027 | 30.880 | 13.431 | 11.033 | 3.361 |

H0 | A | R | R | R | R | A | R | A | R | R | R | A | ||

VIOL | 8 | 11 | 9 | 11 | 14 | 7 | 11 | 7 | 19 | 13 | 12 | 8 | ||

linear (VaR 1%) | LR | 0.987 | 6.779 | 0.289 | 4.955 | 11.033 | 0.229 | 0.289 | 2.027 | 30.880 | 3.361 | 11.033 | 0.987 | |

H0 | A | R | A | R | R | A | A | A | R | A | R | A | ||

VIOL | 6 | 10 | 5 | 9 | 12 | 3 | 5 | 7 | 19 | 8 | 12 | 6 | ||

$\mathsf{\Lambda}$VaR 1.5% (increasing) | linear (VaR 5%) | LR | 0.229 | 1.143 | 0.229 | 0.003 | 0.289 | 1.143 | 0.003 | 0.987 | 0.289 | 0.003 | 0.289 | 0.003 |

H0 | A | A | A | A | A | A | A | A | A | A | A | A | ||

VIOL | 3 | 2 | 3 | 4 | 5 | 2 | 4 | 6 | 5 | 4 | 5 | 4 | ||

linear (VaR 1%) | LR | 0.229 | 1.143 | 0.229 | 0.003 | 0.289 | 1.143 | 0.003 | 0.987 | 0.289 | 0.003 | 0.289 | 0.003 | |

H0 | A | A | A | A | A | A | A | A | A | A | A | A | ||

VIOL | 3 | 2 | 3 | 4 | 5 | 2 | 4 | 6 | 5 | 4 | 5 | 4 |

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1 | See Corbetta and Peri (2017) for a detailed study on the backtesting of the $\mathsf{\Lambda}VaR$. The authors improved on the backtesting of the $\mathsf{\Lambda}VaR$ and based their empirical findings on the $\mathsf{\Lambda}VaR$ estimation proposal introduced in the current paper. |

**Figure 1.**Increasing and decreasing $\mathsf{\Lambda}VaR$. $\mathsf{\Lambda}VaR$ coincides with the smallest intersection between the P&L distribution and the $\mathsf{\Lambda}$ function. $\mathsf{\Lambda}VaR$ is able to capture different tail behaviour of return distributions better than $VaR$. For instance, in the figure on the top, Total has thicker tails than Microsoft. They have the same $2\%$ $VaR$ ($\cong 0.075$) but Total’s $2\%$ $\mathsf{\Lambda}VaR$ ($\cong 0.1$) is higher than Microsoft’s $2\%$ $\mathsf{\Lambda}VaR$ ($\cong 0.0875$). In the figure at the bottom, the same happens for Telefonica and Unilever.

**Figure 2.**Different reactivity to market fluctuations of $VaR$, $ES$ and $\mathsf{\Lambda}VaR$ under historical simulation. The $\mathsf{\Lambda}VaR$ is the most conservative measure and has higher reactivity to adverse market conditions than the $VaR$ and the $ES$. Its behaviour is in line with the other risk measures during stable periods.

**Table 1.**Time evolution of the average number of violations and Kupiec test under the historical simulation assumption. The table shows the evolution over the global financial crisis of the average number of violations and the percentage of portion of failure (POF) acceptance, aggregated at the level of the $VaR$, as well as the increasing and decreasing $\mathsf{\Lambda}VaR$ models.

Violations (Historical Simulation) | Kupiec’s POF-Test (Historical Simulation) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2006 ( T = 261) | 2007 ( T = 259) | 2008 ( T = 260) | 2009 ( T = 261) | 2010 ( T = 263) | 2011 ( T = 264) | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | ||

VaR | 1% | 3.42 | 5.33 | 11.58 | 0.75 | 3.08 | 6.83 | 100% | 83% | 0% | 100% | 92% | 50% |

2% | 5.25 | 9.17 | 16.50 | 1.17 | 5.50 | 10.25 | 100% | 83% | 0% | 67% | 92% | 50% | |

3% | 9.33 | 15.42 | 22.58 | 2.92 | 7.58 | 14.83 | 92% | 50% | 0% | 50% | 83% | 42% | |

6 | 9.97 | 16.89 | 1.61 | 5.39 | 10.64 | 97% | 72% | 0% | 72% | 89% | 47% | ||

$\mathsf{\Lambda}VaR$ $1\%$ (decr) | linear (VaR 5%) | 2.25 | 3.67 | 7.00 | 0.67 | 2.00 | 4.25 | 100% | 83% | 42% | 100% | 100% | 83% |

linear (VaR 1%) | 2.17 | 2.33 | 5.75 | 0.67 | 1.58 | 4.00 | 100% | 83% | 67% | 100% | 100% | 83% | |

2.21 | 3.00 | 6.38 | 0.67 | 1.79 | 4.13 | 100% | 83% | 54% | 100% | 100% | 83% | ||

$\mathsf{\Lambda}VaR$ $1\%$ (incr) | linear (VaR 5%) | 1.17 | 1.00 | 3.92 | 0.42 | 0.92 | 2.75 | 100% | 100% | 100% | 100% | 100% | 100% |

linear (VaR 1%) | 1.17 | 1.00 | 3.92 | 0.42 | 1.00 | 2.75 | 100% | 100% | 100% | 100% | 100% | 100% | |

1.17 | 1.00 | 3.92 | 0.42 | 0.96 | 2.75 | 100% | 100% | 100% | 100% | 100% | 100% | ||

$\mathsf{\Lambda}VaR$ $1.5\%$ (decr) | linear (VaR 5%) | 3.33 | 5.00 | 10.83 | 0.75 | 3.00 | 6.67 | 100% | 83% | 33% | 100% | 100% | 58% |

linear (VaR 1%) | 2.92 | 3.67 | 8.50 | 0.67 | 2.33 | 5.92 | 100% | 83% | 58% | 100% | 100% | 67% | |

3.13 | 4.33 | 9.67 | 0.71 | 2.67 | 6.29 | 100% | 83% | 46% | 100% | 100% | 63% | ||

$\mathsf{\Lambda}VaR$ $1.5\%$ (incr) | linear (VaR 5%) | 1.17 | 1.00 | 3.92 | 0.42 | 0.92 | 2.75 | 100% | 100% | 100% | 100% | 100% | 100% |

linear (VaR 1%) | 1.17 | 1.00 | 3.92 | 0.42 | 1.00 | 2.83 | 100% | 100% | 100% | 100% | 100% | 100% | |

1.17 | 1.00 | 3.92 | 0.42 | 0.96 | 2.79 | 100% | 100% | 100% | 100% | 100% | 100% |

**Table 2.**Time evolution of the average number of violations and Kupiec test under the Monte Carlo Normal and GARCH models. This table details the evolutions over the global financial crisis of the average number of violations and the percentage of POF acceptance, aggregated at the level of the $VaR$ and the increasing and decreasing $\mathsf{\Lambda}VaR$ models.

Violations (Montecarlo Normal) | Kupiec’s POF-Test (Montecarlo Normal) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | ||

VaR | 1% | 4.42 | 6.92 | 15.17 | 1.5 | 4.25 | 9.5 | 83% | 50% | 0% | 100% | 83% | 25% |

2% | 6.92 | 10 | 19.58 | 3 | 5.92 | 13.58 | 100% | 50% | 0% | 83% | 83% | 33% | |

3% | 9.08 | 12.5 | 22.67 | 3.92 | 8 | 17.17 | 83% | 58% | 0% | 58% | 75% | 25% | |

6.81 | 9.81 | 19.14 | 2.81 | 6.06 | 13.42 | 89% | 53% | 0% | 81% | 81% | 28% | ||

$\mathsf{\Lambda}VaR$ $1\%$ (decr) | linear (VaR $5\%$) | 4.33 | 6.58 | 14.33 | 1.5 | 3.83 | 9.08 | 92% | 50% | 0% | 100% | 83% | 25% |

linear (VaR $1\%$) | 4.17 | 5.5 | 13 | 1.17 | 3.33 | 8.75 | 92% | 83% | 0% | 100% | 92% | 33% | |

4.25 | 6.04 | 13.67 | 1.33 | 3.58 | 8.92 | 92% | 67% | 0% | 100% | 88% | 29% | ||

$\mathsf{\Lambda}VaR$ $1\%$ (incr) | linear (VaR $5\%$) | 1.83 | 2.67 | 8.33 | 0.75 | 1.58 | 5.17 | 100% | 92% | 25% | 100% | 100% | 58% |

linear (VaR 1%) | 1.92 | 3.83 | 10.17 | 1.08 | 2 | 5.58 | 100% | 75% | 8% | 100% | 92% | 58% | |

1.88 | 3.25 | 9.25 | 0.92 | 1.79 | 5.38 | 100% | 83% | 17% | 100% | 96% | 58% | ||

$\mathsf{\Lambda}VaR$ $1.5\%$ (decr) | linear (VaR $5\%$) | 5 | 7.92 | 16.17 | 2.17 | 4.75 | 11.08 | 92% | 58% | 0% | 100% | 92% | 33% |

linear (VaR $1\%$) | 4.83 | 6.67 | 13.67 | 1.75 | 4.08 | 10.42 | 92% | 83% | 8% | 100% | 100% | 33% | |

4.92 | 7.29 | 14.92 | 1.96 | 4.42 | 10.75 | 92% | 71% | 4% | 100% | 96% | 33% | ||

$\mathsf{\Lambda}VaR$ $1.5\%$ (incr) | linear (VaR $5\%$) | 1.83 | 3.25 | 8.67 | 0.75 | 1.67 | 5.25 | 100% | 92% | 58% | 100% | 100% | 92% |

linear (VaR 1%) | 2.42 | 4.92 | 11.58 | 1.08 | 2.5 | 5.83 | 100% | 75% | 33% | 100% | 92% | 83% | |

VaR | 1% | 3 | 6.83 | 15.17 | 0.25 | 0.75 | 4.17 | 100% | 75% | 0% | 100% | 100% | 67% |

2% | 5 | 11.17 | 19.58 | 0.92 | 2.08 | 7.83 | 83% | 58% | 0% | 75% | 92% | 67% | |

3% | 7.5 | 14.08 | 22.67 | 1.92 | 3.08 | 12.08 | 100% | 58% | 0% | 50% | 75% | 67% | |

5.17 | 10.69 | 19.14 | 1.03 | 1.97 | 8.03 | 94% | 64% | 0% | 75% | 89% | 67% | ||

$\mathsf{\Lambda}VaR$ $1\%$ (decr) | linear (VaR $5\%$) | 2.83 | 5.58 | 14.33 | 0.25 | 0.33 | 4.08 | 100% | 83% | 0% | 100% | 100% | 75% |

linear (VaR $1\%$) | 2.67 | 4.75 | 13 | 0.17 | 0.17 | 3.75 | 100% | 83% | 0% | 100% | 100% | 75% | |

2.75 | 5.17 | 13.67 | 0.21 | 0.25 | 3.92 | 100% | 83% | 0% | 100% | 100% | 75% | ||

$\mathsf{\Lambda}VaR$ $1\%$ (incr) | linear (VaR $5\%$) | 0.5 | 0.75 | 8.33 | 0 | 0.25 | 0.58 | 100% | 100% | 25% | 100% | 100% | 100% |

linear (VaR 1%) | 0.5 | 1.17 | 10.17 | 0 | 0.5 | 0.75 | 100% | 100% | 8% | 100% | 100% | 100% | |

0.5 | 0.96 | 9.25 | 0 | 0.38 | 0.67 | 100% | 100% | 17% | 100% | 100% | 100% | ||

$\mathsf{\Lambda}VaR$ $1.5\%$ (decr) | linear (VaR $5\%$) | 3.92 | 7.58 | 16.17 | 0.5 | 0.92 | 5.75 | 100% | 75% | 0% | 100% | 100% | 75% |

linear (VaR $1\%$) | 3.33 | 5.83 | 13.67 | 0.25 | 0.5 | 5.42 | 100% | 83% | 8% | 100% | 100% | 83% | |

3.63 | 6.71 | 14.92 | 0.38 | 0.71 | 5.58 | 100% | 79% | 4% | 100% | 100% | 79% | ||

$\mathsf{\Lambda}VaR$ $1.5\%$ (incr) | linear (VaR $5\%$) | 0.5 | 0.83 | 8.67 | 0 | 0.5 | 0.67 | 100% | 100% | 58% | 100% | 100% | 100% |

linear (VaR 1%) | 0.5 | 1.58 | 11.58 | 0 | 0.67 | 0.83 | 100% | 100% | 33% | 100% | 100% | 100% | |

0.5 | 1.21 | 10.13 | 0 | 0.58 | 0.75 | 100% | 100% | 46 % | 100% | 100% | 100% |

© 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/).

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**MDPI and ACS Style**

Hitaj, A.; Mateus, C.; Peri, I.
Lambda Value at Risk and Regulatory Capital: A Dynamic Approach to Tail Risk. *Risks* **2018**, *6*, 17.
https://doi.org/10.3390/risks6010017

**AMA Style**

Hitaj A, Mateus C, Peri I.
Lambda Value at Risk and Regulatory Capital: A Dynamic Approach to Tail Risk. *Risks*. 2018; 6(1):17.
https://doi.org/10.3390/risks6010017

**Chicago/Turabian Style**

Hitaj, Asmerilda, Cesario Mateus, and Ilaria Peri.
2018. "Lambda Value at Risk and Regulatory Capital: A Dynamic Approach to Tail Risk" *Risks* 6, no. 1: 17.
https://doi.org/10.3390/risks6010017