# On Double Value at Risk

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. VaR

#### 2.1.1. Definitions and Basic Descriptions

**Definition**

**1.**

**Remark**

**1.**

#### 2.1.2. Properties of VaR

## 3. Double-VaR

#### 3.1. Introduction of Double-VaR

**Remark**

**2.**

#### 3.2. Double-VaR Model with Respect to $(\mu ,{\sigma}^{2})$

#### 3.2.1. Two-Dimensional Likelihood Ratio Argument

**Definition**

**2.**

**Remark**

**3.**

#### 3.2.2. Solution to Joint Confidence Region on $(\mu ,{\sigma}^{2})$

#### 3.3. Double-VaR Model Based on $(\mu ,Va{R}^{2})$

#### 3.3.1. Accuracy Measurement of VaR

#### 3.3.2. $(\mu ,Va{R}^{2})$-Model

**Definition**

**3.**

**Remark**

**4.**

**Theorem**

**1.**

**Theorem**

**2.**

## 4. Empirical Analysis

#### 4.1. Description of Sample Data

#### 4.2. $(\mu ,{\sigma}^{2})$-Model

#### Effect between Ideal Point Method and the Method of the Smallest

#### 4.3. $(\mu ,Va{R}^{2})$-Model

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Ideal Point (Confidence Level: 99%) | Ideal Point (Confidence Level: 95%) | |||||||
---|---|---|---|---|---|---|---|---|

n | a | b | c | zz | a | b | c | zz |

31 | 3.6185 | 13.0228 | 52.5124 | 0.0675 | 3.2248 | 15.9692 | 45.8943 | 0.0402 |

32 | 3.6284 | 13.6810 | 53.8337 | 0.0625 | 3.2340 | 16.7118 | 47.1456 | 0.0373 |

33 | 3.6373 | 14.3476 | 55.1539 | 0.0580 | 3.2427 | 17.4595 | 48.3940 | 0.0348 |

34 | 3.6454 | 15.0216 | 56.4719 | 0.0540 | 3.2510 | 18.2119 | 49.6392 | 0.0325 |

35 | 3.6529 | 15.7018 | 57.7869 | 0.0504 | 3.2590 | 18.9685 | 50.8812 | 0.0305 |

36 | 3.6601 | 16.3877 | 59.0984 | 0.0471 | 3.2666 | 19.7289 | 52.1197 | 0.0286 |

37 | 3.6668 | 17.0785 | 60.4060 | 0.0441 | 3.2740 | 20.4929 | 53.3550 | 0.0269 |

38 | 3.6734 | 17.7739 | 61.7097 | 0.0414 | 3.2812 | 21.2604 | 54.5870 | 0.0253 |

39 | 3.6797 | 18.4736 | 63.0094 | 0.0390 | 3.2882 | 22.0311 | 55.8159 | 0.0239 |

40 | 3.6858 | 19.1774 | 64.3053 | 0.0367 | 3.2949 | 22.8048 | 57.0417 | 0.0226 |

41 | 3.6918 | 19.8851 | 65.5974 | 0.0347 | 3.3015 | 23.5816 | 58.2646 | 0.0214 |

42 | 3.6975 | 20.5965 | 66.8858 | 0.0328 | 3.3079 | 24.3612 | 59.4847 | 0.0203 |

43 | 3.7032 | 21.3115 | 68.1707 | 0.0311 | 3.3141 | 25.1435 | 60.7021 | 0.0193 |

44 | 3.7086 | 22.0299 | 69.4522 | 0.0295 | 3.3202 | 25.9285 | 61.9168 | 0.0183 |

45 | 3.7140 | 22.7516 | 70.7304 | 0.0280 | 3.3261 | 26.7160 | 63.1291 | 0.0175 |

n | a | b | c | zz |
---|---|---|---|---|

31 | 3.0444 | 17.6622 | 42.7490 | 0.0301 |

32 | 3.0539 | 18.4464 | 43.9610 | 0.0281 |

33 | 3.0629 | 19.2348 | 45.1702 | 0.0262 |

34 | 3.0715 | 20.0270 | 46.3766 | 0.0245 |

35 | 3.0799 | 20.8225 | 47.5800 | 0.0230 |

36 | 3.0879 | 21.6213 | 48.7805 | 0.0217 |

37 | 3.0957 | 22.4231 | 49.9782 | 0.0204 |

38 | 3.1033 | 23.2277 | 51.1732 | 0.0192 |

39 | 3.1106 | 24.0351 | 52.3654 | 0.0182 |

40 | 3.1178 | 24.8451 | 53.5552 | 0.0172 |

41 | 3.1247 | 25.6575 | 54.7424 | 0.0163 |

42 | 3.1314 | 26.4724 | 55.9273 | 0.0155 |

43 | 3.1380 | 27.2895 | 57.1099 | 0.0147 |

44 | 3.1444 | 28.1089 | 58.2902 | 0.0140 |

45 | 3.1507 | 28.9304 | 59.4685 | 0.0134 |

Min-Area (Confidence Level: 99%) | Min-Area (Confidence Level: 95%) | |||||||
---|---|---|---|---|---|---|---|---|

n | a | b | c | zz | a | b | c | zz |

31 | 2.9246 | 14.1512 | 63.8981 | 0.0492 | 2.3414 | 17.1303 | 56.5295 | 0.0275 |

32 | 2.9215 | 14.8181 | 65.1849 | 0.0457 | 2.3384 | 17.8670 | 57.7614 | 0.0256 |

33 | 2.9185 | 15.4917 | 66.4687 | 0.0425 | 2.3356 | 18.6088 | 58.9919 | 0.0239 |

34 | 2.9152 | 16.1680 | 67.7464 | 0.0396 | 2.3328 | 19.3513 | 60.2205 | 0.0224 |

35 | 2.9123 | 16.8527 | 69.0219 | 0.0370 | 2.3301 | 20.0984 | 61.4518 | 0.0210 |

36 | 2.9100 | 17.5339 | 70.3055 | 0.0347 | 2.3279 | 20.8511 | 62.6723 | 0.0198 |

37 | 2.9070 | 18.2239 | 71.5778 | 0.0326 | 2.3255 | 21.6032 | 63.8980 | 0.0186 |

38 | 2.9050 | 18.9163 | 72.8532 | 0.0306 | 2.3233 | 22.3595 | 65.1167 | 0.0176 |

39 | 2.9024 | 19.6137 | 74.1233 | 0.0289 | 2.3212 | 23.1191 | 66.3364 | 0.0166 |

40 | 2.9005 | 20.3139 | 75.3969 | 0.0272 | 2.3192 | 23.8815 | 67.5544 | 0.0157 |

41 | 2.8969 | 21.0252 | 76.6403 | 0.0257 | 2.3175 | 24.6494 | 68.7771 | 0.0149 |

42 | 2.8968 | 21.7419 | 77.9333 | 0.0244 | 2.3155 | 25.4145 | 69.9847 | 0.0141 |

43 | 2.8946 | 22.4362 | 79.1883 | 0.0231 | 2.3137 | 26.1851 | 71.2024 | 0.0134 |

44 | 2.8902 | 23.1570 | 80.4262 | 0.0219 | 2.3121 | 26.9580 | 72.4080 | 0.0128 |

45 | 2.8918 | 23.8752 | 81.7268 | 0.0209 | 2.3105 | 27.7325 | 73.6185 | 0.0122 |

n | a | b | c | zz |
---|---|---|---|---|

31 | 2.0457 | 18.7961 | 53.0159 | 0.0198 |

32 | 2.0428 | 19.5671 | 54.2198 | 0.0185 |

33 | 2.0401 | 20.3418 | 55.4228 | 0.0173 |

34 | 2.0375 | 21.1193 | 56.6243 | 0.0162 |

35 | 2.0351 | 21.9005 | 57.8240 | 0.0152 |

36 | 2.0328 | 22.6834 | 59.0225 | 0.0143 |

37 | 2.0307 | 23.4694 | 60.2189 | 0.0135 |

38 | 2.0286 | 24.2587 | 61.4147 | 0.0128 |

39 | 2.0267 | 25.0498 | 62.6071 | 0.0121 |

40 | 2.0248 | 25.8426 | 63.7978 | 0.0114 |

41 | 2.0230 | 26.6392 | 64.9906 | 0.0109 |

42 | 2.0213 | 27.4365 | 66.1760 | 0.0103 |

43 | 2.0197 | 28.2368 | 67.3628 | 0.0098 |

44 | 2.0182 | 29.0392 | 68.5481 | 0.0093 |

45 | 2.0167 | 29.8436 | 69.7319 | 0.0089 |

Min-Area (Confidence Level: 99%) | Min-Area (Confidence Level: 95%) | |||||||
---|---|---|---|---|---|---|---|---|

n | a | b | c | zz | a | b | c | zz |

46 | 3.0064 | 21.7720 | 78.4073 | 0.0253 | 2.3977 | 27.4369 | 72.2900 | 0.0128 |

47 | 3.0024 | 22.4891 | 79.6858 | 0.0239 | 2.3941 | 28.2136 | 73.5112 | 0.0122 |

48 | 2.9983 | 23.2131 | 80.9560 | 0.0227 | 2.3910 | 28.9950 | 74.7342 | 0.0116 |

49 | 2.9944 | 23.9403 | 82.2217 | 0.0215 | 2.3880 | 29.7792 | 75.9545 | 0.0111 |

50 | 2.9882 | 24.6870 | 83.4624 | 0.0204 | 2.3853 | 30.5680 | 77.1736 | 0.0106 |

60 | 3.1182 | 30.7638 | 87.6602 | 0.0145 | 2.3613 | 38.5214 | 89.2508 | 0.0071 |

70 | 2.9378 | 39.7032 | 108.3873 | 0.0091 | 2.3443 | 46.6338 | 101.1698 | 0.0051 |

80 | 3.2519 | 34.5137 | 108.8108 | 0.0132 | 2.3312 | 54.8651 | 112.9622 | 0.0038 |

90 | 1.1088 | 24.1613 | 27.8069 | 0.0018 | 1.0750 | 24.1328 | 27.6881 | 0.0017 |

100 | 5.5409 | 23.9661 | 26.2147 | 0.0059 | 5.5450 | 23.9705 | 26.2556 | 0.0060 |

n | a | b | c | zz |
---|---|---|---|---|

46 | 2.0912 | 30.2984 | 69.2514 | 0.0089 |

47 | 2.0884 | 31.1074 | 70.4505 | 0.0085 |

48 | 2.0857 | 31.9191 | 71.6481 | 0.0081 |

49 | 2.0831 | 32.7310 | 72.8427 | 0.0078 |

50 | 2.0806 | 33.5453 | 74.0353 | 0.0074 |

60 | 2.0598 | 41.7726 | 85.8687 | 0.0050 |

70 | 2.0447 | 50.1308 | 97.5564 | 0.0036 |

80 | 2.0333 | 58.5941 | 109.1282 | 0.0027 |

90 | 1.0479 | 24.0945 | 27.5455 | 0.0016 |

100 | 5.5482 | 23.9796 | 26.3103 | 0.0061 |

Sample Data | Sample Data | Sample Data | ||||||
---|---|---|---|---|---|---|---|---|

Y-M | price | yield | Y-M | price | yield | Y-M | price | yield |

0901 | 13.51 | - | 1001 | 15.17 | −0.173826 | 1101 | 12.63 | −0.014151 |

0902 | 14.27 | 0.054729 | 1002 | 15.90 | 0.046999 | 1102 | 12.87 | 0.018824 |

0903 | 15.93 | 0.110045 | 1003 | 16.28 | 0.023618 | 1103 | 14.09 | 0.090566 |

0904 | 15.50 | −0.027364 | 1004 | 14.27 | −0.131778 | 1104 | 14.46 | 0.025921 |

0905 | 16.86 | 0.084104 | 1005 | 13.23 | −0.075672 | 1105 | 13.91 | −0.038778 |

0906 | 22.41 | 0.284563 | 1006 | 13.01 | −0.016769 | 1106 | 13.02 | −0.066121 |

0907 | 19.64 | −0.131939 | 1007 | 14.52 | 0.109809 | 1107 | 12.35 | −0.052831 |

0908 | 13.63 | −0.365295 | 1008 | 13.54 | −0.069879 | 1108 | 11.85 | −0.041328 |

0909 | 14.78 | 0.081002 | 1009 | 12.95 | −0.044552 | 1109 | 11.06 | −0.068993 |

0910 | 17.77 | 0.184237 | 1010 | 14.57 | 0.117869 | 1110 | 12.10 | 0.089870 |

0911 | 17.29 | −0.027383 | 1011 | 13.05 | −0.110176 | 1111 | 11.21 | −0.076399 |

0912 | 18.05 | 0.043017 | 1012 | 12.81 | −0.018562 | 1112 | 11.87 | 0.057208 |

- | - | - | - | - | - | 1201 | 12.65 | 0.063643 |

- | - | - | - | - | - | 1202 | 12.87 | 0.017242 |

- | - | - | - | - | - | 1203 | 11.90 | −0.078361 |

- | - | - | - | - | - | 1204 | 12.20 | 0.024898 |

Confidence Level 99% | Confidence Level 95% | |||||||
---|---|---|---|---|---|---|---|---|

Method | a | b | c | zz | a | b | c | zz |

ideal point | 3.6797 | 18.4736 | 63.0094 | 0.0390 | 3.2882 | 22.0311 | 55.8159 | 0.0239 |

Min-area | 2.9024 | 19.6137 | 74.1233 | 0.0289 | 2.3212 | 23.1191 | 66.3364 | 0.0166 |

Method | a | b | c | zz |
---|---|---|---|---|

ideal point | 3.1106 | 24.0351 | 52.3654 | 0.0182 |

Min-area | 2.0267 | 25.0498 | 62.6071 | 0.0121 |

$\mathit{\alpha}$ | a | b | c | zz |
---|---|---|---|---|

1% | 2.9024 | 19.6137 | 74.1233 | 0.0289 |

5% | 2.3212 | 23.1191 | 66.3364 | 0.0166 |

10% | 2.0267 | 25.0498 | 62.6071 | 0.0121 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, W.; Zhang, S.; Zhao, P.
On Double Value at Risk. *Risks* **2019**, *7*, 31.
https://doi.org/10.3390/risks7010031

**AMA Style**

Zhang W, Zhang S, Zhao P.
On Double Value at Risk. *Risks*. 2019; 7(1):31.
https://doi.org/10.3390/risks7010031

**Chicago/Turabian Style**

Zhang, Wanbing, Sisi Zhang, and Peibiao Zhao.
2019. "On Double Value at Risk" *Risks* 7, no. 1: 31.
https://doi.org/10.3390/risks7010031