# 14.6 Example: Backtesting a One-Day 95% EUR Value-at-Risk Measure

Assume a one-day 95% EUR value-at-risk measure was used for a period of 125 trading days. Data gathered for backtesting is presented in Exhibit 14.8. We have already used the data from the second and third columns to construct Exhibit 14.1. We will now use the data to apply coverage, distribution and independence backtests.

###### 14.6.1 Example: Applying Coverage Tests

To apply a coverage test, we need

- the quantile of loss the value-at-risk measure is intended to measure:
*q*= 0.95, - the number of observations: α + 1 = 125, and
- the number of exceedances
*x*= 10.

The last value is obtained by summing the 0’s and 1’s in the fourth column of Exhibit 14.8.

It can also be obtained by visual inspection of Exhibit 14.1.

In Exhibit 14.3, we find that our recommended standard coverage test’s non-rejection interval for *q* = 0.95 and α + 1 = 125 is [2, 11]. Since our number of exceedances falls in this interval, we do not reject the value-at-risk measure.

In Exhibit 14.4, we find that the PF test’s non-rejection interval for *q* = 0.95 and α + 1 = 125 is [2, 12]. Since our number of exceedances falls in this interval, we do not reject the value-at-risk measure.

We cannot use the Basel Committee’s traffic light coverage test because it applies only to 99% value-at-risk measures.

###### 14.6.2 Example: Applying Distribution Tests

For distribution testing, we apply [14.10] to the loss quantiles * ^{t}u* and arrange the results in ascending order to obtain the

*n*. Values for the are obtained from [14.11], with α + 1 = 125. Values for

_{j}*n*and are presented in Exhibit 14.9.

_{j}*n*and calculated for the data of Exhibit 14.8.

_{j}These are plotted in Exhibit 14.10.

*n*, ) from Exhibit 14.9.

_{j}The graphical results are inconclusive. The points do fall near a line of slope one passing through the origin, but the fit isn’t particularly good. Is this due to the small sample size, or does it reflect shortcomings in the value-at-risk measure? For another perspective, we calculate the sample correlation between the *n _{j}* and as .997. Consulting Exhibit 14.6, we do not reject the value-at-risk measure at either the .05 or the .01 significance levels.

###### 14.6.3 Example: Applying Independence Tests

Starting with Christoffersen’s test for independent * ^{t}I*, we use the data of Exhibit 14.8 to calculate α

_{00}= 105, α

_{01}= α

_{10}= 9 and α

_{11}= 1. From these, we calculate = 0.9211, = 0.9000 and = 0.9194. Our likelihood ratio is

[14.23]

so –2*log*(Λ) = 0.0517. This does not exceed 3.814, so we do not reject the value-at-risk measure.

Next, applying our recommended standard independence test, we use [14.21] to calculate values * ^{t}n *from the loss quantiles

*. Results are indicated in Exhibit 14.11.*

^{t}u*for our basic independence test. They are identical to those in the first column of Exhibit 14.9, only they are ordered by time while those in Exhibit 14.9 are ordered by magnitude.*

^{t}nWe calculate the sample autocorrelations of the * ^{t}n* for lags 1 through 5 as indicated in Exhibit 14.12.

Our test statistic—the largest absolute value of the autocorrelations—is 0.132. This is less than the non-rejection value 0.274 obtained from Exhibit 14.7, so we do not reject the value-at-risk measure at the .05 significance level.

###### Exercises

time | 99% VaR | P&L | time | 99% VaR | P&L | time | 99% VaR | P&L |
---|---|---|---|---|---|---|---|---|

t | at t – 1 | at t | t | at t – 1 | at t | t | at t – 1 | at t |

-124 | 3.468 | -2.107 | -82 | 4.693 | 0.252 | -40 | 1.401 | 0.683 |

-123 | 3.095 | -0.143 | -81 | 3.789 | -0.074 | -39 | 1.282 | 0.241 |

-122 | 3.245 | 0.894 | -80 | 4.897 | -0.153 | -38 | 1.524 | 0.118 |

-121 | 2.969 | 0.990 | -79 | 4.256 | 0.267 | -37 | 1.834 | -0.810 |

-120 | 3.472 | -0.060 | -78 | 4.537 | 1.804 | -36 | 1.534 | -0.455 |

-119 | 4.513 | -1.123 | -77 | 4.508 | -0.196 | -35 | 1.839 | -0.612 |

-118 | 3.418 | 1.090 | -76 | 5.010 | 0.887 | -34 | 1.585 | -0.108 |

-117 | 3.641 | -0.948 | -75 | 4.308 | 0.385 | -33 | 1.178 | 0.197 |

-116 | 3.226 | 0.230 | -74 | 5.361 | 0.030 | -32 | 0.801 | 0.136 |

-115 | 3.282 | 0.887 | -73 | 3.940 | -0.356 | -31 | 1.021 | 0.078 |

-114 | 3.047 | 0.352 | -72 | 2.890 | -0.279 | -30 | 0.848 | -0.041 |

-113 | 2.765 | -1.060 | -71 | 3.625 | -1.376 | -29 | 0.937 | 0.517 |

-112 | 2.437 | 0.113 | -70 | 3.332 | 1.031 | -28 | 1.194 | 0.053 |

-111 | 3.093 | 0.475 | -69 | 3.655 | -0.721 | -27 | 1.283 | -0.709 |

-110 | 2.407 | -1.587 | -68 | 3.857 | -0.465 | -26 | 1.362 | 0.189 |

-109 | 2.687 | -0.537 | -67 | 3.646 | 1.189 | -25 | 1.455 | 0.681 |

-108 | 2.326 | -0.854 | -66 | 3.611 | 1.787 | -24 | 1.280 | 0.079 |

-107 | 2.722 | 0.021 | -65 | 5.304 | -1.618 | -23 | 1.619 | -0.809 |

-106 | 2.699 | -0.762 | -64 | 4.849 | -1.711 | -22 | 1.901 | -0.018 |

-105 | 2.887 | 0.619 | -63 | 5.160 | 2.407 | -21 | 1.920 | -0.041 |

-104 | 2.168 | -0.414 | -62 | 4.643 | 1.974 | -20 | 2.114 | -0.714 |

-103 | 1.989 | -1.242 | -61 | 4.784 | 2.092 | -19 | 2.042 | 0.052 |

-102 | 1.987 | -0.375 | -60 | 3.804 | -0.861 | -18 | 1.852 | -2.103 |

-101 | 1.714 | -0.198 | -59 | 4.492 | 2.870 | -17 | 1.662 | 1.062 |

-100 | 2.315 | -0.231 | -58 | 4.701 | -2.246 | -16 | 2.310 | -1.014 |

-99 | 2.788 | 0.528 | -57 | 4.721 | 1.669 | -15 | 2.078 | -0.988 |

-98 | 2.855 | -1.024 | -56 | 4.446 | 1.352 | -14 | 2.460 | 2.662 |

-97 | 3.726 | 0.796 | -55 | 3.793 | -1.976 | -13 | 2.594 | -1.405 |

-96 | 2.734 | -0.057 | -54 | 3.833 | 0.022 | -12 | 1.609 | 2.165 |

-95 | 3.482 | -3.851 | -53 | 3.707 | 0.340 | -11 | 1.970 | -0.034 |

-94 | 3.342 | 0.914 | -52 | 3.805 | -5.143 | -10 | 1.776 | 1.260 |

-93 | 2.486 | -3.966 | -51 | 3.507 | 0.202 | -9 | 2.341 | 2.799 |

-92 | 3.455 | -1.853 | -50 | 3.158 | -0.411 | -8 | 2.335 | 1.797 |

-91 | 3.602 | 3.909 | -49 | 2.688 | 0.606 | -7 | 2.868 | 2.224 |

-90 | 4.021 | -3.818 | -48 | 2.308 | 0.169 | -6 | 2.866 | 2.663 |

-89 | 3.927 | -3.043 | -47 | 2.404 | 1.254 | -5 | 2.843 | -2.600 |

-88 | 3.929 | 0.624 | -46 | 2.079 | 0.010 | -4 | 2.380 | 0.403 |

-87 | 4.805 | -2.384 | -45 | 2.000 | 0.030 | -3 | 2.195 | -1.043 |

-86 | 3.857 | -1.463 | -44 | 1.446 | 0.399 | -2 | 2.107 | -2.325 |

-85 | 3.701 | -0.355 | -43 | 1.533 | 0.034 | -1 | 1.789 | -0.238 |

-84 | 3.481 | -5.738 | -42 | 1.412 | -0.498 | 0 | 2.107 | -1.145 |

-83 | 4.617 | -3.076 | -41 | 1.229 | -0.092 |

In this exercise you will perform several coverage backtests.

- Use the data of Exhibit 14.13 to calculate exceedence data
. Save your results, as you will need them again in Exercise 14.11.^{t}i - Use the data of Exhibit 14.13 to construct a graphical backtest similar to Exhibit 14.1.
- Apply our recommended standard coverage test at the .05 significance level using your results from part (a).
- Apply Kupiec’s PF coverage test at the .05 significance level using your results from part (a).
- Apply the Basel Committee’s traffic light coverage test using your results from part (a).

In this exercise, you will perform the graphical and recommended standard distribution tests of Section 14.4 using the data of Exhibit 14.13.

- Our value-at-risk measure is a linear value-at-risk measure that assumes
is conditionally normal with^{t}L^{t}^{–1}*E*() = 0. Use this information and the data of Exhibit 14.13 to calculate loss quantile data^{t}L.^{t}u - Apply the inverse standard normal CDF to your loss quantile data
to obtain values^{t}u. Save your results, as you will need them again in Exercise 14.11.^{t}n - Order your values
by magnitude. Denote the ordered valued^{t}n*n*._{j} - Calculate values as describe in Section 14.4.2.
- Plot of the points (
*n*, ) in a Cartesian plane. Interpret the result._{j} - Calculate the sample correlation between the
*n*and . Does the value-at-risk measure pass or fail our recommended standard distribution test at the .05 significance level?_{j}

In this exercise, you will perform Christoffersen’s exceedences independence test using the data of Exhibit 14.13.

- Retrieve the exceedence
data you calculated for Exercise 14.8, and use it to calculate values α^{t}i_{00}, α_{01}, α_{10}and α_{11}. - Use your results from part (a) and formulas [14.16], [14.17] and [14.18] to calculate values , and .
- Use your results from parts (a) and (b) and [14.20] to calculate the log likelihood ratio –2
*log*(Λ). What conclusion do you draw?

In this exercise, you will perform our recommended standard loss quantile independence test using the data of Exhibit 14.13.

- Retrieve the
data you calculated for Exercise 14.9, and calculate its sample autocorrelations for lags 1 through 5.^{t}n - Take the absolute value of each sample autocorrelation, and then take the maximum of the five results. Based on this, does the value-at-risk measure pass or fail the recommended standard independence test at the .05 significance level?