# 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.

Exhibit 14.8: Backtesting data for a one-day 95% EUR value-at-risk measure compiled over 125 trading days. Value-at-risk (VaR) and P&L values in the second and third columns are expressed in millions of euros. The exceedance column has a value of 1 if the portfolio realized a loss exceeding the 0.95 quantile of loss, as determined by the value-at-risk measure. Otherwise it has a value of 0. The last column indicates the specific quantile of loss for each P&L result, again, as determined by the value-at-risk measure.

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 tu and arrange the results in ascending order to obtain the nj. Values for the  are obtained from [14.11], with α + 1 = 125. Values for nj and  are presented in Exhibit 14.9.

Exhibit 14.9: Values nj and calculated for the data of Exhibit 14.8.

These are plotted in Exhibit 14.10.

Exhibit 14.10: A plot of the points (nj , ) from Exhibit 14.9.

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 nj 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 tI, we use the data of Exhibit 14.8 to calculate α00 = 105, α01 = α10 = 9 and α11 = 1. From these, we calculate = 0.9211,  = 0.9000 and   = 0.9194. Our likelihood ratio is

[14.23]

so –2log(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 tn from the loss quantiles tu. Results are indicated in Exhibit 14.11.

Exhibit 14.11: Example values tn 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.

We calculate the sample autocorrelations of the tn for lags 1 through 5 as indicated in Exhibit 14.12.

Exhibit 14.12: Example sample autocorrelations for use with our basic independence test.

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
Exhibit 14.13 below presents 125 days of performance data for a one-day 99% USD value-at-risk measure for use in Exercises 14.8 through 14.11. For convenience, you should be able to cut and paste it from this webpage into a spreadsheet.
time99% VaRP&Ltime99% VaRP&Ltime99% VaRP&L
tat t – 1at ttat t – 1at ttat t – 1at t
-1243.468-2.107-824.6930.252-401.4010.683
-1233.095-0.143-813.789-0.074-391.2820.241
-1223.2450.894-804.897-0.153-381.5240.118
-1212.9690.990-794.2560.267-371.834-0.810
-1203.472-0.060-784.5371.804-361.534-0.455
-1194.513-1.123-774.508-0.196-351.839-0.612
-1183.4181.090-765.0100.887-341.585-0.108
-1173.641-0.948-754.3080.385-331.1780.197
-1163.2260.230-745.3610.030-320.8010.136
-1153.2820.887-733.940-0.356-311.0210.078
-1143.0470.352-722.890-0.279-300.848-0.041
-1132.765-1.060-713.625-1.376-290.9370.517
-1122.4370.113-703.3321.031-281.1940.053
-1113.0930.475-693.655-0.721-271.283-0.709
-1102.407-1.587-683.857-0.465-261.3620.189
-1092.687-0.537-673.6461.189-251.4550.681
-1082.326-0.854-663.6111.787-241.2800.079
-1072.7220.021-655.304-1.618-231.619-0.809
-1062.699-0.762-644.849-1.711-221.901-0.018
-1052.8870.619-635.1602.407-211.920-0.041
-1042.168-0.414-624.6431.974-202.114-0.714
-1031.989-1.242-614.7842.092-192.0420.052
-1021.987-0.375-603.804-0.861-181.852-2.103
-1011.714-0.198-594.4922.870-171.6621.062
-1002.315-0.231-584.701-2.246-162.310-1.014
-992.7880.528-574.7211.669-152.078-0.988
-982.855-1.024-564.4461.352-142.4602.662
-973.7260.796-553.793-1.976-132.594-1.405
-962.734-0.057-543.8330.022-121.6092.165
-953.482-3.851-533.7070.340-111.970-0.034
-943.3420.914-523.805-5.143-101.7761.260
-932.486-3.966-513.5070.202-92.3412.799
-923.455-1.853-503.158-0.411-82.3351.797
-913.6023.909-492.6880.606-72.8682.224
-904.021-3.818-482.3080.169-62.8662.663
-893.927-3.043-472.4041.254-52.843-2.600
-883.9290.624-462.0790.010-42.3800.403
-874.805-2.384-452.0000.030-32.195-1.043
-863.857-1.463-441.4460.399-22.107-2.325
-853.701-0.355-431.5330.034-11.789-0.238
-843.481-5.738-421.412-0.49802.107-1.145
-834.617-3.076-411.229-0.092
Exhibit 14.13: Daily performance data (in USD millions) for a one-day 99% USD value-at-risk measure for use in Exercises 14.8 through 14.11.
14.8

In this exercise you will perform several coverage backtests.

1. Use the data of Exhibit 14.13 to calculate exceedence data ti. Save your results, as you will need them again in Exercise 14.11.
2. Use the data of Exhibit 14.13 to construct a graphical backtest similar to Exhibit 14.1.
3. Apply our recommended standard coverage test at the .05 significance level using your results from part (a).
4. Apply Kupiec’s PF coverage test at the .05 significance level using your results from part (a).
5. Apply the Basel Committee’s traffic light coverage test using your results from part (a).

14.9

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

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

14.10

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

1. Retrieve the exceedence ti data you calculated for Exercise 14.8, and use it to calculate values α00,  α01,  α10 and  α11.
2. Use your results from part (a) and formulas [14.16], [14.17] and [14.18] to calculate values  and  .
3. Use your results from parts (a) and (b) and [14.20] to calculate the log likelihood ratio –2log(Λ). What conclusion do you draw?

14.11

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

1. Retrieve the tn data you calculated for Exercise 14.9, and calculate its sample autocorrelations for lags 1 through 5.
2. 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?