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 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.
These are plotted in Exhibit 14.10.
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
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.
We calculate the sample autocorrelations of the tn 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.
|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|
In this exercise you will perform several coverage backtests.
- 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.
- 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 VaR measure is a linear VaR 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.
- 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.
- Order your values tn by magnitude. Denote the ordered valued nj.
- Calculate values as describe in Section 14.4.2.
- Plot of the points (nj, ) in a Cartesian plane. Interpret the result.
- Calculate the sample correlation between the nj and . Does the VaR measure pass or fail our recommended standard distribution test at the .05 significance level?
In this exercise, you will perform Christoffersen’s exceedences independence test using the data of Exhibit 14.13.
- Retrieve the exceedence ti data you calculated for Exercise 14.8, and use it to calculate values α00, α01, α10 and α11.
- 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 –2log(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 tn data you calculated for Exercise 14.9, and calculate its sample autocorrelations for lags 1 through 5.
- Take the absolute value of each sample autocorrelation, and then take the maximum of the five results. Based on this, does the VaR measure pass or fail the recommended standard independence test at the .05 significance level?