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 n_j. Values for the  are obtained from [14.11], with α + 1 = 125. Values for n_j and  are presented in Exhibit 14.9.

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

These are plotted in Exhibit 14.10.

Exhibit 14.10: A plot of the points (n_j , ) 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 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 ^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.