1. CHAPTER 8 Testing VaR Results to Ensure Proper Risk Measurement

2. INTRODUCTION Up to this point, we have discussed how positions change value, and have used relatively complicated math to calculate the statistics of those changes This has enabled us to construct estimates of the probability distributions of the future losses and therefore estimate VaR Now testing is required to tie the results back to reality and give confidence that VaR is a true measure of the risks

3. INTRODUCTION This is especially important now that the Basel Committee allows banks to use their own VaR models to assess the amount of regulatory capital that they hold for market risks The goal of this chapter is to detail the tests that should be carried out on VaR calculators to ensure their validity

4. VAR-TESTING METHODOLOGIES There are three different types of tests –Software-installation test –Profit-and-Loss (P&L) reconciliation test –Modeled-probability-distribution back-test

5. Back-Testing the Modeled Probability Distribution Back-testing requires many days of data The purpose of this test is to make sure that the probability distribution (e.g., the VaR) is consistent with actual losses Back-testing compares the loss on any given day with the VaR predicted for that day. Figure 8-1 illustrates VaR and the experienced losses over 100 days.

6. Back-Testing the Modeled Probability Distribution

7. The VaR changes slowly from day to day as positions change and as the market volatility changes In 100 trading days, we would expect one exception (as on day 73 in the figure) In a year of 250 trading days, we would expect 2 to 3 exceptions.

8. Back-Testing the Modeled Probability Distribution If it was the case that we always got a representative sample, then we could say that our VaR was a good representation of the actual distribution if we only experience exceptions 1% of the time If we experience exceptions more or less often, we would conclude that the VaR was not an accurate representation of the distribution of losses

9. Back-Testing the Modeled Probability Distribution Unfortunately, there is additional complication because the number of exceptions is in itself a random number Sometimes the bank will be lucky and the random market movements will cause fewer losses than usual; sometimes they will be unlucky and suffer many losses This uncertainty in sampling means that it is difficult to tell whether the experienced number of exceptions is due to a poor model or to bad luck.

10. Back-Testing the Modeled Probability Distribution Fortunately, there is a framework to calculate the probability of having a given number of exceptions The exceptions are a binomial variable Binomial variables are those that can have a value of zero or one Exceptions are binomial because on any given day there either is or is not an exception

11. Back-Testing the Modeled Probability Distribution If the VaR calculator is correct, then on each day there is a 1% chance of an exception and a 99% chance of there being no exception

12. Back-Testing the Modeled Probability Distribution The number of exceptions over 250 days has a Bernoulli distribution The Bernoulli distribution describes the probability of having a given number of outcomes that are equal to one if a binomial variable is sampled multiple times From the Bernoulli distribution, we can calculate the probability of a given number of exceptions occurring, as shown in Table 8-1

13. Back-Testing the Modeled Probability Distribution From this table, we can see that if the VaR calculator is correct, there is a 13% chance of having 4 exceptions in 250 trading days and an 89% chance that there will be 0 to 4 exceptions We can also see that there is only a 0.01% chance of there being 10 or more exceptions.

14. Back-Testing the Modeled Probability Distribution We can interpret this by saying that it is very unlikely to get 10 or more exceptions if the VaR model is correct i.e., if 10 or more exceptions do occur, it is likely that the model is incorrect.

15. Back-Testing the Modeled Probability Distribution

16. This principle is used by the Basel Committee to check that a bank's VaR calculator is performing well If more than 4 exceptions have occurred in the last 250 trading days, the Capital Accords for market risk require that the bank should hold additional capital to compensate for the possible unreliability of the bank's calculator

17. Back-Testing the Modeled Probability Distribution Table 8-2 shows that each number of exceptions puts the calculator into a green, yellow, or red "zone.“ Corresponding to each number of exceptions, there is a multiplier by which the amount of market-risk capital must be increased

19. Back-Testing the Modeled Probability Distribution We investigate capital further in the next chapter Back-testing should not only be carried out for the whole portfolio, but also for subportfolios

20. Assessment A stock portfolio=stock A+ stock B Using parametric VaR method One-year learning window: using one-year (250 trading days) historical data to estimate the parameters, such as variances and correlation Five back-testing periods with 250 trading days for each period

21. Assessment