10-days VaR means 1-day*10^.5 or total of 10 days loss?

[Steve Jobs]

Hi,
In q 57.2, where it says that the bank's 10 day's 99% VaR is 1 million and actual loss exceeded the VaR in 25 out of 1000 observations, does it mean that:

1. Each of the 1000 observation is the [daily loss*10^.5] and the result of this calculation exceeded 1 million in 25 samples?
2. or, that the original sample was 10,000, divided into groups of 10 days and so we have 1000 groups, and 25 groups exceeded 1 million?

 

[David Harper CFA FRM]

Hi @Steve Jobs

Good questions about the methodology, but I think my question (P2.T5.57) deliberately avoids such nuances such that the question itself is not troubled by the several nuances of backtesting. Here's text of my question:

57.2 Basel II requires a backtest of a bank's internal value at risk (VaR) model (IMA). Assume the bank's ten-day 99% VaR is $1 million (minimum of 99% is hard-wired per Basel). The null hypothesis is: the VaR model is accurate. Out of 1,000 observations, 25 exceptions are observed (we saw the actual loss exceed the VaR 25 out of 1000 observations)

The essence of the backtest is: out of a total number of observations, how many exceptions are observed? Is this reasonable or concerning; i.e., green/yellow/red?
("observations" and "exceptions" are the terms actually in the Basel document). This, strictly, can be expressed as a binomial; e.g., we observe (X) exceptions out of total (T) days, where each has a (Bernouilli) probability (p% = 1 - VaR confidence) such that we except P*T. The methodological details are "delegated," as far as I am concerned.

Okay but in regard to Basel's actual backtest methodology: while the VaR is of course a 99.0% confident 10-day VaR, the backtest is only of a daily VaR; i.e., out of 250 days (observations), on how many days was the daily VaR exceeded? Although this helps avoid the challenge of requiring 10 years of 250 non-overlapping 10-day observations, it is actually justified by portfolio composition changes:

" ... comparing the ten-day, 99th percentile risk measures from the internal models capital requirement with actual ten-day trading outcomes would probably not be a meaningful exercise. In particular, in any given ten day period, significant changes in portfolio composition relative to the initial positions are common at major trading institutions. For this reason, the backtesting framework described here involves the use of risk measures calibrated to a one-day holding period. Other than the restrictions mentioned in this paper, the test would be based on how banks model risk internally

There are (several) nuances with respect to both scaling the VaR (i.e., the model) and with computing the actual horizon P/L (i.e., the observations in the estimation sample); e.g., the backtest can be overlapping or non-overlapping. But Basel's one-day backtest basically avoids them!

 

[Steve Jobs]

Thanks David, I'm just a little worried that some one might ask me such questions during job interviews.

 

[David Harper CFA FRM]

@Steve Jobs I understand and candidly I think that is a fantastic perspective, but I'd argue that we already enough to answer some such questions intelligently; i.e., sometimes the smartest answer is "it depends and here are some of the considerations. BTW, here is Basel's approach ... " Forget my defense of my question (but don't miss the value of it, either: the backtest itself, as we study it, is a remarkably simple application of binomial model--which can delegate the particulars much like a code function--that concerns observed exceptions to expected number of exceptions where exceptions are simple yes/no).

From my perspective:

• With respect to the VaR model, common practice is to scale the 1-day to a 10-day VaR with *SQRT(10). The chief problem with this acceptable practice is that it assumes i.i.d.
• With respect to backtesting 10-day VaR, regulatory (Basel) approach is to resort to testing a 1-day VaR. However, if we want to backtest a 10-day VaR, from my perspective, we should test the actual 10-day loss; i.e., the point of backtest is to observe what actually happens, to it doesn't really make sense to scale an actual 1-day loss to 10-days with respect to the observations. Rather, we should observe actual 10-day losses. The chief disadvantages are (i) if non-overlapping, this requires a long estimation sample and (ii) it does not necessarily deal with the portfolio composition issue cited by Basel (above)

 

[Steve Jobs]

Thanks for the points, now I have a better understanding and something to say if I was asked. People will have high expectation if I tell them that I passed the exams; I've been through this in my current career.

Thanks again,