## BACK TESTING, EXTREME VAR AND STRESS TESTING

Category: Risk Management in Banking

It is important to make a distinction with day-to-day operations VaR and extreme VaR used for capital adequacy purposes. The second form follows the view that VaR is the economic capital that protects the bank from extreme market movements.

**VaR and Economic Capital**

Day-to-day VaR measures sometimes use low confidence levels, such as the 2.5% level. There are several rationales for adopting low confidence levels. Tight confidence levels result in a higher usage of capital and traders reach their limit VaR quicker than with loose VaR measures, thereby limiting the volume of business. At the extreme, since the VaR is a Profit and Loss (P&L), using a tight VaR implies low P&L, eventually inconsistent with the trading goals. Loose confidence levels make sense if we consider that a single trader might exceed risk limits without deteriorating the credit standing of the bank (although there are examples not supporting this view). However, the entire portfolio VaR cannot rely on a loose confidence level, since it represents the default probability of the trading unit. The regulator recommends the 99% level and makes the measure more conservative by using a multiple to obtain capital. Related issues are the VaR reliability measure, and the extreme VaR or stress testing.

**Back Testing**

Back testing aims to check that measures are in line with actual portfolio variations of value. Since the portfolio structure changes constantly, the exercise requires using the crystallized structure of the portfolio, looking for what the actual deviations were and comparing them to VaR measures. This allows us to check that the number of outliers is in line with the confidence level. With a 2.5% confidence level, the number of deviations beyond VaR should be less than 2 or 3 out of 100 observations. Running a number of sets of historical parameter values allows us to obtain the distribution of the portfolio values, and check the number of outliers at constant portfolio structure. The test compares the frequency of outliers with the confidence level.

The number of outliers embeds a random error because it results from sampling past values. Therefore, the test is subject to traditional errors of rejecting a good model and accepting a bad one because of the sampling error. This requires using threshold values of the number of outliers, limiting the probability of such errors.

Other techniques compare the actual distribution of portfolio returns, from which the VaR derives the actual distribution, or compare the volatilities of these two distributions.

**Extreme VaR and Stress Testing**

Stress testing aims to investigate the possibility of exceptional losses by stressing the value of the risk drivers. With a portfolio, it is not simple to identify which deviations of which parameters result in extreme losses because asset value changes are offset within the portfolio.

Extreme VaR measures involve extreme scenarios. Such scenarios might be judgmental and selective. For instance, if the management fears wide variations of selected parameters, these wide variations would serve to revalue the portfolio and see what would be the magnitude of adverse effects. Using the extreme conditions that prevailed historically over a long period is a common way to stress test a portfolio. The idea is to see how it would behave under similar conditions.

There are various practical techniques to explore stressed conditions. A simple one consists of stressing the main risk drivers one at a time. Using a sensitivity analysis allows us to determine which parameters generate the largest changes, and focus on those. Another technique is to select those parameters with the highest values. Cumulating the effects of stressed values of these main parameters might result in worse losses, or losses not depending on the portfolio structure. Losses will increase when all selected parameter deviations alter adversely the values of the assets of the portfolio. Otherwise, they do not.

Extreme VaR techniques differ in that they attempt to model extreme situations, using distributions with fat tails. A common technique uses the Pareto distribution family to fit the fat tail of the distribution. The technique involves smoothing the tail to obtain better estimates of the value percentiles^{12}. It allows us to use a known distribution instead of the modelled one, to determine loss percentiles at low confidence levels without requiring calculation intensive simulations. Extreme value theory also provides some techniques for capturing tail effects.