1. To address supervisors' prudential concerns and in order to ensure that the dispersion between the results of different models for a uniform set of positions are confined to a relatively narrow range, banks which use models as a basis for calculating market risk capital will be subject to a number of parameters governing the way in which models are specified. The way in which these parameters are selected and the manner in which the capital charge is to be calculated are discussed in this section. **(a) The ***holding period* for calculating potential changes in the value of the bank's trading portfolio

2. In selecting a holding period over which price changes are measured, a number of considerations need to be balanced. Save in exceptional circumstances, the longer the holding period the greater is the expected price change and consequently the measured risk. Many banks' models used for trading purposes currently employ a one-day holding period for the measurement of potential changes in position values. This approach is not unreasonable in the context of a trading environment under normal market conditions, where trading managers can take day-to-day decisions to adjust risk. For capital purposes, however, it seems prudent to consider potential changes in value over somewhat longer horizons. In large measure, the use of a longer holding period reflects the possibility that markets may become illiquid, preventing market participants from being able to trade out of losing positions quickly. In addition, a longer holding period will take greater account of instruments with non-linear price behaviour, such as options. At the same time, the holding period should not be so long as to be unrealistic in the light of banks' past experience in winding down positions.

3. The market risk proposal of April 1993 envisaged a holding period of at least two weeks in order to guard against the consequences of banks being locked into unprofitable positions. The Committee continues to believe that a two-week holding period is necessary for the reasons explained above. **Thus the Committee has concluded that the holding period used to measure value-at-risk for market risk capital purposes should be two weeks (ten business days), taking the bank's trading positions as fixed for this interval. **In computational terms, the intention is to hold the bank's trading positions fixed and to apply changes in risk factors that are based on movements over ten-day intervals. Nonetheless, except for their options positions, banks would be free to continue to use risk factor changes based on shorter holding periods, as long as the resulting figures are scaled up to a ten-day holding period. For example, the value-at-risk calculated according to a one-day holding period could be scaled-up by the "square root of time" method by multiplying by 3.16 (the square root of ten trading days). However, this extension is not suitable for options for the reasons explained in sub-section (e) below.

**(b) The ***observation period* over which historical changes in prices are monitored and their volatilities and correlations measured

4. The historical sample period (or "time horizon") over which past changes in prices are observed varies among banks according to each bank's general strategy. A bank which wants its model to be responsive to short-term market trends and volatilities may apply a relatively short horizon. Banks wishing to evaluate their risk in the light of a medium-term evolution of volatility may look back over a historical period of several years. The question of data availability is also relevant since for many instruments with relatively short lives, lengthy historical data are non-existent and proxies must be used.

5. At any given point in time the choice of historical sample period can have a significant impact on the size of the estimated value-at-risk produced by an internal model. Short sample periods are more sensitive to recent events than long sample periods, but this very sensitivity means that for a fixed set of positions a short sample period leads to greater variability in the measure of value-at-risk relative to a longer measurement horizon. Although a longer time horizon may sound more conservative, the value-at-risk depends on how rapidly prices have changed in different time periods. If recent price volatility has been high, a measure based on a short horizon could lead to a higher risk measure than a horizon covering a longer but overall less volatile period. The disadvantage of a short time horizon is that it captures only recent "shocks", and it could lead to a very low measure of risk if it coincides with an unusually long stable period in the markets. The disadvantage of a longer time horizon is that it does not respond rapidly to changes in market conditions: in this case, the value-at-risk will react only gradually to periods of high volatility, and the reaction may be small if the period of high volatility is relatively brief.

6. Recognising that different time horizons may legitimately reflect individual banks' assessments of how best to measure their risk under current conditions, the Committee does not feel it would be desirable to impose a fixed historical sample period for all institutions that use the internal models approach. On the other hand, the testing exercise conducted in 1994 indicated that the use of widely different time horizons contributes importantly to the variability in measured value-at-risk that may occur for a given set of positions across banks. **The Committee has concluded that a constraint should be set on banks' choice of time horizon.** Accordingly, banks will at the least be required to apply a minimum historical sample period of one year for calculating value-at-risk (with freedom to opt for longer periods if they so wish).** **The Committee is also reviewing the possibility of requiring banks to calculate value-at-risk according to the higher of two value-at-risk numbers obtained by using two historical sample periods to be determined individually by each bank, one long-term (at least one year) and one short-term (less than one year, with, for example, a difference of at least six months between them). Using a "dual" sample period of this kind would on average introduce an additional layer of conservatism in banks' value-at-risk estimates by capturing short-term volatility, albeit at the cost of a greater processing burden. Comment is invited on the validity and technical feasibility of these two alternatives.

7. The Committee is also aware of the existence of methods that do not weight all past observations equally. While such methods do not initially seem to fit easily within the proposed scheme, the Committee is confident that a way can be found to incorporate such schemes (possibly with modification) into the spirit of the proposed limitation. Comment is specifically invited on possible approaches to this issue.

8. Dispersion of results can also arise from banks' choice of historical data used to observe past price movements. The Committee doubts whether it is practical to seek to steer banks toward uniform data sets, but the data clearly need to be subject to a strong control process. In this context, it is essential that the data be updated and the correlations and volatilities recalculated at frequent intervals. The Committee has decided to set a maximum interval of three months for such recalculation, but banks should also reassess their data sets whenever market conditions are subject to material changes.

**(c) The supervisory*** confidence level *for potential value-at-risk loss amounts

9. In specifying a value-at-risk model, one of the variables that has to be determined is the level of protection judged to be prudent. The confidence intervals used by banks typically range from 90% to 99%. As a prudential matter, the Committee feels it is appropriate to specify a common and relatively conservative confidence level. **It is therefore specifying** **that all banks using the models approach employ a 99% one-tailed confidence interval**. A confidence level of 99% means that there is a 1% probability based on historical experience that the combination of positions in a bank's portfolio would result in a loss higher than the measured value-at-risk.

**(d) Limits on ***aggregation* methods

10. In measuring the risk in a portfolio, it is a standard statistical technique to take account of the fact that the price movements of certain instruments (e.g. debt securities with similar coupons or closely-correlated currency pairs) tend to move together. However, observed correlation among some instruments (e.g. foreign exchange rates and equities), while at times perhaps significant, may be unstable; in unusual market conditions, some of the assumed correlations may break down, occasioning losses that greatly exceed measured risk. The Committee has therefore given careful consideration to the possibility of disallowing certain correlations for the purposes of calculating regulatory capital.

11. This is a complex issue because it is difficult to determine in advance which correlation assumptions are or are not prudent. One correlation assumption is not always more conservative than another. For example, an assumption of independence (i.e. zero correlation)^{4} between interest rates and equity prices may not be conservative if a bank holds long positions in both equities and bonds. In practice, most models calculate the correlations within risk factor categories but differ in their treatment of correlations across broad groups of risk factors.

12. The Committee believes that attempts to stipulate detailed and specific correlation assumptions would be difficult and, for certain portfolio compositions, could lead to an underestimation of risk. However, the disadvantage of relying solely on past historical relationships to determine prudential capital standards is also recognised. Of particular concern is the reliance on historical correlations across broad risk factor categories where the interrelationships of market factors may be more tenuous.** **Given its desire to reduce the potential for dispersion and to address the prudential concerns addressed above, the Committee favours an approach which gives banks flexibility on the use of correlation assumptions but limits correlations across risk factor categories:

**within each risk factor category (e.g. interest rates, foreign exchange rates, equity prices and commodity prices, including related options volatilities in each risk factor category), a bank would have the flexibility to use correlations it deems appropriate, provided that its supervisor is satisfied that the process for calculating correlations is carried out with integrity**;^{5}

**across risk factor categories, value-at-risk numbers should be aggregated on a simple sum basis**.

13. The Committee recognises that this treatment is conservative in that it assumes that the "worst case" outcomes for each risk factor category occur simultaneously. However, of the fifteen major market banks which participated in the 1994 testing exercise, more than half used a simple sum approach to aggregate value-at-risk across risk factors, while the others used either a root-sum-of-squares method or empirical correlations. Clearly, therefore, a common industry practice for the treatment of correlations across risk factor categories has yet to emerge. The simple sum approach is preferred by the Committee to other alternatives (such as the root sum of squares approach) because it does not incorporate correlation assumptions that might prove lenient in the event of severe or prolonged market movements.

**(e) Accurate measurement of ***options *and other instruments that display option-like behaviour

14. Currently, there are differences in the degree to which banks are able to incorporate options risk into their market risk models. Some banks rely on approximations of options price movements, which may fail to take account of the fact that options are non-linear, i.e. their prices do not move proportionately with the price of the underlying. However, a number of large banks are moving towards more sophisticated simulation techniques that would more fully account for non-linear price behaviour. In order to encourage a movement over time to more sophisticated risk measurement techniques, it is important to establish requirements concerning the treatment of options that provide strong incentives for banks to update and to refine their risk measurement systems in this area.

**15. Against this background, the Committee has come to the conclusion that banks' internal risk measurement systems should capture the non-linear behaviour of options prices with respect to changes in underlying rates or prices. At a minimum, banks' internal risk measurement systems should incorporate option price behaviour through a non-linear approximation approach involving higher-order risk factor sensitivities (such as gamma)**. The direct use of options risk management models to calculate all possible changes in option values - which would more fully capture the non-linearity inherent in options positions, at a somewhat higher computational cost - should be considered as a longer term goal for banks' market risk systems.

16. It is also important that banks calculate changes in option values based on movements in underlying risk factors measured on relatively long holding periods, because "scaling up" the value-at-risk figures generated by a one-day holding period assumption would fail to capture non-linearity, which is more pronounced for larger changes in underlying risk factors. The two-week holding period suggested in (a) above seems to be adequate for this purpose. This means that banks would *not* be permitted to scale up by the square root of time their value-at-risk for options positions.

17. In contrast to most other instruments, options values are affected by the volatility of the underlying rates and prices as well as by changes in the level of these factors. As a result, banks' risk measurements systems should evaluate the impact of changes in volatility on option values (vega). In practice, this can be accomplished by modelling volatilities as additional risk factors and including them in the overall set of risk factors affecting the value of the bank's trading positions. Banks with relatively large or complex options portfolios should also measure volatilities across different points along the yield curve.

**(f) Calculation of the capital charge**

18. The Committee has examined carefully how banks' value-at-risk measures based on the parameters described above can be converted into a capital requirement that appropriately reflects the prudential concerns of supervisors. One of the problems of recognising banks' value-at-risk measures as an appropriate capital charge is that the assessments are based on historical data and that, even under a 99% confidence interval, extreme market conditions are excluded. The Committee does not believe that a ten-day value-at-risk measure provides sufficient comfort for the measurement of capital for a number of reasons, which include:

- the past is not always a good guide to the future;

- the assumptions about statistical "normality" built into some models may not be justified, i.e. there may be "fat tails" in the distribution curve;

- the correlations assumed in the model may prove to be incorrect;

- market liquidity may become inadequate to close out positions.

19. Many of the factors listed above are very difficult to quantify. Even if they were capable of quantification, a judgement would still have to be made as to how far it is necessary to guard against rare market occurrences. **The conclusion of the Committee is that supervisors would not have sufficient comfort unless the value-at-risk measure, calculated according to the quantitative standards set out in this section, were to be multiplied by an appropriate factor. **Such a multiplication factor would provide a means of adjusting the value-at-risk numbers (using the parameters set out earlier) generated by banks' internal models to produce an enhanced level of capital coverage against losses that banks might sustain in the event of severe or prolonged market movements.** **The Committee, however, emphasises that the multiplication factor is *not* meant to substitute for regular stress testing (see Section V below) by market participants themselves.

20. The *multiplication factor* will be set by individual supervisors on the basis of their assessment of the quality of the bank's risk management system, subject to an absolute minimum of 3 (although this minimum number may be reviewed in light of additional experience). **The Committee has agreed that banks should be required to add to this factor a "plus" directly related to the ex-post performance of the model, thereby introducing a built-in positive incentive to keep high the predictive quality of the model** (e.g. it could be derived from the outcome of so-called "back-testing" and be zero when such results are satisfactory). More work will be done during, and on the basis of, the consultation to check further the feasibility of the "plus" and to arrive at a more precise definition of it.

21. The question of the appropriate capital charge is also related to the accepted rule that banks' capital requirements should be met on a continuous basis. One of the characteristics of market risk is that it is far more volatile than credit risk. The value-at-risk measure produced by a model will change not only when the bank's positions move, but also when the market moves sharply (especially the risk in the options book). **The Committee recommends that banks should be required to meet, on a daily basis, a capital requirement expressed as the ***higher* of :

** the previous day's value-at-risk number calculated according to the parameters established in sections (a) to (e) above;**

** an average of the value-at-risk measures on each of the last sixty business days, multiplied by the multiplication factor designated by the national supervisor.**

22. Basing the capital requirement on the higher of these two measures has the advantage of placing a lower limit on the capital requirement. When the bank's value-at-risk measure, which can fluctuate on a day-to-day basis, produces a relatively low number on a given day, the sixty-day average multiplied by the multiplication factor effectively becomes the capital requirement, thus imposing a certain level of stability and providing a cushion for potential losses that could arise during periods of greater stress. At the same time, banks must also maintain on a continuous basis a sufficient level of capital to guard against peak levels of value-at-risk, as measured by the previous day's value-at-risk number calculated according to the quantitative standards set out in Part B of the Supplement. Banks therefore also need to evaluate whether the sixty-day average scaled up by the multiplication factor produces a sufficient capital cushion for such potential upsurges in measured value-at-risk over short periods of time.

23. Basing the use of internal models on a series of rigorous qualitative standards and ensuring that these are upheld on a continuous basis through the external validation process should give supervisors comfort about the accuracy of banks' internal models, including the principle that banks conduct back-testing. This is done by comparing ex post the risk measure generated by their internal models against actual daily profits and losses over longer periods of time, as well as looking at hypothetical profits and losses generated by the (end-of-day) portfolio used for the value-at-risk calculation. If supervisors fail to gain sufficient comfort, they may either wish to demand that the model specifications be tightened or may increase the bank's multiplication factor (or, in an extreme case, disallow the model altogether). The supervisors might also wish to compare the results of stress tests with the level of capital produced under the requirements laid down. In any event, they will have a number of means of checking that a bank's model is providing an accurate measure of risk.

**Footnotes:**

4. This is done by using the "square root of the sum of the squares" method to aggregate across risk factor groups.

5.However, as explained in Section V, banks' stress testing ought to include the effect of a breakdown of historical correlations within risk factors as part of its on-going risk management process.