P&L distribution is based on only a few points, in which case VaR estimates will not be robust to changes in the data. Readers should see this for themselves by choosing only a short historical period in the historical VaR workbook. It may be too long if there were many very extreme returns in the risk factors a long time ago which have not been repeated during the more recent past.

When attempting to validate a VaR model the risk control function will need to examine many VaR tables, along the lines of that shown in Table 9.1, where the model parameter assumptions are displayed with the table. So for each VaR table the type of covariance matrix should be clearly stated, as should the type of cash-flow maps or factor models if these are employed. In historical VaR models the historical look back period should be stated, and in Monte Carlo VaR models the number of simulations, and any advanced sampling techniques should also be mentioned.

It is the role of risk control to investigate how robust the VaR estimates are to different choices of data and parameters

It is the role of risk control to investigate how robust the VaR estimates are to different choices of data and parameters. This can be achieved using a simple visual analysis - for example, a plot of historical VaR estimates as the historical data period is increased, or a series of covariance VaR plots as certain correlations are varied (cf. Figure 5.4). Since so many assumptions will need to be made about the values of the VaR model parameters this type of sensitivity analysis is a crucial part of model development. Even if the VaR model appears to perform well in backtests, if it does not prove to be robust to small changes in parameter assumptions there is a good chance that future backtesting results will be poor, and the model will be moved into a different Basel zone.

9.6 Scenario Analysis and Stress Testing

The final part of this chapter will examine how the risk characteristics of portfolios can be assessed through scenario analysis and stress testing. Scenario analysis examines the value of a portfolio as the underlying risk factors are perturbed from their current values. Stress testing is really a part of scenario analysis, but instead of considering the sort of perturbations that are expected in normal market circumstances, one looks at the portfolio value when risk factors are moved to extreme positions.

There is much to be said for evaluating portfolios using scenario analysis. Scenario-based methods such as the mark-to-future framework (Dembo, 2000; Dembo et al., 2000) do not depend on distributional assumptions and can incorporate the path-dependent behaviour of any type of security.21 It is therefore not surprising that many banks prefer to quantify their MRR using scenario-based calculations rather than internal VaR models.

21 More details are available from www.mark-to-future.com.

9.6.1 Scenario Analysis

As if the building of a historic database on asset prices and risk factors that is A functionai n

updated in real time is not enough of a challenge, a fully functional risk management capability

management capability needs to provide a library of scenarios for use by needs to provide a

traders and managers. The role of this library should be to allow managers to library of scenarios for

assess all risk characteristics of existing positions and to allow traders to use by traders and

examine the risk and return of potential trades under various scenarios. The managers library should consist of

covariance matrices (so that the portfolios can be stress-tested for extreme market conditions as described in §9.6.3);

joint distributions on risk factors and volatilities (which are used to

compute the expected loss of a portfolio, as described in §9.6.2);

smile surfaces, including those that are particularly relevant to current

market conditions (these can be generated using the method outlined in

§6.3).

The 1996 Amendment to the Basel Accord laid down various guidelines if MRR is being assessed via scenario analysis.22 Regulators require that portfolios are valued on a grid that is defined according to both country and asset class. The scenarios are for underlying risk factors and risk factor volatilities, so the grid may be envisaged in two dimensions, as shown in Figure 9.9.

Figure 9.9 shows the type of grid that could be used to revalue the net UK equity option positions. The underlying risk factors are the FTSE 100 index and the at-the-money FTSE 100 implied volatility. Their current values are

22 These scenarios are much less comprehensive that those that should be considered by internal risk management.

taken as the origin of coordinates and the boundary of the grid is defined as ±8% for the value of the FTSE 100 and ±25% for ATM volatility. Thus if the FTSE index is currently at 6000, and ATM implied volatility is currently 20%, the boundaries are set at (5520, 6480) for the index and (15%, 25%) for the ATM volatility. The volatility shifts are imposed by the regulators and applied to the volatility term structure; large shifts (±30%) are taken for short maturities and smaller shifts are taken for longer maturities (e.g. ±8% for 1 year and ±3% for 5 years).

A fairly coarse grid is defined in this range, say at intervals of 100 points for the index and 1% for the implied volatility. In fact usually the grid will contain only 7x3 points (seven divisions for the underlying and three divisions for the implied volatility). The regulator does not impose more points but will want some justification that the maximum loss found on the grid cannot be too far from the actual maximum loss. Therefore the portfolio is revalued at each point on this grid and then potential hot-spots are discovered, where the portfolio makes large losses; the grid is then refined about these points and the portfolio is revalued in these regions using the finer grid. Eventually the maximum loss of the portfolio over all possible scenarios is recorded.23

If all positions are cash or futures there is no need to revalue the portfolio over a grid or to account for implied volatilities, because in this case the portfolio value will be a linear function of the underlying and the maximum loss will occur at one or other extreme. Interest-rate-dependent products are treated in this way. They are banded into maturity buckets, and the net positions in each bucket are subject to yield variations from ±100 basis points for the short buckets going down to ±60 basis points for the long buckets. If the maximum loss occurs at +100 basis points for the 6-9-month bucket and -100 basis points in the 9-12-month bucket, this is a highly unlikely scenario, so there is also some netting across adjacent buckets. The rules for interest-rate-dependent products are quite complex and the interested reader should consult the current FSA Guidelines for Banking Supervision,24 or equivalent, for further details.

This type of scenario analysis is performed for all asset types and all countries. There is netting within each asset class but not across countries, and the MRR is then the sum of all the maximum losses that are recorded from each scenario analysis.

9.6.2 Probabilistic Scenario Analysis

If a portfolio is properly hedged then the maximum loss over a scenario grid will occur at a very unlikely scenario. Suppose the maximum loss occurs when

Useful software for performing this type of analysis is available from www.fea.com. See www.fsa.gov.uk/pubs/supervisor.