An intermediate output of any VaR or ES application is a set of simulated P&L vectors at trade level. VaR applications will typically allow the user to aggregate P&Ls along various dimensions i.e. desk/business/asset class levels in addition to reporting the overall portfolio VaR. VaR reported at any sub-level, but typically at individual trade level, is known as Standalone VaR and represents the undiversified risk coming from the trade positions at that level. Unlike P&L, standalone VaRs will not sum to the VaR of the overall portfolio as the diversification benefit due to hedging of positions is not captured. In fact, the difference between sum of standalone VaRs, possibly at the business level, and the overall portfolio VaR may be a measure of diversification benefit.
An alternative trade level metric which takes diversification into account, and whose sum is the portfolio VaR is known as Component-VaR (CVaR). Unlike VaR, CVaR can be positive and negative, and the sign reflects the offsetting nature of trades/asset classes within a portfolio hedge each other. For example, equity and rates portfolios will have CVaRs of opposite sign typically. A commonly used approach to CVaR calculation is the kernel estimation methodology described in . There are some caveats to be considered in using CVaR: CVaR is a mathematical construct – it does not tell you the contribution a trade has to the overall VaR, rather it is an indicator to the direction of hedging and the relative contribution to VaR hedging benefit from a trade. CVaR tends to be useful mostly for trades associated with risk factors which are the driving factors behind a portfolio’s P&L. CVaR from trades associated with other risk factors tend to be noisy and so difficult to interpret. Finally, due to the requirement that sum of CVaRs is equal to the overall portfolio VaR a scaling factor is used in the CVaR calculation. In cases where a portfolio is well hedged this scaling factor can dramatically inflate CVaR values causing the magnitude of the CVaR value to lose any economic interpretation. For example, in an extreme situation, a portfolio that is almost perfectly hedged can have CVaRs approaching infinity. In conclusion, CVaR figures should be taken as indicators and not be used as an input to risk limits or inputs to other models.
Incremental VaR is simply the difference in portfolio VaR with and without a given trade. Like VaR, the sum of incremental VaRs does not sum to the overall VaR. Incremental VaR may be used for pre-trade analysis for example.
Another commonly seen metrics is Stressed VaR. Stressed VaR is simply VaR but calibrated to a period of historical stress. The challenge with stressed VaR is in determining which historical period to use, since current regulatory requirements specify that the period to used is the one that maximises the VaR value. In theory it is not possible to know this without computing VaR on every historical period, and so in practice in industry models are employed to answer this question. Another challenge with stressed VaR is availability of historical data. Note that under FRTB, VaR and Stressed-VaR are replaced with a single Expected Shortfall metric that is calibrated to a period of significant market stress.
Weighted VaR is like VaR except that in simulations a greater probability is assigned to more recent market behaviour. For example, if the market suddenly experiences a volatile period, the simulation of risk factors quickly adjusts resulting in larger simulated shocks. This allows VaR to be reactive and captures real-world features such as volatility clustering but may diminish extreme events further into the past which should still be considered. In addition, having a low weight applying to the most distant historical observation ensures that the removal of that last point in the rolling historical observation windows does not cause a jump in VaR. It is common that either both VaR and weighted VaR are both reported, or a hybrid methodology is used, essentially taking the worst of VaR and weighted VaR.
FRTB introduces the Expected Shortfall (ES) as the standard measure of market risk, replacing VaR and stressed-VaR in the internal model approach (IMA). A benefit of ES is that it is sub-additive i.e. that it will correctly capture diversification benefit. ES is the average of losses beyond a given percentile, and so will capture extreme events that are missed by VaR.
All the above metrics are easily computable for the P&L vector output from any simulation engine. Once P&Ls are available the only remaining decision will be the level (levels) of aggregation. Ideally all VaR at any level of aggregation i.e. trade/desk/business, or at product level or risk factor level should be available. The granularity available will depend on the VaR methodology being used, for example, a sensitivity based P&L will naturally allow P&Ls from certain risk factor types to be easily segregated and the resulting VaR determined, whereas in a full repricing approach, changes in all risk factors are simultaneously used and so risk factor level granularity is not an immediate output. Therefore, risk reporting requirements should be considered at the outset of choosing the VaR methodology.