VaR Exceptions in the Ashcombe Rates Portfolio
Ashcombe Asset Management runs a £250m portfolio of UK gilts and high-quality corporate bonds. The portfolio's modified duration is 6.2 and its convexity is 45. Its internal market risk model is a one-day 99% VaR of £6.6m, estimated from 500 days of equally weighted historical returns. Over the last 250 business days the model produced four exceptions, three of them concentrated in a single fortnight in June when a surprise inflation print moved the curve. The front office argues the model is sound because four exceptions is within the accepted green zone, and that the June losses were a one-off. The CRO is uneasy: the portfolio has never been tested against a 100 basis point parallel shift, the VaR limit has been raised once in the year, and the model has not been independently validated for two years. The risk committee wants a view on whether the model remains fit for purpose and what to do before the next regulatory capital submission.
1. Calculate the DV01 and the estimated profit or loss from a 100 basis point parallel rise in yields, explaining the role of convexity and the limitations of the estimate.
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DV01. The change in value for a one basis point move is: £250m × 6.2 × 0.0001 = £155,000 per basis point.
100bp parallel rise. Duration alone gives: £250m × 6.2 × 0.01 = £15.5m, or 6.2% of the portfolio.
Convexity adjustment. The fuller approximation is ΔP/P ≈ −D×Δy + ½×C×(Δy)². For a 100bp rise:
- Duration term: −6.2 × 0.01 = −0.062
- Convexity term: ½ × 45 × 0.0001 = +0.00225
- Net: −0.05975, i.e. about −£14.94m, a loss roughly £0.56m smaller than the duration-only figure.
Convexity works in the holder's favour for large moves in either direction - it cushions losses when yields rise and adds gains when they fall - which is why a purely linear duration estimate overstates losses on a rate rise.
Limitations. The approximation is second-order and degrades for very large shocks. More importantly, it assumes a parallel shift; a steepening or flattening, or a move concentrated at the short end, produces a different P&L, which key-rate durations or full revaluation would capture. Credit spread moves are not modelled at all, and the corporate holdings carry spread duration on top of rate duration. The 100bp scenario is also not a severe stress by historical standards; the committee should see 200bp and curve-twist scenarios alongside it.
2. Interpret the backtesting result. Is four exceptions in 250 days evidence that the model is inadequate? Explain your statistical reasoning.
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Start with the statistic. At 99% one-day VaR, the expected number of exceptions over 250 days is 2.5. Observing four is entirely plausible under a correctly calibrated model: the binomial probability of four or more exceptions is roughly 24%. Four or fewer is conventionally treated as the green zone, so the count alone does not establish model failure and does not trigger an automatic multiplier. The front office is right on that narrow point.
But the count is not the whole test. Three features undermine the model:
- Clustering. Exceptions are assumed to be independent. Three in a single fortnight is not consistent with independence; it indicates the model reacts too slowly to volatility regime shifts. An equally weighted 500-day history gives a recent shock roughly 0.2% weight, so the model is still calibrated to a calmer past precisely when volatility has risen.
- Boundary position. Four is the top of the green zone. One more exception moves the model into the yellow zone and attracts a scaling factor; the CRO should treat the current position as a warning, not a pass.
- Governance evidence. A limit raised once in the year, no independent validation for two years, and no stress testing all point to a model whose limits are being adjusted to fit outcomes rather than the reverse.
Conclusion. The model is not demonstrably broken, but it is not demonstrably fit for purpose either. It is unvalidated, untested against plausible severe scenarios, and its historical weighting is structurally slow. The committee should not accept 'within the green zone' as an answer.
3. Recommend actions to the risk committee ahead of the regulatory capital submission, prioritising them and explaining the reasoning.
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1. Commission an independent validation immediately. Two years without validation is a governance failure independent of the backtesting result. The validation should cover data, methodology, the historical weighting choice, the one-day horizon (capital uses a ten-day holding period, so the scaling assumption must be tested), and P&L attribution for each exception.
2. Investigate the June exceptions specifically. Determine whether they arose from a genuine volatility regime shift, from curve movements the model under-represents, or from desk activity outside the model (new positions, spread exposure). The answer determines whether the fix is recalibration or re-scoping.
3. Test alternative weightings. Compare equally weighted, exponentially weighted and a conditional-volatility approach. If EWMA materially improves the exception clustering without inflating VaR unreasonably, adopt it with the validation evidence.
4. Introduce stress testing and scenario analysis now. Run 100bp and 200bp parallel shifts, curve steepeners and flatteners, a spread-widening scenario, and a historical replay of a high-volatility episode. Stress results should be reported alongside VaR, not instead of it, and should inform limits.
5. Set explicit triggers and report. Define an escalation rule - for example, mandatory committee notification at five exceptions in 250 days, and automatic review at any cluster of two or more in ten days. Report VaR, exceptions, stress results and limit utilisation monthly.
6. Review the limit increase. Require that any future VaR limit rise is justified by portfolio change, not by limit utilisation, and approved with the validation evidence attached. For the capital submission, assume the model may attract a supervisory add-on and plan for it.