Stress testing the trading book

The desk's one-day 99% VaR is £1.94m. Its worst day in three years of history lost £4.98m, and its worst run of ten days £7.44m. VaR says how often a bad day comes; it says nothing about how bad the worst ones can be, or about the day the desk's hedges stop working. Stress tests answer those questions, and banks run them routinely on every trading desk.

Historical: replay the stressed spell

A historical scenarioA stress test that replays the factor moves of a real past episode on today's positions. takes the factor moves of a real episode and applies them to today's book. Its strength is that nobody had to imagine it: the correlations really happened. The desk's history has a stressed spell from day 480 to day 539.

Three things stand out. On day 486, a 6% equity fall with volatility up 7 points, the puts earned £2.78m and cut the loss to £2.34m. Over the worst ten days, the sensitivity estimate (−£16.18m) is more than twice the true loss: with the UK index down 22%, the puts' gamma dominates, so full revaluation is not optional for a book with options. And the whole 60-day spell, applied in one go, makes £6.18m, because rates fell more than 240bp and the gilt and the puts gained more than equities lost. Replaying a long window can net out its worst stretch; replay the worst sub-periods too.

Hypothetical: write the story, then the numbers

History holds only what has happened. A hypothetical scenarioA stress test built from a story about what could happen, with each factor's move chosen to fit the story. starts with a story and turns it into a move for every factor, keeping them consistent: if the story is inflation, short rates rise more than long ones, equities and credit suffer and oil rallies. The risk team, the desk and economists agree the stories; the risk team owns the numbers. Guess each one's sign before you look:

The gilt sell-off is the worst, at −£9.01m, and it is the least dramatic story for global markets. It hits the desk's biggest single exposure, the long 10-year gilt, while the pound's fall barely helps. A scenario hurts in proportion to what the book holds, not to how frightening it sounds. The risk-off scenario, by contrast, is cushioned by the gilt rally and the puts; the oil shock makes money on the Brent futures.

Reverse: start from the loss

A reverse stress testA stress test that starts from an unacceptable loss and searches for the scenarios that would cause it. fixes the loss the desk cannot accept, say £10m, and asks which scenarios cause it. Infinitely many do, so pick the most plausible: the one closest to today in Mahalanobis distanceA distance that measures a scenario in standard deviations, allowing for the volatilities and correlations of the factors: √(x′Σ⁻¹x)., which counts standard deviations while allowing for correlations. For a linear book the answer has a closed form (the formal section derives it): move every factor in proportion to its covariance with the desk's P&L.

For the desk, the most likely £10m loss is 12.0 standard deviations away under calm markets, and 4.9 under the stressed spell's covariance. The first is effectively impossible in the model; the second is a bad week in a crisis. Read the scenario, not just its distance: UK equities −9.9%, US −10.3%, 2y +20bp, 10y +44bp, GBP/USD +0.8%, Brent −12.9%, vol +5.9 pts, IG +17bp. Equities fall and gilt yields rise. The desk is safe as long as government bonds rally when stocks fall; the reverse stress test points straight at that assumption.

The most likely scenario. Let $x$ be the vector of factor moves, $\Sigma$ their covariance matrix and $\delta$ the desk's sensitivities, so the linear P&L is $\delta^\top x$. Among all scenarios that lose $L$, find the one with the smallest Mahalanobis distance. Use a Lagrange multiplier, as in L3.1.07: Mean-variance optimisation:

Each factor moves in proportion to $(\Sigma\delta)_i$, its covariance with the desk's P&L, the same vector that gives marginal VaR in L3.2.09: Parametric VaR for a portfolio. The distance is the loss divided by the P&L's standard deviation: the same number of sigmas as a one-factor problem.

Back to full revaluation. The book is not linear, so scale the scenario, $x = k\,x^{*}$, until the full revaluation loses exactly $L$. A book that is long options (positive gamma) needs $k > 1$; the second-order term from L3.2.06: Option Greeks across a book, $\tfrac12\,x^\top\Gamma x$, works in its favour. A full non-linear search over all directions is a job for an optimiser; scaling the linear answer is the usual first step.

Start from any preset, then move one factor at a time. Try the gilt sell-off with the 10-year rate set to zero: most of the loss disappears. Then take the risk-off scenario and push equities to −30%: watch the puts' bar grow faster than the futures' and the gap between full revaluation and the sensitivity tick widen.

Banks that use internal models for market risk have been required to run regular, rigorous stress tests since the 1996 Market Risk Amendment. The Basel Committee's Stress testing principles (October 2018) ask for scenarios that are severe but plausible, cover the firm's material risks, are reviewed and challenged, and feed into decisions, and they name reverse stress testing as one of the tools. The trading book also carries a stressed VaR or stressed expected shortfall, measured over a period of significant stress (L2.5.07: Historical VaR). Supervisory stress tests run by the Bank of England, the EBA and the Federal Reserve are in M4.9: Supervision, stress testing and disclosure.

• Stress tests fix a scenario and ask for the loss; VaR fixes a probability and asks for the loss. Desks need both.
• Historical scenarios replay real moves on today's positions; take the worst sub-periods, not only whole episodes.
• Hypothetical scenarios start from a story and move every factor consistently; the worst is the one that hits what the book holds.
• Revalue in full: on the worst 10 days the sensitivity estimate is more than twice the true loss because of the puts.
• The most likely scenario for a loss L is x* = −LΣδ/(δ′Σδ), L/σ standard deviations away. Read it for the assumption it breaks.

Historical scenario

A stress test that replays the factor moves of a real past episode on today's positions.

Hypothetical scenario

A stress test built from a story about what could happen, with each factor's move chosen to fit the story.

Reverse stress test

A stress test that starts from an unacceptable loss and searches for the scenarios that would cause it.

Mahalanobis distance

A distance that measures a scenario in standard deviations, allowing for the volatilities and correlations of the factors: √(x′Σ⁻¹x).

Full revaluation

Pricing every position again at the stressed market levels, rather than multiplying sensitivities by the moves.

Instantaneous shock

A shock applied in one go to today's positions, with no time for the desk to trade or hedge.