Value at Risk vs. Stress TestingHard

One asks how bad an ordinary bad day is and answers with a probability. The other asks what a specific catastrophe would cost and refuses to say how likely it is. Neither is the other's approximation.

5 min read · 897 words

Two questions that sound like one

• Value at risk asks a question about the distribution. Over one day, at 99% confidence, what loss will not be exceeded? The answer is a number with a probability attached, estimated from history or from a model of it.
• A stress test asks a question about a case. If rates rose 200 basis points while spreads widened 150 and equities fell a fifth, what would we lose? The answer is a number with no probability attached, and that absence is deliberate.
• They are not two precisions of the same measurement. One describes the middle of the distribution with statistical machinery; the other describes one specific corner of the tail with a story. Risk measures covers the first, stress testing the second.

What each one is silent about

• Value at risk says nothing about the size of the losses beyond it. A 99% one-day figure is the threshold of the worst one day in a hundred; on the day it is exceeded it makes no claim at all about by how much. That is not a flaw to be corrected but the definition, and it is why expected shortfall — the average loss given that the threshold is breached — exists beside it.
• A stress test says nothing about likelihood. Attaching a probability to a scenario that has never occurred would be the least defensible number in the exercise, so none is attached — which means a stress loss can never be compared with a value-at-risk figure as though they were the same kind of quantity.
• The most common error is exactly that comparison: reading a scenario loss as "our 99.9% figure". It is not a percentile of anything. It is one path, priced.

Where the arithmetic actually differs

• A scenario adds. It says both things happen, so the losses sum, and there is no diversification because none was assumed.
• A distribution-based measure asks how often they happen together, and below a correlation of one that returns a smaller number for the same two positions. The gap between the two figures is the correlation assumption, priced.
• Which is why a scenario loss is normally the larger number, and why comparing them tells you something useful: the difference is exactly what the firm is assuming about co-movement, made visible. The calculator on the stress-testing page moves both.
• And correlation is the parameter least stable under stress, estimated from calm periods and drifting towards one when it matters — see diversification.

How each fails

• Value at risk fails by being calibrated on a sample that does not contain the event. A model fitted to two quiet years has no view on a shock it never saw, and it says so nowhere in its output. Backtesting — counting how often the threshold was actually breached — is the honest check, and a model breaching more often than its confidence level allows is telling you something.
• A stress test fails by testing what somebody thought of. The search is bounded by memory and by what is defensible in a meeting, so the scenario library systematically excludes the scenario nobody imagined. Reverse stress testing — start from failure, work backwards — exists precisely to search where the library does not.
• Both fail together on the second round. Forced selling, funding withdrawal, liquidity evaporating in the same instant: a naive version of either revalues today's positions under new inputs and stops, and the losses that turn a bad quarter into a failure are in what happens next.
• Every case study on this site is one of these two failures — LTCM the first, Archegos and LDI the second.

The comparison

Why a firm needs both, said precisely

• They are complements rather than a primary and a sanity check. One is continuous, comparable across desks and testable against what actually happened; the other reaches places the first cannot see and cannot be tested at all.
• A firm with only value at risk has a number every day and no view on the event that would end it.
• A firm with only stress tests has several vivid stories and no way to set a limit, aggregate across desks, or discover that its model has been wrong for six months.
• And both should be quoted with their conditions attached. A stress number stripped of its assumptions becomes a forecast; a value-at-risk figure stripped of its confidence level and horizon means nothing at all.