Core Path · L2.5.09

Stress testing

10 minMarket riskStress testing

By the end you will be able to

  1. 1Tell sensitivity stresses, scenario stresses and reverse stress tests apart.
  2. 2Explain how stress testing complements VaR and expected shortfall.
  3. 3Run a simple reverse stress test: find the move that breaks a portfolio.
  4. 4Describe what supervisory stress tests do, and what they cannot do.

Why it matters

The fund's 99% one-day VaR is $148,044. Its worst day in our data lost $412,600. But the board's real question is different: what would it take to lose $2,000,000, a fifth of the fund, and could it happen?

Core explanation

Severe, but plausible

VaR and expected shortfall (L2.5.06: Value at risk: the idea, L2.5.08: Expected shortfall) describe the bad days a model expects. Stress testing asks about the days it does not: a shock severe enough to threaten the firm, but plausible enough to take seriously. It comes in three forms:

Sensitivity stress

One factor, a long way

Equities −30%. Yields +300 bp. Everything else fixed. Quick, and shows which factor hurts most.

Scenario stress

A story, many factors

The scenario analysis of L2.5.05, made severe: a 2008 replay or an inflation shock, with all factors moving together.

Reverse stress test

Start from failure

Fix the outcome that would break you, then search for the scenarios that cause it. Then ask how plausible they are.

Sort itWhich kind of stress test?

5 items

  1. Move every yield curve up 200 bp in parallel and revalue the bond book.

  2. Replay the autumn of 2008: equities, credit spreads, rates and currencies as they moved then.

  3. Find the combination of market moves that would cut the bank's capital ratio below its minimum.

  4. A recession: unemployment up 5 points, house prices down 25%, equities down 40%, over three years.

  5. Ask which client defaults, alone or together, would exhaust the clearing house's default fund.

Sensitivity stresses on the fund. The index alone, with option volatility rising 1 point per 1% fall, without and with the 12 puts from L2.5.04: Sensitivities:

IndexVolWithout putsWith puts
−10%+10−$600,000−$232,318
−20%+20−$1,200,000−$325,837
−30%+30−$1,800,000−$372,709
−40%+40−$2,400,000−$400,053

And yields alone, on the bond fund: +100 bp −$231,455, +200 bp −$446,591, +300 bp −$646,715. The puts turn a straight-line equity loss into a capped one. That changes where the danger lies, as the reverse stress test below shows.

Firms, not just portfolios. For a bank or insurer, a stress test runs over months or years: losses on loans and trading, lower income, higher funding costs, and the capital ratio at each step. Supervisors now run such tests across whole banking systems (see the regulatory lens).

The formal version

A scenario stress computes, with full revaluation as in L2.5.05: Scenario analysis, the loss L(Δx)=−∑i[Vi(x+Δx)−Vi(x)]L(\Delta\mathbf{x}) = -\sum_i \big[V_i(\mathbf{x}+\Delta\mathbf{x}) - V_i(\mathbf{x})\big]. A reverse stress test turns this around. For a breaking-point loss L∗L^{*}, it looks for the most plausible scenario that reaches it:

find Δx with L(Δx)≥L∗, as plausible as possible\text{find } \Delta\mathbf{x} \text{ with } L(\Delta\mathbf{x}) \ge L^{*}, \text{ as plausible as possible}

With many factors there are infinitely many such scenarios. A common simplification, used here, fixes the direction of travel (how rates and volatility move with equities) and searches along that path for the smallest equity fall ee that reaches L∗L^{*}. Because the loss is not always monotonic with options in the book, the search first scans the path, then narrows the crossing by bisection.

Conventions on this page

Shocks are instantaneous on today's positions, revalued in full: the bond fund as a 7-year 4% bond, the puts by Black-Scholes with three months left. Option volatility rises 1 point for each 1% fall in the index, up to 40 points. Yields move in parallel. A loss is a positive amount; P&L is shown signed.

Try it

The tool opens on the worked example: a $2,000,000 breaking point, rates unchanged, no hedge. Change the path: under an inflation shock the fund breaks sooner; in a flight to quality, later. Then switch the hedge on and see the breaking point vanish.

Try itReverse stress test

As equities fall

$0m$1m$2m$3m0%−10%−20%−30%−40%−50%−60%breaking point

Curve: the fund's loss as the index falls, with yields do not move and option volatility up 1 point per 1% fall (at most 40). Full revaluation; instantaneous.

Index fall that breaks the fund
−33.3%
Yields0 bp
Volatility+33.3 pts
Equity index fund−$2,000,000
Bond fund$0
Total−$2,000,000

Worked example

What would break the fund?

Illustrative dataThe Module 2.5 data set

500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.

The board sets the breaking point at a 20% loss, $2,000,000: beyond it, investors would pull their money. Try each step before revealing it.

A reverse stress test

Step 1 of 5
  1. 1

    Rates unchanged, no hedge

    Only the equity fund loses: $6,000,000×e=−$2,000,000\$6{,}000{,}000 \times e = -\$2{,}000{,}000, so e=−33.3%e = -33.3\%. The bond fund is unchanged ($0).

Regulatory lens

Stress tests are now central to bank supervision. In the US, the 2009 Supervisory Capital Assessment Program (SCAP) led to annual tests under the Dodd-Frank Act (DFAST) and the Federal Reserve's Comprehensive Capital Analysis and Review (CCAR): banks project losses, income and capital over nine quarters of a severely adverse scenario, and since 2020 the result sets each large bank's stress capital buffer. The European Banking Authority runs an EU-wide test, and the Bank of England has run concurrent stress tests of the largest UK banks since 2014. Each compares the capital ratio at the worst point with a hurdle rate. The Basel Committee's Stress testing principles (October 2018, replacing its May 2009 principles) set the expectations, and UK and EU rules also require reverse stress tests. The detail is in M4.9: Supervision, stress testing and disclosure, part of Level 4: Regulation, Capital & Accounting.

Sparks

SparkThe test that calmed a crisis
In May 2009 US supervisors published the results of SCAP for the 19 largest bank holding companies. Ten were told to raise a total of $74.6 billion of capital. Publishing bank-by-bank results was a gamble; it worked, because markets finally had numbers they could trust, and the banks raised the money.
SparkPassed in July, rescued in October
Dexia, a Franco-Belgian bank, passed the European Banking Authority's stress test in July 2011 with room to spare. In October 2011 it needed a state rescue and was broken up. The test had focused on capital; Dexia ran out of funding as the euro area sovereign crisis deepened.

Key takeaways

  • Stress tests ask whether you survive severe but plausible events, without needing a probability.
  • Sensitivity stresses move one factor; scenario stresses move many together; reverse stress tests start from failure.
  • A reverse stress test often reveals the assumption you rely on, such as a hedge being in place.
  • Supervisory tests (CCAR/DFAST, EBA, Bank of England) project capital over years against a hurdle rate.
  • A stress test covers only what its scenario includes: liquidity and risks left out can still break a firm.

Reaction

Check what stuck

Question 1 · objective 3

A fund holds $5 million of an equity index fund and $3 million of bonds with a DV01 of $1,500. A loss of $1.5 million would force it to close. With yields unchanged, how far must the index fall, in per cent, to cause that loss?

%

Question 2 · objective 3

Same fund, but now yields rise 100 bp at the same time. How far must the index fall now, in per cent?

%

Question 3 · objective 1

A bank asks: “Which combination of events would make us fail our minimum capital requirement?” What kind of test is this?

Question 4 · objective 4

Dexia passed the EBA's July 2011 stress test and needed rescue three months later. What is the main lesson?

Question 5 · objective 2

The fund's 99% one-day VaR is $148,044, yet its stress tests look at losses of $2,000,000. Why use both?

Glossary

Stress test
Applying a severe but plausible shock to a portfolio or a firm to see whether it survives.
Sensitivity stress
A stress that moves one risk factor a long way, holding the others fixed.
Scenario stress
A stress that moves several risk factors together, following a story (historical or hypothetical).
Reverse stress test
A stress test run backwards: start from an outcome that would break the firm and find the scenarios that cause it.
Supervisory stress test
A stress test designed and run by a regulator across many banks at once, with a common scenario over several years.
Hurdle rate
The minimum capital ratio a bank must keep even at the worst point of the stress scenario.

Deep Dives & Origins

Deep Dives

  • Basel Committee on Banking Supervision, Stress testing principles, October 2018. Introductory.
  • T. Schuermann, “Stress testing banks”, International Journal of Forecasting 30(3), 2014, 717–728. Intermediate.

Origins

  • Board of Governors of the Federal Reserve System, The Supervisory Capital Assessment Program: Overview of Results, 7 May 2009.
  • Basel Committee on Banking Supervision, Principles for sound stress testing practices and supervision, May 2009.
  • European Banking Authority, 2011 EU-wide stress test aggregate report, July 2011.

Module complete

You have finished M2.5: First risk measures. Try the Lab Test for M2.5: First risk measures →

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Expected shortfall

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Four responses to risk