The Evolution of Stress Testing Counterparty Exposures: formula sheet

Key formulas

Counterparty exposure on default

Exposure = max(V, 0)

V is the net value of the netting set to you. Only positive value is at risk; collateral is then subtracted to get net exposure.

Net exposure after collateral

Net exposure = max(V − C, 0)

C is collateral you hold, after haircuts. In stress, V can rise and C can fall in value or be disputed.

Loss given counterparty default

Loss = Net exposure × LGD

Stress tests shock both exposure and LGD, and consider that LGD may rise with exposure in wrong-way cases.

Rule: lessons to fixes

Weak aggregation → firm-wide view; calm calibration → severe scenarios; ignored wrong-way risk → joint exposure and default stress; static collateral → stressed margin and haircuts

A memory aid, not a formula. Use it to link each crisis failure to the post-crisis expectation.

Current exposure

CE = max(V, 0)

V is the netted mark-to-market value of the trades with the counterparty today. Zero if you owe the counterparty.

Exposure at future date t

E(t) = max(V(t), 0)

V(t) is a random value at date t, so E(t) has a distribution.

Expected exposure

EE(t) = mean of E(t) = average of max(V(t), 0) across scenarios

Average includes zeros from scenarios where V is negative. Do not average only the positive paths.

Potential future exposure

PFE(t) = the α-percentile of E(t), e.g. 95% or 99%

Choose the percentile the question gives. PFE is not an average.

Expected positive exposure

EPE = (1 ÷ T) × ∫ EE(t) dt ≈ Σ EE(tᵢ) × Δtᵢ ÷ T

A time-weighted average of EE. With equal time steps it is the simple average of the EE values.

Effective EE

Effective EE(tₖ) = max(Effective EE(tₖ₋₁), EE(tₖ))

Non-decreasing in time. Captures rollover of short-dated trades.

Effective EPE

Effective EPE = time-average of Effective EE over the first year (or to maturity of the longest trade if under one year)

Basel measure used in exposure at default for internal model method.

Peak exposure

Peak PFE = max over t of PFE(t)

Used for limit monitoring. It can fall at a different date from the peak EE.

Stressed exposure

Stressed exposure = max(Stressed portfolio MtM − Collateral held after haircut, 0)

For a netting set, net the trades first, then subtract collateral. Exposure cannot be negative.

Stressed loss at default

Loss = Stressed exposure × LGD

Use stressed LGD if recoveries or collateral values fall in the scenario.

Stressed expected loss

EL = PD(stressed) × LGD(stressed) × EAD(stressed)

Under wrong-way risk, PD and EAD rise together, so do not treat them as independent.

Collateral after haircut

Collateral value = Market value × (1 − haircut)

Haircuts may widen in stress.

Reverse stress test logic

Fix loss target L*, then find shocks s such that Loss(s) ≥ L*

The outcome is fixed first; the scenario is the output.

Netted exposure

Net exposure = max(Σ Vi, 0) versus Gross exposure = Σ max(Vi, 0)

Valid only for trades in one legally enforceable netting set. Netting never increases exposure.

Netting benefit

Netting benefit = Gross exposure − Net exposure

Ratio forms (net-to-gross ratio = net ÷ gross) are common. A lower ratio means more benefit.

Collateralised exposure

Exposure after collateral = max(V + posted collateral − C, 0)

V is net mark-to-market, C is collateral held after haircuts. Posted collateral, including IM posted, counts only if it is not segregated. If everything you posted is segregated, it is zero and the formula reduces to max(V − C, 0).

Residual exposure over MPOR

Exposure at default ≈ max(V(t + MPOR) + posted collateral − C, 0), where C is collateral held at the last margin call

The key stress lever. Longer MPOR means larger potential price move. Posted collateral is again only the non-segregated part.

Square-root-of-time scaling

Move over MPOR ≈ σ_daily × √(MPOR days)

Rule of thumb. It assumes independent returns and no jumps, so it understates gap risk in stress.

Haircut-adjusted collateral

Collateral value = Market value × (1 − haircut)

Haircut covers collateral price and FX risk over the MPOR. Stress raises haircuts.

Threshold and minimum transfer amount

Uncollateralised exposure can be as high as threshold + minimum transfer amount

Collateral is called only above the threshold, and only in chunks of at least the minimum transfer amount.

Quick revision

• Counterparty credit risk is the risk that a counterparty defaults before final settlement of a transaction's cash flows, causing a loss if the transaction has positive value at default.
• Exposure is floored at zero: you only lose when the contract has positive value to you.
• EE is the mean of the exposure distribution (floored at zero) at a future date across scenarios; EPE is the time-weighted average of EE over a given horizon.
• PFE is a high percentile (such as 95% or 99%) of the exposure distribution at a future date, so it is a tail measure.
• Static limits based only on current exposure miss how exposure changes as markets move.
• Good stress scenarios are severe but plausible and should reflect the actual portfolio and its concentrations.
• Wrong-way risk arises when exposure to a counterparty rises as its credit quality worsens.
• Netting reduces exposure only where it is legally enforceable across the trades in the netting set.
• Collateral reduces exposure but is limited by the margin period of risk, haircuts, thresholds and disputes.
• Stress can raise margin calls, creating liquidity strain for the firm posting collateral.
• Results should inform limits, capital and business decisions, with senior management and board oversight.

Common mistakes

• Saying pre-crisis stress tests did not exist. Fix: Remember that basic scenario analysis existed. It was narrow, siloed and weakly linked to decisions.
• Mixing up the AIG and Archegos lessons. Fix: AIG is wrong-way risk and collateral triggers on a protection seller. Archegos is hidden concentration through swaps across several prime brokers.
• Averaging only the positive scenarios when computing EE. Fix: Floor at zero but keep every scenario in the denominator. Zero-exposure paths count.
• Treating PFE and EE as the same thing or saying PFE is the average. Fix: EE is the mean, PFE is a percentile. PFE is at least as large as EE for the same date when the percentile is high.
• Saying a reverse stress test starts with a scenario and measures its loss. Fix: Remember it starts with the bad outcome and works back to the scenarios that cause it.
• Shocking one risk factor at a time. Fix: Shock related factors together with consistent correlations, since crises move many factors at once.
• Applying netting across all trades with a counterparty regardless of legal agreements Fix: Net only within one enforceable netting set. Trades in different sets or jurisdictions without enforceable netting stay gross.
• Assuming fully collateralised means zero exposure Fix: Residual exposure remains from the MPOR, thresholds, minimum transfer amounts, collateral haircuts and disputes.
• Treating stress testing as a purely quantitative modelling task. Fix: Remember supervisors judge ownership, challenge, documentation and use of results as much as the maths.
• Assuming stress results are informational only. Fix: Choose answers where stress outcomes drive limits, margin terms, hedges or risk appetite changes.

Exam tips

• Link every case to one core lesson: Lehman to speed and aggregation, AIG to wrong-way risk, Archegos to hidden concentration and margin.
• Expect scenario-style MCQs asking which weakness caused the failure, so name the weakness before reading options.
• For calculations, subtract stressed collateral, not original collateral.
• Reject options claiming stress testing eliminates counterparty risk or that single-factor shocks are sufficient.
• Remember that supervisors expect results to feed limits, capital and senior management decisions.
• Read the metric name carefully. Questions often swap EE for PFE or EPE in the options to test whether you know mean versus percentile versus time-average.
• Always floor at zero before averaging or ranking. Many numeric traps use negative values.
• For stress questions, think direction first: PFE and peak exposure rise most, and wrong-way risk can make exposure and default probability rise together.