Counterparty Risk and Beyond: formula sheet

Key formulas

Exposure

Exposure = max(V, 0)

V is the netting-set mark-to-market from your side. A negative value means no credit exposure on that set.

Current exposure

CE = max(V today, 0)

Also called replacement cost. Backward-looking, zero for a freshly struck at-market trade.

Potential future exposure

PFE(t) = the α-percentile of exposure at date t

Typically 95% to 99%. It is a quantile, so it is not additive and not an average.

Expected exposure

EE(t) = E[max(V(t), 0)]

Mean of the positive part, not the mean of V, so it is always at least as large as the expected value of V.

Expected positive exposure

EPE = Σ EE(tₖ) × Δtₖ ÷ T

Time-weighted average of EE over the horizon T.

Effective EE

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

Non-decreasing over the first year. Captures rollover risk of short-term trades.

Effective EPE

Effective EPE = Σ Effective EE(tₖ) × Δtₖ ÷ T, over the first year (or to maturity if all trades mature sooner)

Basel exposure at default for internal models = alpha × effective EPE, with alpha 1.4 as the supervisory default.

Maximum PFE

Max PFE = max over t of PFE(t)

The peak of the PFE profile, often compared with the counterparty limit.

Exposure without netting

Exposure = Σ max(Vi, 0)

Each trade is treated alone. Only positive values count.

Exposure with close-out netting

Net exposure = max(Σ Vi, 0)

Valid only if netting is legally enforceable. Always ≤ the no-netting figure.

Net-to-gross ratio (NGR)

NGR = max(Σ Vi, 0) ÷ Σ max(Vi, 0)

Ranges from 0 to 1. Lower means more netting benefit.

Collateralised exposure

Exposure after collateral = max(Net exposure − Collateral held, 0)

Collateral held should be after haircuts. Add any unsecured amount from the threshold.

Haircut value

Collateral value = Market value × (1 − haircut)

Haircuts cover price and FX moves of the collateral asset.

CSA call amount

Call = Net exposure − Threshold − Collateral held (made only if ≥ MTA)

Where the call is positive. Rounding rules may also apply.

Exposure over the margin period of risk

Residual exposure ≈ max(V(t) + ΔV over MPOR − Collateral, 0)

Under a normal model the std dev of ΔV scales with √MPOR.

Unilateral CVA (discrete)

CVA = LGD × Σ [ EE*(ti) × PD(ti-1, ti) ]

EE* is discounted expected positive exposure. LGD = 1 − recovery rate. Assumes exposure and default are independent.

DVA (discrete)

DVA = LGD(own) × Σ [ ENE*(ti) × PD(own)(ti-1, ti) ]

ENE is expected negative exposure, taken as a positive amount owed by you.

Bilateral CVA

Bilateral CVA = CVA − DVA

Adjusted value = risk-free value − CVA + DVA.

Marginal default probability

PD(ti-1, ti) = Q(ti-1) − Q(ti)

Q(t) is the survival probability.

Credit triangle approximation

Hazard rate λ ≈ CDS spread ÷ LGD

Survival Q(t) = e^(−λt). A rough link between spread and default probability.

Risky value

Risky value = Risk-free value − CVA

A long-only receivable has CVA charged. A payable has no CVA from your side.

Novation

Original trade A↔B becomes A↔CCP and CCP↔B

The CCP is buyer to every seller and seller to every buyer. Its net position is zero when no member defaults.

Variation margin

VM = change in mark-to-market value of the position

Paid at least daily. It removes accumulated exposure, so only the margin period of risk remains.

Initial margin

IM ≈ loss at a high confidence level (e.g. 99%) over the margin period of risk

Set on a portfolio basis. Longer margin period or more volatile products need more IM.

Default waterfall order

Defaulter IM → defaulter default fund → CCP own capital → survivors' default fund → further recovery tools

Typical order. Use the order stated in the question if it differs.

Netting benefit

Net exposure = max(Σ trade values, 0) per netting set; gross exposure = Σ max(trade value, 0)

Multilateral netting at a CCP lowers exposure compared with bilateral gross positions.

Price with adjustments

Price = Risk-free value − CVA + DVA − FVA − KVA − MVA

Signs show a cost to the dealer. Some costs can be benefits, such as FVA on a liability position. Check the sign convention in the question.

CVA (discrete)

CVA = LGD × Σ [EE(ti) × PD(ti-1, ti) × DF(ti)]

EE is the discounted or undiscounted expected positive exposure. Use one convention consistently. LGD = 1 − recovery rate.

FVA (simple)

FVA ≈ Σ [EE(ti) × s_f × Δt × DF(ti)]

s_f is the funding spread over the risk-free rate. Use expected positive exposure for the funding cost and expected negative exposure for the funding benefit.

KVA (simple)

KVA ≈ Σ [Expected capital(ti) × hurdle rate × Δt × DF(ti)]

The hurdle rate is the cost of capital. Some versions use the excess over the risk-free rate, so read the wording.

MVA (simple)

MVA ≈ Σ [Expected IM(ti) × funding spread × Δt × DF(ti)]

IM is initial margin. The spread is the cost of funding the margin, net of any interest earned on it.

Approximate CVA, flat exposure

CVA ≈ EE × LGD × PD

A shortcut for a single period or constant exposure and hazard rate.

Quick revision

• Counterparty exposure is the positive value of the portfolio to you; it is floored at zero, so it behaves like an option.
• EE(t) is the mean of max(V(t), 0) at date t, so it already uses the positive part of the value. EPE is the time-average of EE(t). PFE(t) is a high percentile (for example 95% or 99%) of exposure at date t.
• Wrong-way risk arises when exposure rises as the counterparty's credit quality falls.
• Netting applies only under an enforceable legal agreement and reduces exposure to the net, not gross, value.
• Variation margin covers current mark-to-market changes; initial margin covers possible moves during close-out.
• The margin period of risk is the time between the last margin exchange and when the position is closed out.
• CVA is the market value of counterparty credit risk, approximately the discounted sum of expected exposure × default probability × loss given default.
• DVA reflects your own default risk and gains value when your credit worsens, which is why its use is criticised.
• A CCP becomes the buyer to every seller and the seller to every buyer, which concentrates risk in the CCP.
• A CCP loss waterfall typically uses the defaulter's initial margin first, then the defaulter's default fund contribution, then the CCP's own capital (skin in the game), and only then the surviving members' default fund contributions and further loss allocation.
• FVA captures funding costs of uncollateralised trades; KVA captures cost of regulatory capital; MVA captures cost of funding initial margin.
• Collateral reduces credit risk but adds liquidity and operational risk.

Common mistakes

• Treating EE as the average of V rather than the average of max(V, 0). Fix: Floor each scenario at zero before averaging. EE is never negative.
• Confusing PFE with EE. Fix: PFE is a high percentile (worst-case style, used for limits). EE is the mean (used for pricing and capital).
• Netting trades without checking legal enforceability. Fix: Netting works only under an enforceable master agreement and jurisdiction. Without it, use the sum of positive values.
• Including negative trade values in gross exposure. Fix: Gross exposure sums only positive values. Net exposure sums all values and then floors at zero.
• Using net expected exposure instead of positive exposure Fix: CVA uses only the positive part, because you owe the full amount if the trade is negative.
• Forgetting to multiply by LGD Fix: Always apply LGD = 1 − recovery. Write it as the final step.
• Saying a CCP eliminates counterparty risk. Fix: A CCP transforms and concentrates risk. Members still face CCP default and loss mutualisation risk.
• Putting survivors' default fund contributions before the defaulter's own resources. Fix: The defaulter pays first: its IM, then its default fund contribution. Survivors come later.
• Using LGD and default probability to compute FVA. Fix: FVA depends on the funding spread, not on counterparty default. Use exposure × funding spread × time.
• Treating FVA as always a cost. Fix: Positive exposure creates a funding cost. Negative exposure creates a funding benefit. Net the two as the question directs.

Exam tips

• Read the metric name twice. Many wrong options are the right number for a different metric, such as the mean instead of the percentile.
• Always floor at zero before averaging. This is the most frequent calculation trap.
• For effective EPE, write the running maximum line explicitly and only within the first year.
• Link metric to use: PFE to limits, EE and EPE to CVA and capital, CE to current replacement cost.
• In conceptual questions, mention netting set and enforceability. Exposure without legally enforceable netting is measured gross, trade by trade.
• Read for the word 'enforceable'. It decides whether you net.
• Do the arithmetic in the order: net, haircut, threshold, MTA.
• For concept questions, link IM to MPOR and VM to current value. This is the most testable distinction.