Derivatives: formula sheet

Key formulas

Exposure of a netting set

Exposure = max(V, 0)

V is the net market value to you. Floored at zero because you cannot gain from a counterparty default.

Current exposure

CE = max(V₀, 0)

Today's replacement cost, ignoring collateral unless stated.

Expected exposure

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

Mean of the positive part, not max of the mean. Includes the zeros from negative scenarios.

Potential future exposure

PFE(t) = the α-percentile of max(V(t), 0)

α is typically 95% or 99%. Compare with VaR, but PFE looks at the positive side only and at future dates.

Expected positive exposure

EPE = Σ EE(tₖ) × Δtₖ ÷ T

Time-weighted average of EE over [0, T]. With equal time steps it is the simple average of EE.

Effective EE

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

Non-decreasing profile. Captures rollover of short-dated trades.

Effective EPE

Effective EPE = Σ Effective EE(tₖ) × Δtₖ over the first year (or to maturity if shorter) ÷ the length of that period

Average over one year, or until the longest-maturity contract in the netting set matures if that is sooner.

Regulatory EAD (internal model method)

EAD = α × Effective EPE, with α = 1.4

Alpha is the regulatory default; supervisors may allow banks to estimate their own, subject to a floor.

Net exposure with close-out netting

Netted exposure = max(Σ Vi, 0)

Vi are trade values in one netting set. The formula holds only if close-out netting is legally enforceable. If it is not, exposure is Σ max(Vi, 0). Close-out costs can also make the actual claim differ from this figure.

Netting benefit

Netting benefit = Σ max(Vi, 0) − max(Σ Vi, 0)

It is never negative, so netting never raises exposure. It is zero when all trades have the same sign, whether all positive or all negative. This assumes netting is enforceable.

Collateralised exposure (simple)

Exposure = max(V − C, 0)

V is net mark-to-market, C is collateral held. Haircuts reduce C for non-cash collateral.

Collateral call with threshold and MTA

Required collateral = max(V − Threshold, 0); a call is made only if the change in required collateral ≥ MTA

Residual exposure can be up to threshold + MTA before any market move during MPOR.

Exposure at default with MPOR

Exposure ≈ max(V(t + MPOR) − C, 0)

The relevant risk is the change in value over MPOR, not over one day.

Square-root-of-time scaling

σ(MPOR) ≈ σ(1 day) × √(MPOR in days)

Used as an approximation, assuming independent, identically distributed daily moves.

Unilateral CVA (discrete)

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

Assumes exposure is independent of default. PD is the marginal default probability in each interval. LGD = 1 − R.

Marginal default probability

PD(tᵢ₋₁, tᵢ) = Q(tᵢ₋₁) − Q(tᵢ)

Q(t) is survival probability to t. With constant hazard rate λ, Q(t) = e^(−λt).

Credit triangle approximation

λ ≈ s ÷ LGD

s is the CDS spread as a decimal. A rough way to get the hazard rate from a spread.

DVA

DVA = LGD(own) × Σ [ ENE(tᵢ) × DF(tᵢ) × PD(own)(tᵢ₋₁, tᵢ) ]

ENE uses the absolute size of negative exposure. It is a positive number that adds to value.

Bilateral CVA and value

Bilateral CVA = CVA − DVA; Value = Risk-free value − CVA + DVA

The bilateral adjustment is the net of both.

Rough unilateral CVA for a flat exposure

CVA ≈ EPE × LGD × (1 − Q(T))

Ignores discounting. Good for quick estimates.

CVA under independence

CVA ≈ LGD × Σ EE(tᵢ) × PD(tᵢ₋₁, tᵢ) × DF(tᵢ)

Uses expected exposure EE, the marginal default probability in each period and discounting. Assumes exposure and default are independent.

Wrong-way effect on exposure

E[Exposure | default] > E[Exposure] under WWR

Conditional exposure at default exceeds unconditional exposure. Under right-way risk the inequality reverses.

Direction of dependence

WWR: exposure ↑ as credit quality ↓; RWR: exposure ↓ as credit quality ↓

Use this to classify any scenario quickly.

Exposure at default multiplier (alpha)

EAD = α × Effective EPE, with α = 1.4 as the supervisory default under the internal model method

Supervisors may allow a bank's own alpha estimate, subject to a floor of 1.2. Alpha reflects wrong-way risk, model uncertainty and portfolio granularity, among other factors. It is not a pure wrong-way risk adjustment.

Novation

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

CCP is buyer to every seller and seller to every buyer; its book is matched.

Typical default waterfall order

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

Defaulter pays first; the order can vary by CCP, so use the order given.

Initial margin purpose

IM ≈ potential loss over margin period of risk at a high confidence level

Longer margin period or higher confidence means higher IM.

Variation margin

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

Paid at least daily; it removes current exposure, not future exposure.

Netting benefit

Net exposure = max(Σ values, 0) within a netting set

Less netting when trades are split across sets, so the net exposure can be higher.

Credit triangle

CDS spread ≈ λ × (1 − R)

λ is the annual hazard rate and R the recovery rate. It is an approximation that is good for flat curves.

Hazard rate from spread

λ ≈ s ÷ (1 − R)

Use s in decimals, e.g. 120 bp = 0.0120.

Survival probability (constant hazard)

Q(t) = e^(−λt)

The cumulative default probability is 1 − e^(−λt).

Protection leg payout

Notional × (1 − R)

Paid by the protection seller on a credit event.

Fair CDS spread

s = PV(protection leg) ÷ risky annuity (PV of 1 per year paid while the entity survives)

Premium accrued up to the default date adds a small correction.

CDS basis

Basis = CDS spread − bond spread

A negative basis means that the bond spread is above the CDS spread.

Value of an existing CDS to the buyer

(Current market spread − contract spread) × risky annuity × notional

It is positive for the buyer if spreads have widened.

SA-CCR exposure at default

EAD = α × (RC + PFE), with α = 1.4

Calculated per netting set. The 1.4 scaling factor applies to the sum, not just to PFE.

Replacement cost, unmargined netting set

RC = max(V − C, 0)

V is the netting set mark-to-market and C is net collateral held. Negative values are floored at zero.

Replacement cost, margined netting set

RC = max(V − C, TH + MTA − NICA, 0)

TH is the threshold, MTA the minimum transfer amount and NICA net independent collateral amount. This captures exposure that can arise before the next margin call.

Potential future exposure

PFE = multiplier × AddOn(aggregate)

The aggregate add-on is the sum of the add-ons across the five asset classes: interest rate, FX, credit, equity and commodity.

PFE multiplier

multiplier = min{1, 5% + 95% × exp[(V − C) ÷ (2 × 95% × AddOn(aggregate))]}

Never above 1 and never below 5%. It falls when the netting set is out-of-the-money or over-collateralised, which gives credit for excess collateral or negative MtM.

Trade-level add-on (linear trades)

AddOn = SF × adjusted notional × supervisory delta × maturity factor

SF is the supervisory factor for the asset class. Delta is +1 for long and −1 for short positions in linear products. Interest rate trades use a supervisory duration in the adjusted notional.

Maturity factor

Unmargined: √(min(M, 1 year) ÷ 1 year), with M floored at 10 business days. Margined: 1.5 × √(MPOR ÷ 1 year)

M is remaining maturity and MPOR is the margin period of risk. For unmargined sets, M has a floor of 10 business days and the factor is capped at 1 (a maturity of 1 year or more gives 1). The margined factor uses MPOR, not maturity.

Reduced BA-CVA aggregation

K_reduced = √[(ρ × Σ SCVA_c)² + (1 − ρ²) × Σ (SCVA_c)²], with ρ = 50%. Capital charge = DS × K_reduced, with DS = 0.65. Stand-alone CVA: SCVA_c = (1 ÷ α) × RW_c × Σ (M_NS × EAD_NS × DF_NS), with α = 1.4

SCVA_c is the stand-alone CVA capital for counterparty c. In the first term you add up the SCVA values and then square the total. In the second term you square each SCVA_c first and then add them up. ρ = 50% is the supervisory correlation between a counterparty's credit spread and the systematic factor, which is why the first term is the systematic part and the second the idiosyncratic part. In SCVA_c, RW_c is the supervisory risk weight for the counterparty's sector and credit quality. The sum runs over the netting sets NS with that counterparty, using each netting set's effective maturity M_NS, its EAD and a supervisory discount factor DF_NS. DS = 0.65 is the supervisory discount scalar applied to K_reduced. Do not confuse it with the 1.4 in SCVA_c, which divides rather than multiplies.

Risk weights for CCP exposures

Clearing member trade exposures to a QCCP: 2%. Client exposures (through a clearing member): 2% if the specified conditions are met, otherwise 4%

The 2% for client exposures needs the conditions to be met. Broadly, the client must be protected from losses of the clearing member and the CCP. If they are not met, the weight is 4%. Default fund contributions are capitalised by a separate formula. Exposures to non-qualifying CCPs are treated as bilateral exposures, and their default fund exposures get a much harsher treatment.

Quick revision

• Counterparty risk on derivatives is examined within the Credit Risk Measurement and Management topic; there is no separate Derivatives topic in the six Part II topics.
• Current exposure = max(V, 0), where V is the trade's mark-to-market value to you.
• Expected exposure (EE) is the average positive exposure at a future date; EPE is the time average of EE. Regulatory capital uses effective EPE, based on non-decreasing effective EE, averaged over the first year, or until the maturity of the longest-dated contract in the netting set if all contracts mature within a year.
• PFE is a high percentile of the exposure distribution at a future date, so at a high confidence level it is typically larger than EE at the same date.
• Netting applies only where it is legally enforceable, and it can only reduce exposure, never increase it.
• Collateral reduces exposure but leaves a gap from the margin period of risk, thresholds and minimum transfer amounts.
• CVA is the market value of expected counterparty credit loss; DVA reflects your own default risk and gives a gain when your credit worsens.
• CVA rises with higher exposure, higher default probability and higher loss given default.
• Wrong-way risk means exposure rises as the counterparty's credit quality falls; right-way risk is the opposite.
• A CCP becomes the buyer to every seller and the seller to every buyer, and it uses initial margin, variation margin and a default fund.
• A CDS buyer pays a periodic premium and receives protection on default, so the buyer is short credit risk.
• CDS protection bought from a weak seller carries counterparty risk and may carry wrong-way risk.
• Basel capital for counterparty risk includes a default risk charge and a separate CVA risk charge.

Common mistakes

• Computing EE as max(average V, 0) instead of average of max(V, 0). Fix: Floor each scenario at zero first, then average. Because max(V, 0) is convex (an option-like payoff), EE is always greater than or equal to max(mean V, 0).
• Confusing EE with PFE. Fix: EE is the mean of positive exposure. PFE is a high percentile. At high confidence levels such as 95% or 99%, PFE is normally above EE at the same date. At a low percentile, or with a long right tail, PFE can be below EE.
• Netting trades across different master agreements or counterparties. Fix: Net only within one enforceable netting set. Treat each set separately, then add the results.
• Treating the threshold as collateral that is posted. Fix: The threshold is unsecured exposure. Collateral is due only on the amount above it.
• Using total exposure including negative values in CVA. Fix: CVA uses positive exposure only. Negative values feed DVA through ENE.
• Forgetting to multiply by LGD, or using recovery rate in its place. Fix: Always compute LGD = 1 − R first and write it down.
• Treating any correlation between market variables and credit as specific WWR. Fix: Specific needs a direct trade-level link to the counterparty itself or a legally or structurally linked affiliate. Macro, market or purely statistical links are general.
• Saying wrong-way risk lowers CVA. Fix: WWR raises exposure when default is likely, so CVA computed with independence is understated.
• Saying a CCP eliminates counterparty risk. Fix: It concentrates and mutualises risk. The CCP itself can fail if losses exceed its resources.
• Putting survivors' default fund before the defaulter's contributions. Fix: The defaulter pays first: its IM, then its default fund share, before mutualised resources.

Exam tips

• Write the zero floor on every scenario first. Most wrong options come from skipping it.
• Check ordering: at 95% or 99%, PFE is normally above EE, and effective EPE is at least EPE over the same horizon and grid. Use it to remove options fast.
• Know which measure serves which purpose: PFE for limits, EE and EPE for pricing and CVA, effective EPE for regulatory EAD.
• Read whether the question asks for a time-average over one year or over the life of the trade.
• Remember alpha = 1.4 for the regulatory default, and apply it to effective EPE, not to EE.
• Read the question for the netting set first. A different agreement means no netting.
• When both threshold and MTA appear, apply the threshold to size collateral, then check whether the MTA blocks the call.
• Expect conceptual options on MPOR: a longer MPOR means higher exposure, and disputes lengthen it.