FRM Part II · FRM Exam Part II

Credit Value Adjustment: formula sheet

Full chapter guide

Key formulas

Exposure at date t: E(t) = max(V(t), 0); V is the portfolio value to you. For a netting set, net the trades first, then take the max.
Expected exposure: EE(t) = E[max(V(t), 0)]; Includes the zero outcomes. It is always at least max(E[V(t)], 0).
Potential future exposure: PFE(t) = the α-percentile of E(t), i.e. P(E(t) ≤ PFE) = α; Common α is 95% or 97.5%. Peak PFE is the maximum over time.
Expected positive exposure: EPE = (1 ÷ T) × ∫ EE(t) dt ≈ Σ EE(tᵢ) × Δtᵢ ÷ T; A time-weighted average of EE. Use time weights if the dates are unevenly spaced.
Effective EE: Effective EE(tₖ) = max(Effective EE(tₖ₋₁), EE(tₖ)); Running maximum, so the profile is non-decreasing.
Effective EPE: Effective EPE = Σ Effective EE(tₖ) × Δtₖ ÷ T, over the first year; Basel horizon is the shorter of one year and the longest maturity in the netting set.
Normal exposure percentile: If V ~ N(μ, σ²) and PFE > 0: PFE = μ + z × σ; z = 1.645 for 95% and 1.96 for 97.5%. Floor at zero.
Regulatory EAD (internal model method): EAD = α × Effective EPE, with α = 1.4 unless supervisors approve otherwise; α is a multiplier calibrated by Basel. State it as the default.
Exposure without netting: Exposure = Σ max(Vi, 0); Sum over each trade i. Only positive-value trades count. Used when netting is not enforceable.
Exposure with close-out netting: Exposure = max(Σ Vi, 0); Sum over trades in one legally enforceable netting set. Always ≤ the no-netting exposure.
Netting benefit: Benefit = Σ max(Vi, 0) − max(Σ Vi, 0); Zero if all trades have the same sign. Largest when trades offset.
Net-to-gross ratio (NGR): NGR = max(Σ Vi, 0) ÷ Σ max(Vi, 0); Between 0 and 1. Lower means more netting benefit.
Exposure with collateral: Exposure = max(V − C, 0); V is net value at close-out, C is collateral held after haircuts.
Collateral call with threshold and MTA: Call = max(V − Threshold − Collateral held, 0), made only if the amount ≥ MTA; If the calculated amount is below the MTA, no transfer happens. Practical CSAs also apply rounding.
Maximum uncollateralised gap (simple view): Residual exposure ≈ Threshold + MTA + change in V over MPOR; A rule of thumb for the worst case just before default. Initial margin or independent amount reduces it.
Volatility over the MPOR: σ(MPOR) = σ(1 day) × √(MPOR in days); Square-root-of-time scaling under independent, identically distributed changes. Use only as an approximation.
Haircut collateral value: Collateral value = Market value × (1 − haircut); Haircut reflects the price volatility and liquidity of non-cash collateral.
Unilateral CVA (discrete): CVA = LGD × Σ [ EE*(tᵢ) × PD(tᵢ₋₁, tᵢ) ]; EE* is discounted expected exposure, EE(tᵢ) × DF(tᵢ). Assumes exposure is independent of default.
Loss given default: LGD = 1 − R; R is the recovery rate. Use the same R for all dates unless told otherwise.
Marginal default probability: PD(tᵢ₋₁, tᵢ) = Q(tᵢ₋₁) − Q(tᵢ); Q(t) is the survival probability to time t. Q(0) = 1.
Survival probability from constant hazard: Q(t) = e^(−λt); λ is the constant default intensity. Approximately λ ≈ credit spread ÷ LGD (credit triangle).
Adjusted value: Risky value = Risk-free value − CVA; Unilateral CVA is a non-negative number that reduces the value of your claim on the counterparty.
Single-period shortcut: CVA ≈ PD × LGD × discounted EE; Use only when the question gives one period or one average exposure.
Bilateral value: Value = Risk-free value − CVA + DVA; CVA lowers value. DVA raises value. Bilateral CVA charge = CVA − DVA.
Unilateral CVA (discrete): CVA = LGD_C × Σ EPE(tᵢ) × DF(tᵢ) × PD(tᵢ₋₁, tᵢ); EPE is the expected positive exposure. PD is the counterparty's marginal default probability in the period. LGD_C is the counterparty's loss given default.
DVA (discrete): DVA = LGD_own × Σ ENE(tᵢ) × DF(tᵢ) × PD_own(tᵢ₋₁, tᵢ); ENE is the expected negative exposure, taken as a positive amount. Uses your own default probability and your own LGD.
First-to-default adjustment: CVA uses PD_C × survival of own; DVA uses PD_own × survival of counterparty; Each default probability is multiplied by the other party's survival probability to that time. This avoids double counting.
Credit triangle (approximation): Spread ≈ PD (annual hazard) × LGD; Use it to turn a CDS spread into a default probability when the question gives spreads.
Simple CVA (independence): CVA ≈ LGD × Σ EE(tᵢ) × PD(tᵢ₋₁, tᵢ) × DF(tᵢ); Uses discounted expected exposure EE and marginal default probability PD, with exposure and default assumed independent.
Wrong-way effect: Positive dependence ⇒ CVA with dependence > CVA under independence; Exposure conditional on default exceeds unconditional expected exposure.
Right-way effect: Negative dependence ⇒ CVA with dependence < CVA under independence; Exposure conditional on default is below unconditional expected exposure.
Conditional exposure view: CVA ≈ LGD × Σ E[Exposure | default at tᵢ] × PD(tᵢ) × DF(tᵢ); Replace EE by exposure given default. WWR means this is higher than EE.
Unilateral CVA (discrete sum): CVA = LGD × Σ [ DF(tᵢ) × EE(tᵢ) × PD(tᵢ₋₁, tᵢ) ]; LGD = 1 − recovery. EE is expected exposure (positive part only). PD is the marginal default probability in each interval, under risk-neutral measure.
Credit triangle: λ ≈ s ÷ LGD; s is the CDS spread, λ the hazard rate. Use for a quick risk-neutral PD when the question gives a spread.
Marginal PD from a hazard rate: PD(tᵢ₋₁, tᵢ) = e^(−λ·tᵢ₋₁) − e^(−λ·tᵢ); Gives default probability within one period, for use in the CVA sum.
Approximate CVA with flat inputs: CVA ≈ s × EPE × risky annuity; Flat spread and flat exposure. It is the basis for sizing a CDS hedge quickly.
CDS hedge notional (spread hedge): Notional ≈ CS01 of CVA ÷ CS01 per unit of CDS notional; With a maturity-matched CDS and flat inputs this is roughly the EPE. It is only a hedge of spread risk, not of exposure changes.
BA-CVA reduced version: Capital = DS × K_reduced, with DS = 0.65; DS is the discount scalar. The reduced version gives no credit for hedges.
FVA (approximate): FVA ≈ funding spread × EPE × T − funding spread × ENE × T; The first term is the funding cost on positive exposure. The second is the benefit on negative exposure (ENE). Discounting ignored.
MVA (approximate): MVA ≈ funding spread × average expected initial margin × T; Applies to margin posted that the bank must fund.
KVA (approximate): KVA ≈ hurdle rate × average expected capital × T; Capital is regulatory capital over the trade life. The hurdle rate is the shareholders' required return, often net of what capital earns.

Quick revision

Current exposure is the greater of the current replacement value and zero.
Expected exposure is the average positive exposure at a future date; PFE is a high percentile of exposure at that date.
Netting applies only under an enforceable netting agreement and lowers exposure to the net positive amount.
Collateral lowers exposure but leaves a residual risk over the margin period of risk.
Unilateral CVA ≈ LGD × Σ discounted EE × marginal default probability, where LGD = 1 − recovery rate.
Unilateral CVA is a cost to you; it reduces the value of the portfolio.
Bilateral CVA = CVA − DVA, where DVA is the gain from your own default risk.
DVA rises when your own credit spread widens, which is why it is criticised.
Wrong-way risk means exposure rises as the counterparty's credit quality falls; right-way risk is the reverse.
Wrong-way risk makes CVA larger than an independence-based calculation would show.
A CDS hedge covers credit spread risk in CVA but not the exposure (market) risk.
Funding adjustments reflect the cost of funding uncollateralised positions and are separate from CVA.

Common mistakes

Averaging only the positive outcomes to get EE, or averaging V itself. Fix: EE is E[max(V, 0)] over all scenarios. Negative scenarios count as zero, not dropped.
Treating PFE as an average. Fix: PFE is a percentile. It is used for limits. EE is a mean. It is used for pricing CVA.
Netting positive and negative values across different netting agreements. Fix: Net only within one enforceable master agreement. Floor each set at zero, then add the sets.
Forgetting to floor the net value at zero. Fix: Exposure is always max(net value, 0). A negative net means you owe the counterparty, so your credit exposure is zero.
Using cumulative default probability for every bucket Fix: Use the marginal PD for each interval, the difference between consecutive cumulative values. Using cumulative values counts the same default several times.
Forgetting to multiply by LGD Fix: PD × EE is expected exposure at default. Multiply by 1 − R to get the loss.
Adding DVA to CVA as if both are losses. Fix: CVA is a cost and DVA is a benefit. Bilateral CVA charge = CVA − DVA.
Using the counterparty's default probability in the DVA calculation. Fix: For DVA use your own default probability, your own LGD and expected negative exposure.
Treating wrong-way risk as a higher default probability alone. Fix: Remember it is the dependence between exposure and default. Exposure must rise as default likelihood rises.
Mixing up specific and general wrong-way risk. Fix: Specific has a direct link in the trade, such as a put on the counterparty's own stock. General comes from broad factors such as macro or sector conditions.

Exam tips

Read the metric name twice. Mean, percentile, time average and running-maximum average are four different answers.
Wrong options often come from skipping the zero floor or the running maximum. Check both before choosing.
Know the uses: EE feeds CVA, PFE feeds limits, effective EPE feeds regulatory EAD.
Expect interpretation questions, such as why a swap's exposure profile rises then falls while an option's is highest at start. Link the shape to time to maturity and volatility.
Keep the sanity checks ready: PFE ≥ EE at the same date, and effective EPE ≥ EPE.
Always check whether netting is stated to be legally enforceable. If the question says it is not, compute exposure trade by trade.
Read the CSA terms in order: threshold first, then MTA, then independent amount. Many MCQ distractors come from skipping one.
For MPOR questions, expect square-root-of-time scaling. Match the horizon to the days given, such as 10 business days for daily margining of liquid OTC trades under many regimes, only if the question states it.