FRM Part II · FRM Exam Part II

Credit Risk Management: formula sheet

Full chapter guide

Key formulas

Expected loss: EL = PD × LGD × EAD; Use the same horizon for PD, usually one year. PD and LGD as decimals.
Loss given default: LGD = 1 − Recovery rate; Recovery rate is a share of EAD. Check whether the question gives recovery or loss.
Expected loss rate: EL rate = EL ÷ EAD = PD × LGD; Useful when exposure is not given.
Exposure at default for a credit line: EAD = Drawn + CCF × Undrawn; CCF is the credit conversion factor, the share of the undrawn limit expected to be drawn by default.
Portfolio expected loss: EL(portfolio) = Σ EL(i); Expected losses add up exactly. Unexpected losses do not, because of correlation.
Unexpected loss of one exposure (fixed LGD and EAD): UL = EAD × LGD × √(PD × (1 − PD)); This is the standard deviation of loss when default is a Bernoulli event and LGD is a known constant.
Economic capital: Economic capital = Credit VaR at confidence level − EL; Capital covers the gap between a tail loss and the average loss.
Recovery by seniority: Absolute priority: senior claims are paid in full before junior claims receive anything; In practice deviations from strict priority occur, but this is the exam default.
Row sum condition: Σj P(i → j) = 1 for every starting rating i; Includes the default column. Use it to find a missing entry.
Absorbing default state: P(D → D) = 1, P(D → any other) = 0; Default has no exit in a standard matrix.
Two-year transition (Markov, constant matrix): P(i → k, 2 years) = Σj P(i → j) × P(j → k); This is the row of the matrix times the column of the matrix.
n-year matrix: Pⁿ = P × P × ... × P (n times); Needs the Markov and time-homogeneity assumptions.
Cumulative default probability: CPD(i, n) = Pⁿ(i → D); The default column of the n-year matrix.
Marginal (conditional) default probability in year n: CPD(n) − CPD(n − 1) = unconditional default probability in year n; Dividing by the survival probability (1 − CPD(n − 1)) gives the conditional rate.
Survival probability: S(n) = 1 − CPD(n); Probability of no default through year n.
Equity as a call option: E = V·N(d1) - D·e^(-rT)·N(d2); V = asset value, D = face value of debt, T = debt maturity.
d1 and d2: d1 = [ln(V ÷ D) + (r + σ²÷2)T] ÷ (σ√T); d2 = d1 - σ√T; σ is asset volatility, not equity volatility.
Risk-neutral default probability: PD = N(-d2); Use μ in place of r for the real-world probability.
Risky debt value: B = V - E = D·e^(-rT) - Put(V, D); Lenders are long a risk-free bond and short a put on assets.
Equity and asset volatility link: σE · E = N(d1) · σ · V; Used to back out unobserved asset value and volatility.
Distance to default (simple form): DD = (V - default point) ÷ (V · σ); Approximate, with drift ignored. Higher DD means lower default risk.
Survival probability, constant hazard: Q(t) = e^(-λt); PD(t) = 1 - e^(-λt); λ is the annual hazard rate.
Spread approximation: λ ≈ s ÷ (1 - R); s = credit spread, R = recovery rate. This is the risk-neutral intensity.
Expected loss: EL = PD × LGD × EAD; Portfolio EL is the sum of individual ELs. It does not depend on correlation.
Vasicek worst-case default rate: WCDR(X) = N[ (N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1 − ρ) ]; N is the standard normal CDF, N⁻¹ its inverse. X is the confidence level, e.g. 99.9%. Valid for a large homogeneous portfolio with one factor.
Credit VaR (Vasicek): Credit VaR = WCDR × LGD × EAD; This is the total loss quantile, assuming LGD is fixed and known.
Unexpected loss / capital: UL = Credit VaR − EL = (WCDR − PD) × LGD × EAD; Basel IRB capital follows this idea, with extra adjustments for maturity.
Asset correlation and default correlation: Asset value Aᵢ = √ρ × M + √(1 − ρ) × Zᵢ; M is the systematic factor, Zᵢ is idiosyncratic. Both are standard normal. Default correlation is lower than asset correlation for typical PDs.
Two-asset default correlation: ρ_D = (P(both) − PD₁ × PD₂) ÷ √(PD₁(1 − PD₁) × PD₂(1 − PD₂)); P(both) is the joint default probability.
Herfindahl-Hirschman Index: HHI = Σ wᵢ²; wᵢ is each exposure's share of the portfolio. Higher HHI means more concentration. 1/HHI is the effective number of equal-sized exposures.
Exposure: Exposure = max(V, 0); V is the value of the trade or netted portfolio to you at the future date.
Expected exposure: EE(t) = E[max(V(t), 0)]; Average of positive values only. Negative values count as zero.
Potential future exposure: PFE(t) = the α-percentile of max(V(t), 0); Usually 95% to 99%. It is a limit-setting measure and is not an average.
Expected positive exposure: EPE = (1 ÷ T) × ∫ EE(t) dt, or the average of EE over the dates; A time average. Effective EPE uses a non-decreasing EE profile over the first year.
Unilateral CVA: CVA ≈ LGD × Σ [EE*(tᵢ) × PD(tᵢ₋₁, tᵢ)], with EE* discounted; LGD = 1 − recovery. PD is the marginal default probability in each period. Assumes independence of exposure and default.
Marginal PD from a constant hazard rate: PD(t₁, t₂) = e^(−λt₁) − e^(−λt₂); λ is the hazard rate. Approximately λ ≈ spread ÷ LGD.
Net exposure: Net exposure = max(Σ Vᵢ, 0) ≤ Σ max(Vᵢ, 0); Netting never increases exposure.
Collateralised exposure: Exposure = max(V − C, 0); C is collateral held (after haircuts). Risk remains from movement over the margin period of risk.
Expected loss: EL = PD × LGD × EAD; Collateral works on LGD, covenants mainly on PD, netting on EAD.
LGD and recovery rate: LGD = 1 − recovery rate; Recovery is net of costs of selling collateral and of time delay.
Haircut-adjusted collateral value: Adjusted collateral = C × (1 − Hc − Hfx); Hc is the haircut for collateral price volatility; Hfx applies only if collateral currency differs from the exposure currency.
Net exposure after collateral: Net exposure = max(0, E − adjusted collateral); E is the exposure. Exposure cannot be negative for the lender.
Netting exposure: Net exposure = max(0, Σ trade values); With enforceable close-out netting. Without it, exposure = Σ max(0, each trade value).
Guarantee substitution (simple approach): Guaranteed part takes the guarantor's risk weight; the rest keeps the borrower's; Needs an eligible guarantor and a legally enforceable, direct, unconditional guarantee.
Joint default of borrower and guarantor: P(both default) ≤ min(PD borrower, PD guarantor); Equals the product only if defaults are independent. Positive correlation raises it.
Risk-weighted assets: RWA = Exposure × Risk weight; Use exposure after any credit risk mitigation recognised by the framework.
Minimum capital: Capital required = Capital ratio × Total RWA; Basel minimum total capital is 8% of RWA; buffers sit on top of that.
Implied RWA from IRB capital: RWA = 12.5 × K × EAD; K is the capital requirement per unit of exposure. 12.5 = 1 ÷ 8%.
Expected loss: EL = PD × LGD × EAD; Covered by pricing and provisions, not by capital.
RAROC: RAROC = (Revenue − Costs − Expected loss) ÷ Economic capital; Some versions add the return on capital and tax adjustments; follow the question's definition.
Risk-adjusted loan spread (break-even): Required spread ≈ Funding cost + Operating cost + (PD × LGD) + Hurdle rate × Capital ratio − Return earned on capital; Simplified one-year view; the last term is often ignored if not given.

Quick revision

Expected loss = PD × LGD × EAD, using consistent time horizons.
Recovery rate = 1 − LGD when both are expressed as a share of exposure.
Expected loss is a cost of business; unexpected loss is the variability that capital is held against.
Transition matrices: each row sums to 100%, and multi-year transitions come from matrix multiplication under the Markov assumption.
Structural models treat equity as a call option on firm assets; default occurs when asset value falls below debt at the horizon in the basic Merton model.
Reduced-form models treat default as a random event driven by a hazard rate, not tied directly to firm value.
Higher default correlation fattens the tail of portfolio losses without changing expected loss.
Concentration risk comes from large single names, sectors or regions, and diversification reduces it.
Counterparty exposure on derivatives depends on market moves, and netting reduces it only where legally enforceable.
Collateral lowers loss given default but brings its own risks, such as value falls and legal issues.
Credit derivatives transfer credit risk but can create counterparty risk with the protection seller.
Capital rules set minimums, and governance sets limits, approval and monitoring of credit exposure.

Common mistakes

Using the recovery rate as LGD. Fix: Always compute LGD = 1 − recovery before multiplying.
Ignoring the undrawn portion of a credit line. Fix: Apply EAD = drawn + CCF × undrawn whenever a limit and CCF are given.
Confusing TTC and PIT by saying TTC ratings move more with the cycle. Fix: TTC ratings ignore the cycle, so they are stable. Their realised default rates per grade vary with the cycle. PIT ratings migrate more.
Leaving out the path through default when computing two-year default. Fix: Include the term P(i → D) × 1. Default is absorbing, so those borrowers stay defaulted.
Using equity volatility in d1 and d2 Fix: Always use asset volatility σ in the formulas. Convert with σE·E = N(d1)·σ·V.
Treating N(-d2) as the real-world default probability Fix: N(-d2) is risk-neutral because it uses r. Replace r with the asset drift μ for the real-world PD.
Saying correlation changes portfolio expected loss Fix: EL is additive. Correlation changes only the shape and tail of the loss distribution, so it affects UL and VaR.
Forgetting to subtract EL when UL or capital is asked Fix: Read the last line of the question. Capital under Basel IRB covers UL, so use (WCDR − PD) × LGD × EAD.
Averaging signed values to get EE. Fix: Floor at zero first. EE is the mean of max(V, 0), so it is never negative.
Treating PFE as the average of the worst cases. Fix: PFE is a single percentile of the exposure distribution. It is not a tail average.

Exam tips

Read the horizon. If PD is given for several years, match it to the loss period asked for.
Watch for questions that give recovery rate while asking for loss, and the reverse.
Conceptual questions often test the split: EL for pricing and provisions, UL for capital.
For default versus spread risk questions, look for whether a default actually occurred.
When ranking recovery across debt classes, apply seniority and collateral first.
Know the TTC and PIT direction cold. Questions often ask which approach gives stable ratings and which gives stable PDs per grade.
In matrix questions, check the row sum first. A missing entry is often 1 minus the others.
For two-year questions, use the row-times-default-column shortcut. It saves time.