FRM Part II · FRM Exam Part II

Fundamentals of Credit Risk: formula sheet

Full chapter guide

Key formulas

Expected loss: EL = PD × LGD × EAD; EL is in currency. PD and EAD must refer to the same horizon. LGD is a fraction of EAD.
LGD and recovery rate: LGD = 1 − RR and RR = 1 − LGD; Both are stated as a share of EAD. Recovery rate is often quoted per 100 of face value.
EAD for a credit line: EAD = Drawn + CCF × Undrawn; CCF is the expected share of the undrawn limit that is drawn by default. It lies between 0% and 100%.
Expected loss rate: EL% = EL ÷ EAD = PD × LGD; Useful for comparing loans of different size.
Portfolio expected loss: EL(portfolio) = Σ ELᵢ; Expected losses add up across exposures. Unexpected losses do not add up, because of diversification.
Unexpected loss for one exposure, fixed EAD and LGD: UL = EAD × LGD × √(PD × (1 − PD)); Applies when EAD and LGD are fixed and default is a Bernoulli event. Standard deviation of loss.
Expected loss (single exposure): EL = PD × LGD × EAD; LGD = 1 − recovery rate. Use the same time horizon for PD, usually one year.
Unexpected loss, fixed LGD: UL = EAD × LGD × √[PD × (1 − PD)]; Standard deviation of loss for one loan when LGD is known with certainty.
Unexpected loss, random LGD: UL = EAD × √[PD × σ_LGD² + LGD² × PD × (1 − PD)]; Assumes default and LGD are independent. σ_LGD is the standard deviation of LGD.
Portfolio expected loss: EL_P = Σ EL_i; Expected losses add up directly. No diversification effect.
Portfolio unexpected loss: UL_P = √(Σ Σ ρ_ij × UL_i × UL_j); Diversification arises when default correlation ρ is below 1. For two loans: UL_P² = UL₁² + UL₂² + 2ρUL₁UL₂.
Credit VaR: Credit VaR = quantile of the loss distribution at confidence level α; Some definitions measure it as the quantile minus EL. Check the question.
Economic capital: EC = Credit VaR(α) − EL; Where credit VaR is the absolute loss quantile. Capital covers unexpected loss only.
Normal approximation of capital: EC ≈ k × UL_P; k is the multiplier, e.g. 2.33 at 99%. Only a rough guide, because credit losses are skewed.
Row sum of a transition matrix: Σⱼ P(i → j) = 1 for every starting rating i; Includes the default column. Use it to find a missing cell.
Two-year transition probability: P²(i → k) = Σⱼ P(i → j) × P(j → k); Sum over every intermediate rating j, including default if k is default.
n-year matrix: Pⁿ = P × P × ... × P (n times); Assumes a time-homogeneous Markov chain. Default column gives cumulative PD.
Cumulative PD with absorbing default: Cumulative PD(t) = 1 − Probability of no default by t; Cumulative PD never falls as t increases.
Marginal (unconditional) default probability: Marginal PD in year t = Cumulative PD(t) − Cumulative PD(t−1); This is the unconditional probability of defaulting in year t, not conditional on survival. The conditional PD given survival to t−1 is [Cumulative PD(t) − Cumulative PD(t−1)] ÷ [1 − Cumulative PD(t−1)].
Annualised PD from cumulative PD: Annual PD = 1 − (1 − Cumulative PD)^(1/t); Constant-hazard approximation.
Merton equity value: E = V × N(d1) − D × e^(−rT) × N(d2); Equity is a European call on assets V with strike D, maturing at T.
Merton d1 and d2: d1 = [ln(V ÷ D) + (r + σ²÷2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; σ is asset volatility. Use r for risk-neutral, μ in place of r for real-world.
Merton default probability: Risk-neutral PD = N(−d2) = 1 − N(d2); Real-world PD uses μ: d2 = [ln(V ÷ D) + (μ − σ²÷2) × T] ÷ (σ × √T). N(d2) is the survival probability, not PD.
Equity and asset volatility link: σE × E = N(d1) × σV × V; Used with the equity formula to solve for V and σV from observed equity data.
Risky debt value and spread: B = V − E; yield y = −(1 ÷ T) × ln(B ÷ D); spread = y − r; Spread rises as leverage D ÷ V and asset volatility rise.
KMV distance to default: DD = (V − default point) ÷ (V × σV) (one-year, simple form); Default point = short-term debt + 0.5 × long-term debt. EDF comes from an empirical mapping of DD.
Reduced-form survival and default probability: Q(t) = e^(−λt); cumulative PD = 1 − e^(−λt); Holds for a constant hazard rate λ. Time-varying: Q(t) = e^(−∫λ(s)ds).
Spread–hazard relation: λ ≈ s ÷ (1 − R); Approximation for a constant hazard rate. s is the credit spread, R the recovery rate.
Credit spread: s = y(risky) − y(risk-free); Use bonds of the same maturity, currency and coupon structure where possible.
Spread approximation: s ≈ λ × (1 − R) = λ × LGD; λ is the annual risk-neutral hazard rate. Approximation for small values and a flat hazard rate.
Hazard rate from spread: λ ≈ s ÷ (1 − R); Gives the risk-neutral default intensity. Also holds approximately for CDS spreads.
Cumulative default probability: PD(T) = 1 − e^(−λT); Assumes a constant hazard rate. Survival probability is e^(−λT).
Annual PD (approximate): PD ≈ 1 − e^(−λ) ≈ λ for small λ; Hazard rate and annual probability are close only when λ is small.
Risky bond price (one-year zero): P = [(1 − PD) × 100 + PD × R × 100] ÷ (1 + r); Expected payoff discounted at the risk-free rate gives the risk-neutral PD.
Risk-neutral to real-world gap: λ(Q) = λ(P) × multiplier, usually multiplier > 1; Multiplier is a descriptive ratio, not a fixed rule. Not always the same across ratings.
Expected loss: EL = PD × LGD × EAD; Collateral and guarantees mainly cut LGD. Netting cuts EAD. Covenants aim to cut PD.
Collateral value after haircut: Adjusted collateral = Market value × (1 − haircut); The haircut reflects price volatility, illiquidity and currency mismatch.
Net exposure with collateral: Exposure after collateral = max(0, Exposure − Adjusted collateral); The result cannot be negative for the lender's credit exposure.
Exposure with and without netting: Without netting = Σ max(Vi, 0); with netting = max(Σ Vi, 0); Vi is the value of trade i to the lender. Netting never raises exposure.
Netting ratio: Net-to-gross ratio = Net exposure ÷ Gross exposure; A lower ratio means a larger netting benefit.
Guarantee substitution: Protected amount takes the guarantor's risk; the remainder keeps the borrower's risk; Only the covered part is substituted, and only if the guarantee is legally valid.
CDS hedge: Payout on credit event = Notional × (1 − Recovery rate); Assumes cash settlement at the recovery rate. Premium is paid until the credit event or maturity.

Quick revision

Expected loss = PD × LGD × EAD.
LGD = 1 − recovery rate.
Expected loss is the average loss; it is priced and provisioned.
Unexpected loss is the variability of loss around the expected loss, usually measured by standard deviation.
Credit VaR is a loss quantile at a chosen confidence level; it is often quoted relative to expected loss to give the unexpected part.
Transition matrix rows show migration probabilities from a starting rating and sum to 1.
Multi-year migration probabilities come from multiplying the matrix by itself, assuming the process is Markov.
Structural models tie default to asset value falling below debt; reduced-form models use a default intensity (hazard rate).
Approximate credit spread ≈ PD × LGD, under simplifying assumptions.
Risk-neutral PDs are usually higher than real-world PDs because spreads include a risk premium.
Netting reduces exposure only where it is legally enforceable.
Collateral and guarantees reduce loss but add risks, such as collateral value falling or guarantor default.

Common mistakes

Using the recovery rate in place of LGD in the EL formula. Fix: Always convert first: LGD = 1 − RR. A high recovery means a low LGD.
Using only the drawn balance as EAD for a credit line. Fix: Add CCF × Undrawn. If the question gives a CCF, it is telling you to use it.
Treating EL as a risk that needs capital. Fix: EL is covered by pricing and provisions. Capital covers only the unexpected part, credit VaR minus EL.
Using UL = EAD × LGD × PD. Fix: UL needs the square root of PD × (1 − PD). Check that PD appears under a root.
Adding annual default probabilities to get cumulative PD. Fix: Use the matrix: cumulative PD(2) = PD₁ + Σ P(start → j) × PD₁(j). Add the first-year default and the second-year defaults of survivors, weighted by migration.
Leaving out the default state's own contribution in the second year. Fix: Include the one-year default probability of the start row, or use P²(start → D) with P(D → D) = 1.
Using N(d2) as the default probability. Fix: N(d2) is the probability of finishing above D, so it is survival. Default is N(−d2) = 1 − N(d2).
Mixing risk-neutral and real-world probabilities. Fix: Check the wording. Use r for pricing and spreads, and μ for real-world PD. Spread-implied PDs are risk-neutral and are usually higher than real-world PDs.
Using the spread as the default probability. Fix: Always divide by LGD: λ ≈ s ÷ (1 − R).
Using R instead of 1 − R in the denominator. Fix: Write LGD = 1 − R first. A higher recovery rate means a higher implied hazard rate for the same spread.

Exam tips

When recovery rate is given, convert to LGD before anything else. Distractors are built on this slip.
For credit lines, look for the undrawn amount and CCF. EAD is often the hidden step.
Know the roles: EL is priced and provisioned, UL is capitalised. Conceptual questions test this.
Remember LGD and PD tend to rise together in downturns, so a simple product can understate stressed loss.
Check that the horizon of PD matches the question before multiplying.
Always compute EL even if the question asks for UL. Options often include EL as the trap answer.
Read the definition of credit VaR in the stem. Whether it is measured from zero or from the mean changes the capital answer.
When a question mentions skewness or fat tails, the expected conclusion is that a normal multiple of UL understates tail loss.