Introduction to Credit Risk Modeling and Assessment: formula sheet

Key formulas

Expected loss

EL = PD × LGD × EAD

PD and LGD as decimals, EAD in currency. The PD horizon sets the EL horizon.

Loss given default from recovery

LGD = 1 − Recovery rate

Recovery rate is a share of EAD recovered, net of costs if the question says so.

Exposure at default with undrawn limit

EAD = Drawn amount + CCF × Undrawn commitment

CCF is the credit conversion factor, the share of the undrawn limit expected to be drawn at default.

Expected loss rate

EL ÷ EAD = PD × LGD

Useful when exposure is not given or you compare loans.

Portfolio expected loss

EL(portfolio) = Σ EL(i)

Expected loss is additive across exposures. No correlation is needed.

Unexpected loss of one exposure with fixed LGD and EAD

UL = EAD × LGD × √(PD × (1 − PD))

Standard deviation of loss when LGD is certain, treating default as a Bernoulli event. UL does not add across exposures unless defaults are perfectly correlated.

Expected loss (single exposure)

EL = PD × LGD × EAD

LGD is a fraction of EAD. Use the same horizon for PD, usually one year.

Portfolio expected loss

EL_p = Σ EL_i

EL is additive across exposures, regardless of correlation.

Unexpected loss, single exposure (fixed LGD and EAD)

UL = EAD × LGD × √(PD × (1 − PD))

Default is a Bernoulli event, so its standard deviation is √(PD(1−PD)). Assumes LGD is known with certainty.

Unexpected loss with random LGD

UL = EAD × √(PD × σ_LGD² + LGD² × PD × (1 − PD))

Adds recovery uncertainty. σ_LGD is the standard deviation of LGD. Assumes LGD is independent of default.

Portfolio unexpected loss, two exposures

UL_p = √(UL₁² + UL₂² + 2ρ × UL₁ × UL₂)

ρ is the correlation between the two loss outcomes. UL is not additive unless ρ = 1.

Credit VaR

Credit VaR = quantile of the loss distribution at confidence level α

Measured as total loss, including EL, unless the question defines it relative to the mean.

Economic capital

EC = Credit VaR − EL

Also called the unexpected loss at the chosen confidence level. Some questions define EC as the full quantile, so read the wording.

Row sum condition

Σj P(i → j) = 1 for every starting rating i

Includes the default column. Use it to find a missing entry.

Default as absorbing state

P(D → D) = 1; P(D → any other) = 0

The default row never changes in matrix multiplication.

n-period transition matrix

P(n) = P(1)^n (Markov, time-homogeneous)

Multiply the matrix by itself. For two years, P(2) = P × P.

Two-year cumulative PD from rating i

PD2(i) = Σk P(i → k) × P(k → D)

Sum over all intermediate ratings k, including D itself where P(D → D) = 1.

Survival probability

S(n) = 1 − cumulative PD(n)

Probability of no default through year n.

Marginal PD in year n

marginal PD(n) = cumulative PD(n) − cumulative PD(n−1)

Unconditional probability of defaulting in exactly year n.

Conditional PD in year n

conditional PD(n) = marginal PD(n) ÷ S(n−1)

Probability of default in year n given survival to the end of year n−1.

Equity as a call

E = V·N(d1) − D·e^(−rT)·N(d2)

Black-Scholes applied to assets V with strike D.

d1 and d2

d1 = [ln(V/D) + (r + σ_V²/2)T] ÷ (σ_V√T); d2 = d1 − σ_V√T

Use r for risk-neutral, μ in place of r for real-world.

Default probability

PD = N(−d2)

Probability that V < D at time T in the Merton model.

Distance to default (simple)

DD = [ln(V/D) + (μ − σ_V²/2)T] ÷ (σ_V√T)

Number of standard deviations from the default point.

Risky debt value

B = V − E = D·e^(−rT) − Put(V, D)

Debt equals risk-free debt minus a put on assets.

Credit spread

s = −(1/T)·ln(B ÷ (D·e^(−rT)))

Yield on risky debt minus r, with B the value of the debt.

Equity volatility link

σ_E·E = N(d1)·σ_V·V

Used to solve for unobserved V and σ_V.

Survival probability (constant hazard)

Q(τ > t) = e^(−λt)

τ is the default time. λ is a constant annual rate. Use the same time unit for λ and t.

Cumulative default probability

PD(t) = 1 − e^(−λt)

For small λt, PD(t) ≈ λt.

Time-varying hazard rate

Q(τ > t) = exp(−∫0→t λ(s) ds)

With piecewise-constant λ, add λ × length across each interval inside the exponent.

Credit spread and hazard rate

s ≈ λ × (1 − R)

Approximation under risk-neutral pricing with a flat curve and continuous compounding. λ = s ÷ (1 − R). R is the recovery rate.

Average (annualised) hazard from cumulative PD

λ = −ln(1 − PD(t)) ÷ t

Use this to convert a cumulative default probability into a constant annual intensity.

Conditional default probability in interval

P(default in (t, t+dt) | survival to t) ≈ λ(t) × dt

Defines the hazard rate. It is a conditional measure, not unconditional.

Risk-neutral vs real-world

λQ ≥ λP in practice

A rule of thumb from observed markets, not a mathematical law. The difference reflects risk premia.

Altman Z-score (original, listed manufacturers)

Z = 1.2·X1 + 1.4·X2 + 3.3·X3 + 0.6·X4 + 1.0·X5

X1 = working capital ÷ total assets; X2 = retained earnings ÷ total assets; X3 = EBIT ÷ total assets; X4 = market value of equity ÷ book value of total liabilities; X5 = sales ÷ total assets.

Altman Z-score zones (original)

Z > 2.99 safe; 1.81 ≤ Z ≤ 2.99 grey; Z < 1.81 distress

The grey zone is not a prediction. It means the model cannot separate the firm clearly.

Altman Z'-score (private firms)

Z' = 0.717·X1 + 0.847·X2 + 3.107·X3 + 0.420·X4 + 0.998·X5

X4 uses book value of equity ÷ total liabilities. Cut-offs are about 2.9 (safe) and 1.23 (distress). Use only if the question gives these weights.

Logistic score

S = β0 + β1·x1 + β2·x2 + … + βk·xk

S is the log-odds of default, so it can be any real number.

Logistic probability of default

PD = 1 ÷ (1 + e^(−S))

PD is always between 0 and 1. S = 0 gives PD = 50%.

Odds and log-odds

Odds = PD ÷ (1 − PD) = e^S; ln(Odds) = S

A one-unit rise in xi multiplies the odds by e^βi, holding other variables fixed.

Key ratios

Leverage = Debt ÷ EBITDA; Interest cover = EBIT ÷ Interest expense

Higher leverage and lower cover signal higher default risk.

Quick revision

• Expected loss = PD × LGD × EAD; it is the average loss and is a cost of doing business.
• Unexpected loss is the variability of loss around expected loss; capital covers it.
• Credit VaR = loss at a chosen confidence level minus expected loss, under the common definition. Check the question's wording.
• LGD = 1 − recovery rate, when recovery is stated as a share of exposure.
• EAD includes drawn amounts plus an expected share of undrawn commitments.
• Transition matrices show the chance of moving between ratings over a period; default is an absorbing state.
• Merton: equity is a call option on firm assets, with strike equal to the debt face value.
• In Merton, higher asset volatility or leverage raises the PD. N(−d2) is the risk-neutral PD; the real-world PD uses the asset drift instead of the risk-free rate.
• KMV uses distance to default and maps it to an expected default frequency from historical data.
• Reduced-form models: with constant hazard rate λ, survival probability to time t is e^(−λt).
• Approximate credit spread ≈ PD × LGD, with a risk-neutral PD that is usually higher than the real-world PD because of risk premia.
• Scoring models rank borrowers; check discrimination and calibration separately.

Common mistakes

• Using the recovery rate as LGD Fix: Always write LGD = 1 − recovery. A 35% recovery means a 65% LGD.
• Ignoring the undrawn part of a credit line Fix: Check for a limit and a CCF. EAD = drawn + CCF × undrawn.
• Adding unexpected losses across exposures to get portfolio UL. Fix: Only EL adds. Combine UL with the correlation formula. Simple addition applies only when ρ = 1.
• Using a normal multiple such as 2.33 × UL for every credit VaR. Fix: Credit losses are skewed with a fat right tail. Use the quantile given, or the multiple the question states.
• Doubling the one-year PD to get the two-year PD. Fix: Use matrix multiplication. The second year adds default only from borrowers still alive, and they may have migrated to a different rating.
• Forgetting that default is absorbing, so leaving out the D term when summing two-year PD. Fix: Include P(i → D) × 1. This captures the one-year defaults already booked.
• Using N(d2) as the default probability. Fix: Default probability is N(−d2) = 1 − N(d2).
• Mixing risk-neutral and real-world drift. Fix: Use r for risk-neutral PD and pricing spreads. Use μ for real-world PD and DD.
• Treating the hazard rate as the unconditional annual default probability. Fix: Remember that λ is conditional on survival and is an intensity. Convert with 1 − e^(−λt). For small λ the two are close, but they are not the same thing.
• Using spread = λ without dividing by (1 − R). Fix: Always write λ = s ÷ (1 − R). With R = 40%, λ is the spread divided by 0.6.

Exam tips

• Read for a hidden step: a recovery rate, an undrawn limit or collateral that changes LGD or EAD.
• Check which horizon the PD covers. Questions often give a cumulative multi-year PD.
• When a question mentions capital, it is usually testing unexpected loss, not expected loss.
• Know the Basel link: IRB risk weights use PD, LGD, EAD and maturity as inputs.
• Estimate the rate PD × LGD first. It lets you reject wrong options quickly.
• Read whether 'credit VaR' is total loss or loss relative to the mean. This decides whether you subtract EL.
• Expect conceptual options on why credit VaR differs from market VaR: longer horizon, skewed tail, illiquidity, model dependence.
• When correlation changes, track UL only. EL does not move with correlation.