Credit Scoring and Rating: formula sheet

Key formulas

Altman Z-score (original, public manufacturers)

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

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 model)

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

Revised versions for private firms or non-manufacturers use different coefficients and cut-offs. Use the ones given in the question.

Logit probability

PD = 1 ÷ (1 + e^(−z)), where z = β0 + Σ βi·xi

A higher z means a higher PD when coefficients are set so that default is the outcome being modelled.

Log-odds and odds

ln[PD ÷ (1 − PD)] = z; odds = PD ÷ (1 − PD) = e^z

A one-unit rise in xi multiplies the odds by e^βi. It does not add βi to the PD.

Probit probability

PD = N(z), where N is the standard normal CDF

Same linear index as logit but a different link function.

Scorecard points scaling

Score = Offset + Factor × ln(odds good:bad); Factor = PDO ÷ ln 2

PDO is points to double the odds. Each PDO points added doubles the good:bad odds.

Gini coefficient from AUC

Gini = 2 × AUC − 1

AUC of 0.5 means no discrimination (Gini 0). AUC of 1 means perfect ranking (Gini 1).

Investment-grade boundary

S&P / Fitch: BBB- and above. Moody's: Baa3 and above.

Anything below is speculative grade (high yield). Bonds are one notch from the boundary at BB+ / Ba1.

Scale mapping (full letter grades)

S&P/Fitch: AAA, AA, A, BBB, BB, B, CCC, CC, C, D. Moody's: Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C.

Moody's uses 1, 2, 3 modifiers (Aa1 = AA+, Aa3 = AA-). S&P and Fitch use + and -.

Issue rating via notching

Issue rating = Issuer rating ± notches for seniority, security, guarantees and expected recovery

Senior secured is notched up or equal. Subordinated is notched down. Number of notches differs by agency and by issuer grade.

Rating philosophy

Through-the-cycle = stable, looks through the cycle. Point-in-time = responsive to current conditions.

Agency ratings are mainly through-the-cycle. This makes transitions slow and causes cliff effects.

Rating meaning

Rating = ordinal ranking of credit risk, not a fixed PD

Default rates rise as ratings fall, but the PD for a grade changes with the cycle.

Expected loss

EL = PD × LGD × EAD

Covered by provisions and pricing. Under IRB, the capital requirement targets unexpected loss.

LGD and recovery

LGD = 1 − Recovery rate

Express both as a percentage of EAD. Basel requires LGD to reflect downturn conditions.

Grade PD from history

PD (grade) = Defaults in year ÷ Obligors in grade at start of year; long-run PD = average over many years

Use obligors that were in the grade at the start of the period. Use a long-run average for TTC-style estimates.

Number of defaults implied

Expected defaults = PD × Number of obligors

Useful for backtesting a grade against observed defaults.

Basel PD floor

PD ≥ 0.05% for corporate and bank exposures under the finalised Basel III framework (Basel II floor: 0.03%)

Retail classes have their own floors, for example 0.05% for mortgages and 0.10% for QRRE revolvers. Defaulted exposures are assigned PD = 100%.

Basel IRB estimates by approach

F-IRB: bank estimates PD, with supervisory LGD and EAD; maturity generally fixed at 2.5 years unless national supervisors permit effective maturity. A-IRB: bank estimates PD, LGD, EAD and maturity

In F-IRB, supervisory LGD for senior unsecured corporate claims was 45% under Basel II. The finalised Basel III framework sets it at 40% for non-financial corporates and 45% for financial institutions. Large corporates (revenue above €500 million) and banks are limited to F-IRB.

IRB capital charge

Capital requirement = K × EAD; RWA = K × 12.5 × EAD; K depends on PD, LGD, maturity and asset correlation, with a 99.9% one-year confidence level

You need the structure, not the full formula. Higher PD or LGD raises K. Higher EAD raises the capital amount.

Row sum

Σj P(i → j) = 1

Each row of the matrix sums to 1, including the default column.

Multi-year matrix

P(n) = P^n (matrix power)

Valid under the time-homogeneous Markov assumption. Use matrix multiplication, row times column.

Two-year transition probability

P2(i → k) = Σj P(i → j) × P(j → k)

Sum over every intermediate rating j, including paths through default.

Cumulative default probability

CPD(n) = P^n(i → Default)

Probability of defaulting at any time up to year n.

Marginal default probability

MPD(n) = CPD(n) − CPD(n−1)

Unconditional probability of defaulting in year n only.

Conditional (hazard) default rate

h(n) = MPD(n) ÷ [1 − CPD(n−1)]

Default in year n given survival to the start of year n.

Cumulative from hazards

1 − CPD(n) = Π [1 − h(t)], t = 1 to n

Survival is the product of one-year survival rates.

Hit rate (true positive rate)

Hit rate = defaulters flagged ÷ total defaulters

Vertical axis of the ROC curve.

False alarm rate (false positive rate)

False alarm rate = non-defaulters flagged ÷ total non-defaulters

Horizontal axis of the ROC curve.

Gini from AUC

Gini = AR = 2 × AUC − 1

Inverse: AUC = (Gini + 1) ÷ 2.

Benchmarks

Random model: AUC = 0.5, Gini = 0. Perfect model: AUC = 1, Gini = 1

AUC below 0.5 means the ranking is reversed.

Accuracy ratio

AR = area between model CAP and random line ÷ area between perfect CAP and random line

The perfect CAP depends on the portfolio default rate.

Binomial back-test

Observed defaults ~ Binomial(n, PD); z ≈ (D − n × PD) ÷ √(n × PD × (1 − PD))

Normal approximation for large n. Assumes independent defaults.

Quick revision

• A scorecard converts borrower characteristics into a score that ranks default risk.
• Rating systems rank borrowers; they do not by themselves give a default probability until calibrated.
• Agency ratings are usually described as through-the-cycle, while many internal ratings lean point-in-time.
• Under IRB, banks estimate risk inputs such as probability of default; expected loss = PD × LGD × EAD.
• Each row of a transition matrix sums to 100%, and the default column is the probability of ending in default.
• Default is typically treated as an absorbing state in a transition matrix.
• Default rates generally rise as ratings worsen; a reversal in the pattern is a warning sign.
• Discrimination is ranking power; calibration is whether predicted rates match realised rates.
• Backtesting compares predicted default rates with observed outcomes over time.
• A model can discriminate well yet be badly calibrated, and the reverse.
• Ratings can be procyclical: they tend to be downgraded in downturns, which can tighten capital and credit.
• Criticisms include rating lag, conflicts of interest, cliff effects and over-reliance by investors and regulators.

Common mistakes

• Treating the Z-score as a probability of default Fix: A Z-score is a ranking score. You can map it to a PD only with a separate calibration. Zones are safe, grey and distress.
• Mixing up the Altman X4 definition Fix: In the original model X4 is market value of equity ÷ book value of total liabilities. Revised models for private firms use book equity instead.
• Treating issuer and issue ratings as the same thing Fix: Remember the issue rating adjusts the issuer rating for the instrument's seniority, security and recovery.
• Mapping Moody's Baa3 to S&P BBB+ Fix: Moody's 1 = high end (+), 2 = middle, 3 = low end (-). Baa3 = BBB-.
• Treating TTC ratings as unaffected by borrower quality changes Fix: TTC ratings ignore cyclical swings but still migrate when a borrower's own creditworthiness changes.
• Saying PIT ratings reduce procyclicality Fix: PIT PDs rise in downturns, so capital requirements rise when capital is scarce. That increases procyclicality.
• Squaring the matrix element by element instead of using matrix multiplication. Fix: Use row-times-column multiplication. Sum over all intermediate ratings.
• Doubling the one-year PD to get the two-year PD. Fix: Use CPD(2) = PD1 + (1 − PD1 paths that survive and then default). Include migration to weaker ratings.
• Treating Gini and accuracy ratio as different numbers Fix: Remember they are equal. Both equal 2 × AUC − 1.
• Saying AUC of 0.5 is a Gini of 0.5 Fix: AUC 0.5 gives Gini 0. Always compute 2 × 0.5 − 1.

Exam tips

• Memorise the five Altman coefficients and the 1.81 and 2.99 cut-offs. Questions often give ratios and expect you to supply the weights.
• Check the sign of z first in logit questions. It tells you whether PD is above or below 50% and removes trap options.
• Know the difference between discrimination (AUC, Gini, KS) and calibration. Case questions often ask which test addresses which weakness.
• Expect conceptual items on logit versus probit versus linear probability, and on why scorecards are favoured for retail portfolios: transparency and ease of explaining decisions.
• Read what the question asks: a score, a PD, odds, or a zone. Each needs a different final step.
• Practise converting between S&P, Fitch and Moody's including the modifiers. Questions often hide the answer in the boundary notch.
• When a question asks why a bond is rated differently from its issuer, think seniority, security, guarantees and recovery first.
• Watch for words like 'through-the-cycle', 'stable' and 'outlook'. They point to agency philosophy, not market-implied measures.