Credit Scoring and Retail Credit Risk Management: formula sheet

Key formulas

Expected loss

EL = PD × LGD × EAD

Applies to both. In retail it is estimated per pool; in corporate it is estimated per borrower or facility.

Loss rate on a retail pool

Loss rate = (Defaults ÷ Number of accounts) × LGD

A simple pool view when EAD is similar across accounts. Use it for pool-level questions.

Roll rate

Roll rate = Accounts moving to the next delinquency bucket ÷ Accounts in the starting bucket

Retail monitoring tool. Higher buckets usually roll to default at higher rates.

Concentration intuition

For N equal, independent exposures, relative loss volatility ∝ 1 ÷ √N

Explains why granular retail books diversify idiosyncratic risk. It does not remove systematic risk.

Logistic regression

ln(p ÷ (1 − p)) = β0 + β1x1 + β2x2 + … + βkxk

p is the probability of default. Left side is the log-odds.

Probability from score

p = 1 ÷ (1 + e^−(β0 + Σβixi))

Always lies between 0 and 1, unlike a linear probability model.

Weight of evidence

WoE = ln(% of goods in bin ÷ % of bads in bin)

Sign conventions vary. Be consistent. Here positive WoE means the bin is better than average.

Information value

IV = Σ (% goods − % bads) × WoE

Measures a variable's predictive power. A common rule of thumb treats IV below about 0.02 as weak and above about 0.5 as suspiciously strong.

Score scaling

Score = Offset + Factor × ln(odds), Factor = PDO ÷ ln 2

Odds here are good:bad. Score rises by PDO points each time odds double.

Offset

Offset = Base score − Factor × ln(base odds)

Anchors the scale at a chosen score and odds.

Gini coefficient

Gini = 2 × AUC − 1

Equivalent to AUC. AUC = (Gini + 1) ÷ 2. Ranges 0 to 1 for a model that ranks in the right direction.

True positive rate (sensitivity)

TPR = bads below cut-off ÷ total bads

Y-axis of the ROC curve. Defined with low score meaning high risk.

False positive rate

FPR = goods below cut-off ÷ total goods

X-axis of the ROC curve.

KS statistic

KS = max over scores | F_bad(s) − F_good(s) |

F is the cumulative share of each group at or below score s. Equals the maximum of TPR − FPR.

Population stability index

PSI = Σ (A_i − E_i) × ln(A_i ÷ E_i)

A_i is the actual (recent) share in bucket i, E_i the expected (development) share. Shares are fractions that sum to 1.

PSI rule of thumb

< 0.10 stable; 0.10 to 0.25 some shift; > 0.25 significant shift

Industry convention, not a regulatory rule. Cut-offs vary by firm.

AUC interpretation

AUC = P(score of random bad < score of random good)

Ties are usually counted as half.

Expected loss

EL = PD × LGD × EAD

Used in pricing and provisioning for each retail segment.

Net charge-off rate

Net charge-off rate = (Gross charge-offs − Recoveries) ÷ Average balances

Usually annualised; check whether the question gives a monthly figure.

Roll rate

Roll rate = Balances moving from bucket n to bucket n+1 ÷ Balances in bucket n at start

Measures how many dollars worsen from one delinquency bucket to the next.

Risk-based break-even margin

Required margin ≈ Expected loss rate + Funding cost + Operating cost + Capital charge

Rough guide; price above this to earn profit.

Net recovery on collections

Net benefit = Amount recovered − Collection cost

Compare strategies by net benefit, not gross recovery.

Application vs behavioral score

Application score: new applicants. Behavioral score: existing accounts.

A rule to recall which model fits which decision.

Expected loss

EL = PD × LGD × EAD

Average loss over the horizon. Express PD and LGD as decimals. EL rate = PD × LGD.

LGD from recovery

LGD = 1 − Recovery rate

Recovery must be net of collection costs and discounted to default date for economic LGD.

Unexpected loss (standard deviation form)

UL = EAD × √(PD × σLGD² + LGD² × PD × (1 − PD)), with LGD treated as a fixed mean and independent of default

If LGD is fixed, UL = EAD × LGD × √(PD × (1 − PD)). Applies to a single exposure.

Roll rate chain

P(current → default) ≈ r(C→30) × r(30→60) × r(60→90) × r(90→charge-off)

Assumes the rates are stable and each step is a one-period transition. Rates are measured on balances or accounts.

Cumulative default rate for a vintage

Cumulative default rate at age t = defaults to age t ÷ original number of loans (or balance)

Use the original cohort size as denominator, not surviving loans.

Capital requirement (IRB)

Capital = EAD × (Unexpected-loss capital rate K)

K = LGD × [conditional PD at 99.9% − PD], with a supervisory correlation. Expected loss is covered separately by provisions.

Model risk definition

Model risk = risk of adverse outcomes from decisions based on incorrect or misused model outputs

Two sources: fundamental errors in the model, and incorrect or inappropriate use.

Adverse impact ratio (selection rate test)

Impact ratio = approval rate of protected group ÷ approval rate of reference group

A common screen; a ratio well below 1 flags possible disparate impact. The 0.80 level (four-fifths rule) is a US employment-testing rule of thumb, not a universal lending law. It is a screen, not proof.

Disparate treatment vs disparate impact

Treatment = intent or direct use of protected trait; Impact = neutral rule with unjustified disproportionate effect

Impact can arise through proxy variables even when the protected trait is excluded.

Core validation components

Conceptual soundness + ongoing monitoring + outcomes analysis (backtesting)

These are the three pillars of model validation in supervisory guidance such as SR 11-7.

Population stability index (PSI)

PSI = Σ (Actual% − Expected%) × ln(Actual% ÷ Expected%)

Summed across score bands. Higher values mean a larger shift in the score distribution; common rules of thumb treat below 0.10 as stable and above 0.25 as a major shift.

Quick revision

• Retail credit: many small, homogeneous loans, managed statistically at pool level. Corporate credit: fewer, larger exposures, assessed individually.
• A scorecard ranks borrowers by risk. Ranking power is not the same as accurate PD calibration.
• Expected loss = PD × LGD × EAD, with all inputs on the same time horizon.
• KS measures the maximum separation between the cumulative distributions of good and bad accounts.
• AUC of 0.5 means no discrimination. Higher means better ranking. Gini = 2 × AUC − 1.
• Population stability checks whether current applicants resemble the development sample. A shift can weaken a scorecard.
• Reject inference addresses the bias from only observing outcomes for approved applicants.
• Validate on out-of-time or hold-out data, not only on the development sample, to detect overfitting.
• Lifecycle: origination, account management, collections, recovery. Scores guide limits, pricing and treatment at each stage.
• Through-the-cycle PD is more stable across the cycle. Point-in-time PD moves with current conditions.
• Downturn LGD reflects recoveries in weak conditions and is usually higher than average LGD.
• Fair lending: models must not produce unlawful discrimination, so check variables and outcomes for bias and keep them explainable.

Common mistakes

• Saying retail portfolios have no concentration risk. Fix: Retail removes idiosyncratic risk but keeps systematic and geographic or product concentration, such as one region's housing market.
• Assuming retail loans are always lower risk than corporate loans. Fix: Compare risk by PD, LGD and correlation. Unsecured cards can have high loss rates even though each loan is small.
• Saying discriminant analysis and logistic regression make the same assumptions. Fix: Remember that discriminant analysis assumes multivariate normal predictors with equal covariance. Logistic regression does not.
• Treating the score as a probability of default. Fix: A score is a scaled log-odds. Convert it back through the scaling formula to get a probability.
• Treating Gini and AUC as two different measures of performance. Fix: Remember Gini = 2 × AUC − 1. AUC 0.80 means Gini 0.60. They carry the same information.
• Saying AUC of 0.5 is the worst possible result. Fix: AUC 0.5 is random ranking. AUC below 0.5 means the scores are inverted, which you can fix by flipping the sign.
• Using application scores to manage limits on existing accounts. Fix: Remember that behavioral scores use account performance and are updated often; application scores are for decisions at origination.
• Ignoring EAD when raising credit limits. Fix: A higher limit raises potential exposure. Include EAD in any limit increase analysis.
• Using recovery rate in place of LGD in the EL formula. Fix: Compute LGD = 1 − recovery first. A 35% recovery means a 65% LGD.
• Multiplying only some roll rates, or using the wrong starting bucket. Fix: Write the chain of buckets first and match each rate to a transition. Start where the question's balance sits.

Exam tips

• Expect pairing questions: match the characteristic to retail or corporate. Use the many-small versus few-large test.
• Watch for absolute words such as always, never or only. Most are wrong.
• Know that Basel treats retail as its own IRB asset class with pool-based risk estimation.
• For numeric questions, convert pool default rates to EL with PD × LGD × EAD and check units.
• Expect conceptual questions on logistic regression versus discriminant analysis. Lead with assumptions and the probability output.
• For scaling, practise the doubling-of-odds shortcut until it is automatic.
• Link reject inference to selection bias in the through-the-door population.
• Watch wording such as always or guarantees. Scorecard methods rarely guarantee anything.