FRM Exam Part II · Introduction to Credit Risk Modeling and Assessment

Credit Scoring and Quantitative Assessment of Borrowers

Q: What is the Altman Z-score formula and how do I interpret it?

The original formula is Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5, using working capital, retained earnings, EBIT, market equity to liabilities and sales, each scaled as the ratios above. Z above 2.99 is safe, between 1.81 and 2.99 is grey, and below 1.81 is distress. Lower scores mean higher distress risk.

Q: How does logistic regression work in credit scoring?

It builds a linear score from borrower ratios, then converts it to a probability with PD = 1 ÷ (1 + e^(−S)). The score is the log-odds of default. Coefficients are estimated from historical default data, and each one shows how a ratio changes the odds of default.

Q: Why use logistic regression instead of a simple linear model for default?

A linear model can produce values below 0 or above 1, which are not valid probabilities. The logistic function keeps the output between 0 and 1. It also suits a yes or no outcome such as default or no default.

Q: How do I assess the creditworthiness of a corporate borrower?

Calculate leverage, coverage, liquidity and profitability ratios, and compare them with peers and past trends. Add a scoring model such as Altman Z or a logistic PD. Then weigh qualitative factors such as management, industry outlook, covenants and collateral before deciding.

Updated 11 October 2026 · Fact-checked

Credit scoring turns a borrower's financial data into a number that ranks default risk. The Altman Z-score is a weighted sum of five ratios, with safe, grey and distress zones. Logistic regression converts ratios into a default probability between 0 and 1. To solve questions, compute the ratios, apply the model, then interpret.

Understand Credit Scoring and Quantitative Assessment of Borrowers

Credit scoring uses past data to rank borrowers by how likely they are to default. A lender collects financial ratios for borrowers that defaulted and borrowers that did not. A statistical model then finds which ratios separate the two groups. The output is a score or a probability of default (PD).

The Altman Z-score is a discriminant model. It combines five ratios into one number. Higher Z means lower distress risk. The original 1968 model was built for listed manufacturing firms. Variants exist for private firms and for non-manufacturers and emerging-market firms, with different weights and cut-offs. Always check which version the question gives you.

Logistic regression fits a different shape. A linear score is built from the ratios: score = β0 + β1x1 + β2x2 + and so on. The score is then passed through a logistic function so the result is always between 0 and 1. That result is read as a PD. The score itself is the log of the odds of default. This is why each coefficient has a clean meaning: a one-unit rise in a ratio multiplies the odds by e raised to its coefficient.

Beyond the models, analysts read key ratios in four groups: leverage (debt to EBITDA, debt to equity), coverage (EBIT or EBITDA to interest), liquidity (current ratio, working capital to assets) and profitability or efficiency (EBIT to assets, sales to assets). Quantitative scores are only one input. Judgement on management, industry and covenants still matters.

Models have limits. They rely on historical data, accounting figures can be stale or managed, and the grey zone gives no clear answer. A model built in one period or sector may perform badly in another. That is why validation matters.

Key formulas to remember

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.

How to solve Credit Scoring and Quantitative Assessment of Borrowers questions

Use this order for any scoring question, whether it asks for a number or an interpretation.

1Identify the model: Altman Z (original, private or non-manufacturer version) or logistic regression. Note the weights or coefficients given.
2List the inputs and check units. Convert everything to the same currency and period, and note whether equity is market or book value.
3Compute each ratio first, as a decimal. Write them down before applying weights.
4Apply the model. For Z, multiply each ratio by its weight and add. For logistic, compute S, then PD = 1 ÷ (1 + e^(−S)).
5Check the sign and size. A negative S must give PD below 50%. A very high Z should sit in the safe zone.
6Compare with the cut-offs or a threshold and name the result: safe, grey or distress, or the PD level.
7Interpret in words: which ratio drives the result, what the lender should do, and what limits the model.

Quickest way: Ratio-first shortcut

When to use it: Use when time is short and the options are far apart, which is common in an 80-question paper.

Compute all five products in the original Z-score, each as weight × ratio. X3 has the largest weight (3.3), but weight alone does not set the contribution. A ratio with a smaller weight but a large value, often X4 or X5, can dominate the score.
Add the products and compare with 1.81 and 2.99. You often need only the zone, not the exact value.
For logistic questions, find the sign of S. Negative S means PD below 50%, positive S means above 50%. Eliminate options on the wrong side.
Remember benchmarks: S = 0 is 50%, S = −2 is about 12%, S = 2 is about 88%.
For coefficient questions, use e^β directly. A β of 0.7 roughly doubles the odds, since e^0.7 is about 2.01.

Common mistakes in Credit Scoring and Quantitative Assessment of Borrowers

Using the wrong equity value in X4 of the original Z-score.
Students see 'equity' and use book value from the balance sheet.
Fix: Original Z uses market value of equity ÷ book value of total liabilities. Book equity is used only in the private-firm and non-manufacturer versions.
Treating the grey zone as a default signal or as safe.
Students want a yes or no answer.
Fix: Call it indeterminate. Say the firm needs further analysis, such as cash flow review and covenant checks.
Reading the logistic score S as the probability of default.
The score looks like the final answer.
Fix: S is the log-odds. Always convert with PD = 1 ÷ (1 + e^(−S)).
Confusing odds with probability when interpreting coefficients.
Both measure likelihood, and e^β is easy to misread.
Fix: A one-unit rise in x multiplies the odds by e^β. It does not multiply the PD by e^β, and it does not add β percentage points to PD.
Dropping the sign of a coefficient.
Coverage ratios have negative coefficients because higher cover lowers default risk.
Fix: Plug in the sign exactly as given. A negative coefficient times a positive ratio reduces S.
Applying the manufacturer Z-score cut-offs to a bank or service firm.
Students memorise one set of numbers.
Fix: Z-scores rely on asset-heavy industrial ratios. Use the version stated in the question and note the model may not suit financial firms.

Worked examples

Example 1

A listed manufacturer has total assets of $500 million, working capital of $60 million, retained earnings of $100 million, EBIT of $40 million, market value of equity of $300 million, total liabilities of $250 million and sales of $450 million. Using the original Altman Z-score, what is Z and which zone is the firm in? Options: A) 1.81 B) 2.31 C) 2.99 D) 3.45

Show the solution

X1 = 60 ÷ 500 = 0.12.
X2 = 100 ÷ 500 = 0.20.
X3 = 40 ÷ 500 = 0.08.
X4 = 300 ÷ 250 = 1.20.
X5 = 450 ÷ 500 = 0.90.
Z = 1.2(0.12) + 1.4(0.20) + 3.3(0.08) + 0.6(1.20) + 1.0(0.90).
Z = 0.144 + 0.280 + 0.264 + 0.720 + 0.900 = 2.308, about 2.31.
2.31 lies between 1.81 and 2.99, so the firm is in the grey zone.

Answer: B) Z ≈ 2.31, which is in the grey zone. The model neither flags distress nor confirms safety, so further credit analysis is needed.

Example 2

A bank's logistic model gives the log-odds of default as S = −4.0 + 0.6 × (Debt ÷ EBITDA) − 0.5 × (Interest cover). A borrower has Debt ÷ EBITDA of 5.0 and interest cover of 2.0. What is its PD? Options: A) 2.3% B) 11.9% C) 26.9% D) 88.1%

Show the solution

S = −4.0 + 0.6(5.0) − 0.5(2.0).
S = −4.0 + 3.0 − 1.0 = −2.0.
PD = 1 ÷ (1 + e^(−S)) = 1 ÷ (1 + e^2).
e^2 = 7.389, so PD = 1 ÷ 8.389 = 0.1192.
Check: S is negative, so PD must be below 50%. That removes D. The odds are e^(−2) = 0.135, which is about 1 to 7.4, consistent with PD near 12%.

Answer: B) PD ≈ 11.9%. Option D comes from dropping the minus sign in S.

Exam tips

Know the five Altman ratios, their order and their weights cold. Questions often give the weights but expect you to know what each ratio is.
Expect questions that ask which ratio or input is the weakness. In the original Z-score, EBIT ÷ assets has the largest weight (3.3), so a change in it moves Z most per unit of ratio. The actual contribution depends on the ratio's size as well as its weight. In the worked example, X3 adds only 0.264, while X4 adds 0.720 and X5 adds 0.900.
Be ready to interpret a logistic coefficient as a change in odds, and to say whether a sign is sensible (more leverage should raise PD).
Link models to their limits: backward-looking data, accounting distortion, sector fit and the need for validation and back-testing against actual defaults.
In case-style questions, finish with a recommendation. Say what the score implies and what other information you would still want.

Practice questions from Introduction to Credit Risk Modeling and Assessment