A.2.6 Advanced Extensions

Basic Prerequisites: 02 · Profitability and 04 · Liquidity & Leverage.

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1. Intuition & Practical Objective

The single-ratio pages asked how hard the capital earned and how well the balance sheet stood. This page is the launchpad into composite and decomposed instruments - where isolated ratios get combined into predictive scores and broken apart into their drivers. Three tools, three distinct jobs:

• Altman Z-score (1968): combines five balance-sheet/income ratios into one distress composite that separates bankrupt from healthy firms. The canonical proof that a small set of ratios carries real predictive power.
• ROIC / ROCE decomposition (Penman Ch 11; DuPont): breaks the headline return into operating-margin × asset-turnover × leverage terms so you can see where a return comes from and whether it is repeatable.
• Piotroski F-score (2000): turns nine binary fundamental signals (profitability, leverage/liquidity, operating efficiency) into a 0–9 quality score - a working composite of exactly the ratios on this folder’s hub table, and the most-cited bridge from statement analysis to a mechanical value screen.

The through-line (Penman’s whole thesis): a ratio is only as good as the decomposition that explains it and the score that aggregates it. Each tool below is the answer to “so what does this one number mean?“

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2. Mathematical Ground Truth & Derivations

Altman Z (1968, discriminant function). With $X_1$ = working capital / total assets, $X_2$ = retained earnings / total assets, $X_3$ = EBIT / total assets, $X_4$ = market value of equity / book value of total liabilities, $X_5$ = sales / total assets:

$$Z=1.2X_1+1.4X_2+3.3X_3+0.6X_4+1.0X_5.$$

Zones: $Z>2.99$ safe · $1.81\le Z\le 2.99$ grey · $Z<1.81$ distress. (The weights assume $X_1..X_3$ are decimal fractions - see the hub’s scaling caveat.)

ROIC / ROCE decomposition (Penman Ch 11; DuPont). The operating return is the product of margin and turnover, and the shareholder return adds the leverage effect:

$$\text{RNOA}=\underset{\text{operating margin}}{\underbrace{\frac{\text{NOPAT}}{\text{S}}}}\times \underset{\text{asset turnover}}{\underbrace{\frac{S}{\text{avg NOA}}}},\text{ROCE}=\text{RNOA}+[\text{FLEV}\times (\text{RNOA}-\text{NBC})].$$

Every term is independently attackable: margin is pricing power, turnover is asset productivity, and the bracket is the leverage effect that is additive only while RNOA > NBC.

Piotroski F-score (2000). Nine binary signals summed to a 0–9 score. Profitability: ROA $>0$, CFO $>0$, ΔROA $>0$, CFO $>$ ROA (accrual quality). Leverage/liquidity/funding: ΔLeverage $<0$ (long-term debt fell), ΔCurrent-ratio $>0$, no equity issuance. Efficiency: ΔGross-margin $>0$, ΔAsset-turnover $>0$. Piotroski applies it inside high book-to-market (value) firms, where a high F-score separates the financially strong from the distressed - value + quality in one mechanical screen.

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3. Computational Implementation - Z-score and F-score, stdlib only

Runs on the standard library. Computes the Altman Z for the sample firm (returns “Safe”), then the full 9-signal Piotroski F-score (returns 7/9), using both current- and prior-year statement data.

(Altman’s 1968 sample separated bankrupt from healthy firms on exactly this weighted set; Piotroski’s 2000 high-book-to-market sample showed high-F-score firms beat low-F-score firms decisively in subsequent returns.)

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4. Failure Modes & First-Principles Breakdowns

1. Altman Z is a 1968 manufacturing model. Its weights and thresholds were estimated on listed manufacturers; private firms, financials, and young growth firms distort it (the original paper itself, and Altman’s 2006 Corporate Financial Distress treatment, adapt the coefficients for private/non-manufacturing firms). Use the shape of the signal, not the literal cutoff, outside manufacturing.
2. The F-score rewards any improvement, even from a terrible base. A signal of “loss narrowed from −40% to −10%” still scores F_ROA = 1; and because it is designed for distressed value firms, the same signals are ambiguous for healthy growth firms. It is a distressed-value screen, not a universal quality score (Piotroski’s own conditioning on high book-to-market is the point).
3. Both tools are as good as the inputs. A Z-score built on restated, smoothed, or fraudulently adjusted statements inherits all of 05 · Failure Modes. The accrual signal inside the F-score (CFO vs. ROA) exists precisely to catch that.
4. Decomposition ≠ causality. Showing ROE = margin × turnover × leverage explains arithmetically where a return came from; it does not tell you which driver is sustainable. Penman’s entire forecasting apparatus is about projecting which of the three drivers persists.

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5. References

• Altman, Edward I.: “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy” (JF, 1968)
• Ohlson, James: “Financial Ratios and the Probabilistic Prediction of Bankruptcy” (JAR, 1980)
• Piotroski, Joseph: “Value Investing: The Use of Historical Financial Statement Information to Separate Winners from Losers” (JAR, 2000)
• Penman, Financial Statement Analysis and Security Valuation
• Subramanyam, Financial Statement Analysis

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Screen building: Quantitative Fundamental Investing · Fundamental Analysis & Screening
• Distress/theory: Capital Structure & Corporate Finance
• Defense: Accounting Quality & Red Flags