Diluted Earnings Per Share – Comprehensive Study Notes

Technical Competencies & Learning Outcome

• Competency 1.2.2: Ability to evaluate the accounting treatment for routine transactions.
• Competency 1.3.2: Ability to prepare routine financial-statement note disclosures.
• Learning outcome: Correctly determine diluted earnings per share (EPS).

Key Terminology & Concepts (EPS Focus)

• Basic EPS
  • Formula: \text{Basic EPS}=\dfrac{\text{Net earnings available to common shareholders}}{\text{Weighted-average common shares outstanding (WACSO)}}
• Diluted EPS
  • Reflects the impact of all potentially dilutive securities (PCS): instruments that can be converted into, or allow issuance of, common shares (convertible debt, convertible preferred shares, stock options, warrants, etc.).
  • General computation steps:
  1. Calculate incremental EPS for each PCS.
     \text{Incremental EPS}=\dfrac{\text{Income effect}}{\text{Share effect}}
  2. Rank PCS from most to least dilutive (lowest incremental EPS first).
  3. Sequentially include PCS in diluted EPS calculation only if each inclusion reduces (or does not increase) the running provisional EPS; otherwise, the PCS is anti-dilutive and excluded.
  • Final disclosure: Basic EPS + Diluted EPS (highlighting anti-dilutive securities in note form).
• Treasury-stock method (TSM) for options/warrants
  • Assumes options/warrants are exercised; cash proceeds repurchase shares at average market price.
  • New shares added: \text{Shares issued} - \dfrac{\text{Proceeds}}{\text{Average market price}}
• If-converted method for convertibles
  • Assume conversion at beginning of period (or issue date).
  • Income effect:
  • Debt: add back after-tax interest \big(\text{Interest} \times (1-\text{tax rate})\big).
  • Preferred: add back dividends (cumulative or declared).
  • Share effect: add conversion shares.

Summary Problem – Kingman Crown Inc. (KCI)

Capital structure at Jan 1

• Common shares: 1,000,000 issued; \text{WACSO}_{\text{year-end}}=1,100,000.
• Convertible debt: Face \$2{,}500{,}000, coupon 6 %, each \$100 bond → 5 common shares.
• Class B non-cumulative preferred: 20,000 shares, dividend \$3.00/share, each preferred → 2 common shares.
• CEO options: 220,000 options, exercise price \$6.00.

Current-year share events

• May 1 – Issued 150,000 common shares.
• Jul 1 – CEO exercised 100,000 options when price \$6.50.
• Sep 1 – Redeemed 150,000 common shares.

Dividends & earnings

• Preferred dividend declared: 20{,}000 \times \$3 = \$60{,}000.
• Common dividend declared: \$0.90/share.
• After-tax net earnings: \$1,800,000.
• Net earnings available to common: \$1,740,000 (already net of preferred dividend).
• Average market price of common: \$7.
• Tax rate: 30 %.

Step 1 – Basic EPS

• \text{Basic EPS}=\dfrac{1{,}740{,}000}{1{,}100{,}000}=\$1.58 per share.

Step 2 – Incremental EPS for each PCS

1. Options (TSM)

   • Outstanding entire year: 120,000 (220k – 100k exercised).

   • Proceeds: 120{,}000 \times \$6 = \$720{,}000.

   • Shares repurchased: \dfrac{720{,}000}{7}=102{,}857.

   • Incremental shares: 120{,}000-102{,}857 \approx 17{,}143.

   • Exercised batch (100,000) outstanding Jan 1–Jun 30 (6/12):
     • Proceeds: 100{,}000 \times 6 = 600{,}000.
     • Shares repurchased: 600{,}000/7\approx85{,}714.
     • Incremental shares: 14{,}286 then weighted 6/12 = 7,143.

   • Total incremental shares from options: 17{,}143+7,143=24,286.

   • Income effect: None (purely equity instrument) → 0 earnings added.

   • Incremental EPS: 0 / 24,286 = \$0.00 (most dilutive).

2. Convertible debt (if-converted)

   • Interest saved: 2,500,000 \times 6\% = 150,000 pre-tax.
   • After-tax interest: 150,000\times(1-0.30)=105,000.
   • Shares issued: {2,500,000 \over 100} \times 5 = 125,000.
   • Incremental EPS: \dfrac{105,000}{125,000}=\$0.84.

3. Convertible preferred

   • Dividend saved: 60,000.
   • Shares issued: 20,000 \times 2 = 40,000.
   • Incremental EPS: \dfrac{60,000}{40,000}=\$1.50.

Step 3 – Rank PCS (lowest → highest incremental EPS)

1. Options (0.00) – most dilutive.
2. Convertible debt (0.84).
3. Convertible preferred (1.50).

Step 4 – Sequential inclusion & diluted EPS

• Diluted EPS disclosed: \$1.48.
• Basic EPS disclosed: \$1.58.
• Note disclosure: Convertible preferred shares are anti-dilutive; excluded from diluted EPS.

Conceptual Takeaways from KCI Example

• Options are always tested first because TSM normally makes them most dilutive (zero income effect).
• A PCS can be dilutive in isolation but anti-dilutive once other PCS are added. Test sequentially.
• Non-cumulative preferred dividends are only subtracted from earnings if declared. For conversion tests, add back only declared amounts.
• After-tax interest applies to debt; no tax effect on preferred dividends (dividends are distributions, not tax-deductible).

Practice Problem 1 – Jerry Ltd. (Outline Only)

• Opening shares: 700,000 common; 150,000 \$6 cumulative convertible preferred; 200,000 \$3 non-cumulative preferred.
• Share issue Sept 30: 120,000 common for \$2,100,000.
• Stock options: 100,000 @ \$12; average price \$18.
• Convertible bonds: \$5,000,000 face, 8 %, 50 shares per \$1,000 bond (total potential shares = 5,000\times50=250,000).
• Net earnings: \$3,200,000; preferred dividends declared: 900,000 (cumulative) + 300,000 (non-cum) = 1,200,000.
• Tax rate: 30 %.

Tasks (student practice):

1. Compute WACSO (factor in Sept 30 issue at 3/12 weighting).
2. Basic EPS: \dfrac{Earnings - \text{all preferred dividends}}{WACSO}.
3. Test dilutive securities: options (TSM), bonds (if-converted), cumulative preferred (if-converted) – rank & compute diluted EPS.

Practice Problem 2 – Kerti Inc. (Outline Only)

• Opening shares: 400,000 common.
• Cumulative conv. preferred: 11,000 shares, \$3.50 dividend, 2.5 share conversion.
• 11 % convertible bonds: \$2,000,000 at par, 37 shares per \$1,000 bond (potential shares = 2,000 \times 37 = 74,000).
• Warrants: 22,000 @ \$61.75.
• Stock options: 40,000 @ \$30.
• Two-for-one stock split on Mar 1 (all share data already split-adjusted).
• Additional issues: 2,000 shares on Apr 1; 40,000 shares on Oct 1.
• After-tax income: \$1,490,000.
• Tax rate: 40 %; average price \$60.
• No preferred dividends for two years → cumulative arrears exist but only current-year dividend subtracted in basic EPS.
• No conversions in the year.

Student steps:

1. Compute WACSO (split considered). Remember partial-year weightings:
   • Jan 1–Feb 28: 400,000 shares (pre-Apr issue)
   • Apr 1 issue 2,000 shares (9/12)
   • Oct 1 issue 40,000 shares (3/12).
2. Basic EPS: subtract current-year preferred dividend (even if not declared) because they are cumulative.
3. Potential dilutive instruments (options, warrants, bonds, preferred) – compute incremental EPS, rank, sequentially include.

Ethical & Practical Considerations

• Transparency: Users rely on diluted EPS to understand worst-case share dilution; aggressive exclusion of PCS undermines comparability.
• Forecasting & valuation: Analysts model fully diluted share counts; mis-stated dilution affects valuation multiples.
• Executive compensation: Options granted to CEOs influence dilution; exercising mid-year impacts WACSO and must be precisely timed in computations.
• Convertible instruments as financing strategy: Trade-off between lower interest/dividend costs and potential dilution; accounting highlights this impact rather than hiding it.

Quick-Reference Formulas & Reminders

• Interest (after-tax): \text{Interest} \times (1-\text{tax rate}).
• Dividends on cumulative preferred: subtract regardless of declaration for basic EPS; add back for if-converted method.
• Treasury-stock shares repurchased: \dfrac{\text{Proceeds (options price × shares)}}{\text{Average market price}}.
• Incremental shares (options): \text{Shares issued} - \text{Shares repurchased}.
• Weighted shares for partial-year events: multiply by fraction of year outstanding.
• Dilutive test: Include only if incremental EPS < provisional EPS.

Real-World Applications

• Capital-raising decisions: Companies compare fully diluted EPS pre- and post-transaction.
• Debt covenants & performance bonuses: Often tied to EPS targets; correct dilution accounting crucial.
• Market perception: Announcements of conversions or option exercises can trigger price adjustments as investors update share-count expectations.

Study Tips & Common Pitfalls

• Forgetting to tax-effect interest on convertible debt.
• Applying TSM using year-end instead of average market price.
• Ignoring the time weighting for options exercised or shares issued mid-year.
• Mis-classifying cumulative preferred dividends (must subtract even when skipped).
• Failing to re-rank PCS after each inclusion step.

Practice Drill

1. Re-work KCI example using a different average market price (e.g., \$8$$) to see how options’ dilutive effect changes.
2. For Jerry Ltd., test scenario where bonds are converted on Jul 1; compute separate diluted EPS.
3. Build a template spreadsheet with columns for Income adjustment, Share adjustment, Incremental EPS, automatic ranking & inclusion testing.