Diluted EPS: if-converted and treasury stock methods

What is diluted EPS?

Diluted EPS is EPS if every security that could become a common share, and would lower EPS, had been converted or exercised. It is the conservative figure, so it is never above basic EPS.

$$EPS_d = \frac{NI - D_p + D_{conv} + A}{W + C + \Delta S}$$

Here $D_p$ is all preferred dividends, $D_{conv}$ those on convertible preferred being assumed converted, $A$ the after-tax interest on convertible debt being assumed converted, $C$ the conversion shares and $\Delta S$ the net new option shares. Only securities that dilute enter.

How does the if-converted method work?

Memorize

The if-converted method pretends the convertible was converted at the start of the year. The payments you would no longer make go back into the numerator, and the new shares go into the denominator.

• Convertible preferred stock : add back its preferred dividends, add the conversion shares.
• Convertible debt : add back the after-tax interest, add the conversion shares. Interest saves tax, so the add-back is interest × (1 − tax rate).

$$A = F \times c \times (1 - t)$$

How does the treasury stock method work?

Memorize

The treasury stock method handles options and warrants . Assume they are exercised, then assume the exercise cash is used to repurchase shares at the average market price. Only the net new shares dilute. The numerator does not change.

$$\Delta S = N - \frac{N \times K}{P_{avg}} = N \times \frac{P_{avg} - K}{P_{avg}}$$

The options count only when they are in the money (K below the average price). Under International Financial Reporting Standards (IFRS) the shortfall is described as shares issued for no consideration, which gives the same number.

Exam tip The year-end price is a decoy. Every treasury stock question wants the average price for the period.

What does a full calculation look like?

Compute basic EPS, test each security, include only those that dilute, then divide.

How do you spot an antidilutive security?

Antidilutive securities are those that would raise EPS if included, so leave them out. For a convertible, divide its add-back by its new shares and compare with basic EPS. Above basic EPS means antidilutive.

$$q = \frac{A \text{ or } D_{conv}}{C}, \quad \text{include only if } q < \text{basic EPS}$$

Complex capital structurediluted EPS assumes the dilutive ones are issued

• If-converted method Memorize convertibles
  • Convertible preferred add back dividends, add shares
  • Convertible debt add back after-tax interest, add shares
• Treasury stock method Memorize options and warrants
  • In the money (K below average price) net new shares, no numerator change
  • Out of the money (K above average price) antidilutive, excluded

Does including the security raise EPS?

• No, EPS falls: dilutive Memorize include it
• Yes, EPS rises: antidilutive Memorize exclude it
  • Convertibles add-back per new share above basic EPS
  • Options exercise price above average price

Options are antidilutive when K is above the average price. Because every antidilutive security is left out, diluted EPS can never exceed basic EPS.

Go deeper Several convertibles: rank them

With more than one convertible, the strict procedure ranks securities from the lowest add-back per new share to the highest and adds them in that order, stopping when EPS stops falling. At Level I you can usually test each against basic EPS on its own. The ranked procedure matters when including a security changes the EPS against which the next one is tested.

Go deeper Securities issued or converted during the year

A security outstanding all year is treated as converted at the start of the year. One issued during the year is included from its issue date, weighted by the fraction of the year. One converted during the year has its effect for the part before conversion in the numerator and denominator, while the actual shares are already in the weighted average count after conversion.

Go deeper Why the average price and not the year-end price

Diluted EPS is an earnings-for-the-period figure. The share price the options would have been tested against over the period, and the price at which the company would have repurchased shares through the period, is the average. A year-end price would reflect a single day, so a rally in December would overstate dilution for the whole year.