Good afternoon everyone!

This is my first blog post from Tel Aviv, Israel, where I decided to go to take some time to work on my CFA Level I curriculum as the exam scheduled on June 2nd is getting closer and closer.

The first topic I wanted to come back on is part of the “Understanding Income Statements” and in particular the LOS 25.g and 25.h where we have to learn how to compute Earnings per Share (EPS). This topic first looked quite cumbersome to me at the beginning as the formula can look pretty awful at first glance, and different criteria may look as if you just have too learn thing by heart, a method which I try to avoid anytime I can; if you understand a formula, you’ll remember it much better than if you just swallow and spit it stupidly.

## Basic EPS

The first thing to know about EPS is that it is computed from the point of view of the common stock holder. Depending on the securities issued by the company, you may have to alter its computations, but in its simplest form called Basic EPS, you just assume that the company has common stock and preferred stock. It is defined as follows:

$$\text{Basic EPS} = \frac{\text{revenue} – \text{preferred dividends}}{\text{weighted average of common stocks outstanding}} $$

This is pretty easy to understand, the nominator is the revenue minus the preferred dividends (as we are looking from the common stock holder’s point of view). The denominator seems dodgy, but it simply a weighted average because the company might have added some stocks during the year, so you just weight the stocks count by the portion of the year is been active in. For example if 50’000 stocks where added on July 1, you add $50000 \cdot \frac{1}{2}$ to the yearly stocks count.

## Diluted EPS

The problem arise when the company issued securities which my be dilutive, that is, that might decrease the Basic EPS if some right is exercised. There are 3 cases considered where this *might* be the case:

- Convertible preferred stocks
- Convertible bonds
- Stock Warrants

In general the Diluted EPS will be of the following form:

$$\text{Diluted EPS} = \frac{\text{revenue} – \text{preferred dividends} + \text{adjustment}}{\text{wgt. arg. of common stocks} + \text{additional created stocks}} $$

The procedure to compute the Diluted EPS is first to know whether the exercise of the holder rights would indeed *lower* the Basic EPS and if that’s the case, to compute the new EPS using the formula above. Otherwise, the EPS is still computed with the basic approach.

### Convertible Preferred Stocks

Convertible preferred stocks are preferred stocks which holders might convert into common stocks. To know whether the conversion would be dilutive, we compute the following value:

$$\frac{\text{dividends of convertible preferred stocks}}{\text{number of convertible preferred stocks}}$$

The criteria is simple, if this value is below the basic EPS, then it will be dilutive. It’s in fact simple to understand, if the portion of dividends added back to the dividends available for common stocks (by being removed from the preferred dividend pool upon conversion) compensates or betters the dividend per common stock decrease due to the increased number to common stocks, then it will not be dilutive.

If the conversion is dilutive, you simply add back the amount of preferred dividends that were subtracted in the Basic EPS method, and you add the additional number of common stock outstanding to the denominator.

$$\text{Diluted EPS} = \frac{\text{rev} – \text{pref. dividends} + \text{converted pref. dividends}}{\text{wgt. arg. com. stocks} + \text{nbr of converted stocks}} $$

### Convetible bonds

Convertible bonds are bonds which can be converted into a given number of common shares $n$. The thing to understand is that the interest that would have been paid to the bonds holders is added back to the EPS nominator (because it’s nod paid anymore), BUT, you have to subtract the tax deduction that was allowed on these interests so you just add $\text{interests on the convertible bonds} \cdot (1-t)$ where $t$ is the tax rate.

The criteria for a dilutive effect is very similar to previously explained:

$$\frac{\text{interests on convertible bonds} \cdot (1-t)}{\text{number of convertible bonds} \cdot n} < \text{Basic EPS}$$

The Diluted EPS is then computed as follows:

$$\text{Diluted EPS} = \frac{\text{rev} – \text{pref. dividends} + \text{interests on bonds} \cdot (1-t)}{\text{wgt. arg. com. stocks} + \text{number of convertible bonds} \cdot n} $$

### Stock Warrants

This is the last possibility we consider, and it’s basically when the company might have to issue new shares to warrants holders if they decide to exercise their options at a strike price $K$. The key thing to understand here is that the company could use the money collected from the exercise price from the warrants to buy back shares in the market and provide them to the warrants holders. So we take into account the average market price of the stock $\text{AMP}$ and the criteria for a dilutive effect is as follows:

$$\frac{\text{AMP} – K}{\text{AMP}} > 0$$

Basically the criteria gives the numbers of shares that cannot be bought back by the firm using the exercise price. Trivially, if $K=AMP$, then the company simply uses the money to buy back the stock and sells it making a profit of 0. If $K > \text{AMP}$, then the company makes a profit by buying the shares in the market and selling them at the market price, so it’s clearly not dilutive. So, the criteria might even be reduced to

$$ K < \text{AMP}$$

In case of dilution the Diluted EPS is computed as follows, assuming $n$ warrants are outstanding:

$$\text{Diluted EPS} = \frac{\text{rev} – \text{pref. dividends}}{\text{wgt. arg. com. stocks} + \frac{\text{AMP} – K}{\text{AMP}} \cdot n}$$

The three dilution effects can be combined in CFA exercises, but the approach should be split and applied to every sub-case presented above, and the global formula simply add ups the adjustments to the numerator and denominator of the Diluted EPS formula I gave at the top of the section. It’s is very important to always check whether a conversion/exercise is dilutive; it even spares some calculation.

Looking at it this way, there is no big formula to be remembered, just basic understand of what’s going on.

I’ll be back with more!

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