# Earnings per Share (EPS) computations in CFA Level I

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!

# Logarithmic approximation: application in CFA Level I and interview questions

Good afternoon everyone,

I’m almost done going through the CFA material for the Level I curriculum and I came across what they simply call an approximation in the Schweser resources but what actually is in fact a logarithmic approximation of numbers close to 1.

## What is the logarithmic approximation for numbers close to 1?

Assuming you have a number which is relatively close to 1, say, 1.02, then you can write the following:

$$x \simeq\ln (1+x)$$

Let’s plot two functions $f(x)=x$ and $g(x)=\ln(1+x)$ to see how they behave when $x$ is small:

As you cane see both function yield approximately the same value but $\ln(1+x) \leq x$; let’s visualize the error of the approximation (in absolute values):

The error is pretty small! As we can see on the graph it gets larger as $x$ gets away from 0, but this the important thing is to see that, as long as $x$ is close to 0, the approximation is pretty good.

## Application to CFA Level I: Fixed Income

At some point of the curriculum they discuss about, they talk about the term structure of the interest rates. As this term structure is not flat; higher interest rates are (usually) required for bonds with higher maturity. This is why you have different yield for bonds with the same features but different maturity $T$. Let’s yields defined at time $t$ for maturity is defined as $R(t,T)$, they are the spot yields at time $t$.

There is also the possibility to enter a forward agreement, called forward yield, which allows you at time $t$ to have a certain rate between time $T$ and $S$, which we denote by $F(t,T,S)$.

Now, if you now the term structure of the interest rates at time $t$, for $T$ and $S$ (i.e. you know $R(t,T)$ and $R(t,S)$), you can then find the forward rate $F(t,T,S)$ as follows:

$$F(t,T,S)=\left( \frac{(1+R(t,S))^{S-t}}{(1+R(t,T))^{T-t}} \right)^{\frac{1}{S-T}}-1$$

Let’s take an example, you know that $R(0,2)=4\%$ and $R(0,3)=5\%$ and you want to calculate $F(0,2,3)$. By applying the formula above, you find that $F(0,2,3)=7.03\%$.

You could also have used the logarithmic approximation by noticing that $1+F(0,2,3)$ if close to 1, and since $ln(1+x) \simeq x$, well you can write:

$$\ln(1+F(0,2,3))=\ln \left( \left(1+ \frac{(1+R(0,3))^3}{(1+R(0,2))^{2}} \right)^{1}-1 \right)$$

$$\ln(1+F(0,2,3))=\ln \left( \frac{(1+R(0,3))^3}{(1+R(0,2))^{2}} \right)$$

Using the properties of logarithms, you can write that:

$$\ln(1+F(0,2,3))=\ln\left( (1+R(0,3))^3\right) – \ln \left( (1+R(0,2))^{2}\right)$$

$$\ln(1+F(0,2,3))=2 \ln\left(1+R(0,3)\right) – 3 \ln \left( 1+R(0,2)\right)$$

Now you can apply the logarithmic approximation here : $\ln (1+R(t,T)) \simeq R(t,T)$ and it follows that:

$$\ln(1+F(0,2,3))=2 R(0,3)- 3 R(0,2)=0.07 \simeq F(0,2,3)$$

As you can see, both the real value, 7.03% and 7.00% are pretty close!

For the general case, we can write:

$$\ln \left(1+F(t,T,S)\right)=\ln \left(1+ \left( \frac{(1+R(t,S))^{S-t}}{(1+R(t,T))^{T-t}} \right)^{\frac{1}{S-T}}-1\right) \simeq F(t,T,S)$$

$$F(t,T,S)=\frac{1}{S-T} \left( (S-t)\ln \left(1+R(t,S)\right) – (T-t)\ln \left(1+R(t,T)\right) \right)$$

$$F(t,T,S) \simeq \frac{1}{S-T} \left( (S-t) R(t,S) – (T-t) R(t,T) \right)$$

For $t=0$, we get:

$$F(0,T,S) \simeq \frac{1}{S-T} \left( S \cdot R(0,S) – T \cdot R(0,T) \right)$$

## Application to potential interview questions

In interviews, especially for a quantitative position, the interviewer might want to know if you’re aware of this logarithmic approximation. Therefore, they would ask you to compute without a calculator the following in a short period of time:

$$\frac{(1.02)^6 \cdot (1.01)^4}{(1.03)^3 \cdot (1.05)^2 }$$

The only way to perform this computation properly is to nice that result would be close to 1, and hence look like $1+x$. You can then approximate $x$ using the logarithmic approximation $\ln (1+x) \simeq x$:

$$x \simeq \ln \left( \frac{(1.02)^6 \cdot (1.01)^4}{(1.03)^3 \cdot (1.05)^2 } \right)$$

$$x \simeq \ln \left( (1.02)^6 \cdot (1.01)^4 \right) – \ln \left( (1.03)^3 \cdot (1.05)^2 \right)$$

$$x \simeq 6\ln (1.02) + 4 \ln (1.01) – \left( 3 \ln (1.03) + 2\ln (1.05) \right)$$

$$x \simeq 6 \cdot 2\% + 4 \cdot 1\% – 3 \cdot 3\% – 2 \cdot 5\% = -3\%$$

Hence, you estimate that :

$$\frac{(1.02)^6 \cdot (1.01)^4}{(1.03)^3 \cdot (1.05)^2 } \simeq 1+x = 0.97$$

If you use a calculator, you find that the result is 0.9727388, which is pretty close to 0.97.

Pretty cool huh?

I’ll be back with more CFA footage soon!

# Introduction to Risk-Neutral Pricing Theory

I recently came across this QuantSE post where the author of the post tries to compute an expectation under the risk-neutral measure $\mathbb{Q}$.

Risk-neutral pricing is a technique widely use in quantitative finance to compute  the values of derivatives product and I thought I could write a post  explaining what the theory is and how it can be used to compute a simple option’s price.

## What is the Risk-Neutral Probability Measure?

Usually, probabilities on events are expressed in terms of the so-called “real world” probability measure $\mathbb{P}$, i.e, if a stock can either move up or down, and that you think that there is an equal chance for it to go either way, you would say that it will go up with probability $p=\frac{1}{2}$.

However, when you want to compute the price of a financial asset $X$ at time $t=0$, you would do it through the computation of its expected value of its discounted future cash flows. The problem is that investors discount risk with different rates depending on their risk aversion (they require a risk-premium), and you would need to perform an adjustment which is very difficult to estimate.

Therefore we would like to be able to use a probability measure $\mathbb{Q}$, equivalent to $\mathbb{P}$ (i.e that agrees on events that cannot happen) under which the investor is insensitive to risk. This means that when computing expectations using $\mathbb{Q}$,  we can discount cash flows using the risk-free rate $r$.

Mathematically, this is described by saying that under the risk-neutral probability measure $\mathbb{Q}$, discounted prices are martingales:

$$P(0,t) X_t = E_\mathbb{Q}[ P(0,T) X_T | \mathcal{F}_t]$$

Rearranging a bit and using $P(0,t)=\frac{1}{(1+r)^t}$, you get:

$$(1+r)^{T-t}= E_\mathbb{Q}[ \frac{ X_T}{X_t}| \mathcal{F}_t]$$

That is, under $\mathbb{Q}$, the expected value of the return on a asset $X$ from $t$ to $T$ is the risk-free rate $r$ (compounded from $t$ to $T$).

The risk-neutral probability measure is the probability measure that makes return on an investment the risk-free rate. It is “built” for that purpose.

The Fundamental Theorem of Asset Pricing (referred as FTAP thereafter) states that if markets are arbitrage-free and complete, then there exists a risk-neutral measure and it is unique  "A general version of the fundamental theorem of asset pricing" (Freddy Delbaen and Walter Schachermayer, 1994).

## How do we characterize the risk-neutral measure?

Let’s take a simple example. Assume a stock $S$ in a single step framework, where the initial price is $S_0$ . We define that after the single step, the price of the stock is either going up (in state $u$) $S_1=S_0 \cdot u$ with probability $p$ or going down  (in state $d$) $S_1=S_0 \cdot d$ with probability $1-p$ . Such a framework needs that $d < 1+r < u$.

Let’s find the probability $q$ such that discounted prices are martingales:

$$\frac{1}{(1+r)^0} S_0 = E_\mathbb{Q}\left(\frac{S_1}{(1+r)^1} | \mathcal{F}_0 \right) = \frac{1}{1+r} \left( S_0 u q + S_0 d (1-q) \right)$$

Dividing by $S_0$ and multiplying by $1+r$ on both sides , we get

$$1+r= uq + d(1-q) = uq+ d -qd = d+q(u-d)$$

And we can find easily that:

$$q = \frac{1+r-d}{u-d}$$

Ok, so we found the risk-neutral measure $\mathbb{Q}$ for $S$ by finding that $\mathbb{Q}(S_1 = S_0 u) = q$ and hence $\mathbb{Q}(S_1 = S_0 d) = 1-q$ .

## How do we use the risk-neutral measure?

Now, assume we want to price a derivative product $X$ which pays 1 if $S$ goes in state $u$ and 0 otherwise.

Using the FTAP, we can write

$$X_0 = E_\mathbb{Q} \left(\frac{X_1}{(1+r)^1} | \mathcal{F}_0 \right) = \frac{1}{1+r} E_\mathbb{Q} \left(1_{\{S_1 = S_0 u\}} \right) =\frac{1}{1+r} \mathbb{Q} \left( S_1 = S_0 u \right)$$

Note that by developing the possible outcomes of $X_1$, we did not have to characterize the risk-neutral measure for $X$.

Indeed, as we characterized $\mathbb{Q}$ for $S$, we can write:

$$X_0=\frac{1}{1+r} \mathbb{Q} \left( S_1 = S_0 u \right)=\frac{1}{1+r} \frac{1+r-d}{u-d}$$

This way, we easily managed to find the value of the derivative product $X_0$ without having to worry about risk-aversion and without having to even know the real-world probability measure $\mathbb{P}$.

I hope you enjoyed this, I’ll be back with more CFA footage another time.

## References

1. Freddy Delbaen and Walter Schachermayer (1994), A general version of the fundamental theorem of asset pricing, (MATHEMATISCHE ANNALEN), , 463-520.