Category Archives: Level I

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!

CFA Level 1 Review, Quantitative Finance: Sampling & Estimation and Hypothesis Testing (Part I)

Good evening everybody,

First of all, I would like to thank you all for the feedback you provided me with on Facebook and personally. I understand that this kind of reviews have been quite appreciated, even for people who are actually only considering taking the CFA in the future.

I decided to skip the part of Quantitative Finance dedicated to probabilistic concepts and well-known probability distribution, as I reckon that it would be too basic to really be of any interest in here.

However, I though that the “Sampling & Estimation” and “Hypothesis Testing” chapters were good refreshers and could be of some value for some of you. I’ll start with the first one and carry on another day with the second.

Sampling & Estimation

The Central Limit Theorem

The key concept to understand for these topics is the Central Limit Theorem (CLT) which is well known to anybody who basically went through high-school. Let’s restate it for the sake of completeness of the post:

For a simple random samples of size \(n\) from a population with mean \(\mu\) and variance \(\sigma^2\), the sampling distribution of the sample mean \(\bar{x}\) approaches a normal probability distribution with mean \(\mu\) and variance \(\frac{\sigma^2}{n}\) as the sample size become large.

Or, more formally, let \(S_n = \frac{1}{n} \sum_{i=1}^n X_i\), then

$$S_n \quad \overset{n \to \infty}{\longrightarrow_d} \quad \mathcal{N}(\mu,\frac{\sigma^2}{n})$$

In the CFA, they assume that \(n\) is large enough when \(n \geq 30 \).

Note that the CLT is valid for any population probability distribution.

Estimating the mean of a population with a sample

In the exam, you will be using the CLT to compute confidence intervals of population parameter estimates.

The main parameter you will have to estimate is the mean of a population using a given  sample. The way to compute an estimate of the mean is to compute the sample mean as follows:

$$ \bar{x}=\frac{1}{n}  \sum_{i=1}^n x_i $$

If you were to have several samples from a population, and you computed several estimates of the mean, the distribution of the different sample mean would be \(\bar{x} \sim \mathcal{N}(\mu,{\sigma_{\bar{x}}}^2)\) where \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\). This assume that you know the population variance \(\sigma^2\). If you do not know the variance of the population, you can compute the standard error of the sample mean \(s_{\bar{x}} = \frac{s}{\sqrt{n}}\) where \(s=\sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i – \bar{x})^2}\) is the sample standard deviation.

To compute the confidence interval of a mean estimator you use the following formula:

$$\text{point estimate} \pm \text{reliability factor} \times \text{standard error} (1)$$

The point estimate is actually your estimate of the mean \(\bar{x}\). The standard error is computed as explained above. The reliability factor depends on the level of confidence \({\alpha}\) of the confidence interval with level \(\alpha\) (\(CI_\alpha\)) you wish to compute.

When the variance of the population is known, the reliability factor is taken as the point of the standard normal distribution \(z \sim \mathcal{N}(0,1)\) where the value below the curve is equal to half the level of confidence \(z_{\alpha/2}=\Phi^{-1}(\alpha/2)\). The level of confidence is divided by 2 because you take half of the confidence for each side of the curve. Looking back at the formula to compute the the confidence interval, you will see that  it is easy to understand. Basically, you take your estimate as the central point of confidence interval, and the you use the standard normal distribution to determine how much of the values below the curve on each side you need to take, and you scale it with the standard deviation of the mean estimates.

When the population variance is unknown, instead of using the standard normal distribution, you would use the Student-t distribution. This distribution has a single parameter, the degrees of freedom \(\text{df}\). Basically, the Student-t distribution looks like a standard normal distribution but with fatter tails. As \(\text{df} \longrightarrow \infty\), the student-t distribution evolves towards the standard normal distribution. So, when you do not know the variance of the population, the reliability factor is a strudent-t variable with \(n-1\) degrees of freedom : \(t_{\alpha/2}\).

So the only thing that is difficult here is basically to know when to use the right reliability factor. This depends on whether the population variance is known, whether its distribution is known and whether the number of observations was large enough. The right strategy is summarized below:

Population distribution \(n < 30\) \(n \geq 30\)
Normal distribution with known variance z-statistic z-statistic
Normal distribution with unknown variance t-statistic t-statistic
Unknown distribution with known variance \(\emptyset\) z-statistic
Unknown distribution with unknown variance \(\emptyset\) t-statistic

Finally, it’s important to note that when \(n \geq 30\), you could possibly always use the z-statistic, but when variance is unknown, using t-statistic is safer.

I’ll be back with the review over “Hypothesis Testing” soon.

Thanks for reading!

CFA Level 1 Review: Quantitative Finance, Time Value of Money.

In this post, I will present a summary of the quantitative finance part of the CFA Level 1 exam.

The time value of money is a trivial concept in Finance, which can be summarized as “one dollar today is better than one dollar tomorrow“, because of the risk-free interest rate \(r\). The following formulas allow to compute the present value \(PV\) of a future cash flow \(FV\) (in case of a single cash flow) or \(CF_i\) (in case of several future cash flows).
$$PV=FV (1+r)^{-t}=\frac{FV}{(1+r)^t} \quad (1)$$

$$PV=\sum_{i=1}^N \frac{CF_i}{(1+r)^i} \quad (2)$$

That’s all trivial.

If there are infinite cash flows of constant value c, the product is called a perpetuity and its values is computed as follows:


The only tricky part comes when you have multiple cash flows coming at the beginning \(PV_b\) or at the end of the period \(PV_e\). To compute the value of the investment, you just have to know that \(PV_b = (1+r) PV_e\) as the two computation are basically shifted one period from one to the other.

When trying to compute the value of a project, the idea is to compute the NPV (net present  value) of the project which is the present value of the project cash flows which can be computed easily using (2). Any project with \(NPV < 0\) should be rejected.

That was pretty easy anyway.

I’ll be back with more.