# CFA Level I: DuPont Analysis

Hello again,

Back to reality after the events of last week in the Premier League and last night in the Champions League with some … accounting, in a sense.

The CFA Level I curriculum has a lot of question about ratios and a lot of questions ask how the return on equity ROE is affected by different properties of the firm. Let’s first look at the basic formula for ROE:

$$\text{ROE} = \frac{\text{net income}}{\text{equity}}$$

Ok that’s quite intuitive, but it doesn’t really provide a lot of insight on how different properties of the company will affect the ROE. This is where DuPont analysis is useful. The idea is pretty simple, we will introduce factors in the formula which will cancel out each other but will help us understand what is happening underneath the ROE. Let’s look at the traditional DuPont equation:

$$\text{ROE} = \frac{\text{net income}}{\text{sales}}\frac{\text{sales}}{\text{assets}}\frac{\text{assets}}{\text{equity}}$$

We now have a multiplication of three ratios, respectively:

• Net Profit Margin
• Assets Turnover
• Leverage Ratio

This decomposition is very useful to answer questions. Indeed, you know that if ROE is low, it is because either the net profit margin is poor or because the asset turnover is poor or because the firm is under-leveraged.

An easy way to remember the formula is just to remember two of the terms (for me, I remembered net profit margin and financial leverage); the third term can be found again using simple math.

Now there is an extended DuPont equation which further decomposes the net profit margin as follows:

$$\text{ROE} = \frac{\text{net income}}{\text{EBT}}\frac{\text{EBT}}{\text{EBIT}} \frac{\text{EBIT}}{\text{sales}} \frac{\text{sales}}{\text{assets}}\frac{\text{assets}}{\text{equity}}$$

Again, you see ratios coming up:

• Tax Burden
• Interests Burden
• EBIT Margin
• Asset Turnover
• Leverage Ratio

You can perform an analysis on how the ROE would change by having a look at each ratio.

Finally, there is an interesting thing about ROE that also comes up in the Corporate Finance topic of the CFA Level I curriculum: it is related to growth and dividend payout ratio. Indeed, we have the following relation:

$$\text{g} = \text{ROE} \cdot (1-\text{dividend payout ratio}) = \text{ROE} \cdot \text{retention rate}$$

This is quite simple to understand and it allows you to solve questions when the growth rate is not explicitly given. The growth rate is in fact determined as the amount of return on equity that is not given away to shareholders as dividends. This amounts allows the company to grow and produce more net income in the following period.

That’s it for today. I’ll try to add more content soon!

# CFA Level I: Hypothesis Testing

Good evening,

As I keep practicing towards the Level I exam, I want to finish my review of the  Quantitative Finance material. This post will hence be the follow up of my previous post. I will here discuss how you test a hypothesis on some statistical measure.

## General Concept

The main concept is as follows, you make an initial hypothesis which is called the     null hypothesis, $H_0$, and which is the statement you want to reject. If $H_0$ is rejected, we hence can accept the alternative hypothesis $H_a$. Of course in statistics, you can never be sure of anything. Hence, you can only reject the null hypothesis with a certain confidence level $\alpha$. It is important to understand that if you can’t reject the null hypothesis, it does not mean that you can accept it! Hence, rejecting the null hypothesis is more powerful than failing to reject it. So, if you want to prove some statement, you should test for its opposite as the null hypothesis; if you can reject it, then you can accept the alternative hypothesis which is your original statement. A test can either be one-tailed or two-tailed, depending on what you want to test.

## First example

Let’s take a simple example: you measure a sample of the returns of the S&P that you assume to be normal. You measure a sample volatility $\mu_s$ and a sample standard deviation $s$. By the central limit theorem, we now that the estimate of the mean follows a law $\mathcal{N}(\mu_0, \frac{\sigma^2}{n})$ where $\mu_0$ is the population mean and $\sigma$ is the population standard deviation. Now, you cannot use your measure $\mu_s$ to say that you found the true population mean, because it’s just a sample statistic. However, what you can say is that, given the fact that you found the sample statistic $\mu_s$ and a given confidence interval $\alpha$, it is highly improbable that the population mean $\mu_0$ was equal to some value $x$. Hence the null hypothesis is $\mu_0=x$.

Let’s assume that you found that $\mu_s=0.1$, $s=0.25\%$ and $n=250$.  You want to show that $\mu_0 \neq 0$, so you null hypothesis is $\mu_0 = 0$ and with a significance level $\alpha=5\%$. You can compute the z-statistic as follows:

$$z= \frac{\mu_s-\mu_0}{\frac{s}{\sqrt{n}}} \sim \mathcal{N}(0,1)$$

Now if $\mu_0$ was equal to 0, you know that there is 5% chance that the estimate $\mu_s$ would lie outside the range $\mu_0 \pm z_{2.5\%} = 0 \pm 1.96 = \pm 1.96$ (because the sample statistic can lie on both side of the distribution). The z-statistic was computed at 6.33, which is more than 1.96, so you can reject the null hypothesis $\mu_0=0$ and accept the alternative hypothesis $\mu \neq 0$.

This was a two-tailed hypothesis. One-tailed hypothesis would be looking at only one side of the distribution: $H_0 = \mu_0 \geq 0$ or $H_0 = \mu_0 \leq 0$. As a rule of thumb, you can remember the the null hypothesis always contains the “equal” sign.

## P-values

The p-value is the probability (assuming the null hypothesis is true) to have a sample statistic at least as extreme as the one being measure. You compute it by look at the test statistic (6.33 in the previous example) and you find the probability (using the Z-table) that a test statistic can be above that value (and you multiply it by 2 if it is two-tailed). In this case, the p-value is very close to 0. Now, if the p-value of the test statistic is below the significance level of the test, you can reject the null hypothesis. This is useful if you want to discuss statistics without having to impose a certain significance level $\alpha$ to the reader; you can just display the p-value and let him decide whether it’s good enough or not.

There are some other hypothesis tests presented in the curriculum, but this is the main framework to remember. It’s pretty easy, and it allows you to score a lot of points in the quantitative finance part of the exam.

# CFA Level I: Buying stocks on margin.

Hi everybody,

Today I’ll try to write several posts on the different things I looked at during my time in Tel Aviv, which will come to an end very early tomorrow morning.

I will also setup soon a page where I will display my progress in terms of score percentage in my practice sessions as time flows towards the exam day.

The posts I will be adding to the blog mostly concern things you can learn quickly and that will allow you to score easy points in the exam by just applying the formula. The first one of this series has actually been written already, it was concerning Basic and Diluted EPS.

In this post, I will talk about buying stocks on margin. This might seem at first glance quite complicated, but it’s in fact a quite simple concept. When you buy stocks, brokers will allow you to buy them on margin, that is, they will allow you not to deposit the full amount you would require to buy the stocks and lend you to rest of the amount for a given interest rate $r$. This allows you to gain leverage on the investment. As always when it comes to leverage, it’s a double-edged sword; your gains will be magnified by the leverage effect, but so will your losses.

Let’s take an example which I will you throughout the post: I want to buy 100 stocks priced today at 10. Assuming my broker allow be to buy using an initial margin $m_i$ of 40%, I will have to deposit only $100 \cdot 10 \cdot 40\% = 400$. Now if the price in a year is 11 (that is, a return of 10%) and I decide to sell the stock, I will make a profit on the stock price of $(11-10)*100=100$. The simple case assumes that the broker lend my the extra $600$ for an interest rate $r=0$. My return on the investment is hence $\frac{400+100}{400} = 25\%$. This is much bigger that the initial stock return. Another way to compute this simple case is to compute the leverage ratio $LR=\frac{1}{m_i} = \frac{1}{0.4}=2.5$ and then multiply it by the return of the stock to get the leveraged return: $10\% \cdot 2.5 = 25\%$. Okay so that’s easy. The CFA Institute might be willing to make this a bit more complex by adding dividends $d=0.5$ to the stock, and setting $r=5\%$ (the values are taken as examples). The reasoning is still the same:How much was I required to invest? still 400. How much do I get from the increase in price? still 100. How much dividend do I get? $100 \cdot 0.5=50$. How much interests do I have to pay to the broker? $100 \cdot 10 \cdot (1-40\%) \cdot 5\% = 30$. So you can compute the return on the investment as follows: $$\frac{400+100 + 50 – 30}{400} -1 = \frac{520}{400}-1 = 30\%$$ Finally, the last typical question is about margin calls. The idea is that the broker (and the regulators) would want to make sure that you are able to pay your debt if the investment goes bad, as you did not deposit the full amount of the investment. Hence they have a maintenance margin $m_m$ which is a “limit” under which they do not want your margin to go. The price under which you will get a margin call (that is when the broker requires you to refill the account back to the initial margin requirement) is computed as follows: $$P_0 \frac{1-m_i}{1-m_m}$$ So if we assume in our example that $m_m=25\%$, then you will have a margin call when the price reaches $10 \frac{0.6}{0.75} = 8$ So, with basically one formula to remember, you will get some very easy points in the exam!