Slide 1: Lecture 1 - Du Pont Identity
Welcome to Lecture 1 on the Du Pont Identity.
Slide 2: Financial ratios
What is a Du Pont Identity? Basically, Du Pont Identity is a way for us to look at the
different components of the financial ratios: Return on Assets (ROA) and Return on
Equity (ROE).
The Return on Assets (ROA) is calculated as:
ROA = Net Income / Total Assets = NI / A
The Return on Equity (ROE) is calculated as:
ROE = Net Income / Total Equity = NI / E
It is good to always memorize these two formulas for ROA and ROE, because they are
the basic formulas from which all the Du Pont Identities are derived. And so, if you
know these two basic formulas by heart, you will be able to derive the Du Pont Identities
yourself, without having to memorize all the identities.
Slide 3: Du Pont Identity (DPI) for ROA
Let’s start with the Du Pont Identities for the ROA.
How do we get from the basic formula of ROA = NI/A to a more involved
formula/identity for the ROA. We do this by formula manipulation. Taking the basic
formula:
ROA = NI/A
We multiply and divide the right-hand-side of the formula by S, giving us
ROA = (NI/A) x (S/S)
(Note that S/S = 1, which means that we are multiplying the right-hand-side by 1, which
gives us the original formula of NI/A.)
Switching the denominators on the right-hand-side of the new formula, we get
ROA = (NI/S) x (S/A)
This is the first identity for the ROA formula.
Looking closely at this new formula, we see that (NI/S) is what we normally call the
profit margin, and (S/A) is what we normally call the asset-use efficiency ratio (AUE).
Therefore, ROA can also be written as:
ROA = Profit margin x Asset-use efficiency
That’s it! With a few strokes of the pen, we now have two identities for the ROA
formula.
Slide 4: Numerical Example
Let’s work through a simple numerical example of calculating ROA using one of the
identities. Given the following data:
NI/S = 0.1
S/A = 2
Calculate the ROA.
ROA = (NI/S) x (S/A)
Plugging the numbers for NI/S and S/A into the ROA identity, we get
ROA = 0.1 x 2 = 0.2
This gives us a Return on Assets of 20%.
Slide 5: What is S/A?
Remember when we were looking at the first identity for ROA, we have
ROA = (NI/S) x (S/A)
What does the second term on the right-hand-side of the formula, S/A, represent?
Again, S/A is what is referred to as Asset-use efficiency. What it literally means is the
dollar amount of sales generated per dollar amount invested in assets.
Slide 6: DPI for ROE
Next, let’s move on to the Du Pont Identities for ROE. How do we get from the very
basic ROE formula of ROE = NI/E to a more involved form of the formula such as this
one:
ROE = (NI/S) x (S/A) x (1 + D/E)?
Again, this is done by formula manipulation. First, multiply and divide the right-handside of the basic ROE formula by S:
ROE = (NI/E) x (S/S)
Switch the denominators in this new formula, and we get
ROE = (NI/S) x (S/E)
This is the first identity for the ROE formula.
Going one step further, multiply and divide the right-hand-side of this first identity by A:
ROE = (NI/S) x (S/E) x (A/A)
Switch the denominators between (S/E) and (A/A), and we get
ROE = (NI/S) x (S/A) x (A/E)
This is the second identity for the ROE formula.
An aside: Why do we call these formulas identities? This is because we should get the
same answer for ROE when we calculate it using any of these formulas/identities.
Looking closely at the second identity for ROE, we see that NI/S is what is commonly
referred to as the profit margin, S/A represents asset-use efficiency, and A/E (Total
Assets / Total Equity) is what is commonly referred to as the equity multiplier.
Therefore, we now have the third identity for ROE:
ROE = Profit margin x Asset-use efficiency x Equity multiplier.
Again, with just a few strokes of the pen, we have derived three different identities for
the ROE formula.
Slide 7: Variables in ROE
Let’s take a closer look at the second ROE identitiy.
ROE = (NI/S) x (S/A) x (A/E)
What are the variables we will need in order to calculate the ROE using this formula? We
will need the values of Net Income (NI), Sales (S), Total Assets (A), and Total Equity
(E).
Slide 8: One step further
We could stop at three identities for the ROE and be satisfied, but we finance people are
adventurous. So, we go on.
We know from the accounting balance sheet identity that the value of the total assets
(A) must equal the sum of the values of total debt (D) and total equity (E):
A=D+E
So, given this balance sheet identity, we can rewrite the equity multiplier, A/E, as
follows:
A/E = (D + E)/E = (D/E) + (E/E) = (D/E) + 1 = 1 + D/E
Slide 9: One step further (cont.)
Going one step further with the second ROE identity of
ROE = (NI/S) x (S/A) x (A/E),
we plug A/E = 1 + D/E into this ROE formula to get
ROE = (NI/S) x (S/A) x (1 + D/E)
This is the fourth identity for ROE.
Now, looking closer at this new identity, we see that NI/S is profit margin, S/A is assetuse efficiency, and D/E is the familiar debt-equity ratio. Therefore, we have our fifth
identity for ROE:
ROE = Profit margin x Asset-use efficiency x (1 + Debt-equity ratio)
Slide 10: Summary – ROA DPI
To sum up, we have found three Du Pont Identities for calculating ROA:
1. ROA = NI/A (basic formula)
2. ROA = (NI/S) x (S/A)
3. ROA = Profit margin x Asset-use efficiency
Slide 11: Summary – ROE DPI
As well, we have found seven Du Pont Identities for calculating ROE:
1. ROE = NI/E (basic formula)
2. ROE = (NI/S) x (S/E)
3. ROE = (NI/S) x (S/A) x (A/E)
4. ROE = Profit margin x Asset-use efficiency x Equity multiplier
5. ROE = ROA x (A/E)
6. ROE = (NI/S) x (S/A) x (1 + D/E)
7. ROE = Profit margin x Asset-use efficiency x (1 + Debt-equity ratio)
Note:
The fifth identity is derived by plugging ROA = (NI/S) x (S/A) into the third identity: ROE
= (NI/S) x (S/A) x (A/E).
Slide 12: What does it mean?
What does all this mean? It means that, when we calculate ROA and ROE, we get to see
the different components that contribute to the return on assets and return on equity.
For example, with the ROE formula:
ROE = Profit margin x Asset-use efficiency x (1 + Debt-equity ratio)
we see that the return on equity is affected by three variables: the profit margin, the
asset-use efficiency, and the debt-equity ratio. If a company wants to increase its
return on equity, it will have to increase profit margin, improve asset-use efficiency, or
increase debt-equity ratio. In fact, given that the debt-equity ratio is calculated as the
value of total debt divided by the value of total equity, to increase ROE, the company will
have to increase debt or reduce equity.
So, the model says, to increase ROE, we must do one or more of the followings:
Increase profit margin
Increase asset-use efficiency
Increase debt
Decrease equity
Slide 13: Numerical Example - ROA
Now, without further ado, let’s work through a numerical example. Let’s start with one
of calculating the ROA.
Given the following information, find the return on assets.
Net Income = $1,000
Total Assets = $10,000
Sales = $15,000
Writing down the information given in the symbols we have used in this lecture, we have
NI = $1000
A = $10000
S = $15000
The ROA identity we could use is:
ROA = (NI/S) x (S/A)
Plugging the numbers for NI, A, and S into this formula, we get
ROA = (1000/15000) x (15000/10000) = 0.1 = 10%
The return on asset is therefore 10%.
Slide 14: Numerical Example - ROE
Now let’s move on to a numerical example for ROE.
Given the following information, calculate the ROE.
Profit margin = 8%
Sales = $20,000
Total Assets = $50,000
Total Equity = $30,000
Net Income = $1,600
Rewriting the information given in the symbols we have used in this lecture, we have
NI/S = 0.08
S = 20000
A = 50000
E = 30000
NI = 1600
Which ROE identity should we use? The decision is usually based on what information
we have on hand. In this case, because we have the numbers for A and E, the easiest
ROE formula to use is:
ROE = (NI/S) x (S/A) x (A/E)
Plugging the numbers we know into this formula, we get
ROE = 0.08 x (20000/50000) x (50000/30000) = 0.05333333
This gives us a return on equity of approximately 5.33%.
Slide 15: Numerical Example - ROE calculation (cont.)
Left blank – unused as the space was not needed during the lecture.
Slide 16: Numerical Example – missing variable
Let’s move on to the next numerical example. Here, we are given an ROE and a bunch
of information, and we are then asked a slightly different question: What is the value of
one of the missing variables?
Given the following information, calculate the value of total debt, D.
Profit margin = 8%
Total equity = $30,000
Net Income = $5,000
ROE = 16.67%
Asset-use efficiency = 0.5
Writing down the information in the symbols we have been using, we have
NI/S = 0.08
E = $30000
NI = $5000
ROE = 0.1667
S/A = 0.5
Where do we start? First, we know the value for ROE, and we also know how ROE is
calculated. Which ROE identity to use though? The ROE formula that is often used is:
ROE = (NI/S) x (S/A) x (A/E)
We know that ROE = 0.1667, NI/S = 0.08, S/A = 0.5, and E = $30000. With these
numbers, we can figure out the value for A, the total assets. Once we obtained A, we
can calculate the value for debt, D, by using the balance sheet identity: A = D + E.
So, without further ado, let’s do this. Plugging the numbers for all the known variables
into the ROE formula, we get
0.1667 = 0.08 x 0.5 x (A / 30000)
Dividing both sides by 0.08 x 0.5 = 0.04, we get
0.1667 / 0.04 = A / 30000
Multiplying both sides by 30000, we get
(0.1667 / 0.04) x 30000 = A
This gives us A = 125000.
Slide 17: Numerical Example - Missing variable (cont.)
We now have
A = $125000
E = $30000
We also have the balance sheet identity:
A=D+E
Plugging the numbers for A and E into this formula, we get
125000 = D + 30000
Subtracting 30000 from both sides, we get
D = 125000 – 30000 = 95000
Therefore, the total debt value is $95,000.
Slide 18: Numerical Example - Missing variable (cont.)
Left blank – unused as the space was not needed during the lecture.
Slide 19: Practice, Practice, Practice
Practice is the secret to passing any finance course. In this practice problem, you are
given a whole bunch of information.
Net Income = $5,000
Sales = $62,500
Total Assets = $125,000
Total Equity = $30,000
Total Debt = $95,000
ROE = 16.67%
The exercise is to pretend that you do not have the value for one of the variables. For
example, let’s say that we forgot what the value of Net Income is. Can you find it using
the ROE Du Pont Identity? The answer is, as always, yes we can!
Try this exercise on your own, and see if you can derive each of the numbers by using
the other numbers given and the ROE Du Pont Identities.
Here ends the lecture on the Du Pont Identities. Thank you for attending.