Athabasca University



Slide 1: Lecture 2 – Internal growth rate vs. Sustainable growth rate

Welcome to Lecture 2: internal growth rate versus sustainable growth rate.

Slide 2: What’s the diff?

What’s the difference, you might ask? The answer is defined by the definitions of the two growth rates.

An internal growth rate requires no outside financing whatsoever. That means, no new debt and no new equity, and the capital budget depends entirely on internally generated funds.

On the other hand, a sustainable growth rate requires no external equity financing, but debt financing is okay in order to make the debt-equity ratio constant due to addition to retained earnings.

Slide 3: Remember these formulas

The basic formulas for internal growth rate and sustainable growth rate are:

Ig = (ROA x R) / [1 – (ROA x R)]

and

Sg = (ROE x R) / [1 – (ROE x R)]

Where

Ig = Internal growth rate

Sg = External growth rate

ROA = Return on Assets

ROE = Return on Equity

R = Retention Ratio

The ROA, ROE, and R are calculated as follows:

ROA = Net Income / Total Assets = NI/A (or using other Du Pont Identities[1])

ROE = Net Income / Total Equity = NI/E (or using other Du Pont Identities)

R = Retention ratio = 1 – (Dividends / Net Income) = 1 – Dividend payout ratio

It is highly recommended that you memorize the two formulas for internal growth and sustainable growth rates. They are fairly easy to remember, as each equation only requires two variables. The internal growth rate formula requires only knowledge of ROA and R, and the sustainable growth rate formula requires only knowledge of ROE and R.

Slide 4: Numerical Example – Internal growth rate

Let’s go through a numerical example. Given an ROA of 15% and R of 0.5, what is the internal growth rate (i.e., the growth rate that is possible without any new debt or equity)?

Using the basic internal growth rate formula:

Ig = (ROA x R) / [1 – (ROA x R)]

We can make things easy by doing the calculations in two blocks:

ROA x R

[1 – (ROA x R)]

We first calculate ROA x R and then calculate [1 – (ROA x R)]. We can then put them together to calculate the internal growth rate.

Plugging the numbers into the two blocks, we get

ROA x R = 0.15 x 0.5 = 0.075

[1 – (ROA x R)] = 1 – 0.075 = 0.925

Then we put them together in the internal growth rate formula:

Ig = 0.075/0.925 = 0.081081081

That is, the internal growth rate, without any external financing at all, is approximately 8.11%. This is not a bad growth rate, considering that the company will only be using internally generated funds to pay for capital budgeting projects.

Slide 5: Numerical Example – Sustainable growth rate

Next, we do a numerical example on sustainable growth rate. Again, the formula is:

Sg = (ROE x R) / [1 – (ROE x R)]

and again, we do the calculations by blocks, with the first block being ROE x R,

and the second block being [1 – (ROE x R)].

We are given the information:

ROE = 12%

R = 0.7

What is the sustainable growth rate?

We plug in the numbers to our two blocks:

ROE x R = 0.12 x 0.7 = 0.084

[1 – (ROE x R)] = 1 – 0.084 = 0.916

Putting everything together in the sustainable growth rate formula, we get

Sg = 0.084/0.916 = 0.09170306

Therefore, the sustainable growth rate is approximately 9.17%, a rate achievable without any external equity financing.

Slide 6: What about the Du Pont Identities?

You may ask, “How do the Du Pont Identities relate to the calculations of internal growth rate and sustainable growth rate? “

We know that one of the ROA identities is

ROA = (NI/S) x (S/A)

We also know that one of the ROE identities is

ROE = (NI/S) x (S/A) x (A/E)

With our new knowledge of the internal growth rate and sustainable growth rate, we know that:

1. the three variables in the internal growth rate formula are the internal growth rate, the ROA, and R.

2. the three variables in the sustainable growth rate formula are the sustainable growth rate, the ROE, and R.

Therefore, given the values of any two of the three variables in the internal growth rate formula, we can derive the value of the third variable. The same is true for the sustainable growth rate formula.

For example, given R and internal growth rate, we can find ROA. Also, given ROA and internal growth rate, we can find R.

Slide 7: Numerical Example – ROE

Let’s do a numerical example to prove our point from the previous slide. Given the following information, find ROE.

Sustainable growth rate = Sg = 0.1; R = 0.6

We know that the sustainable growth rate formula is:

Sg = (ROE x R) / [1 – (ROE x R)]

Plugging in the known numbers to this formula, we get

0.1 = (ROE x 0.6) / [1 – (ROE x 0.6)]

Since this is one equation with one unknown, we can always solve for ROE. Simplifying the formula, we get

0.1 = 0.6ROE / [1 – 0.6ROE]

This formula is the same as the one above it; it is just easier to look at when arranged this way.

Now the real work begins. Multiply both sides by (1 – 0.6ROE), and we will get

0.1 x (1 – 0.6ROE) = 0.6ROE

Multiplying out the left-hand-side of the formula, we get

0.1 – 0.06ROE = 0.6ROE

This makes things easier. Now we add 0.06ROE to both sides, yielding

0.1 = 0.6ROE + 0.06ROE = 0.66ROE

Dividing both sides by 0.66, we get

ROE = 0.1/0.66 = 0.15151515

That is, the return on equity is approximately 15.15%.

Slide 8: Numerical Example (cont.)

At any time, you can check that an ROE of 0.15151515 will indeed give us the sustainable growth rate of 10%. We plug in the numbers: ROE = 0.15151515 and R = 0.6 to the sustainable growth rate formula:

Sg = (ROE x R) / [1 – (ROE x R)]

Let’s do this in two blocks again.

ROE x R = 0.15151515 x 0.6 = 0.09090909

[1 – (ROE x R)] = 1 – 0.09090909 = 0.90909091

Plugging these back in to get Sustainable growth rate:

Sg = 0.09090909 / 0.90909091 = 0.1

which is exactly the number we had for the sustainable growth rate!

Slide 9: ROE = (NI/S) x (S/A) x (1 + D/E)

Remember how all these discussions started with the allusion to Du Pont Identities? We know that one of the ROE identities is:

ROE = (NI/S) x (S/A) x (1 + D/E)

Where

NI/S = profit margin

S/A = asset-use efficiency

D/E = debt-equity ratio

So, what if we know the values for asset-use efficiency, debt-equity ratio, retention ratio, and the sustainable growth rate? Can we find the profit margin by using the sustainable growth rate formula and the ROE Du Pont Identity?

This involves, essentially, the derivation for a missing variable. So… the answer to this question is “Yes”.

Slide 10: Numerical Example - ROE Du Pont Identity

Let’s illustrate this idea with a numerical example. Given the following information, find the ROE and the profit margin.

Sg = 0.11

R = 0.5

S/A = 0.8

D/E = 1.5

First, we want to figure out what formula is appropriate for our purposes here. Since the sustainable growth rate (Sg) is given, as well as the retention ratio (R), we know that the sustainable growth rate formula is relevant:

Sg = (ROE x R) / [1 – (ROE x R)]

And, since S/A and D/E are given, we know that the Du Pont Identity for ROE relevant here is:

ROE = (NI/S) x (S/A) x (1 + D/E)

Let’s call the Sg formula ‘Equation I’ and the ROE formula ‘Equation II.’

Plugging in the numbers for Sg and R to Equation I, we get

0.11 = (ROE x 0.5) / [1 – (ROE x 0.5)]

Plugging in the numbers for S/A and D/E to Equation II, we get

ROE = (NI/S) x 0.8 x (1 + 1.5)

Looking at these two equations, we can see that if we can get ROE from Equation I, we can plug that value into Equation II and solve for NI/S, which is the profit margin.

Slide 11: Numerical Example (cont.)

We first solve for ROE using Equation I:

0.11 = (ROE x 0.5) / [1 – (ROE x 0.5)]

Multiplying both sides by the denominator term of [1 – (ROE x 0.5)], we get

0.11 x [1 – (ROE x 0.5)] = ROE x 0.5

Multiplying out the terms on the left-hand-side, we get

0.11 – 0.11(0.5)ROE = ROE x 0.5

Simplifying the formula, we get

0.11 – 0.055ROE = 0.5ROE

Adding 0.55ROE to both sides, we get

0.11 = 0.5ROE + 0.055ROE = 0.555ROE

Dividing both sides by 0.555, we get

ROE = 0.11/0.555 = 0.1981982

Slide 12: Numerical Example (cont.)

Now that we have ROE, we can plug this ROE number into Equation II:

ROE = (NI/S) x 0.8 x 1.5

We then have:

0.1981982 = (NI/S) x 0.8 x (1 + 1.5)

Simplifying the right-hand-side of the formula, we get

0.1981982 = (NI/S) x 2

Dividing both sides by 2, we get

NI/S = 0.1981982 / 2 = 0.0990991

Therefore, the profit margin is approximately 9.91%.

You can check that you have the correct answer for NI/S by plugging this number back into the ROE formula:

ROE = 0.0990991 x 0.8 x (1 + 1.5) = 0.1981982

You can then plug the ROE number back into the sustainable growth rate formula:

Sg = (0.1981982 x 0.5) / [1 – (0.1981982 x 0.5)]

This should come out to Sg = 0.11.

Slide 13: Steps for figuring out unknown variables

To summarize, these are the steps for figuring out unknown variables:

1. Write down all the available information on known variables.

2. Figure out what formula patterns are relevant. In our example, the relevant formulae are the sustainable growth formula and the ROE Du Pont Identity formula.

3. Plug the known numbers into the relevant formulas.

4. Figure out which variable(s) is (are) missing.

5. Work backward to solve for the missing variable(s).

Slide 14: Be the boss of numbers and formulas

My advice for the end of this lecture is: Be the boss of the numbers and formulas. Do not be intimidated by them. Formulas and numbers are just tools, and you can make them dance for you. Do not fear them; they should fear you!

Here ends Lecture 2 on internal growth rate and sustainable growth rate.

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[1] See Lecture 1 on Du Pont Identities.

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