Chapter 4, Problem 32QP

Sustainable Growth Rate [LO3] In the chapter, we discussed the two versions of the sustainable growth rate formula. Derive the formula ROE × b from the formula given in the chapter, where ROE is based on beginning of period equity. Also, derive the formula ROA × b from the internal growth rate formula.

Summary Introduction

To determine: The internal growth rate is $\left(\text{ROA}\times b\right)$ and sustainable growth rate is

$\left(\text{ROE}\times b\right)$

Introduction:

ROA (return on assets) indicates the operating efficiency of the firm and using ROA the internal growth of firm can be determined. ROE (return on equity) measures the amount of profit, which can be derived from the shareholder’s equity and using ROE the sustainable growth rate of the firm can be determined.

Explanation of Solution

Given information:

ROE is based on the beginning of period equity.

Assumption: The beginning of period equity as x, which represents (ROE_x) and the end of period equity as y, which represents (ROE_y)

Formulae:

The formula to compute the sustainable growth rate:

$\text{Sustainable growth rate}=\frac{\left({\text{ROE}}_y\times b\right)}{1-\left({\text{ROE}}_y\times b\right)}$

Where,

b refers to the retention ratio

The formula to compute the internal growth rate:

$\text{Internal growth rate}=\frac{\left({\text{ROA}}_y\text{×}b\right)}{\left(\text{1 – }\left({\text{ROA}}_y\text{×}b\right)\right)}$

Derive $\left(\text{ROE}\times b\right)$ using sustainable growth rate:

$\text{Sustainable growth rate}=\frac{\left({\text{ROE}}_y\times b\right)}{1-\left({\text{ROE}}_y\times b\right)}$ , let this be equation 1

Here, NI is net income and TE is total equity.

${\text{ROE}}_{\text{y}}=\frac{\text{NI}}{{\text{TE}}_y}$

Substitute ${\text{ ROE}}_y$ in equation 1

$\text{Sustainable growth rate}=\frac{\left(\frac{\text{NI}}{{\text{TE}}_{\text{y}}}\times b\right)}{1-\left(\frac{\text{NI}}{{\text{TE}}_{\text{y}}}\times b\right)}$

Multiply $\frac{{\text{TE}}_y}{{\text{TE}}_y}$ where TE is total equity

$\text{Sustainable growth rate}=\frac{\left(\frac{\text{NI}}{{\text{TE}}_y}\times b\right)}{1-\left(\frac{\text{NI}}{{\text{TE}}_y}\times b\right)}\times \frac{{\text{TE}}_y}{{\text{TE}}_y}$

$\text{Sustainable growth rate}=\frac{\left(\text{NI}\times b\right)}{\left({\text{TE}}_y-\left(\text{NI}\times b\right)\right)}$ , let this be equation 2

Where, the beginning of period equity is $\left({\text{TE}}_x-\left(\text{NI}\times b\right)\right)$

$\left({\text{TE}}_y-\left(\text{NI}\times b\right)\right)$= ${\text{TE}}_x$, let this be equation 3

The below equation is arrived by substituting equation 3 in equation 2.

$\text{Sustainable growth rate =}\frac{\text{NI}\times b}{{\text{TE}}_x}$ , let this be equation 3

Rewrite it as $\frac{\text{NI}}{{\text{TE}}_x}\times b$

$\text{ROE}=\frac{\text{NI}}{{\text{TE}}_x}$

Therefore, ${\text{Sustainable growth rate =ROE}}_x\times b$

Hence, the sustainable growth rate equation is ${\text{ROE}}_x\times b$

Compute $\left(\text{ROA}\times b\right)$using internal growth rate:

Where, TA is total assets

$\text{ROA=}\frac{\text{NI}}{{\text{TA}}_y}$

$\text{Internal growth rate}=\frac{\left(\frac{\text{NI}}{{\text{TA}}_y}\text{×}b\right)}{\left(\text{1 – }\left(\frac{\text{NI}}{{\text{TA}}_y}\text{×}b\right)\right)}$

Multiply, internal growth rate with $\left(\frac{T{A}_y}{T{A}_y}\right)$

$\begin{array}{c}\text{Internal growth rate}=\frac{\left(\frac{\text{NI}}{{\text{TA}}_y}\text{×}b\right)}{\left(\text{1 – }\left(\frac{\text{NI}}{{\text{TA}}_y}\text{×}b\right)\right)}\times \frac{{\text{TA}}_y}{{\text{TA}}_y}\\ =\frac{\left(\text{NI }\times b\right)}{\left({\text{TA}}_y–\left(\text{NI}\times b\right)\right)}\end{array}$

Where,

The beginning of period equity is $\left({\text{TA}}_{\text{y}}-\left(\text{NI}\times b\right)\right)$

Therefore, $\left({\text{TA}}_y–\left(\text{NI }\times b\right)\right)={\text{TA}}_x$

Substitute, ${\text{TA}}_x$ in equation

$\begin{array}{c}\text{Internal growth rate}=\frac{\left(\text{NI }\times b\right)}{\left({\text{TA}}_y–\left(\text{NI}\times b\right)\right)}\\ =\left(\frac{\text{NI}}{{\text{TA}}_x}\right)\times b\end{array}$

${\text{ROE}}_x=\left(\frac{\text{NI}}{{\text{TA}}_x}\right)$

$\text{Internal growth rate}={\text{ROE}}_x\times b$

Hence, the equation of internal growth rate is ${\text{ROE}}_x\times b$