Chapter 3, Problem 45QP

Summary Introduction

To discuss: The internal growth rate and sustainable growth rate and calculate the rates, growth rate by using the starting of period equity, the reason for the similar growth rates and the best estimate the internal and sustainable growth rates.

Introduction:

The rate of internal growth is the highest growth rate which can be achieved without any kind of external financing. The rate of sustainable growth is the highest growth rate which can be achieved without external equity financing.

Answer to Problem 45QP

The ROA in the beginning is 6.98%and in the end is 7.15%respectively. The ROE in the beginning is 20.26%and the ROE in the end is 19.34%.

Explanation of Solution

Given information:

The net income of Company B for the year 2012 is $1,277. The dividend in the year 2012 is 237. The total assets for the year 2011 and 2012 are $18,302 and $17,849 respectively. The total equity for 2011 and 2012 are $6,302 and $6,602 respectively.

Formula to compute the ROA in the beginning:

${\text{ROA}}_{\text{Beginning}}=\frac{\text{Net income}}{\text{Total assets}}$

Compute the ROA in the beginning:

$\begin{array}{c}{\text{ROA}}_{\text{Beginning}}=\frac{\text{Net income}}{\text{Total assets in 2011}}\\ =\frac{\$1,277}{\$18,302}\\ =0.0698\text{ or 6}\text{.98\%}\end{array}$

Hence, the ROA in the beginning is 6.98%.

Compute the ROA in the end:

$\begin{array}{c}{\text{ROA}}_{\text{End}}=\frac{\text{Net income}}{\text{Total assets in 2012}}\\ =\frac{\$1,277}{\$17,849}\\ =0.07154\text{ or 7}\text{.15\%}\end{array}$

Hence, the ROA in the end is 7.15%.

Formula to compute the ROE using the starting period equity:

${\text{ROE}}_{\text{Beginning}}=\frac{\text{Net income}}{\text{Total equity}}$

Compute the ROE using the starting period equity:

$\begin{array}{c}{\text{ROE}}_{\text{Beginning}}=\frac{\text{Net income}}{\text{Total equity in 2011}}\\ =\frac{\$1,277}{\$6,302}\\ =0.2026\text{ or 20}\text{.26}\%\end{array}$

Hence, the ROE in the beginning is 20.26%.

Compute the ROE using the end of period equity:

$\begin{array}{c}{\text{ROE}}_{\text{End}}=\frac{\text{Net income}}{\text{Total equity in 2012}}\\ =\frac{\$1,277}{\$6,602}\\ =0.1934\text{ or 19}\text{.34}\%\end{array}$

Hence, the ROE in the end is 19.34%.

Formula to compute the retention ratio:

$b=1-\frac{\text{Dividends}}{\text{Net income}}$

Compute the retention ratio:

$\begin{array}{c}b=1-\frac{\text{Dividends}}{\text{Net income}}\\ =1-\frac{\$237}{\$1,277}\\ =0.8144\text{ or 81}\text{.44}\%\end{array}$

Hence, the retention ratio is 81.44%.

Formula to compute the internal growth rate:

$\text{Internal growth rate}=\left[\frac{\left(\text{ROA}\right)\left(b\right)}{1-\left(\text{ROA}\right)\left(b\right)}\right]$

Where,

ROA denotes the return on assets.

b denotes the retention or plowback ratio.

Calculate the internal growth rate:

$\begin{array}{c}\text{Internal growth rate}=\left[\frac{\left(\text{ROA}\right)\left(b\right)}{1-\left(\text{ROA}\right)\left(b\right)}\right]\\ =\left[\frac{0.0715\left(0.8144\right)}{1-0.0715\left(0.8144\right)}\right]\\ =0.06183\text{ or 6}\text{.18}\%\end{array}$

Hence, the internal growth rate is 6.18%.

Formula to calculate the sustainable growth rate:

$\text{Sustainable growth rate}=\frac{\text{ROE}\times b}{1-\text{ROE}\times b}$

Where,

ROE denotes the return on equity.

b denotes the retention or plowback ratio.

Compute the sustainable growth rate:

$\begin{array}{c}\text{Sustainable growth rate}=\frac{\text{ROE}\times b}{1-\text{ROE}\times b}\\ =\frac{0.1934\times 0.8144}{1-0.1934\times 0.8144}\\ =0.1869\text{ or 18}\text{.69}\%\end{array}$

Hence, the sustainable growth rate is 18.69%.

Formula to compute the internal growth rate using ROA at the end:

$\text{Internal growth rate}={\text{ROA}}_{\text{END}}\times b$

Compute the internal growth rate using ROA at the end:

$\begin{array}{c}\text{Internal growth rate}={\text{ROA}}_{\text{END}}\times b\\ =0.07154\times 0.8144\\ =0.05823\text{or 5}\text{.82\%}\end{array}$

Hence, the internal growth rate is 5.82%.

Formula to compute the sustainable growth rate using ROE at the end:

$\text{Sustainable growth rate}={\text{ROE}}_{\text{END}}\times b$

Compute the sustainable growth rate using ROE at the end:

$\begin{array}{c}\text{Sustainable growth rate}={\text{ROE}}_{\text{END}}\times b\\ =0.1934\times 0.8144\\ =0.15750\text{or }15.75\text{\%}\end{array}$

Hence, the sustainable growth rate is 15.75%.

Formula to compute the internal growth rate using ROA at the beginning:

$\text{Internal growth rate}={\text{ROA}}_{\text{Begin}}\times b$

Compute the internal growth rate using ROA at the beginning:

$\begin{array}{c}\text{Internal growth rate}={\text{ROA}}_{\text{Beginning}}\times b\\ =0.0698\times 0.8144\\ =0.0568\text{or 5}\text{.68\%}\end{array}$

Hence, the internal growth rate is 5.68%.

Formula to compute the sustainable growth rate using ROE at the beginning:

$\text{Sustainable growth rate}={\text{ROE}}_{\text{Begin}}\times b$

Compute the sustainable growth rate using ROE at the beginning:

$\begin{array}{c}\text{Sustainable growth rate}={\text{ROE}}_{\text{Beginning}}\times b\\ =0.2026\times 0.8144\\ =0.1650\text{ or 16}\text{.50\%}\end{array}$

Hence, the sustainable growth rate is 16.50%.