Suppose that assets and costs maintain a constant ratio to sales. The ﬁrm retains 30% of earnings. If the ﬁrm is producing at full capacity, what is the maximum growth rate (assuming no equity sales)
That will maintain a constant debt-equity ratio?

Question

Quizplus · Accepted Answer

To maintain a constant debt-equity ratio, the firm's growth rate cannot exceed the maximum sustainable growth rate. The maximum sustainable growth rate can be calculated using the formula:

Maximum sustainable growth rate = Retention ratio x Return on equity

Retention ratio = 1 - Dividend payout ratio = 1 - 0.3 = 0.7

Return on equity is not given, but we know that assets and costs maintain a constant ratio to sales. This indicates that the firm's profit margin is constant, and therefore return on equity will also be constant. Let's call this constant return on equity "r".

Then, the maximum sustainable growth rate can be calculated as:

Maximum sustainable growth rate = 0.7 x r

To maintain a constant debt-equity ratio, the firm's growth rate cannot exceed this maximum sustainable growth rate. Therefore, we need to solve for "r" in the following equation:

Debt-equity ratio = Total debt / Total equity

Since costs and assets maintain a constant ratio to sales, we can assume that total assets are also constant. Then, we can simplify the above equation:

Debt-equity ratio = Total liabilities / (Total assets - Total liabilities)

Debt-equity ratio = 1

Total liabilities / (Total assets - Total liabilities) = 1

Total liabilities = Total assets - Total liabilities

Total equity = Total assets - 2 x Total liabilities

Since assets and costs maintain a constant ratio to sales, we can assume that total assets and total liabilities increase at the same rate as sales, which we can call "g". Then:

Total assets = A0 x (1+g)^t

Total liabilities = L0 x (1+g)^t

Total equity = E0 x (1+g)^t

where A0, L0, and E0 are the total assets, total liabilities, and total equity at time t=0.

Using the above equations, we can substitute into the debt-equity ratio equation:

1 = L0 x (1+g)^t / [A0 x (1+g)^t - L0 x (1+g)^t]

1 = L0 / (A0 - L0)

L0 = A0 / 2

Total equity = A0 / 2

Now we can solve for "r":

r = Net income / Total equity

Net income / Total assets = Profit margin

Profit margin x Sales = Net income

Sales = Total assets x Sales turnover

Therefore:

r = Profit margin x Sales / Total equity

r = Profit margin x Total assets x Sales turnover / Total equity

r = Profit margin x 2 x Sales turnover

r = Constant

Now we can calculate the maximum sustainable growth rate:

Maximum sustainable growth rate = 0.7 x r

Maximum sustainable growth rate = 0.7 x Constant

Maximum sustainable growth rate = Constant

Therefore, the maximum growth rate that can be sustained while maintaining a constant debt-equity ratio is equal to the constant return on equity, which we have assumed is constant at "r".

Suppose That Assets and Costs Maintain a Constant Ratio to Sales