Question

Is the following statement accurate? Please provide a numerical reasoning to support your claim. A firm that is debt-using with a negative leverage gain will always have a return on equity that is negative.

Answer 1

Negative leverage gain means

Lets assume intially the firm has only equity E and sales S

then return on equity=S*Gross Profit margin*(1-tax rate)/Equity

Now lets say Debt of D is taken and sales increase to S'=S+S*D/E

return on equity=(S*(1+D/E)*Gross profit margin-Debt*interest rate)*(1-tax rate)/E

We know that S'*Gross profit margin-Debt*interest rate<S*Gross Profit margin

or, S/E*gross profit margin<Interest rate then only negative leverage would occur

But negative leverage does not necessarily imply negative income or loss or negative RoE. The RoE would decrease but for it to become negative, Sales*Assets/Equity*Gross Profit Margin<Interest expense

Which may not always happen