Question

Can someone please help me with this question?
Firm 1 has a capital structure with 20% debt and 80% equity. Firm 2’s capital structure consists of 50% debt and 50% equity. Both firms pay 7% annual interest on their debt. Finally suppose both firms have invested in assets worth $100 million. Calculate the return on equity (ROE) for each firm assuming the following:
a. The return on assets is 3%
b. The return on assets is 7%
c. The return on assets is 11%
What general pattern do you observe?

Answer 1

If i may try to answer the question, (cmiiw):
first thing first, the ROA = 3% means, the Return compared to Total Assets = 3:100 which gives us 3 million as a Return.

Usually there are 2 approaches to compute ROA, one is using the net income, and the other is using the EBIT (or so i was told)

condition 1 (ROA = Net Income)
Net Income = 3,000,000 while Assets = 100,000,000.
We know that Total Assets are equal to Total Liabilities + Total Equities. Given the proportion of the Debt&Equity, we can compute the ROE for each Firm:
Firm 1 with 20% Debt -80% Equity, that means:
Total Debt = 20 million
Total Equity = 80 million
With the Net Income of 3,000,000 the ROE will be 3 million / 80 million = 3.75%
Firm 2 with 50% Debt -50% Equity, that means:
Total Debt = 50 million
Total Equity = 50 million
With the Net Income of 3,000,000 the ROE will be 3 million / 50 million = 6%

however, given the information about the interest rate of 7%, i assume that the ROA is computed using the EBIT, hence the computation will be slight different, which is:
3,000,000 - Interest Expense - Tax (assuming there is no tax) = Net Income
so for the Firm 1
3,000,000 - (7% x 20,000,000) = 2,600,000
with that amount of NI, the ROE is now:
2,600,000 / 80,000,000 = 3.25%
for the Firm 2
3,000,000 - (7% x 50,000,000) = -500,000
and with that amount of NI, the ROE is:
-500,000 / 50,000,000 = (10%)

and i believe the other conditions will use the same approach...
about the intepretation, i honestly can't do the intepretation since i'm not quite sure which assumption of mine, is correctly used in the calculation.. hence i'll leave the rest to you ^^ i hope i'm not misleading you..