1. General Considerations

1.2. A Levered Firm

Now let's consider the case of an otherwise identical levered firm, and let's ignore taxes for the moment. Because the levered firm has the exact same business activities as the previously considered unlevered firm, \(k_A\) is still 20%, since what matters for the firm's overall cost of capital is its operating risk, and we are not changing that. Suppose the firm issues zero-coupon debt with a face value of 800'000, repayable at year end, and uses the proceeds to repurchase a fraction of the shares outstanding. Also, suppose the cost of debt (the rate of return required by debtholders,  \(k_D\)) is 10%. The market value of the firm's debt therefore equals:

  

Market value of debt =\( \frac{800'000}{1.1} \)= 727'273 = 0.727 million.

  

Put differently, the firm receives 0.727 million today in exchange for the promise to pay back 0.8 million in one year. The following table summarizes the relevant scenarios and cash flows at year end for the levered firm:

   

State I

State II

Average
(millions)

Free cash flow

1 million

2 million

1.5 million

Firm value (Present value)

 

 

1.5/1.2 = 1.25

Cash flows to debt

800'000

800'000

800'000

Present value of Debt

0.8/1.1 = 0.727

Cash flows to equity

200'000

1'200'000

700'000

Equity value (Present value)

 

 

1.25-0.727 = 0.523

 

 

Assuming debt leaves firm value unaffected (no taxes and no other side effects of financing), we can compute:

  

Market value of equity = 1'250'000 - 727'273 = 522'727= 0.523 million.

  

Consequently, the original shareholders' total wealth is unchanged at 1.25 million: 0.727 million cash (from the repurchase) plus 0.523 million invested in the firm.

  

But what return do the shareholders of the levered firm earn under the two states of the world, and on average? Since shareholders receive what is left after debtholders have been paid (see table above), their return will be:

  

State I State II Average
Return on the market value of equity \( \frac{0.2}{0.52273}-1=-0.62 \) \( \frac{1.2}{0.52273}-1=1.30\) 33.91%

   

Shareholders earn a return of -62% in State I and 130% in State II. On average, they earn a return of 34%, which is more than the 20% they receive if the firm does not lever up.

  

Since the total wealth of shareholders is unaffected by the debt issue (remember that we are ignoring taxes), it must be the case that the higher average return on equity is a compensation for risk. Hence, the higher return on equity corresponds to a higher required return on equity. Let's consider this issue in further detail in the following section.