Question

Consider a single period problem where the riskless interest rate is zero, and there are no taxes. A firm consists of a machine that will produce cash flows of $210 if the economy is good and $80 if the economy is bad. The good and bad states occur with equal probability and the covariance of these states with the market portfolio is zero (no systematic risk). Initially, the firm has 100 shares outstanding and debt with a face value of $50 due at the end of the period.

a.) What is the share price of this firm? (Hint: you need to calculate cash flows to equity holders in the good state and at the bad state of the economy separately. After that, you compute the expected cash flow. Also, don’t forget that debt holders are paid first in both states of the world.)

Suppose the firm unexpectedly announces that it will issue additional debt, with the same seniority as existing debt and a face value of $50. The firm will use the entire proceeds to repurchase some of the outstanding shares.

b.) What is the market price of the new debt? (Hint: in case of bankruptcy, old and new debt holders will split the cash flow proportional to the face value because they have debt of the same seniority.)

c.) Just after the announcement, what will happen to the price of the equity shares?

d.) What has happened to the total value of the firm? Does the Modigliani-Miller (MM) theorem hold in this case?

Answer 1

If the economy is good then the cash flow is $210 with a probability of 0.50
If the economy is bad, then the cash flow is $80 with a probability of 0.50
The firm has 100 shares outstanding with a debt having face value of $50 due at the end of the period.
Since the covariance of the states is zero with the market portfolio, we can discount the cash flow with risk free rates

a) value of the stock = 0.5*(210-50) + 0.5*(80-50)
   = 0.5*160 + 0.5*30
= 80 + 15
=$95
Price per share = 95/100 (Value of the stock/No.of shares outstanding)
   = $0.95

b) Market price of the new debt

When the state is good, the firm is able to repay both the debts ( new and old debt) but when the state is bad, then the debt is repaid equally between the new and the old debt, since the new debt also has equal prioroty leading to $80 being repaid equally as $40.

Thus, Value of the new debt = 0.5*(50) + 0.5*(40)
      = 25 + 20
      = $45
c) Just after the announcement, the expected value of the equity share is as follows

New value of the share= 0.5*[210 -(50+50)]+ 0.5*[80 - (40+40)]
= 0.5*[210-100] + 0.5*[0]
      = $55.

Now, we need to calculate the no. of shares repurchased and hence the number of shares outstanding

let n be the number of shares repurchased,
(Price of the share) P*n = 45
and (100-n)*P = 55
Solving the two equations
(100-n)*45/n = 55
100*45 - 45n = 55n
4500 = 100n
n= 45 (no.of shares repurchased)
Therefore, P = $1
Therefore no. of shares outstanding = 100-45 = 55shares

Total of 45 shares at a price of $1 is repurchased from the proceeds of new debt.

Thus, the price has increased from $0.95 to $1.

d)
The total Value of the firm remains unchanged.
Value of the firm = Debt value +equity value
Before the issue of new debt,
Total Value of the firm was = D1+E = 50+95 = $145
After the issue of new debt,
Total Value of the firm is D1(value of old debt)+D2(value of new debt)+E(Value of equity posy capital restructure) = 45+45+55 = $145

Thus, MM theory holds true in this case as the total market value of the firm remains unchanged even after capital restructuring.