Question

In the MM perfect markets world, consider a firm with a single project that will worth either $100 or $200 in one year with equal probability. If the firm issues debt with a face value (promised payment) of $55(LD firm). The appropriate discount rate for the project is 10%. What is the value of the firm’s equity? What is the value of the equity if the firm were to issue debt with a face value of $110 (HD firm)? How would I buy the equivalent of )10% of the HD firm equity if the only securities for sale are the debt and equity of the LD firm (in other words ow much LD equity and LD debt would I need to own)?

Answer 1

Value of equity = max (V - D, 0)

Case 1: D = 55

Hence, E1 = Max (V1 - D, 0) = max (100 - 55, 0) = 45; p1 = 50%

E2 = max (V2 - D, 0) = max (200 - 55, 0) = 145; p2 = 50%

the value of the firm’s equity = (p1 x E1 + p2 x E2) / (1 + r) = (50% x 45 + 50% x 145) / (1 + 10%) = 86.36

Case 2: D = 110

Hence, E1 = Max (V1 - D, 0) = max (100 - 110, 0) = 0; p1 = 50%

E2 = max (V2 - D, 0) = max (200 - 110, 0) = 90; p2 = 50%

the value of the firm’s equity = (p1 x E1 + p2 x E2) / (1 + r) = (50% x 0 + 50% x 90) / (1 + 10%) = 40.91

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Let's say the replicating portfolio is A of LD and amount A of debt.

Payoff when the project is worth $ 100:

10% of HD = 10% x max(V - D, 0) = 10% x max (100 - 110, 0) = 10% x 0 = 0

A% of LD and amount A of debt = A x max (100 - 55, 0) - B x r = A x 45 - B x 10% = 45A - 0.1B

The two payoff should be same, hence 45A - 0.1B = 0, Or 0.1B = 45A

Payoff when the project is worth $ 200:

10% of HD = 10% x max(V - D, 0) = 10% x max (200 - 110, 0) = 10% x 90 = 9

A% of LD and amount A of debt = A x max (200 - 55, 0) - B x r = A x 145 - B x 10% = 145A - 0.1B

The two payoff should be same, hence 145A - 0.1B = 9,

Or, 145A - 45A = 100A = 9

Hence, A = 9/100 = 9%; B = 450 x A = 450 x 9% = 40.50

I buy the equivalent of )10% of the HD firm equity by

Owning / buying 9% of LD and
Borrowing 40.50