Finance-Theory · Unit 12 · Video 6 · Interactive Practice
Equity Is a Call Option: Options in Capital Structure
IKey Formulas
Formula
Name
What it says
E=Max[0,V−B]
Equity as a call
A call on the firm's assets V with strike equal to the face value of the debt B. With B = $100 million: V = $150 million leaves $50 million, V = $60 million leaves nothing
E=V−B+Max[0,B−V]
Protected levered put
Own the assets, borrow B, and hold a put on the assets with strike B: the same payoff as the call
D=Min[V,B]=B−Max[0,B−V]
Debt as lending minus a put
Riskless lending of B, combined with a put on the assets (strike B) written by the bondholders
V=D+E
The whole capital structure
The B terms cancel and the two puts cancel: the put the bondholders wrote is the put the shareholders hold
Key Insight: Shareholders stand second in line and can never be made to pay more, so their claim is Max[0,V−B], the call payoff Max[0,ST−K] with the firm's assets as the underlying and the debt's face value as the strike.
IIWho Gets Paid Out of the Firm's Assets
At maturity the assets V split between bondholders, paid first up to B, and shareholders, who keep the rest.
IIITwo Readings of the Same Payoffs
Equity is assets, minus borrowing, plus a put with strike B; debt is riskless lending minus that same put.
IVWhy Levered Shareholders Like Risk
Spreading the possible asset values moves expected payoff from bondholders to shareholders, even when expected firm value falls.
💡 These are expected payoffs at maturity, not prices: an expected payoff is not what an option is worth today. The direction is the same, though, because a call's price rises with the volatility of its underlying.
VQuiz Questions
Problem 1 · Equity at Maturity
Given: a firm has a single debt issue with face value $70 million, due on one date. On that date its assets are worth $115 million. Find the payoff to the shareholders.
✅ Correct!V≥B, so the bondholders get their $70 million in full and the shareholders keep the remaining $45 million: Max[0,115−70]=45.
❌ Not quite. That is the whole firm. The bondholders stand first in line and are paid their $70 million before the shareholders see anything.
❌ Not quite. $70 million is what the bondholders receive. The shareholders get what is left after them.
❌ Not quite. The debt is a claim on the assets, not an addition to them: it is paid out of the $115 million.
❌ Not quite. Equity is Max[0,V−B].
Show solution
Equity is a call on the assets with strike equal to the face value of the debt. In $ millions:
E=Max[0,V−B]=Max[0,115−70]=45
The shareholders receive $45 million; the bondholders receive Min[115,70]=70, and 70+45=115=V.
Problem 2 · When the Assets Fall Short
Given: the same firm owes $70 million at maturity, but its assets turn out to be worth only $55 million. Find the payoffs to the shareholders and to the bondholders.
Payoff to the shareholders?
Payoff to the bondholders?
✅ Correct!V<B: the shareholders get nothing and the assets pass to the bondholders, who receive $55 million of the $70 million they were promised.
❌ Not quite. $15 million is the shortfall B−V, which is owed to the bondholders and never paid. It is not something the shareholders receive: V−B is negative, so equity is zero.
❌ Not quite. $55 million is all the firm has, and the bondholders are first in line for it. Nothing is left for the shareholders.
❌ Not quite. $0 is the shareholders' payoff. The bondholders are not wiped out: the assets pass to them, so they receive what the firm is worth.
❌ Not quite. Like a call, equity's payoff never falls below zero: the shareholders cannot be made to pay in the shortfall. They walk away with nothing.
❌ Not quite. $70 million is what was promised, but there is only $55 million to pay it with. Debt is Min[V,B], not B.
❌ Not quite. $15 million is the shortfall B−V, the amount the bondholders do not get. Their payoff is B minus the shortfall.
❌ Check the equity. Equity is Max[0,V−B], with V−B=55−70.
❌ Check the debt. Debt is Min[V,B]: the smaller of what the firm is worth and what is owed.
The shareholders receive $0 and the bondholders $55 million, which is B minus the $15 million shortfall. Check: D+E=55+0=55=V.
Problem 3 · The Put Inside the Bond
Given: a firm's debt has face value $110 million, and at maturity its assets are worth $85 million. Write the debt as D=B−Max[0,B−V]. Find the payoff of the put term, then the payoff to the bondholders.
Payoff of the put Max[0,B−V]?
Payoff to the bondholders?
✅ Correct! The put finishes $25 million in the money, and because the bondholders wrote it, it comes off their $110 million: 110−25=85=Min[85,110].
❌ Not quite. $85 million is V, the value of the assets. A put with strike B pays the amount by which the assets fall short of the strike, B−V.
❌ Not quite. $110 million is the strike B itself. The put pays Max[0,B−V], the shortfall below the strike.
❌ Not quite. $0 is Max[0,V−B], the call (equity) term. The put pays Max[0,B−V], and here B>V.
❌ Not quite. $110 million is only the riskless-lending term. The bondholders also wrote the put, and its $25 million payoff comes out of their pocket.
❌ Check the sign. The put is subtracted: the bondholders wrote it, so its payoff is a loss to them, not a gain. They cannot receive more than the firm is worth.
❌ Not quite. $25 million is the put's payoff, which goes to the other side. The bondholders get B minus that.
❌ Check the put. Evaluate Max[0,B−V] with B=110 and V=85.
❌ Check the debt.D=B−Max[0,B−V]: riskless lending minus the written put.
Check against the direct form: Min[V,B]=Min[85,110]=85. The shareholders hold the same put inside E=V−B+Max[0,B−V]=85−110+25=0, and D+E=85=V: the put the bondholders wrote is the put the shareholders hold.
Problem 4 · A Riskier Plan
Given (illustrative numbers): a firm owes $100 million at maturity. Under its current plan its assets will be worth $120 million for sure. A riskier plan would leave the assets worth either $180 million or $40 million, each with probability ½, so an expected $110 million. Find the expected payoffs at maturity under the riskier plan.
Expected payoff to the shareholders?
Expected payoff to the bondholders?
✅ Correct! Against $20 million and $100 million under the safe plan, the shareholders' expected payoff doubles while the bondholders' falls by $30 million, even though the expected value of the firm drops by $10 million. Levered equity is biased toward risk, and the bias can outweigh net present value.
❌ Not quite. $20 million is the shareholders' payoff under the safe plan, 120−100. Under the riskier plan, work out each outcome and weight it by ½.
❌ Not quite. $40 million is what the bondholders get in the bad outcome only. In the good outcome they are paid their $100 million in full, and each outcome has probability ½.
❌ Not quite. $110 million is the expected value of the whole firm. The shareholders' expected $40 million comes out of it too.
❌ Not quite. That applies the payoff to the average, Max[0,110−100]. Because of the kink you must average the payoffs instead: the $80 million upside counts, but the downside stops at zero.
❌ Not quite. $80 million is the payoff in the good outcome only. It happens with probability ½, and the bad outcome pays $0.
❌ Not quite. In the bad outcome the assets are worth only $40 million, so that is all the bondholders get. They are paid in full only half the time.
❌ Check the equity. Compute Max[0,V−100] in each outcome, then weight each by ½.
❌ Check the debt. Compute Min[V,100] in each outcome, then weight each by ½.
Show solution
In $ millions, with B=100:
Outcome
V
E=Max[0,V−100]
D=Min[V,100]
Good (½)
180
80
100
Bad (½)
40
0
40
Expected
110
40
70
21⋅80+21⋅0=40,21⋅100+21⋅40=70,40+70=110
The safe plan gives E=120−100=20 and D=100. Switching raises the shareholders' expected payoff by $20 million and cuts the bondholders' by $30 million; the firm as a whole loses $10 million in expectation. The extra upside goes to the shareholders, and the extra downside falls on the bondholders.