Finance-Theory · Unit 9 · Video 1 · Interactive Practice
Why Common Stock Finances Innovation: Residual Claims, Limited Liability, Votes and Short Sales
IKey Formulas
Formula
Name
What it says
bondholders=min(V,F)
Capped first claim
V is what the firm is worth when the debt falls due and F the promised payments, face value plus coupons (our notation for the video's payoff picture). Bondholders are paid first, and never more than F
shareholders=max(V−F,0)
Residual claim with limited liability
Everything left once the bondholders are paid, with no ceiling; the floor at 0 means the most a shareholder can lose is what they put in
equity value=dividends+capital gains
Two parts of a share's value
Cash or stock dividends (growth firms often pay none) plus price appreciation, collected by selling; both uncertain in size and timing, unlike a bond's promised coupons
short-sale gain=Psell−Pbuy back
Short sale, per share
Borrow shares, sell them, buy them back later to return them: a gain when the price falls, which is one way bad news reaches the price
Key Insight: The two claims split the firm exactly, min(V,F)+max(V−F,0)=V. Capping the bondholders hands all of the upside to the shareholders, and limited liability caps the shareholders' downside at their stake. Add votes for the many owners and public markets to bring them in, and equity is built to fund risky, ambitious ventures.
IICapped Claim, Residual Claim
Bondholders never get more than they were promised; shareholders keep everything above it and, with limited liability, lose at most their stake.
IIIEight Bankruptcies, Then a Ninth Company
Limited liability let Lo's speaker walk away from eight bankruptcies; a one-failure rule would have stopped him at the first.
IVShort Sales Let Bad News Into the Price
A trader who knows the stock is worthless but cannot sell short runs out of shares before the price gets anywhere near zero.
💡 The SEC's September 2008 short-sale ban, covered earlier in the course, applied to financial stocks only; for those shares it switched this channel off.
VQuiz Questions
Problem 1 · Who Gets What at Maturity
Given: a firm's bondholders are owed a face value of $100 million plus a final coupon of $10 million, due today. The firm's assets are worth $500 million. Find what the shareholders receive.
✅ Correct! The bondholders take their promised 100+10=110 (in millions) first; the residual 500−110=390 belongs to the shareholders.
❌ Close, but… the bondholders' claim is the face value plus the coupons along the way. Subtract all $110 million, not just the $100 million of face value.
❌ Not quite. Shareholders are the residual claimants: they stand behind the bondholders, who are paid first.
❌ Not quite. $110 million is the bondholders' capped claim. The shareholders get everything else.
❌ Not quite. Pay the bondholders their promised payments first; the shareholders get what remains.
Show solution
In millions of dollars, the promised payments are F=100+10=110 and the firm is worth V=500.
The shareholders receive $390 million. Had the firm been worth $5 billion instead, the bondholders would still get $110 million and every extra dollar would go to the shareholders: the bondholders' claim is capped, the residual is not.
Problem 2 · The Most You Can Lose
Given: you buy $50,000 of common stock in a start-up. The start-up goes bankrupt owing its creditors $3 million, with assets worth only $1 million. Find the most you can lose as a shareholder.
✅ Correct! Limited liability: the most a shareholder can lose is everything they put in, not everything they have. The creditors take the $1 million of assets and bear the rest of the loss.
❌ Not quite. The $2 million shortfall is the creditors' loss. Shareholders owe nothing beyond their stake.
❌ Not quite. That is the loss under unlimited liability, the world before the modern corporation, where owners had to make good the firm's debts.
❌ Not quite. "Everything" means everything you put in, not your house, savings or freedom: that is what limited liability guarantees.
Show solution
In millions of dollars, the creditors are owed F=3 and the firm is worth V=1:
creditors=min(1,3)=1,shareholders=max(1−3,0)=0
Your shares pay nothing, so you lose the $50,000 you paid for them and not a cent more. The creditors absorb the 3−1=2 million shortfall. Under unlimited liability you would have owed that $2 million personally, and defaulting on it could once have landed you in prison.
Problem 3 · Upside With No Ceiling, Upside Capped
Given: a founder puts $2 million of equity into a firm, and a bank lends it $5 million for one year at 8%, so the firm must repay $5.4 million. The firm has limited liability. Find each return below.
If the firm is worth $12.4 million in a year, the founder's return is
If instead the firm is worth $4.0 million, the bank's return is
✅ Correct! In the good state the founder earns +250% while the bank earns its capped +8%; in the bad state the bank takes the whole $4.0 million and loses 20%, while the founder loses the $2 million stake and no more.
❌ Close, but… the founder receives $7.0 million, which is 3.5× the stake; the return subtracts the $2 million put in: (7.0−2.0)/2.0=250%.
❌ Not quite.12.4/2.0−1 treats the whole firm as the founder's. The bank is paid its $5.4 million first.
❌ Not quite.12.4/5.0−1 divides by the loan. The founder's return is the residual, 12.4−5.4, measured against the founder's own $2 million.
❌ Not quite.+8% is the promised return, and the most the bank can ever earn. With only $4.0 million in the firm, it cannot be paid in full.
❌ Close, but…4.0/5.4−1 measures the payoff against what was promised. A return is measured against what the bank actually lent, $5.0 million.
❌ Not quite.−100% is the founder's return in this state. The bank, first in line, takes all $4.0 million.
❌ Check the founder's return. Pay the bank $5.4 million first, then compare what is left with the $2 million stake.
❌ Check the bank's return. The bank receives min(V,5.4); compare that with the $5.0 million it lent.
Show solution
In millions of dollars, F=5.0×1.08=5.4.
Firm worth 12.4: the bank gets min(12.4,5.4)=5.4, a return of 5.4/5.0−1=8%. The founder gets
max(12.4−5.4,0)=7.0,rE=2.07.0−2.0=2.50=250%
Firm worth 4.0: the bank gets everything, min(4.0,5.4)=4.0:
rB=5.04.0−5.0=−0.20=−20%
The founder gets max(4.0−5.4,0)=0, a return of −100%: the loss stops at the $2 million stake. The bank's return can never beat +8%; the founder's has no ceiling.
Problem 4 · Profiting From Bad News
Given: a trader borrows 200 shares and sells them short at $25 each. Bad news reaches the market, the price falls to $5, and the trader buys the 200 shares back and returns them. Find the trader's gain or loss.
✅ Correct! Sold for 200×25=5000, bought back for 200×5=1000: a $4,000 gain. Short selling pays when the price falls, which is why it pushes bad news into the price.
❌ Check the direction. A falling price hurts someone who owns the shares. A short seller sold first and buys back later, so a fall is a gain.
❌ Not quite. $5,000 is what the short sale raised, but the borrowed shares must be bought back and returned, which costs $1,000.
❌ Not quite. $1,000 is the cost of buying the shares back. The gain is the sale proceeds minus that cost.
❌ Not quite. Gain per share is Psell−Pbuy back; multiply by the number of shares.
Show solution
gain per share=Psell−Pbuy back=25−5=20total gain=200×20=4000
The trader gains $4,000. Each short sale adds selling pressure on a stock the trader believes is overpriced, so a trader who owns no shares can still move the price toward the bad news. Under a short-sale ban, this trader could not have acted on the news at all.