Finance-Theory · Unit 10 · Video 2 · Interactive Practice

Should a Firm Hedge? Irrelevance, Distress Costs and What Firms Actually Do

IKey Formulas

Formula	Name	What it says
$\text{NPV}_{\text{hedge}} = \text{PV}(\text{hedge payoff}) - \text{price paid} = 0$	Zero-NPV hedge (View 1)	At fair market prices the firm pays exactly what the hedge is worth, so the hedge adds no value on its own
$V_{\text{hedged}} = V_{\text{unhedged}} + \text{NPV}_{\text{hedge}} = V_{\text{unhedged}}$	Hedging irrelevance (Modigliani-Miller)	In a frictionless market, anything the firm can do its shareholders can do for themselves, so hedging leaves firm value unchanged
$V = \mathbb{E}[\text{CF}] - \Pr(\text{distress}) \times \text{distress cost}$	Distress costs (View 2, our illustrative model)	A zero-NPV hedge leaves $\mathbb{E}[\text{CF}]$ unchanged, but by lowering $\Pr(\text{distress})$ it cuts the expected deadweight cost, which raises $V$

Key Insight: Neither view wins in general. In a frictionless market hedging is irrelevant; with financial distress costs and projects that need funding, hedging can add value. Firms weigh those frictions differently, which is why their policies range from no hedging to aggressive hedging.

IIDo-It-Yourself Hedging (View 1)

With illustrative numbers: a shareholder can rebuild the firm's full hedge herself by selling gold forward.

Ounces she sells forward 0 oz

💡 Challenge: make her payoff match the hedged firm in every state. The video's counterargument is that a real shareholder rarely knows the firm's exposure well enough to compute that position, and is not set up to trade forwards.

IIIWhen Distress Is Costly (View 2)

With illustrative numbers: a costly distress state makes hedging add value; without that friction, hedging changes nothing.

Share of output sold forward 0% Distress cost $30M

IVHow Much Did Firms Actually Hedge?

For Guay and Kothari's 413 large non-financial firms: what would a three-standard-deviation price move have paid?

Your guess: cash payoff of a 3σ move $150M

💡 Use has spread since: 57% of the 1997 sample used derivatives, while a March 2025 review found 86.6% of non-financial companies in seven major stock indices using over-the-counter derivatives (87.1% with financial firms included). The samples differ, so the comparison is approximate.

VQuiz Questions

Problem 1 · A Fairly Priced Hedge

Given: An unhedged copper miner is worth $500 million. It sells next year's output forward at the fair forward price and pays no fees. Markets are frictionless and its shareholders are diversified. Find the firm's value right after it hedges.

✅ Correct! A hedge bought at fair value is a zero-NPV transaction, so in a frictionless market it leaves firm value unchanged.

❌ Not quite. Lower risk is not extra value here: diversified shareholders do not care about firm-specific risk, and the firm paid exactly what the hedge is worth.

❌ Not quite. The firm gives up the upside but also sheds the downside, and at the fair price the two are worth the same, so nothing is lost on average.

❌ Not quite. That describes the hedge's payoff after the fact. Its value today, at the fair price, is zero whichever way copper later moves.

❌ Not quite. Ask what the hedge is worth net of what the firm paid for it.

Show solution

At the fair forward price the hedge's present value equals what the firm pays for it:

\text{NPV}_{\text{hedge}} = \text{PV}(\text{payoff}) - \text{price} = 0

V_{\text{hedged}} = V_{\text{unhedged}} + 0

So the firm is still worth $500 million. This is View 1 (Modigliani-Miller): with no frictions, the hedge moves risk around without creating value, and diversified shareholders do not pay extra for the firm-specific risk it removes.

Problem 2 · Hedging It Yourself

Given: A shareholder owns 4% of an unhedged gold miner that produces $60{,}000$ ounces a year. The miner could sell its whole output forward, but she wants to get the same result herself. Find how many ounces she should sell forward.

✅ Correct! Her exposure is her share of the firm's output,

0.04 \times 60{,}000 = 2{,}400

ounces.

❌ Not quite. That is the firm's entire output. She owns only 4% of it, so selling it all forward would leave her heavily short gold.

❌ Not quite. Selling nothing leaves her fully exposed to the gold price. That is fine if she wants the risk, but it does not reproduce the hedged firm.

❌ Check the decimal. 4% is

0.04

, not

0.004

❌ Not quite. Match her share of the firm's ounces.

Show solution

Owning 4% of the firm means owning 4% of its gold exposure:

0.04 \times 60{,}000 = 2{,}400 \text{ oz}

Selling 2,400 ounces forward at the fair price gives her exactly the payoff of 4% of a fully hedged miner. This is the "do it yourself" half of View 1. It also shows the practical catch: she can only do it if she knows the firm's exposure.

Problem 3 · Paying for Distress

Given (illustrative): A miner must pay $45 million of costs and debt service at year end. Unhedged, its revenue is $40 million or $80 million with equal probability. A forward sale of a fraction $h$ of output locks in the fair price, so revenue in the bad state becomes $40(1-h) + 60h$ million dollars. If revenue falls short of the $45 million, financial distress destroys a further $12 million. Ignore discounting.

What is the smallest hedge ratio that avoids distress in both states?

By how much does hedging at that ratio raise firm value compared with no hedge?

✅ Correct! A 25% hedge closes the bad-state shortfall, and removing a 50% chance of a $12 million loss is worth $6 million.

❌ Not quite. 50% would work, but so does less. Solve

40 + 20h \ge 45

❌ Not quite. The $5 million shortfall has to be divided by what each unit of hedge adds in the bad state (

60 - 40 = 20

million), not by the bad-state revenue of 40.

❌ Not quite. A full hedge avoids distress, but it is not the smallest ratio that does.

❌ Not quite. That is the frictionless answer. Here distress costs real money, so lowering its probability adds value even though the hedge itself is zero NPV.

❌ Not quite. Distress happens only half the time, so its expected cost is

0.5 \times 12

million.

❌ Not quite. $15 million is the hedged firm's value, not the increase over the unhedged firm.

❌ Check the hedge ratio. Set bad-state revenue equal to the obligation and solve for

h

❌ Check the value change. Compare

\mathbb{E}[\text{CF}]

minus the expected distress cost with and without the hedge.

Show solution

Step 1: Hedge ratio. Bad-state revenue is $40(1-h) + 60h = 40 + 20h$ million. It must cover 45:

40 + 20h \ge 45 \quad\Longrightarrow\quad h \ge \tfrac{5}{20} = 25\%

The good state is never a problem: with $h \le 1$ , revenue $80 - 20h \ge 60 > 45$ .

Step 2: Values (in millions of dollars). The hedge is zero NPV, so expected revenue is $\tfrac{1}{2}(40) + \tfrac{1}{2}(80) = 60$ either way, and expected cash flow after the obligation is $60 - 45 = 15$ .

V_{\text{unhedged}} = 15 - 0.5 \times 12 = 9

V_{\text{hedged}} = 15 - 0 \times 12 = 15

Hedging raises value by $15 - 9 = 6$ , that is $6 million, all of it from the distress cost avoided.

Problem 4 · Where Hedging Pays

Given: The four firms below each consider a hedging program. Find the one for which the video's reasoning makes the strongest case that hedging adds value.

✅ Correct! Here the frictions are real: hedging lowers the chance of costly distress and keeps cash available for positive-NPV investment without turning to external finance.

❌ Not quite. That is exactly the Modigliani-Miller setting, where hedging is irrelevant: the shareholders can do anything the firm can.

❌ Not quite. Gold price risk is what these shareholders chose to bear. The video gives little reason to hedge a firm's own output unless management has a view that today's locked-in price is unusually high.

❌ Not quite. Paying above fair value makes the hedge negative NPV, so it destroys value before any benefit is counted.

❌ Not quite. Look for the frictions: distress costs and projects that need funding.

Show solution

View 2 says hedging creates value when frictions exist. It keeps cash available for positive-NPV investments, reduces the need for external finance and lowers the chance of costly financial distress. Firm (b) has two of these frictions at once, so (b) has the strongest case.

Firm (a) is the frictionless world of View 1, where hedging is irrelevant. Firm (c) would be hedging the very risk its shareholders bought. Firm (d) overpays, so its hedge has a negative NPV.

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