Finance-Theory · Unit 11 · Video 6 · Interactive Practice

Hedging 25% Means Shorting Futures Worth 25%: Five Contracts Do What Selling 500 Stocks Would

IKey Formulas

Formula	Name	What it needs
$N = h \times V_0$	Notional to hedge	The fraction $h$ you want hedged and the exposure's current value $V_0$ : 25% of $5,000,000 is $1,250,000
$N_{\text{contract}} = 250 \times S_0$	Notional of one S&P 500 futures contract	$250 times the index level $S_0$ : at 1,000, one contract carries $250,000
$n = \dfrac{h\,V_0}{250\,S_0}$	Contracts to short	Notional to hedge divided by one contract's notional: 1,250,000 / 250,000 = 5
$V_1 = (1-h)\,V_0\,\dfrac{S_1}{S_0} + h\,V_0$	Hedged value at index level $S_1$	Stocks plus the short futures' gain or loss, which equals holding $(1-h)$ in stocks and $h$ in cash; it assumes the portfolio moves one-for-one with the index and ignores interest and dividends

Key Insight: To hedge X% of an exposure, take a futures or forward position whose notional is X% of the exposure's current value. The hedged share then behaves like cash; the other $(1-h)$ still carries its market risk, so the hedge dampens a move without eliminating it.

IIFrom a Percentage to a Number of Contracts

The futures notional must equal the hedged share of the portfolio, and each contract carries $250 times the index.

Portfolio ($M) 5.0 Hedge 25% Index level 1,000

IIIShort Futures = Part of the Portfolio in Cash

When the portfolio tracks the index one-for-one, shorting futures on X% gives exactly the wealth of moving X% into cash.

Contracts short 5 Index at the horizon 1,100

💡 Like the video's table, this ignores interest on the cash and dividends. Counting them shifts both routes by the same amount, because the futures price already builds in the risk-free rate minus the dividend yield.

IVQuiz Questions

Problem 1 · Size the Hedge

Given: you hold an $8,000,000 portfolio that moves one-for-one with the S&P 500, the index stands at 1,200, and one S&P 500 futures contract has a notional value of $250 times the index. Find the futures position that hedges 30% of the portfolio.

✅ Correct! 30% of $8,000,000 is $2,400,000 of notional, and at 1,200 one contract carries $300,000, so you short 8.

❌ Not quite. That uses $250,000 per contract, the value when the index was at 1,000 (2,400,000 / 250,000 = 9.6, rounded to 10). A contract's notional moves with the index: at 1,200 it is $250 × 1,200 = $300,000.

❌ Right size, wrong side. A long position gains when the index rises, which adds to the exposure you already have. To soften a fall you need a position that gains when the market falls: a short.

❌ Not quite. 8,000,000 / 300,000 ≈ 26.7, rounded to 27, hedges the whole portfolio. Hedging 30% needs a notional of only 30% of the portfolio's value.

❌ Not quite. Divide the notional to hedge (30% of the portfolio) by one contract's notional ($250 times 1,200), and sell.

Show solution

Notional to hedge (in dollars):

0.30 \times 8{,}000{,}000 = 2{,}400{,}000

One contract's notional at index 1,200 (in dollars):

250 \times 1{,}200 = 300{,}000

Number of contracts:

n = \frac{2{,}400{,}000}{300{,}000} = 8

You want a position that gains when the index falls, so you short 8 contracts. Because the portfolio moves one-for-one with the index, this hedges exactly 30% of it.

Problem 2 · What the Hedge Does on a Rise

Given: a $6,000,000 portfolio that moves one-for-one with the S&P 500, with the index at 1,500. You hedge 50% by shorting S&P 500 futures (notional $250 times the index per contract). The index then rises to 1,650. Find the value of stocks plus futures, ignoring interest and dividends.

✅ Correct! The stocks gain $600,000 and the 8 short contracts lose $300,000, so you keep half of the rise: the same result as holding $3,000,000 in stocks and $3,000,000 in cash.

❌ Check the sign. You added the futures result. A short position loses when the index rises: 8 × $250 × 150 = $300,000 comes off the stocks' value.

❌ Not quite. $6,600,000 is the stocks alone. The 8 short contracts lose $300,000 on the 150-point rise, and that loss belongs in the total.

❌ Not quite. Only a 100% hedge would hold the value at $6,000,000. At 50%, half the portfolio still rides the 10% rise.

❌ Not quite. Find the number of contracts first, then add the stocks' new value and the short futures' loss.

Show solution

Contracts: the notional to hedge is 50% of $6,000,000 = $3,000,000; one contract carries $250 × 1,500 = $375,000:

n = \frac{3{,}000{,}000}{375{,}000} = 8 \text{ contracts, short}

Stocks: the index rises $\frac{1{,}650}{1{,}500} = 1.10$ , a 10% rise, so the stocks are worth $6,000,000 × 1.10 = $6,600,000.

Futures: each contract changes by $250 × 150 = $37,500 for the long side; short 8, you lose 8 × $37,500 = $300,000.

Total: $6,600,000 − $300,000 = $6,300,000.

Check with the cash route: selling 50% leaves $3,000,000 in stocks, worth $3,300,000 after the rise, plus $3,000,000 in cash: $6,300,000.

Problem 3 · Share Hedged and the Dampened Swing

Given: a $10,000,000 portfolio that moves one-for-one with the S&P 500, with the index at 1,400, and a position of 10 short S&P 500 futures (notional $250 times the index per contract). Interest and dividends are ignored. Find the share of the portfolio hedged, and the change in the hedged position's value if the index falls 280 points.

What share of the portfolio is hedged?

How does the hedged position's value change?

✅ Correct! 35% is hedged, so the position moves by 65% of the unhedged swing: 0.65 × $2,000,000 = $1,300,000.

❌ Check the contract's notional. 25% uses $250,000 per contract, the value at index 1,000. At 1,400 each contract carries $250 × 1,400 = $350,000.

❌ That is the other part. 65% is the share still exposed to the market; the share hedged is the futures notional divided by the portfolio value.

❌ Check the share hedged. Divide the futures notional (10 contracts × $250 × 1,400) by the portfolio's value.

❌ Not quite. $2,000,000 is the stocks' loss alone. The 10 short contracts gain when the index falls, and that gain offsets part of the loss.

❌ Not quite. $700,000 is only the futures' gain. The stocks still lose $2,000,000 on the 20% fall; the hedge dampens the loss without eliminating it.

❌ Check the sign. A short position gains when the index falls, so the futures' $700,000 reduces the loss rather than adding to it.

❌ Check the change in value. Add the stocks' change and the short futures' change.

Show solution

Share hedged: one contract carries $250 × 1,400 = $350,000, so 10 contracts carry $3,500,000 of notional:

h = \frac{3{,}500{,}000}{10{,}000{,}000} = 0.35 = 35\%

Stocks: a 280-point fall from 1,400 is a 20% fall, so the stocks lose 0.20 × $10,000,000 = $2,000,000.

Futures: each contract changes by $250 × 280 = $70,000; short 10, you gain 10 × $70,000 = $700,000.

Net: −$2,000,000 + $700,000 = a fall of $1,300,000.

Check: the hedged position moves by $(1 - h) = 0.65$ of the unhedged swing, and 0.65 × $2,000,000 = $1,300,000.

Problem 4 · The Same Rule for a Company

Given: a gold miner will sell 400,000 ounces of gold next year. At today's price of $1,500 an ounce that output is worth $600,000,000, and its value moves one-for-one with the gold price. One gold futures contract covers 100 ounces. Find the futures position that hedges 60% of this exposure.

✅ Correct! The notional is 60% of $600,000,000 = $360,000,000; each contract carries $150,000 of gold, so the miner shorts 2,400. The other 40% of its revenue still moves with the gold price.

❌ Right size, wrong side. The miner will sell gold, so it loses when the price falls. It needs a position that gains when the price falls: a short, like the portfolio's hedge.

❌ Not quite. 4,000 contracts cover all 400,000 ounces, a 100% hedge. For 60%, the notional is 60% of the exposure's current value.

❌ That is the other part. 1,600 contracts cover 40% of the output, the share the miner leaves unhedged.

❌ Not quite. Notional to hedge = 60% of the exposure's current value; divide by one contract's notional (100 ounces at today's price).

Show solution

Notional to hedge (in dollars): 60% of the exposure's current value,

0.60 \times 600{,}000{,}000 = 360{,}000{,}000

One contract's notional (in dollars): 100 ounces at today's price,

100 \times 1{,}500 = 150{,}000

Number of contracts:

n = \frac{360{,}000{,}000}{150{,}000} = 2{,}400

The miner is long gold (it will sell its output), so it shorts 2,400 contracts, a position that gains when the gold price falls. This is the video's rule in a new setting: the notional sets the percentage hedged, and the unhedged 40% still carries the price risk.

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