Finance-Theory · Unit 10 · Video 3 · Interactive Practice

A Forward Contract Fixes the Price at Signing; the Spot Price at Settlement Decides Who Gains

IKey Formulas

Formula	Name	What it says
$F_{0,T}$	Forward price	Fixed at date 0 when the contract is signed, paid at the settlement date $T$ ; it does not change afterward
$V_{\text{buyer}} = S_T - F_{0,T}$	Long payoff	The buyer pays $F_{0,T}$ and receives an asset worth the spot price $S_T$
$V_{\text{seller}} = F_{0,T} - S_T$	Short payoff	The seller receives $F_{0,T}$ and gives up an asset worth $S_T$
$V_{\text{buyer}} + V_{\text{seller}} = 0$	Zero sum	Whatever one side gains, the other loses

Key Insight: Of the three prices — today's spot $S_0$ , the forward price $F_{0,T}$ and the future spot $S_T$ — only $S_T$ is uncertain. The long gains when $S_T \gt F_{0,T}$ and the short when $S_T \lt F_{0,T}$ ; where $S_T$ ends relative to $S_0$ does not enter either payoff. A forward is an obligation: absent default, both sides must perform.

IIWho Gains at Settlement

At date $T$ the long receives $S_T - F_{0,T}$ and the short exactly the opposite, whatever $S_T$ turns out to be.

Forward price F $110 Spot at settlement S_T $125

IIIThree Prices, and Only One Moves

Between signing and settlement only the spot price moves; the payoff is its final gap to the fixed $F_{0,T}$ .

Spot path Time day 126 of 126

IVQuiz Questions

Problem 1 · The Long's Payoff

Given: A jeweller goes long a gold forward with forward price $2,050 per ounce. At the settlement date the spot price of gold is $1,980 per ounce. Find the jeweller's payoff per ounce.

✅ Correct! The long pays the fixed $2,050 for gold worth only $1,980, so it loses $70 an ounce — and the short gains exactly that.

❌ Check the sign. +$70 is the short's payoff,

F_{0,T} - S_T

. The spot price ended below the forward price, and the long is locked into paying the higher price.

❌ Not quite. $1,980 is what the gold received is worth. The long also pays the forward price for it, so the payoff is

S_T - F_{0,T}

, not

S_T

❌ Not quite. Taking delivery is how the contract settles, but the long pays the fixed forward price for an asset worth the spot price; the gap between the two is a real gain or loss.

❌ Not quite. Use

V_{\text{buyer}} = S_T - F_{0,T}

Show solution

The jeweller is the buyer, so it is long. In dollars per ounce:

V_{\text{buyer}} = S_T - F_{0,T} = 1{,}980 - 2{,}050 = -70

The jeweller loses $70 per ounce. The seller's payoff is $F_{0,T} - S_T = 2{,}050 - 1{,}980 = 70$ , a gain of $70 per ounce, and the two sum to zero.

Problem 2 · Measured From Which Price?

Given: Natural gas trades today at $3.00 per MMBtu. A utility signs a 3-month forward to buy gas at $3.40 per MMBtu. At settlement the spot price is $3.25 per MMBtu. Which statement is correct, per MMBtu?

✅ Correct! The spot price did rise, but it ended below the $3.40 written into the contract, so the long loses.

❌ That's the rise from today's spot price. $3.25 − $3.00 measures

S_T - S_0

, but the long pays

F_{0,T}

, not

S_0

. The payoff is

S_T - F_{0,T}

❌ Check the sign.

S_T

ended below

F_{0,T}

, so the long, locked into paying the higher price, is the side that loses.

❌ Not quite. The two payoffs always sum to zero:

V_{\text{buyer}} + V_{\text{seller}} = 0

. One side's gain is the other's loss.

❌ Not quite. Compare

S_T

with

F_{0,T}

, not with today's spot price.

Show solution

Three prices appear: $S_0 = 3.00$ , $F_{0,T} = 3.40$ and $S_T = 3.25$ (dollars per MMBtu). Only $F_{0,T}$ and $S_T$ enter the payoffs:

V_{\text{buyer}} = S_T - F_{0,T} = 3.25 - 3.40 = -0.15, \qquad V_{\text{seller}} = F_{0,T} - S_T = 0.15

The utility loses $0.15 per MMBtu and the seller gains $0.15. That the spot price rose $0.25 from $S_0$ is irrelevant: the long's break-even is $S_T = F_{0,T}$ , and $3.25 falls short of it.

Problem 3 · Back Out the Forward Price

Given: On a forward contract, the long's payoff at settlement would be +$8 per unit if the spot price at $T$ were $57. Find the short's payoff per unit if the spot price at $T$ is instead $44.

✅ Correct! The forward price is $49, and $44 is $5 below it, so the short gains $5.

❌ Check the sign. A spot price below

F_{0,T}

means the short delivers an asset worth less than the price it receives, so the short gains:

V_{\text{seller}} = F_{0,T} - S_T

❌ $57 is not the forward price. $57 is one possible spot price at

T

. The forward price is fixed, and the first scenario pins it down:

S_T - F_{0,T} = 8

with

S_T = 57

❌ That's the first scenario. −$8 is the short's payoff if the spot price were $57. The forward price carries over between scenarios, but the spot price does not: recompute

F_{0,T} - S_T

with

S_T = 44

❌ Not quite. First find

F_{0,T}

from the long's payoff, then use

V_{\text{seller}} = F_{0,T} - S_T

Show solution

Step 1: Find the forward price. The long's payoff is $S_T - F_{0,T}$ , and $F_{0,T}$ is the same in every scenario:

S_T - F_{0,T} = 57 - F_{0,T} = 8 \quad\Longrightarrow\quad F_{0,T} = 49

Step 2: The short's payoff at the new spot price.

V_{\text{seller}} = F_{0,T} - S_T = 49 - 44 = 5

The short gains $5 per unit (and the long loses $5). Only $F_{0,T}$ carries over from the first scenario; the payoff is recomputed from the new spot price.

Problem 4 · An Obligation, Not an Option

Given: An airline is long a forward on 50,000 gallons of jet fuel at $2.60 per gallon. At the settlement date jet fuel trades at $2.36 per gallon, and a manager suggests skipping the contract and buying the fuel in the spot market instead. The seller does not default. What happens at settlement, and what is the airline's total payoff on the forward?

✅ Correct! The contract is binding: the airline pays $130,000 for fuel worth $118,000, a $12,000 loss that is the seller's $12,000 gain.

❌ That describes an option. An option is a right without an obligation, and its holder may let it lapse. A forward is an obligation: once signed it is legally binding, and absent default both sides must perform.

❌ Check the sign. The spot price ended below the forward price, so the long pays more than the fuel is worth; +$12,000 is the seller's payoff.

❌ That's the payoff per gallon. −$0.24 is

S_T - F_{0,T}

for one gallon; the contract covers 50,000 gallons, so scale it up.

❌ Not quite. Use

V_{\text{buyer}} = S_T - F_{0,T}

per gallon, then scale by the contract size.

Show solution

Step 1: What happens. A forward is an obligation, so the airline must buy the 50,000 gallons at the forward price of $2.60, whatever the spot price.

Step 2: Payoff per gallon (dollars):

V_{\text{buyer}} = S_T - F_{0,T} = 2.36 - 2.60 = -0.24

Step 3: Scale by the contract size.

-0.24 \times 50{,}000 = -12{,}000

Equivalently, the airline pays $F_{0,T} \times 50{,}000 = 2.60 \times 50{,}000 = 130{,}000$ dollars for fuel worth $S_T \times 50{,}000 = 2.36 \times 50{,}000 = 118{,}000$ dollars. It loses $12,000 and the seller gains $12,000.

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