Finance-Theory · Unit 10 · Video 4 · Interactive Practice

Why Every Forward Contract Starts at Zero, and What Its Price Forecasts

IKey Formulas

FormulaNameWhat it needs
Vbuyer=PV0[ST−F0,T],  Vseller=−VbuyerV_{\text{buyer}} = \text{PV}_0\big[S_T - F_{0,T}\big], \ \ V_{\text{seller}} = -V_{\text{buyer}}Value of a forwardThe payoff at settlement, valued today; what one side gains, the other gives up
V0=0  ⟺  PV0[ST]=PV0[F0,T]V_0 = 0 \iff \text{PV}_0[S_T] = \text{PV}_0[F_{0,T}]Forward price at signingThe price haggled out between producers (short) and users (long)
Vbuyer(F)=F0,T−F(1+rf)TV_{\text{buyer}}(F) = \dfrac{F_{0,T} - F}{(1+r_f)^T}Off-market price (our step, not in the video)Any other price FF: its gap to the haggled F0,TF_{0,T}, discounted at the risk-free rate because FF is a known amount
F0,T>S0,  F0,T=S0,  F0,T<S0F_{0,T} > S_0, \ \ F_{0,T} = S_0, \ \ F_{0,T} < S_0Forward vs spotAbove, at par or below: the market's forecast of STS_T, alongside the cost of carrying the asset

Key Insight: Any price below the haggled one hands the buyer money and any price above hands it to the seller, so bargaining stops only where the contract is worth zero to both. Value appears only afterwards, as prices move.

IIHaggling to the Zero-Value Price

At every forward price but one, one side's gain is the other's loss, and the losing side refuses.

💡 $110 is the video's example price; the formula that pins down F0,TF_{0,T} comes in a later video. Any other price is valued with our step from the Key Formulas table.

IIILock In Rather Than Forecast

The longer the horizon, the less accurate a price forecast; a forward replaces the whole band with one known price.

IVWhat the Forward Price Forecasts

Whether the six-month forward sits above, at or below today's spot is the market's forecast of where oil is headed.

💡 It is the same phenomenon as the yield curve: implied forward rates carry the market's forecast of future borrowing costs, and forward prices carry its forecast of future spot prices.

VQuiz Questions

Problem 1 · Who Walks Away?

Given: A grain user and a farmer have haggled a three-month wheat forward to $7.20 per bushel, the price at which the contract is worth zero to both. The user then proposes $6.50 per bushel instead. What happens?

✅ Correct! Lowering the price by $0.70 moves the present value of $0.70 per bushel from the farmer to the user, so the farmer will not sign.
❌ Check the direction. The buyer pays FF and receives wheat worth STS_T; a lower FF makes ST−FS_T - F larger, which is better for the buyer, not worse.
❌ Not at any price. The value is zero only at the haggled price. It is zero because both sides bargained to it; a contract at any other price has value to one side and an equal loss to the other.
❌ Not quite. The contract's value to the buyer is PV0[ST−F]\text{PV}_0[S_T - F]. Compare it at $6.50 with its value at the haggled $7.20.
Show solution

At the haggled price, PV0[ST−7.20]=0\text{PV}_0[S_T - 7.20] = 0. At the proposed price:

Vbuyer=PV0[ST−6.50]=PV0[ST−7.20]⏟= 0+PV0[0.70]>0V_{\text{buyer}} = \text{PV}_0[S_T - 6.50] = \underbrace{\text{PV}_0[S_T - 7.20]}_{=\,0} + \text{PV}_0[0.70] > 0

So Vseller=−Vbuyer<0V_{\text{seller}} = -V_{\text{buyer}} < 0. Selling at $6.50 would amount to handing the user money, and the farmer refuses. Zero value is a property of the haggled price, not of every price.

Problem 2 · Value at Signing

Given: Gold trades at $2,000 per ounce today. A jeweller and a mine agree on a one-year gold forward at the market's forward price, $2,080 per ounce. What is the contract worth to the jeweller (the buyer) at signing?

✅ Correct! At the market's forward price the contract is worth zero to both sides; the $80 gap to spot reflects the market's forecast, alongside the cost of carrying gold; it is not a gain or a loss.
❌ The gap is not a value. The buyer's value is PV0[ST−F0,T]\text{PV}_0[S_T - F_{0,T}], which compares the forward price with the future spot price, not today's. At the haggled price that value is zero.
❌ Discounting the gap does not help. F0,T−S0F_{0,T} - S_0 is not a payoff anyone receives. The payoff is ST−F0,TS_T - F_{0,T}, and at the haggled price its present value is zero.
❌ A premium over spot is not a loss. The jeweller receives gold worth STS_T in a year, not today's $2,000. Bargaining set $2,080 exactly where paying it is worth what the gold will be worth, so the value is zero.
❌ Not quite. What is the defining property of the price two sides haggle to?
Show solution

$2,080 is the market's forward price, the price at which neither side would rather walk away, so by construction

V0=PV0[ST−F0,T]=0⟺PV0[ST]=PV0[F0,T].V_0 = \text{PV}_0\big[S_T - F_{0,T}\big] = 0 \quad\Longleftrightarrow\quad \text{PV}_0[S_T] = \text{PV}_0[F_{0,T}].

The $80 difference from today's spot tells you something else: read as a forecast, the market expects gold to be worth more in a year, and the forward price also reflects the cost of carrying gold until delivery (storage, financing), which the next video takes up. Neither side gains or loses anything at signing; value appears only as prices move afterwards.

Problem 3 · Pricing an Off-Market Deal

Given: The haggled one-year crude oil forward price is $80 per barrel. A refinery asks a producer to sell it 1,000 barrels one year forward at $74 per barrel instead. Take the one-year risk-free rate to be 5% (an assumption for this problem). What is the contract worth to the producer (the seller) at signing? Use Vbuyer(F)=(F0,T−F)/(1+rf)TV_{\text{buyer}}(F) = (F_{0,T} - F)/(1+r_f)^T.

✅ Correct! The $6 per barrel shortfall is paid a year from now, so its value today is $6,000 ÷ 1.05, and it is the producer's loss.
❌ Right sign, missing the discount. The $6,000 difference is settled in one year; its value today is $6,000 divided by 1.05.
❌ Check whose value this is. That is the buyer's value. The seller delivers oil for $74 when the fair price is $80, so Vseller=−VbuyerV_{\text{seller}} = -V_{\text{buyer}} is negative.
❌ Discount, don't compound. Multiplying by 1.05 moves the amount forward in time. A value today divides the future amount by 1.05.
❌ Not quite. Find the buyer's value per barrel, scale by 1,000 barrels, then flip the sign.
Show solution

Per barrel, for the buyer (the refinery):

Vbuyer=80−741.05=61.05≈5.7143V_{\text{buyer}} = \frac{80 - 74}{1.05} = \frac{6}{1.05} \approx 5.7143

For 1,000 barrels the buyer's value is 6,0001.05≈5,714.29\tfrac{6{,}000}{1.05} \approx 5{,}714.29 dollars, and

Vseller=−Vbuyer≈−5,714.29 dollars.V_{\text{seller}} = -V_{\text{buyer}} \approx -5{,}714.29 \text{ dollars}.

The formula is our step: at the haggled price PV0[ST]=801.05\text{PV}_0[S_T] = \tfrac{80}{1.05}, and the known payment of 74 is worth 741.05\tfrac{74}{1.05} today. The producer would be handing the refinery about $5,714.29 of value, which is why it refuses unless the price moves back to $80.

Problem 4 · Reading a Forward Below Spot

Given: Natural gas trades at $3.00 per MMBtu today, and the one-year forward price is $2.40. Which reading is consistent with the video?

✅ Correct! A forward below spot carries the market's forecast of a lower future spot price, just as $90 against $100 did for oil in the video.
❌ Carrying costs push the other way. Storage and financing are costs of holding the asset until delivery (the next video takes them up); they would add to the forward price, not subtract from it. They cannot explain a forward below spot.
❌ Every forward starts at zero. At the market's forward price, PV0[ST−2.40]=0\text{PV}_0[S_T - 2.40] = 0. The buyer gains only if the spot price in a year ends up above $2.40.
❌ Today's price is not the alternative. The producer cannot sell next year's output at today's $3.00; the alternative is next year's spot price. A producer who expects prices to fall may be glad to lock in $2.40 and know its revenue.
❌ Not quite. What does a forward price below today's spot price tell you about where the market thinks prices are heading?
Show solution

Forward prices carry the market's forecast of the future spot price, alongside the cost of carrying the asset. Here

F0,T−S0=2.40−3.00=−0.60<0,F_{0,T} - S_0 = 2.40 - 3.00 = -0.60 < 0,

a drop of 20% of spot. Carrying costs would lift the forward price rather than lower it (our step ahead of the next video), so read as the video reads it, the gap points to expectations: either large new supply is expected, or demand is forecast to fall.

The contract is still worth zero to both sides at signing, and a producer that does not want to forecast gas prices can remove the uncertainty by locking in $2.40.

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