Finance-Theory · Unit 11 · Video 3 · Interactive Practice

Pricing a Forward by Replication: The Cost of Carry

IKey Formulas

FormulaNameMeaning
St,  Ft,T≈Ht,TS_t,\ \ F_{t,T} \approx H_{t,T}NotationSpot price at tt; forward and futures prices priced at tt for settlement at TT
F0,T≈H0,T=(1+rf)TS0+FVT(net storage costs)F_{0,T} \approx H_{0,T} = (1+r_f)^T S_0 + \text{FV}_T(\text{net storage costs})Cost of carry (future-value form)Forward price = replication cost: financed spot price plus net storage
F0,T(1+rf)T≈H0,T(1+rf)T=S0+PV0(net storage costs)\dfrac{F_{0,T}}{(1+r_f)^T} \approx \dfrac{H_{0,T}}{(1+r_f)^T} = S_0 + \text{PV}_0(\text{net storage costs})Cost of carry (present-value form)Both sides discounted back to date 0
net storage costs=storage costs−convenience yield\text{net storage costs} = \text{storage costs} - \text{convenience yield}Net storageCumulative costs of holding the asset, less the benefit of having it on hand

Key Insight: No forecast of STS_T appears anywhere — only S0S_0, rfr_f, TT and storage. If F0,TF_{0,T} rose above the replication cost, selling forward and carrying the asset would be a riskless profit.

IIVisualization 1 — Two Routes to the Asset at Date T

Both routes cost nothing at date 0 and deliver the asset at TT, so their date-TT costs must match.

IIIVisualization 2 — Arbitrage Enforcement

A forward price away from the replication cost invites a riskless trade — but not every gap can be closed.

IVQuiz Questions

Problem 1 · Forward Price by Cost of Carry

Given (illustrative figures): soybeans trade at a spot price of $25.00 per bushel. The risk-free rate is rf=4%r_f = 4\% a year (annual compounding), delivery is in T=2T = 2 years, and the future value at TT of net storage costs is $0.75 per bushel — find the no-arbitrage forward price F0,2F_{0,2}.

✅ Correct! Financed spot price $27.04 plus $0.75 of net storage gives $27.79.
❌ Not quite. That is only the loan repayment (1.04)2×25.00=27.04(1.04)^2 \times 25.00 = 27.04. Route two also pays the net storage costs at TT.
❌ Check the sign. Net storage is a cost of carrying the asset, so it is added to the financed spot price, not subtracted.
❌ Close, but… 25.00×(1+2×0.04)+0.75=27.7525.00 \times (1 + 2 \times 0.04) + 0.75 = 27.75 uses simple interest. The video compounds annually: (1+rf)T(1+r_f)^T.
❌ Not quite. Use F0,T=(1+rf)TS0+FVT(net storage costs)F_{0,T} = (1+r_f)^T S_0 + \text{FV}_T(\text{net storage costs}).
Show solution

Route two borrows the spot price for two years and pays net storage at delivery:

F0,2=(1+rf)2S0+FV2=(1.04)2×25.00+0.75=1.0816×25.00+0.75=27.04+0.75=27.79F_{0,2} = (1+r_f)^2 S_0 + \text{FV}_2 = (1.04)^2 \times 25.00 + 0.75 = 1.0816 \times 25.00 + 0.75 = 27.04 + 0.75 = 27.79

The forward price is $27.79 per bushel.

Problem 2 · Where Is the Forecast?

Given: A commodity with no storage costs and no convenience yield trades at a spot price of $40.00. The risk-free rate is rf=6%r_f = 6\% a year, and delivery is in T=1T = 1 year. Analysts expect the spot price in one year to be $45.00 — find the forward price F0,1F_{0,1}.

✅ Correct! The forward price is the replication cost 1.06×40.00=42.401.06 \times 40.00 = 42.40; the $45.00 forecast plays no role.
❌ Not quite. The cost-of-carry relation has no E[ST]E[S_T] term. Replication pins the price using only S0S_0, rfr_f, TT and storage.
❌ Close, but… Buying at $40.00 today means borrowing $40.00, and the loan must be repaid with interest at TT.
❌ Not quite. Compute the cost of borrowing to buy and hold the asset until TT.
Show solution

With no net storage costs, route two costs only the loan repayment:

F0,1=(1+rf)1S0+0=1.06×40.00=42.40F_{0,1} = (1+r_f)^1 S_0 + 0 = 1.06 \times 40.00 = 42.40

The forward price is $42.40. If the forward traded at the $45.00 forecast, you could sell forward, borrow $40.00 and buy the asset, then deliver at TT for a riskless $2.60 — whatever the spot price turns out to be.

Problem 3 · A Forward Priced Too High

Given: Crude oil trades at a spot price of $60.00 per barrel, rf=5%r_f = 5\% a year, and T=1T = 1 year. Storing a barrel until TT costs $2.50 and the convenience yield is worth $0.50 (both as date-TT amounts). The one-year forward trades at $66.40 — find the arbitrage trade and its riskless profit per barrel at TT.

Which trade earns a riskless profit?

What is the profit per barrel at TT?

✅ Correct! The replication cost is $65.00, so selling forward at $66.40 and carrying the oil locks in $1.40 per barrel.
❌ Not quite. Compare F0,TF_{0,T} with the replication cost (1+rf)TS0+FVT(net storage costs)(1+r_f)^T S_0 + \text{FV}_T(\text{net storage costs}). When the forward is too high, sell it and replicate the long position.
❌ Not quite. Profit at TT = forward price received − loan repayment − net storage costs.
❌ Close, but… 66.40−63.00=3.4066.40 - 63.00 = 3.40 ignores storage. Carrying the barrel costs $2.50 − $0.50 = $2.00 net.
❌ Not quite. The $60.00 was borrowed, so you repay 1.05×60.00=63.001.05 \times 60.00 = 63.00, not $60.00, and you also pay net storage.
❌ Close, but… You subtracted the full $2.50 storage cost. Net storage costs are storage minus convenience yield: $2.00.
Show solution

Step 1: Replication cost. Net storage costs are $2.50 − $0.50 = $2.00, so

(1+rf)TS0+FVT=1.05×60.00+2.00=63.00+2.00=65.00(1+r_f)^T S_0 + \text{FV}_T = 1.05 \times 60.00 + 2.00 = 63.00 + 2.00 = 65.00

Step 2: Compare. The forward trades at $66.40, above $65.00, so sell the forward and carry the oil.

Step 3: Cash flows. Date 0: sell forward ($0), borrow $60.00, buy one barrel — net $0. Date TT: deliver the barrel and collect $66.40, repay $63.00, pay $2.00 net storage:

66.40−63.00−2.00=1.4066.40 - 63.00 - 2.00 = 1.40

Riskless profit: $1.40 per barrel, from a position that cost nothing today.

Problem 4 · A Forward Priced Too Low

Given: An asset with no storage costs and no convenience yield, which can be borrowed and sold short, trades at $100.00. The risk-free rate is rf=3%r_f = 3\% a year and the two-year forward trades at $104.00 — find the arbitrage trade and its riskless profit per unit at T=2T = 2.

Which trade earns a riskless profit?

What is the profit per unit at TT?

✅ Correct! The replication cost is $106.09; buying forward at $104.00 against a short sale keeps the $2.09 gap.
❌ Not quite. Compare the forward with (1+rf)TS0(1+r_f)^T S_0, not with S0S_0.
❌ Not quite. Selling forward and carrying pays 104.00−106.09<0104.00 - 106.09 < 0 at TT: that trade enforces the upper bound, and here the forward is too low.
❌ Not quite. The benchmark is the replication cost (1.03)2×100.00=106.09(1.03)^2 \times 100.00 = 106.09, not the spot price. The forward sits $2.09 below it.
❌ Not quite. That is long the asset twice, once through the forward and once outright, so nothing offsets and the payoff depends on STS_T. To exploit a forward below the replication cost, buy it and short the asset.
❌ Not quite. Profit at TT = what the loan returns − forward price paid.
❌ Close, but… 100.00×(1+2×0.03)=106.00100.00 \times (1 + 2 \times 0.03) = 106.00 is simple interest. With annual compounding the loan returns (1.03)2×100.00(1.03)^2 \times 100.00.
❌ Not quite. 104.00−100.00104.00 - 100.00 ignores the interest earned on the lent proceeds, and has the direction backwards.
❌ Not quite. 106.09−100.00106.09 - 100.00 is the interest earned; you still pay the $104.00 forward price to take delivery, not the spot price.
Show solution

Step 1: Replication cost (no storage costs):

(1+rf)TS0=(1.03)2×100.00=1.0609×100.00=106.09(1+r_f)^T S_0 = (1.03)^2 \times 100.00 = 1.0609 \times 100.00 = 106.09

Step 2: Compare. The forward at $104.00 is below $106.09, so reverse every leg of the replication.

Step 3: Cash flows. Date 0: buy the forward ($0), short the asset (+$100.00), lend $100.00 at rfr_f — net $0. Date TT: the loan returns $106.09; pay $104.00, take delivery, and return the borrowed asset:

106.09−104.00=2.09106.09 - 104.00 = 2.09

Riskless profit: $2.09 per unit. This lower-bound trade is guaranteed here because the asset can be shorted and has no storage costs.

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