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CFA Level II Exam · Pricing and Valuation of Forward Commitments

Pricing and Valuation of Forward Contracts for CFA Level II

Updated 7 October 2026 · Fact-checked

The no-arbitrage forward price is the spot price grown at the risk-free rate, adjusted for carry: subtract the present value of benefits (dividends, coupons, convenience yield) and add the present value of costs (storage). The value of a forward at time t is the present value of the difference between the new forward price and the original price.

Understand Pricing and Valuation of Forward Contracts

A forward contract fixes a price today for buying an asset at a later date. At initiation the forward price is set so the contract has zero value to both sides. Nobody pays anything up front, so the price must leave no free profit.

The logic is replication. You can own the asset at expiry in two ways: buy a forward, or buy the asset now and hold it. Buying now costs the spot price, which you fund at the risk-free rate. So the base forward price is the spot price grown at the risk-free rate. Holding the asset also has side effects. Benefits such as dividends, coupons and convenience yield reduce the net cost of holding. Costs such as storage and insurance increase it. This is the cost-of-carry model: forward price = spot grown at the risk-free rate, minus benefits, plus costs.

If the market forward price is above the no-arbitrage price, you do a cash-and-carry trade: sell the forward, buy the asset with borrowed money, and deliver at expiry. If it is below, you do a reverse cash-and-carry: short the asset, invest the proceeds, and buy the forward. Both lock in a risk-free profit with no net investment.

After initiation the forward price changes as spot moves. The contract you hold was signed at the old price, so it now has positive or negative value. The long's value at time t is the present value of the gap between today's forward price for the same expiry and the original price. The short's value is the negative of that.

Convenience yield is a benefit of holding the physical commodity, such as keeping a plant running during a shortage. You cannot see it in cash flows. It is usually inferred from market prices. It lowers the forward price, just like a dividend. Because it only benefits holders of the physical asset, the reverse cash-and-carry trade can fail for commodities.

Key formulas to remember

Forward price, no carry (discrete compounding)
F0(T) = S0 × (1 + r)^T
Use for assets with no benefits or costs. T is in years. r is the annual risk-free rate.
Forward price with benefits and costs
F0(T) = (S0 − γ0 + θ0) × (1 + r)^T
γ0 is the PV of benefits (dividends, coupons, convenience yield). θ0 is the PV of costs (storage). Discount each cash flow from its payment date back to time 0.
Forward price, continuous compounding
F0(T) = S0 × e^((r + θ − γ) × T)
Here r, θ and γ are annual continuous rates. For a stock index with dividend yield δ, use e^((r − δ) × T).
Value of a long forward at time t
Vt(T) = [Ft(T) − F0(T)] ÷ (1 + r)^(T − t)
Ft(T) is the current no-arbitrage forward price for the same expiry. The short's value is −Vt(T).
Value of a long forward, spot form
Vt(T) = (St − γt + θt) − F0(T) ÷ (1 + r)^(T − t)
γt and θt are PVs at time t of benefits and costs still to come. Benefits already paid are excluded.
Arbitrage rule
Market F > no-arbitrage F: sell forward, buy asset, borrow. Market F < no-arbitrage F: buy forward, short asset, lend.
The profit at expiry equals the difference between the two forward prices, ignoring transaction costs.

How to solve Pricing and Valuation of Forward Contracts questions

Use the same sequence for any forward pricing or valuation question. Do the work in the vignette's time units.

  1. 1Identify what is asked: initial forward price, value at time t, or an arbitrage trade. Note the expiry T and the current time t.
  2. 2Pull the inputs from the vignette: spot price, risk-free rate, compounding convention (discrete or continuous), and every benefit and cost with its timing.
  3. 3Decide how each carry item enters. Dividends, coupons and convenience yield are benefits and reduce the price. Storage and insurance are costs and increase it.
  4. 4Compute the forward price. For discrete cash flows, discount each one to time 0 and subtract or add it to S0, then grow at (1 + r)^T. For rates, use the continuous formula.
  5. 5If valuing at time t, update the spot price and keep only the carry items still to come. Compute Ft(T) with the remaining term T − t.
  6. 6Value the long as [Ft(T) − F0(T)] ÷ (1 + r)^(T − t). Flip the sign for the short.
  7. 7For arbitrage, compare the market forward price with your price. If market is higher, sell the forward and buy the asset. If lower, do the reverse. State the profit at expiry.
  8. 8Sanity-check the sign: a higher spot should raise the long's value, and a larger dividend should lower the forward price.

Quickest way: Subtract-and-grow shortcut

When to use it: Use it when the vignette gives a few discrete cash flows and asks for a price or value, and you want to avoid building a long table.

  1. Compute the net carry in future-value terms: FV of costs minus FV of benefits, both to expiry.
  2. Forward price = S0 × (1 + r)^T − FV(benefits) + FV(costs). The FV of a dividend paid at time s is the dividend × (1 + r)^(T − s).
  3. For value at t, a fast check is Vt = St − PV(remaining benefits) + PV(remaining costs) − F0 ÷ (1 + r)^(T − t). You do not need Ft first.
  4. Eliminate options by sign. If the spot has risen above the level implied by the original forward, the long's value must be positive.

Common mistakes in Pricing and Valuation of Forward Contracts

  • Adding dividends to the forward price instead of subtracting them.

    Students think of dividends as extra value and add them.

    Fix: The forward buyer does not receive the dividends, so the forward price is lower. Benefits are subtracted; costs are added.

  • Subtracting the full dividend without discounting it, or discounting it for the wrong period.

    Rushing past the timing of the cash flow.

    Fix: Discount each dividend from its payment date to time 0 (or to t when valuing later). Or use the FV method: grow each dividend from its payment date to expiry.

  • Using the original forward price when valuing at time t, or using full T instead of T − t.

    Confusing the price locked in at initiation with the current forward price.

    Fix: Compute Ft(T) with the new spot and the remaining term. The original F0(T) only enters as the price you locked in. Discount the difference for T − t.

  • Including a dividend that was already paid before time t.

    Copying the initial pricing setup into the valuation step.

    Fix: At time t, only benefits and costs still to come count. If the dividend has been paid, γt for it is zero.

  • Treating convenience yield like a visible cash flow that can be arbitraged.

    It is listed beside dividends in the formula.

    Fix: Convenience yield is inferred and only accrues to holders of the physical asset. Reverse cash-and-carry may not be possible when it is high.

  • Mixing continuous and discrete compounding in one calculation.

    Rates are quoted in different forms across the vignette.

    Fix: Check the stated convention first. Use the e^ form only when rates are continuous.

Worked examples

Example 1

Vignette: An analyst prices a 9-month forward on a stock. The spot price is $80. The risk-free rate is 4% a year, effective annual (discrete). The stock pays one dividend of $1.00 in 3 months and none after. Q1: What is the no-arbitrage forward price? Q2: Three months later, just after the dividend, the stock trades at $82. What is the value to the long of the original forward? Q3: If the forward had been quoted at $82.00 at initiation, what trade earns an arbitrage profit?

Show the solution
  1. Q1: PV of the dividend = 1.00 ÷ 1.04^0.25 = 1.00 ÷ 1.009853 = $0.9902.
  2. Net spot = 80 − 0.9902 = $79.0098.
  3. Growth factor for 0.75 years = 1.04^0.75 = 1.029852.
  4. F0 = 79.0098 × 1.029852 = $81.37.
  5. Q2: Time t = 3 months, so 6 months remain. The dividend has been paid, so only the spot matters.
  6. Ft = 82 × 1.04^0.5 = 82 × 1.019804 = $83.62.
  7. Vt = (83.62 − 81.37) ÷ 1.019804. Using unrounded values: (83.6239 − 81.3684) ÷ 1.019804 = 2.2555 ÷ 1.019804 = $2.21.
  8. Q3: Market price $82.00 is above the no-arbitrage price $81.37. Sell the forward, buy the stock with borrowed money, collect the dividend and use it to repay part of the loan, then deliver at expiry. Profit at expiry is about 82.00 − 81.37 = $0.63 per share.

Answer: Q1: $81.37. Q2: $2.21 to the long. Q3: Sell the forward, buy the stock financed by borrowing (cash-and-carry), locking in about $0.63 per share at expiry.

Example 2

Vignette: A trader analyses a 6-month forward on a commodity with spot price $100. The continuously compounded risk-free rate is 5%. Storage costs are 2% of the price per year and the convenience yield is 1% per year, both continuous. Q1: What is the no-arbitrage forward price? Q2: The market forward price is $101.00. What convenience yield is implied? Q3: Why might the trader be unable to profit from a forward priced below the cost-of-carry value?

Show the solution
  1. Q1: Net carry rate = r + θ − γ = 5% + 2% − 1% = 6%.
  2. F0 = 100 × e^(0.06 × 0.5) = 100 × e^0.03 = 100 × 1.030455 = $103.05.
  3. Q2: Set 100 × e^((0.07 − y) × 0.5) = 101. So e^((0.07 − y) × 0.5) = 1.01.
  4. Take logs: (0.07 − y) × 0.5 = ln 1.01 = 0.009950.
  5. 0.07 − y = 0.019901, so y = 0.050099, about 5.01% a year.
  6. Q3: A forward priced below the cost-of-carry value would call for a reverse cash-and-carry: short the commodity, invest, and buy the forward. Holders of the physical commodity value having it on hand, so they will not lend it for shorting. The convenience yield is not a cash flow you can capture by trading, so the apparent mispricing may not be an arbitrage.

Answer: Q1: $103.05. Q2: Implied convenience yield about 5.01% a year. Q3: Short selling the physical commodity is hard because holders value it, so the arbitrage cannot be executed and the price gap reflects convenience yield.

Exam tips

  • Read the vignette for the compounding convention and the exact timing of each dividend or storage payment. Most lost marks come from timing, not from the formula.
  • For valuation questions, check whether a dividend falls before or after time t. Drop any carry item already paid.
  • Use sign logic to eliminate options. If spot has risen since initiation and there are no other changes, the long must show a gain.
  • When a question asks for an implied convenience yield, rearrange the continuous formula using natural logs. Keep four decimals until the end.
  • Know which arbitrage trade goes with which price gap, and be ready to explain why a reverse cash-and-carry can fail for commodities.

Pricing and Valuation of Forward Contracts in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Pricing and Valuation of Forward Contracts: frequently asked questions

What is the forward price formula with dividends?

Subtract the present value of the dividends from the spot price, then grow the result at the risk-free rate to expiry: F0(T) = (S0 − PV of dividends) × (1 + r)^T. With a continuous dividend yield δ, use F0 = S0 × e^((r − δ) × T).

How do I value a forward contract at time t?

Compute the new forward price Ft(T) using the current spot and the remaining term. Then take the present value of the difference from the original price: Vt = [Ft(T) − F0(T)] ÷ (1 + r)^(T − t). That is the long's value. The short's value is the negative.

How do storage costs and convenience yield affect the forward price?

Storage costs raise the forward price because holding the asset costs money. Convenience yield lowers it because holding the physical asset has a benefit. In the continuous formula they appear as r + θ − γ.

Why does a forward contract have zero value at initiation?

The forward price is set so that neither side pays anything and neither expects a gain. If the price were off, arbitrageurs would trade until it matched the cost-of-carry value. The contract only gains or loses value afterwards as spot and rates move.