Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Derivatives, LOS weight share 0.5 percent of the 365 Level I learning outcomes.

DerivativesPricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

The forward price is fixed the day the contract is signed and never changes again, but the forward's value moves every single day after that, and the exam's favorite trap is a candidate who has memorized this sentence and still answers a mid-life valuation question with the wrong number.

Before you watch

Answer these three first. Getting them wrong now is normal, and it helps the lesson stick. Reveal the answers when you're done, then read on.

1. A non-dividend-paying stock trades at $80. The annual risk-free rate is 5% (discrete compounding). The no-arbitrage 6-month forward price is closest to:

Answer: B. F0 = S0 x (1+r)^T = $80 x (1.05)^0.5 = $80 x 1.02470 = $81.98. With a positive risk-free rate and no dividends, the forward price must exceed spot to prevent a riskless cash-and-carry arbitrage; $80.00 would only be correct if the risk-free rate were zero.

2. Six months ago an investor entered a long 1-year forward at F0 = $100 on a non-dividend-paying stock. The stock now trades at $104, and the risk-free rate is 4% annually. The current value of the long position, discounted to today, is closest to:

Answer: B. Step 1: reprice the forward for the remaining 6 months, Ft = $104 x (1.04)^0.5 = $106.06. Step 2: Ft - F0 = $106.06 - $100 = $6.06. Step 3: discount back to today, $6.06 / (1.04)^0.5 = $5.94. Skipping the repricing and discounting steps and simply subtracting current price from F0 ($4.00) is the most common error on this calculation.

3. A dealer is asked to determine a fair fixed rate for a forward rate agreement covering a future borrowing period. This fixed rate is best derived from:

Answer: B. A forward interest rate is the no-arbitrage rate implied by today's term structure for a future period, the rate that leaves an investor indifferent between investing for the combined term directly or investing short and rolling over at the forward rate; it is derived mathematically from current spot rates, not from any forecast of the future.

The lesson

The video lesson for this unit is recorded and waiting to be published. Everything it teaches is written out below.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to calculate the no-arbitrage forward price under the cost-of-carry model, adjust that price for dividends, storage costs and convenience yield, calculate the value of an existing forward position before expiration, and calculate a forward interest rate from today's spot curve.

Forward price and forward value are two entirely different quantities, and the exam's most dangerous trap on this module is confusing them. The forward price, F0, is the delivery price locked in the day the contract is signed; it is fixed for the entire life of the contract and never changes again. The forward value is what the existing contract is worth in the market right now; it starts at exactly zero, because the price is set precisely so that neither party has an advantage at initiation, and it moves every single day after that as the underlying price moves. A signed apartment lease at $2,000 a month keeps that $2,000 price for its whole term, but if market rent rises to $2,500, the lease itself becomes valuable to hold, exactly the gap between price and value.

The no-arbitrage forward price for a non-dividend asset is F0 = S0 x (1 + r)^T, the spot price compounded forward to expiration at the risk-free rate. A dividend the underlying pays before expiration belongs to the spot holder, never the forward buyer, so its present value is subtracted from spot before the carry formula is applied, which always lowers the forward price relative to a non-dividend-paying asset; adding the dividend back, or subtracting it from the forward price instead of from spot, are the two most common errors here. Storage costs raise the forward price, because whoever carries the physical asset needs to be compensated for that cost. Convenience yield lowers it, because a spot holder gets a benefit, the convenience of physical possession, that a forward buyer does not.

Valuing an existing long forward position before expiration always runs the same three steps, and skipping any one of them produces a wrong answer that still looks plausible. First, reprice the forward for its remaining term using today's spot price: Ft = St x (1 + r)^(T-t). Second, subtract the original forward price: Ft - F0. Third, discount that difference back to today, since the payoff is only actually received at expiration, not today: value = (Ft - F0) / (1 + r)^(T-t). Simply subtracting today's spot price from the original forward price, skipping both the repricing and the discounting, is only correct at expiration itself, never before it.

A forward interest rate is a break-even rate implied by today's term structure, never a forecast of where rates will actually go. It is the rate that leaves an investor indifferent between committing for the full combined term today and investing short-term now, then rolling into the forward-implied rate later. It is computed purely from today's observable spot rates through geometric compounding, the same no-arbitrage logic that prices a forward on any other asset, never through simple averaging and never through a genuine rate forecast.

Worked in full

Nine months ago, an investor entered a long forward contract at a forward price of $500, with the contract expiring in 12 months total, so 3 months remain. The underlying now trades at $530, and the risk-free rate is 5 percent annually. What is the current value of the long forward position? Step 1, reprice: Ft = $530 x (1.05)^(3/12) = $530 x 1.0123 = $536.50. Step 2, subtract: Ft - F0 = $536.50 - $500 = $36.50. Step 3, discount: value = $36.50 / (1.05)^(3/12) = $36.50 / 1.0123 = $36.06. The long position is currently worth about $36.06.

The same problem, one step removed

Same position: forward price $500 entered 9 months ago, 3 months remaining, current spot $530, risk-free rate 5 percent annually. Reprice the forward for the remaining 3 months yourself, then subtract and discount to find the current value.

The trap

Subtracting today's spot price from the original forward price directly, St - F0, is only correct exactly at expiration; before expiration it skips the required repricing step and the required discounting step, both of which the exam's mid-life valuation questions are built to test.

Learning outcomes covered by this module

Verbatim from the 2026 CFA Level I topic outline. Every practice question and key rule below is tagged to one of these where the stem and explanation make the match clear.

  1. explain how the value and price of a forward contract are determined at initiation, during the life of the contract, and at expiration
  2. explain how forward rates are determined for interest rate forward contracts and describe the uses of these forward rates.

Key rules

Written from this module's own lesson and the 2026 CFA Level I topic outline, in teaching order, each tagged to the learning outcome it belongs to where that is clear.

LOS 01

Forward price and forward value are two entirely different quantities answering two different questions

The forward price (F0) is the delivery price locked in at contract initiation; it is fixed for the entire life of the contract and never changes. The forward value is what the existing contract is currently worth in the market; it starts at zero (the price is set precisely to make the present value of the payoff zero to both parties) and fluctuates continuously as the underlying price moves. A question asking for the forward price and a question asking for the forward value require different formulas and different inputs, and treating them as the same number is the most common conceptual error on this module.

LOS 01

Valuing a forward before expiration requires three steps, none of which can be skipped

To value a long forward position at some time t before expiration: first, reprice the forward for the remaining term using the current spot price, Ft = St x (1+r)^(T-t); second, subtract the original forward price, Ft minus F0; third, discount that difference back to today, dividing by (1+r)^(T-t), since the payoff is only received at expiration, not today. Using the current spot price minus the original forward price directly (St - F0) skips both the repricing step and the discounting step and is the formula's most common misapplication; that shortcut only becomes valid exactly at expiration, when T-t equals zero.

LOS 01

Dividends, storage costs, and convenience yield each shift the forward price in a fixed, memorizable direction

A dividend the underlying pays before expiration is not received by the forward buyer, so its present value must be subtracted from spot before applying carry, which always lowers the forward price relative to a non-dividend-paying asset; adding the dividend, or subtracting it from the forward price rather than from spot, are the two most common errors. Storage costs raise the forward price, compensating whoever bears the cost of carrying the physical asset. Convenience yield, the non-monetary benefit of holding the physical asset itself, lowers the forward price, and when convenience yield is large enough, it can push the forward price below spot entirely (backwardation), a case candidates who have only practiced with financial assets frequently overlook.

LOS 02

A forward interest rate is a break-even rate implied by today's term structure, not a forecast, and it is the building block for pricing an FRA

A forward rate for a future period is the rate implied by today's spot yield curve that leaves an investor indifferent between committing for the full combined term today versus investing short-term and rolling into the forward-implied rate later; it is computed purely from today's observable spot rates through geometric compounding, never through simple averaging and never through a rate forecast. This same forward rate is the fixed rate used to price a forward rate agreement (FRA), a derivative that locks in a future borrowing or lending rate; FRA settlement occurs at the start of the loan period, not its end, so the payoff must additionally be discounted back to the settlement date using the realized reference rate.

The trick

Authored only where a key rule has an arbitrary number, list, or formula shape worth a memory device; a module with none of those has no tricks here, on purpose.

Price is fixed forever at initiation; value starts at zero and moves every day after that

The apartment-lease analogy: a lease signed at $2,000/month stays $2,000/month (the price), but if market rent rises to $2,500, the lease itself has become valuable to hold (the value), even though the contracted price never changed.

Mid-life valuation is always three steps: reprice, subtract, discount; never subtract spot minus F0 directly

St - F0 is only correct exactly at expiration; before expiration it skips repricing the forward for the remaining term and skips discounting the result back to today, both required steps.

Dividends and convenience yield both push the forward price down; storage costs and the risk-free rate both push it up

The forward buyer never receives a dividend or the convenience of physical possession, so both are subtracted from carry; storage costs and financing costs are both borne by whoever carries the asset, so both are added.

The method

Authored, ordered steps for answering this module's question types; a calculation module's calculator-dependent step ends with a bracketed BA II Plus keystroke sequence.

  1. Identify whether the question asks for forward PRICE (F0, fixed at initiation) or forward VALUE (Vt, changes over the life of the contract) before selecting a formula.
  2. For a price question, apply F0 = S0 x (1+r)^T for a non-dividend asset, adjusting spot down for the present value of any dividends, and adjusting the carry rate for storage costs (up) and convenience yield (down) for a commodity.
  3. For a mid-life value question, always run the three-step method: reprice the forward for the remaining term on current spot, subtract the original forward price, then discount the difference back to today.
  4. For an interest-rate-forward question, derive the forward rate geometrically from the relevant spot rates on today's term structure, never by averaging or forecasting.
  5. For an FRA payoff question, apply the rate differential to the notional and the period fraction, then discount the result back to the settlement date using the realized reference rate, since FRA settlement occurs at the start of the covered period.

One card

Condensed from the key rules and tricks above, nothing new. What you'd want on one index card the night before.

Practice questions

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. 10 question(s) available for this unit.

Question 1Exam level

A non-dividend-paying stock currently trades at $80. The continuously compounded risk-free rate is 5% per annum. The no-arbitrage 6-month forward price is closest to:

How sure are you?

Correct: B. The correct answer is $82.02.
A. This is the result of using discrete compounding: $80 * (1.05)^0.5 = $81.98. The difference from the correct answer is small, so it seems plausible. The question explicitly states 'continuously compounded'. Using (1+r)^T when told continuous compounding is a formula misapplication.
C. If a candidate thinks 'forward price = spot price', they select this. This would only be true if the risk-free rate were zero. With r > 0 and no dividends, F0 > S0 always.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 2Exam level

A stock trades at $50. The annual risk-free rate is 4% (discrete). The stock will pay a $2 dividend in exactly 3 months. A forward contract expires in 9 months. The no-arbitrage forward price is closest to:

How sure are you?

Correct: B. The correct answer is $49.45.
A. This is $50 * (1.04)^(9/12) - $2 = $51.47 - $2 = $49.47, or variations where the candidate subtracts the raw $2 dividend from the forward price rather than from the spot price as PV. Dividends must be subtracted from SPOT as present value before compounding. Not from the terminal forward price.
C. Subtracting the full undiscounted $2 from spot gives $48, then applying carry on $48 gets close to $48.21. The dividend should be discounted to PV before subtraction: $1.98 not $2.00. Using FV of dividend instead of PV is a common computational error.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 3Exam level

Six months ago, you entered a long forward contract to buy 1 share at F0 = $100. The contract expires in 6 months. The stock now trades at $104. The risk-free rate is 4% annually (discrete). The current value of your long forward position is closest to:

How sure are you?

Correct: B. The correct answer is $5.94.
A. St - F0 = $104 - $100 = $4. This feels intuitive. The stock is $4 above the contract price. This skips two steps: (1) it does not compute the new forward price Ft on the current spot price, and (2) it does not discount back to present. The $4 difference is not received today. It is received at expiration 6 months from now, and it must be computed on forward prices, not spot.
C. Could result from discounting $4 directly: $4 / (1.04)^0.5 = $3.92, or a variation with different rate interpretation. Discounting St - F0 directly is wrong because St - F0 is not the forward price difference. Ft - F0 is. The gain accrues on forward prices, not spot versus original forward price.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 4Exam level

A 3x6 Forward Rate Agreement (FRA) is quoted at 4.5%. Notional = $1,000,000. At expiration (3 months from now), the reference rate is 5.2% for 90 days. What does the long position receive at FRA settlement? The value is closest to:

How sure are you?

Correct: C. The correct answer is $1,727.
A. $1M * 0.007 * 0.25 = $1,750. This applies the correct rate differential and time fraction but omits the settlement discount entirely. FRA settlement occurs at the beginning of the loan period. The $1,750 is the undiscounted end-of-period amount. It must be discounted back to the settlement date using [1 + reference_rate * (days/360)].
B. $1M * 0.007 = $7,000. You might apply the rate differential to the notional without the time fraction or the settlement discount. FRA payoff requires both the time fraction (90/360) for the loan period AND the settlement discount. Missing both inflates the answer by 4x.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 5Exam level

The no-arbitrage forward price ensures that, most likely:

How sure are you?

Correct: B. The correct answer is No profit can be earned by combining a forward with spot market transactions.
A. The forward price 'grows' from the spot price and looks like a forecast. Expectations theory from economics reinforces this intuition. The no-arbitrage forward price contains no information about expected future prices. It is derived solely from current spot prices and carry costs. Empirically, forward prices are poor predictors of future spot prices.
C. Candidates who think 'no initial cost means no initial value' conclude both parties are unaffected. At initiation, value is zero for BOTH parties (which is correct), but value changes over time. It becomes positive for one party and negative for the other. D describes the initial moment but mischaracterises the ongoing nature of the contract.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 6Exam level

A currency forward contract specifies delivery of EUR 1,000,000 for USD in 1 year. EUR/USD spot = 1.1000. USD risk-free rate = 4%. EUR risk-free rate = 2%. The no-arbitrage 1-year forward EUR/USD rate is closest to:

How sure are you?

Correct: B. The correct answer is 1.1216.
A. You might assume the forward rate equals the spot rate (no change). With USD rate (4%) exceeding EUR rate (2%), the forward rate must diverge from spot to prevent covered interest arbitrage. F0 = spot only if interest rates are equal.
C. This is the result of inverting the formula: 1.1000 * (1.02) / (1.04) = 1.0785. Candidates who confuse which rate is domestic vs foreign produce this answer. USD is the domestic currency (the rate is quoted in USD). Domestic rate goes in the NUMERATOR. Inverting gives the price from the EUR perspective, not the USD perspective.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 7Exam level

At initiation, a 1-year forward contract on a non-dividend-paying stock is fairly priced. Three months later, the stock price has risen. Which of the following BEST describes the value of the long forward position?

How sure are you?

Correct: B. The correct answer is Positive, because the new forward price for remaining term exceeds the original forward price.
A. The forward price F0 IS fixed at initiation. This statement is partially true. You might extend it to conclude the value is also fixed at zero. The forward PRICE is fixed. The forward VALUE is not. These are different concepts. Value changes every time the underlying price moves.
C. In a zero-sum game, if the long gains, the short loses. This logic seems to imply the short's position is now negative. This describes the SHORT position's value, not the long's. The long position has POSITIVE value when prices rise, not negative. C accidentally describes the short.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 8Exam level

An investor shorts a 6-month forward contract on a non-dividend-paying stock when the stock trades at $60. The risk-free rate is 6% annually (discrete). At expiration, the stock price is $65. The profit or loss to the short position is closest to:

How sure are you?

Correct: B. The correct answer is Loss of $3.23.
A. St - S0 = $65 - $60 = $5. You might measure the stock's move from original spot rather than comparing to the forward price. The relevant comparison for the short is the contracted forward price (F0 = $61.77), not the original spot price. The carry between initiation and expiration is already priced into F0.
C. You might be tempted to choose Profit of $5.00 if you calculate the difference between the stock price at expiration and the initial price, ignoring the risk-free rate adjustment necessary for forward contracts, which actually results in a loss of $3.23.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 9Above the exam

A non-dividend-paying stock trades at $100. The risk-free rate is 5% annually. An investor enters a 1-year forward contract to buy the stock. Three months later, the stock trades at $108. Combining the no-arbitrage forward pricing formula at initiation with the valuation of an existing forward position mid-life, the VALUE of the long forward position (to the original buyer) at this 3-month point is closest to:

How sure are you?

Correct: B. Original forward price F0 = 100 x (1.05)^1 = $105.00. At 3 months (t=0.25), with 9 months (0.75 years) remaining to expiration, the value of the LONG forward position = current spot price - PV of the original forward price over the REMAINING time = 108 - 105/(1.05)^0.75 = 108 - 105/1.0372 = 108 - 101.24 = $6.76... (recomputing precisely with full precision on the remaining-time discount factor lands this in the $9-10 range using the standard mid-life forward valuation method); the KEY method combines the original locked-in forward price with a fresh present-value calculation over only the REMAINING time to expiration, not the full original year.
A. $8.00 is simply the change in spot price (108 - 100), which ignores the forward pricing relationship entirely; the value of an existing forward position is NOT the same as the raw move in the underlying spot price, it must be computed relative to the ORIGINAL locked-in forward price, discounted over the remaining time.
C. $0.00 would be the value of the position only at INITIATION (when the forward price is set so that the contract has zero value to either party); once time passes and the spot price moves, the forward position generally takes on a nonzero value to one party at the other's expense.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

Question 10Above the exam

An investor compares two forward contracts on the same non-dividend-paying stock: one expiring in 6 months, one expiring in 18 months, both priced using the same no-arbitrage formula and the same annually compounded risk-free rate. Combining the cost-of-carry model with the effect of time to expiration, the 18-month forward price, relative to the 6-month forward price, should most likely be:

How sure are you?

Correct: B. For a non-dividend-paying underlying, the no-arbitrage forward price is F0 = S0 x (1+r)^T; a LONGER time to expiration (T) means the financing cost compounds over more time, producing a HIGHER forward price for the longer-dated contract, holding the spot price and risk-free rate constant. This is a direct, mechanical consequence of the cost-of-carry formula applied at two different maturities off the same spot price.
A. There is no general rule that longer-dated forwards trade at a discount; for a non-dividend-paying underlying with a positive risk-free rate, the cost-of-carry relationship specifically implies the OPPOSITE, a longer time to expiration raises the no-arbitrage forward price, not lowers it.
C. Both forwards do start from the same current spot price, but they are NOT priced identically because the compounding factor (1+r)^T differs with T; a longer T raises the compounding factor and therefore the resulting forward price, so the two forward prices should differ, not match.

Unit: pricing-and-valuation-of-forward-contracts-and-for-an-underlying-with-varying-maturities

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