CFA Level I Exam · Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Forward Pricing Across Varying Maturities Using the Term Structure

Updated 7 October 2026 · Fact-checked

Forward pricing across maturities uses spot rates from the term structure. For a forward expiring at T, compound the spot price at the T-year spot rate. For a bond, first subtract the present value of coupons paid before expiry, then compound. Implied forward rates link spot rates of different maturities and give the same answer.

Understand Forward Pricing Across Varying Maturities

A forward price is set so that the contract has zero value at the start. No-arbitrage pricing gives the answer: buy the underlying today with borrowed money, hold it, and deliver it at expiry. The forward price must equal what that strategy costs at expiry. The cost depends on the interest rate for the period from today to expiry.

The interest rate is not the same for every horizon. The yield curve gives a spot rate for each maturity. A forward that expires in 1 year uses the 1-year spot rate. A forward that expires in 3 years uses the 3-year spot rate. Using one flat rate for all maturities is the main error to avoid.

An implied forward rate is the rate for a future period that is consistent with today's spot rates. If you invest for 3 years at the 3-year spot rate, you must end up with the same money as investing for 1 year and then rolling over at the 1-year-forward-2-year rate. This gives: (1 + z3)^3 = (1 + z1) × (1 + f(1,2))^2. The forward rate is a break-even rate, not a forecast of the future spot rate.

For a fixed income forward, the underlying is a bond that usually matures after the forward expires. At expiry, the buyer receives a bond with a shorter remaining life. Any coupon paid before expiry goes to the current holder, so you subtract its present value from the bond price today. Then you compound the result forward to expiry. Equivalently, the forward price is the value at expiry of the bond's remaining cash flows, discounted at the implied forward rates.

Once the contract is running, its value changes. The long position's value is the present value of the difference between the new forward price and the original forward price, discounted over the time left at the spot rate for that remaining time.

Key formulas to remember

Forward price, asset with no cash flows: F0(T) = S0 × (1 + r_T)^T; Use the spot rate r_T for the forward's expiry T, with annual compounding unless told otherwise.
Implied forward rate: (1 + z_(A+B))^(A+B) = (1 + z_A)^A × (1 + f(A,B))^B; f(A,B) is the rate for a period of B years starting A years from now. Solve: f(A,B) = [(1 + z_(A+B))^(A+B) ÷ (1 + z_A)^A]^(1/B) − 1.
Forward price of a zero-coupon bond: F0 = P0(A+B) ÷ P0(A) = P0(A+B) × (1 + z_A)^A; P0(n) is today's price of a zero maturing in n years. The contract expires at A and the bond matures at A+B.
Forward price of a coupon bond: F0(T) = (B0 + AI0 − PVCI) × (1 + r_T)^T − AI_T; B0 is today's flat price, AI is accrued interest, PVCI is the present value of coupons paid before T. Discount each coupon at the spot rate for its own date. If the quoted price is flat, remove AI_T from the result.
Value of a long forward during its life: V_t(long) = [F_t(T) − F_0(T)] ÷ (1 + r_(T−t))^(T−t); F_t(T) is the new forward price for the same delivery date. The short position has the opposite value.

How to solve Forward Pricing Across Varying Maturities questions

Use this order for any question that gives a term structure and asks for a forward price, an implied forward rate, or a contract value.

1Write down the timeline: today, forward expiry T, and the maturity of the underlying. Mark every coupon date.
2Pick the correct spot rate for each cash flow. Use the T-year spot rate to compound to expiry, and each coupon's own spot rate to discount it.
3If the underlying is a bond, find today's price (given, or the sum of cash flows discounted at spot rates). Add accrued interest if the price is flat.
4Subtract the present value of any coupons paid before delivery. Treat a coupon paid on the expiry date as stated in the question (usually it is paid to the holder, so it is subtracted).
5Compound the remaining value to expiry: multiply by (1 + r_T)^T.
6Subtract accrued interest at expiry if the forward is quoted as a flat price.
7For an implied forward rate, divide the longer growth factor by the shorter one, then take the root of the length of the forward period.
8For valuation after initiation, compute the new forward price for the remaining term, subtract the original, and discount at the spot rate for the time left. Check the sign for long and short.

Quickest way: Growth-factor shortcut for forwards on bonds

When to use it: Use it when spot rates are given for several maturities and the question asks for a forward price or forward rate. It lets you work straight from growth factors.

Forward price of a zero = today's zero price × (1 + spot rate to expiry)^expiry.
Forward rate for years A to A+B: divide the two growth factors, then take the B-th root.
For a coupon bond, discount each cash flow after expiry back to today at its own spot rate, then compound the sum to expiry at r_T. This gives the same result as subtracting PVCI from today's price first, because both approaches discount every cash flow at its own spot rate.
BA II Plus: 1.04 [y^x] 3 [=] gives 1.124864. Use [1/x] for the discount factor. Store values with [STO] to reuse them.
HP 12C: 1.04 [ENTER] 3 [y^x] gives 1.124864. Press [1/x] for the discount factor.
Eliminate answers: the forward price on a bond with no coupons before expiry must exceed today's price, because you compound it at a positive rate.

Common mistakes in Forward Pricing Across Varying Maturities

Using the bond's yield to maturity to compound to expiry.
The yield is the most familiar rate, and the question may show it prominently.
Fix: Use the spot rate whose maturity equals the forward expiry. Use YTM only if the question tells you to treat it as the single rate.
Subtracting the coupon at face value instead of its present value.
Students remember that coupons before expiry reduce the forward price and forget to discount them.
Fix: Discount each coupon at the spot rate for its payment date, then subtract the sum from the price today before compounding.
Forgetting the exponent when finding a forward rate over several years.
Students divide the growth factors and stop, so the answer is a multi-year factor, not an annual rate.
Fix: After dividing, take the root equal to the length of the forward period, then subtract 1.
Treating the implied forward rate as the expected future spot rate.
The forward rate looks like a prediction.
Fix: It is the break-even rate that makes two investment paths equal today. Actual future spot rates can differ.
Ignoring accrued interest when the bond is quoted flat.
Questions often give a clean price, and students treat it as the full price.
Fix: Add accrued interest to get the full price before compounding, and remove accrued interest at expiry if the answer must be a flat price.
Valuing an existing forward by discounting the original forward price alone.
Students mix up pricing at the start with valuation later.
Fix: Value equals the PV of the difference between the new and original forward prices. Check the sign: the long gains when the forward price rises.

Worked examples

Example 1

The annual-compounding spot rates are 2.0% for 1 year, 3.0% for 2 years and 4.0% for 3 years. What is the price of a 1-year forward contract on a 3-year zero-coupon bond with par 100? A) 88.90 B) 90.68 C) 95.22

Show the solution

Today's price of the 3-year zero: 100 ÷ 1.04^3 = 100 ÷ 1.124864 = 88.8996.
The forward expires in 1 year, so compound at the 1-year spot rate of 2.0%.
F = 88.8996 × 1.02 = 90.6776.
Check with the forward rate: 1.124864 ÷ 1.02 = 1.102808, so the 2-year growth factor from year 1 to year 3 is 1.102808. F = 100 ÷ 1.102808 = 90.6776.
The implied annual forward rate is f(1,2) = √1.102808 − 1 = 1.050147 − 1, about 5.01%.
Option A is today's spot price, which ignores compounding. Option C divides 100 by 1 plus the annual forward rate only once: 100 ÷ 1.050147 = 95.22 (95.2248 before rounding). That discounts for one year instead of the two-year forward period, so it treats the forward period as one year long.

Answer: B) 90.68

Example 2

Use the same spot curve: 2.0% for 1 year, 3.0% for 2 years and 4.0% for 3 years. A 3-year bond pays an annual coupon of 5% on par of 100 (first coupon in 1 year). What is the price of a forward contract expiring in 1 year, right after that coupon is paid, on this bond? Ignore accrued interest at delivery. A) 98.06 B) 100.02 C) 102.96

Show the solution

Find today's bond price: 5 ÷ 1.02 + 5 ÷ 1.03^2 + 105 ÷ 1.04^3.
5 ÷ 1.02 = 4.9020. 5 ÷ 1.0609 = 4.7130. 105 ÷ 1.124864 = 93.3446.
B0 = 4.9020 + 4.7130 + 93.3446 = 102.9596 (about 102.96).
The question says the contract expires right after the year-1 coupon is paid, so that coupon is paid before delivery and goes to the current holder. PVCI = 4.9020.
Subtract: 102.9596 − 4.9020 = 98.0576.
Compound for 1 year at the 1-year spot rate: 98.0576 × 1.02 = 100.0188.
Independent check with forward rates: f(1,1) = 1.0609 ÷ 1.02 − 1 = 4.0098%. The year-1 value of the remaining cash flows is 5 ÷ 1.040098 + 105 ÷ 1.102808, where 1.102808 is the growth factor from year 1 to year 3. This is 4.8072 + 95.2115 = 100.0187, which matches 100.0188 apart from rounding.
Option A forgets to compound. Option C is today's bond price.

Answer: B) 100.02

Exam tips

The questions are three-option MCQs, so work out the direction first. A forward on a bond with no coupons before expiry costs more than the spot price when rates are positive.
Check which rate the question gives. Spot rates compound directly. Forward rates must be chained: multiply growth factors of consecutive periods.
Watch for flat versus full price. If accrued interest is mentioned anywhere, expect it to affect the answer.
Distractors often include today's price, an uncompounded price, or a price using the wrong maturity's rate. Compute once, then match.
For valuation after initiation, decide the sign by asking if the forward price has risen or fallen since the start. A higher forward price benefits the long.

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

Forward Pricing Across Varying Maturities: frequently asked questions

How do you find an implied forward rate from spot rates?

Divide the longer spot growth factor by the shorter one, then take the root equal to the length of the forward period and subtract 1. For example, f(1,2) from 1-year and 3-year spot rates is [(1 + z3)^3 ÷ (1 + z1)]^(1/2) − 1.

Why is the 1-year forward on a 3-year zero priced at P(3) × (1 + z1)?

Buying the 3-year zero today and financing it for 1 year costs the zero's price compounded at the 1-year rate. The forward price must equal that cost, otherwise an arbitrage exists.

Do I subtract coupons when pricing a forward on a coupon bond?

Subtract the present value of coupons paid before the forward expires, because the forward buyer does not receive them. Coupons paid after expiry stay in the bond and are not subtracted.

Is the implied forward rate a forecast of future interest rates?

No. It is the rate that makes rolling over short investments equal to a longer investment given today's curve. It is a break-even rate and the future spot rate can turn out higher or lower.