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CFA Level I Exam · The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Forward Rates and the Forward Curve Explained

Updated 7 October 2026 · Fact-checked

A forward rate is the interest rate agreed today for borrowing or lending over a future period. You derive it from spot rates by no-arbitrage: (1 + z_B)^B = (1 + z_A)^A × (1 + f)^(B − A). Solve for f. The forward curve plots these rates across future start dates.

Understand Forward Rates and the Forward Curve

A spot rate is the yield today on a zero-coupon bond for a given maturity. A forward rate is the rate for a loan that starts at a future date and runs for a set length. For example, the 2y1y forward rate is the one-year rate that starts in two years.

Forward rates are not forecasts you must believe. They are the rates that make you indifferent between two strategies. Strategy one: invest for B years at the B-year spot rate. Strategy two: invest for A years at the A-year spot rate, then reinvest for the remaining B − A years at the forward rate. If the two ended with different amounts, an arbitrage would exist. So the two must give the same growth factor.

This is why the forward rate comes straight from the spot curve. If the spot curve slopes upward, the forward rates sit above the spot rates. If it slopes downward, forward rates sit below spot rates. The forward curve is the set of forward rates for a fixed loan length (for example, all one-year rates) across different start dates.

Forward rates also value bonds. Discounting each cash flow at its spot rate gives the same price as discounting year by year at successive one-year forward rates. The discount factor for year n is the product of 1 ÷ (1 + forward rate) for each year up to n.

A forward contract on a zero-coupon bond uses the same idea. Its forward price per 1 of par is the discount factor for the later date divided by the discount factor for the earlier date. In the exam, you must keep notation straight: f(A, B−A) means start at time A, loan length B − A.

Key formulas to remember

Spot-forward relationship
(1 + z_B)^B = (1 + z_A)^A × (1 + f(A, B−A))^(B−A)
z is the annual spot rate. A is the start of the forward period and B−A is its length in years. Works for annual compounding.
Implied forward rate
f(A, B−A) = [ (1 + z_B)^B ÷ (1 + z_A)^A ]^(1 ÷ (B−A)) − 1
Always take the root of the ratio over the loan length, B − A, not over B.
One-year forward from adjacent spots
f(n−1, 1) = (1 + z_n)^n ÷ (1 + z_(n−1))^(n−1) − 1
Use this to build the one-year forward curve step by step.
Spot rate from forward rates
(1 + z_n)^n = (1 + f(0,1)) × (1 + f(1,1)) × … × (1 + f(n−1,1))
The spot rate is the geometric average of the one-year forward rates. f(0,1) equals z_1.
Bond value using forward rates
PV = CF_1 ÷ (1 + f(0,1)) + CF_2 ÷ [(1 + f(0,1))(1 + f(1,1))] + …
Gives the same value as discounting each cash flow at its spot rate.
Forward price of a zero-coupon bond
F = DF_B ÷ DF_A, where DF_t = 1 ÷ (1 + z_t)^t
Price per 1 of par, for a contract made today to buy at time A a bond that matures at B (A < B).

How to solve Forward Rates and the Forward Curve questions

Use this method for any question on forward rates, forward curves or bond values from the curve.

  1. 1Read the notation. Write down A (start of the forward period), B (end) and B − A (loan length).
  2. 2List the spot rates z_A and z_B. If you are given a par or forward curve instead, convert first so you work with spot rates or discount factors.
  3. 3Compute the growth factors (1 + z_B)^B and (1 + z_A)^A.
  4. 4Divide the longer growth factor by the shorter one. This gives (1 + f)^(B − A).
  5. 5Take the root of order B − A, then subtract 1. For a one-year forward, just subtract 1.
  6. 6For bond valuation, discount each cash flow at its spot rate, or chain the forward rates. Both give the same price.
  7. 7Check sense: an upward-sloping spot curve should give a forward rate above z_B. A downward-sloping one should give a forward rate below.
  8. 8Match the answer to the three options. Remove options that reflect a standard slip, such as dividing by the wrong root.

Quickest way: Growth-factor shortcut for one-year forwards

When to use it: Use when the question asks for a one-year forward rate starting at a whole-year date, which is the most common case.

  1. Compute (1 + z_B)^B on the calculator and store it.
  2. Compute (1 + z_A)^A.
  3. Divide the first by the second and subtract 1.
  4. On the BA II Plus, key 1.03 [y^x] 2 [=] [÷] 1.02 [=] [−] 1 [=] to get 0.0401 for a 1y1y forward from z1 = 2% and z2 = 3%.
  5. For multi-year forwards, after the division press [y^x], then 1 [÷] n [=] (or use [1/x] on the result of n) and subtract 1.
  6. Sense-check: if the spot curve rises, the forward must be above the longer spot rate.

Common mistakes in Forward Rates and the Forward Curve

  • Averaging spot rates instead of using growth factors

    The forward rate looks like a simple difference between two rates, for example 3% and 2%.

    Fix: Always compound. Use (1 + z_B)^B ÷ (1 + z_A)^A. A simple difference understates the forward rate in an upward-sloping curve.

  • Taking the root over B instead of B − A

    Students remember a root and use the final maturity.

    Fix: The root order equals the length of the forward loan. For a 2y3y forward, take the 3rd root.

  • Confusing the notation 2y1y with 1y2y

    The first number is the start date and the second is the loan length, but this is easy to flip under time pressure.

    Fix: Write A and B − A explicitly. 2y1y starts in 2 years and runs 1 year, so B = 3.

  • Treating forward rates as certain predictions

    Students think the forward rate is the expected future spot rate.

    Fix: If the future spot rate ends up higher than the forward rate, a borrower who locked in the forward rate gains and a lender who locked it in loses. The forward rate is a break-even rate, not a forecast.

  • Using the par (coupon) rate as a spot rate

    Par rates and spot rates are both quoted as yields, and the question may not label them clearly.

    Fix: Check whether the given curve is spot, par or forward. Convert par rates into spot rates by bootstrapping before computing forwards.

  • Discounting a bond's cash flows with a single yield after being given a curve

    Yield to maturity is familiar, so students apply it by habit.

    Fix: When spot or forward rates are given, discount each cash flow at its own spot rate (or chain the forward rates). The result will differ slightly from a flat-yield price.

Worked examples

Example 1

The one-year spot rate is 2.0% and the two-year spot rate is 3.0% (annual compounding). What is the one-year forward rate starting one year from now, f(1,1)? A: 2.51% B: 3.00% C: 4.01%

Show the solution
  1. Here A = 1, B = 2, so the loan length is B − A = 1 year.
  2. Growth factor to year 2: (1.03)^2 = 1.0609.
  3. Growth factor to year 1: 1.02.
  4. (1 + f) = 1.0609 ÷ 1.02 = 1.040098.
  5. f = 1.040098 − 1 = 0.040098, or about 4.01%.
  6. Sense-check: the spot curve slopes upward, so the forward rate must exceed the longer spot rate of 3.0%. That removes A (2.51%) and B. B (3.00%) is just the longer spot rate itself, and on an upward-sloping curve the forward rate must be above it, not equal to it.

Answer: C: 4.01%

Example 2

Spot rates are z1 = 3%, z2 = 4% and z3 = 5% (annual compounding). What is the value per 100 of par of a 3-year, 6% annual-pay bond? A: 100.00 B: 102.72 C: 102.94

Show the solution
  1. Discount factors: DF1 = 1 ÷ 1.03 = 0.970874; DF2 = 1 ÷ 1.04^2 = 1 ÷ 1.0816 = 0.924556; DF3 = 1 ÷ 1.05^3 = 1 ÷ 1.157625 = 0.863838.
  2. PV of year 1 coupon: 6 × 0.970874 = 5.8252.
  3. PV of year 2 coupon: 6 × 0.924556 = 5.5473.
  4. PV of year 3 cash flow (106): 106 × 0.863838 = 91.5668.
  5. Sum: 5.8252 + 5.5473 + 91.5668 = 102.9393, about 102.94.
  6. Check with forward rates: f(1,1) = 1.0816 ÷ 1.03 − 1 = 5.01%, and f(2,1) = 1.157625 ÷ 1.0816 − 1 = 7.03%. Chaining these gives the same discount factors, so the same price.
  7. Option B (102.72) is what you get by discounting every cash flow at the 3-year spot rate of 5%: 6 ÷ 1.05 + 6 ÷ 1.1025 + 106 ÷ 1.157625 = 5.7143 + 5.4422 + 91.5668 = 102.7233, about 102.72. That is a flat-yield shortcut and ignores the lower 3% and 4% spot rates for the earlier cash flows.
  8. Option A (100.00) is the par value. A 6% bond is worth par only if all its cash flows are discounted at a flat 6%. Every spot rate here is below 6%, so the price must be above par.

Answer: C: 102.94

Exam tips

  • Most questions give two spot rates and ask for a forward rate. Write A and B − A before touching the calculator.
  • Use direction to eliminate options. With an upward-sloping spot curve, the forward rate is above the longer spot rate. With a downward-sloping curve, it is below.
  • Expect the forward-rate bond valuation to match the spot-rate valuation. If a question asks you to choose between the two, the prices are equal.
  • Options are listed from smallest to largest. A naive answer, such as the simple difference of two spot rates, is often placed as an option, so do not stop at the first value that looks right.
  • At about 90 seconds per question, practise the keystroke chain for the growth-factor ratio until it takes under 30 seconds.

Practice questions from The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Forward Rates and the Forward Curve in other exams

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

Forward Rates and the Forward Curve: frequently asked questions

How do I calculate a forward rate from spot rates?

Divide the growth factor of the longer spot rate by that of the shorter one: (1 + z_B)^B ÷ (1 + z_A)^A. Then take the root of order B − A and subtract 1. For a one-year forward, you only subtract 1.

What is the difference between a spot rate and a forward rate?

A spot rate applies to a loan that starts today. A forward rate applies to a loan that starts at a future date. Spot rates are observed from zero-coupon bond prices, and forward rates are derived from them by no-arbitrage.

Does the forward rate predict the future spot rate?

No. The forward rate is the break-even rate that removes arbitrage between investing long and rolling over shorter investments. Actual future spot rates can end up higher or lower.

Can I value a bond using forward rates?

Yes. Discount each cash flow by the product of (1 + forward rate) for every year up to that cash flow. This gives exactly the same value as discounting each cash flow at its spot rate.