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CFA Level II Exam · The Arbitrage-Free Valuation Framework

Spot Rates, Forward Rates and the Forward Rate Model

Updated 7 October 2026 · Fact-checked

A spot rate is the yield on a zero-coupon bond paid today for a given maturity. A forward rate is the rate for a loan starting at a future date, implied by the spot curve. Solve by equating two investment routes: (1+z_(A+B))^(A+B) = (1+z_A)^A × (1+f)^B.

Understand Spot Rates, Forward Rates and the Forward Rate Model

A spot rate is the yield to maturity on a zero-coupon bond. The 3-year spot rate is the annual return you lock in by paying today and receiving one payment in 3 years. The set of spot rates across maturities is the spot curve. You use it to discount each cash flow of a bond at the rate that matches its own date.

A forward rate is an interest rate for a loan that starts at a future date. Written f(A,B), it is the rate for a period of B years that starts A years from now. It is not a forecast. It is the rate that makes you indifferent between two routes: invest for the full term at the long spot rate, or invest for a shorter term and then roll over at the forward rate. If the two routes gave different results, an arbitrage would exist.

That is the whole logic. Money grown over A+B years at the long spot rate must equal money grown over A years at the short spot rate, then over B years at the forward rate. Rearrange this and you get any forward rate. Rearrange it differently and you get the implied spot rate from a set of forwards.

The forward rate model uses forward rates for valuation and trading. A bond's value at a future date can be found by discounting with forward rates, and it equals the forward price when the future spot rates equal today's implied forwards. If you expect the future spot rate to differ from the forward rate, you expect the bond's future price to differ from its forward price. That is the basis for deciding whether to buy or sell.

In the exam, the numbers sit in a vignette, usually as a table of spot rates or forward rates. Your job is to pick the right rates, apply the equation and read the result correctly.

Key formulas to remember

Bond value using spot rates
PV = CF₁ ÷ (1+z₁) + CF₂ ÷ (1+z₂)² + … + CFₙ ÷ (1+zₙ)ⁿ
Each cash flow is discounted at the spot rate for its own maturity. Rates are annual effective, with annual cash flows.
Forward rate from spot rates
(1+z_(A+B))^(A+B) = (1+z_A)^A × (1+f(A,B))^B
f(A,B) is the B-year rate starting A years from now. This is the no-arbitrage link between spot and forward rates.
Solved for the forward rate
f(A,B) = [ (1+z_(A+B))^(A+B) ÷ (1+z_A)^A ]^(1/B) − 1
Compute the ratio first, then take the 1/B power. For B = 1, there is no root to take.
Spot rate from one-year forward rates
(1+z_T)^T = (1+f(0,1)) × (1+f(1,1)) × … × (1+f(T−1,1))
f(0,1) equals the 1-year spot rate z₁. The spot rate is the geometric average of the one-period forwards.
Forward price of a bond
F(j,k) = P_(j+k) ÷ P_j, where P_t = 1 ÷ (1+z_t)^t
Price per 1 of face value for a zero-coupon bond. F(j,k) is the forward price at time j of a zero maturing at j+k. Multiply by face value for a price.
Forward rate model decision rule
Expected future spot rate < forward rate → expected price > forward price
Then the bond is expected to be worth more than the forward contract price, so buying is favoured. The reverse holds if the expected spot rate is above the forward rate.

How to solve Spot Rates, Forward Rates and the Forward Rate Model questions

Use this method for any question that gives you spot or forward rates and asks for a forward rate, a bond value or a trading view.

  1. 1Read the vignette and write down the rates given. Note whether each is a spot rate, a forward rate or a yield to maturity, and the compounding basis.
  2. 2Identify what is asked: a forward rate f(A,B), a spot rate, a bond price today, or a forward price at a future date.
  3. 3For a forward rate, set A as the start date and B as the length. The long spot maturity is A+B and the short one is A.
  4. 4Compute (1+z_(A+B))^(A+B) and (1+z_A)^A separately, then divide the first by the second.
  5. 5Raise the ratio to the power 1/B and subtract 1. Convert to a percentage only at the end.
  6. 6For a bond value, discount every cash flow at the spot rate for its maturity, or discount step by step with one-year forwards. Both give the same answer.
  7. 7For a trading question, compare the rate you expect with the forward rate. Lower expected rate means higher expected price, so the bond looks cheap versus the forward price.
  8. 8Check that the answer is sensible. A forward rate is above the later spot rate when the curve is upward sloping, and below it when the curve is downward sloping.

Quickest way: Growth-factor shortcut

When to use it: Use when the vignette gives a table of spot rates and you need one forward rate quickly.

  1. Work with growth factors, not rates. For example, 3.0% over 3 years is 1.03³.
  2. Divide the long growth factor by the short growth factor. The result is the growth factor for the forward period.
  3. If B = 1, the result minus 1 is your answer. No root is needed.
  4. If B > 1, take the B-th root (use the y^x key with 1/B) and subtract 1.
  5. Sanity check: on an upward sloping curve the forward rate must exceed the long spot rate. If it does not, you have mixed up A and B.
  6. Only the long and short spot rates matter. Ignore other rows in the table.

Common mistakes in Spot Rates, Forward Rates and the Forward Rate Model

  • Using the wrong periods, for example using the 4-year and 2-year spot rates but taking a 1/4 root.

    Students confuse the total term A+B with the forward period B.

    Fix: The exponent on the long spot is A+B, the exponent on the short spot is A, and the root is 1/B only.

  • Finding a forward rate by subtracting spot rates, or averaging them.

    It feels natural to treat rates as additive, but compounding makes them multiplicative.

    Fix: Always divide growth factors. The forward rate is a geometric result, not an arithmetic one.

  • Treating the forward rate as a prediction of the future spot rate.

    The word 'implied' sounds like a forecast.

    Fix: The forward rate is a no-arbitrage rate. The actual future spot rate can differ, and the forward rate model is about what happens if it does.

  • Discounting all bond cash flows at the bond's yield to maturity when the question gives spot rates.

    Students are used to the YTM method from Level I.

    Fix: When the vignette gives a spot curve, discount each cash flow at its own spot rate.

  • Getting the trade direction backwards in the forward rate model.

    Students remember 'rates down, prices up' but do not link it to the forward rate comparison.

    Fix: Compare your expected future spot rate to the forward rate. If yours is lower, you expect a higher price than the forward price, so buy.

  • Mixing up f(A,B) notation, such as reading f(2,1) as a 2-year rate starting in 1 year.

    The order of the two numbers is easy to flip under time pressure.

    Fix: Say it as 'start in A, last for B'. Write the timeline on scratch paper if needed.

Worked examples

Example 1

Vignette: An analyst has these annual-pay spot rates: 1-year 2.00%, 2-year 2.50%, 3-year 3.00%, 4-year 3.40%. Q1. What is the one-year forward rate two years from now, f(2,1)? Options: A. 3.50%, B. 4.01%, C. 4.50%. Q2. What is the price of a 3-year 5% annual-coupon bond with face value 100? Options: A. 105.75, B. 102.80, C. 100.00.

Show the solution
  1. Q1: f(2,1) starts in 2 years and lasts 1 year, so A = 2, B = 1, and the long spot is the 3-year rate.
  2. Long growth factor: 1.03³ = 1.092727.
  3. Short growth factor: 1.025² = 1.050625.
  4. Ratio = 1.092727 ÷ 1.050625 = 1.04007. Since B = 1, no root is needed.
  5. f(2,1) = 1.04007 − 1 = 4.01% (rounded).
  6. Q2: discount each cash flow at its own spot rate.
  7. Year 1: 5 ÷ 1.02 = 4.9020.
  8. Year 2: 5 ÷ 1.050625 = 4.7591.
  9. Year 3: 105 ÷ 1.092727 = 96.0899.
  10. Sum = 4.9020 + 4.7591 + 96.0899 = 105.75 (rounded). The price is above par because the coupon of 5% exceeds every spot rate.

Answer: Q1: B, 4.01%. Q2: A, 105.75.

Example 2

Vignette: Spot rates are 1-year 3.0%, 2-year 3.5%, 3-year 4.0% (annual compounding). A portfolio manager looks at a zero-coupon bond with face value 100 that will have 2 years to maturity one year from now. Q1. What is the implied forward rate for a 2-year loan starting in one year, f(1,2)? Options: A. 4.00%, B. 4.50%, C. 5.01%. Q2. The manager expects the 2-year spot rate in one year to be 4.00%. Given the forward price of this bond is 91.57, what does the forward rate model suggest? Options: A. Sell the bond forward, because expected price is below the forward price. B. Buy the bond, because the expected price is above the forward price. C. Take no position, because the forward price always equals the future price.

Show the solution
  1. Q1: A = 1, B = 2, long spot is the 3-year rate.
  2. Long growth factor: 1.04³ = 1.124864.
  3. Short growth factor: 1.03¹ = 1.03.
  4. Ratio = 1.124864 ÷ 1.03 = 1.092101.
  5. Take the square root: √1.092101 = 1.04504. So f(1,2) = 4.50%.
  6. Q2: the forward price is 100 ÷ 1.092101 = 91.57 (this matches the figure given).
  7. Expected price in one year at a 4.00% spot rate = 100 ÷ 1.04² = 100 ÷ 1.0816 = 92.46.
  8. Expected spot rate 4.00% is below the forward rate 4.50%, so the expected price 92.46 is above the forward price 91.57.
  9. The manager expects a gain over the forward price, so buying (or going long the forward) is favoured.

Answer: Q1: B, 4.50%. Q2: B, buy the bond, because the expected price of 92.46 exceeds the forward price of 91.57.

Exam tips

  • Write the timeline first. Mark A, A+B and the period B. This prevents the most common error, which is mixing up the exponents.
  • Keep at least five decimals in growth factors. Rounding early can push your answer onto a wrong option when the options are close.
  • If the vignette gives forward rates and asks for a spot rate, multiply the growth factors and take the root. Do not average the forward rates.
  • For trading questions, decide the direction before computing. Expected rate below the forward rate means expected price above forward price.
  • Check which compounding the vignette uses. Stay with annual compounding unless it says otherwise, and do not switch bases halfway through.

Spot Rates, Forward Rates and the Forward Rate Model in other exams

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

Spot Rates, Forward Rates and the Forward Rate Model: frequently asked questions

How do I calculate a forward rate from spot rates in CFA Level II?

Divide the growth factor of the longer spot rate by the growth factor of the shorter spot rate. Then take the root equal to the forward period and subtract 1. For f(A,B), the formula is [(1+z_(A+B))^(A+B) ÷ (1+z_A)^A]^(1/B) − 1.

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

A spot rate applies to a loan or zero-coupon bond starting today. A forward rate applies to a loan starting at a future date. The forward rate is implied by the spot curve through no-arbitrage.

What is the forward rate model?

It values a bond at a future date using forward rates, and compares the expected future spot rate with the forward rate. If the future spot rate turns out equal to the forward rate, the bond's price equals its forward price. If your expected rate is lower than the forward rate, you expect a price above the forward price.

Is a forward rate a forecast of the future spot rate?

No. It is the rate that rules out arbitrage given today's spot curve. The future spot rate can differ, and the forward rate model asks what follows if it does.

Can I price a bond with forward rates instead of spot rates?

Yes. Discount the cash flows step by step using each one-year forward rate in sequence. This gives the same price as discounting with the spot curve, because spot rates are the geometric average of the one-year forwards.