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CFA Level II Exam · Pricing and Valuation of Forward Commitments

Forward Rate Agreements: Pricing, Payoff and Valuation

Updated 7 October 2026 · Fact-checked

A forward rate agreement (FRA) fixes an interest rate today for a future loan period. Its price is the implied forward rate from the spot Libor curve. At expiry, the long receives the floating-minus-fixed difference, scaled by days and discounted. Before expiry, value the FRA using the new forward rate.

Understand Forward Rate Agreements (FRAs)

A forward rate agreement is an over-the-counter contract on an interest rate. One side agrees to pay a fixed rate, the FRA rate, on a notional amount for a future period. The other side pays a floating rate, usually a Libor-type rate, set at the start of that period. The notional is never exchanged. Only the interest difference is settled.

The notation g x h (for example 3x6) uses months. The FRA expires in g months, and the underlying rate is the one that starts at month g and ends at month h. So a 3x6 FRA expires in 3 months and refers to a 3-month rate (the period from month 3 to month 6). A 2x8 FRA refers to a 6-month rate that begins in 2 months. Read the first number as the expiry and the gap between the numbers as the rate's term.

The long is the party who pays fixed and receives floating. It is a borrower's position and gains when the floating rate at expiry is above the FRA rate. The short pays floating and receives fixed, and gains when rates fall. Think of the long as having locked in a borrowing rate.

The FRA rate is set so the contract has zero value at initiation. That rate is the implied forward rate: the rate that makes investing for the long period equal to investing for the short period and then rolling over at the forward rate. This is plain no-arbitrage logic. If the forward rate were anything else, you could lock in a risk-free profit.

Settlement happens at expiry (month g), but the interest period ends at month h. So the interest difference is discounted from h back to g using the floating rate at expiry. This is called settlement in advance. Before expiry, the FRA gains or loses value as the forward curve moves. You value it by comparing the new forward rate with the original FRA rate and discounting back from the end of the underlying period.

Key formulas to remember

Implied forward rate (FRA rate)
(1 + L(h) × days_h ÷ 360) = (1 + L(g) × days_g ÷ 360) × (1 + F × (days_h − days_g) ÷ 360), so F = [ (1 + L(h) × days_h ÷ 360) ÷ (1 + L(g) × days_g ÷ 360) − 1 ] × 360 ÷ (days_h − days_g)
L(g) and L(h) are spot Libor rates for g and h days. Use the day-count in the vignette (usually 360). Solve for F. Do not average the rates.
FRA settlement at expiry (long)
Payoff = Notional × (Floating − FRA rate) × (days ÷ 360) ÷ (1 + Floating × days ÷ 360)
Floating is the underlying rate observed at expiry; days is the length of the underlying period. A negative payoff means the long pays. The short's payoff is the opposite sign.
FRA value before expiry (long)
Value = Notional × (New FRA rate − Old FRA rate) × (days ÷ 360) ÷ (1 + L_t(h − t) × (h − t) ÷ 360)
New FRA rate is the forward rate now for the same remaining period. Discount with the spot rate from today to the end of the underlying period (time h). The short's value is the negative.
Notation
g x h FRA: expires at g months; underlying rate covers (h − g) months starting at g
For example, 3x6 means a 3-month rate starting in 3 months. 2x8 means a 6-month rate starting in 2 months.

How to solve Forward Rate Agreements (FRAs) questions

Use this sequence for any FRA item. Most errors come from mixing up time points, so mark them first.

  1. 1Read the notation. Write down expiry g, end of the underlying period h, and the days in the underlying rate.
  2. 2Check which position the question asks about. Long pays fixed and receives floating. Short is the reverse.
  3. 3Identify which rate you need: the original FRA rate, the rate at expiry, or a new forward rate from the current curve.
  4. 4For a forward rate, pull the two spot rates for g and h from the exhibit. Solve the no-arbitrage equation for F, then annualise with 360 ÷ underlying days.
  5. 5For settlement at expiry, compute the interest difference on the notional for the underlying period. Then divide by 1 + floating × days ÷ 360.
  6. 6For valuation before expiry, compute the new FRA rate for the remaining period, take the difference from the old rate, scale by days ÷ 360, and discount using the rate to time h.
  7. 7Check the sign and the position. Rates up since initiation means the long gains. Write the answer with the right sign and currency.

Quickest way: Compare, scale, discount

When to use it: Use when the vignette gives you a new rate and you only need the direction and size of the payoff or value.

  1. Decide the sign first. Floating or new forward rate above the old FRA rate means long positive and short negative.
  2. Take the rate difference in decimals, multiply by days ÷ 360 and by the notional.
  3. Divide by 1 + (discount rate × days ÷ 360). The discount rate is the floating rate at expiry for settlement, or the spot rate to time h for valuation.
  4. Compare with the answer options. Because discounting only shrinks the number slightly, the undiscounted figure helps you eliminate options quickly.

Common mistakes in Forward Rate Agreements (FRAs)

  • Reading 3x6 as a 6-month rate or a contract that expires in 6 months.

    The two numbers look like a start and a term.

    Fix: The first number is the expiry. The rate term is the second minus the first. A 3x6 is a 3-month rate starting in 3 months.

  • Computing the forward rate by subtracting or averaging the spot rates.

    Rates are quoted as annual percentages, so simple arithmetic feels natural.

    Fix: Compound each spot rate over its own number of days, divide the long growth factor by the short one, subtract 1, then annualise.

  • Forgetting to discount the settlement payoff.

    The interest formula looks complete after the rate difference and the day fraction.

    Fix: The interest is notionally paid at time h, but cash settles at g. Divide by 1 + floating × days ÷ 360.

  • Discounting the pre-expiry value with the wrong rate or period.

    Students use the rate to expiry instead of the rate to the end of the underlying period.

    Fix: Use the current spot rate for the period from now to h, with the days from now to h.

  • Getting the sign wrong for long and short.

    Students confuse the FRA long with owning a bond, which gains when rates fall.

    Fix: The long pays fixed, so it gains when rates rise. Remember: long FRA is like being a borrower who locked in a rate.

  • Using the old spot rates when valuing mid-life.

    The initial curve is in the first exhibit and is easy to reuse.

    Fix: Use the current curve for the remaining times. The old FRA rate is used only as the fixed rate in the difference.

Worked examples

Example 1

Vignette: A treasury analyst prices a 3x6 FRA on 90-day Libor with a notional of ₹10,00,00,000. Spot rates are 90-day Libor 4.0% and 180-day Libor 4.4%. Use a 360-day year. (1) What is the FRA rate? (2) At expiry, 90-day Libor is 5.20%. What is the settlement to the long? (3) Who pays at settlement?

Show the solution
  1. Part 1: 180-day growth = 1 + 0.044 × 180 ÷ 360 = 1.022.
  2. 90-day growth = 1 + 0.040 × 90 ÷ 360 = 1.010.
  3. Ratio = 1.022 ÷ 1.010 = 1.011881. Subtract 1 to get 0.011881.
  4. Annualise: 0.011881 × 360 ÷ 90 = 0.047525, or about 4.752%.
  5. Part 2: Rate difference = 0.0520 − 0.047525 = 0.004475.
  6. Interest difference = ₹10,00,00,000 × 0.004475 × 90 ÷ 360 = ₹1,11,880 (approximately).
  7. Discount: 1 + 0.052 × 90 ÷ 360 = 1.013. Payoff = ₹1,11,880 ÷ 1.013 ≈ ₹1,10,444.
  8. Part 3: Floating is above the FRA rate, so the short pays the long.

Answer: (1) FRA rate ≈ 4.752%. (2) The long receives about ₹1,10,444. (3) The short pays the long.

Example 2

Vignette: The investor in the previous example is long the 3x6 FRA at a rate of 4.75%, notional ₹10,00,00,000. Thirty days later, the 60-day spot rate is 4.2% and the 150-day spot rate is 4.6%. Use a 360-day year. (1) What is the new FRA rate for the remaining contract? (2) What is the value of the long position? (3) What would the short's value be?

Show the solution
  1. Part 1: The FRA still expires in 60 days and the underlying 90-day period ends at day 150.
  2. 150-day growth = 1 + 0.046 × 150 ÷ 360 = 1.019167.
  3. 60-day growth = 1 + 0.042 × 60 ÷ 360 = 1.007.
  4. Ratio = 1.019167 ÷ 1.007 = 1.012082. Subtract 1 to get 0.012082.
  5. Annualise: 0.012082 × 360 ÷ 90 = 0.048328, or about 4.833%.
  6. Part 2: Difference from the old rate = 0.048328 − 0.0475 = 0.000828.
  7. Interest difference = ₹10,00,00,000 × 0.000828 × 90 ÷ 360 ≈ ₹20,712.
  8. Discount with the rate to day 150: divide by 1.019167. Value ≈ ₹20,322.
  9. Part 3: The short has the opposite value.

Answer: (1) New FRA rate ≈ 4.833%. (2) Long value ≈ ₹20,320 (a gain, because rates rose). (3) Short value ≈ −₹20,320.

Exam tips

  • Mark g, h and the underlying days on the vignette before calculating. Most wrong answers come from a wrong time point.
  • Expect the same FRA to be used for two or three questions: price at initiation, payoff at expiry, and value mid-life. Keep your intermediate results.
  • Settlement is discounted using the floating rate at expiry. Valuation before expiry is discounted using the current spot rate to time h. Do not swap them.
  • Check the day-count convention in the vignette. Some exhibits give 365 or actual days, but you must use what is stated.
  • Use the direction of rates to sanity-check the sign. If rates rose after the FRA rate was set, the long should be positive.

Forward Rate Agreements (FRAs) in other exams

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

Forward Rate Agreements (FRAs): frequently asked questions

What does 3x6 mean in a forward rate agreement?

It means the FRA expires in 3 months and the underlying rate is a 3-month rate that starts at month 3 and ends at month 6. The first number is the expiry. The difference between the two numbers is the term of the rate.

How do you calculate the implied forward rate for an FRA?

Compound the longer spot rate over its full period and divide by the compounded shorter spot rate. Subtract 1 from the result, then multiply by 360 ÷ the days in the forward period. This gives the annualised FRA rate that leaves no arbitrage.

Why is the FRA payoff discounted?

The interest difference relates to a period that ends at time h, but the FRA settles at time g, the start of that period. Dividing by 1 + floating × days ÷ 360 converts the amount to its value at settlement.

How do you value an FRA before it expires?

Find the new forward rate for the remaining period from the current spot curve. Take the difference from the original FRA rate, scale by days ÷ 360 and the notional, and discount with the spot rate to the end of the underlying period. The long gains if the new rate is higher.