Skip to content

FRM Exam Part I · Properties of Interest Rates

Forward Rates and Forward Rate Agreements for FRM Part I

Updated 11 October 2026 · Fact-checked

A forward rate is the interest rate for a future period that is implied by today's zero rates. You get it by comparing growth to the later date with growth to the earlier date. A forward rate agreement (FRA) locks in that rate. You value it by discounting the difference between the FRA rate and the forward rate.

Understand Forward Rates and Forward Rate Agreements

A zero rate (spot rate) is the annual rate earned on money invested from today to a single future date, with no payments in between. You can read zero rates off the zero curve for 1 year, 2 years and so on.

A forward rate is the rate for a period that starts in the future, such as year 2 to year 3. It is not a forecast. It is the rate that makes you indifferent between two routes: invest to year 3 directly, or invest to year 2 and then reinvest for one more year at the forward rate. If the forward rate differed, you could earn a risk-free profit.

This gives the core idea. Growth to T2 equals growth to T1 times growth from T1 to T2. With annual compounding: (1 + R2)^T2 = (1 + R1)^T1 × (1 + F)^(T2 − T1). With continuous compounding it is simpler: R2 × T2 = R1 × T1 + F × (T2 − T1). When the zero curve slopes up, forward rates sit above zero rates. When it slopes down, they sit below.

A forward rate agreement (FRA) is an OTC contract. One side agrees to pay a fixed rate R_K, and the other pays a floating rate (such as a reference rate) on a notional amount for a future period T1 to T2. Only the interest difference is exchanged, not the notional. The FRA is worth zero at inception when R_K equals the forward rate. Later, its value depends on how the current forward rate compares with R_K.

A standard FRA pays the difference at the start of the period (T1) in practice, discounted from T2, but the textbook valuation often treats the payoff as received at T2. Read the question to see which one it asks for.

Key formulas to remember

Forward rate, continuous compounding
F = (R2 × T2 − R1 × T1) ÷ (T2 − T1)
R1 and R2 are zero rates to T1 and T2. Use decimals and years.
Forward rate, annual compounding
F = [(1 + R2)^T2 ÷ (1 + R1)^T1]^(1 ÷ (T2 − T1)) − 1
Use when the question gives annually compounded zero rates.
Forward rate from discount factors (simple rate)
F = (DF1 ÷ DF2 − 1) ÷ (T2 − T1)
Gives the simple, non-compounded forward rate for the period. Common for FRAs.
FRA payoff at T2 (receive fixed, pay floating)
Payoff = L × (R_K − R) × (T2 − T1)
L is notional, R is the actual rate for the period. Flip the sign for the floating receiver.
FRA value (receive fixed R_K)
V = L × (R_K − R_F) × (T2 − T1) × DF(T2)
R_F is the current forward rate and DF(T2) is the discount factor to T2. Both rates must use the same compounding.
FRA settlement at T1
Settlement = L × (R_K − R) × τ ÷ (1 + R × τ)
τ is the accrual fraction. This is the market practice of paying at the start of the period.

How to solve Forward Rates and Forward Rate Agreements questions

Use this method for any question on forward rates or FRAs. Most errors come from mixing compounding bases or time periods.

  1. 1Identify the period. Write T1 and T2 in years and the length T2 − T1.
  2. 2Identify the compounding of the given rates: continuous, annual, or simple. Keep it the same throughout.
  3. 3Compute the forward rate with the matching formula, using total growth to T2 divided by growth to T1.
  4. 4Check reasonableness: the forward rate should be above the zero rates if the curve slopes upward.
  5. 5For an FRA, decide your position: receive fixed gains when the realised rate is below R_K, and pay fixed gains when it is above.
  6. 6Compute the payoff as notional × rate difference × period length.
  7. 7If valuing, discount from T2 (or settle at T1 using the 1 + Rτ adjustment) as the question states.
  8. 8Check the sign and the units of your final answer before choosing an option.

Quickest way: Total growth over earlier growth

When to use it: Use for any forward rate question with zero rates, especially when options are close together.

  1. Continuous compounding: multiply each zero rate by its time, subtract, divide by the gap in years.
  2. Annual compounding: divide (1 + R2)^T2 by (1 + R1)^T1, then take the root of the gap and subtract 1.
  3. For FRA value, multiply notional by the rate gap by the period length, then by DF(T2).
  4. Sanity check: a one-year forward from year 1 to 2 on a rising curve must exceed R2.

Common mistakes in Forward Rates and Forward Rate Agreements

  • Averaging zero rates instead of using the growth identity

    It feels natural to treat the forward rate as a midpoint of two spot rates.

    Fix: Always take R2 × T2 minus R1 × T1, then divide by T2 − T1. Never average.

  • Mixing compounding conventions

    Zero rates may be given as continuous while the FRA rate is quoted as annual or simple.

    Fix: Convert everything to one basis before comparing. Note the compounding stated in the question.

  • Forgetting to discount the FRA payoff

    Students stop after computing notional × rate difference × period.

    Fix: Multiply by DF(T2), or use the 1 + Rτ adjustment if settlement is at T1.

  • Using the wrong time period

    A 2 × 5 FRA covers months 2 to 5, a 3-month period, not 5 months or 2 months.

    Fix: Write T1 and T2 first, then subtract to get the accrual length.

  • Getting the sign of the payoff wrong

    Fixed receiver and fixed payer are confused.

    Fix: Fixed receiver gains when the market rate falls below R_K. Fixed payer gains when it rises above R_K.

  • Treating the forward rate as a prediction of future spot

    The word 'forward' suggests a forecast.

    Fix: It is an arbitrage-implied rate. Actual future rates can differ and that gap is the FRA profit or loss.

Worked examples

Example 1

Continuously compounded zero rates are 4.0% for 1 year and 4.5% for 2 years. What is the 1-year forward rate from year 1 to year 2, continuously compounded?

Show the solution
  1. T1 = 1, R1 = 4.0%. T2 = 2, R2 = 4.5%.
  2. R2 × T2 = 0.045 × 2 = 0.090.
  3. R1 × T1 = 0.040 × 1 = 0.040.
  4. Difference = 0.090 − 0.040 = 0.050.
  5. Divide by T2 − T1 = 1: F = 0.050 ÷ 1 = 5.0%.

Answer: 5.0% per year, continuously compounded.

Example 2

A bank enters a 1 × 2 FRA as the fixed receiver at 5.0% (annual, simple) on a notional of USD 10,000,000 for the period from year 1 to year 2. The realised rate for that period turns out to be 4.0%. Ignoring discounting, what is the payoff at the end of year 2? Then find the settlement value at year 1 if settled in advance.

Show the solution
  1. Payoff at T2 = L × (R_K − R) × (T2 − T1).
  2. = 10,000,000 × (0.05 − 0.04) × 1.
  3. = 10,000,000 × 0.01 = USD 100,000 received at T2.
  4. For settlement at T1, discount by (1 + R × τ) = 1 + 0.04 × 1 = 1.04.
  5. Settlement at T1 = 100,000 ÷ 1.04 = USD 96,153.85.

Answer: Payoff at year 2 is USD 100,000. Settled at year 1 it is USD 96,153.85.

Exam tips

  • Check the compounding in the question first. Most wrong answers use the right formula on the wrong basis.
  • Read FRA names like 3 × 6 as start month × end month. The accrual period is the difference.
  • Know the direction: rising zero curve means forward rates above zero rates.
  • Remember an FRA has zero value at inception when R_K equals the forward rate.
  • Use a calculator for powers and roots under annual compounding, and keep four decimals to avoid rounding errors between close options.

Practice questions from Properties of Interest Rates

Forward Rates and Forward Rate Agreements 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 Forward Rate Agreements: frequently asked questions

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

A spot (zero) rate applies from today to a future date. A forward rate applies to a period that starts in the future. The forward rate is implied by two spot rates through no-arbitrage.

How do I calculate a forward rate from zero rates?

Compare growth to the later date with growth to the earlier date. With continuous compounding, F = (R2 × T2 − R1 × T1) ÷ (T2 − T1). With annual compounding, divide the two growth factors and take the root for the gap.

What is the FRA payoff formula?

For a fixed receiver, payoff = notional × (R_K − R) × period length, where R is the realised rate. The sign flips for the fixed payer. If settled at the start of the period, divide by 1 + R × period length.

When is an FRA worth zero?

An FRA has zero value at inception when its fixed rate equals the current forward rate for that period. Afterwards its value changes as the forward rate moves.