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

Forward Rate Agreements (FRAs) Pricing and Valuation for CFA Level 1

Updated 7 October 2026 · Fact-checked

A forward rate agreement fixes an interest rate for a future loan period. Its price is the implied forward rate from today's spot rates. After initiation, you value the long by taking the rate difference times notional and period fraction, adjusting it for the loan period, then discounting from expiry to today.

Understand Forward Rate Agreements (FRAs) Pricing and Valuation

A forward rate agreement (FRA) is an over-the-counter contract. One side agrees to pay a fixed rate on a notional amount for a future period. The other pays a floating rate, usually a reference rate such as a term benchmark rate. No principal changes hands. Only the net interest difference is paid.

The notation tells you the timing. A 3x6 FRA starts in 3 months and ends in 6 months, so it covers a 3-month loan that begins 3 months from now. The first number is when the contract period starts. The second is when it ends. The period length is the difference. A 1x4 FRA is also a 3-month rate, starting in 1 month. The long side pays fixed and receives floating, so it gains if rates rise.

Pricing means finding the fixed rate that gives the FRA zero value at the start. That rate is the implied forward rate. It comes from no-arbitrage. Borrowing for the full term must cost the same as borrowing for the short term and then rolling into the forward period. So (1 + long rate × long time) = (1 + short rate × short time) × (1 + forward rate × forward period), using simple add-on rates as for money-market rates.

Valuation after initiation is a different task. The contract rate is now fixed. Market rates move, so the current forward rate for the same period changes. The interest difference, (new forward rate − contract rate) × notional × period fraction, would be due at the loan end. But the FRA settles at the start of the loan period (expiry). So you first divide that amount by (1 + new forward rate × period fraction) to put it at expiry. Then you discount from expiry to today using the current spot rate for the time left to expiry. The long gains if the new forward rate is higher than the contract rate.

Settlement happens at the start of the loan period, not the end. Because the interest would normally be paid at the end, the payoff is discounted back to the start date using the floating rate at that time. At expiry, the payment equals notional × (floating rate − FRA rate) × period fraction ÷ (1 + floating rate × period fraction), paid to the long.

Key formulas to remember

Implied forward rate (add-on rates): F(A,B) = [(1 + R_B × t_B) ÷ (1 + R_A × t_A) − 1] ÷ (t_B − t_A); A is the start, B is the end, in years. For a 3x6 FRA, A = 0.25 and B = 0.5. R_A and R_B are the spot rates for those terms.
No-arbitrage link: (1 + R_B × t_B) = (1 + R_A × t_A) × (1 + F × (t_B − t_A)); Rearrange this to get the forward rate. It is the idea behind the formula, so you can rebuild it under pressure.
FRA settlement at expiry (paid at start of loan period): Payoff to long = Notional × (L − K) × (m ÷ 12) ÷ (1 + L × m ÷ 12); L is the floating rate at expiry. K is the contract (FRA) rate. m is the number of months in the loan period. Positive means the long receives.
FRA value before expiry (long position): V = Notional × (F_new − K) × (m ÷ 12) ÷ [(1 + F_new × m ÷ 12) × (1 + R_A × t_A)]; F_new is the current forward rate for the same loan period. The (1 + F_new × m ÷ 12) term moves the amount from the loan end to expiry. The (1 + R_A × t_A) term discounts from expiry to today, where R_A is the current spot rate for the remaining time t_A (in years) to expiry. Short value is the negative.
Long and short value: Value of short = − Value of long; An FRA is a zero-sum contract. The two sides have equal and opposite values.

How to solve Forward Rate Agreements (FRAs) Pricing and Valuation questions

Use this order for any FRA question. First decide whether you are pricing at initiation or valuing later, then match the timing carefully.

1Decode the notation. For a AxB FRA, the loan starts at A months and ends at B months. The loan length is B − A months.
2Write all times in years: A ÷ 12 and B ÷ 12. Identify which spot rates apply to each date.
3If pricing at initiation, compute the implied forward rate from the formula. The result is the FRA rate K and the initial value is zero.
4If valuing after initiation, recompute the forward rate for the same loan period using current rates and the remaining times. Call it F_new.
5Find the rate difference F_new − K. Multiply by notional and the period fraction (months ÷ 12). Then divide by (1 + F_new × months ÷ 12). This gives the amount at the start of the loan period (expiry).
6Discount that amount from expiry to today by dividing by (1 + R_A × t_A), where R_A is the current spot rate for the time remaining to expiry. Do not use the loan-period rate here.
7Check the sign. The long gains when F_new > K. The short value is the negative of the long value.
8For expiry settlement, use the floating rate L as the discount rate for the period, and pay at the start of the loan.

Quickest way: Forward rate shortcut with a ratio check

When to use it: Use it when the question gives two spot rates and asks for the forward rate, or asks which of the three options is the FRA rate.

Compute the long growth factor 1 + R_B × t_B and the short growth factor 1 + R_A × t_A.
Divide long by short to get the growth factor for the forward period.
Subtract 1, then divide by the forward period in years.
Sanity check: the forward rate lies above the longer spot rate only when the spot curve is upward sloping. It lies below the longer spot rate when the curve is downward sloping. Use this to eliminate an option fast.
On the BA II Plus, use the chain calculation: 1 + .0X × t, then ÷ , then the short factor, then − 1 ÷ period. Store intermediate factors in memory.

Common mistakes in Forward Rate Agreements (FRAs) Pricing and Valuation

Reading 3x6 as a 6-month loan or as a 3-year contract.
The numbers look like a start and a length, or like years.
Fix: Both numbers are months from today. The loan starts at 3 and ends at 6, so it lasts 3 months.
Forgetting to divide the forward rate by the forward period.
The ratio of growth factors minus 1 gives a period return, not an annual rate.
Fix: Always divide by (t_B − t_A) in years. For 3 months divide by 0.25, or multiply by 4.
Discounting the FRA value from the loan end date, or skipping a step.
Students treat the interest difference as paid at the loan end, or discount only over the loan period, and forget that the FRA settles at expiry.
Fix: The payment is made at the start of the loan period. Divide the interest difference by (1 + F_new × m ÷ 12), then discount from expiry to today using the current spot rate for the time to expiry.
Getting the sign wrong for the long and short.
Students think the long position is buying a bond, which gains when rates fall.
Fix: The long FRA pays fixed and receives floating. It gains when rates rise above K. Short is the opposite.
Forgetting the settlement discount at expiry.
The interest difference is computed correctly, but it is not adjusted for early payment.
Fix: Divide by (1 + L × m ÷ 12), because the payment is made at the start of the loan, not the end.

Worked examples

Example 1

The 90-day spot rate is 2.00% and the 180-day spot rate is 2.50%, both add-on rates on a 360-day year. What is the fixed rate on a 3x6 FRA?

Show the solution

The loan runs from day 90 to day 180, so the forward period is 90 days, or 0.25 years.
Long factor: 1 + 0.025 × 0.5 = 1.0125.
Short factor: 1 + 0.02 × 0.25 = 1.005.
Ratio: 1.0125 ÷ 1.005 = 1.0074627.
Subtract 1: 0.0074627.
Divide by 0.25: 0.0298507, or about 2.985%.

Answer: The 3x6 FRA rate is about 2.99% (2.985%). It sits above the 2.50% 180-day spot rate, as expected for an upward-sloping curve.

Example 2

A fund is long a 3x6 FRA on a notional of USD 10,000,000 at a contract rate of 3.00%. At expiry (3 months from now), the 90-day floating rate is 3.60%. What does the long receive, and when?

Show the solution

Rate difference: 3.60% − 3.00% = 0.60%.
Interest difference over 3 months: 10,000,000 × 0.006 × 0.25 = 15,000.
Discount for early settlement: 1 + 0.036 × 0.25 = 1.009.
Payoff: 15,000 ÷ 1.009 = 14,866.20.
The payment is made at the start of the loan period, which is expiry (month 3).

Answer: The long receives about USD 14,866 at expiry. The result is positive because the floating rate is above the contract rate.

Exam tips

Options are three numbers from smallest to largest. If you can say whether the answer must be above or below a spot rate, you can often remove two options without a full calculation.
Check whether the question asks for the FRA rate, the payoff at expiry, or the value before expiry. Each uses different discounting.
Watch the day-count basis in the stem. Many FRA questions use 30/360 or 360-day add-on rates, so do not switch to 365.
If the question gives the current forward rate for the same period, you may skip the pricing step and go straight to the difference and discounting.

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

Forward Rate Agreements (FRAs) Pricing and Valuation 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) Pricing and Valuation: frequently asked questions

What does 3x6 mean in a forward rate agreement?

It means the contract period begins 3 months from today and ends 6 months from today. The underlying rate is therefore a 3-month rate, starting in 3 months. Both numbers are months from the initiation date.

How do you calculate the implied forward rate from spot rates?

Divide the growth factor of the longer spot rate by the growth factor of the shorter spot rate, subtract 1, then divide by the forward period in years. Use add-on rates for money-market terms. The result is the FRA rate that gives zero initial value.

Is an FRA settled at the start or end of the loan period?

In the standard FRA, settlement is at the start of the loan period, which is the FRA expiry date. Because the interest would normally be due at the end, the payment is discounted using the floating rate. This is why the denominator appears in the payoff formula.

Who gains when interest rates rise on an FRA?

The long position gains because it pays the fixed contract rate and receives the higher floating rate. The short position loses by the same amount. The reverse holds when rates fall.