Price the payments of a forward rate agreement (FRA) by working out the value of its cash flows.
The idea
A forward rate agreement (FRA) fixes an interest rate RK for a future period from T1 to T2 on a notional principal L. One party receives interest at RK and pays the floating rate, or the reverse. RF denotes the forward rate for the period. If the floating rate equals RF, the receiver of fixed gets a net payment at T2. Net payment = L × (RK − RF) × (T2 − T1). The FRA's value today is that payment discounted at the T2 zero rate. Value = L × (RK − RF) × (T2 − T1) × discount factor to T2. It is zero at inception when RK equals RF. The receiver of fixed gains when RK exceeds the forward rate. In practice, FRAs are often settled at T1, with the difference discounted back from T2 at the reference rate. Typical mistakes are discounting to the wrong date, flipping the sign, and ignoring the period length.
Formulas
Net FRA payment to the receiver of fixed
net payment = L × (RK − RF) × (T₂ − T₁)
- L
- the notional principal
- RK
- the fixed rate agreed in the FRA
- RF
- the floating rate for the period, or the forward rate when valuing
- T₁
- the start of the period, in years
- T₂
- the end of the period, in years
Paid at T₂. The payer of fixed has the opposite sign, which is the most common slip.
Value of an FRA to the receiver of fixed
FRA value = L × (RK − RF) × (T₂ − T₁) × DF₂
- L
- the notional principal
- RK
- the fixed rate agreed in the FRA
- RF
- the forward rate for the period implied by today's curve
- T₁
- the start of the period, in years
- T₂
- the end of the period, in years
- DF₂
- the discount factor to T₂ from the T₂ zero rate
Zero at inception when RK equals RF. Discount to T₂, not T₁.
A practice question
A company enters a forward rate agreement (FRA) in which it receives a fixed rate and pays the floating reference rate on a notional principal for a future period. At the start of the FRA the fixed rate equals the forward rate implied by the current spot curve. What is the value of the FRA at inception?
- Positive, because the company is guaranteed a fixed rateWhy not: The fixed rate is set at the forward rate, so it is not better than market and carries no value at inception.
- Zero, because the fixed rate equals the forward rate and the expected net payment is zero - correct
- Negative, because the company gives up the chance to benefit if rates riseWhy not: Giving up that chance is exactly compensated by the protection; at the forward rate the value is nil.
- Equal to the notional principal times the forward rateWhy not: Confuses the FRA's value with an interest amount on the notional; the FRA exchanges only the difference in interest.
The fixed rate in a new FRA is the forward rate. Valuing the floating payment at the forward rate makes the fixed and floating legs worth the same, so the net value is zero. Later, the FRA gains or loses as the forward rate moves away from the contract rate.
A harder one
A 3x9 FRA (starting in 3 months and covering 6 months) has the holder pay a fixed 3.2% on a notional principal of USD 20 million. At the start date the 6-month reference rate is fixed at 3.8% (simple interest). The FRA is cash-settled at the start of the period, discounting the interest difference with the reference rate. The settlement amount received is closest to:
- USD 60,000Why not: Does not discount the interest difference, which is payable at the end of the period, back to the settlement date.
- USD 61,140Why not: Compounds forward instead of discounting back.
- USD 58,881 - correct
- USD 117,763Why not: Uses a full year of interest instead of the 6-month period.
Interest difference at the end of the period = 20,000,000 x 0.006 x 0.5 = 60,000. Discounted to the start at 3.8% for half a year: 60,000 / 1.019 = 58,881.
The hardest tier
The 1-year and 2-year continuously compounded spot rates are 3.0% and 3.5%. A company receives a fixed 5.0% (continuously compounded) and pays the floating rate on a notional principal of USD 1 million for the year from t = 1 to t = 2. Use the standard approximation: value = L x (RK - RF) x (T2 - T1) x e^(-R2 x T2). The value of the FRA is closest to:
- USD 9,324 - correct
- USD 10,000Why not: Does not discount the net payment, which is made at t = 2.
- USD 9,704Why not: Discounts at the 1-year rate for one year, but the payment is at t = 2 and requires e^(-R2 x 2).
- USD 13,986Why not: Uses the 2-year spot rate in place of the forward rate when valuing the floating leg.
Forward rate = (3.5 x 2 - 3.0 x 1)/1 = 4.0%. Value = L x (RK - RF) x (T2 - T1) x e^(-R2 x T2) = 1,000,000 x 0.01 x e^(-0.07) = 9,324.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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