Forward Rate Agreement (FRA) – Solutions
Given Information
Date of agreement: June 15, 2000
FRA start date: December 15, 2000
FRA end date: June 15, 2001
Length of underlying loan: 6 months (0.5 years)
6-month zero-coupon bond price: $95.75
1-year zero-coupon bond price: $92.50
Assume both bonds have a face value of $100.
For parts C–F, assume a $100 notional principal and that we are receiving the floating rate and paying the fixed rate.
A) What is the agreed-upon forward rate?
The forward rate is calculated using the prices of the two zero-coupon bonds.
Formula:
Forward rate = [(6-month bond price / 1-year bond price) - 1] / 0.5
Substitute:
Forward rate = [(95.75 / 92.50) - 1] / 0.5
Forward rate = (1.035135 - 1) / 0.5
Forward rate = 0.07027
Answer A: 7.027% annually, or approximately 3.514% for the six-month period.
This is the interest rate agreed upon when the FRA is established.
B) What is the value of the forward at inception?
At inception, a fairly priced forward rate agreement has a value of zero.
This is because the agreed-upon forward rate equals the market forward rate.
Neither party has an initial advantage.
Answer B: $0
C) What is the value of the FRA on September 15, 2000?
Three months have passed.
The FRA now begins in 3 months and ends in 9 months.
From the September 15 table:
3-month discount factor = 0.9844
9-month discount factor = 0.9531
First, calculate the new forward rate.
Formula:
New forward rate = [(0.9844 / 0.9531) - 1] / 0.5
New forward rate = 0.06568
New forward rate = 6.568%
The original agreed-upon forward rate was 7.027%.
Because the new market forward rate is lower than the original rate, the position receiving floating and paying fixed has lost value.
To calculate the FRA's value:
Value = Notional × [P(0,0.25) - (1 + Original Rate × 0.5) × P(0,0.75)]
Substitute:
Value = 100 × [0.9844 - (1 + 0.07027027 × 0.5) × 0.9531]
Value = 100 × [0.9844 - 0.98659]
Value = approximately -$0.219
Answer C: Approximately -$0.22 per $100 of notional principal.
The FRA has a negative value because the market forward rate has fallen below the original contracted rate.
D) What is the value of the FRA on December 15, 2000?
December 15 is the FRA's start date.
The underlying interest period now runs from December 15, 2000, to June 15, 2001.
From the December 15 table:
6-month discount factor = 0.9692
Original agreed-upon forward rate = 7.027%
Formula:
Value = Notional × [1 - (1 + Original Rate × 0.5) × 6-month discount factor]
Substitute:
Value = 100 × [1 - (1 + 0.07027027 × 0.5) × 0.9692]
Value = approximately -$0.325
Answer D: Approximately -$0.33 per $100 of notional principal.
The FRA has a negative value for the party receiving floating and paying fixed.
E) What is the 6-month semiannual rate?
We use the December 15 discount factor for a 6-month maturity.
6-month discount factor = 0.9692
Formula:
6-month interest rate = (1 / Discount Factor) - 1
Substitute:
6-month interest rate = (1 / 0.9692) - 1
6-month interest rate = 0.031779
Answer E: 3.178% for six months.
If expressed as an annualized rate with semiannual compounding:
Annualized rate = 3.178% × 2
Annualized rate = 6.356%.
F) What will the balance to be paid be at the end of the FRA?
We compare the original contracted interest rate with the market interest rate.
Original contracted six-month rate:
7.027% / 2 = 3.514%
Current market six-month rate:
3.178%
Difference:
3.178% - 3.514% = -0.336%
For a $100 notional principal:
Payment = $100 × (0.031779 - 0.035135)
Payment = approximately -$0.336
Answer F: Approximately $0.34 per $100 of notional principal is owed by the floating-rate receiver to the fixed-rate receiver.
The negative amount indicates that the party receiving floating and paying fixed owes money.
Note: In a standard FRA, the net settlement is generally paid at the beginning of the underlying interest period (December 15), discounted from the end-of-period interest difference. That settlement is approximately $0.325 per $100, consistent with Part D. The $0.336 amount represents the equivalent undiscounted payment at June 15, 2001.
Final Answers
A) Agreed-upon forward rate: 7.027% annually
B) Value at inception: $0
C) Value on September 15: Approximately -$0.22 per $100 notional
D) Value on December 15: Approximately -$0.33 per $100 notional
E) Six-month semiannual interest rate: 3.178% (6.356% annualized)
F) End-of-period net payment: Approximately $0.34 per $100 notional, paid by the floating-rate receiver
Important: The problem does not specify the notional principal or which side of the FRA was entered. These numerical values assume a $100 notional and a position receiving floating and paying fixed. For the opposite position, the signs in C, D, and F reverse.
