Forward-Forward Agreements

A forward-forward agreement is a contract that guarantees a certain interest rate on an investment or a loan for a specified time interval in the future, that begins on one forward date and ends later. It is called a forward-forward interest rate because it is for a time period that both begins and ends in the future. Hence, a forward-forward contract protects against market changes in the interest rates.

A forward-forward is different from other interest-rate derivatives. Both forward rate agreements and short-term interest rate futures can protect against market changes in the interest-rate, but they do so by paying the difference between the contract rate and the reference market rate, such as the libor. There are also forward-forward currency swaps, involving the swapping of 1 currency for another at the beginning of the forward period, which is then reversed at maturity.

Forward-forwards have a special notation to designate the future term. For instance, a term that begins in 6 months and ends 1 year later, would be designated as 6 v 18 or 6 × 18. To designate years, a forward-forward term that starts in 2 years and ends 1 year later would be designated as 2 years v 3 years, or 2 years × 3 years.

The forward-forward interest-rate is the forward rate for the term of the contract. How is the forward rate determined? Banks generally set forward-forward rates by checking the prices of short-term interest rate futures, which will allow banks to hedge their interest-rate risk. Or they can check the yields on zero-coupon bonds for the forward period.

The forward-forward rates for a range of maturities can be represented by the forward-forward yield curve. The actual spot rates for forward periods cannot be known in advance, but implied forward-forward rates can be constructed by bootstrapping, which starts with short-term market yields of money market instruments and futures, then uses those values to calculate yields for later periods. From this implied forward-forward yield curve, formulas can be used to calculate forward-forward rates for those periods covered by the yield curve. Forward-forward interest rates covering full years can be calculated by the following formula:

Calculate Forward-Forward Rates Using Zero-Coupon Yields
Forward-Forward Rate=(1 + Zero-Coupon Rate for k Years)k
(1 + Zero-Coupon Rate for k + 1 Years)(k+1)

A forward-forward rate can also be calculated with discount rates for zero-coupon bonds. The discount rate = 1 ÷ (1 + Yield) raised to a power equal to the number of years till maturity.

 Discount Rate Formula
Discount Rate=1
(1+Yield)(Term in Years)

Thus, the discount rate for a 2-year zero with a 2% yield would be:

Example: Discount Rate Calculation
Discount Rate=1
(1+.02)2
=1/1.04040.961169
Calculate Forward-Forward Rates Using Discount Factors
Forward-Forward Rate=Discount Rate for k Years
Discount Rate for k +1 Years
1
Example: Calculate Forward-Forward Yield with Discount Factors
Zero-Coupon Yields
Year 14.0% 
Year 24.3% 
Year 34.6% 
Year 45.0% 
Zero-Coupon Discount Rates
Year 10.961538462= 1/(1+Year 1 Yield)
Year 20.919245226= 1/(1+Year 2 Yield)2
Year 30.873785727= 1/(1+Year 3 Yield)3
Year 40.822702475= 1/(1+Year 4 Yield)4
Forward-Forward Rates
1 v 20.046008654= 4.6% = (Year 1 Discount) / (Year 2 Discount) – 1
2 v 30.052025912= 5.2% = (Year 2 Discount) / (Year 3 Discount) – 1
3 v 40.062092012= 6.2% = (Year 3 Discount) / (Year 4 Discount) – 1

If the forward-forward term is less than 1 year, as it often is, then the above rate can be modified by multiplying by the number of days in the period divided by the number of days in a year, according to the day-count convention that applies, which is 365 days for sterling and 360 days for all other currencies:

Calculate Forward-Forward Rates Using Money Market Rates
   Calculate Using Interest Rates for the Forward-Forward Term    Then Annualize the Interest Rates
Forward-Forward Rate=[(1 + (Interest Rate till 2nd Date × Days till 2nd Date/Year))
(1 + (Interest Rate till 1st Date × Days till 1st Date/Year))
– 1]xDays in Year
Days in Forward-Forward Term

Note that the interest rates in the above formula are annualized.

In this example, market interest rates are used, and since the term is less than 1 year, money market instruments, such as commercial paper, are appropriate. In this example, the interest-rate for the 1st date is determined by the market yield on commercial paper with the term equal to the number of days from the agreement date until the 1st date. Likewise, for the 2nd date, using the market yield for commercial paper with the term equal to the number of days from the agreement date until the 2nd date.

Example 2: Forward-Forward Rate for Term < 1 Year
Interest Rate till Start of Forward Term3.0%
Number of Days till Start of Forward Term91
Interest Rate till End of Forward Term3.3%
Number of Days till End of Forward Term183
Days in Year (Day-Count Convention)360
Days in Forward-Forward Term92
Forward-Forward Interest Rate3.57%
Example: Calculate Forward-Forward Rates Using Money Market Rates
Forward-Forward Rate=[(1 + (.033 × 183/360))
(1 + (.03 × 91/360))
– 1]x360
92
=3.57%