# 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:

 Forward-ForwardRate = (1 + Zero-Coupon Rate for k Years)k(1 + Zero-Coupon Rate for k + 1 Years)(k+1) Year = 365 for sterling, 360 days for all other currencies Based on the day-count convention.

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 = 1(1+Yield)k k = Term in Years

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

 Discount Rate = 1(1+.02)2 = 1/1.0404 ≈ 0.961169
 Forward-ForwardRate = Discount Rate for k YearsDiscount Rate for k +1 Years – 1
Example: Calculate Forward-Forward Yield with Discount Factors
Zero-Coupon Yields
Year 1 4.0%
Year 2 4.3%
Year 3 4.6%
Year 4 5.0%
Zero-Coupon Discount Rates
Year 1 0.961538462 = 1/(1+Year 1 Yield)
Year 2 0.919245226 = 1/(1+Year 2 Yield)2
Year 3 0.873785727 = 1/(1+Year 3 Yield)3
Year 4 0.822702475 = 1/(1+Year 4 Yield)4
Forward-Forward Rates
1 v 2 0.046008654 = 4.6% = (Year 1 Discount) / (Year 2 Discount) – 1
2 v 3 0.052025912 = 5.2% = (Year 2 Discount) / (Year 3 Discount) – 1
3 v 4 0.062092012 = 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 Using Interest Ratesfor theForward-Forward Term Then Annualize theInterest Rates Forward-ForwardRate = [ FF2 FF1 – 1 ] x Days in YearDays inForward-ForwardTerm Year = 365 for sterling, 360 days for all other currencies, based on the day-count convention. FF1 = 1 + IR1 × T1/Year FF2 = 1 + IR2 × T2/Year IR1 = Interest Rate during T1 IR2 = Interest Rate during T2 T1 = Days from Agreement Date till Start of Forward Term T2 = Days from Agreement Date till End of 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.

 Interest Rate till Start of Forward Term 3.0% Number of Days till Start of Forward Term 91 Interest Rate till End of Forward Term 3.3% Number of Days till End of Forward Term 183 Days in Year (Day-Count Convention) 360 Days in Forward-Forward Term 92 Forward-Forward Interest Rate 3.57%
 Forward-ForwardRate = [ (1 + (.033 × 183/360))(1 + (.03 × 91/360)) – 1 ] x 36092 = 3.57%

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