Forward Transactions on Interest Rates

Abstract

A forward rate agreement, abbreviated as FRA, is a contract in which both contract parties agree to a fixed rate K on a principal N to be paid for some future interest period between T and T′. An FRA can be interpreted as an agreement loan to be made in the future with an interest rate already fixed today. The party receiving the loan makes the fixed interest payments. In contrast to bonds, we will refer to this party’s position as a long position in the FRA, whereas the counterparty receiving the interest payments is short in the FRA. A (long) FRA can thus be interpreted as an agreement on two future cash flows: a receipt of the principal N at time T (the loan is made) and a payment at maturity T′ of the FRA in the amount of the principal N compounded at the agreed rate K over the period T′ — T (the loan plus interest is paid back). Both of these cash flows discounted back to time t yield the present value of the FRA at time t:

$$\begin{gathered} {F_R}\left( {T,T',\left. K \right|t} \right) = {B_R}\left( {t,T} \right)\underbrace N_{Cash\;Flow\;at\;Time\;T} - {B_R}\left( {t,T'} \right)\frac{N}{{\underbrace {{B_K}\left( {T,T'} \right)}_{Cash\;Flow\;at\;Time\;T'}}} \hfill \\ \quad \quad \quad \quad \quad \;\; = N\,{B_R}\left( {t,T} \right)\left[ {1 - \frac{{{B_R}\left( {T,T'\left| t \right.} \right)}}{{{B_K}\left( {T,T'} \right)}}} \right] \hfill \\ \end{gathered} $$

((17.1))

where the definition of the forward rate, Equation 3.3, was used in the last step. It is common practice in the market for forward transactions, the interest rate K is chosen so that the contract is worthless at the time it is concluded. As can be deduced from Equation 17.1, this condition is satisfied when the fixed rate K agreed upon in the FRA equals the forward rate at time t corresponding to the FRA period from T until T′.