Forward rates are interest rates for money to be borrowed between any two dates in the future, but determined today. For example, I don't know what the one-year spot rate will be next year. I do know, however, what the one-year spot rate is today. I also know the two-year spot rate today. So, I could do one of two things:
The one-year spot rate, one year hence, is unobserved. Let's denote this forward rate by f. Then it must be true that the following relation holds:
The rate f is referred to as the one-year forward rate. The presence of s1 and s2 imply a forward rate in this case. It is:
For example, if and , then f = 6.6 percent . That is,
which means that the one year forward implied by s1 and s2 must equal 6.6 percent for us to be indifferent between a two-year loan at s2 and two consecutive one-year loans at s1 and f (assuming no transactions costs).
In sum, the forward rate structure is implied by the existing term structure of spot rates. Let's elaborate some on this concept. The notation fij is the forward rate between times ti and tj. So f12 is the forward rate between periods one and two; it is the forward one-year spot rate in the second year; f13 is the two-year forward implied to hold between the first and third year's known spot rates; f23 is the one-year forward in the third year; and so on. Generally then, we have:
With continuous compounding, the math is easier. We've already established that
where we assume that si is the spot rate over ti years. Therefore the forward/spot relationship given by:
is now equivalent to:
Taking natural logarithms of both sides:
Here tj and ti are integers. For example, with and , then the nine-year forward rate (to appear as implied one year from now) is the difference between the 10 times the current 10-year and 1-year continuous spot rates divided by the integer nine.