Forward Rates

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 f_ij is the forward rate between times t_i and t_j. 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 s_i is the spot rate over t_i years. Therefore the forward/spot relationship given by:

is now equivalent to:

Taking natural logarithms of both sides:

Here t_j and t_i 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.

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Example 3.3

Compute the forward curve using the following table of spot rates:

Example 3.3 Table A

Why do you think that these forward rates are plausible forecasts of future rates? The answer is that the forward rates are consistent with the current spot rate structure, and unless spot rates change over time, then these forwards will have to hold. Let's now fill out a matrix for all the possible forward rates implied by this term structure.

Example 3.3 Table B

Interpreting, 6.605 is the one-year forward rate (the rate to hold for one year beginning one year from now), 7.047 is the two-year forward (again, one year from now), and so on. The third row shows the implied forward rates, consistent with the current term structure, that will hold two periods from now, the fourth row shows forward rates implied to hold three periods from now, and so on.

Go to the companion website for more details (see Forward Rates under Chapter 3 Examples).

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