How this calculator works
A forward rate is the interest rate implied today for a loan or investment that starts later. It comes from the no-arbitrage condition: investing for the whole second period at the second spot rate must give the same result as investing for the first period at the first spot rate and then rolling over at the forward rate. Solving that equality for the unknown forward rate produces the formula used here. The spot rates are treated as annual rates with annual compounding, so the growth factor for t years is (1 + s)^t. The result is the rate for the gap between the two maturities, not for the whole second period.
Formula
(1 + S2)^t2 = (1 + S1)^t1 x (1 + f)^(t2 - t1). Solving, f = [(1 + S2)^t2 / (1 + S1)^t1]^(1 / (t2 - t1)) - 1, where S1 and S2 are the spot rates for t1 and t2 years and f is the forward rate between them.
Worked example
A one-year spot rate of 4% and a two-year spot rate of 4.5% imply a one-year forward rate of (1.045^2 / 1.04)^1 - 1 = 5.0024%. Investing for one year at 4% and then one more year at 5.0024% gives the same return as investing at 4.5% for two years.
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Frequently asked questions
What compounding does this assume?
Annual compounding throughout, so (1 + s)^t is the growth factor. Markets often quote spot rates on a semi-annual bond basis or a continuously compounded basis; convert them to the same convention before using the formula.
Is the forward rate a forecast?
No. It is the rate at which a forward contract can be struck today, implied by the spot curve. It may or may not match the spot rate that actually occurs later.
What if the second spot rate is lower?
The forward rate will be lower still if the curve slopes down. A downward-sloping curve implies forward rates below the long spot rate, which is the market's way of pricing a decline in rates.
Can I use this between non-annual periods?
Yes, if the times are fractions of a year and the rates are annual rates with annual compounding. For example, 0.5 and 1.0 years would use the same formula with t1 = 0.5 and t2 = 1.0.