A forward rate is the interest rate implied by the current yield curve for a future period. It is derived from the no-arbitrage condition: investing for two years at the 2-year spot rate must produce the same return as investing for one year at the 1-year spot rate and then reinvesting for one year at the 1-year rate, one year forward.
Mathematically, the 1-year rate, 1-year forward (denoted 1y1y) satisfies:
(1 + r_2)^2 = (1 + r_1)(1 + f_{1,1})
where r_1 and r_2 are the 1-year and 2-year spot rates, and f_{1,1} is the forward rate.
Forward rates serve multiple purposes:
The forwards tool on this site computes and visualizes the implied forward curve for any horizon, allowing users to see what the current par curve implies about future rate levels. A steep forward curve suggests the market prices higher future rates (or positive term premium), while a declining forward curve suggests expectations of lower rates.
The forward rate follows from two simple rules.
The first rule is that money earned is reinvested. An investor who holds a bond for one year and then buys another one year bond compounds the two returns. The second rule is that there is no free lunch. Two strategies that carry the same risk over the same period must return the same amount.
Put the rules together and the forward rate becomes the only rate that satisfies them. It is the rate that makes an investor indifferent between locking in a single long investment today and rolling a series of shorter ones.
The general relationship uses spot rates. The forward rate covering the period between year m and year n solves:
(1 + z_n)^n = (1 + z_m)^m x (1 + f)^(n-m)
Traders name a forward by its start and its length. A 1y1y forward is the one year rate that starts one year from now. A 3y7y forward is the seven year rate that starts three years from now. The first number is always the start.
One practical warning. The formula above uses whole years. Real markets use day count conventions, so an accurate forward divides actual days by the convention basis rather than counting years. The whole year form is close enough to build intuition and wrong enough to matter in a trade ticket.
The most useful job a forward rate does is set a hurdle.
A trader who buys a bond because it yields more than a shorter bond has not yet made an argument. The curve already tells everyone that the longer bond yields more. The forward rate states the price of that advantage. It says how far yields must rise before the extra yield is given back.
The same logic applies to a curve trade. A steepener does not profit because the spot spread is narrow. It profits because the spread ends up wider than the forward spread. If the spot 5s10s spread is 50 bps and the forward spread in three months is 40 bps, then the curve must steepen past 40 bps for the trade to make money. Comparing the outcome to the spot spread of 50 bps would flatter the trade and hide a loss.
This is the point that separates a curve view from a curve trade. The view is about where the spread goes. The trade is about whether it goes further than the market has already priced.
A forward rate is often described as the market's expectation of a future rate. That description is close but not correct, and the gap matters.
A forward rate contains two things. One is the expected path of short rates. The other is the term premium, which is the extra yield investors demand for accepting interest rate risk. A forward can therefore sit above the expected rate simply because investors want to be paid for uncertainty.
The historical record supports the distinction. Forward rates have on average implied more increase in short rates than actually occurred. If forwards were pure forecasts, that pattern would be a long standing forecasting error. Read as expectations plus a premium, it is exactly what theory predicts.
It is the five year rate that begins five years from today, implied by the current curve. Analysts watch it because it strips out the near term policy cycle. Whatever the Federal Reserve does in the next two years has mostly washed out by the time that period starts, so the 5y5y rate is read as a view on longer run inflation and growth.
Because they are the market's price, and a trade is settled against prices rather than opinions. A forecast that differs from the forward is the only kind of forecast that can make money. If your view matches the forward exactly, there is nothing to trade.
Yes. If the spot curve is inverted steeply enough over a segment, the arithmetic can produce a negative forward for that period. It means the market is pricing rates far enough below current levels that the implied rate for that window falls below zero.