A corporate treasurer knows she will need to borrow $10 million for six months, starting in three months' time — perhaps to bridge a receivable. Today's 3-month and 9-month rates are both quoted, but neither is the rate for her loan, which starts three months from now and runs for six. The zero curve she already has turns out to pin that rate down exactly, with no guesswork required — and a contract exists to lock it in today. This lesson derives the forward rate the curve implies, and introduces the forward rate agreement (FRA) that trades it.

The forward rate: what the curve already knows

Suppose you can invest risk-free out to T_1 at rate R_1, or out to a later date T_2 at rate R_2. Consider two strategies for turning $1 into cash at T_2:

• invest directly to T_2: grows to e^{R_2 T_2};
• invest to T_1 (growing to e^{R_1 T_1}), then re-invest that from T_1 to T_2 at whatever rate R_F you can lock in today for that future period.

No arbitrage means both strategies must produce the same terminal wealth — exactly the same logic that pins down a forward price from a spot price and a cost of carry, with money itself now playing the role of the asset being carried forward. Setting the two strategies equal,

and solving for the only unknown gives the forward rate implied by the curve — no forecasting involved, just algebra on two rates the market already quotes:

Worked example. The curve quotes R_1 = 5.0\% at T_1 = 0.25 (3 months) and R_2 = 5.6\% at T_2 = 0.75 (9 months). The forward rate for the 6-month period starting in 3 months (market shorthand: the "3×9" FRA rate) is

Notice 5.90\% > 5.6\% > 5.0\% — with an upward-sloping curve, the market is quietly pricing in even higher rates for the future 6-month stretch than either quoted rate alone.

The FRA: trading that forward rate

A forward rate agreement lets two parties lock in exactly that forward rate for a notional principal L, over a future period from T_1 to T_2, without either side actually lending or borrowing the notional — only the difference between the agreed rate and the rate that actually prevails changes hands. The party long the FRA (who is hedging a future borrowing need, like our treasurer) is protected if rates rise above the agreed rate R_K; the party short is protected if rates fall.

At the start of the period, the reference floating rate R_M (historically LIBOR, now a SOFR-based rate) is observed, and the long side receives

— positive if the actual rate came in above what was locked in, negative otherwise. This is exactly the extra interest the long side would have paid on a real loan at the higher market rate, handed over in cash instead: economically identical to having actually borrowed at R_K. In practice this amount is usually settled early, at T_1 (when R_M becomes known), discounted back one period at the now-known market rate.

Worked example, continued. Our treasurer locks in the FRA at R_K = 5.90\% on notional L = \$10{,}000{,}000. Three months later, the realized 6-month rate turns out to be R_M = 6.20\% — rates rose, so the hedge pays off:

payable, in principle, at T_2. Settled instead at T_1 (the market norm), the payment is discounted one period at the now-known rate R_M:

Either way, the treasurer's real borrowing cost is topped up (or trimmed) by exactly the gap between what the market charged and the 5.90% she locked in — the loan itself, wherever she actually takes it out, now effectively costs her 5.90%.