Forward Rates and Implied Forwards

The yield curve gives you spot rates: what it costs to borrow for one year, two years, five years, starting today. But it also secretly tells you something about the future — the rate the market implies for borrowing money starting later, not today. That embedded number is a forward rate, and it comes out of the curve through simple no-arbitrage logic, without anyone needing to forecast anything.

Where it comes from

Suppose you can invest for 1 year at rate $r_1$, or for 2 years at rate $r_2$. A third route: invest for 1 year at $r_1$, then reinvest the proceeds for a second year at whatever the forward rate $f_{1,2}$ turns out to be. For there to be no arbitrage, compounding through the 2-year spot rate must equal compounding through the 1-year rate followed by the forward rate:

$$(1 + r_2)^2 = (1 + r_1) \times (1 + f_{1,2})$$

In words: locking in the 2-year rate today must produce the same total growth as locking in 1 year today and pre-agreeing now on the rate for the second year — otherwise you could borrow on one path and lend on the other for a riskless profit.

Worked example

The 1-year spot rate is 4.0%, and the 2-year spot rate is 4.6%. What forward rate is implied for year 2 (i.e., a 1-year loan starting one year from now)?

1. Grow through 2-year spot: $(1.046)^2 = 1.09412$.
2. Grow through 1-year spot: $1.040$.
3. Solve for the forward: $1 + f_{1,2} = 1.09412 / 1.040 = 1.05204$, so $f_{1,2} \approx 5.20\%$.

Even though no rate on the curve directly says 5.20%, that's the rate embedded between year 1 and year 2 — and it's noticeably higher than either the 1-year or 2-year spot rate, because a rising curve implies rates are expected to keep climbing further out.

What this means in practice

Forward rates are how traders price forward-starting instruments — a forward rate agreement, a deferred bond purchase, the floating leg of a swap — without needing a model of future rates, just today's curve. They're also the benchmark for judging whether holding a longer bond is worth it: if realized future rates come in lower than the forward the curve implied, longer bonds outperform; if realized rates come in higher, you'd have been better off staying short and rolling. This comparison — forward versus realized — is the basis of carry-and-rolldown analysis.