Annually compounded zero rates are 2.00% at one year and 3.00% at three years. What is the two year forward rate starting in one year?

Annually compounded zero rates are 2.00% at one year and 3.00% at three years. What is the two year forward rate starting in one year?

Approach: Require that rolling a one year deposit into the forward matches the three year deposit, then solve for the forward.

3.5037%. No arbitrage requires (1 + z1)*(1 + f)^2 = (1 + z3)^3, so (1 + f)^2 = 1.03^3/1.02 = 1.092727/1.02 = 1.071301 and f = 3.5037%. The forward sits above both zero rates because it has to carry the whole difference between a three year term at 3% and a one year term at 2% across the remaining two years. Any two year rate quoted for delivery in one year away from this level is arbitraged by borrowing at one maturity, lending at the other and locking the gap with a forward rate agreement. The forward curve is therefore a restatement of the zero curve under no arbitrage, and reading it as a forecast of future spot rates additionally assumes a zero term premium.

Follow-up: What forward rate does the market imply for the third year alone, and what would a 50 basis point term premium do to it?

Key concepts: forward rate, zero rate, no arbitrage, forward rate agreement.