# Deriving the Forward Rate Formula from the Expectation Hypothesis

The Expectation Hypothesis (EH) states that the current spot yield for any of the maturities is the geometric average of current and future short rates. $$\Big(1 + y(t=0, m=\mu) \Big)^{\mu} = \prod_{t=0}^{\mu-1}\Big(1 + y(t, m=1)\Big)$$ What are the steps to arrive from EH to Forward Rate? Most learning resources seem to use some naive concrete 3-period example, without really showing the proper mathematical derivation from EH.

We start with the expectation hypothesis (current spot rate is the product of all future short spot rates): $$\Big(1+y(t=0, m = \mu)\Big)^{\mu} = \prod_{t=0}^{\mu-1}\Big(1 + y(t, m =1)\Big)$$ We then factor out the last element of the multiplication series: $$\Big(1+y(t=0, m = \mu)\Big)^{\mu} = \prod_{t=0}^{\mu-2}\Big(1 + y(t,m=\mu-1)\Big) \cdot \Big(1 + y(t=\mu-1, m=1)\Big)$$ To finally arrive at the forward rate: $$\Big(1 + y(t=\mu-1, m=1)\Big) = \frac{\Big(1+y(t=0, m = \mu)\Big)^{\mu}}{\prod_{t=0}^{\mu-2}\Big(1 + y(t,m=\mu-1)\Big)}$$ Forward rate is essentially just a future spot rate.