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.
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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.
answered Jan 15, 2018 at 16:55