Deriving Forward Rates from the Expectation Hypothesis

Summary

The document derives a forward rate from the expectation hypothesis under annually compounded rates. The hypothesis equates the compounded return from investing over a longer maturity at today’s spot yield with the product of the successive one-period spot rates across that horizon.

To isolate the final one-period rate, divide the compounded long-maturity spot return by the product of the earlier short-rate returns. The resulting expression identifies the implied future spot rate as the forward rate for that final period. This provides an algebraic derivation rather than relying on a numerical multi-period example.

The explanation is brief and assumes the expectation hypothesis as its starting point. It does not discuss risk or term premiums, market frictions, or whether forward rates accurately predict future realized spot rates; those limits matter when applying the relationship to observed yield curves.

Key ideas

• The expectation hypothesis equates a compounded long-term spot return with the product of successive short-rate returns.
• Dividing out earlier period returns isolates the implied rate for the final period.
• Under the stated setup, that implied future one-period spot rate is the forward rate.
• The derivation assumes the expectation hypothesis and does not address term premiums or predictive accuracy.

Tags

# Deriving the Forward Rate Formula from the Expectation Hypothesis


# 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.

## Answer by A.L. Verminburger (score 0, accepted)

https://quant.stackexchange.com/a/37745

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.