Deriving a One-Year Forward Rate from Spot Bond Yields

Summary

The document shows how to infer the rate for a one-year investment beginning one year in the future from the yields of one-year and two-year bonds. It equates the compounded return from holding the longer bond with the sequential return from holding the shorter bond for the first year and then reinvesting for the next year. Solving that relationship gives the implied forward rate.

The explanation interprets the equality through expected interest parity or the expectations hypothesis: under the stated assumption, neither investment path has an advantage. This is a basic no-arbitrage style calculation and illustrates why compounding periods matter. The document gives the setup and equation but does not discuss term premiums, credit differences, taxes, liquidity, or other reasons observed bond yields may not translate directly into an expected future short rate. Its result should therefore be read as an implied rate under the stated framework, rather than a guaranteed forecast.

Key ideas

• The implied forward rate links yields on bonds with different maturities.
• Compare compounded returns from holding a longer bond and rolling shorter bonds.
• Under the stated expectations hypothesis, the two investment paths have equal returns.
• The calculation omits term premiums and other market frictions that can affect observed yields.

Tags

# Implied rate of a bond question


# Implied rate of a bond question












A 2 year bond, yield 6%. A 1 year bond, yield 4%. What's the implied rate for the bond that starts one year from now?

## Answer by Alex C (score 2, accepted)

https://quant.stackexchange.com/a/20882

(1+0.06)^2 = (1+0.04)*(1+x) Solve for x

The left hand side represents the gross return on a 2 year 6% bond

The right hand side represents two gross returns in sequence: first we hold a 1 year bond, then (for the next year) we hold a 1 year bond (that starts a year from now) with unknown return x

By expected interest parity (or by "the Expectations Hypothesis") the two sides have to be equal, there is no advantage (or disadvantage) to holding the 2 year bond or the two 1-year bonds in sequence.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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