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Deriving a Forward-Rate Identity from Zero-Coupon Bond Prices

Article Quant Q&A · Author: Lucas Morin
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Summary

The note examines an identity equating ratios of forward rates observed at different dates. It rewrites each ratio using zero-coupon bond prices and the time intervals, then applies the no-arbitrage relationship between bond prices across successive periods: investing to an intermediate date and then onward is equivalent to investing directly to the later date. This establishes the identity under the stated pricing setup.

A second answer offers a simpler demonstration for flat interest rates by adjusting both forward rates by the same accumulation factor. The discussion is theoretical and brief; it does not specify market conventions, compounding details, or assumptions such as deterministic rates and frictionless trading. Those details matter when applying the relation to a particular forward-rate definition or real-world data.

Key ideas

  • Forward-rate ratios can be expressed in terms of zero-coupon bond prices and time intervals.
  • Bond prices across successive periods multiply to the price for the combined investment horizon.
  • That bond-price relationship supplies the key step in proving the forward-rate identity.
  • A flat-rate case can also be checked by applying a common accumulation factor to both rates.

Tags

Fixed incomeDerivatives pricing
Full text
# Forward rates formulae


# Forward rates formulae












I am now working with forward rates and have somehow been asked to use an "intuitive" formula for forward rates.

$$ \frac{F(0,s,T)}{F(0,t,T)} = \frac{F(s,s,T)}{F(s,t,T)} $$

I can understand the logic behind it but i am failling at proving/disproving it. I've tried to rewrite it in term of Zero Coupon Bond Price, in short term rates, but the equation are not working.

Is it because the previous equation does not hold ? Or is this because I am lacking some argument ?

## Answer by KaapstadKwant (score 4, accepted)

https://quant.stackexchange.com/a/12969

Note that $\frac{F(0,s,T)}{F(0,t,T)} = \frac{T-t}{T-s}\frac{B(0,s)-B(0,T)}{B(0,t)-B(0,T)}$ and $\frac{F(s,s,T)}{F(s,t,T)} = \frac{T-t}{T-s}\frac{B(s,s)-B(s,T)}{B(s,t)-B(s,T)}$. Multiplying the numerator and denominator of the last expression with $B(0,s)$ and noting that $B(0,s)B(s,u)=B(0,u)$ (investing one Dollar for $s$ years and then for another $u-s$ years is equivalent to investing one Dollar for $u$ years) leads to the required expression.

## Answer by KaapstadKwant (score -1)

https://quant.stackexchange.com/a/12968

Here is a simple (trivial) non-finance answer for the case of flat interest rates. Multiply both the denominator $F(s,t, T)$ and the numerator $F(s,s,T)$ with $e^{rs}$.

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.