Deriving Forward Rates from the Spot Yield Curve

Summary

The document derives the instantaneous forward rate from the continuously compounded zero-coupon spot yield curve. It equates a bond’s price expressed using its maturity-specific spot rate with the same price expressed as the accumulation of forward rates over time. Taking logarithms gives spot yield multiplied by maturity as the integral of the forward curve; differentiating with respect to maturity yields the spot yield plus maturity times its first derivative.

This is an analytical identity, not an empirical result or a calibration procedure. It explains why a forward rate is not generally just the spot rate: it also reflects how the spot yield changes with maturity. The derivation assumes the stated continuous-compounding conventions and smoothness sufficient to differentiate the curve. The discussion offers no numerical example or further caveats about curve construction or market conventions.

Key ideas

• A zero-coupon bond price can be written using either its spot yield or the integrated forward curve.
• The maturity-scaled spot yield equals the integral of forward rates from today to maturity.
• Differentiating that identity gives the forward rate as spot yield plus maturity times its maturity derivative.
• The result depends on the yield and forward-rate conventions used.

Tags

# Relation Between Yield Curve, First Order Derivative of YC and Forward Rate


# Relation Between Yield Curve, First Order Derivative of YC and Forward Rate












I'm reading the book, "Derivatives Analytics with Python" by Yves Hilpisch. In an application of calibration of CIR85 process for the short-term interest rate. I found some codes which can be interpreted as which follows. The forward rate = spot rate + first order derivative of spot yield curve w.r.t. time horizon * time horizon. The full python script can ben found in the following link, http://www.riskreversal.net/calibration-of-cir85-model-to-euribor-rates/, and the formula in question is located in line 35. Could anyone give some additional explanation about the logic of the formula or indicate some reference regarding the issue. Thanks a lot!

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/42083

You know the zero coupon bond price can be written in terms of spot rate, say y, or forward rates:

$B(0,t)=e^{-y(0,t) \, t}=e^{-\int_0^t{f(0,u) \, du}}$

Which means:

$y(0,t) \, t=\int_0^t{f(0,u) \, du}$

Differentiating with respect to t, you get your answer:

$y(0,t)+ y\prime (0,t) t=f(0,t)$

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