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Converting Continuous Zero Rates to a Forward Bond Yield

Article Quant Q&A · Author: A.Oreo
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Summary

The document addresses how to interpret the yield of a forward bond price when the available input is a zero curve quoted with continuous compounding. The example concerns a European put on a coupon bond and a relation that uses duration to translate forward price volatility into yield volatility. The question is how to reconcile a semiannual-compounded forward yield with a continuously compounded flat zero curve.

The answer gives the conversion principle: match the one-year accumulation produced by the two compounding conventions. A continuous rate grows an investment exponentially, while a nominal annual rate compounded semiannually grows it through two half-year periods. Equating those accumulation factors gives the equivalent rate under the other convention. The note supplies the method rather than a full derivation of the forward bond yield or the duration-based volatility calculation. Its stated example and quoted yield are context; the explanation does not discuss curve construction, bond cash-flow pricing, or model assumptions behind the volatility relation.

Key ideas

  • A yield must be expressed using the same compounding convention as the calculation that uses it.
  • Equivalent continuous and semiannual rates can be found by matching their one-year accumulation factors.
  • Duration links proportional forward bond price changes to forward yield changes in the stated relation.
  • The answer explains rate conversion but does not fully derive the forward bond yield or volatility calculation.

Tags

Fixed incomeOptionsDerivatives pricingVolatility
Full text
# How to calculate the yield of a forward bond price from the zero curve


# How to calculate the yield of a forward bond price from the zero curve












We want to use the `Duration` to convert `forward price volatility` to `yield volatility` with following relation $$\dfrac{\Delta F_B}{F_B} = -D \Delta y_F.$$ But how to calculate the the `yield of forward bond price?` if we know the zero curve $Z(t^*,T).$ In the example it seems directly use the rate of zero curve. I only know the how to calculate the yield of a bond with following relation: $$\dfrac{\Delta B}{B} = -D \Delta y$$ here $B$ is the bond price.

Example: Consider a `European put option` on a 10-year bond with a principal of 100.

The coupon is 8% per year payable semiannually.

The life of the option is 2.25 years

The strike price of the option is 115.

The forward yield volatility is 20%.

The zero curve is flat at 5% with continuous compounding.

Here is some computation. What I don't understand is the `yield with semiannual compounding` $5.0630\%.$ I know how to calculate the yield of a coupon bond, but here how to obtain the yield of a forward $5.0630\%?$

## Answer by Bram (score 0)

https://quant.stackexchange.com/a/35604

You need to convert from continuous compounding to simple compounding. With a continuous rate $r_c$, \$1 grows in one year to $exp(r_c)$. With semi-annual compounding at an annualized rate $r_s$ it grows to $(1+r_s/2)^2$. Equating the two you can convert one into an equivalent yield on the other.

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.