Modified Duration and Bond Price Across Compounding Conventions

Summary

The question compares a bond valuation and duration calculated under continuous compounding with a modified-duration calculation under semiannual compounding. It asks why the initial price in the second calculation remains the same when the yield is expressed using a different compounding convention. The accepted response says that recalculating the price with the converted yield and semiannual compounding produces the same value, because the yield representation has been converted to preserve the underlying discounting.

The example also illustrates that modified duration adjusts ordinary duration for the compounding frequency when approximating the percentage price change for a yield move. Duration provides a local, first-order estimate, rather than an exact repricing rule. The brief answer does not show the conversion algebra or discuss convexity, so readers should distinguish equivalence of the starting valuation from the approximation used for the subsequent price change.

Key ideas

• A yield can be converted between compounding conventions while preserving the same bond valuation.
• Repricing under semiannual compounding with the equivalent yield gives the same starting price.
• Modified duration adjusts duration for the compounding frequency in a first-order price-change estimate.
• The duration approximation does not account for convexity or provide an exact repricing.

Tags

# If we modify duration, should we modify bond price? Options Futures and Other Derivatives


# If we modify duration, should we modify bond price? Options Futures and Other Derivatives












In Example 4.5 of Section 4.8 on Duration of Options, Futures and Other Derivatives (p.92), a bond's price and duration are computed assuming continuous compounding where the bond yield is y = 12%. The price is B $\approx$ 94.213, and the duration is D $\approx$ 2.653. Then, the formula $\frac{\Delta B}{B} = -D\Delta y$ is tested for accuracy. The claim is that if bond yield increases from 12% to 12.1%, then the bond price will decrease from 94.213 to 93.963.

Next, Hull talks about Modified Duration in Example 4.6 where semiannual compounding is instead used. y = 12% is converted to y = 12.3673%. The modified duration is $D* = \frac{D}{1+\frac{y}{m}} = \frac{2.653}{1+\frac{12.3673%}{2}} = 2.4985$. The next claim is that if bond yield increases from 12.3673% to 12.4673%, then the bond price will decrease from 94.213 to 93.978 using the formula $\frac{\Delta B}{B} = -\frac{D\Delta y}{1+\frac{y}{m}} = -D*\Delta y$.

My question is regarding the second claim. Why still 94.213 in "94.213 to 93.978" ? Shouldn't we recompute the bond price using semiannual compounding?

## Answer by BCLC (score 0, accepted)

https://quant.stackexchange.com/a/11065

Got this. Recomputing the bond price would just give the same bond price XD

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