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?
