# % price change of a bond // Calculate PV vs using ModDur

An 8-year, 3.5% annual coupon bond is priced at 92.1492, with a yield to maturity of 4.7% and a Macaulay duration of 7.0705. If rates decrease by 75 bps, the percentage price change of the bond is closest to:

1. A.β5.30%.
2. B.5.07%.
3. C.5.30%.

They give al the data needed to calculate the PV of the bond with the rate decrease:
N=8, PMT=3.5, I/Y= 4.7-0.75=3.95, FV=100
Therefore PV= 96.96

Change in price= 96.96/92.149 -1 = 5.2%

Why does the answer(below) calculate ModDur using MacDur and YTM ? ModDur= Mac/(1+Rf) , not YTM (???)

1. Correct because to determine the percentage price change of a bond for a given change in yield, first convert Macaulay duration (7.0705) to modified duration by dividing Macaulay duration by 1 plus yield per period.ModDur = 7.07051.047=6.75317.07051.047=6.7531Next, multiply annual modified duration by the change in yield.%ΞPVFullββ6.7531Γ(β0.0075)=0.0507%π₯PVπΉπ’ππββ6.7531Γ(β0.0075)=0.0507
2. Incorrect because the price change is incorrectly calculated by using Macaulay duration, rather than modified duration. β7.0705 x (β0.0075) = β0.05303 β 5.3%.

Thatβs the trick of the question: they are looking to see if you remember to use modified duration instead of Macaulay. As well, they are not asking for the exact answer; they want you to use the approximation. It makes me happy that you used your calculator, but ya hafta answer the question as posed!!

To calculate modified duration, you divide Macaulay by (1 + periodic YTM). Since the coupon is annual, you divided by 1 + (0.047/1). Your formula should have defined Rf as the periodic YTM.