问题如下：
An investor buys a three-year bond with a 5% coupon rate paid annually. The bond, with a yield-to-maturity of 3%, is purchased at a price of 105.657223 per 100 of par value. Assuming a 5-basis point change in yield-to-maturity, the bond’s approximate modified duration is closest to:
选项：
A. 2.78.
B. 2.86.
C. 5.56.
解释：
A is correct.
The bond’s approximate modified duration is closest to 2.78. Approximate modified duration is calculated as:
ApproxModDur= [(PV−) − (PV+)] / [2×(ΔYield)×(PV0)]
Lower yield-to-maturity by 5 bps to 2.95%: PV-=105.804232
PV−= 5 / (1+0.0295) + 5/(1+0.0295)^2 + 105/ (1+0.0295)^3 =105.804232
Increase yield-to-maturity by 5 bps to 3.05%: PV+=105.510494
PV+= 5/ (1+0.0305) + 5/ (1+0.0305)^2 + 105/ (1+0.0305)^3 =105.510494
PV0 = 105.657223, ΔYield = 0.0005
modified duration = (105.804232 − 105.510494)/(2×0.0005×105.657223) = 2.78
题目没有交代是不是含权债券，怎么想到是求approximate duration？？课堂讲只有含权债券才用这个方法
