BP change

Can someone explain the BP explanation? I dont get the math. Said another way, I dont get the 4.625 to 4.875. If you could clear this up – that would be helpful!

Consider a 25-year, $1,000 par semiannual-pay bond with a 7.5% coupon and a 9.25% YTM. Based on a yield change of 50 basis points, the approximate modified duration of the bond is closest to:

a) 10.03

b) 8.73

c) 12.5

Calculate the new bond prices at the 50 basis point change in rates both up or down and then plug into the approximate modified duration equation:

Current price: N = 50; FV = 1,000; PMT = (0.075/2) × 1,000 = 37.50; I/Y = 4.625; CPT → PV = $830.54.

+50 basis pts: N = 50; FV = 1,000; PMT = (0.075/2)1,000 = 37.50; I/Y = 4.875; CPT → PV = $790.59.

–50 basis pts: N = 50; FV = 1,000; PMT = (0.075/2)1,000 = 37.50; I/Y = 4.375; CPT → PV = $873.93.

Approximate modified duration = (873.93 – 790.59) / (2 × 830.54 × 0.005) = 10.03.