Question

Both Bond Sam and Bond Dave have 7 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 5 years to maturity, whereas Bond Dave has 12 years to maturity. (Do not round your intermediate calculations.) Requirement 1: (a) If interest rates suddenly rise by 4 percent, what is the percentage change in the price of Bond Sam? (b) If interest rates suddenly rise by 4 percent, what is the percentage change in the price of Bond Dave? Requirement 2: (a) If rates were to suddenly fall by 4 percent instead, what would the percentage change in the price of Bond Sam be then? (b) If rates were to suddenly fall by 4 percent instead, what would the percentage change in the price of Bond Dave be then?

Answer 1

Let the par value of bonds be $1000. So, par value = market price = $1000

Since par value is equal to the market price, current YTM = coupon rate = 7%

Bond price is the present value of inflows from the bond, i.e., present value of coupon payments and maturity value (par) using YTM as the discount rate. Since we are given semi - annual payments, we need semi - annual rates and time periods.

1) a) Bond Sam

New YTM = 7% + 4 % = 11%

Semi - annual YTM = 11% / 2 = 5.5%, no. of semi - annual periods till maturity = 5 x 2 = 10

Semi - annual coupon payment = $1000 x 7% x 6/ 12 = $35

New bond price = $35 x PVIFA (5.5%, 10) + $1000 x PVIF (5.5%, 10) = $35 x 7.53762582854 + $1000 x 0.58543057939 = $849.247483388

Change in Price = (New price - Old price) / Old price = ($849.247483388 - $1000) / $1000 = (-)0.1507525 or (-)15.08%

b) Bond Dave

Everything is same except no. of semi - annual periods = 12 x 2 = 24

New bond price = $35 x PVIFA (5.5%, 24) + $1000 x PVIF (5.5%, 24) = $35 x 13.1516989523 + $1000 x 0.27665655754 = $736.96602087

Change in bond price = ($736.96602087 - $1000) / $1000 = (-)0.263034 or (-)26.30%

2) a) Bond Sam

New YTM = 7% - 4% = 3%

Semi - annual YTM = 3% / 2 = 1.5%

Everything else is the same as part 1a.

New bond price = $35 x PVIFA (1.5%, 10) + $1000 x PVIF (1.5%, 10) = $35 x 9.22218455 + $1000 x 0.86166723168 = $1184.44369093

Change in Bond price = ($1184.44369093 - $1000) / $1000 = 0.1844437 or 18.44%

b) Bond Dave

New YTM = 3%, Semi - annual YTM = 1.5%

Everything else is the same as part 1b.

New bond price = $35 x PVIFA (1.5%, 24) + $1000 x PVIF (1.5%, 24) = $35 x 20.030405362 + $1000 x 0.69954391941 = $1400.60810708

Change in bond price = ($1400.60810708 - $1000) / $1000 = 0.4006081 or 40.06%

NOTE :

PVIF = 1 / (1 + r)ⁿ