Both Bond Sam and Bond Dave have 9 percent annual coupons, but make semiannual p

Question

Both Bond Sam and Bond Dave have 9 percent annual coupons, but make semiannual payments. The bonds' yield to maturity is equal to the coupon rate, so the bonds and are priced at par value. Bond Sam has five years to maturity, whereas Bond Dave has 18 years to maturity. To see how changes in interest rates affect bond prices, assume that interest rates suddenly rise by 2 percent. What is the percentage change in the price of Bond Sam and Bond Dave? (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) Percentage change in price of Bond Sam Percentage change in price of Bond Dave -4.27 % -11.24 % If rates were to suddenly fall by 2 percent instead of rising, what would be the percentage change in the price of Bond Sam and Bond Dave? (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) Percentage change in price of Bond Sam Percentage change in price of Bond Dave 4,52 % 13.19 %

Explanation / Answer

Solution: a. If interest rate suddenly rise by 2% % change in price of Bond Sam = -7.54% % change in price of Bond Dave = -15.54% b. If interest rate suddenly fall by 2% % change in price of Bond Sam =8.32% % change in price of Bond Dave =20.29% Working Notes: AS both the bonds sells at par value, YTM of both bonds is equal to Coupon Rate that is 9% 1st case is of interest rate suddenly rise by 2% means YTM becomes 9%+2%=11% then price of both bond will fall Bond Sam price =$924.62374 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond Coupon Rate = 9% Annual coupon = Face value of bond x Coupon Rate = 1,000 x 9 % = $90 Semi annual coupon = Annual coupon / 2 = $90/2=$45 YTM= 11% p.a (annual)   Semi annual YTM= 11%/2 = 5.5% n= no.of coupon = No. Of years x no. Of coupon in a year = 5 x 2 = 10 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond = $45 x Cumulative PVF @ 5.5% for 1 to 10th + PVF @ 5.5% for 10th period x 1,000 = 45 x 7.537625829 + 1000 x 0.585430579 =$924.62374 Cumulative PVF @ 5.5 % for 1 to 10th is calculated = (1 - (1/(1 + 0.055)^10) ) /0.055 = 7.537625829 PVF @ 5.5% for 10th period is calculated by = 1/(1+i)^n = 1/(1.055)^10 =0.585430579 Bond Dave price =$844.63930 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond Coupon Rate = 9% Annual coupon = Face value of bond x Coupon Rate = 1,000 x 9 % = $90 Semi annual coupon = Annual coupon / 2 = $90/2=$45 YTM= 11% p.a (annual)   Semi annual YTM= 11%/2 = 5.5% n= no.of coupon = No. Of years x no. Of coupon in a year = 18 x 2 = 36 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond = $45 x Cumulative PVF @ 5.5% for 1 to 36th + PVF @ 5.5% for 36th period x 1,000 = 45 x 15.536068 + 1000 x 0.145516236 =$844.63930 Cumulative PVF @ 5.5 % for 1 to 36th is calculated = (1 - (1/(1 + 0.055)^36) ) /0.055 = 15.536068 PVF @ 5.5% for 36th period is calculated by = 1/(1+i)^n = 1/(1.055)^36 =0.145516236 Percentage change in price = (New price – Original price) / Original price % change in price Bond Sam=($924.62374 -1000)/1000 =-7.54% =-0.07537626 % change in price Bond Dave=($844.63930 -1000)/1000 =-15.54% =-0.1553607 If interest rate suddenly fall by 2% % change in price of Bond Sam =+8.32% % change in price of Bond Dave =+20.29% Working Notes: 2nd case is of interest rate suddenly fall by 2% means YTM becomes 9%-2% =7% then price of both bond will rise Bond Sam price =$1,083.16605 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond Coupon Rate = 9% Annual coupon = Face value of bond x Coupon Rate = 1,000 x 9% = $90 Semi annual coupon = Annual coupon / 2 = $90/2=$45 YTM= 7% p.a (annual)   Semi annual YTM= 7%/2 = 3.5% n= no.of coupon = No. Of years x no. Of coupon in a year = 5 x 2 = 10 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond = $45 x Cumulative PVF @ 3.5% for 1 to 10th + PVF @ 3.5% for 10th period x 1,000 = $45 x 8.316605323 + 1000 x 0.708918814 =$1,083.16605 Cumulative PVF @ 3.5 % for 1 to 10th is calculated = (1 - (1/(1 + 0.035)^10) ) /0.035 = 8.316605323 PVF @ 3.5% for 10th period is calculated by = 1/(1+i)^n = 1/(1.035)^10 =0.708918814 Bond Dave price =$1202.90494 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond Coupon Rate = 9% Annual coupon = Face value of bond x Coupon Rate = 1,000 x 9 % = $90 Semi annual coupon = Annual coupon / 2 = $90/2=$45 YTM= 7% p.a (annual)   Semi annual YTM= 7%/2 = 3.5% n= no.of coupon = No. Of years x no. Of coupon in a year = 18 x 2 = 36 Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond = $45 x Cumulative PVF @ 3.5% for 1 to 36th + PVF @ 3.5% for 36th period x 1,000 = 45 x 20.29049381 + 1000 x 0.289832717 =$1202.90494 Cumulative PVF @ 3.5% for 1 to 36th is calculated = (1 - (1/(1 + 0.035)^36) ) /0.035 = 20.29049381 PVF @ 3.5% for 36th period is calculated by = 1/(1+i)^n = 1/(1.035)^36 =0.289832717 Percentage change in price = (New price – Original price) / Original price % change in price Bond Sam=($1,083.16605-1000)/1000 =0.08316605 =+8.32% % change in price Bond Dave=($1202.90494-1000)/1000 =0.20290494 =+20.29% Please feel free to ask if anything about above solution in comment section of the question.

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