Question

Bond A has a coupon rate of 4.22 percent, a yield-to-maturity of 9.3 percent, and a face value of 1,000 dollars; matures in 11 years; and pays coupons annually with the next coupon expected in 1 year. What is (X + Y + Z) if X is the present value of any coupon payments expected to be made in 6 years from today, Y is the present value of any coupon payments expected to be made in 8 years from today, and Z is the present value of any coupon payments expected to be made in 14 years from today?

Answer 1

Coupon rate = 4.22%, face value = $1,000, YTM = 0.093

Coupon payment = face value*coupon rate = $1,000*4.22% = $42.20

X = {Coupon payment in year 1/(1+YTM)}+{Coupon payment in year 2/[(1+YTM)^2]}+{Coupon payment in year 3/[(1+YTM)^3]}+{Coupon payment in year 4/[(1+YTM)^4]}+{Coupon payment in year 5/[(1+YTM)^5]}+{Coupon payment in year 6/[(1+YTM)^6]}

= {42.2/(1+0.093)}+{42.2/[(1+0.093)^2]}+{42.2/[(1+0.093)^3]}+{42.2/[(1+0.093)^4]}+{42.2/[(1+0.093)^5]}+{42.2/[(1+0.093)^6]}

= {42.2/1.093}+{42.2/[1.093^2]}+{42.2/[1.093^3]}+{42.2/[1.093^4]}+{42.2/[1.093^5]}+{42.2/[1.093^6]}

= 38.6093+{42.2/1.194649}+{42.2/1.305751}+{42.2/1.427186}+{42.2/1.559915}+{42.2/1.704987}

= 38.6093+35.3242+32.3186+29.5687+27.0528+24.7509 = $187.6245

Y = {Coupon payment in year 1/(1+YTM)}+{Coupon payment in year 2/[(1+YTM)^2]}+{Coupon payment in year 3/[(1+YTM)^3]}+{Coupon payment in year 4/[(1+YTM)^4]}+{Coupon payment in year 5/[(1+YTM)^5]}+{Coupon payment in year 6/[(1+YTM)^6]}+{Coupon payment in year 7/[(1+YTM)^7]}+{Coupon payment in year 8/[(1+YTM)^8]}

= {42.2/(1+0.093)}+{42.2/[(1+0.093)^2]}+{42.2/[(1+0.093)^3]}+{42.2/[(1+0.093)^4]}+{42.2/[(1+0.093)^5]}+{42.2/[(1+0.093)^6]}+{42.2/[(1+0.093)^7]}+{42.2/[(1+0.093)^8]}

= {42.2/1.093}+{42.2/[1.093^2]}+{42.2/[1.093^3]}+{42.2/[1.093^4]}+{42.2/[1.093^5]}+{42.2/[1.093^6]}+{42.2/[1.093^7]}+{42.2/[1.093^8]}

= 38.6093+{42.2/1.194649}+{42.2/1.305751}+{42.2/1.427186}+{42.2/1.559915}+{42.2/1.704987}+{42.2/1.86355}+{42.2/2.036861}

= 38.6093+35.3242+32.3186+29.5687+27.0528+24.7509+22.645+20.7182 = $230.9877

Z = {Coupon payment in year 1/(1+YTM)}+{Coupon payment in year 2/[(1+YTM)^2]}+{Coupon payment in year 3/[(1+YTM)^3]}+{Coupon payment in year 4/[(1+YTM)^4]}+{Coupon payment in year 5/[(1+YTM)^5]}+{Coupon payment in year 6/[(1+YTM)^6]}+{Coupon payment in year 7/[(1+YTM)^7]}+{Coupon payment in year 8/[(1+YTM)^8]}+{Coupon payment in year 9/[(1+YTM)^9]}+{Coupon payment in year 10/[(1+YTM)^10]}+{Coupon payment in year 11/[(1+YTM)^11]}+{Coupon payment in year 12/[(1+YTM)^12]}+{Coupon payment in year 13/[(1+YTM)^13]}+{Coupon payment in year 14/[(1+YTM)^14]}

= {42.2/(1+0.093)}+{42.2/[(1+0.093)^2]}+{42.2/[(1+0.093)^3]}+{42.2/[(1+0.093)^4]}+{42.2/[(1+0.093)^5]}+{42.2/[(1+0.093)^6]}+{42.2/[(1+0.093)^7]}+{42.2/[(1+0.093)^8]}+{42.2/[(1+0.093)^9]}+{42.2/[(1+0.093)^10]}+{42.2/[(1+0.093)^11]}+{42.2/[(1+0.093)^12]}+{42.2/[(1+0.093)^13]}+{42.2/[(1+0.093)^14]}

= {42.2/1.093}+{42.2/[1.093^2]}+{42.2/[1.093^3]}+{42.2/[1.093^4]}+{42.2/[1.093^5]}+{42.2/[1.093^6]}+{42.2/[1.093^7]}+{42.2/[1.093^8]}+{42.2/[1.093^9]}+{42.2/[1.093^10]}+{42.2/[1.093^11]}+{42.2/[1.093^12]}+{42.2/[1.093^13]}+{42.2/[1.093^14]}

= 38.6093+{42.2/1.194649}+{42.2/1.305751}+{42.2/1.427186}+{42.2/1.559915}+{42.2/1.704987}+{42.2/1.86355}+{42.2/2.036861}+{42.2/2.226289}+{42.2/2.433333}+{42.2/2.659633}+{42.2/2.906979}+{42.2/3.177328}+{42.2/3.47282}

= 38.6093+35.3242+32.3186+29.5687+27.0528+24.7509+22.645+20.7182+18.9553+17.3425+15.8669+14.5168+13.2816+12.1515 = $323.1023

X+Y+Z = $187.6245+$230.9877+$323.1023 = $741.7145