Question

Jason has just bought a bond that pays 2% annual coupons with $1,000 face value and 30 years to maturity. (a) If the yield of the bond bought today was 3%, what was its purchase price? (b) One year later, the bond's YTM has dropped to 2.5%. If you sell the bond immediately after receiving the coupon, i) what is the bond’s current yield? ii) what is the bond’s capital gains yield (CGY)? iii) what is the bond’s total (holding period/1-year total) yield? (1 mark) (c) Now suppose another year has passed and the bond’s YTM remains unchanged at the previous year’s (Year one) level. If you sell the bond immediately after receiving the second year’s coupon, calculate i) the 2-year CGY ii) the total interest incomes (coupon and reinvestment of coupons) for the two years iii) the 2-year holding period/total yield.

Answer 1

Solution a)

The purchase price of the bond at t=0 will be PV of all the coupons + PV of maturity value discounted at YTM@3%

= PV of all the coupons + PV of maturity value discounted at YTM@3%

= Coupon * Cumulative PV discount factor for 1-30 years at YTM + Face Value * PV discount factor for 30th year at YTM

= $1000*2% * 1-1/(1+YTM)^T/YTM + $ 1000 * 1/(1+YTM)^T

= $20 * 1-1/(1+3%)³⁰/3% + $ 1000 * 1/(1+3%)³⁰

= $ 20 * 19.600 + $ 1000 * 0.412

= $ 804.00

Solution b)

The price of the bond at t=1 will be PV of all the coupons + PV of maturity value discounted at [email protected]%

= PV of all the coupons + PV of maturity value discounted at [email protected]%

= Coupon * Cumulative PV discount factor for 29 years at YTM + Face Value * PV discount factor for 29th year at YTM

= $1000*2% * 1-1/(1+YTM)^T/YTM + $ 1000 * 1/(1+YTM)^T

= $20 * 1-1/(1+2.5%)²⁹/3% + $ 1000 * 1/(1+2.5%)²⁹

= $ 20 * 20.4535 + $ 1000 * 0.4887

= $ 897.73

i) Current Yield = Annual Coupon / Bond Price = $ 20 / $897.73 = 2.23%

ii) Capital Gain Yield = (Price at t=1 - Price at t=0) / Price at t=0 = ($897.73 - $ 804.00) / $804.00 = 11.66%

iii) Total Yield = Current Yield + Capital Gain Yield = 2.23% + 11.66% = 13.89%

Solution c)

The price of the bond at t=2 will be PV of all the coupons + PV of maturity value discounted at [email protected]%

= PV of all the coupons + PV of maturity value discounted at [email protected]%

= Coupon * Cumulative PV discount factor for 28 years at YTM + Face Value * PV discount factor for 28th year at YTM

= $1000*2% * 1-1/(1+YTM)^T/YTM + $ 1000 * 1/(1+YTM)^T

= $20 * 1-1/(1+2.5%)²⁸/3% + $ 1000 * 1/(1+2.5%)²⁸

= $ 20 * 19.9649 + $ 1000 * 0.5009

= $ 900.18

i) Capital Gain Yield = (Price at t=1 - Price at t=0) / Price at t=0 = ($900.18 - $ 804.00) / $804.00 = 11.96%