Question

3-year horizon

10-year bond

face value 100

5% coupon

initial y-t-m 5.75%

first reinvestment rate 4%

second reinv rate 4.2%

y-t-m in three years 5.8%

rather than reinvesting the first coupon at 4% for two years, invest it for one year at 4% and then combine those proceeds with the second coupon to reinvest at 4.2%. Calculate ROR

Answer 1

No of periods = 10 years

Coupon per period = (Coupon rate / No of coupon payments per year) * Par value

Coupon per period = (5% / 2) * $100

Coupon per period = $5

Let us compute the current Bond price at YTM = 5.75%

Bond Price = Coupon / (1 + YTM)^period + Par value / (1 + YTM)^period

Bond Price = $5 / (1 + 5.75%)¹ + $5 / (1 + 5.75%)² + ...+ $5 / (1 + 5.75%)¹⁰ + $100 / (1 + 5.75%)¹⁰

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $5 * (1 - (1 + 5.75%)^-10) / (5.75%) + $100 / (1 + 5.75%)¹⁰

Bond Price = $37.24 + $57.17

Current Bond Price = $94.41

Let us compute the Bond price after 3 years at YTM = 5.8%

Bond Price = Coupon / (1 + YTM)^period + Par value / (1 + YTM)^period

Bond Price = $5 / (1 + 5.8%)¹ + $5 / (1 + 5.8%)² + ...+ $5 / (1 + 5.8%)⁷ + $100 / (1 + 5.8%)⁷

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $5 * (1 - (1 + 5.8%)^-7) / (5.8%) + $100 / (1 + 5.8%)⁷

Bond Price = $28.11 + $67.39

Bond Price after 3 years = $95.50

Reinvestment amount of coupons = Coupon 1 * (1 + first reinvestment rate) * (1 + second reinvestment rate) + Coupon * (1 + second reinvestment rate) + Coupon 3

Reinvestment amount of coupons = $5 * (1 + 4%) * (1 + 4.2%) + $5 * (1 + 4.2%) + $5  

Reinvestment amount of coupons = $15.6284

Rate of return = ((Bond Price after 3 years + Reinvestment amount of coupons) / Current Bond Price)^{(1 / Investment horizon)} - 1

Rate of return = (($95.50 + $15.6284) / $94.41)^{(1 / 3)} - 1

Rate of return = 5.5850%