Question

A bond with exactly five years until maturity paying 6% p.a. coupons semi-annually and with a face value of $100 was purchased at a yield of 6.5% p.a. The bond was sold exactly two years later for a yield of 5% p.a. All coupons were reinvested at 6% p.a. Calculate the realised yield-to-maturity on this bond.A bond with exactly five years until maturity paying 6% p.a. coupons semi-annually and with a face value of $100 was purchased at a yield of 6.5% p.a. The bond was sold exactly two years later for a yield of 5% p.a. All coupons were reinvested at 6% p.a. Calculate the realised yield-to-maturity on this bond.

Answer 1

First we need to find the purchase price of bond

Face value = $100, Coupon rate = 6% p.a semi annually, Years to maturity = 5, Yield to maturity = 6.5%

As the bond pays coupons semi-annually, therefore

Semi annual coupon payment = (Coupon rate x face value) / 2 = (6% x 100) / 2 = 6/2 = 3

Semi annual yield to maturity = Yield to maturity / 2 = 6.5% / 2 = 3.25%

No of half years to maturity = 2 x years to maturity = 2 x 5 = 10

We can find the purchase price of the bond using PV function in excel

Formula to be used in excel: =PV(rate,nper,-pmr,-fv)

Using PV function in excel, we get purchase price of bond = 97.8944

Now we will find the selling price of bond after 2 years

After 2 years Yield to maturity = 5% p.a , No of years to maturity = 3, Semi annual coupon = 3 , face value = $100

Semi annual yield to maturity = 5%/2 = 2.5%, No of half years to maturity = 2 x 3 = 6

We can find the selling price of bond by using PV function in excel

Formula to be used in excel: =PV(rate,nper,-pmr,-fv)

Using PV function in excel, we get selling price of bond after 2 years = $102.7540

Now we will find the Future value at end of year 2 from reinvestment of coupons

Reinvestment rate = 6% p.a , Semi annual reinvestment rate = 6% / 2 = 3%

Future value at end of year 2 from reinvestment of coupons = 3(1+3%)³ + 3(1+3%)² + 3(1+3%) + 3 = 3.2781 + 3.1827 + 3.09 + 3 = 12.5508

Semi annual Realized yield = [(Selling price of bond after 2 years + Future value at end of year 2 from reinvestment of coupons) / Purchase Price]^1/4 - 1 = [(102.7540 + 12.5508) / 97.8944]^1/4 - 1 = [115.3048 / 97.8944]^1/4 - 1 = [1.177848]^1/4 - 1 = 1.041771 - 1 = 0.041771 = 4.1771%

Realized Yield to maturity = 2 x semi annual realized yield = 2 x 4.1771% = 8.3542%

Hence Realized Yield maturity = 8.3542%